The Iordanskii Fore in Helium II
Jrn Inge Vestgarden
Thesis submitted for the Degree of
Candidatus Sientiarum
Department of Physis
University of Oslo
May 2001
2
Aknowledgments
First of all thanks to my supervisor professor Jon Magne Leinaas for guidane and
support through the whole work with this thesis, and for seleting the fasinating
topi of quantized vorties and fores in helium II.
Also thanks to my father, Jrgen Vestgarden, for orreting the language.
Jrn Inge Vestgarden
Oslo, May 2001
3
4
Contents
Introdution
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Bakground
1.1 Classial Hydrodynamis . . . . . . . . . .
1.1.1 The Cirulation . . . . . . . . . . .
1.1.2 The Magnus Fore . . . . . . . . .
1.2 Ginzburg-Landau Theory . . . . . . . . . .
1.2.1 Hydrodynamial form . . . . . . .
1.2.2 Plane Waves . . . . . . . . . . . . .
1.2.3 Stationary Vortex . . . . . . . . . .
1.2.4 Moving Vortex . . . . . . . . . . .
1.3 Vortex Motion . . . . . . . . . . . . . . . .
1.3.1 A Vortex in a Plane Wave . . . . .
1.4 Analogy to Eletro-Magnetism . . . . . . .
1.4.1 2-D Eletro-Magnetism . . . . . . .
1.4.2 The Analogy to Eletro-Magnetism
1.4.3 The Lorentz Fore . . . . . . . . .
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2.1 The Sattering Equation . . . . . . . . . . . . . .
2.2 Analogy to Aharonov-Bohm Sattering . . . . . .
2.3 Solution by Born Approximation . . . . . . . . .
2.3.1 The Sattering Amplitude . . . . . . . . .
2.3.2 The Hidden Terms . . . . . . . . . . . . .
2.3.3 Small Angle Sattering . . . . . . . . . . .
2.3.4 Disussion of the Born Approximation . .
2.4 Solution by Partial Wave Analysis . . . . . . . . .
2.4.1 Boundary Conditions Near the Vortex . .
2.4.2 Boundary Conditions Far From the Vortex
2.4.3 The Aharonov-Bohm Sum . . . . . . . . .
2.4.4 The Solution Far From the Flux-Point . .
2.4.5 Disussion of the Partial Wave Analysis .
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2 Phonon Sattering
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7
9
11
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18
19
20
21
22
23
24
25
27
27
29
31
32
33
36
37
38
39
41
43
45
46
6
CONTENTS
2.5 Density Corretions . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 The Phonon Fore
3.1 The Momentum-Flux Tensor . . . .
3.2 Calulation of the Fore . . . . . .
3.2.1 Ciruit Far From the Vortex
3.2.2 Ciruit Near the Vortex . .
3.2.3 Numerial Calulations . . .
3.3 Conlusion . . . . . . . . . . . . . .
3.4 The Bakground Fluid Fore . . . .
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4.1 Phenomenology . . . . . . . . . . . . . . .
4.1.1 Exitations . . . . . . . . . . . . .
4.1.2 Superuidity . . . . . . . . . . . .
4.1.3 The Equations . . . . . . . . . . .
4.2 Derivation from Ginzburg-Landau Theory
4.2.1 The Mass Flux . . . . . . . . . . .
4.2.2 The Energy . . . . . . . . . . . . .
4.3 The Cirulation . . . . . . . . . . . . . . .
4.3.1 Numerial Calulation . . . . . . .
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4 The Two-Fluid Desription
5 Fores on a Vortex
5.1 The Iordanskii Fore . . . . . . .
5.2 The Superuid Magnus Fore . .
5.3 The Eetive Magnus Fore . . .
5.3.1 Appliation on Helium II .
5.3.2 Berry's Phase . . . . . . .
5.4 Disussion of the Controversy . .
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Conlusion
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51
52
54
56
60
61
65
66
69
69
70
71
72
73
76
77
79
83
85
87
88
91
94
95
97
99
A Some Useful Integrals
101
B Vetor Calulus in 2-D
103
C The Two-Dimensional Green's Funtion
105
Introdution
Among the most ontroversial topis in superuid helium II, is the Iordanskii
fore. This is the phonon ontribution to the fore perpendiular to the normal
uid veloity. It is an apparent onit between diret alulations, whih get
a Iordanskii fore, and an indiret determination, whih gets none. The goal of
this thesis is to oer a areful presentation of the topi and the importane is
attahed to the diret alulations.
We will study phonons sattered on a vortex. The theory used is the quantum mehanial Ginzburg-Landau theory, and the equations are solved by both
Born approximation and partial wave analysis. To omplete the understanding
of the problem, the analogy to Aharonov-Bohm sattering is disussed. The vortex will be seen to impose a hange of the inoming waves, in addition to the
sattered wave. The region right behind the vortex is of speial interest and will
be disussed separately.
The transverse fore from one phonon is found, by onsidering the momentum
balane on a ontour surrounding the vortex. Analytial answers are found in the
limits when the iruit is far from the vortex and lose to the vortex. Numerial
studies will be done in the intermediate regions. The ontribution from the region
right behind the vortex will prove to be essential to the fore. The Iordanskii
fore is found as the fore from a thermal assembly of phonons.
The indiret determination of the Iordanskii fore omes through the eetive
Magnus fore, whih is the fore perpendiular to a moving vortex, in a stationary liquid. The eetive Magnus fore is found from a many-partile quantum
mehanis expression, and is assoiated with the geometrial Berry phase. Applied on helium II, the result implies no Iordanskii fore, opposed to the diret
alulations.
Most of the topis disussed in this thesis are built on the tension between
these oniting results.
The ontents of the hapters are:
Chapter 1: Presentation of Ginzburg-Landau theory, whih is the formalism used in most of the thesis. Quantized vorties and plane wave solutions
are introdued. The hapter also inludes two appliations whih have
nothing to do with the Iordanskii fore: the motion of a free vortex, and
7
8
INTRODUCTION
the analogy between Ginzburg-Landau theory and two-dimensional eletromagnetism.
Chapter 2: Calulations of a phonon sattered on a vortex. The methods used are the Born approximation, and the partial wave analysis. The
analogy to Aharonov-Bohm sattering is disussed.
Chapter 3: The results from hapter 2 is used to alulate the transverse
fore from one phonon, by onsidering the momentum balane on a ontour
enlosing the vortex. The fore is found both analytially and numerially.
In addition the bakground uid fore is found.
Chapter 4: A presentation of the two-uid model and how the model an be
onstruted from Ginzburg-Landau theory in the low temperature regime.
The normal uid irulations is alulated from the sattering results from
hapter 2.
Chapter 5: Disussion of the dierent fores on a vortex in helium II, at
nite temperature. Presentation of the Iordanskii fore, the superuid Magnus fore and the eetive Magnus fore. Derivation of the eetive Magnus
fore and disussion of the disagreement with the diret alulations.
9
NOTATION
Notation
In Ginzburg-Landau theory, dimensionless units are used. This means that the
speed of sound is = 1, and the onstant density of the bakground uid is
0 = 1. The irulation of a q fold vortex is = 2q .
Some of the symbols used are not standard in ondensed matter physis, but
we have tried to be onsequent in our notation throughout the thesis.
(r; t)
(r; t)
(r; t)
(r)
(r; t)
(r)
q = 1
= qk
f (r ) p
l = l2 + 2l
Phase of Ginzburg-landau wavefuntion
Density of Ginzburg-landau wavefuntion
Phase utuation
Time-independent phase utuation
Density utuation
Time-independent density utuation
Vortex winding number
Aharonov-Bohm ux & sattering parameter
Density prole of stationary vortex
Phase shift in aousti Aharonov-Bohm sattering
10
INTRODUCTION
Chapter 1
Bakground
1.1 Classial Hydrodynamis
The language used in most of this thesis is the quantum mehanial GinzburgLandau theory, but sine many of the quantities used there have lassial analogies, it is useful rst to write down the equations ruling a lassial uid. A
more areful treatment of lassial uids an be found in a textbook, for example [LL87℄.
The rst equation to mention is the ontinuity equation. This is a statement
of mass onservation
+ r (v) = 0
t
(1.1)
The quantities used here is whih is the uid mass density and v whih is the
veloity. Conservation of momentum gives the important Navier-Stoke's equation
dv v
1
=
+ ( v r) v = rP
dt t
1
r
Vex + r2 v
m
(1.2)
where P is the pressure, is the dynamial visosity and Veq an external potential.
The Navier-Stoke's equation is diÆult to solve and in many ases it is enough
to onsider a ase without visosity. Then the simpler Euler's equation an be
used instead
1
dv v
=
+ ( v r) v = rP
dt t
1
rV
m ex
(1.3)
In many uids Euler's equation gives an adequate desription away from boundaries and turbulent regions. Most pratial solutions in hydrodynamis are done
by tting solutions of Navier-Stoke's equation near boundaries with solutions of
Euler's equation in the rest of the uid.
11
12
CHAPTER 1.
BACKGROUND
1.1.1 The Cirulation
In lassial hydrodynamis the irulation is dened as a losed ontour integral
of the veloity. As the name implies, the irulation tells us how muh the uid
is rotating in the area enlosed by the ontour. It an be shown ([LL87℄) that
in ase of inompressible lassial uid, the irulation is onserved when the
ontour follows the ow of the uid. Classially the irulation an take any
value
=
I
C
v dl = onstant
(1.4)
Another lassial quantity, is the vortiity whih is dened as the url of the
veloity ! = r v. If Stoke's theorem is applied to the denition of the irulation (1.4), the line integral of the veloity an be transformed to a surfae
integral of the vortiity
=
Z
S
dS ! = onstant
(1.5)
whih shows that vortiity is irulation per. unit area. A point vortex situated
at the origin has vortiity ! = Æ (r).
The irulation of ordinary uids an take any value. In superuids this is no
longer true, and the the irulation is quantized in integer steps of
h
(1.6)
m
where h is Plank's onstant and m is the physial atom/moleule mass. The
irulation of a superuid annot be ontinuously hanged and irulation of the
whole uid, N , is thus a topologial quantity; quantized vortex solution an
not be found by perturbation of non-vortex solutions. The quantized vorties, as
they appear in Ginzburg-Landau theory, are disussed in setion 1.2.3.
The idea of quantization of irulation in superuids an be asribed to Onsager [Ons49℄. In a footnote(!) he writes: \Vorties in a superuid are presumably
quantized; the quantum of irulation is h=m, where m is the mass of a single
moleule."
=
1.1.2 The Magnus Fore
The main attention of this thesis will be devoted to the transverse fore on a
vortex in superuid helium II. This fore will prove to have similarities to the
fore on a body in a onventional uid, so it an be useful rst to study the
lassial situation. A lassial uid passing a solid body will in many ases exert
a fore on the body. The fore an be split in one parallel and one transverse
to the initial uid veloity. These omponents are of dierent natures. In the
1.1.
13
CLASSICAL HYDRODYNAMICS
ases when the longitudinal fore is non-zero it is found to be strongly dependent
on the shape and the surfae of the body. In ase of potential ow there is no
longitudinal fore on the body, whih is a statement of d'Alembert's paradox.
The transverse fore, when it appears, is on the ontrary not dependent on the
details of the body, but only on the irulation in the uid. The transverse fore
on a body from a uid is alled the Magnus fore, after the German physiist
Magnus who disovered it in 1852, on spinning bodies in a liquid.
In the following we will work in two dimensions. The result an easily be
generalized to three dimensions by letting the fore be the fore per unit length.
To nd the fore, let us suppose that the body is at rest and the uid has the
veloity v. The fore on the body an be found by onsidering the momentumux tensor
ij = P Æij + vi vj
(1.7)
The momentum-ux tensor is interpreted as the density of j -momentum in the
diretion +i. The fore on the uid from the body is found by integrating the
tensor around a surfae enlosing the body. From Newtons third law the fore
on the body from the uid is the same, but in opposite diretion ( i-diretion).
The omponents of the fore are then
I
Fi =
dSj ij
(1.8)
Let us suppose that far from the body solutions of the hydrodynamial equations
with onstant pressure P and density an be found. Besides that, the veloity
must be very near a onstant veloity whih we put along the x-axis, v = v1 ex +
Æ v. The onstant terms with P and v1i v1j do not ontribute to the fore. If
the seond order term in the veloity dierene Æ v is ignored, we get for the
transverse fore
F? =
I
dSj v1 Ævj
(1.9)
With onstant , the integral is nothing but the irulation (1.4). In vetor
notation, the Magnus fore is
FM = v1 (1.10)
The diretion of the vetor is out of the plane. In this ase with onstant and
P , the longitudinal fore is zero. With the above hoie of solutions for P and v,
only the vi vj term in the momentum-ux tensor is non-vanishing, but in general
both terms will ontribute. On a airplane wing, for example, the transverse
fore is aused by the pressure dierene on the two sides of the wing. With
other solutions a longitudinal fore omponents ould have been experiened, in
addition.
14
CHAPTER 1.
BACKGROUND
If the body moves with veloity u, v1 u an be substituted for v1 in the
Magnus fore (1.10). For a body moving along with the veloity of the uid,
there is no Magnus fore.
1.2 Ginzburg-Landau Theory
The understanding of superuid helium has developed in many steps and there
are many ontributors. The theory whih is presented here is alled GinzburgLandau theory. The name is often used on the more general theory of superondutors as well, but here it will be applied to superuids only (also referred
to as Ginzburg-Pitaevskii theory). The historial development of the theories on
liquid helium will not be presented here, but an be found in many text-books,
e.g. [Gly94℄, [NP90℄ and [WB87℄. This presentation will just serve as a brief introdution and the main point is to establish the formalism whih will be used
for most of the alulations in this thesis.
In nature there are two speies of helium atoms 4 He, whih are Bosons and
3 He whih are Fermions. For low temperatures these isotopes have very dierent behavior so that eah of them must be disussed separately. This thesis will
be restrited to bosoni helium only. With normal pressure we talk about three
dierent phases of helium. For high temperatures it is of ourse a gas and below
the boiling point it is an ordinary visid liquid alled helium I. In bosoni helium we an in addition experiene a seond order phase transition at a ritial
temperature T . Below this temperature we speak of helium II, whih is partly
superuid. At the absolute zero, helium is a pure superuid. Helium II is more
arefully desribed in hapter 4. bosoni helium is liquid down to the absolute
zero and solid helium is only found at high pressure.
The theory desribing liquid helium an start with the grand anonial Hamiltonian for a D-dimensional interating system
Kb =Hb Nb
Z
2 r2
}
D
y
b (r)
b r)
= d r
(
(1.11)
2
m
Z
Z
1
b y(r)
b y(r0 )V (r r0)(
b r0 )(
b r)
dD r dD r0 +
2
b y(r) and (
b r) are the helium atom reation and annihilation operators
where 0
and V (r r ) is the interation potential. The hemial potential is . The elds
satisfy the Bose ommutation rule
h
i
b r) ; b y(r0) = Æ (r
(
r0)
(1.12)
To get a reliable theory of the liquid, an appropriate interation must be found.
Interations whih are realisti at a mirosopi level, do often distort the mathematial handling and hide the marosopi behavior. The Ginzburg-Landau
1.2.
GINZBURG-LANDAU THEORY
15
theory is, on the ontrary, a simple model where the uid is treated as onsisting
of weakly repelling Bosons, i.e. the only interation entering into our alulations
is the repulsion when two atoms happen to be at the same point. The potential
is then
V (r r0 ) = gÆ (r r0 )
(1.13)
Suh a theory is not valid for small length-sales, but it an serve a good desription at a larger level. The mathematial simpliation of the model is overwhelming and makes it possible to arry out more detailed alulations. The equation
of motion is
i
b r) h
(
b r) ; K
b
i}
= (
t
(1.14)
}2 2
y
b
b
b
=
r + g (r)(r) (r)
2m
Here m is the helium mass, g is the interation strength and is the hemial
potential. The interpretation of the hemial potential is that the system has a
density 0 = m=g whih is energetially favorable. At temperatures near the
b i = .
absolute zero, the quantized elds an be replaed by their mean elds, h
b
Sine is a lassial funtion, it is muh easier to handle than the operator .
Mean eld in this ontext means the o-diagonal matrix element
h(N 1)jb j(N )i, where (N ) is the N partile wavefuntion [NP90℄. This
is alled the Gross-Pitaevskii approximation, whih will be used in most of the
alulations done later. The equation of motion for the lassial eld is
}2 2
2
r + gjj (1.15)
i} =
t
2m
whih is often in literature referred to as the nonlinear Shrodinger equation. A
Lagrangian formalism an also be used instead of the Hamiltonian formalism.
The Ginzburg-Landau Lagrangian density is
}2
g
_ _ L = i2} jr
j2
(mjj2 0 )2
(1.16)
2m
2m
The theory as desribed above ontains many parameters. These are important when the thermodynamial properties of the system is onsidered, but are
just distorting the equations when working at zero temperature. The elds and
oordinates an be resaled to dimensionless form, denoted with primes
r
pg0
0
g0
0
0
0
(1.17)
=
t =
t
x =
x
m
}m
}
This transformations orrespond to setting all the onstants }; g; m and 0 to
unity in our equation. Most alulations done in the rest of this hapter and
in hapter 2 and 3 are in dimensionless units. The speed of sound is (when the
veloity
p is small) equal to unity in dimensionless units. In dimensional units it is
= g0 =m.
16
CHAPTER 1.
BACKGROUND
1.2.1 Hydrodynamial form
The Ginzburg Landau theory is a quantum theory. The appliation of the theory
is however on a liquid whih in many ases an be treated like a lassial liquid.
The analogy between the lassial equations in setion 1.1 and the GinzburgLandau theory is best seen by writing the wavefuntion on polar form
=
r
i
e
m
(1.18)
where and are real funtions. The wavefuntion is a solution of the Shrodinger
equation
}2 2
g
i}_ =
r
+ (mjj2 0 )
(1.19)
2m
m
The standard interpretation of quantum mehanis is that jj2 is the probability
density. The quantity = mjj2 is then mass density. The phase of the wavefuntion will be a potential for the veloity, v = m} r. By separating out the
real and imaginary part of (1.18), we get two equations
_ +
_
}
r (r) = 0
p }
} r2 1g
+ (r)2 +
(mjj2
p
m 2 2m
}m
(1.20)
m
0 ) = 0
(1.21)
The rst equation is the ontinuity equation. If we take the gradient of the seond
equation it begins to look like the Euler equation. Supposing that is nonsingular, the order of time derivative and spae derivative an be interhanged.
This is however not true in the ase of vorties as we will see later. Expressing
the equations by the density and veloity, we get
_ + r (v) = 0
v_ + v(r v) =
g
1
r
m m
}2 r2
p
2m (1.22)
(1.23)
where we have used the mathematial identity 21 r(v2) = v(r v). The last
equation is a variant of Euler's equation (1.3), where the quantum eets are
hidden in the peuliar looking potential on the right hand side.
1.2.2 Plane Waves
The physial ground-state of the Ginzburg-Landau liquid is a state minimizing the
interation energy, thus having density = 1 in dimensionless units. Considering
small utuations around this density, plane wave solutions an be found. In
1.2.
17
GINZBURG-LANDAU THEORY
addition it is supposed that the bakground uid has the onstant veloity v0 =
r0. If the veloity is small, the orretion to the density of order v02, an be
ignored. Seeking solutions of the Ginzburg-Landau equations near the ground
state, the eld equations an be linearized in the small quantities = 1 and
= 0 .
1 2
r
+ v0 r = 0
_ + r2 + v0 r = 0
(1.24)
4
Plane wave solutions for and are speially important, sine other non-vortex
solutions an be expressed as superpositions of suh waves. Plane waves are of
the form
_ + k (x; t) = k eikx
i!t
k (x; t) = k eikx
i!t
(1.25)
where ! is the frequeny and k is the wave-number of the wave. In dimensionless
units ! is also equal to the wave energy. It must be pointed out that all physial
solutions of and must be real, so we must be ready taking the real part of our
solutions at any time. Inserting (1.25) into one of the linearized eld equations
leads to the following relations between the onstants k and k
!
k = i 20 k
(1.26)
k
where !0 = ! k v0 is the energy in the rest frame where ondensate veloity
is zero. From the other eld equation we get
k2
!0 = k 2 (1 + )
(1.27)
4
with k = jkj. For low k the ordinary linear phonon energy ! k is obtained,
but for higher momenta the energy
deviates from this. For k 1 it gives the
2
k
peuliar phonon energy ! 2 + 1 and the speed of sound is also frequeny
dependent k = !0 =k k=2. We must be areful to use this energy spetrum for
higher energies in quantum liquids, sine it does not t the experimental urve,
gure 4.1. In this thesis all alulations are in the low energy regime, with a
linear
pphonon energy !0 k and a onstant speed of sound , = 1 (In full units
= g0 =m).
An important quantity is the time average mass ux1 of a plane wave. The
denition is
hji = h(1 + )(v0 + r)i
(1.28)
The plane waves and are real, and only the real part of (1.25) ount. The
produt of the real parts are <( )<(r) = 21 <( r + r). The rst term,
1 In
our dimensionless units this is also equal to the probability ux and the energy ux.
18
CHAPTER 1.
BACKGROUND
<(r), is time-independent and survives the time averaging, while the latter
has time dependeny e2i! , and vanishes.
2
hjk i = 12 !k jkj2 k nk k
k
(1.29)
where nk , as the the number density of the plane wave, is dened. In a thermal
assembly of phonons this will appear as the distribution funtion.
1.2.3 Stationary Vortex
In Ginzburg-Landau theory the quantized vorties appear as multi-valued phases
of the wavefuntion. Sine the wavefuntion itself must be single-valued, the
only transformations allowed for the phase are ! + 2q , with q as an integer.
There is no way to hange it ontinuously from one value to another, and q an
then be seen as a topologial quantity of the uid; it an only hange in disrete
steps.
If the vortex is at rest at the origin the q -fold vortex-solution is = q' where
' = artan(y=x) is the angle between y and x. The wave funtion on polar form
is now
= feiq'
(1.30)
where f is the square root of the density prole (or just the density prole for
short) of the vortex. The symbol f = f (r) will in the rest of the thesis be used to
q} e .
denote the density prole of a stationary vortex. The veloity eld is v = rm
'
The vortiity is
q
h
r
r
' = q Æ (r)
m
m
}
(1.31)
where the delta funtion is reognized from the formula in appendix B. This
tells us that the irulation is an integer multiple q of the irulation quantum
h=m = if the integral is enlosing the vortex, in all p
other ases it is zero.
The density prole is found by substituting f = in the equations (1.20)
and (1.21), whih here are presented on dimensionless form. The rst equation
is now simply r' rf , whih means that f is a funtion of the radius alone,
f = f (r). From the seond we obtain the dierential equation
q2
2f 1 f
+
+
(2
)f 2f 3 = 0
(1.32)
r2 r r
r2
The boundary ondition far from the vortex is that the density must reah the
density of the rest of the uid, whih in dimensionless units means that
f ! 1. Near the origin we must have f ! 0 to avoid a singular wave-funtion.
