JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 44, NUMBER 9
SEPTEMBER 2003
Classical transport of charged particles in a magnetic field
Noel Corngolda)
Watson Laboratories, California Institute of Technology, Pasadena, California 91125
共Received 8 September 2002; accepted 23 April 2003兲
We examine the traditional transport equation for classical, charged particles diffusing in a cold, absorbing medium subject to a uniform magnetic field and in
which scattering is isotropic. Steady-state solutions in plane geometry are examined
in some detail; we make no expansion about an isotropic angular distribution.
Restricting the motion to two dimensions captures most of the interesting features;
there is some discussion of the three-dimensional case. © 2003 American Institute
of Physics. 关DOI: 10.1063/1.1591994兴
I. INTRODUCTION
The motion of ions and electrons in a weakly ionized gas subjected to external electric and
magnetic fields is an ancient and ‘‘classical’’ problem in kinetic theory.1,2 The analysis of these
systems 共‘‘swarms’’兲 is based, almost always, upon a linear ‘‘Boltzmann,’’ or transport equation.
The equation is treated in its most general form so that one can be as faithful as possible to the
experiments. Most often the velocity variation of the distribution function is expressed in terms of
a series of tensorial spherical harmonics, and the evolution of the density described in ‘‘hydrodynamic’’ and ‘‘nonhydrodynamic’’ terms. 共Here, the interested reader should consult the impressive
publications of Australian physicists.3– 6兲 Such expansions—which may be viewed as a generalization of the traditional, Chapman–Enskog treatment of Boltzmann’s equation—generate an infinite hierarchy of ‘‘moment-equations,’’ a hierarchy which is truncated at some level, for reasons
which are physical or practical. This procedure, with its advantages and its limitations, has been
part of transport theory almost since its creation. Since these expansions are usually asymptotic, at
best, one welcomes comparison with exact solutions of related transport equations which are not
trivial.
This paper is concerned with the solution of such a relevant model—it describes the timeindependent transport of charged test-particles in a cold medium which scatters isotropically, may
capture particles, and is immersed in a uniform magnetic field. Since no electric field is present,
the distribution in velocity is simpler, but still interesting. In the solution, the most important
dimensionless parameter is ␻, the ratio of cyclotron frequency to collision frequency. If one
wishes to go beyond treating the analysis as a mathematical exercise, merely, one notes that ␻ is
trivially small for ions in realistic situations, but can be interestingly large for electrons. Then, the
共asymptotic兲 velocity distribution deviates considerably from the isotropic form about which the
simplest ‘‘hydrodynamic’’ models are centered. Other results are—apart from an array of attractive
equations involving Bessel functions—共1兲 that the magnetic field shatters the famous continuous
spectrum associated with the ‘‘one-speed transport equation7’’ into an infinity of discrete eigenvalues. 共We discuss the behavior of these eigenvalues as field and absorption strength are varied;
the limit ␻→0 is quite singular.兲 共2兲 The magnetic field induces a flow parallel to the source plane.
This ‘‘diamagnetic drift’’ is well known to plasma physicists.8 Here we treat it in some detail.
Most important is that none of our results rely upon an assumption of ‘‘small gradients.’’
Justifying the irreversible, transport—rather than the reversible, Liouville treatment of a problem in kinetic theory—poses famous and delicate problems. In the case of an isolated and simple
gas one justifies mathematically the transition from Liouville to transport by a limiting
a兲
Electronic mail: nrdc@caltech.edu
0022-2488/2003/44(9)/4057/21/$20.00
4057
© 2003 American Institute of Physics
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4058
J. Math. Phys., Vol. 44, No. 9, September 2003
Noel Corngold
process—by invoking the ‘‘Grad-limit.9’’ To the physicist, this means, roughly, that the gas should
be sufficiently dilute for transport theory to be useful. When the system is subject to external
fields, matters become more difficult. Recently, a group of scholars10 has analyzed the limiting
process for the Lorentz model—which describes charged particles diffusing through a static array
of scatterers—immersed in a magnetic field. Their analysis considers how unusual trajectories—a
particle whirling in circular orbit in the space between two scatterers, for example—are to be
treated in the construction of the ‘‘correct,’’ coarse-grained transport equation. Our paper avoids
these difficult issues. We simply study the ‘‘traditional’’ transport equation as a mathematical
object, acknowledging that it describes the physics correctly in some regime where the magnetic
field is not too strong and that often, continuing results obtained from a sanctioned region, say
␻⬍1, into a questionable one, say ␻Ⰷ1, has some value.
Most of the paper is devoted to motion in two dimensions 共2D兲—in a plane perpendicular to
the uniform magnetic field. We give, briefly, expressions for densities and currents in the threedimensional case as well. In treating the equation, its features and consequences, we allow the
strength of the uniform magnetic field to vary arbitrarily. The fact that a steady source of particles
in an absorbing medium produces a steady distribution indicates that the traditional equation gives
no weight to particles whirling, collisionless in tight circles,
II. ANALYSIS OF A SIMPLE MODEL
We begin with a simple model: noninteracting charged particles of fixed kinetic energy move
in a plane perpendicular to an uniform magnetic field. The particles scatter elastically and isotropically from host atoms 共‘‘neutrals.’’兲 We are in steady state, there is a plane source (y – z plane兲
at the origin, and the medium, which is unbounded, may absorb particles. The distribution function
for position and velocity, F(x, ␾ ), depends then upon two variables, and we face the kinetic
equation
␮
⳵
⳵
c
F 共 x, ␾ 兲 ⫺ ␻
F 共 x, ␾ 兲 ⫽
⳵x
⳵␾
2␲
冕
2␲
0
d␾ F 共 x, ␾ 兲 ⫺F 共 x, ␾ 兲 ⫹Q 共 ␾ 兲 ␦ 共 x 兲 .
共1兲
This familiar equation, with ( v •x)⫽ ␮ ⫽cos ␾, is written in dimensionless variables. The constant
speed is taken to be unity, distance is scaled by the total mean-free-path for encounters, time by
the corresponding mean-free-time, and ␻ represents the cyclotron frequency qB/mc divided by
the collision frequency. ␻ may also be seen as the ratio of mean-free-path to Larmor radius. c, the
number of secondaries produced per collision, is limited to 1⬎c⭓0. As we see later, transport in
three dimensions—with plane symmetry—is not that much more difficult.
After Fourier transformation in x we have
Of 共 k, ␾ 兲 ⫽⫺ ␻
⳵
c
f 共 k, ␾ 兲 ⫹ 共 1⫹ik ␮ 兲 f 共 k, ␾ 兲 ⫽
⳵␾
2␲
冕
2␲
0
d␾ f 共 k, ␾ 兲 ⫹Q 共 ␾ 兲
共2兲
or
共 O⫺cP兲 f 共 k, ␾ 兲 ⫽Q 共 ␾ 兲 .
A new notation, Dirac-like, helps. Our operators act in function spaces whose elements are distribution functions 兩 f 典 . and 具 g 兩 . There is a complex inner product. Thus,
具g兩 f 典⬅
冕
2␲
0
d␾ g * 共 ␾ 兲 f 共 ␾ 兲 ⫽ 具 f 兩 g 典 *
and
共 O⫺cP兲 兩 f 典 ⫽ 兩 Q 典 .
共3兲
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4059
P is the projection operator,
P兩 f 典 ⬅ 兩 a 典具 a 兩 f 典 ,
具 a 兩 a 典 ⫽1
with
兩a典⫽
1
,
冑2␲
具a兩 f 典⫽
冕
2␲
0
d␾
1
f 共 ␾ 兲.
冑2␲
We may also use 兩 1 典 ⫽ 冑 2 ␲ 兩 a 典 , so that 具 1 兩 f 典 ⫽ 兰 20 ␲ d␾ f (k, ␾ ) gives n(k), the angle-integrated
density. The particle currents are
Jx,y 共 k 兲 ⫽
冕
2␲
0
d␾ 关 cos ␾ ,sin ␾ 兴 f 共 k, ␾ 兲 .
We use the special notation Jx,y (k)⫽ 具 j x,y 兩 f 典 for the currents.
Two results follow quickly; the solution to the kinetic equation, Eq. 共3兲, is
1
1
兩 f 典 ⫽ 兩 Q 典 ⫹c 兩 a 典
O
O
冋
1
兩Q典
O
,
1
1⫺c 具 a 兩 兩 a 典
O
具a兩
册
共4兲
and if we take the particularly simple case of an isotropic source of strength 兩 Q 典 ⫽ 兩 a 典 we find
兩 f 典⫽
具a兩 f 典⫽
冋
冋
1
1⫺c 具 a 兩
1
兩a典
O
1
1⫺c 具 a 兩
1
兩a典
O
册
册
1
1
1
兩a典⫽
兩a典,
O
D共 k,c 兲 O
共5兲
1
1
1
具a兩 兩a典⫽
具a兩 兩a典.
O
D共 k,c 兲
O
The last equation also states that with unit source, 兩 Q 1 典 ⫽1/2␲ ,
n共 k 兲⫽
1
1
K共 k 兲
K共 k 兲
⫽
.
具a兩 兩a典⫽
D共 k,c 兲
O
D共 k,c 兲 1⫺cK共 k 兲
There are corresponding expressions for the currents,
Jx,y 共 k 兲 ⫽
1
1
具 j x,y 兩 兩 a 典
D共 k,c 兲
O
for source 兩 Q 典 ⫽ 兩 a 典 .
An alternative expression for Jx (k) appears if we integrate Eq. 共1兲 directly, to get the equation
of continuity,
⳵
J 共 x 兲 ⫹ 共 1⫺c 兲 n 共 x 兲 ⫽Q ␦ 共 x 兲 .
⳵x x
We find
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4060
J. Math. Phys., Vol. 44, No. 9, September 2003
ikJx 共 k,c 兲 ⫽
Noel Corngold
1⫺K共 k 兲
D共 k,1兲
⫽
,
1⫺cK共 k 兲 D共 k,c 兲
共6兲
for unit source 兩 Q 1 典 . That the various expressions are related by partial integration will be evident
later, as will the expression
iJy 共 k,c 兲 ⫽
␻
⳵
1
K共 k 兲 .