An analytial solution of (1.32) with these boundary onditions is not possible
1.2.
19
GINZBURG-LANDAU THEORY
1
f~
0.9
f
0.8
0.7
0.6
0.5
0.4
0.3
0.2
pr2
0.1
0
0
1
2
3
4
5
6
7
Figure 1.1: Numerial solution ofp the density prole f , and the analytial approximation f~, as funtions of r= 2.
to ahieve, so the best we an do is to nd the asymptoti behavior and plot
numerial solutions.
By doing series expansions near the origin we easily nd f rjqj , but the
proportionality fator is still undetermined. It must be found by tting solutions
near the vortex to those far from it. Far from the vortex the prole an be expanded in powers of r 1 , giving f 1 41r2 . Sine the gradient of the phase
goes as r 1 the tail of the vortex prole far from the vortex an in many ases be
ignored. This is alled the point vortex approximation. These two asymptoti expressions an, when q = 1, be brought together in an analytial approximation
for f . The funtion
r
f~ = q
(1.33)
r2 + 12
whih was found by Fetter [Fet65℄ will serve as an adequate approximation to the
full funtion f . The gure 1.1 shows a plot omparing the analytial approximation (1.33) and the numerial solution of Kawatra and Pathria [KP66℄.
For the rest of the thesis we will suppose that vorties have winding-numbers
jqj = 1, where a solution with q = 1 will be alled a vortex, and a solution with
q = 1 will be alled an anti-vortex.
1.2.4 Moving Vortex
From stationary vortex solutions, it is possible to nd approximate solutions for
a slowly moving vortex, by assuming that the solutions of the moving ase an be
seen a small perturbation of the stationary
The phase of the wave funtion
ase.
y
y
is thought to be = q'0 = q artan x x00 , where r0 = r0 (t) is the vortex
20
CHAPTER 1.
BACKGROUND
position. For the phase we write = f 2 (1 + Æ), where f = f (jr r0 j) is the
density prole of a stationary vortex, dened in the last setion. By inserting this
into the Ginzburg-Landau equation of hydrodynamial form, (1.20) and (1.21),
the denition of f , (1.32), an be used to get rid of some terms. By ignoring
seond order terms in Æ, we get the following two equations
2f f_ + f 2 Æ _ + qf 2rÆ r'0 = 0
(1.34)
1
1
r
f rÆ
rÆ 0
(1.35)
q '_0 + f 2 Æ
f
2
The time derivatives of f and '0 are entirely through the vortex position r0, so
that f_ = rf r_ 0 and '_ 0 = r'0 r_ 0 . When making a rough estimate of Æ far
from the vortex, it is enough to work to order jr r0 j 1 , thus putting f = 1.
Seeking slowly hanging solutions, the r2Æ term an be ignored giving a simple
equation for the density orretion
Æ q r'0 r_ 0
(1.36)
The orretion is of order jr r0j 1 and thus dominating ompared to the
orretion in the onstant density prole 1 f jr r0 j 2 . If the vortex moves
with onstant veloity along the x-axis, so that x0 = vt and y0 = 0, the density
orretion is Æ = qv (x vty)2 +y2 . The orretion is anti-symmetri about the
x-axes, and proportional to the vortex winding number q .
This result an also be applied to the situation when the bakground uid is
not at rest, but moving with some veloity vs , by substituting r_ 0 ! r_ 0 vs . The
fore on the vortex in this situation will be alulated in setion 3.4.
1.3 Vortex Motion
In setion 1.2.4 the hange of the elds far from the vortex, due to the vortex
motion, was studied. Now we want to examine the equations inside the vortex
ore, in order to nd the vortex equations of motion. The derivation will follow
that of E. Shroder and O. Tornkvist [ST97℄, exept that their derivation is
done for a vortex string in three dimensions while we only will onsider a vortex
point in two dimensions. We start with the Ginzburg-Landau eld equations on
hydrodynamial form
_ + r (r) = 0
(1.37)
1 1 2p
1
_ + (r)2
(1.38)
p r + 1=0
2
2 where and is the density and the phase of the wave funtion. The ontributions
originating from the vortex an be separated out
! q' + ~
! f 2 ~
1.3.
21
VORTEX MOTION
where the density prole of the vortex is f = f (R). We use R = jr r0 j and
' = artan( xy xy00 ) whih is the polar oordinate system entered at the moving
vortex point r0 = r0 (t). The remaining funtions ~ and ~ are supposed to be
non-singular and well behaved near the vortex point. The eld equations an
now be written as
f_ ~_ 2
1
2 + + rf r~ + r~ r(q' + ~) + ~r2~ = 0
(1.39)
f ~ f
~
2 1 1
1
2(f p~) = 0
p
q '_ + ~_ + q' + r~
r
(1.40)
2
2 f ~
where the identities rf r' = 0 and r2' = 0 have been used. All the time
dependeny in f and ' is through r0, so that '_ = r' r_ 0 and f_ = rf r_ 0.
We now do expansions in powers of R 1 . The gradient of the phase is r' = R1 e'
and the density prole is proportional to R, f R. All the terms of power R 2
fall out and the the terms proportional to R 1 are
q
~
r r_ 0 + 2 r ln(~) ez eR = 0
(1.41)
r~ r_ 0 + 2q r ln(~) ez e' = 0
(1.42)
In the limit where R ! 0 these solutions are exat. The following equation
relates the motion of the vortex to the elds ~ and ~
i
h
q
~
(1.43)
r_ 0 = r + r ln(~) ez
2
r=r0
The exat vortex motion is found as self-onsistent solutions for the vortex equation of motion (1.43) and the elds. And approximation of the vortex motion
an be found by inserting the vortex into solutions of the eld equation without
bothering about the inuene of the vortex on the solutions. The only appliation whih will be onsidered in this thesis, is the vortex in a plane wave, in
setion 1.3.1.
1.3.1 A Vortex in a Plane Wave
One appliation of the vortex motion formula, (1.43), is to see how a vortex
behaves in a plane wave. This means that the modiation of the wave beause
of the vortex (whih is so arefully disussed in hapter 2) is ignored.
The linearized eld equations have plane wave solutions, whih were studied
in setion 1.2.2. To avoid onfusion, let us suppose that k is real. The plane
wave solutions of the phase and density = 1 are then
1
= k sin(k r !t)
k
= k os(k r !t)
(1.44)
22
CHAPTER 1.
BACKGROUND
where the frequeny, in the low energy limit is, ! = k . If the plane wave is
propagating along the x-axes the equations of motion (1.43) for the vortex are
x_ 0 = k os(kx0 !t)
(1.45)
q
y_0 = k sin(kx0 !t)
(1.46)
2
This is in qualitatively agreement with what is earlier done by
p L. K. Myklebust [Myk96℄ in his thesis for Cand. Sient. He uses f~ = r= r2 + 1=2 as an
approximation for the vortex prole, and in the small k limit, he gets the same
k -dependeny as above. But his result diers from ours in that he gets 4q instead
of q2 as the fator in the y_ 0 equation.
A very rough estimate of the solution of the above equations an be found if
k r0 is small. If the kx0 terms are ignored on the right hand side, we get
x_ 0 k os(!t)
(1.47)
q
(1.48)
y_0 kk sin(!t)
2
The boundary onditions an be hosen so that y0(0) = 2q k and x0(0) = 0,
whih gives
1
(1.49)
x0(t) = k sin(!t)
k
q
y0(t) = k os(!t)
(1.50)
2
whih desribes an ellipse. The relation between x0 and y0 is then
x 2 y 2
0 + 0 =1
(1.51)
a
b
with a = k =k and b = 2q k . When k gets smaller, the motion gets more and more
2
in the x-diretion. The area of p
the ellipse is ab = q
2k k . The area is independent
of k if the normalization k k is hosen.
In this estimate of the motion, a lot of eets have been ignored. The sattering of the wave on the vortex will, as seen in hapter 2 and 3, exert a fore
on the vortex. If the vortex has a mass2 it would give a drift of the vortex in
the diretion of the fore. If the analogy to eletro-magnetism is applied (setion
1.4), where vorties are interpreted as harges, one ould expet a radiation of
energy from the aelerating vortex. The vortex losing energy will spiral inwards.
1.4 Analogy to Eletro-Magnetism
For a long time there has been known analogies between two-dimensional eletromagnetism and hydrodynamis on a thin lm. Before Maxwell's equations were
2 The
question of the vortex mass is still unsolved.
1.4.
ANALOGY TO ELECTRO-MAGNETISM
23
well established people ould gain insight on eletro-magnetism from suh analogies by experimenting on uids. An introdution to the analogies between a
lassial uid and eletro-magnetism an be found in [Myk96℄. Today the situation is the other way round: eletro-magnetism is very well understood and
studied, while there still exist phenomena in uid mehanis whih are still not
entirely understood. One suh phenomenon is quantized vorties in superuid
lms, whih an be shown to have some of the same behavior as (quantized)
point harges in an eletro-magneti theory. A referene starting with GinzburgLandau theory is an artile by D.F. Arovas and J.A. Freire [AF96℄. They work
diretly with the Ginzburg-Landau Lagrangian density. By requiring that the
partition funtion shall be invariant, they an integrate out some of the variables
and obtain a Lagrangian whih is to seond order in the elds equal to the QEDLagrangian. We will do a more simple and diret approah instead, by starting
with the eld equations on polar form, (1.20) and (1.21), and try to transform
them into Maxwell's equations.
There also exists an analogy to the potential formulation of eletro-dynamis,
whih is desribed in [Myk96℄. In hapter 2 the analogy between phonon sattering on a vortex and Aharonov-Bohm sattering is disussed. This analogy omes
from terms quadrati in the elds and will not be transparent in the linearized
equations onsidered here.
1.4.1 2-D Eletro-Magnetism
First we will take brief look at two-dimensional eletro-magnetism, where the
E-eld is lying in the plane while the B -eld is perpendiular to the plane. This
B -vetor an hene be treated as a salar, sine it has a xed diretion. In two
dimensions there are only three Maxwell's equations, sine r B = 0 always
holds.
r E = E
+ rB ez = J
t
B
+rE =0
t
(1.52)
(1.53)
(1.54)
The equations are Gauss' law, Ampere's law and Faraday's law. The density is the harge density and J is the urrent density. Instead of using the elds B
and E, we ould use a potential formulation and introdue a Lagrangian density
as a funtion of the potential and A. Then the rst two equations would be
the eld equations, while Faraday's law would be a onsequene of the denitions
B = r A and E = r A_ .
Charged partiles moving in an eletri and magneti eld will also experiene
a fore; the Lorentz fore. If the partile position is r0 and it has mass m and
24
CHAPTER 1.
BACKGROUND
harge Q, the fore is simply
mr0 = Q(E + B r_ 0 ez )
(1.55)
where E and B is the eletri and magneti eld evaluated at the partile position.
These are all the basi equations needed in lassial eletro-magnetism in two
dimensions.
1.4.2 The Analogy to Eletro-Magnetism
In dimensionless units, the Ginzburg-Landau eld equations on hydrodynamial
form, (1.20) and (1.21) are
_ + r (r) = 0
1 1 2p
1
_ + (r)2
p r + 1=0
2
2 We want to do the identiation
Q = 2q
E = r ez
B=1 (1.56)
(1.57)
(1.58)
where Q will be identied as harge, B as the magneti eld and E as the eletri
eld. The most ommon way is to dene the eletri eld as proportional to
r ez . The ontinuity equation, (1.56), is then Faraday's law exatly. The
reason why we hoose to do it dierently, is that our hoie makes the E-eld
singular. This makes it possible to draw the analogy to point harges and still
keep information from the density prole3 . If there are N vorties present with
positions rk (t), Gauss' law is diretly found from the denition of E
r E = r (r ez ) =
X
k
2qk Æ (r
rk ) v
(1.59)
where the delta funtions are identied from r r' = 2Æ (r) found in appendix B. The quantized vorties appear as point harges in the eletro-magneti
formulations! The harges are given by Q = 2q . In full units this would have
been proportional to the quantized irulation of the vortex.
In the rst equation (1.56) the substitutions an be done without muh trouble. In the seond (1.57) we have to take the gradient of the whole equations and
take the ross produt with ez to be able to do the identiations. The gradient
and the time derivative do not ommute,
sine
the E-eld is singular. From ap
pendix B the ommutator is found r ; t '0 ez = 2 r_ 0 Æ (r r0). The vortex
urrent an be identied
X
r ; t ez = 2qkr_ k Æ(r rk ) Jv
k
3 In
most analogies only point vorties are onsidered.
1.4.
25
ANALOGY TO ELECTRO-MAGNETISM
The two hydrodynamial eld equations are now
B_ + r E = B r E E (rB ez )
2 p1 B r
1
E_ + rB ez = Jv + r E2 + p
ez
2
1 B
(1.60)
(1.61)
To ompare this with the last two of Maxwell's equations we must linearize in B
and E. It must be noted that linearization is only valid far from the vortex ores,
sine B ! 1 near the vortex. After the linearization the equations are still not
exat Maxwell's equations, sine (1.61) has an additional term r2B . In the
low energy limit, where the B eld is slowly varying, this term an be negleted.
The analogy to eletro-magnetism is hene valid just in the low energy limit.
1.4.3 The Lorentz Fore
The last equation needed is the Lorentz fore, or the fore on a moving harge in
an eletro-magneti eld. In hydrodynamis this must be analogous to the fore
on a moving vortex in a Bose-liquid. In setion 1.3 an equation for the motion of
a vortex (1.43) was derived
h
r_ 0 = r + q r ln ez
2
i
r=r0
(1.62)
where r0 is the vortex position. Applying the analogy (1.58), this an be written
Q [E + r_ 0 Bex ℄ = rB
(1.63)
where a large onstant external magneti eld is dened to be Bex = ez . The
evaluations of E and rB are still at the vortex position r0. The left hand side
of (1.63) looks pretty muh as a Lorentz fore. The term r_ 0 Bex is also equal
to the bakground uid Magnus fore, whih is disussed in setion 3.4.
The right hand side is not proportional to the vortex aeleration as expeted
for a massive vortex, nor zero, as for a massless vortex. From the form of the
equation (1.63) we an thus not draw any onlusion about the vortex mass.
In [AF96℄ the right hand side of the equation was zero, implying a massless vortex.
The reason why we arrived on dierent looking result, is that the derivation done
in setion 1.3 kept information from the density prole of the vortex, while most
analogies are done within the point vortex approximation.
26
CHAPTER 1.
BACKGROUND
Chapter 2
Phonon Sattering
A well-known phenomenon in quantum physis is the Aharonov-Bohm eet,
whih says that an eletron beam is aeted by a magneti ux-string, even if
the eletron itself is not in diret ontat with the ux. The whole eet is due
to the vetor potential around the ux-string and hene a pure quantum phenomenon. In this hapter, we will see that this is to some extent analogous to
the eet of a stationary vortex string on a phonon wave, whih will be alled the
aousti Aharonov-Bohm eet. We will exploit this analogy by using the well
known solutions of the Aharonov-Bohm eet to disuss our aousti sattering
problem. In resent time there has been a lot of interest in this subjet, sine it is
still not entirely understood and it is important when disussing the fores ating
in helium II. A more omplete disussion of the fores in helium II is held in hapter 5. In this hapter only the eet of a single plane wave on a stationary vortex
is onsidered. The disussion will mainly follow the arguments of Sonin [Son96℄
and Stone [Sto99℄. The original paper by Aharonov and Bohm [AB59℄ will also
prove a useful soure. The summation in the partial wave analysis will be done
aording to Sommereld and Minakata [SM00℄.
2.1 The Sattering Equation
There are at least two possible starting points when disussing the sattering of
a plane wave on a vortex. The rst one is the lassial hydrodynamial equation
for a superuid; the seond is the quantum mehanial Ginzburg-Landau theory.
We will use the latter in the semi-lassial Gross-Pitaevskii approximation. If
the vortex line is straight, we have ylindrial symmetry and eetively a twodimensional problem.
We start with the Shrodinger equation on polar form. With one vortex
situated at the origin, = +q' and = f (1+ ) are substituted. If the deviation
from a stationary vortex solution is small, the equations (1.20) and (1.21) an be
27
28
CHAPTER 2.
linearized in and , to obtain
1 2
_ + r = qr' r + (1
4
PHONON SCATTERING
1
f 2 ) + r ln(f ) r
2
(2.1)
and
_ + r2 = f q r' r 2r ln(f ) rg
(2.2)
where the parameter = 1 is introdued to ease the book-keeping. Notie
that = 0 orrespond to no vortex present, giving the wave equations studied
in (1.2.2). In the ase of plane waves, without disturbing vorties, r2 = k 2 .
In the long wavelength limit, where k 1, we thus have
jr2j jj
(2.3)
We will now suppose that this ondition holds with a vortex in the bakground
too. The r2 term in (2.1) will thus be ignored. It is not obvious that small
k always ensures that r2 is small when the vortex is present, but this an be
expliitly heked later when the solutions are found (see setion 2.4.5).
The equations (2.1) and (2.2) will now be solved by treating the appearane
of the vortex as a perturbation. We want the solutions to be as lose as possible
to plane waves, sine suh waves later enable us to introdue temperature in the
uid. In the point vortex approximation f = 1, and q ould have been used as the
perturbation parameter. But sine we will try to extrat some information from
the density prole, the new parameter = 1 is introdued. Both and q have
the disadvantage that they are not small numbers themselves. The perturbation
must then in some sense reet that the operators appearing with the vortex
shall not be too inuential. This an happen in at least two ways. First: sine
the operators ontain derivatives, their eet an be small in the low energy
limit. Seond: The operators an be small far from the vortex. Typially the
phase ontribution goes as r 1 and the density as r 2 . In full units we
would in addition have notied that the vortex ontributions are proportional to
the ux quantum mh , and the perturbation ould also be seen as a semi-lassial
approximation in powers of h. Formally all this redues to series expansions
in . From the rst equation (2.1) the density an be expressed by the phase
exlusively
1
2
= _ + q r' r
1 f + r ln(f ) r
+ O 2
(2.4)
2
t
Inserting this into the equation (2.2), the sattering equation is found
2
2
r t2 = 2qr' r t (2.5)
1 2
2
2
2
)r ln(f ) r + O (1 f ) 2 2(1
t
4 t2
2.2.
29
ANALOGY TO AHARONOV-BOHM SCATTERING
Further we assume a harmoni time dependeny
(r; t) = (r)e
i!t
(r; t) = (r)e
i!t
(2.6)
In the low energy regime the frequeny is ! jkj. After dropping a higher order
term in k and putting bak to unity, the sattering equation is
2
r + k2 = 2ikqr' r (1 f 2)k2 2r ln(f ) r (2.7)
In most ases only the phase ontributions to the equations will be onsidered,
sine it is probably dominant. The funtion f will then be put to unity.
The ontents of the rest of this hapter are disussions and solutions of the
sattering equation (2.7). In setion 2.2 the analogy between the sattering equation and Aharonov-Bohm sattering is onsidered. In setion 2.3 the equation is
solved by using the Born approximation, while in 2.4 the tools of partial wave
analysis is used. The density orretions are studied in setion 2.5.
When the phase is known, the density utuation an be found from (2.4).
2.2 Analogy to Aharonov-Bohm Sattering
Consider a magneti ux-string piering a plane while the magneti eld is zero
anywhere else. The vetor potential generating the magneti eld is non-zero
however, and it will interat with a passing eletron beam, even if the beam is
not in any diret ontat with the ux itself. The eet is then totally due to
the vetor potential, and this is what is alled the Aharonov-Bohm eet. The
time-independent Shrodinger equation for an eletron interating with a vetor
potential is
h
e i2
r + i A = 2mE
(2.8)
where E = 2pm is the eletron energy and } = 1. A ommon example of suh a
potential is one giving a magneti eld proportional to a delta funtion
A = 1 mr'
(2.9)
2
where m is the total magneti ux and r' = r1 e'. It is onvenient to dene
e . In their artile [AB59℄, Aharonov and Bohm solved this equation by
= 2
m
partial wave analysis and from that solution they extrated, in the limit of large
r, the famous expression1
2
eikr os(')
1 In
i(' ) +
aAB (') ikr
pr e
their paper, the inoming wave was from the right, not left as in our ase.
(2.10)
30
CHAPTER 2.
PHONON SCATTERING
where the Aharonov-Bohm sattering amplitude is
aAB (') =
'
sin() e i 2
p sin ' 2ik
2
(2.11)
Notie that the rst term in (2.10) solves the Aharonov-Bohm equation (2.8) but
this solution does not satisfy the right boundary onditions. Aharonov and Bohm
solved the equation (2.8) by partial wave analysis, and this method is presented
in setion 2.4. One problem with partial wave analysis is to determine the right
boundary onditions. In the original paper [AB59℄ the onstraint was that the
urrent density should be along the x-axis asymptotially far from the vortex.
Exatly the same solution was found by M.V. Berry [Ber80℄ by requiring that
the wavefuntion should be single-valued. The single-valuedness of the solution
is not transparent in the asymptoti form (2.10) but is obvious in the partial
wave expression. The partial wave expression for the wavefuntion is nite and
well behaved at all distanes r from the origin and at all angles '. The asymptoti expression for the sattering amplitude (2.11) is on the ontrary singular
as ' ! 0. This singularity appears beause the attempt to extrat a sattering amplitude fails near the forward diretion! A dierent expression is needed.
A good treatment of the near forward diretion is found in an artile by C.M.
Sommereld and H. Minakata [SM00℄. E.B. Sonin has also derived a version for
the aousti problem [Son96℄. It turns out that the forward diretion will be
extremely important when the fore on a vortex is found in hapter 3.
An analogy to the magneti Aharonov-Bohm sattering an be found in superuid helium, where the time independent phase variation of the Ginzburg-Landau
wavefuntion in (2.7) plays the role of the eletron wavefuntion in (2.8). The
stati eld around the point vortex plays the role of the vetor potential. In the
point vortex approximation, the aousti sattering equation (2.7) is
r2 + k2 = 2ir' r
(2.12)
where the vortex ux is = qk , and q is quantized, q = 1, with sign dependent
whether we have the vortex, or the anti-vortex solution. Working to rst order in
the vortex ontribution, a seond order term in might as well be added, giving
[r + ir'℄2 k 2
(2.13)
whih is similar to the original Aharonov-Bohm equation. The rewriting an be
done sine r2' = 0. In ontrast to the original Aharonov-Bohm problem, where
the magneti ux is an external parameter whih an be hanged independently
of the eletron energy, the total ux in the aousti ase is dependent of the
frequeny ! of the inoming wave. For a situation with a xed wave number this
will not ause any problems, but as soon as we have to manipulate the boundary
onditions in a speied situation this dierene will be visible.
2.3.