2 D共 k,c 兲 ⳵ k
共7兲
The quantity
K共 k 兲 ⫽ 具 a 兩
1
兩a典
O
contains the essence of the system, generating the relaxation lengths, for example. This kernel is
the Fourier transform of the kernel controlling the Peierls integral equation11 for n(x) 共see the
following兲. We shall refer to D(k,c)⫽1⫺cK as the dispersion function. This article is concerned
with the features of density and current. To get at them, we learn as much as we can about K(k)
and D(k,c).
A. Evaluating the kernel
Of the several ways to evaluate the kernel; two are particularly helpful. In the first, we make
use of the eigen-vectors of the operator O,
O兩 n 典 ⫽␭ n 兩 n 典 .
One finds
兩n典⫽
1
冑2 ␲
冋
exp关 in ␾ 兴 exp i
册
k
sin ␾ ,
␻
共8兲
␭ n ⫽1⫺in ␻ .
The 兩 n 典 are complete and orthonormal, with the usual complex inner product. The wave number
k appears parametrically in the eigenvector but not in the eigenvalue. The 兩 n 典 form a useful basis
for expansion. Since
具a兩n典⫽
1
2␲
冕
2␲
0
冋
d␾ exp关 in ␾ 兴 exp i
冉 冊
册
k
k
sin ␾ ⫽ 共 ⫺ 兲 n J n
⫽具n兩a典.
␻
␻
共9兲
Bessel functions make their expected appearance. A quite straightforward calculation leads then to
the first of several expressions for the kernel,
具a兩
冋 冉 冊册
1
k
兩a典⫽ J0
O
␻
兺 冋 冉 冊册 冒
⬁
2
⫹2
n⫽1
Jn
k
␻
2
共 1⫹n 2 ␻ 2 兲 .
共10兲
共In the following text, we may denote k/ ␻ by ␬, which is proportional to the ratio of cyclotron
radius to wavelength of the k mode.兲
Another straightforward calculation, making use of J⫾ ⫽Jx ⫾iJy and recursion relations,
leads to Eq. 共7兲 for the transverse current—the ‘‘diamagnetic drift’’ of the plasma physicists. We
note an alternate derivation ahead.
A more powerful and elastic representation of the kernel stems from an ‘‘algebraic’’ method,
which is a simplified version of the method of characteristics.
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4061
冊
共11兲
Write
1
兩 f 典⫽
O
冕
⬁
0
dt e ⫺t e ⫺t 共 A⫹B 兲 兩 f 典 ,
冉
⳵
⳵
⫽e i ␬ sin ␾ ⫺ ␻
e ⫺i ␬ sin ␾ .
共 A⫹B 兲 ⫽ik cos ␾ ⫺ ␻
⳵␾
⳵␾
Multiplication by t and constructing the exponential series gives
冉 冊
e ⫺t 共 A⫹B 兲 ⫽e i ␬ sin ␾ exp ␻ t
⳵
e ⫺i ␬ sin ␾ ,
⳵␾
and we can write, quite generally,
具g兩
1
兩Q典⫽
O
冕
⬁
dt e ⫺t
0
冕
2␲
0
d␾ g 共 ␾ 兲 * e i ␬ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴 Q 共 ␾ ⫹ ␻ t 兲 .
共12兲
In particular, our kernel is
具a兩
冕
1
1
兩a典⫽
O
2␲
⬁
dt e ⫺t
0
冕
2␲
0
d␾ e i ␬ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴 ,
共13兲
and inversion of the Fourier transform gives the Peierls integral equation which corresponds to Eq.
共1兲. One finds—for an isotropic source—
n共 x 兲⫽
1
2␲
冕
⬁
⫺⬁
dx ⬘ K共 x⫺x ⬘ 兲关 n 共 x ⬘ 兲 ⫹q 共 x ⬘ 兲兴
with
K共 x⫺x ⬘ 兲 ⫽
冕
⬁
dt e ⫺t
0
冕
2␲
0
d␾ ␦ 共 x⫺x ⬘ ⫹ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴 兲 .
After some reduction, using the fact that the integral of a periodic function over its period is
unaltered by a shift in the variable of integration, we find
K共 x⫺x ⬘ 兲 ⫽
1
1⫺e ⫺4 ␲ / ␻
冕 ␪冕
2␲
2␲
d
0
0
d␾ e ⫺2 ␪ / ␻ ␦
冉
冊
␻
共 x⫺x ⬘ 兲 ⫺sin ␪ sin ␾ .
2
K is even, in its argument. In our dimensionless units ␻, the ratio of cyclotron frequency to
collision frequency is the reciprocal of the ratio of cyclotron radius to mean-free-path. Thus, the
constraint expressed by the ␦-function, that 兩 x⫺x ⬘ 兩 ⭐2/␻ , is precisely the statement that the
farthest-ranging particle is found at one ‘‘cyclotron-diameter’’ from a plane source, having left the
source traveling parallel to it. The Peierls’ kernel, K(x), has ‘‘compact support’’; it has no exponential tail. Further reduction will be remarked upon later. 关See Eq. 共24兲.兴
Returning to the kernel, note the limit ␻→0 when
具a兩
1
1
兩a典→
O
2␲
冕
2␲
0
d␾
1
1
⫽
.
1⫹ik cos ␾ 冑1⫹k 2
共14兲
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4062
J. Math. Phys., Vol. 44, No. 9, September 2003
Noel Corngold
In the field-free case, the kernel and the dispersion function are analytic in a cut, complex-k plane,
and spatial relaxation is described by a discrete and a continuous spectrum of relaxation lengths.
The Fourier inverse of the kernel,
冕
1
2␲
⬁
⫺⬁
1
1
兩 a 典 ⫽ K 0共 兩 x 兩 兲 ,
O
␲
dk e ikx 具 a 兩
expresses the density of uncollided particles as a Bessel function. Another useful limiting expression holds when ␻Ⰷ1. Then Eq. 共10兲 informs us that
冋 冉 冊册
1
k
兩a典→ J0
O
␻
具a兩
2
共15兲
.
The kernel, as expressed by Eq. 共13兲, may be simplified if we expand the exponential, noting
that only even powers survive the angle-averaging. One finds that when n⫽2m,
1
2␲
冕
2␲
0
d␾ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴 n ⫽
冉 冊冋 冉 冊册
2m
m
sin
␻t
2
2m
.
Now, two paths are open; one can use
1
共 2m 兲 !
冕
⬁
冉 冊
␻
t
2
dt e ⫺t sin
0
m
2m
⫽ ␻ 2m
1
,
兿
r⫽1 1⫹r 2 ␻ 2
to get the useful series
冉 冊冉 冊 兿
⬁
具a兩
1
2m
兩 a 典 ⫽1⫹
共 ⫺ 兲m
m
O
m⫽1
兺
k2
4
m m
r⫽1
1
1⫹r 2 ␻ 2
,
共16兲
or eschew the t-integration, recognizing the series as generating the Bessel function J 0 , and hence
that
K共 k 兲 ⫽ 具 a 兩
1
兩a典⫽
O
冕
⬁
0
冉
dt e ⫺t J 0 2
冊
␻t
k
sin
.
␻
2
共17兲
共The Appendix contains a shorter derivation of this result.兲
Equation 共17兲 may be rearranged by exploiting periodicity, and using
冕
␲
0
d␾ g 共 ␾ 兲 F 共 sin ␾ 兲 ⫽
冕
␲ /2
0
冋 冉 冊 冉 冊册
d␾ g
␲
␲
⫺ ␾ ⫹g
⫹␾
2
2
F 共 cos ␾ 兲 ,
to get
K共 k 兲 ⫽
⫽
冕
1
2
␻ 1⫺e ⫺2 ␲ / ␻
2
␻
1
sinh
冉冊
␲
␻
␲
0
冕
␲ /2
0
冉 冊
冉 冊冉 冊
d␾ e ⫺2 ␾ / ␻ J 0 2
d␾ cosh
k
sin ␾
␻
k
2␾
J 0 2 cos ␾ ,
␻
␻
共18兲
共19兲
all of which have been found useful. A nice connection with the eigenfunction expansion, Eq. 共10兲,
is made via Neumann’s addition formula
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4063
FIG. 1. K2 , the 2D kernel 共Peierls兲 for the integral equation for density, n(x), with ␻⫽0.5, shown as a function of its
argument, y⫽ ␻ 兩 x 兩 /2. The kernel is zero when y⬎1 while K2 (1 ⫺ )⫽1/关 2 sinh(␲/␻)兴.
⬁
兺
J 0 共 2t sin ␺ 兲 ⫽J 0 共 t 兲 ⫹2
2
n⫽1
J n 共 t 兲 2 cos 2n ␺
in concert with Eq. 共17兲.
Finally, we may Fourier-invert Eq. 共19兲 to get another representation of the Peierls kernel,
namely
K共 x 兲 ⫽
1
␲
1
sinh
⫽0,
冉冊
␲
␻
2
x⬎ ,
␻
冕
␾ 共␻x兲
0
*
cosh
d␾
2
冑 cos
冉 冊
2␾
␻
␾ ⫺cos ␾ 共 ␻ x 兲
*
2
,
0⬍x⬍
2
␻
共20兲
where ␾ is defined through cos ␾ ⫽(␻/2)x. 共See Fig. 1.兲
*
*
A final comment for this section: There is still another way to solve our transport equation,
Fourier expansion in ␾. That approach, which is more natural when the scattering is anisotropic,
leads to difference equations, to solutions in terms of continued fractions and to the traditional,
truncated solutions which we wish to avoid.