31
SOLUTION BY BORN APPROXIMATION
2.3 Solution by Born Approximation
A standard method of solving sattering equations is the Born approximation. It
gives the solution of the equation to rst order in the sattering parameter. The
sattering equation (2.7) an be written as
(k 2 + r2) = O
(2.14)
where the operator O is
O=
2ikq r' r
(1
f 2 )k 2 2r ln(f ) r
(2.15)
The solution of the equation to rst order in is
= k eikr + Æ + O 2
(2.16)
where k is a onstant. The sattered wave an be expressed by the Green's
funtion, whih is a solution of the equation
(r2 + k 2 )D(r) = Æ 2(r)
(2.17)
The two-dimensional time-independent Green's funtion is
i
D(r) = H0(1) (kr)
4
(2.18)
found in appendix C. The funtion H0(1) (kr) is the zeroth order Hankel funtion
of the rst kind. The Born solution for the sattered wave is
i
Æ =
4
Z
d2 r0 N (r0 ) H0(1)(k jr r0j)
(2.19)
The funtion N (r) is
N (r) =
Oeikx
sin(')
= 2k
r
k 2 (1
0
f
f 2 ) + 2 ik os(') eikr os(')
f
(2.20)
where = qk and f = f (r) is the density prole of the vortex. Notie that
in a point vortex desription, with f = 1, the last two terms disappear, leaving
only the rst term, whih we will all the phase ontribution. Most likely this is
the most important term, but we are interested in seeing if the other two terms
give any onsiderable ontribution, so they will be kept until a good reason to
throw them away has been found. The last two terms will be alled the density
ontributions sine they are the ontributions from the density prole of the
vortex.
32
CHAPTER 2.
PHONON SCATTERING
2.3.1 The Sattering Amplitude
In sattering problems it is often possible to separate the r dependeny and the
angle dependeny in the sattered wave. The fator whih is a funtion of ' is
then the sattering amplitude. The sattering amplitude for a phonon sattered
on a vortex will now be found.
The phase ontribution to the sattering equation is usually onsidered as
most important, so in the following we will only use the rst term in (2.20),
giving the sattered wave as
Z
ik
sin('0 ) (1) 0
Æ =
d2 r0
H0 (k jr rj)eikr os(' )
(2.21)
2
r0
The eet of the density prole of the vortex will be studied in setion 2.5. The
Born integral is diÆult to arry out in general and we need to do dierent
approximations. The ordinary sattering amplitude is found by assuming that
the main ontribution to the integral omes from the region where k jr r0 j 1.
Doing series expansions in r 1 , we get
0
jr r0j = r
r0 os('0
0
') + O r 1
(2.22)
The expansion of the Hankel funtion for large arguments is
H0(1) (z ) z!1
!
r
2 iz
e
iz
(2.23)
A trigonometri identity is also needed
'
'
)
') = 2 sin( ) sin('0
2
2
Notie that if ' 0 the '0 dependeny in the exponent in the integrand disappears and therefore more terms are needed in the expansion to get a orret
result for small angles (setion 2.3.3). If the integration is performed in polar
angles (r0 ; '0), the integral is
os('0 )
os('0
r
Z
Z
2 ikr 1 0 2 0
ik
d' sin('0)e 2ikr sin('=2)sin(' '=2)
dr
Æa = e
2 ikr
0
0
First the integration with respet to '0 is done. Sine the integration is over a
whole period, the substitution '0 ! '0 + '2 an be made without hanging the
integration limits
0
Z 2
0
=
Z 2
0
0
d'0 sin('0 + '=2)e i2kr sin('=2)sin(' )
0
0
d'0 [sin('0 ) os('=2) + os('0 ) sin('=2)℄ e i2kr sin('=2)sin(' )
0
0
2.3.
SOLUTION BY BORN APPROXIMATION
33
By substituting u = sin('0) it is easily seen that the seond integral is zero. The
rst integral an be rewritten to a form reognizable as a Bessel funtion (see
appendix A)
Z 2
d'0 os('0 ) e i2kr sin('=2)sin(' ) = 2i J1 ( 2kr0 sin('=2))
0
0
The r integration is now only the integral over a Bessel funtion. The ondition
that ' shall be non-zero is now learly seen to be essential to the r0 integration
Z 1
1
dr0 J1 ( 2kr0 sin('=2)) =
2k sin('=2)
0
The nite angle sattered wave an far from the vortex be separated in a ' part
and a r part. When all onstants are gathered, the phase ontribution to the
sattered wave an be written as
a(')
Æa = p eikr
(2.24)
r
0
0
where the sattering amplitude is
'
a(') = p
(2.25)
ot( )
2
2ik
This is the most ommon form seen for the sattered wave and it is valid for nite
angles. The index a, at the sattered wave, is to remind us that this is the term
whih an be expressed by the sattering amplitude. The result deviates from
the original Aharonov-Bohm sattering amplitude (2.11) expanded to rst order
in , by an additional s-wave. The partial wave analysis (see setion 2.4.2) will
show that this s-wave omes from the seond order term in whih marks the
dierene between the aousti and magneti Aharonov-Bohm sattering.
The Aharonov-Bohm sattering amplitude is divergent and beomes innitely
large as the angle reahes the forward diretion. In the setion about small angle
sattering (setion 2.3.3), we will see that this apparent singularity is not physial,
but a onsequene of how approximations are dealt with.
2.3.2 The Hidden Terms
The asymptoti wave in the original Aharonov-Bohm result (2.10) is, opposed
to the Born solution, not a plane wave but of the form eikx i(' ) . The
missing fator is the twist of the inoming wave, but sine it is proportional to
the perturbation parameter , it is still hope that it is not gone for good, but has
been lost in the approximations done when extrating the sattering amplitude
(setion 2.3.1). The sattered wave in the Born approximation was
Z
ik
y 0 ikx (1) 0
2
0
(2.26)
d r 0 2 e H0 (k jr rj)
2
r
0
34
CHAPTER 2.
PHONON SCATTERING
40
20
0
-20
-40
-40
-20
0
20
40
Figure 2.1: The real part of the sattered wave (2.24) expressed by the Born
sattering amplitude (2.25). The plot learly shows the phase dierene at the
positive x-axis. What is not lear from the plot, however, is that the sattered
wave beomes innitely large when ' ! 0.
The region in r0 spae giving the sattering amplitude was r0 r. The hidden
terms must then be found by some other approximation. We will now searh in
the region where jy y 0j jx x0 j and jy j jy y 0j. Sine these ontributions
are entered around the point r0 r, it is onvenient to do a hange of variables
ik
eikx
2
Z
y 00 + y ikx (1) 00
d2 r00 00
jr + rj2 e H0 (kr )
00
(2.27)
where we have substituted r00 = r0 r. In last setion the asymptoti form of the
Hankel funtion was used on the basis of r r0 . Now the same approximation
is used, with the assumption that jy 00j jx00j. The Hankel funtion is then
H0(1) (kr00) s
ik
2
2
eikjx j+ 2 x (y )
00
ik jx j
00
j 00 j
00
(2.28)
Supposing that jy j jy 00j, gives us a Gaussian integral in y 00, whih by the
ordinary formula (appendix A) gives
s
2
ik jx00 j
Z 1
1
dy 00 e
k
2ijx00 j
(y )2 = 2
00
k
2.3.
35
SOLUTION BY BORN APPROXIMATION
The ondition that jy j jy 00 j makes the result just valid for nite angles. Introduing the step funtion jx00j + x00 = 2(x00 ) gives us the hidden terms as
Æhidden =
ieikx
Z 1
y
e2ikx (x )
dx00 2
00
2
y
+
(
x
+
x
)
1
00
00
This integralan be solved by doing a trik.
We have
y = artan( y ) sign(y )(x). The step funtion (x) does not onx
y2 +x2
x
tribute to the derivative, but it makes the whole funtion ontinuous sine it
anels the singularity in the Artan( xy ) funtion.
Z 1
y
e2ikx (x )
dx00 2
00
2
y
+
(
x
+
x
)
1
x =1
y
00
2
ikx
(
x
)
sign(y )(x + x) e
=
artan 00
x +x
x = 1
Z 1
y
00
2
ikx
00
+
sign(y )(x + x)
e
dx artan
x + x00
x00
0
00
00
00
00
00
00
00
where the integration limit in the last integral has hanged beause of the step
funtion in the exponent. The rst term vanishes. The artan( x y+x ) is zero in
both limits and in the 1 limit is the step funtion zero, while the e2ikx (x )
term disappears
in the 1 limit beause of rapid osillations. The funtion
y
sign(y )(x00 + x) is ontinuous and slowly varying. Opposed
artan x+x
to this, the last term is rapidly osillating, and most ontribution to the integral
omes from small x00 . An approximation is to put the slowly varying term outside
the integral, evaluated in x00 = 0. The remaining integral just gives 1, so the
hidden terms are
h
y
i
Æhidden = ieikx artan
sign(y )(x)
(2.29)
x
00
00
00
00
The angle ' 2 (0; 2 ) is ' = artan( xy ) + ( x) + 2 (x)( y ). This gives
the hidden terms as
Æhidden = iekx(' )
(2.30)
To rst order in , this is the lost fator in eikx i(' ) from the original AharonovBohm solution. In the Born approximation, this term appears as a part of the
sattered wave sine it is proportional to . But it is proportional to exp(ikx),
not exp(ikr), and it does not disappear far from the vortex, whih makes it more
natural to interpret it, in aordane with Aharonov and Bohm, as a modiation
of the inoming wave. The Born approximation is usually applied on sattering
problems where the sattering enter is well loalized so that the inoming waves
are just plane waves. In the sattering of a phonon on a vortex this is impossible,
36
CHAPTER 2.
PHONON SCATTERING
whih is a manifestation of the long range of the vortex; there is no loalized
sattering enter. Despite this, the Born approximation gives the orret answer,
and the reason why these non-sattering terms are usually not found in the Born
approximation, is just beause of the mathematial approximation done in the
evaluation of the integral.
The sattered wave found in setion 2.3.1 together with these hidden terms,
form the solution for nite angles. Near the forward diretion this solution is not
valid, whih leads us on a searh for a spei small angle result in setion 2.3.3.
2.3.3 Small Angle Sattering
In setion 2.3.1 an expression for the sattered wave was found, but the solution
diverged for small angles. By doing dierent approximations an expression whih
an desribe the important forward diretion will now be looked for. Sine our
inoming wave is propagating in diretion of positive x, the small angle region is
the area entered at the positive x-axis. The integration will here be performed in
Cartesian oordinates x0 and y 0 . The main ontributions to the integral is thought
to ome from the terms where r0 is small, so that we an do series expansion of
jr r0j with x y; x0; y0 as a ondition. This derivation relies on one done by
Sonin [Son96℄. The only dierene is that he uses polar oordinates desribing r.
We use the following expansion
jr r0j = x
x0 +
1
(y
2x
y 0 )2 + O x 2
Compared with the expansion done in setion 2.3.1 one extra term must be kept,
sine the x0 term anels against the +x0 term from the inoming wave. The
extra terms also saves us from a breakdown when x0 x, so in the small angle
sattering there is no need to searh for hidden terms as in setion 2.3.2. The
asymptoti expansion of the Hankel funtion for large arguments (2.23) gives
H0(1)(k jr
r0j)eikx
0
r
2 ikx+ 2ikx (y y )2
e
kxi
0
The small angle sattered wave an hene be written
r
2 ikx
ik
Æs e
2 ikx
Z
y 0 ik
2
d2 r0 0 2 e 2x (y y )
r
0
(2.31)
There is no x0 dependeny in the exponent, so the x0 integration an simply be
done
Z 1
dx0
= 0
2
2
0
0
jy j
1 x +y
2.3.
37
SOLUTION BY BORN APPROXIMATION
The y 0 integration is a bit more bothersome. It an not be arried out ompletely, but by some manipulations it an be transformed to an integral where
all y -dependeny is in the integration limits.
Z 1
y 0 ik
2
dy 0 0 e 2x (y y)
1 jy j
Z 1
u + y 2ikx u2
=
du
e
1 ju + y j
=
=
0
Z
y
1
Z 1
=2
Z y
0
y
+
+
Z 1
y
Z 1
y
ik 2
du e 2x u
ik 2
du e 2x u
ik 2
du e 2x u
The small angle result an then be expressed as the integral
r
Z
ik 2
2ik ikx y
(2.32)
Æs = e
du e 2x u
x
0
It is not a problem that the nal result an not be expressed by elementary
funtions, sine usually only the derivative with respet to y is needed. The
answer an be written more ompat with the error funtion
Æs = i eikx
r
k
erf y
2xi
!
(2.33)
R
with the error funtion dened as erf(z ) = p2 0z dt e t2 . This integral is well
dened in the limit y ! 0, so that there is no trouble with singularities in ontrast
to the sattering amplitude expression (2.25).
2.3.4 Disussion of the Born Approximation
From the Born approximation three dierent results have been found: two being valid for nite angles and one whih is valid just for small angles. All the
terms originate in the same integral and they appear as a result of the dierent approximations. If one is lever enough it should be possible to extrat one
asymptoti expression for all angles, but for so long the dierent ases must be
treated separately. The Born approximation result an be summarized as
"
r
k
Born
=k eikx + i erf y
s
2xi
!#
(2.34)
38
CHAPTER 2.
PHONON SCATTERING
for small angles and
a(')
i(' )) + p eikr
=k
(2.35)
r
for nite angles. The sattering amplitude is
'
a(') = p
ot( )
(2.36)
2
2ik
Beause of the long range of the vortex, it is a twist on inoming waves. Both
solutions are ontinuous in their domains. The error funtion is well dened for
small argument and atually it approahes zero at the positive x-axis, leaving
just a plane
wave there. The argument of the error funtion goes to innity for
p
x y x, whih are the largest angles where the small angle approximation
is still valid. The asymptoti expansion of the error funtion for large arguments
is
erf(z ) z!1
! sign(<(z)) p1 z1 e z2
(2.37)
Born
a
eikx (1
For small angles, the approximations
' y=x and r x(1 + 2yx2 ) are valid. In
p
the region ' 1 and r' 1, the small angle expression (2.34) an thus be
written
eikr 2
ikx
Born
p
s k e [1 + i sign(y )℄ + p
(2.38)
2ik r '
The rst term is the plane wave, while the seond omes from the twist of the
inoming wave. The last is the sattered wave. The expression is exatly the
same as the nite angle expression (2.35), expanded for small angles. The Born
solution is thus ontinuous.
The onlusion of the Born approximation is that it gives a ontinuous solution
to the sattering problem. The solution seems to be in agreement with the
original Aharonov-Bohm result, to rst order in . In addition a speial small
angle result is developed, giving a ontinues solution at all angles. The hidden
term found in 2.3.2, giving the twist of the inoming wave, was essential to get
a ontinuous solution. The solution presented here is the math of dierent
solutions in dierent regions, but sine all these solutions ome from the same
integral it should be theoretially possible to get one solution valid at all angles,
just as done in the partial wave expansion (setion 2.4.3).
2
2.4 Solution by Partial Wave Analysis
In the original artile by Aharonov and Bohm [AB59℄, the Shrodinger equation
was solved by partial wave analysis. This means that the solution is an expansion
2.4.
SOLUTION BY PARTIAL WAVE ANALYSIS
39
in eigenstates of the angular momentum operator. To write down a general
expression for the wavefuntion is very easy, so the worry is how to treat the
boundary onditions right, and later how to sum up. We will see that there
are dierenes between the aousti and magneti Aharonov-Bohm sattering
in deiding the boundary onditions, but that the result is mainly the same.
Aharonov and Bohm also provided a method for doing the summation when kr
is large, but their result is not satisfatory right behind the vortex, and we will
instead present a solution found by Sommereld and Minakata [SM00℄. This is
done in setion 2.4.3.
In ontrast to the Born approximation, the partial wave analysis is nonperturbative by nature and the alulations are best done by keeping all orders
of . But the sattering equation was derived by perturbation, whih means that
the answer obtained is just valid to lowest order. The sattering equation for a
point vortex is
(2.39)
r2 + k2 = 2ir' r
where = qk is still supposed small ompared to unity. The partial wave expansion is
(r) =
1
X
l= 1
l (r)eil'
Sine l is just running over integers we have no problems with multivaluedness.
Substituting this into our equation all derivatives of '2 just give the
eigenvalues
2
il
1
l
2
2
il, whih means that r' r ! r2 and r ! rl = r2 + r r r2 . Demanding
that the equation holds for eah l, we get the Bessel equations
2 1 l2 + 2l
2
k + 2+
l = 0
(2.40)
r r r
r2
where the ux = qk . Note that in the magneti Aharonov-Bohm problem, we
would have got an additional term 2 =r2 . The total solution an be written as
superpositions of Bessel and Neumann funtions
l = l J
l (kr) + l N
l (kr)
(2.41)
p
where k = l2 + 2l, and the onstants l and l are dependent of k . The
Bessel funtions J l (kr) with negative subsripts ould have been used instead
of the Neumann funtions as well, but sine this, in the aousti ase, ause a
degeneration for l = 0, the Neumann funtions are hosen.
2.4.1 Boundary Conditions Near the Vortex
In the magneti Aharonov-Bohm sattering, all the onstants l in the general
partial wave analysis solution (2.41) are zero. This an be found by examining a
40
CHAPTER 2.
PHONON SCATTERING
ux tube of nite ore size, with the onstraint that the total wavefuntion shall
be normalizable. Sine Neumann funtions get arbitrarily large for small arguments, a wave-funtion ontaining the Neumann funtions is not normalizable
when the size of the ux tube is shrunk to zero.
The solution of the aousti Aharonov-Bohm problem presented in this setion, is a solution within the point vortex approximation. Opposed to the magneti problem, it is in the aousti ase not possible to make an arbitrary solution
with nite ore size, sine the density prole of the vortex is determined of the eld
equations. Instead we will try to derive advantage from the known behavior of the
vortex prole. The sattering equation (2.7) was derived by requiring that presene of the vortex ould be treated as a perturbation of a wave equation. Near,
or inside, the vortex ore this ondition fails, sine the free wave ontribution
in the equations are vanishing ompared to those terms ontaining ontributions
from the vortex. However, the original linearized eld equations (2.1) and (2.2)
an be used inside the vortex ore, where the vortex prole an be approximated
with f r. If the elds and and their derivatives are non-singular, the only
singular terms are r' r 1 and r ln(f ) r 1 . So the terms ruling inside the
vortex ore are
1
q r' r + r ln(f ) r 0
2
q r' r 2r ln(f ) r 0
Atually these are the same equations that were found in setion 1.3, about
vortex motion, applied on a stationary vortex. In this setion the main onern,
thought, is the boundary onditions in the partial wave analysis. Let us then use
the partial wave expression for the elds, r' ! ril2 e' and r ln(r) ! r1 er . The
above equations then look like
iql
1 l
l +
=0
2
r
2r r
2 l
iql
l
=0
2
r
r r
If l is isolated from the seond equations and put into the rst, the result is a
seond order ordinary dierential equation in l alone.
2l
+ r l l 2 l = 0
(2.42)
2
r
r
As all seond order partial dierential equations, the general solution an be
written as a superposition of two funtions. In this ase the funtions are l rjlj . Beause of degeneray there is for l = 0 an additional solution 0 ln(r).
The total solution for must be nonsingular when r ! 0, whih means that
r2
l r+jlj
(2.43)
2.4.
41
SOLUTION BY PARTIAL WAVE ANALYSIS
40
20
0
-20
-40
-40
-20
0
20
40
Figure 2.2: The real part of the partial wave analysis solution to the sattering
problem (2.48). The plot is generated with jlj 60 and = 0:25.
The partial wave expression (2.41) should be t to the onstraint (2.43). Taking
the limit r ! 0 of (2.41) is not allowed, sine it is a point vortex result. But kr
an be taken to zero, by keeping r xed and letting k ! 0. For small angles the
Bessel funtion goes as J (z ) z while the behavior of the Neumann funtions
are N (z ) z when 6= 0 and N0 (z ) ln(z ). We thus have the ondition
k!!0 0
(2.44)
l
In the low energy limit the onstants l are vanishing, just as in the magneti ase.
The fat that our point vortex solution just satises the onstraints for small k ,
an be seen as a lear indiation that the point vortex approximation is just valid
in the long wavelength limit. This is natural sine waves with shorter wavelengths
will have more interferene with the vortex ore. Pratially this means that the
point vortex approximation works ne together with long wavelength solutions,
but if the intention is to keep higher orders in k , it is more dubious if the point
vortex approximation holds. This would for example have onsequenes for the
alulation of the longitudinal fore on a vortex, sine it is higher order in k .
2.4.2 Boundary Conditions Far From the Vortex
In last setion restritions were put on the general partial wave solution (2.41)
from onditions on the behavior lose to the vortex point. Now the behavior
at a large distane from the vortex is onsidered. The normal uid in helium II
42
CHAPTER 2.
PHONON SCATTERING
(hapter 4) is onstruted as a weighted integral over plane waves. To identify the
normal uid it is then a need for solutions whih are as lose as possible to plane
waves, far from the vortex. The ideal situation must then be a solution whih far
from the vortex an be separated in an inoming plane wave, and an outgoing
sattered wave. We will thus apply the onstraint that eah partial wave far
from the vortex an be divided in one term belonging to the inoming plane wave
and one term proportional to exp(ikr), demanding that the terms proportional
to exp( ikr) are zero. But this does not mean that the total solution an be
divided in a plane wave and a term proportional to exp(ikr). Beause of the
summing of l to innity the total wave looks dierent, as we will see.
For simpliity, the x-axis is put along the diretion of the plane wave, whih
in partial wave expansion looks like
eikr os ' =
X
l
il Jl (kr)eil'
(2.45)
The asymptoti expansions of the Bessel funtions for large kr are also needed
r
2
os(kr
)
(2.46)
kr
2
4
An expansion like this proves to be problemati when summing up, sine it is just
valid for kr. It is, however, working ne when applied at one partial wave
at a xed l. The partial waves of the sattered phonons are l = l J
l (kr) where
the onstants l are to be determined. Subtrating the partial waves of exp(ikx),
writing out the asymptoti expressions and deoupling the osine terms, leads to
J (kr)
kr!1
!
r
2 1 i(kr 4 ) e
l e i 2 l
r kr 2
2 1 i(kr 4 ) i 2 l
e
+
l e
kr 2
l (kr) ilJl (kr) =
1
eil
In the partial wave analysis the overall onstant k will be put to unity. It an
easily be put bak in the nal answers. Our boundary onditions are fullled if
we let the oeÆient before exp( ikr) be zero, whih leads to
(2.47)
l = eil i 2 l
The onstants are then determined so that the total wave an be written as
X
= ( 1)l e i 2 l J
(kr)eil'
(2.48)
l
l
The solution is plotted in gure 2.2. The anti-symmetry between the upper and
lower half plane is learly seen. We also notie that even though the tear right
behind the vortex gets broader far from the vortex, the angle it oupies gets
2.4.
43
SOLUTION BY PARTIAL WAVE ANALYSIS
smaller. This leads to the apparent singularity in polar oordinates at an innite
distane behind the vortex. The plot an be ompared to the photographs of
water waves sattered on a lassial vortex in [BCL+ 80℄.
p
The subsripts of the Bessel funtions are in the aousti ase l = l2 + 2l.