B. The complex k plane
The Fourier inversion of 具 a 兩 f 典 and 具 j x,y 兩 f 典 will be controlled by singularities in the complex
k plane. The series in Eq. 共16兲 converges throughout the k plane for all nonzero ␻, assuring us that
our kernel is an entire function of k. Thus, the only singularities are the zeros of the dispersion
function. The magnetic field has shattered the continuous spectrum and it is to the zeros that we
turn. They will describe the relaxation lengths which characterize the system. Since their location
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4064
J. Math. Phys., Vol. 44, No. 9, September 2003
Noel Corngold
depends upon both magnetic field and absorption, 0⭐␻⬍⬁ and 0⭐c⭐1, the picture, overall, is
complicated. But the analysis is eased somewhat by the fact that c and ␻ occur separately in the
dispersion function. We begin by observing the following.
共1兲 Since D is a real function of k 2 , a complex root of the dispersion equation will generate
four zeros in the k plane, of the form ⫾( ␰ ⫾i ␩ ). Thus we may limit our discussion to a single
quadrant of the complex k plane, with the understanding that the other three will be filled by
reflection.
共2兲 If the particle distribution is to be bounded at infinity we expect that D has no zeros on the
real axis of k. 共The exceptional case c⫽1 produces a second-order zero at the origin, and is
‘‘physical’’ when the source is moved to infinity—the Milne case.兲 And inspection of the series for
D shows that when c⫽1, there are no zeros 共poles兲 on the imaginary axis.
共3兲 The dominant zero. When 0⬍c⬍1 there is a dominant zero 共pole兲 on the positive 共and the
negative兲 sections of the imaginary axis, where k⫽i ␩ . This pole, with its associated residue,
describes the dominant, large-x, ‘‘asymptotic’’ behavior of the distribution. There is no other zero
on the axis because the kernel is a function of ␩ which increases smoothly. 共The zeros which are
off-axis and are discussed later, give oscillatory contributions which cannot dominate, asymptotically.兲 Then, for the dominant zero, we have four regimes as follows.
共i兲 Strong field 共␻Ⰷ1, c arbitrary兲. Here, Eq. 共15兲 enables us to write
␩ 0⫽ ␻ y 共 c 兲,
where
1⫺cI 0 共 y 兲 2 ⫽0
共21兲
for all c. 共In practice, the estimate is valuable even at ␻⬇1.兲
共ii兲 Weak absorption (1⫺cⰆ1, ␻ arbitrary兲. Here, the zero lies close to the origin, and the
series, Eq. 共16兲 yields
␩ 0 ⯝ 冑2 共 1⫺c 兲共 1⫹ ␻ 2 兲 ,
共22兲
a result which emerges from Eq. 共21兲, too. It describes a simple, exponential attenuation, which
has been increased by the magnetic field. In the field-free case ␩ 0 ⫽ 冑1⫺c 2 exactly.
共iii兲 Strong absorption (c→0, ␻ arbitrary兲. When field is absent the kernel displays branch
points, and the zero falls into the branch singularity when c⫽0. With field present, there are no
branch points and matters are quite different. The dominant zero moves along the imaginary axis
to infinity as c→0, the dominant relaxation length becoming arbitrarily small. It is the magnetic
field, rather than the mean free-path, which is in control. Equation 共21兲 yields an easy estimate,
when the field is large,
␩ 0⯝
冉冊
␻
1
log
.
2
c
共23兲
In the limit, c⫽0, the distribution becomes that of uncollided particles proceeding from a
plane, isotropic source. It is simply the Peierls kernel, Eq. 共20兲. It helps to write the symmetric
distribution as (x⬎0)
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J. Math. Phys., Vol. 44, No. 9, September 2003
1
n 0 共 x 兲 ⫽K共 x 兲 ⫽
␲
冉
2␾ 共␻x兲
*
␻
␲
sinh
␻
cosh
1
⫹
␲
冉冊
1
sinh
冉冊
␲
␻
冕
Transport in magnetic field
冊冕
␾ 共␻x兲
*
0
␾ 共␻x兲
*
0
d␾
cosh
d␾
1
2
冑 cos
␾ ⫺cos2 ␾ 共 ␻ x 兲
*
冉 冊
冉
2␾
2␾ 共␻x兲
*
⫺cosh
␻
␻
2
冑 cos
␾ ⫺cos ␾ 共 ␻ x 兲
*
2
冊
,
0⬍x⬍
2
x⬎ .
␻
⫽0,
4065
2
␻
共24兲
The key quantity in its description is ␾ ( ␻ x), which descends from 共␲/2兲 at the plane source to
*
zero at the edge of the distribution. We note further that 共a兲 the second term in Eq. 共24兲 is bounded
for all ␾ (x), vanishing at the edge of the distribution, while 共b兲 the integral appearing in the first
*
term may be transformed into the complete elliptical integral K(sin 2 ␾ ). It diverges, logarithmi*
cally at the source plane, and is 共␲/2兲 at the edge. Thus, the distribution of uncollided particles is
discontinuous at its edge. We have the limits 共see Fig. 1兲
n 0 共 x 兲 ⫽K共 x 兲 →
1
冉冊
冉冊 冉冊
2 sinh
␲
␻
,
x→
2
,
␻
共25兲
␲
1
1
log
⫹¯,
n 0 共 x 兲 ⫽K共 x 兲 ⬃ coth
␲
␻
x
x→0.
The size of the step-discontinuity vanishes, conveniently, as ␻→0, while the leading behavior at
the source plane coincides with that of the field-free case.
共iv兲 Weak field 共␻→0, c arbitrary兲. While the large-␻ limit is relatively simple, the small-␻
limit is not. Since singularities which are absent when ␻⫽0 appear 共at ␩⫽⫾1兲 when ␻⫽0, the
behavior in the neighborhood of these points cannot be analytic. The simple expression, Eq. 共22兲,
does not tell the full story. We begin by studying
I共 ␾ , ␩ 兲 ⫽cosh
冉 冊冉
冊
2
2
␾ I 0 ␩ cos ␾ ,
␻
␻
the integrand appearing in Eq. 共19兲, in the regime 共2/␻兲⬅␭Ⰷ1. At the limits of integration we have
I共 0,␩ 兲 ⫽I 0 共 ␭ ␩ 兲 ,
I
冉 冊
冉 冊
␲
␲
,
, ␩ ⫽cosh ␭
2
2
and behavior will be different in different intervals of ␩, for absorption may cause ␩ 0 to be large.
It is no surprise that ␩⫽1, and ␩⫽␲/2, play important roles in the analysis.
共a兲 When ␩⬍1, I共␾,␩兲 is seen to be an increasing function of ␾ in 共0,␲/2兲 and the integral is
controlled by the behavior of I near its maximum—at the edge. If we rearrange Eq. 共19兲 to place
the maximum at the origin we obtain
K共 i ␩ 兲 ⫽
1
2
冕
⌳
0
冉 冉 冊冊
dt I 0 ␭ ␩ sin
t
␭
兵 共 coth共 ⌳ 兲 ⫺1 兲 e t ⫹ 共 coth共 ⌳ 兲 ⫹1 兲 e ⫺t 其 ,
where
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4066
J. Math. Phys., Vol. 44, No. 9, September 2003
Noel Corngold
TABLE I. 1/␩ 0 is the relaxation length, in units of mean-free-path, for the
particle distribution far from its plane source, in the presence of various
amounts of capture and various magnetic field strengths.
Dominant zero, ␩ 0
␻
c⫽0.9
c⫽0.7
c⫽0.5
c⫽0.3
c⫽0.1
0.25
0.5
1
0.46
0.50
0.65
0.78
0.89
1.2
1.0
1.2
1.6
1.2
1.5
2.1
1.5
2.0
3.0
⌳⫽
␲ ␲
⫽ ␭.
␻ 2
A little consideration shows that when ␭ is very large we may write
K共 i ␩ 兲 ⯝
冕
⬁
0
冉 冉 冊冊
dt I 0 ␭ ␩ sin
t
␭
e ⫺t
with exponentially small error, O(e ⫺(1⫺ ␩ )⌳ ). Then, expansion about t⫽0, aided by the relation
1
n!
冕
⬁
0
dt e ⫺t t n J 0 共 kt 兲 ⫽
( P n is the Legendre polynomial兲 yields
K共 i ␩ 兲 ⫽ 具 a 兩
1
2 共 n⫹1/2兲
共 1⫹k 兲
冋 冉冊
1
1
1 ␻
兩a典⫽
1⫺
2
O
2 2
冑 1⫺ ␩
Pn
1
冑1⫹k 2
4⫹ ␩ 2
2
␩2
冉
共 1⫺ ␩ 2 兲 3
冊
共26兲
册
⫹¯ ,
共27兲
in which the singular nature of the small-␻ behavior is displayed.
In the complimentary situation, when ␩⬎1, I共␾,␩兲 posesses a maximum inside the interval
共0,␲/2兲. When that sharp maximum dominates the contribution from the edge, expansion produces
K共 i ␩ 兲 ⯝
1
冑␩
2
⫺1
冋 冉 冊冉
2
␻
exp
1 ␲
sin⫺1 ⫺ ⫹ 冑␩ 2 ⫺1
␩ 2
冊册
,
共28兲
as 共2/␻兲→⬁. Thus, when ␩Ⰷ1 as well,
K共 i ␩ 兲 ⯝
1
␩
冋 冉 冊册
exp
2
␲
␩⫺
␻
2
,
共29兲
and ␻ ‘‘small,’’
共30兲
which leads easily to the estimate
␩ 0⯝
冉冊
␻
1
log
2
c
for c→0
familiar, and in agreement with Eq. 共23兲.
Some dominant zeros are displayed in Table I.
共4兲 Transients. The other zeros, or relaxation lengths, form a complex pattern. They produce
transient, oscillating terms in the spatial distribution. We would like a picture of the pattern, and
how it alters as the magnetic field is altered. In particular, we are interested in the limit ␻→0.
Ultimately, we rely on the summation, numerically, of Eq. 共16兲. But the search for zeros is helped
greatly by asymptotic estimates, to which we turn.