For small values of this is approximately the same as the magneti version,
namely jl + j. An exeption is for the s-wave, or the l = 0 term. To rst order
in , the dierene between aousti and magneti Aharonov-Bohm sattering is
=e
i 2 J
(kr)
J0 (kr)
(2.49)
For large arguments the expansion of the Bessel funtion (2.46) an be safely
used, so that to rst order in r
i
ikr
e
2ikr
(2.50)
whih is exatly the dierene between the found solutions of the magneti (2.11)
and aousti (2.25) sattering problem.
The sum (2.48) is onvergent for all kr 2 , sine for the index muh greater
than kr the Bessel funtions behave as
J (kr) p 1 ekr
2kr 2
This means that when doing numerial alulations there is just need to keep
terms with l just a bit larger than kr.
When expanding the Bessel funtions in terms of osine funtions this onvergene is lost, so that our expression for the wave beomes divergent for all
nite r. This means we have to be extremely areful when using these asymptoti expressions. To avoid problems with singularities, will follow the example
of Sommereld and Minakata [SM00℄ and instead express the Bessel funtions as
ontour integrals.
2.4.3 The Aharonov-Bohm Sum
Aharonov and Bohm found the wavefuntion of an eletron sattered on a magneti ux string as a series of partial waves. Their solution was
AB =
X
l
( 1)l e
i 2 jl+j J
il'
jl+j (kr) e
(2.51)
Most of the artile [AB59℄ is about how to sum up for large r, and they end
up with a solution as a sum of a twisted plane wave exp(ikx i(' )),
there exist terms proportional to the Neumann funtions for all l, the expression for the
wave would be divergent.
2 If
44
CHAPTER 2.
PHONON SCATTERING
and a sattered wave on the form ap('r) exp(ikr). The unfortunate thing about
their solution is that the sattering amplitude a(') beomes innite as the angle
approahes the forward diretion ' 0; 2 . This is in sharp ontrast to the
partial wave expression they started with (2.51), whih is onvergent, ontinuous,
single-valued and well behaved for all nite r and '. The reason for the apparent
singularity, is the use of the asymptoti expansion of the Bessel funtions for
large arguments (2.46), whih is used for all values of = jl + j, while the
approximation is just valid for kr . This is the reason why they end up with
a divergent series.
Even though the expanded series is stritly speaking divergent, Aharonov
and Bohm end up with an expression with turns out to be right for all large
angles, sine they are areful in analyzing the asymp6=(t)
toti expression before summing up. Many referenes
([Son96℄,[WT98℄,[She98a℄) use this expansions of the
C
?
Bessel funtion when disussing Aharonov-Bohm sat6
<-(t) tering.
To do the summation, we will instead use the method
of
Sommereld and Minakata [SM00℄ whih leaves the
Figure 2.3: The integration path C in the omplex sum beautifully onvergent and ontinuous. The most
t-plane.
important step is the rst, expressing the Bessel funtions as ontour integrals
Z
1 (2.52)
J (z ) = ei 2 dt e iz os(t)+it
2
C
The integration ontour C is in the upper half of the omplex plane . The nie
thing about this ontour is that it < 0 for t 2 C . It is this quality whih makes
our series onvergent. The sum is now
Z
X
1
AB
=
dt e ikr os(t) ( 1)l eijl+jt eil'
2 C
l
To ease the notation we will now restrit ourself to positive values of . Negative
values of (anti-vorties) an easily be obtained from the nal answer by letting
! and ' ! ' + 2 . The above sum an be divided in one sum for l 0
and one for l < 0. Both of these series are geometri and by using the ordinary
formula for suh series we arrive at
Z
eit
e it
1
ikr
os(
t
)
AB
dt e
=
2 C
1 + ei('+t) 1 + ei(' t)
whih put on one fration line is
AB
1
=
4
Z
C
dt e
ikr os(t)
it
e (e i' + e it )
e it (e
os(') + os(t)
i' + eit ) 2.4.
45
SOLUTION BY PARTIAL WAVE ANALYSIS
The question now is how to remove the denominator so that (2.52) an be used
to get the answer expressed by Bessel funtions. The trik is to use
Z kr
e ikr(os(t)+os()) 1
dz e iz(ot(t)+os(')) =
i(os(t) + os('))
0
The term 1=( i(ot(t) + os())) will disappear from the nal answer. This an
be seen by noting that the originalRAharonov-Bohm sum (2.51) is zero at kr = 0,
whih also applies to the integral 0kr dz .
Z
i ikr os(') kr
AB
= e
dz e iz os(')
4Z
0
e iz os(t) e i'(eit e it) + e i(1 )t ei(1 )t
C
By using the formula (2.52) we an transform bak to Bessel funtions and hene
get a sum of four Bessel funtions of the rst kind. These an in turn be expressed
by two Hankel funtions of the rst kind
J (z ) e i J (z )
(2.53)
i sin( )
The whole solution an now be written as an integral over two Hankel funtions.
By making use of sin( (1 )) = sin() the solution an be expressed as
H(1)(z ) =
AB = eikr os(') A(kr)
(2.54)
where we, to ease notation, have dened
A(kr) Z
sin() kr
dz e
2
0
h
iz os(')
e
i'+i 2 H (1)(z ) + ei 2 (1 ) H (1) (z )
1 i
(2.55)
What we have managed to do so far is to transform the original Aharonov-Bohm
sum to an integral. This transription is exat. The expression in itself is not
very useful, but it turns out to be possible to do a very nie expansion of it far
from the ux point.
2.4.4 The Solution Far From the Flux-Point
As promised, the expression from the last setion will be simplied in the large
kr limit, and we will start by arrying the integral in (2.55) to innity. The
integration an then be performed exatly with the formula
ei 2 Z 1
0
dz e
iz os(') H (1)(z )
= 2
sin (' )
sin( ) sin(')
(2.56)
46
CHAPTER 2.
PHONON SCATTERING
From now on it is important that the angle ' is running between 0 and 2 to get
the orret result. By doing some trigonometri manipulations we get
sin [(' )℄ sin [(1 )(' )℄
= e i(' )
A(1) = e i'
sin(')
sin(')
This is the asymptoti behavior far from the vortex. The deviation from this
expression an now be expressed as an integral from kr to innity. Now the
asymptoti expansion of the Hankel funtions an be used
r
2 iz i 2 (2.57)
e
iz
Opposed to the sum (2.51), the subsripts of the Hankel funtions are now xed
and small. This means that the rewriting is safe, and the the approximation is
atually very good.
Z 1
sin()
dz i(1 os('))z
i'
p
A(1) A(kr) ( e + 1)
e
2i
z
kr
h
p
' i
'
2ikr sin
= i sin()e i 2 1 erf
2
p
This was obtained by substituting 1R os(') = 2 sin2 ('=2) and u = sin('=2) 2iz .
The error funtion is erf(z ) = p2 0z du exp( u2). Sine sin('=2) is positive for
angles between 0 and 2 we have no problems with multivaluedness. The nal
solution of the Aharonov-Bohm problem far from the ux point and all angles is
' io
n
h
p
'
AB
ikr
os(
'
)
i
(
'
)
2
2ikr sin
=e
e
i sin()e 1 erf
2
(2.58)
H (1) (z ) z!1
!
This is the solution obtained by Sommereld and Minakata. We notie that the
rst term is disontinuous at forward angles, but the singularity anel against
a similar disontinuity in the sattered wave, leaving the full solution ontinuous
and single valued.
2.4.5 Disussion of the Partial Wave Analysis
Far from the ux-point the sum of partial waves (2.48) has been transformed to
the funtion (2.58) whih is valid for all angles. The Born solution was divided in
one expression for nite angles and one for small angles. The partial wave result
an be expanded in eah of these regions to
p see if the solutions are idential.
For nite angles, whih means that sin( '2 ) r 1, the error funtion an be
expanded by the use of (2.37). The total wave (2.58) an hene be written
AB "
'
sin() e i 2 eikr os(') i(' ) + p
2i sin '2
#
(2.59)
2.4.
47
SOLUTION BY PARTIAL WAVE ANALYSIS
40
20
0
-20
-40
-40
-20
0
20
40
Figure 2.4: The real part of Sommereld and Minakata's summation of the
Aharonov-Bohm sum (2.58), for = 0:25. It must be ompared with the diret plot of the partial wave expression (gure 2.2). The phase shift of the wave
is learly seen. The sattered wave is seen as the \dots" distributed over the plot.
-20
1
-10
0
-1
40
0
20
10
0
-20
20-40
Figure 2.5: The same plot as gure 2.4, but seen from another angle, and with
jkxj < 20 and jkyj < 40. This plot learly shows the damping of the wave right
behind the vortex.
48
CHAPTER 2.
PHONON SCATTERING
as expeted. This is the ordinary expression found by Aharonov and Bohm. It
diers from the Born solution in an additional s-wave, whih marks the dierene
between aousti and magneti Aharonov-Bohm sattering.
The other expression to ompare it with, is Sonin's small angle expression,
(2.33). This result is only to rst order in , so we must expand (2.58) in the
same way. In the small angle limit, the angle is ' 0; 2 , so that sin('=2) y=x,
' sign(y ) and exp( i'=2) sign(y ). Then the disontinuous terms
anel and
AB eikx
"
r
k
1 + i erf y
2ix
!#
(2.60)
This is the same as the small angle result found by Born approximation (2.33).
The anellation of the disontinuous terms is just shown to rst order in , but
the full solution is ontinuous at arbitrary order.
It is important that the vortex is not a mass sink or soure. There must be
no net mass ux passing through a ontour surrounding the vortex. We will now
use the original partial wave expression for the mass ux
X
l
er + il e' ei(l m)'
hjk i = 12 m r
r
ml
(2.61)
Integrating the mass transport through a irle surrounding the vortex, means
an integral with line segment dl = r d' er . We need the density found from (2.4)
m = ik 1
m
(kr)2
m J
m (kr)
(2.62)
and the omponent in r-diretion of gradient of the phase
J (kr)
l
= l l
r
r
(2.63)
p
where l = l2 + 2l and Rl = ( 1)l exp( i 2 l ). The integral just gives ontributions for diagonal terms, 02 d' exp(i(l m)') = 2Æml . The net ux through
the ontour is then
#
"
Z 2
X
J
l
l
(2.64)
1
r
d' hjk ir = < ikr
2 J
l r = 0
(
kr
)
0
l
The expression is purely imaginary. The real part, whih is the physial omponent, is zero, and the mass is onserved.
In the derivation of the sattering equations, derivatives of the phase was
negleted to ease the mathematial handling. The ondition jr2 j j j will now
2.5.
49
DENSITY CORRECTIONS
be expliitly heked by inserting the solution (2.62). There are two assumptions:
the wave-number issmall, k 1, and the distane from the vortex is large, r 1.
l
The derivatives of 1 (kr)2 with respet to r will thus be negleted as higher
order in r 1 . The Bessel funtions satises the Bessel equation, whih makes the
derivations easy to perform
2l
2
2
(2.65)
rl J
l (kr) = k + r2 J
l
2 + 1 l2
where rl = r
r r
r2 . The rst terms is seond order in k , and an be
negleted. The seond term is small when r is large. In addition is proportional
to = qk , whih is small. The onlusion is that the assumption jr2 j j j is
justied.
2.5 Density Corretions
So far only the point vortex approximation has been studied, both for the Born
approximation 2.3 and for the partial wave analysis 2.4. Now we will again use
the Born approximation and try to extrat information from the terms ontaining
the density prole f in (2.20). Of speial interest is the long wavelength limit;
to see how the k dependeny is ompared to the phase ontributions. The rst
density term in (2.20) is k 2(1 f 2 ) exp(ikr0 os('0)). If the approximation of
the density prole (1.33) and the asymptoti form of the Hankel funtion (2.23)
is used, we get
r
Z
ik 2
2 ikr
d2 r0 2ikr sin( '2 ) sin(' '2 )
e
Æd1 e
(2.66)
4 ikr
1 + 2r 0 2
After substituting '0 ! '0 + '2 , the angle part of the integral gives a Bessel
funtion 3
Z 2
'
'
d'0 e2ikr sin( 2 ) sin(' ) = 2J0 2kr0 sin( )
2
0
The r0 integration gives a modied Bessel funtion as an answer, so that
0
0
0
Æd1 0
r
'
1 2i 3=2 ikr p
k e K0 2k sin( )
4 r
2
(2.67)
The seond density orretion in (2.20) is ik ff os('0 ) exp(ik os('0)), where f 0
here denotes derivative with respet to r0 . This gives the ontribution
0
Æd2 3 The
k
4
r
2 ikr
e
ikr
Z
d2 r0 os('0)
formulae used an be found in appendix A.
f 0 2ikr sin( '2 ) sin(' + '2 )
e
f
0
0
(2.68)
50
CHAPTER 2.
PHONON SCATTERING
As for the other integral, the angle integration here also gives a Bessel funtion
Z 2
'
'
'
'
d'0 os('0 + )e2ikr sin( 2 )sin(' ) = sin( ) 2i J0 2kr0 sin( )
2
2
2
0
0
0
Using the same approximation for the density prole as above, we have
r0 f 0 (r0 )=f (r0 ) 1+21r 2 . The r0 integration is then an integral of the kind
0
Z 1
dx
x2
J1(ax) =
dx 1
J (ax)
x2 + 12 1
0
0 2x2 + 1
1
1
1
p
=
K1 ( p a)
a
2
2
Z 1
This gives that the seond density orretion is
Æd2 1
2
r
2i 1=2
'
1
k sin( )
r
2 2k sin( '2 )
p
p1 K1 2k sin( '2 )
2
(2.69)
The leading order of the modied Bessel funtions for small arguments are
K0 (z ) ln(z ) and K1 (z ) z1 + 2z ln z . The rst density orretion (2.67) is thus
logarithmially divergent near the forward diretion. As in the ase of the phase
ontributions we must think that this is just a onsequene of the approximations
being made, and not a physial property. If we forget this term as higher order in
k , and take the low k limit, we see that both the density orretions have leading
order terms of order k 3=2 ln(k ). The density orretion is then in the low k limit
Æd 1p
2i (3
8
eikr
os(')) k 3=2 ln(k ) p
r
(2.70)
The ordinary sattering amplitude goes for small wavenumbers as k 1=2. The
density orretions an thus in the small wavelength limit be ignored. But the
dierene is only of order k ln(k ), so that if we shall go up one order from the
leading one, we must also take the density prole into onsideration.
Chapter 3
The Phonon Fore
The goal of this hapter is to provide an expression for the fore on a vortex
from a single sattered wave. Or more aurate: to nd the transverse fore
omponent to order = qk . The answer to this question is of vital importane for the disussion of the Iordanskii fore in helium II, as disussed in hapter 5. There are several artiles available alulating the fore from a single wave.
Sonin [Son96℄[Son01℄ and Shelankov [She98a℄[She98b℄ end up with the onlusion that the fore is proportional to , while Wexler and Thouless [WT98℄ and
Demiran, Ao and Niu [DAN95℄ onlude that it must be higher order in .
Stone's artile [Sto99℄ is also important.
Central in the interpretation of the result is that the fore originates in the
asymmetry of the phonon wave in the presene of a vortex. This seems to be an
aousti analogy to the Aharonov-Bohm eet, disussed in setion 2.2. Central in
the disussion is also the region right behind the vortex, where we remember from
hapter 2 that the formalism with the sattering amplitude failed. Stone [Sto99℄
interprets the fore as the reation of transverse momentum in this region. The
ritiism of the results [WT98℄ and [DAN95℄ also points to the fat that they do
not inlude the eets of small sattering angles.
All alulations in this hapter will use the solutions of the sattering problem
from hapter 2, where a plane wave sattered on a stationary vortex was examined. The fat that the vortex was not allowed to move, was in the sattering
problem presented as a boundary ondition introdued to ease the mathematial
handling, but in this hapter it will be of great physial signiane, sine the
vortex at rest an also be thought of as being held still by an external fore.
There must be a momentum transfer between the vortex and the wave whih is
possible to extrat by examining the momentum-ux tensor. The momentumux tensor Hij is the density of i-momentum in the j diretion, so the fore on
the wave is dSi ij . To get the orret result the momentum transfer must take
plae inside the integration ontour. The region over whih the fore is applied,
orresponds to region whih is kept at rest. If a point vortex is onsidered, it
does not matter where the ontour is, as long as it enloses the vortex. If a larger
51
52
CHAPTER 3.
THE PHONON FORCE
part of the vortex is kept at rest, for example the ore, or the whole prole, we
must be more areful in our hoie of ontour. A safe hoie in all ases is to put
it innitely away from the vortex.
The fore will utuate a lot, but we are only interested in the time-average.
When from now on speaking about the fore on the vortex, the meaning is the
time-average of the fore. The fore on the vortex is then
Fj =
I
dSi hij i
(3.1)
with summation over i = x; y . The dSi is the omponents of the vetor desribing
an innitesimal surfaes segment (line in two dimensions).
The alulations will be done in several ways: In 3.2.1 the sattering result
from the Born approximation will be used to nd the fore by letting the integration ontour be innitely away from the vortex. In the spirit of the Born approximation only terms to rst order in the sattering parameter will be brought
into onsideration. If problems with divergenes appear near ' = 0, the small
angle result for the sattered wave must be used instead.
The limit when the integration ontour is plaed lose to the vortex is studied
in setion 3.2.2. This limit requires that a point vortex is onsidered, sine
all interation between the fore and the wave has to be inside the integration
ontour. The alulation is amazingly simple, and it is not even neessary to
solve the sattering problem to get the result.
Instead of trying to extrat an analytial expression from the partial wave
analysis, the alulation is done numerially in setion 3.2.3 at a nite distane
from the vortex and by keeping a nite number of terms. This result an be
ompared with the two analytial results. The numerial treatment is straightforward, even though the expressions themselves are long. It seems to be onsensus
of what the partial wave expression should look like, and sine muh of the problem is how to take the limits innitely from the vortex, a numerial treatment
at nite distane is a way to get a reliable result, and omplement the analytial
alulations.
3.1 The Momentum-Flux Tensor
In the alulation of the fore the momentum-ux tensor is needed. By Noethers
theorem the momentum-ux tensor an be found from the Lagrangian density. If
the Lagrangian density L is a funtion of the elds and and their derivatives,
the denition of the tensor is
L
L
ij = LÆij
j (3.2)
(i)
(i) j
where i denotes derivation with respet to oordinate i. The symbol L is in
this ontext no longer a funtion of the elds, but the funtion of the spatial
3.1.
53
THE MOMENTUM-FLUX TENSOR
oordinates and time that we get when a speial solution for the elds is inserted.
The Ginzburg-Landau Lagrangian on polar form is
L=
1
(r)2
2
_
1
(r)2
8
1
(
2
1)2
(3.3)
whih, by the denition, gives the energy momentum tensor
ij = LÆij + i j +
1
4 i j
(3.4)
The solutions whih we are going to insert are plane waves sattered on a stationary vortex. In the long wavelength limit it will be assumed that the derivatives
of the density an be ignored, so that suh terms an be dropped both in the
Lagrangian and the momentum-ux tensor. Suppose that the integration ontour
is at a large radius r. The density prole of the vortex goes as 1 41r2 . Sine the
length of the integrations ontour is of dimension r, the density prole an be
ompletely ignored in the momentum-ux tensor, though it an not neessarily
be ignored when solving the eld equations. In the rest of this derivation only a
point vortex will be onsidered, so that the following substitutions are done
=1+
= + q'
(3.5)
The elds and are thought to be small, so that the Lagrangian an be expanded in these elds. In hapter 2 the linearized eld equations were solved, and
we want to use these solutions to get an expression for the fore. Upon thermal
averaging all rst order terms in the momentum-ux tensor vanish and we are
left with an expression to seond order in and . There are at least two ways
of nding the right expressions for the momentum-ux tensor. The rst is to
expand the tensor to seond order in the elds, and then insert the linear elds.
The other way is to expand the Lagrangian to third order so that the eld equations are quadrati in and . Then we must be areful sine the eld equations
are no longer linear, and terms that appear to be linear in the Lagrangian density
an still ontain hidden quadrati terms that will survive the time averaging.
Those two proedures skethed above do anyway give the same result, so we
are free to use the easiest and only onsider the quadrati terms in the Lagrangian
density
L(2) =
_
1
(r)2
2
r q r'
1 2
2
(3.6)
The diagonal term LÆij an be rewritten by using the eld equation
L
= 0 = _ + r q r' (3.7)
54
CHAPTER 3.
THE PHONON FORCE
So that
L = 12 2
1
(r)2
(3.8)
2
The time average of the exitation part of the momentum-ux tensor is then
hij i = P Æij + hij i + hii qj ' + hj i qi'
(3.9)
The oeÆient before the diagonal part of the tensor is what we dene as the
average pressure P = h 21 2 21 (r)2i. Aording to the solutions in hapter 2,
it is useful to assume the solutions to have a harmoni time dependeny
(r; t) = e
i!t (r)
(r; t) = e
i!t (r)
Sine both the phase and density are real funtions, we have to be a bit
areful. In most alulations it is easiest to think of them as the real parts of
omplex funtions, but for produts of funtions, the produts of the real parts
are not the same as the real part of the produt. The formula to use in our
multipliations is
<(ae
i!t)
<(be
i!t )
=
1
<
ab + (a + b)e 2i!t
2
Taking the time average, the e 2i!t term vanishes and
h<(ae
i!t )
<(be
i!t )
i = 12 < (ab)
For onveniene we forget to write < before all terms from now on. The nal
expression for the time average of the momentum-ux tensor is then
hij i = P Æij + 21 ij + q2 ij ' + 2q j i'
(3.10)
where the pressure is
1
P = (j j2 jrj2)
(3.11)
4
The reipes of how to nd the fore is: plug in solutions of and into the
energy momentum tensor (3.10), insert this in the expression for the fore (3.1),
and nally do the ontour integration.
3.2 Calulation of the Fore
In hapter 2 the sattering problem of a phonon on a vortex was onsidered both
with the Born approximation and by partial wave analysis. These two solutions
3.2.
CALCULATION OF THE FORCE
55
will be inserted in the fore expression, and the answers will be ompared. But
before this is done, let us do some more rewriting. With vetor notation the fore
is
I q 1 F=
P dS + r (r dS) + [r(r' dS) + r'(r dS)℄
2
2
(3.12)
where the pressure is P = 41 (j j2 jrj2). If the integration ontour is a irle
with radius r, the surfae segment (a urve in two dimensions) is dS = rd'. The
fore then beomes
Z 2
r (r + q r')
(3.13)
F=
d' P r +
2 r
0
This expression should be valid both for the transverse and the longitudinal fore
omponent. In our thesis only the transverse fore is of interest. If we were
really pedanti in our notation, an index k on the fore ould have been added
to remind us that this is just the eet of a single phonon with wavenumber k .
Z 2
P
F? = r
d' [sin(')P ℄
(3.14)
0
Z
r 2
F? =
d' [r y ℄
(3.15)
2 0
Z
q 2
q
d' [ r os(')℄
(3.16)
F? =
2 0
These formulae are the ommon starting point for our alulations. To go further
separate formulae for the ase of Born approximation and partial wave analysis
must be developed.
As mentioned earlier the fore from a sattered wave is disussed in dierent artiles ([Son96℄,[She98a℄ and [WT98℄). All these referenes agree that the
pressure part of the fore, F?P is zero if the integration ontour is far from the
vortex. Muh attention is due to a term proportional to the transverse sattering
ross-setion
Z 2
? = r
d' sin(')ja(')j2
(3.17)
0
whih an be extrated from (3.15). There are two dierent expressions used
for the sattering amplitude. One is the ordinary Born sattering amplitude
ot('=2), the other is a partial wave expression, whih an be extrated by
using the asymptoti form of the Bessel funtions. Both these expressions reate
problems, sine they make the transverse sattering ross-setion divergent. The
Born result gives a divergent integral near ' = 0 and the partial wave expression
56
CHAPTER 3.
THE PHONON FORCE
gives a divergent series. The artiles [Son96℄, [She98a℄ and [WT98℄ treat the
divergenes in dierent ways and end up with dierent onlusions. However in
this thesis we will not go into this disussion. There are two two reasons for that:
First, we know that the sattering amplitude is not well dened for ' 0. If we
end up with a divergene, there is no need to regularize, we only need to use the
small angle result from setion 2.3.3 instead. This should remove all divergenes
in a painless way. Seond, only the fore to order is of interest. The term ?
is of order 2 , so if we get a result proportional to from this, it seems like the
whole perturbative approah to the problem fails (!) and an no longer be used.
3.2.1 Ciruit Far From the Vortex
In setion 2.3 the sattering of a phonon on a vortex was studied by using the
Born approximation, and now this solution is inserted into the derived formulae
for the transverse fore, (3.14), (3.15) and (3.16). The sattering formalism is
just valid far from the vortex, and the integration urve will thus be put at a
large distane. In a way this is also the most general limit, sine it does not put
any laims on exatly where the momentum transfer between the vortex and the
wave takes plae. It works ne both for point vorties and vorties where the
ore or the whole prole is kept xed.
In the spirit of the Born approximation the alulation will be done just to
rst order in = kq . The orretions from the density prole of the vortex (setion 2.5) will not be taken into aount, sine they are of higher order in k . The
alulation is tedious and involves many steps and has a lot of pitfalls. There is
no need to say that an aurate treatment of the sattering problem in setion 2.3
is required to get the orret result. Sine the alulation of the transverse fore
is the main goal for the thesis, the derivation is presented in great detail.
The solution found to the sattering problem was divided in one expression
valid for nite angles, and one for small angles. These used together gave a
solution whih was ontinuous and single-valued. As disussed by Sonin [Son96℄,
the small angle ontribution is important for the fore, but it is, as we shall see,
not neessary to use the speial solution derived for small angles to nd this, sine
it atually is ontained in the ordinary Aharonov-Bohm sattering amplitude as
well.
The nite-angle solution an be written as the sum of a twisted plane wave,
and a sattered wave disappearing at innity. The twist of the inoming wave
will prove to be essential to the result. No part of the waves or blows up
far from the vortex1. Sine the integration ontour is of length r, all terms of
order r 3=2 an be thrown away when r goes to innity. The nite-angle Born
sattering amplitude a(') blows up near ' 0, but this is not a physial property,
sine the sattering amplitude is just valid for nite angles.
1 The
3.2.
57
CALCULATION OF THE FORCE
solutions is
= k eikx (1 i(' )) + Æa
(3.18)
where Æa = ap('r) exp(ikx) is the sattered wave (2.35) expressed by the sattering
amplitude. Beause of the form of the sattered wave, only derivatives with
respet to r ount far from the vortex
rÆa = ikÆ er + O r 3=2
(3.19)
A rekless expansion in powers of r 1=2 like this, is not to be reommended in
all situations. It does indeed work rather good in the er diretion but in the
e' diretion if fails badly, sine it inludes derivatives of a(') with respet to
', whih gives very nasty divergenes near ' = 0. For the alulation of the
transverse fore this does not matter, sine only the er diretion ounts, but
when for example the irulation of the phonon gas is found in setion 4.3, this is
of importane. Problems like this is also a motivation to do numerial alulations
as well.
The rst part of the fore to be disussed is the one from the pressure, (3.14).
It is often said to vanish, and we shall see that a orret form of the inoming
wave is required for this to happen. Sine the density is P = 41 (j j2 jrj2), the
two things needed are
= k ik
and
eikx (1
r = k ikeikx (1
i('
sin(') ikx
)) + Æa + i
e + O 2
r
i(' )) ir' + ikÆa er + O 2 ; r 3=2
where k = k ex . Next we take the absolute square of these two quantities. The
gradient of the angle ' is r' = r1 ( sin('); os(')), so that
j j2 = jk j2 k2 1 + 2<(e
and
jrj2 = jk j2 k2 1 + 2<(e
ikx Æ ) + 2k sin(')
a
r
+ O 2 ; r 3=2
ikx os(')Æ ) + 2k sin(')
a
r
+ O 2 ; r 3=2
This is all whih is needed to nd the pressure, and we see that it an be expressed by the sattered wave Æa alone, sine the onstant terms and the terms
proportional to sin(') anel.
k2 (3.20)
P = jk j2 < (1 os('))e ikx Æa
2
58
CHAPTER 3.
THE PHONON FORCE
Now the importane of the extra fator exp( i(' )) in the inoming wave
is learly seen. If it was not there, it would not anel the sin(') fator from
the density. In turn this would have given a non-zero ontribution to fore! If
')
a(') 1 sin(
os(') is inserted, the pressure part of the fore (3.14) is proportional to
FP
?
Z p
r d'
sin2 (')eikr(1 os(')) r!1
!0
(3.21)
When r ! 1 the osillations of the integrand beome more and more rapid.
Where the osillations are fast it will average to zero, and only for ' near 0 the
osillations will be slow. On the other hand: near ' = 0 the sin2 (') term tends
to zero. There is then no net ontribution to the integral when r gets large, and
hene no ontribution to the fore oming from the pressure term.
In addition to the pressure term, there are two more terms in the expression
for the transverse fore. First we take a look at (3.15). It ontains a derivative
with respet to y , whih an be expressed by the derivatives of polar angles as
y = sin(')r + r1 os(')'. The plane wave has no dependeny of y , so to rst
order in what we need is
r = k ik os(')eikr os(') + O ()
and
y = k
os(') ikr os(')
i
e
+ ikÆa sin(') + O r 3=2
r
The produt is
r y = jk j2k
os2(')
+ k sin(') os(')e
r
ikr os(')
Æa
From rst term in the produt, the following ontribution to the fore is found
Z 2
1
2
kjk j
(3.22)
d' os2 (') = jk j2 k
2
2
0
This is again a term oming from the twist of the plane wave. The seond term
in r y gives
r 2 2
j j k
2 k
Z d' sin(') os(') e
1 2 2 p j j k r p
2 k
2ik
ikr os(')
Z Æa
os(') sin2 (') ikr(1 os('))
e
1 os(')
sin(') ikr is used.
where the Born result for the sattered wave Æa = p2
ikr 1 os(') e
The integrand is, as in the pressure term, osillating faster and faster as r goes
=
d'
3.2.
CALCULATION OF THE FORCE
59
to innity. Only around ' 0 the osillations are rather slow. The main
ontribution is expeted to ome from from a small region, ( ; ), near zero.
The parameter is thought to be a small number. The integration limit an
nevertheless be dragged to innity, sine the main ontribution anyway omes
from the small values of '. The integral is then Gaussian and an be solved with
the standard formula (appendix A)
Z p
kr 2
2
2
! jkj2k
(3.23)
jk j k r p
d' e 2i ' !1
2ik The above ontribution omes from the small sattering angles. It ould as well
be found from the small angel expression for , just as done by Sonin [Son96℄. In
a way it is surprising that it is ontained in the ordinary sattering amplitude as
well. The two ontributions to F? are added, so that
F? = jk j2 k
(3.24)
2
There is then one ontribution left to study, and that is what we labeled F?q in
(3.16). Unlike the expression for F?, F?q is in itself proportional to . The only
possibility for a rst order result in is when plane waves are inserted for and
, i.e.
= k eikr os(') + O ()
= ikk eikr os(') + O ()
The last ontribution to the fore then beomes
Z 2
1 2
q
F? = jk j k
(3.25)
d' os2(') = jk j2 k
2
2
0
Again Sonin's artile is atually the only one of the disussed artiles mentioning
this ontribution.
So far = qk is used as a sattering parameter, sine that is the natural
hoie when the analogy to Aharonov-Bohm sattering is onsidered. This is
however not the most onvenient hoie when doing a thermal average. Then the
best quantity to use is the average mass ux (1.2.2) of a plane wave, whih is
hjk i = 21 jk j2kk
(3.26)
The irulation is, in dimensionless units, given as q = 2q . Expressed by these
quantities the result from the Born approximation an be summarized as follows
1
1
(3.27)
F?P = 0
F? = q hjk i
F?q = q hjk i
2
2
This is in agreement with Sonin [Son96℄, Stone [Sto99℄ and Shelankov [She98a℄,
but diers from the result by Wexler and Thouless [WT98℄.
60
CHAPTER 3.
THE PHONON FORCE
3.2.2 Ciruit Near the Vortex
In setion 3.2.1 the fore was found by onsidering the asymptoti behavior of
the Born solution at a large distane from the vortex. In this setion we will
turn the other way round, and try to extrat information when the integration
ontour is plaed lose to the vortex. In a way this limit is more sinister than the
large distane limit. When the integration ontour is far from the vortex it does
not matter where the fore on the vortex is applied, sine it anyway is inside the
iruit. If the area enlosed by the integration loop is tiny it is however ruial
that we know exatly where the fore is applied. If the integration radius is taken
to zero, the only possibility is to onsider a point vortex, sine a nite vortex ore
kept at rest must be aeted by a fore over a nite area. So far the point vortex
approximation is presented as something whih is just valid far from the vortex.
This is just partly true, and atually it is just as muh an approximation whih
holds for waves with long wavelengths. The distane r must be well away from
the vortex ore, but at a distane where kr an get arbitrarily small.
As a onsequene of this, the solution of the sattering on a point vortex will
now be used for small integration ontours to nd the fore. Sine the integration
ontour is of length r, only terms of order r 1 an be ontributing when
the integration ontour is small. In the Born approximation, the solution was
only found asymptotially at large distanes from the vortex. But the partial
wave analysis result is valid near a vortex, and the partial wave expression for is nonsingular as kr approahes zero. This means that all terms ontaining just
or derivatives of an be negleted in this limit, and F ! 0. The density
utuation is however singular in the the point vortex approximation
sin(') ikx
= ik q r' r = ik + ik
e + O 2
(3.28)
r
The pressure is P = 41 (j j2 jrj2). In the small r limit the jrj2 term vanishes.
The same happens to the k 2 jj2 part of j j2 . So when the limit is taken we get
1
sin(')
P r!!0 jk j2 k
(3.29)
2
r
This gives
Z 2
1
P
(3.30)
F? = r
d' P sin(') r!!0 q hjk i
2
0
where the irulation quantum is q = 2q and the average mass urrent is
hjk i = 21 jk j2k2. In the last term ,F?q , only the plane waves are inserted for and
to lowest order in . The result is hene independent of r, and the result from
last setion is still valid. The analytial alulation of the fore in the limit near
the vortex an hene be summarized as
1
1
(3.31)
F?q = q hjk i
F? = 0
F?P = q hjk i
2
2
3.2.
61
CALCULATION OF THE FORCE
This must be ompared with the large r limit (3.27). The terms are not the same,
but their sum is equal. The onlusion is that in the point vortex approximation,
the fore on the system is just in one point, and hene it does not matter where
the integration ontour is plaed.
The asymptoti alulation presented here is remarkably simple. It was not
even neessary to solve the sattering problem to get the result. The only onditions it relies on, is that is nonsingular when r ! 0, while is not.
3.2.3 Numerial Calulations
In addition to the Born solution of the Aharonov-Bohm problem, there is also an
expression by partial wave analysis. In the previous setions 3.2.1 and 3.2.2 the
asymptotes when the integration ontour was plaed at large distane and lose
to the vortex, were onsidered. The same ould be done with the partial wave
solution. There should however not be any need to do the same alulations
again2 , so instead we will make formulae valid for integration ontours at all
distanes, and then study them numerially. Our intention is to lutter as little
as possible with the original result, and hene we will just plug in the full partial
wave expression into the formula for the transverse fore. All the formulae, as
presented here, are omplex, but from the derivation of the transverse fore we
remember that the fore was given as the real part of the expressions. All numbers
presented from now on are real parts of the expressions. It must be noted that
the imaginary parts are usually very small. In the analytial alulations done
earlier, they disappeared ompletely.
In the numerial handling of the problem, the substitution z = kr is done.
The wave = (z; ') is
=
X
l
l eil'
(3.32)
where
l = ( 1)l e
i 2 l J
l (z )
(3.33)
The subsriptpof the Bessel funtions are in the aousti Aharonov-Bohm sattering l = l2 + 2l, with = qk . In the alulations the density is also
needed. It is found from the phase in (2.4). In the point vortex approximation
it is
= [ik q r' r℄ X
X
l
(3.34)
il' ik
Rl eil'
e
1
=ik
l
2
z
l
l
2 The
Born and partial wave analysis solutions have been arefully ompared, and found to
be the same.
62
CHAPTER 3.
THE PHONON FORCE
whih denes Rl = Rl (z ). The derivative of l is
l k
(3.35)
= ( 1)l e i 2 l (J
l 1 (z ) J
l +1 (z ))
r 2
The integrals with respet to ' are easy to perform in the partial wave analysis.
They will always be one of the two
Z 2
0
Z
d' sin(')ei(l m)' = i (Æl
2
m;1
Æl
1)
m;
os(')ei(l m)'
(3.36)
d'
= (Æl m;1 + Æl m; 1 )
0
Let us now nd the fore by handling the equations (3.14), (3.15) and (3.16) one
by one. From the alulation of the momentum-ux tensor, the pressure was
found to be P = 41 (j j2 jrj2). In partial wave analysis this is
l m lm i(l m)'
k2 X
RR
e
P=
4 lm l m z z
z2 l m
Integrating this aording to (3.36) the pressure part of the fore is
Z 2
P
F? = r
d' P sin(') = k (S1P + S2P + S3P )
4
0
where the tree sums are
S1P = iz
X
Rl Rl 1
Rl+1
SP
l
X l l
iz
S3P =
1
z
2 =
l
X
l
z
ll (l
z
1
1)l 1
(3.37)
(3.38)
l 1
z
(3.39)
(l + 1)l+1
(3.40)
An important thing to notie is that the only dependeny of k in these sums is
through the ux = kq , so lowest order in q also means lowest order in k .
The partial derivative with respet to y is in polar oordinates
y = sin(')r + 1r os(')' . The integrand in F? is then
X l
sin(') m
= k2
r y
z
z
lm
m
i os(') m ei(l m)'
z
With the integration formulae (3.36) the double sums redue to single sums
Z
1 2
F? =
=
k (S + S2 )
(3.41)
d'
2 0
r y
2 1
3.2.
63
CALCULATION OF THE FORCE
where
l l 1 l+1
+
S1 = iz
z
z
z
l
X l
S2 = i
(l 1)l 1 (l + 1)l+1
z
l
X
(3.42)
(3.43)
The only term left now is the one that was alled F?q . It involved a produt of
the kind
X l
= ik 2
Rm ei(l m)'
r
z
lm
What made this term dierent from the other was that it was in itself proportional
to . Sine we are just keeping rst order terms, plane waves an atually be
inserted for and . It an however be nie to have a formula at the same form
as the other terms, so that
q
? = 2 k S 2 k
"
Fq
i
X
l
#
l Rl 1 + Rl+1
z
(3.44)
The transverse fore an now be found by summing S1P ; S2P ; S3P ; S1; S2 and S q .
The only k dependeny of the sums are through powers of . Sine we are only
interested in the leading order terms, let us do some nal denitions
1 P
S1 + S2P + S3P =0
4 i
1 h A =
S1 + S2
2 =0
AP =
(3.45)
(3.46)
The last sum S q is in itself proportional to so it just needs to be evaluated at
= 0. To rst order in , the fore an then be written as
F?P = q hjk iAP
F? = q hjk iA
F?q =
1
q hjk i S q j=0
2
(3.47)
This is a onvenient way of writing it, when it shall be ompared with the analytial limits (3.27) and (3.31). Both AP and A are funtions of z = kr only.
When z ! 0 it an mean either that r gets small or k gets small (or both).
The above result has been studied analytially in the two limits z ! 1 and
z ! 0. In (3.27) the large ontour limit was found to be
AP
!0
A !
1
2
64
CHAPTER 3.
A(z)
0.6
THE PHONON FORCE
A(z) + AP (z)
0.4
AP (z)
0.2
5
10
15
20
z
Figure 3.1: Shows AP (z ), A (z ) and AP (z ) + A(z ) as funtion of z . The
numerial expression are done with jlj < z + 10. The two terms AP (z ) and
A (z ) osillate around their respetive average values for large z , but swith
plaes near z = 0. The sum AP (z ) + A(z ) is onstant.
The other limit, for small z , was found in (3.31) to be
AP
! 12
A ! 0
The last terms, F q , was the same in both limits and sine only plane waves enter
into the alulation, S q j=0 = 1 always. As said earlier it is expeted that the
sum AP + A should remain onstant in the point vortex approximation. This
beause the momentum transfer between the vortex and the wave is just in one
point, and hene it does not matter at what distane the integration ontour is
plaed. This an be seen in plot 3.1 whih shows numerial values of AP , A
and AP + A as funtions of z .
The numerial handling of the sums is easiest for small z . It is the Bessel
funtions J
l (z ) that make our series onvergent, sine they onverge rapid to
zero as jlj z . This means the further out z is, the more terms must be
evaluated. It is then a question of omputer speed how far out we an go. There
is also a problem with respet to numerial errors. The more terms that are
added, the more numerial errors are aumulated.
The table 3.1 shows some values of AP and A around z = 100, and where all
terms with jlj < 120 are evaluated. The values are not far from the asymptoti
values of AP = 0 and A . They seem to osillate around these values, just as in
the plot 3.1, but with lower amplitude sine they are further out.
3.3.
65
CONCLUSION
AP =
A =
z = 100 z = 101 z = 102 z = 103
0:0025
0:0029
0:0048
0:0011
0:5025
0:5029
0:4952
0:5011
Table 3.1: Some values AP and A near kr = 100.
3.3 Conlusion
In this hapter three dierent alulations of the transverse fore from a sattered wave has been done: one by putting the integration ontour far from the
vortex (setion 3.2.1), one by putting the integration ontour near the vortex (setion 3.2.2) and one numerial at nite distanes (setion 3.2.3). The analytial
limit with the integration ontour at large distane gave the result
F?P = 0
F? =
1
hj i
2 q k
F?q =
1
hj i
2 q k
(3.48)
This limit is the one usually onsidered when olleting the fore. The alulation
was tedious, and strongly dependent of the form of the wave. Both the speial
small angle ontribution and the twist of the inident wave, proved to be ruial
for the result.
Considering a point vortex, the fore ould also be found with an integration
ontour asymptotially near the vortex. This gave the result
F?P =
1
hj i
2 q k
F? = 0
F?q =
1
hj i
2 q k
(3.49)
This alulation was very easy, but the limit is not so safe as the large ontour
limit, sine it is just valid for a point vortex, and for small wavelengths.
The numerial result was obtained by summing up a nite number of terms
from the partial wave analysis. The result is shown in gure 3.1, whih also
veries the two analytial results.
The respetive terms dier in dierent alulations, but the total fore, when
all terms are olleted, is always the same. The onlusion from this hapter is
that the transverse fore from a single sattered wave is
F? = q hjk i
(3.50)
The fore is proportional to the irulation quantum and the mass ux of the
inoming wave. When going the full two-uid expression this gives a fore proportional to the normal uid veloity, whih is in favor of the Iordanskii fore.
The full disussion of the onsequenes of this result, is held in hapter 5.
In the alulations of the fore only the solution of the aousti AharonovBohm sattering problem was used. Some of the literature use the magneti
version instead, and to rst order in the ux , the magneti analogy diers from
66
CHAPTER 3.
THE PHONON FORCE
this by an additional s-wave, = (r), found in (2.49). This is, however, not
problemati to the transverse fore, sine any s-wave gives ontributions antisymmetri in ', and thus vanishing in the integrals.
But even if the s-waves are not ontributing to the transverse fore, they are
essential to the evaluation of the longitudinal fore.
3.4 The Bakground Fluid Fore
In this hapter the fore on a vortex from a plane wave has been arefully disussed. The derivation involved many steps and terms, so to be able to devote
all attention to the wave, only the simplest of all possible bakground uids was
onsidered, namely one vortex at rest. A more general bakground uid ould for
example be the sum of a vortex at rest and a uid moving with onstant veloity
vs1 far from the vortex. It is possible that the asymptoti superuid veloity eld,
ould as the plane waves, exert a fore on the vortex. In setion 1.2.4 a vortex
moving with onstant veloity, was onsidered. Beause of galilean invariane
this is equivalent with onsidering a stationary vortex and a onstant veloity
eld far from the vortex. If we substitute r_ ! vs1 in the density orretion
from setion 1.2.4, we get the solution
vs = qr' + vs1
= 1 q r' vs
(3.51)
The fore on the vortex an be found, in exatly the same manner as previously
in this setion, by an integral of the momentum-ux tensor (3.1). The only
dierene is that the above solutions are inserted for the plane wave solutions.
The integration ontour is a large irle with the vortex in the enter. All terms
of order r 2 an be negleted, and thus all derivatives of the density an be
forgotten too. The two veloity elds r', and vs1 are of very dierent natures.
Far from the vortex the onstant eld vs1 is dominating the loal behavior sine
the vortex ontribution goes as r 1 . The vortex is on the ontrary essential
to the global properties of the uid, suh as the irulation, where the onstant
eld vs1 does not ontribute at all.
The bakground uid fore must be rst order in vs1 and rst order in the
irulation quantum q , if it exists (hopefully it does). Terms of the kind (vs1)2
and (r')2 are symmetri about the origin and do not ontribute. The relevant
terms in the energy momentum tensor (3.4) are
0 = (q r' v1 ) Æij + qv 1j ' + qv 1i '
(3.52)
ij
s
si
sj
The rst term is diagonal so the oeÆient is identied as the pressure. Let us
then let the asymptoti superuid veloity be along the x-axis, vs1 = vs1 ex. The
transverse fore an then be written as
I
0
1
[( x') dSy + y ' dSx + x' dSx ℄
F = qv
?
s
3.4.