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4067
共i兲 Strong field 共␻Ⰷ1, 兩␬兩 arbitrary兲 ( ␬ ⬅k/ ␻ ). We begin with the simplest estimate, ␻Ⰷ1,
when a different approach yields a familiar result,
⬁
K共 k 兲 ⯝1⫹
兺
m⫽1
冉 冊冉
2m
m
⫺
k2
4␻2
冊
m
1
共 m! 兲
⫽
2
2
␲
冕
␲ /2
0
冉
d␾ J 0 2
冊 冉冊
k
k
sin ␾ ⫽J 0
␻
␻
2
.
共31兲
Then, the ␻ dependence is simple, k n ⫽ ␻ ␬ n , where ␬ n (c) is one of the complex zeros of 1
⫺cJ 0 ( ␬ ) 2 . 共In practice, the estimate is valuable even at ␻⬇1.兲 Among these, the ‘‘higher harmonics’’ 共兩␬兩Ⰷ1兲 obey
sin 2x 2y ␲
e ⫽ ,
2x
c
cos 2x⫽2ye ⫺2y
␲
,
c
共32兲
where ␬ ⫽x⫹iy. From these, we infer
再
k n ⬇ ␻ n ␲ ⫹i
冋冉
冊 册冎
1
1 ␲2
log 2n⫹
2
2 c
,
共33兲
n⫽0,1,2,...
for the regime ␻Ⰷ1 and 兩 k 兩 Ⰷ ␻ . The real parts of the relaxation constants increase only logarithmically, the imaginary parts linearly. Note that the dominant zero is contained here, associated
with cⰆ1, and behaves as
k 0 ⯝i
冉冊
␻
1
log
.
2
c
共ii兲 Weak field is included in the regime 共兩␬兩Ⰷ1, ␻ arbitrary兲. We begin with the estimate
冕
␲
0
d␾ e ⫺2 ␾ / ␻ J 0 共 2 ␬ sin ␾ 兲 ⬃
1
关 1⫹e ⫺2 ␲ / ␻ ⫹2e ⫺ ␲ / ␻ sin 2 ␬ 兴 ,
2␬
兩 ␬ 兩 →⬁,
共34兲
derived by the method of stationary phase, including end-point corrections. The derivation proceeds with the assumption that k is real, but suggests strongly that the result holds throughout the
quadrant when Re共␬兲⬎0. This conjecture is supported by the fact that Eqs. 共34兲 and 共31兲 are
identical, namely,
K共 k 兲 ⬇
i
e ⫺2i ␬
2␲␬
when their domains overlap, and by numerical experience. We have, then, the approximate dispersion equation,
1⫹
冉
2e ⫺ ␲ / ␻
1⫹e ⫺2 ␲ / ␻
冊
sin 2 ␬ ⫽
冉
冊
␬ ␻ 1⫺e ⫺2 ␲ / ␻
.
c 1⫹e ⫺2 ␲ / ␻
共35兲
For field strengths e ⫺2 ␲ / ␻ Ⰶ1, and for ␬ large, further simplification enables us to write the
estimate as
冋 冉 冊册
冋 冉 冊册
c ␰ ⫽1⫹exp
2
␲
␩⫺
␻
2
2
␲
␩⫺
c ␩ ⫽exp
␻
2
sin
2␰
,
␻
2␰
cos
␻
共36兲
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4068
J. Math. Phys., Vol. 44, No. 9, September 2003
Noel Corngold
共recall that k⫽ ␰ ⫹i ␩ ).
When ␻ is small these expressions are useful when ␩⬎␲/2. Then, with
␩⫽
␲ ␻
⫹ ␩ 共␻兲
2 2 1
and
␰⫽
␻
␰ 共 ␻ 兲.
2 1
the new functions assumed to be regular, we are led to
k n ⫽ ␻ 关 n ␲ ⫺ ␰ 兴 ⫹i
*
冋
再 冉 冊 冎册
␲ ␻
␲c
⫹ log 1⫹
2 4
2
2
⫹o 共 ␻ 兲 ,
where (n ␲ ⫺ ␰ )⫽2 ␰ 1 (0) is the solution to
*
tan兵 2 ␰ 1 共 0 兲 其 ⫽⫺
2
,
␲c
and n⫽1,2,... is not too large. This pattern of zeros, differing little in their imaginary parts, and
marching into a point on the imaginary axis, is roughly 共‘‘semiquantitatively’’兲 correct. The numerical values it yields are helpful, even though the true point of accumulation is k⫽i rather than
k⫽i( ␲ /2). Of course, the asymptotic expansion fails on the imaginary axis. There, and in an
unknown strip containing it, we return to the expansion used in connection with Eq. 共27兲 to obtain
K共 k 兲 ⫽ 具 a 兩
冋 冉冊
1
1
1 ␻
兩a典⫽ 2
1⫺
O
冑 k ⫹1
2 2
2
k2
k 2 ⫺4
共 k 2 ⫹1 兲 3
册
⫹¯ ,
共37兲
a complicated series which exhibits clearly the branch-point singularity and omits exponentially
small terms 共in ␻兲. Before Eq. 共37兲 is used to estimate zeros computation of the winding-number
suggests that if we consider a small circular region about k⫽i, the number of zeros 共poles兲 present
in the region increases without limit as ␻→0. Analysis of the series suggests that the zeros do not
follow distinct, ray-like paths as they fall into the 共nascent兲 branch point. Rather, their paths
merge. For example the first few terms of Eq. 共37兲 yield a pair of roots and a merged path
described by
k⫺i⬇ ␣ ␻ 2/3 exp
冉 冊冋
␲
i
6
1⫿
冉 冊册
␲
1
冑2 ␣ ␻ 1/3 exp i
3
3
,
共38兲
1
␣ ⫽ 共 5 兲 1/3
4
in the first quadrant. The dependence upon ␻ 2/3 is borne out to a fraction of a percent by numerical
calculation, and the ␲/6, which characterizes the asymptote, appears to be correct, but the numerical coefficients need improvement. Overall the dependence upon ␻ is quite singular, for we have
omitted exponentially singular factors. See Fig. 2.