67
THE BACKGROUND FLUID FORCE
where dS = r(os('); sin('))d' and r' = r1 ( sin('); os(')). Inserting this
leads to
Z 2
0
1
F? = qvs
d' sin2 (') + os2 (') = 2qvs1
0
We notie that there are two equal ontributions to the fore, one from the
pressure and one from the ross term vsi1 j '. With vetor notation this gives
F0 = v1
(3.53)
?
s
At zero temperature this is the only transverse fore, and it is idential to the
lassial Magnus fore (1.10). In full units, the fore is proportional to the total
density .
This derivation was not so areful as the one with a plane wave passing the
vortex. Physial arguments were used instead of mathematial alulations to
get the result as fast as possible. The question now is: is this result still valid at
nonzero temperature? In hapter 4 helium II at nonzero temperature is studied.
The onlusion from that hapter is that with Ginzburg-Landau theory, helium
II looks like a phonon gas on a bakground uid, where the bakground uid is a
lassial solution just like this. The result from this setion is hene not altered
when temperature is introdued. But we might get modiations to the sattering
problem in hapter 2, and this ould hange the fore from the sattered plane
waves alulated earlier in this hapter.
68
CHAPTER 3.
THE PHONON FORCE
Chapter 4
The Two-Fluid Desription
4.1 Phenomenology
With normal pressure helium is the only existing uid at temperatures near
the absolute zero, and onsequently helium is the only possibility of studying
liquids at temperatures where quantum phenomenon beomes ruial. In nature
there are two speies of helium, 3He obeying Fermi statistis and 4 He obeying
Bose statistis, where in this thesis only the latter will be onsidered. Two phase
transitions our in 4 He. The rst is at the boiling point at T = 5:2K . Below this
temperature 4 He behaves as a visous liquid, alled helium I, down to the lambda
point at T = 2:172K . Below this temperature we speak of helium II, in whih
a new phenomenon appears: ow through narrow apillaries without visosity.
This property is alled superuidity, and was rst disovered by P.L.Kapitza in
1938. But regarding helium II as a pure superuid is not orret either, sine
visous phenomena are observed. For example will a rotating disk in helium II
be brought to rest beause of frition with the liquid. These two properties were
rst explained by Landau [Lan41℄ in 1941, by treating helium II as a mixture of
two uids: a superuid with density s and a normal visous uid with density
n , where the physial density of the liquid is the sum of these. There is no
entropy assoiated with the superuid, so all heat transfer is in the normal uid.
This explains the superuid ow through apillaries sine a temperature gradient
is needed to set up a net normal uid ow. The two densities are strongly
dependent of temperature, so at the absolute zero helium II is purely superuid
while at the lambda temperature only the normal uid is present. The total
density is approximately onstant.
The total mass ux of the uid is s vs + n vn with superuid veloity vs
and normal uid veloity vn . The superuid veloity is without rotation so that
r vs = 0 everywhere (exept vortex points).
The mirosopi piture of helium II is that the superuidity is due to the the
Bose-Einstein ondensation below the lambda temperature. Landau [Lan41℄ did
69
70
CHAPTER 4.
THE TWO-FLUID DESCRIPTION
not believe in a relation between superuidity and Bose-Einstein ondensation,
and he ommented that \nothing ould prevent atoms in a normal state from
olliding with exited atoms, i.e. when moving through the liquid they would
experiene a frition, and there would be no superuidity at all". The objetion
is later regarded as false and Bogoliubov [Bog47℄ showed that in the ase of a
weakly interating Bose gas there was no frition between the exitations and
the ground state atoms. This is also believed to be true for strongly interating
helium II 1. In [LL87℄ it is also emphasized that we must be areful not to think of
the two uids as two physial uids existing independently of eah other. Instead
it must be thought of as if the quantum uids possess the property of having two
ows simultaneously. They therefore reommends to use the terms superuid
and normal ow instead of part.
Many textbooks give a brief introdution to the two-uid model. A nie
general desription an be found in [WB87℄, while [LL87℄ gives a marosopi
desription with muh attention to thermodynamis. For our situation with
quantized vorties, [Don91℄ must be onsulted, sine he disuses fores in the
two-uid model, and the two-uid model in presene of vorties.
4.1.1 Exitations
It is possible to begin a disussion of helium II by desribing the individual atoms
and their interations. The interation between the atoms is however strong,
even at zero temperature, so suh an approah is likely to miss the more olletive nature of the exitations in the liquid. Two kinds of elementary olletive
exitations are usually spoken of: phonons and rotons. A disussion of these exitations was rst held by Landau [Lan41℄. Sometimes the phrase phonon means
\sound wave". Speaking of a gas of sound waves is however not too instrutive,
so phonon means in our ontext just a kind of olletive exitation of the uid.
If the exitations have momentum p, their energies are
(p p0 )2
E ph = p
Er =
+
(4.1)
where is the speed of sound, is the eetive mass of the roton and is
the roton energy gap. After the rst artile by Landau it has been pointed
out by Bogoliubov [Bog47℄ that the division in phonons and rotons is artiial.
The phonons should be regarded as dierent region of the same exitation spetrum. The same onlusion was also reahed by Landau in [Lan47℄, where he
also skethes the urve 4.1. Near zero momentum, the urve is almost linear.
This is the phonon region. The roton region is near the minimum of the urve
at Q = 1:9
A 1 . Even though the phonons and rotons an be regarded as different regimes of an exitation urve , it is often onvenient to speak of phonon
1 There
are, as we will see, interations between the normal uid and the superuid, but
these are only through vortex lines.
4.1.
71
PHENOMENOLOGY
20
Energy(K)
18
16
14
12
10
Rotons
8
6
Phonons
4
2
0
0
0.5
1
1.5
2
2.5
Q(
A 1)
3
3.5
4
Figure 4.1: Experimental result for the exitation energy as a funtion of momentum. Found from inelasti neutron sattering data in [DDH81℄.
regions and roton regions. It is still unlear what auses the non-linear form of
the exitation urve. For temperatures above 1K the thermodynamial behavior
is almost ompletely due to the rotons. For low temperatures mainly exitations
with small energies are reated. Beause of the energy gap, , this means that
phonons are dominating for very low temperatures, usually below 0:4K . In the
rest of this thesis we will forget about rotons, and only the phonon region will be
onsidered.
The piture of a gas of exitation on a bakground also leads to the interesting
feature of seond sound. First sound is what previously has been alled just
\sound": waves moving through the ondensate. Seond sound is sound waves
propagating in the gas of exitations. There is no room (nor need) for a disussion
of the properties of seond sound in this thesis. More about exitations in helium
II an be found in [Gly94℄ or other text-books.
4.1.2 Superuidity
Sine superuidity is suh an important phenomenon in helium II a short argument for superuidity, as it was rst done by Landau [Lan41℄, is presented here.
The argument is very simple and it shows that superuidity is just a onsequene
of the exitation spetrum and the galilean invariane of the system. Landau's
argument has been modied and generalized, but the basi idea is always the
same. Let us onsider a superuid at the absolute zero with uniform veloity v,
owing through a narrow apillary. Frition with the walls is, at a mirosopi
level, exitation of phonons and rotons in the uid in ontat with the boundaries. Thus let a phonon be exited with momentum p in the rest system of the
uid, where the walls are moving with veloity v. The total momentum is P0
72
CHAPTER 4.
THE TWO-FLUID DESCRIPTION
and E0 is the total energy of the system (uid+exitation).
P0 = p
E0 = p
(4.2)
In the rest system of the apillary, the energy and momentum is
1
P = P0 + M v
E = E0 + P0 v + M v2
(4.3)
2
where M is the mass of the uid. The last term in the energy is the kineti energy
of the uid. The rst two terms are the exitation energy E = E0 + P0 v in
the rest system of the apillary. For a phonon to be exited, this energy must be
negative
E = p + p v < 0
(4.4)
This means that there annot be any exitation of phonons unless the uid is
moving faster than the speed of sound, i.e.
v>
(4.5)
if the uid veloity is slow, there are no exitations and hene no frition with
the walls. Exatly the same argument
an be applied to the roton exitations. It
q
2
leads to the ondition that v > , where is the energy gap of the roton and
is the hemial potential. As long as the energy gap is non-zero, superuidity
is possible. In ontrast to liquid helium, there is no energy gap for a free Bose
gas. It is thus not a superuid. A disussion of the energy spetrum of a weakly
interating Bose gas is held by GriÆn [Gri96℄
At nite temperatures helium II onsists of a ondensate and partiles in
exited states. The above argument whih shows that for a slowly moving uid,
no new phonons and rotons are exited, should still be valid. There are, however,
interations between the walls and the thermal exitations already existing in the
liquid. The superuid is thus assoiated with the ondensate, while the normal
uid is regarded as an ordinary visid uid.
4.1.3 The Equations
The intention with this introdution to the two-uid model is not to give a omplete disussion of all its properties and peuliarities, but to give enough information to understand the disussion of the fores on a vortex in hapter 5. So
far, the important properties of the two-uid model has been desribed mainly
with words, but it an also be useful to see the equations of motion. As desribed
in [Don91℄, they are
dv
(4.6)
s s = s rP + s S rT + n s r(vn vs )2 Fns
dt
2
d
n s
n n = n rP s S rT
r
(vn vs )2 + Fns + r2vn2
(4.7)
dt
2
4.2.
DERIVATION FROM GINZBURG-LANDAU THEORY
73
So the superuid is desribed by a variant of Euler's equation (1.3) and the
normal uid is desribed by a kind of Navier-Stoke's equations (1.2), as expeted.
In addition eah of the uids respet a ontinuity equation. The most interesting
feature, from our point of view, are the mutual frition terms between the normal
uid and superuid, Fns . They are neessary to get a orret desription of the
two uid model at a marosopi level. The link to the rest of this thesis is that
on a mirosopi level, the mutual frition fores are through vortex lines. The
superuid Magnus fore and the Iordanskii fore disussed in hapter 5 are then
both ontributing to the mutual frition terms.
A detailed desription of the thermodynamis of the two-uid model is either
not needed. One feature worth mentioning, though, is that there is no entropy
assoiated with the superuid ow. This must be kept in mind when disussing
adiabati motion in setion 5.3.
4.2 Derivation from Ginzburg-Landau Theory
The two-uid desription, as skethed in this hapter, is pure phenomenology; this
is how helium II looks in experiments. The point of the following is to link the
two-uid model to Ginzburg-Landau theory. In the hapters 1, 2 and 3 the GrossPitaevskii approximation was onsidered, whih means that the quantum elds
were approximated with lassial funtions. This approximation is valid near the
absolute zero, when the liquid is in a ondensed state. A better approximation
for low temperatures is to onsider small deviations from the lassial elds, and
expand in these quantities.
A seond quantized desription of liquid helium ould start with the GinzburgLandau Hamiltonian density (4.8), whih is a funtion of the atom reation and
b y and .
b For an interating Bose gas, is it onvenient to
annihilation operators, b = + Æ ,
b where = h
b i is the ondensate wavefuntion. In a Bose gas
write the ondensed state is usually the non-interating zero momentum state. The
problem with liquid helium is that it is not easy to identify the ondensate in
suh way, sine even the ground state is strongly interating. A way to avoid
suh problems is to forget about ondensate and non-ondensate, and desribe
the liquid as a phonon gas at the top of a bakground uid instead. In suh
a desription the opportunity to do diret identiation of the ondensate or
normal uid is lost, but the olletive harater of the bakground uid and the
exitations is emphasized.
In pthe hapters 1, 2 and 3 the polar form of the lassial elds is used,
= exp(i), where and are real. We will now quantize these elds.
A quantization of the phonon gas makes it possible to nd the normalization of
the elds, and thus determine the mass ux and the energy of the liquid. The
time averages of the lassial quantities, beome assembly averages in the seond
quantized formalism.
74
CHAPTER 4.
THE TWO-FLUID DESCRIPTION
The simplest ase is to onsider a homogeneous bakground moving with
veloity vs . As in most of the alulations in Ginzburg-Landau theory, dimensionless units are used.
The Hamiltonian is
(4.8)
Hb = 14 b(rb)2 + (rb)2b + 81b(rb)2 + 12 (b 1)2
where b is the onjugate momentum of b. The Hamiltonian is Hermitian sine
the elds are self-onjugate. Working with the full elds is impossible beause of
the non-linear struture of the Hamiltonian density. As desribed above, small
deviations from the lassial solutions will now be onsidered. A very simple
lassial solution of the above equations is = 1 + O (vs2) and = 0 , where the
superuid veloity vs = r0 is small and onstant. The operators are then
b = 1 + b
b = 0 + b
(4.9)
The Hamiltonian (4.8) an now be expanded to seond order in the elds b and
b. The zeroth and rst order terms will not ontribute to the dynamis of the
system, so only the quadrati terms must be taken into aount
Hb 2 = 12 (rb)2 + bjph vs + 18 (rb)2 + 21 b2
(4.10)
where the phonon ux operator is
1
= (brb + rbb)
(4.11)
2
The lassial solutions of the above Hamiltonian are plane waves, as desribed in
(1.25). The elds an hene be expanded by the lassial waves as
b
jph
b(x; t) =
b(x; t) =
X
k
X
k
k eikx
!t bk
+ k e ikx+!t byk
(4.12)
k eikx
!t bk
+ k e ikx+!t byk
(4.13)
where bk and byk are the phonon annihilation and reation operators obeying the
ordinary Bose ommutator
h
i
bk ; byl = Ækl
(4.14)
Aording to the lassial q
solution found in setion 1.2.2, the frequeny is ! =
!0 + k vs , where !0 = k 1 + k42 . The energy ould be expanded in small k
giving !0 k . In a way this is an honest thing to do, sine the expression is not
dealing right with the high momenta. The ost of keeping the full expression is
4.2.
DERIVATION FROM GINZBURG-LANDAU THEORY
75
small, thought, so it will be kept to remind us that the energy spetrum is not
simply linear. An experimental result for the energy spetrum in helium II is
shown in gure 4.1. The eld b and its onjugate momentum b must satisfy
the fundamental ommutator
[b(x; t) ; b(y; t)℄ = iÆ (x
y)
(4.15)
In the thermodynamial limit,
R dD k is taken to innity. The sum over k
P the volume
then beomes an integral, k ! V (2)D , where the dimension D is either 2 or
3. Inserting the elds (4.12) and (4.13) into the fundamental operator leads to
V 2 i=
Z
dD k e ik(x y) = iÆ (x y)
(2 )D k k
(4.16)
where = indiates the imaginary part of the expression. The relation
= (kk) = 21V
(4.17)
is then obtained. Together with the lassial relation k = i !k 0 k , this uniquely,
up to a phase fator, determines the normalization of the elds
2
k =
r
!0
2V k 2
k
k = i p
2V !0
q
(4.18)
where the frequeny is given by !0 = k 1 + k42 . This is the result for the simple
situation of a homogeneous ondensate moving with onstant veloity. There are
many ways to generalize this, with for example a more general bakground uid
or we ould have tried to draw more information from higher order terms, but
the result presented here should be suÆient to reognize the ordinary two-uid
formulae. The most serious objetion, due to the fat that the goal of this thesis
is to determine the Iordanskii fore, is that this desription does not inlude
vorties. A quantization of the phonon wave with a vortex in the bakground, is
a lot more ompliated than the one presented here. Probably it is more easily
ahieved in polar oordinates than in the ordinary Cartesian used here.
Regarding the k -dependeny, a few things must
p be kept in mind. In the low
energy limit the asymptoti k dependeny is k k and k p1k . The phase is
dominating the density, whih is the reason why derivatives of the density is often
ignored. So far it is presented as we do expansion in and . This is not entirely
orret sine the phase is not small for small k . But all dependeny of in
our theory is through the derivatives, r and _ whih are small quantities. The
orret presentation is hene that we do expansion in the density and derivatives
of the phase.
76
CHAPTER 4.
THE TWO-FLUID DESCRIPTION
4.2.1 The Mass Flux
Helium II, as presented here, onsists of a bakground uid, desribed by a lassial wave-funtion, and a bath of thermal exitations around it. To identify the
entral formulae of the marosopi two-uid desriptions from this, translation
and interpretation are needed. The identiation is best done by studying the
mass ux of the whole system. The mass ux operator is dened by
bj =
1 b
b
br + rb
2
(4.19)
The assembly average of the operator is
hbji = vs + hbjphi
(4.20)
sine the rst order terms vanish. The phonon ux operator is dened in (4.11).
In this simplied desription the phonon gas is thought to be homogeneous, so
that
jph
1
=
V
Z
dD xhbjph i
(4.21)
inserting the elds (4.13) and (4.12), only the diagonal terms where k = k0 survive
the integration. Using the normalization from last setion the expeted result is
obtained
jph
1
=
V
X k
1
k N () + 2
(4.22)
where the Bose-Einstein distribution funtion is
N () = hbyk bk i =
1
exp(=kB T )
1
(4.23)
The energy spetrum of phonons on a moving bakground was found in setion 1.2.2. Giving the phonon gas the drift veloity vn , the energy is
=!
k vn = !0 k (vn vs)
p
(4.24)
where the rest frame energy is !0 = k 1 + k 2=4. The energy of the phonon gas
is thus dependent on the relative veloity vn vs . The veloities are thought to
be small, so that the distribution funtions an be expanded. In the thermodynamial limit V ! 1, we obtain
jph
=
Z
dD k
k [k (vn
(2 )D
vs)℄ N =!0
4.2.
DERIVATION FROM GINZBURG-LANDAU THEORY
77
By putting the kx -axis along the diretion of vn vs , the omponents along all
other diretions in k spae disappear and the important formula identifying the
normal uid density is obtained:
jph
= (vn
vs)
Z
dD k 2 N k
(2 )D x =!0
n(vn vs)
(4.25)
The phonon part of the mass ux is proportional to the normal uid density. But
the diretion is along the relative veloity vn vs , not the normal uid veloity
vn. The phonon gas is thus not equivalent to the normal uid. The total mass
ux is
j = vs + n(vn vs) = svs + n vn
(4.26)
The superuid mass-density is dened to be s = 1 n . The relation between
Ginzburg-Landau theory and the two-uid desription should be established. In
Ginzburg-Landau theory we speak about a bakground uid and a phonon gas,
not superuid and normal uid. The density of the bakground uid is idential
to the total density and must not be onfused with the superuid density. The
normal uid is not visible in the total density, whih is onstant, but is identied
from the mass ux, and it will be seen to be proportional to the normal uid rest
energy.
It an be useful to integrate out the angle dependeny, to get a new formula for
the normal density. In the low temperature limit only the small k are relevant, and
we put !k = k . There is no angle dependeny in the distribution funtion, so we
get from the angle integration in two dimensions and 4=3 in tree dimensions.
By doingR a partial integration with respet to k , and transform bak the angle
integral d
= 2(D 1) , a formula for the normal uid density in the low
temperature limit is obtained
D+1
n =
D
Z
dD k
kN0
(2 )D
(4.27)
The temperature dependeny of n is easily found by substituting u = k=kB T .
The integral then beomes independent of T and it is seen that n T D+1 . In
the same way it is immediately seen that integrals of higher orders in k will also
be higher order in T , and in the low temperature limit they an be negleted.
The exat evaluations of the integrals are not important, sine the normal
density will always be identied by one of the integrals (4.25) or (4.27).
4.2.2 The Energy
In the same manner as we found the two-uid expressions for the mass ux the
energy density an be found. The total Hamiltonian of the exitations is an
78
CHAPTER 4.
THE TWO-FLUID DESCRIPTION
integral of the Hamiltonian density over the whole quantization volume
Hb2
=
Z
b2
dD x H
(4.28)
where the Hamiltonian density after a few partial integrations is
(4.29)
Hb 2 = 12 br2b + bjph vs 81 br2b + 21 b2
Inserting the Fourier omponents of (4.13) and (4.12), the Hamiltonian density is given as a double sum. Using that the integration just gives the volume
V for the diagonal elements, and zero for the o-diagonal elements, we an write
the Hamiltonian as
X
1
!k Nbk +
(4.30)
Hb2 =
2
k
where the number operator is Nbk = byk bk . The average energy density of the
system an be found as the operator average of the Hamiltonian divided with
the volume. It an, if the vauum energy is dropped and the ontinuity limit is
taken, be written as
1 E = hH
i=
V 2
Z
dD k
! N ()
(2 )D k
(4.31)
where N () = hNbk i = (exp(=kB T ) 1) 1 is the Bose-Einstein distribution funtion. The phonon frequeny is, as in the last setion, = !0 k (vn vs).
To nd the two-uid expressions for the mass ux the distribution funtion was
expanded to rst order in the veloities vn and vs . In the ase of the energy
we need to go to seond order, sine the rst order terms fall out in the angle
integration. The zeroth order term is the rest energy density of the normal uid
En =
Z
dD k
! N (!0 )
(2 )D 0
(4.32)
whih is proportional to the normal uid density (4.27). So far it has been diÆult
to nd an interpretation of the normal uid density, but it is now lear that it is
assoiated with the thermal energy. The terms of seond order in vn and vs are
#
"
Z
2
1
N () N
(
)
dD k
+ k ((vn vs ) k)2
(vs k)k (vn vs )
(2 )D
=!0 2
2 =!0
where in the low energy limit !0 k . In the last term the the substitution
2
N (k )
N (k )
k 2 N (k ) =
k
+
(4.33)
k
k
k
k
4.3.
79
THE CIRCULATION
an be done. The rst term gives a ontribution of higher order in T . The last an
be identied as proportional to the normal uid density (4.25). The ross-terms
between the veloities anel out, so that the exitation kineti energy density is
1
(v 2 v 2 )
(4.34)
2 n n s
This is the result from the quadrati terms in the Hamiltonian. In addition there
is a zeroth order ontribution from the bakground uid , 21 vs2 , so the total energy
density is exatly what we hoped for, namely
1
1
(4.35)
E = En + s vs2 + n vn2
2
2
where En is the rest energy density of the normal uid and the two other terms
are respetively the superuid and normal uid kineti energy densities.
4.3 The Cirulation
In the two-uid desription of helium II, the irulation of the superuid veloity
vs is quantized. Opposed to this, the normal uid is regarded as an ordinary
visous uid, and the irulation should be allowed to possess any value. With
a relative motion to the walls, the normal uid will experiene frition, so at
equilibrium the normal uid veloity at the boundary must be idential with the
motion of the ontainer. In our thesis we will always use a ontainer at rest, and
hene a normal uid veloity and irulation whih is zero at the boundaries.
When integrating the mass ux along the boundaries, only the superuid
density is visible
I
dl j =
I
dl (svs + n vn) = s
(4.36)
The densities s and n are thought to be onstant by the boundaries. This
formula appears expliitly in the alulation of the eetive Magnus fore in
setion 5.3.
In the phonon sattering in hapter 2 the onstraint was that the sattered
phonons should deviate as little as possible from plane waves. Plane waves give
a non-rotating phonon gas. What is the onsequene of this onstraint for the
normal uid irulation in the presene of a vortex? Is it onsistent with a nonrotating normal uid by the boundaries?
Earlier in this hapter, the two-uid model has been developed in the low
temperature limit from the Ginzburg-Landau theory, without vorties. If, on the
ontrary, one vortex is present, it is known from hapter 2 that plane waves are
not just sattered by the vortex, but the inoming plane waves themselves are
modied as well. When using the same thermal distribution as in the ase of
80
CHAPTER 4.
THE TWO-FLUID DESCRIPTION
no vorties, is it possible that the phonon gas is rotating. Our situation is in a
sense paradoxial: We will nd the irulation of the exitations in the presene
of the vortex, whih orrespond to no irulation when there are no vorties. In
other words it is the equilibrium state of the thermal system with a vortex in the
middle.
As before, the Ginzburg-Landau theory is presented in dimensionless units.
Considering the situation of one vortex at rest at the origin, the integral of the
mass urrent of the bakground uid an easily be found,
I
I
dl j0 = q dl r' = 2q = q
(4.37)
whih is not surprising. To nd the ontribution from the phonons we rst
alulate the ontribution from one single wave sattered on the vortex, and then
generalize the result to be valid for a thermal average of exitations. The mass
ux of a wave with wave-number k is
hjk i = 21 r'
(4.38)
where and are plane waves sattered on a stationary vortex. Beause of
time averaging there are no ontributions from the mixed terms between the
bakground and the exitations, so that the leading order term ontains the
elds to seond order. The phase was found in (2.35). We take the gradient
r = k ikeikx(1
i(' )) ieikx r' + rÆa
(4.39)
where = kq as usual. The other quantity needed is the density whih by (2.4)
is
= k
ikeikx (1
i('
sin(') ikx
)) + ikÆa + i
e + O 2
r
(4.40)
If the integration ontour is a irle, the line segment is dl = re' d' and only the
gradient in the e' diretion is needed in the ontour integral. For large angles,
the sattered wave is expressed by the sattering amplitude (2.35) so that the
gradient is
rÆa = ikÆ er + O
r 3=2
(4.41)
This formula is lying a bit. It is true that the derivative in the e' diretion is
of order r 3=2 , but a pure expansion in powers of r 1=2 misses the rather ugly
singularities in ' oming from the derivatives of a(') near ' = 0. This singularity
will not, and must not, enter into the integrals, but its existene is a motivation
to look for expliit small angle ontributions later.
4.3.
81
THE CIRCULATION
The inoming wave is along the x-axes, whih in polar oordinates is ex =
os(')er sin(')e'. This gives the mass urrent in the e' diretion to be
i
h
(4.42)
hjk i' = 12 jk j2k k sin(') r (1 + sin2(')) k sin(')e ikxÆa
Integrated over a whole period the plane wave part disappears as it should. When
taking out the rst term proportional to and integrating it, we get
Z
i
h 1 2 2
3
j
k j r
(1 + sin2 (')) = jk j2 k
(4.43)
d'
2
r
2
0
This is the ontribution from the term proportional to in the density and from
the twist of the inoming wave in the phase.
We will now nd the ontributions from the sattered wave in (4.42). The
sattered wave is
sin(') ikr
e
(4.44)
Æa = p
2ikr 1 os(')
The term exp(ikr(1 os(')) is more and more rapidly osillating as r ! 1.
Where the osillations are fast, the integral will average to zero, and the major
ontribution will ome for the small angles, where the osillations are alm. We
expand the integrand in this region
Z
sin2 (') ikr(1 os('))
1 2 2
j j r d' k p
e
2 k
2ikr 1 os(')
0
Z p
kr 2
2
= k jk j r p
d' e 2i '
2k This situation is similar to that in the alulation of the transverse fore. The
parameter is in this approximation small. But sine the main ontribution to
the integral is for small ', when r ! 1, the integration limit an still be taken to
innity. The integral is then of a Gaussian kind, and with the standard formula
(appendix A), we end up with the ontribution
jk j2 k
(4.45)
whih must be added to the other ontribution (4.43). But this is not the omplete
result sine so far no ontributions from derivatives of the sattered wave has been
onsidered. The derivative with respet to ' of the nite angle ontribution Æa,
although higher order in r 3=2, had wild divergenes when ' ! 0 and an not be
used.
We will instead onsider the speial small angle expression for the sattered
wave (2.34), whih was
r
Z
2ik ikx y
ik 2
Æs = e
du e 2x u
x
0
(4.46)
82
CHAPTER 4.
THE TWO-FLUID DESCRIPTION
The derivative of this with respet to y is
r
Æs
2ik ikx 2ikx y2
=
e e
y
x
whih is highly nite and well behaved near y = 0. For small angles y r'.
The ontribution to the irulation from this term an then be written as an
integral over a small region near the origin. The integral obtained is again of the
Gaussian kind, where the main ontribution omes from small arguments. The
integration limits an then, in the same manner as above, be taken to innity.
Sine the sattered wave is proportional to , a plane wave an be inserted for r
Z
1 2 1
2ik 2ikx y2
= jk j2k
j
k j
e
dy ( ik )
2
x
1
(4.47)
This is the last ontribution to the integral. The terms (4.43),(4.45) and (4.47)
must be added. The onlusion is that the ontour integral far from the vortex
of one phonon sattered on a vortex is
I
3
3
dl hjk i = 2 jkj2k = 2 q hjk i
(4.48)
where q = 2q is the irulation quantum and hjk i is the average mass ux
of the plane wave. There is a bit abuse of notation in this formula, sine mass
urrent on the left side is the full sattered wave, while the one on the right side
only is the plane wave part. This is not a vetor result, and it is the magnitude of
the mass ux whih is entering into the right hand side. The whole ontribution
from all phonons is found by letting hjk i ! kN (k ), where N (k ) is the BoseEinstein distribution funtion, and integrating over all k . This gives the phonon
ontribution
Z
I
3
d2 k
ph
dl j = q
kN = n
(4.49)
2
(2 )2 0
where the last identiation follows from the Landau expression for the normal
density (4.27). Together with the ontribution from the bakground uid (4.37),
this is the result from the Ginzburg-Landau theory. In our units the total (or
bakground) density is = 1. The superuid density is s = 1 n , so that
I
dl j =
I
dl j0 + jph = q (1 n) = q s
(4.50)
whih proves that it is possible to derive this formula from Ginzburg-Landau
theory. The alulation shows that the phonon gas is indeed rotating, and the
diretion is opposite of the bakground uid. In a way this is analogous to Lenz'
4.3.
83
THE CIRCULATION
law in eletro-magnetism, whih says the urrent indued by a hange is in the
diretion opposing the hange.
The alulation here is only to lowest order in k , whih orresponds to order
T D+1. There is no reason to expet that this result should be valid at arbitrary
power of T . At higher order in k , it is known from setion 2.5 that the ore
struture of the vortex beomes important. The irulation by the boundaries
will then also depend on the vortex ore.
A alulation of the irulation, diretly in the two-uid model, is done
in [TGV+ 01℄.
4.3.1 Numerial Calulation
In the last setion the rotation of the phonon gas was found from the asymptoti
behavior of the sattered phonons. The problem with that alulation was that
it involved ombinations of many terms. A numerial alulation an help us
to show that no terms have been lost and that no ontributions has been overounted. The partial wave expression for the time average of the mass ux is
1 X l
il
hjk i = 2 l r er + r le' ei(l m)'
(4.51)
ml
where the point vortex expressions are
l =( 1)l e 2 l J
l (kr)
(4.52)
l
(4.53)
l =ik 1
(kr)2 l
The subsripts
p of the Bessel funtions are in the aousti Aharonov-Bohm sattering l = l2 + 2l, = kq . An integral over a irle with radius r, gives 2
when l = m, else zero.
I
X l
dl hjk i = l 1 (kr)2 J
2l (kr)
l
This is a rather simple result whih we unfortunately have not found any way
to sum up exatly. The s-wave is not ontributing to the integral, whih is quite
natural sine it is symmetri about the origin. To rst order in there is hene no
dierene between aousti and magneti Aharonov-Bohm sattering sine they
only dier in the s-wave. To rst order in we write the result as
I
3
dl hjk i = q hjk iA(kr)
(4.54)
2
where q = 2q and
2 A(kr) =
3 "
X
l
l 1
#
l
J 2 (kr)
(kr)2 l
=0
(4.55)
84
CHAPTER 4.
1.4
THE TWO-FLUID DESCRIPTION
A(kr)
1.2
1
0.8
0.6
0.4
kr
0.2
5
10
15
20
Figure 4.2: Numerial alulation of A(kr) for kr = 0::20, with jlj < kr + 10.
The analytial result from the last setion an be written limkr!1 A(kr) = 1. A
numerial plot of A(kr) is in gure 4.2. If r = R is the boundary of the system,
no waves an have wavelengths longer that the size of the system and kR > 1
and to avoid ompliation with boundary eets we must assume kR 1. The
plot learly approahes a onstant value in this limit, and veries the analytial
result; the normal uid irulation at the boundary is zero.
Exept from the veriation of the analytial value of the normal uid irulation, the numerial result oers some interesting prospets for the interior
too. The result is only valid for a point vortex, whih means it must be used
well outside the vortex ore. But it is possible to have kr small and r outside
the ore as long as the phonon wave-numbers are small enough. The plot has
the limit limkr!0 = 31 . In this limit the ontour integral of the phonon ux is
H
dl jph = 31 n and the total irulation is
I
dl j0 + jph = q 1 q n = q s + 2 q n
(4.56)
3
3
where the superuid density is s = 1 n . The surprise is that the normal
uid seems to have a maximum irulation n = 23 q in the interior. The normal
uid irulation must hange ontinuously from 32 q somewhere in the interior, to
zero at the boundaries. The phenomenologial two-uid model oers a value for
the irulation by the boundaries, but does not say anything about the interior.
Therefore this result is interesting for the understanding of helium II.
Chapter 5
Fores on a Vortex
In the past years there has been a lot of ontroversy about the fore on a quantized
vortex in helium II, or more preisely: disussion about the existene of the
Iordanskii fore. Before entering into this disussion, let us take a brief look
at the more general properties of fores in helium II (see [Don91℄). The twouid desription, as presented in hapter 4, did not inlude any fores, and there
were no interations between the superuid and the normal uid. Nor did it
inlude vorties, and atually the vorties are ruial for the mirosopi piture
of fores ating in helium II. At a marosopi level, the interation between the
superuid and normal uid appears as mutual frition. When going to a smaller
length sale, the fores between the uid and a single quantized vortex an be
onsidered. This is an intermediate region, where lassial quantities an mostly
be used to desribe the uid, but where on the other hand quantized vorties,
as pure quantum eets, must be inluded as well. The simplest situation to
onsider at this level is the presene of one, straight vortex line. Sine the vortex is
straight uniformity along the z -axis an be assumed, and the system is eetively
two-dimensional. The three-dimensional result an be reovered by letting the
two-dimensional results be regarded as per vortex string length instead.
One of the fundamental assumptions in the two-uid desription of helium II,
is that the superuid and normal uid veloities are independent of eah other.
As lassial uids, the two-uids also obey the galilean invariane priniple. This
puts strong limitations on the form of fore, and the only allowed expressions are
dependent of the veloity dierenes vn vs and vv vn , where vv is the vortex
veloity, vs the superuid veloity and vn the normal uid veloity. With slow
veloities, only ontributions linear in the veloities need to be onsidered. The
fore an be divided in one longitudinal and one transverse omponent to vv ,
where in this thesis only the transverse part will be onsidered. The longitudinal
fore is interesting as well, but there is unfortunately no room to make a disussion
of it here. The transverse fore an be written as
F? = [A(vv vs) + B (vv vn)℄
85
(5.1)
86
CHAPTER 5.
FORCES ON A VORTEX
where A and B are, at this point, unknown oeÆients and = mh ez is the
quantized irulation. At the absolute zero, there is no normal uid, and there
is general agreement that the oeÆient A is the total density of the uid. The
fore is thus fully analogous to the lassial Magnus Fore, setion 1.1.2. At nite
temperature the generally aepted ontinuation of this is that A = s , and this
part of the fore is alled the superuid Magnus fore.
FM = s (vv vs)
(5.2)
The ontroversy is about the seond parameter, B . The normal uid an be
onstruted of thermal exitations: phonons and rotons. The transverse fore for
rotons was found by Lifshitz and Pitaevskii [LP58℄. In this thesis we will however
onentrate on the region dominated by the phonons, whih means temperatures
safely below 1K. The low temperature is also a reason for the disagreements,
sine experimental data are hard to obtain. In 1965 S.V. Iordanskii [Ior65℄ made
a alulation of the transverse normal uid fore in this region, and found it to
be B = n . The phonon ontribution to the fore transverse to vn is named after
him: the Iordanskii fore.
FI = n (vv vn)
(5.3)
The normal uid density goes as T D+1. The Iordanskii fore is hene small
ompared to the superuid Magnus fore for low temperatures1.
Instead of dividing the transverse fore in one term proportional to vs vv
and one proportional to vn vv , the terms proportional to the vortex veloity
an be isolated. This is the eetive Magnus fore
Fv = (A + B ) vv
(5.4)
Iordanskii's alulations was regarded as true for many years, until a paper by P.
Ao and D. Thouless in 1993 [AT93℄. They present a derivation where the eetive
Magnus fore is assoiated with the geometrial phase of the wave funtion, the
Berry phase, whih will be disussed in setion 5.3.2. Applied on the two-uid
model, their result is that the eetive Magnus fore is proportional to s and
hene there is no Iordanskii fore. Later there have been a lot of artiles polishing these arguments, where among these the artiles [TAN96℄ and [GTRV00℄
should be onsidered. The onlusion that there is no Iordanskii fore is often
referred to as the TAN result. As a reation to these topologial arguments, a
lot of artiles have been written where the Iordanskii fore is alulated diretly
from the sattering of phonons on a vortex, and usually these attempts end with
Iordanskii's onlusion. Among these E.B. Sonin's artile [Son96℄ and the artile
by M. Stone [Sto99℄ are authoritative.
1 At
higher temperatures it is small ompared to the transverse fore from rotons.
5.1.
87
THE IORDANSKII FORCE
5.1 The Iordanskii Fore
As mentioned earlier, the Iordanskii fore is the most ontroversial part of the
transverse fore. The general expression for the transverse fore (5.1) has two
parameters where the rst seems to be established as A = s . The other parameter, B , determining the Iordanskii fore, an either be found by alulating the
eetive Magnus fore giving A + B , or it an be alulated diretly. Here only the
diret approahes of determining the Iordanskii fore are disussed. This means
that the fore on a vortex from a single phonon wave, found in hapter 3, will be
generalized to a thermal gas of phonons. There are several artiles doing diret
alulations. Of those onluding in favor of a Iordanskii fore are in addition
to Iordanskii [Ior65℄ himself: Sonin [Son96℄[Son01℄, Stone [Sto99℄ and Shelankov
[She98a℄[She98b℄ the most important. The analogy between phonon sattering
and the Aharonov-Bohm eet is entral in their disussions, as well as in ours
(hapter 2). The Iordanskii fore is interpreted as a onsequene of the left-right
asymmetry of the phonon wave in the presene of a vortex. An imaginative analogy between the veloity eld of the vortex and a spinning osmi string is oered
by Volovik [Vol98℄ (and also disussed by Stone). A relativisti generalization is
disussed by Carter, Langlois and Prix [CLP01℄.
Those doing diret alulations and onluding against the existene of the
Iordanskii fore are mainly involved in the alulations of the eetive Magnus
fore as well. The artiles by Wexler and Thouless [WT98℄ and Demiran, Ao
and Niu [DAN95℄ are not too onvining, sine they do not apply the analogy
to Aharonov-Bohm sattering, and they also miss the important small angle
ontribution to the fore. A better attempt to alulate the fore is done by
Thouless et al. in [TGV+ 01℄. There the alulation is done diretly at the twouid model, without onsidering the Ginzburg-Landau theory.
Our onlusion from hapter 3 was that the fore from one phonon wave ould
be written
hjk i
(5.5)
where the quantized irulation vetor is = mh ez , and hjk i is the time average
of the mass ux of one single plane wave. From the disussion of the two-uid
desription in hapter 4, we remember that the normal uid ould be onstruted
as an integral over plane waves, so that
Z
dD r
hj i = jph = n(vn vs)
(5.6)
(2 )D k
The fundamental assumption is that the presene of the vortex does not hange
the oupation numbers of the waves, so that this formula is still valid. The total
phonon fore is thus the Iordanskii fore
FI = n vn
(5.7)
88
CHAPTER 5.
FORCES ON A VORTEX
The derivation of the normal uid without vorties is not ideal for the diret alulation of the fore. The best approah would of ourse be to do the quantization
of the phonons diretly with the vortex in the bakground. This is a diÆult proedure, and our approah, where the presene of the vortex is thought of only
as a slightly perturbation of the system, should also be reliable. The sattering
problem, and the alulations of the fore from one phonon were all done with a
stationary vortex, vv = 0, and no asymptoti superuid veloity2, vs = 0. The
diret alulations is then in favor of the Iordanskii fore.
The most serious hallenge to this result is not the diret alulations in
[WT98℄, [DAN95℄ and [TGV+ 01℄, but the alulations of the eetive Magnus
fore. A disussion of this will be held in setion 5.3.
5.2 The Superuid Magnus Fore
The superuid Magnus fore is the least ontroversial fore omponent, and most
authors agree that the oeÆient before vv vn in (5.1) is A = s , the superuid
density.
So far in this thesis no alulations of the superuid Magnus fore has been
done at nite temperatures, but in setion 3.4 the bakground uid fore was
found. It orresponds to the superuid Magnus fore at the absolute zero. The
superuid Magnus fore at nite temperatures an be found by ombining the
expression for the bakground uid fore with the fore from the phonon gas.
The fore from the phonon gas has so far just been found with a non-moving
bakground uid whih means that both the sattering problem in hapter 2
and the alulation of the phonon fore in hapter 3 must be reonsidered to
get the orret expression with a bakground uid in motion. But a qualitative
guess is that the hanges of the phonon fore is small, sine the bakground uid
veloity eld ould, as seen in setion 1.2.2, be taken as a harmless modiation
of the inident waves, whih will not inuene the sattering equations, and
the modiations of the momentum-ux tensor in hapter 3 will be ross terms
between k and vs . If we suppose that all terms proportional to k give terms
proportional to jph when the full phonon ontribution is olleted, these ross
terms are seond order in the veloities and an be negleted. These qualitative
arguments save us from a lot of mathematis, but they are not suÆient if the
superuid Magnus fore were to be severely derived.
Let us assume that the result obtained by phonon sattering on a stati bakground an be straightforward generalized to a bakground uid in motion by
substituting vn ! vn vs . We get
F? = F0? + Fph
? = vs n (vn vs )
= (s vs + n vn )
2 The
veloity oming from the vortex itself, vs = r', is ignored
(5.8)
5.2.
89
THE SUPERFLUID MAGNUS FORCE
y
vs
l
R
F
x
Figure 5.1: A superuid streaming between two ylinders.
where the superuid density is s = n . This is the total fore on a stati
vortex, where both the superuid Magnus fore and the Iordanskii fore are
inluded.
Another way of determining the superuid Magnus fore is through an imaginary experiment, as presented by Wexler [Wex97℄. It relies on thermodynamial
properties of helium II. The system whih is onsidered is a double ylinder where
the uid is trapped between the outer and inner wall (see gure 5.1). The radius
of the inner irle R is large ompared with the relative distanes l between the
walls. N quanta of irulation, where N is large, are trapped at the the inner
wall in suh a way that they give rise to a uniform superuid veloity eld
h
vs N
(5.9)
2Rm
where m is the helium atom mass. The normal uid is thought to be at rest,
vn = 0. Suppose that a vortex is reated near the outer wall at the x-axis,
and is slowly dragged along the x-axis until it reahes the inner wall, where it
is annihilated and gives rise to a modiation of the superuid veloity (5.9). If
A is positive, the fore on the vortex is towards the outer wall, and work must
be done to move the vortex. If the motion is so slow that vortex veloity an be
negleted, (5.1) gives the work as
W = FM l = Avs l
(5.10)
On the other hand let us onsider the Helmholtz free energy density f = " T s,
where s is the entropy density and " is the energy density. Sine the normal
uid is at rest, the entropy an be regarded as a onstant, and any hange in free
energy is idential to the hange in energy. The free energy density an also be
written as
f = "0 + fex
(5.11)
90
CHAPTER 5.
FORCES ON A VORTEX
where "0 = 21 vs2 is the energy density of the bakground uid, and fex is the
free energy of the phonon exitations. The ordinary formula for the exitation
free energy density is
fex =
kB T
A
X
k
ln 1
e
=kB T (5.12)
where A 2Rl is the area in whih the uid lives. The exitation energy must
be evaluated in the frame where the uid is at rest. In the laboratory system the
energy is
= }!
}vs k
(5.13)
Aording to the gure 5.1 we let vs be along the y -axis. To seond order in vs
the exitation free energy is
1 2 2fex (5.14)
fex (vs ) = fex (vs = 0) + vs 2 2 vs vs =0
sine the rst order term disappears. When dierentiating the phonon free energy
(5.12) with respet to vs and taking the ontinuity limit, we get
Z
Z
fex
d2 k
d2 k
ky
=}
=
}
k N ()
(5.15)
vs
(2 )2 e =kb T 1
(2 )2 y
where we have identied N () as the Bose-Einstein distribution funtion. By
N = }k N , this gives for the seond derivative evaluated
using the identity v
y s
in vs = 0:
Z
d2 k 2 N (}! )
2 fex
2
=
}
k
= n
(5.16)
vs2
(2 )2 y !
where the normal uid density has been identied from the usual Landau formula
(4.25). Inluding the bakground uid kineti energy, the free energy density is
1
1
f = ( n ) vs2 = s vs2
(5.17)
2
2
When the vortex is pinned to the inner wall it gives rise to a hange of the
superuid veloity, and hene to the free energy. Using the expression for the
superuid veloity eld as a funtion of N , (5.9), the hange in free energy density
is
f = f (N + 1) f (N )
2
(5.18)
h
h
1
s vs
2 s 2Rm (N + 1)2 N 2 2Rm
5.3.
THE EFFECTIVE MAGNUS FORCE
91
The hange in free energy is the same as the work performed on the system,
(5.10), sine the entropy is onstant. This determines the superuid Magnus
fore
We write
(2Rl) f = W =) A = s
(5.19)
FM = s vs
(5.20)
This derivation relies on simple thermodynamial properties of helium II. Again
it is seen that the bakground uid is not idential to the superuid. The bakground uid has density while the superuid has density s . The derivation is
diÆult to generalize to a moving normal uid. What made this derivation easy,
is the onstant entropy making the hange in energy and free energy idential.
With a moving normal uid, an expression for the entropy and entropy hange
must have been needed as well.
A more omplete disussion of the relation between the bakground uid and
the phonon gas on one side, and the superuid and normal uid on the other,
was held in setion 4.2.
5.3 The Eetive Magnus Fore
Muh of the ontroversy about the fores ating on a vortex string in helium
II originates in the artiles by Ao and Thouless [AT93℄ and Thouless, Ao and
Niu [TAN93℄ in 1993, where they derive an expression for Magnus fore in a
superondutor. In their derivation they interpret the fore as a geometrial
phase of the wave funtion, the Berry-phase ([Ber84℄, setion 5.3.2), assoiated
with an adiabati motion of a vortex around a iruit. The alulation was
modied in 1996 [TAN96℄ and applied on superuids only.
The following derivation relies on the papers [TAN96℄ and [Ao96℄. The goal
is to alulate the eetive Magnus fore, and hene to determine the sum A + B
in (5.1). The superuid and normal uid veloities are put to zero, vs = vn = 0.
Consider a vortex moving in system of N partiles in two dimensions. The
number of partiles is thought to be onserved so that ordinary wave mehanis
an be used. We want to nd an expression for the fore on the vortex from the
system, as we push it slowly around. If the vortex is at rest, the whole uid is
supposed to be at rest, and the fundamental assumption needed to perform the
alulation, is that the total time evaluation of the system, is due to the motion
of the vortex. This ondition holds if the vortex motion is so slow that the system
is in equilibrium at any time, and the motion is hene adiabati. The fore on
the vortex is found as
h
i
Fv = p_ = Tr ^_ p^
(5.21)
92
CHAPTER 5.
FORCES ON A VORTEX
P
where the density operator is ^ = n fn jn ihn j and the sum is over all energy
eigenstates of the system. The formalism with the density matrix is a generalization of the Shrodinger theory, and is used on large systems where it is impossible
to nd a wavefuntion. The system is instead desribed by a statisti distribution
over states. At equilibrium with temperature T the fators fn are the Boltzmann
fators fn = exp( En =kb T ). Considering adiabati motion, these equilibrium
fators an be used. The total momentum operator is simply the sum of the
momentum operators for eah partile
p^ = i}
X
i
ri
(5.22)
where ri means derivatives with respet to the position of partile i. When the
density matrix is inserted, the fore beomes
Fv = Tr
"
X
n
n
o
#
fn j_ n ihn j + jn ih_ n j p^
(5.23)
Now the full meaning of the adiabati motion beomes visible. Sine the Boltzmann fators fn are independent of time, thePwhole time evaluation is through
the elds. The trae of an operator is Tr A = k hk jAjk i. The full expression
for the fore is then
o
Xn
X
hk j_ nihn jp^ jk i + hk jnih_ n jp^ jk i
Fv =
fn
n
k
The operator p^ is hermitian, so that it an be taken inside to work on either the
\bra" or the \ket" aording to our hoie. Rearranging
the terms and using that
P
the sum over all k just gives the identity operator k jk ihk j = 1^ , we get
Fv =
X
n
fn
n
hp^ n j_ ni + h_ n jp^ n i
o
Now the expression for the momentum operator and for the time derivative an
be inserted. The wave funtions n = n (r1 ::rN ; rv (t)) are dependent of the
position of the partiles and the vortex. A ruial ondition to get any further,
is that the only dependeny of the vortex position is through the distane from
partiles to the vortex, so that n = n (r1 rv (t); ::; rN rv (t)). This ondition
holds if the ontainer is large, and the vortex is far from the boundaries. The
time derivative an hene be transformed to derivatives of the partile positions
X
_ n = rrv n vv =
rj n vv
(5.24)
j
with vortex veloity vv = r_ v . With the denition of the momentum operator (5.22), the fore an be written as
Fv = i}
X
n
fn
X
ij
[hri n jrj n vv i
hrj n vv jrin i℄
5.3.
93
THE EFFECTIVE MAGNUS FORCE
The seond term in the above expression is the onjugate of the rst, whih
guarantees that the fore is real. The result is now proportional to the vortex
veloity. Only the fore to rst order in vv is of interest, so the wave-funtions
an from now on be evaluated as if the vortex Q
was atR rest, n = n (r1 ::rN ).
^
To express this in partile position spae, 1 = Nk=1 d2 rk jr1 ::rN ihr1 ::rN j is
inserted. Eah of the integrations is over the physial area. The many partile
wave funtion is n (r1 ::rN ) = hr1 ::rN jn i. The dot produts an by the use of a
vetor alulus formula be transformed to
X
ij
[rin (rj n vv )
rin (rj n vv )℄ = vv X
ij
(ri n rj n )
Here the sum over i and j are interhanged in the seond term on the left hand
side. The anti-symmetry of the ross produt guarantees that the result is still
purely imaginary. The fore is here seen to be perpendiular to the vortex veloity.
This is a onsequene of the adiabati motion. If it were a fore omponent along
the vortex path, it would do work on the system. Sine the work annot give a
motion of the uid (the uid is at rest), the energy must be lost as heat in the
system, hanging the entropy. A nal vetor formula gives
X
X1
r
[ri n rj n ℄ =
i (n rj n n rj n )
2
ij
ij
So far no assumptions about the statistial properties of the partiles has been
made. We will now suppose that the system onsists of N idential Bosons. The
partiles an then be interhanged without hanging the sign of the wavefuntion.
The sum over i and j has N diagonal terms where i = j , and N (N 1) o-diagonal
terms with i 6= j . Beause of the integration over all rk , the diagonal terms are
idential to the i = j = 1 ase, while all o-diagonal terms are idential to
i = 1; j = 2. The nal expression for the fore an hene be split in a one-partile
ontribution and a two-partile ontribution,
Fv = F1 + F2
(5.25)
v
where
F1 =vv v
F2 =vv v
Z
Z
v
i
d2 r r (r r0 )(r0; r)
2
r =r
Z
i
2
2
0
0
d r1 d r2 r1 (r2 r2 ) (r1 r2; r1 r2)
2
r2 =r2
(5.26)
0
0
(5.27)
The one-partile ontribution is expressed by the one-partile density matrix,
(r0 ; r) = }N
X
n
fn
N Z
Y
k=2
d2 rk (r0 r2::rN )(rr2::rN )
(5.28)
94
CHAPTER 5.
FORCES ON A VORTEX
while the two-partile ontribution is expressed by the two-partile density matrix
(r0 r0 ; r
1 2 1 r2) = }N (N
1)
X
n
fn
N Z
Y
k=3
d2 rk
(r01r02r3::rN )(r1r2r3::rN )
(5.29)
This is the TAN result. None of the speial properties of the two-uid model has
been used. The only assumptions are that the motion is slow and adiabati, and
the system is large, so that boundary eets an be ignored.
5.3.1 Appliation on Helium II
The TAN result (5.25) is valid for a general N -partile system, but usually is
it applied on either superondutors or superuids. We will use the formula on
superuid helium II only. In the previous setion the one-partile density matrix
was identied from a N -partile quantum mehanis expression. The same matrix
an also be written as
b r)i
b r0 )y(
(r0 ; r) = h(
(5.30)
b y and b are the helium atom reation and annihilation operators. The
where onserved urrent an be expressed by the one-partile density matrix
i}
(r
2
r0)(r0; r)jr =r
0
(5.31)
i} b y
y
b
b
b
b
=
h (r)r(r) r (r)(r)i = hj(r)i
2
This makes it possible to identify the mass urrent in the one partile ontribution
to the fore (5.26). The expression for the fore is proportional to the vortex
veloity, so to rst order in the veloity it is enough to onsider the mass urrent
of a vortex at rest. With Stoke's theorem, the integration an be transformed to
a ontour integral, where the integration ontour is the physial boundary of the
system. The one-partile ontribution to the fore is then
F1v = vv = vv I
I
dr hbj(r)i
dr (svs + n vn) = [s + n n℄ vv
(5.32)
where = mh ez is the quantized superuid irulation and n is the normal uid
irulation. The normal uid irulation is not quantized, and by the boundary
it must at equilibrium be equal to the irulation of the ontainer. This is a
onsequene of the interpretation of the normal uid as an ordinary visous liquid.
5.3.
THE EFFECTIVE MAGNUS FORCE
95
A relative motion between the normal uid and the wall would lead to a frition
fore ounterating the normal uid motion. We will in the following use the
boundary ondition with non-rotating normal uid, n = 0.
For the omparison with the diret alulation, it is ruial that the same
boundary onditions are used. In the Ginzburg-Landau theory, the onstraint
was that the normal uid in the presene of the vortex should be as lose as
possible to the non-vortex situation. In setion 4.3 we derived that this led to
no normal uid irulation. The Boundary onditions in the alulation of the
phonon fore are hene idential to the boundary onditions in the alulation of
the eetive Magnus fore.
The two-partile ontribution (5.27) is diÆult to alulate in general. Using
the quantized elds, as in the one-partile ontribution, leads to a produt of
four eld operators. This produt is hard to handle. We an thus not make a
alulation of it, as in the one-partile ontribution. But there are reasons to think
that it will vanish. The two-partile density matrix desribes the interations and
orrelations between partiles and the internal interation between the partiles
should not ontribute to the fore between the vortex and the liquid.
The TAN result gives then an eetive Magnus fore proportional to s , implying no Iordanskii fore. The onlusion is dependent of the two-partile part
of the fore to vanish, whih we are unable to verify by alulation.
5.3.2 Berry's Phase
Central in the interpretation of the TAN result, is a geometrial phase of the wavefuntion; the Berry phase. The idea of geometrial phases will no be skethed, as
presented by Berry [Ber80℄. Consider a Hamiltonian, Hb (rv (t)), depending on a
time-varying parameter, rv (t). The state j(t)i satises the ordinary Shrodinger
equation
Hb (rv (t))j(t)i = i} j(t)i
(5.33)
t
The Hamiltonian has a set of time-dependent eigenfuntions, with eigenstates En
whih are (for simpliity) independent of time
H (rv (t))jn (rv (t))i = En jn (rv (t))i
(5.34)
Let us put the system in the state jn (rv (0))i at t = 0. The system will ontinue
to be in the state jn (rv (t))i, if the time-evolution is adiabati
j(t)i = e i1} Ent+in (t)jn (rv (t))i
(5.35)
where the exp( i1} En t) is the dynamial Dira phase fator. The attention now is
to the other phase fator, n (t). Inserting the above expression into the timedependent Shrodinger equation (5.33), gives the equation
1
En jn (rv (t))i = i} En + i_ n jn (rv (t))i + jn (rv (t))i
i}
t
96
CHAPTER 5.
FORCES ON A VORTEX
The time dependeny of the eigenstates are only through rv (t) so the time derivative an be found as t = r_ v rrv . If the eigenstates are normalized to unity, this
gives an expression for _ n
_ n = ihn (rv (t))jrrv (rv )i r_ v
The interesting point is the r_ v on the right hand side. When integrating over
t the right hand side transforms to a ontour integral over the path rv (t). Of
speial interest is when the path is losed and rv (t) = rv (0). The phase fator
n is then not zero, but has a value depending on the losed path C . The Berry
phase is
n (C ) =
I
C
drv hn (rv )jrrv n(rv )i
(5.36)
The value of n (C ) is independent of how the iruit C is traversed, as long as
the motion is slow enough to be adiabati.
Now we want to see the onnetion between the Berry phase and the expression for the eetive Magnus fore in setion 5.3. In that alulation the derivatives with respet to the vortex position was hanged to derivatives with respet
to the partile positions. Now we go the other way around, and express
the fore
P
by derivatives of the vortex position only, whih means that i ri = rrv .
Substituting this in the expression for the fore, we get
Fv = i}vv
X
n
fn
rrv hnjrrv ni
(5.37)
This looks like the integrand in the Berry phase. The fore Fv is thought to be
independent of the vortex position rv . An integral of rv over the left hand side
only gives the volume. The right hand side gives, with the use of Stoke's theorem,
a ontour integral. So when integrating over a volume , we get
i}v
Fv = v
I
( )
drv hn jrrv n i = }vv (5.38)
where the Berry phase as funtion of the boundary is
( ) =
X
n
fn n ( )
(5.39)
where n ( ) is the Berry phase of the wavefuntion whih was in state n at t =
0. We have thus two expressions for the eetive Magnus fore. In setion 5.3.1
the fore was found to be proportional to the superuid density and now we have
seen that it an also be expressed by the Berry phase. This relation was also
found by Haldane and Wu [HW85℄. The fore is independent of whih iruit
that is hosen and how it is traversed, giving the result topologial authority.
5.4.
DISCUSSION OF THE CONTROVERSY
97
Impurities and other perturbations an thus not hange the TAN result. The
superuid Magnus fore is hene a topologial quantity. This interpretation of
the TAN result is the same both for superuids and superondutors. Of speial
importane is this for dirty superondutors, where this result says that the
superuid Magnus fore is not inuened by impurities, nor details in the vortex
ore.
5.4 Disussion of the Controversy
Opposed to the diret alulations, the TAN result implies no Iordanskii fore. We
will now make a summary of the assumptions done in both the diret alulations
and the TAN alulation and point to the possible reasons for the oniting
results.
The diret alulations of the Iordanskii fore are arried out in GinzburgLandau theory. There is of ourse a fundamental assumption that GinzburgLandau theory gives an adequate desription of helium II. We saw in hapter 4
that the phenomenologial two-uid desription of helium II ould be derived
from Ginzburg-Landau theory, in the absene of vorties. In the derivation, it
was neessary to make some interpretations and identiations, but most of the
two-uid results ame out rather easily. Thus it seems like the Ginzburg-Landau
theory works, at least at a marosopi length sale.
In the sattering of a phonon on a vortex, the presene of the vortex was
treated as a perturbation of the non-vortex situation. In an ordinary sattering
problem, this means that the solution far from the vortex an be divided in an
inoming plane wave and a sattered wave. But in the sattering of a phonon on
a vortex this was not possible, sine the long range of the vortex fored a twist
on the inoming wave. This twist, although long range, was proportional to the
wave-number k , so in the low energy limit there should still be expeted that
the perturbative approah works. There are hene two ondition on k : First,
the wavelengths of the thermal phonons must be small ompared with the size of
the system, for boundary eets to be negligible. But the phonon wavelengths
must also be large ompared with the vortex ore, for ore eet to be ignored.
Mathematially this means that k 1 and kR 1 when R is the radius of the
ontainer.
The TAN derivation of the eetive Magnus fore starts with a very general N partile system. The fore is found from the response in the liquid, as the vortex
is pushed adiabatially around. The ontainer is, as in the diret alulations,
thought to be large, so that boundary eets an be ignored. The dependeny
of the vortex position is thus only through the distane from the partiles to
the vortex. Beause of the adiabati motion, all time-dependeny is through the
vortex position. Using this, we are able to express the fore by the one- and twopartile density matries. The one-partile ontribution is proportional to s ,
98
CHAPTER 5.
FORCES ON A VORTEX
provided that the normal uid irulation is zero. The normal uid irulation
was also found to be zero in Ginzburg-Landau theory, proving that the GinzburgLandau theory and the TAN derivation have the same boundary onditions. The
ontribution from the two-partile density matrix is diÆult to alulate exatly,
but general arguments indiates that it is zero. The TAN expression for the
eetive Magnus fore is proportional to s , implying no Iordanskii fore.
There are two possible ways out of the disagreement. Of ourse there ould
be some tehnial error in one of the arguments, leading to a wrong answer.
In this thesis we have arefully gone trough the alulations, and not found any
obvious mistakes. We will therefore onentrate about the other possibility, whih
is that both alulations are indeed orret, but alulating dierent quantities.
The diret alulations are of ourse determining the Iordanskii fore. The TAN
alulation nds the adiabati fore as the vortex is pushed around. Is it possible
that the Iordanskii fore is not adiabati? Let us onsider the TAN situation
with a vortex pushed around in a stationary liquid. The liquid is in ontat
with the surroundings at the vortex and by the boundaries. Sine the Iordanskii
fore is perpendiular to the vortex motion, there is no work on the liquid at the
vortex point. But we know that the thermal phonons are sattered on the vortex
and these sattered phonons will in turn interat with the boundaries. There
might be a net heat transfer between the boundaries and the liquid. This an
be the reason why the Iordanskii fore is not visible in the Berry phase and the
adiabati alulation of the fore. If the Iordanskii fore is not adiabati, it is
still possible to assoiate the superuid Magnus fore with the Berry phase. But
it is not possible to nd the eetive Magnus fore, only the superuid Magnus
fore, from the alulation based on adiabati motion.
Conlusion
The purpose of this thesis was to alulate the Iordanskii fore in liquid helium II.
The diret alulations of the fore were arried out in Ginzburg-Landau theory.
In order to nd the fore, the sattering of a phonon on a quantized vortex was
examined, and the analogy to Aharonov-Bohm sattering was disussed. In the
Born approximation, the onventional Aharonov-Bohm sattering amplitude was
obtained. The sattering amplitude was not valid for small angles, though, and
right behind the vortex a speial small angle solution was developed. Hidden
in the Born integral was also an important modiation of the inoming wave,
whih was interpreted as a onsequene of the long range of the vortex. The Born
solution was thus onsisting of three dierent ontributions; a sum of two valid
at nite angles, and one valid for small angles. A omparison in the overlapping
region showed that the Born solution was ontinuous and single valued. The
sattering problem was in addition solved as a sum of partial waves. By expressing
Bessel funtions as ontour integrals, we were able to sum up. The solution, whih
was valid at all angles, was idential to the Born solution when ompared in the
dierent limits.
The transverse fore on the vortex from one phonon was found, by examining
the momentum hange through a ontour enlosing the vortex. Analytial answers were found in two limits: with the integration ontour far from the vortex,
and lose to the vortex (but outside the ore). The Born solution was used far
from the vortex, where the alulation was laborious and involved many terms
and limits. With the integration ontour put near the vortex, the fore was
found almost with no alulations at all. In addition numerial expressions were
obtained at nite distanes from the vortex. All these alulations agreed on a
transverse fore proportional to the phonon mass ux, giving the Iordanskii fore,
when the full phonon ontribution was olleted.
The two-uid desription of helium II was derived from Ginzburg-Landau
theory. In Ginzburg-Landau theory, liquid helium was modeled as a phonon gas
on the top of a bakground uid. The superuid and normal uid were identied
from the expression for the mass ux. From the phonon sattering we were able
to show that the normal uid irulation was zero at the boundaries, proving
that the boundary onditions used in Ginzburg-Landau theory were onsistent
with the two uid model.
99
100
CONCLUSION
We have gone through the alulation of the eetive Magnus fore, found by
pushing the vortex adiabatially around in a general N -partile system. Applied
on the two-uid model it gave a fore proportional to s , implying no Iordanskii
fore. The superuid Magnus fore was identied with the geometrial Berry
phase, giving the fore a topologial harater.
Several ways out of the onit were pointed out. One worth onsidering is
the possibility of Iordanskii fore being non-adiabati, and hene lost in the Berry
phase alulation.
Appendix A
Some Useful Integrals
Here is a list of some integrals whih are used in the thesis. These integrals an
also be found in many integral tables, for example Table of integrals, series, and
produts [GR94℄ or Handbook of Mathematial Funtions [AS68℄.
First some integral representations of the Bessel funtions
Z Z Z d' e
in'+iz sin(')
= 2 Jn (z )
(A.1)
d' sin(n') sin(z sin(')) = [1 ( 1)n ℄ Jn (z )
(A.2)
d' os(n') os(z sin(')) = [1 + ( 1)n ℄ Jn (z )
(A.3)
This leads to two integrals we need
Z Z d' sin(') eiz sin(') = 2i J1(z )
(A.4)
d' os(') eiz sin(') = 0
(A.5)
Another integral we need is
Z 1
dx
=
1 x2 + 2 j j
(A.6)
and the Gaussian integral
Z 1
1
dx e
x2
This formula works for <() 0
101
=
r
(A.7)
102
APPENDIX A.
SOME USEFUL INTEGRALS
We also need integrals of the kind
Z 1
0
dx
x +1 J (ax)
= K (a )
x2 + 2
(A.8)
For the integrals of the Hankel funtions, there are not always available formulas,
but the Hankel funtions of the rst kind an always be expressed by Bessel
funtions as
H(1)(z ) =
J (z ) e i J (z )
i sin( )
(A.9)
For b 1 we have the integral
Z 1
0
J (t)eibtdt =
p
1
1
i arsin(b)
e
b2
(A.10)
Combining these formulas we get
Z 1
2 sin [ (' + 2n)℄
(A.11)
H(1)(t) e it os(t)dt =
sin( ) sin(')
0
On the right hand side we must hoose an integer n aording to the value of '.
ei 2 Appendix B
Vetor Calulus in 2-D
In two dimensions the usual vetor alulus formulae an take some peuliar
shapes. Let the the vetors A = Ax ex + Ay ey and B = Bx ex + By ey live in the
xy-plane, and let ez be the unit vetor in the z-diretion. Then we have
A ez = (Ay ; Ax)
(B.1)
A B = AxBy Ay Bx
(B.2)
r A = r (A e z )
(B.3)
The two dimensional delta funtions an also be identied as
r r' = 2Æ(r)
(B.4)
where ' = artan(y=x). This an be seen by remembering that the url of a
gradient is zero everywhere exept in singularities. An integral on a irle with
radius r gives, when Stoke's theorem is applied
Z
I
Z 2
1
2
d r r r' =
dl r' = r
d' = 2
r
!
!
0
whih proves that this is a delta funtion. The ommutator r ; t '0 will also
be needed when '0 = artan((y y0)=(x x0 )) and r0 = r0 (t). First we notie
that all time dependeny of '0 is through r0, so that t r'0 = (_r0 r)r'0.
Using the ordinary vetor formulae found in text-books we nd
0 ) = r [_r r' ℄
r( '
0
0
t
= (_r0 r)r'0 r_ 0 (r r'0 )
= r'0 (_r0 ez )2Æ (r r0)
t
By taking the ross produt on both sides, we get
(B.5)
r ; t '0 ez = 2 r_ 0Æ(r r0)
103
104
APPENDIX B.
VECTOR CALCULUS IN 2-D
Appendix C
The Two-Dimensional Green's
Funtion
The time-independent two-dimensional Green's funtion D(r) is a solution of the
equation
(k 2 + r2 )D(r) = Æ 2(r)
(C.1)
The Fourier transform of the equation is
(k 2 p2 )D~ (k) = 1
The Green's funtion is then found as the double integral
Z
1
d2 p
D(r) =
eipr
2
2
(2 ) p k 2
If we introdue polar oordinates (p; ) in phase spae, the angle part of the
integral an be easily arried out
Z 2
Z 2
ipr
os(
'
)
d e
=
d eipr sin(') = 2 J0 (pr)
0
0
from equation (A.1). The restoring integral an be found by regularizing
k 2 = ( ik )2 for some small positive , and use formula E.H.II 96(58) in [GR94℄.
Z 1
x
dx 2 2 J0 (ax) = K0 (ab)
x +b
0
where K0 is a Bessel funtion of seond kind. The Green's funtion is then
1
i
D(r) = K0 (r( ik )) !!0 H0(1)(kr)
(C.2)
2
4
The nal result is independent of the hoie of sign. H0(1) is the zeroth order
Hankel funtion of the rst kind.
105
106
APPENDIX C.
THE TWO-DIMENSIONAL GREEN'S FUNCTION
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