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4069
FIG. 2. Three zeros 共poles兲 and their ‘‘motion’’ in the k plane as the magnetic field 共␻兲 is altered. ␻→0 brings them to
k⫽i.
C. Densities and currents
The currents Jx,y (k) may be obtained from the kernel via Eqs. 共6兲 and 共7兲 or directly, from
1
1
具 j ⫾兩 兩 a 典 ⫽
冑2␲
O
冕
⬁
dt e ⫺t
0
冕
2␲
0
d␾ ⫾i ␾ i ␬ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴
e e
,
2␲
which yields, for 兩 Q 1 典 共unit source兲,
iJx 共 k 兲 ⫽
冕
1
D共 k,c 兲
⬁
dt e ⫺t cos
0
1
⫺iJy 共 k 兲 ⫽
D共 k,c 兲
冕
⬁
0
冉
冊
␻t
␻t
J 1 2 ␬ sin
,
2
2
冉
共39兲
冊
␻t
␻t
dt e ⫺t sin J 1 2 ␬ sin
,
2
2
as symmetric alternatives to Eqs. 共6兲 and 共7兲. Also, there is Eq. 共17兲,
n共 k 兲⫽
1
D共 k,c 兲
冕
⬁
0
冉
dt e ⫺t J 0 2
冊
k
␻t
sin
.
␻
2
Equations 共6兲 and 共7兲 follow easily from Eq. 共39兲 by integration by parts.
Since D(k,c) is even in k, we conclude that the components of current change sign when x is
replaced by ⫺x. Since D(k,c) diminishes with increasing k 关see Eq. 共34兲兴, the behavior of density
and current near the source is linked to the behavior of the integrals in Eq. 共39兲. These, which
describe the uncollided particles, may be inverted and reduced to
J0x 共 x 兲 ⫽
1 cos ␾ 共 x 兲
*
␲
␲
sinh
␻
冉冊
冕
␾ 共x兲
0
*
d␾ sinh
⫽ 12 ⫺ 共 1⫺c 兲 x n 0 共 x 兲 ⫹¯,
冉 冊冑
2␾
␻
tan ␾
cos ␾ ⫺cos2 ␾ 共 x 兲
2
*
共40兲
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4070
J. Math. Phys., Vol. 44, No. 9, September 2003
J0y 共 x 兲 ⫽⫺
1 cos ␾ 共 x 兲
␲
sinh
*
␲
冉冊
冕
Noel Corngold
␾ 共x兲
*
0
cosh
d␾
冉 冊
2␾
␻
冑cos2 ␾ ⫺cos2 ␾
␻
*
共x兲
␻
⫽⫺ x n 0 共 x 兲
2
共41兲
near the source-plane. For the behavior 共‘‘asymptotic’’兲 far from the plane, we turn to inversion by
contour integration to get
Jy 共 x 兲 ⫽⫺
Jx 共 x 兲 ⫽⫺
␻
2c
兺n e ik x ,
n
冉 冊兺
1⫺c
c
n共 x 兲⫽
i
c
兺n
n
1
e ik n x ,
k nD n
共42兲
1 ik x
e n ,
Dn
with D n ⬅( ⳵ / ⳵ k)D(k n ,c). Though the summation is over all zeros in the upper half-plane, we are
most interested in the contribution from the dominant zero. Note that the equation of continuity is
satisfied, ‘‘mode by mode,’’ and observe the strangely simple expression for Jy (x), as well as the
singular behavior at x⫽0.
The transverse current, Jy (x), which is zero at the source plane, grows as one proceeds away.
The ratio Jy (x)/Jx (x) is then of some interest. The ratio assumes the value 21 关 k 0 D 0 /(1⫺c) 兴 ␻
asymptotically. One may compute the quantity easily in two limiting cases. In the first, very weak
capture, cⰇ1⫺c, the ratio is simply 共⫺␻兲. In the second, very high frequency, ␻ Ⰷ1 关see Eqs.
共16兲 and 共31兲兴 the ratio is (⫺ ␻ F(c)), where
F共 c 兲 ⫽
冑c
1⫺c
␬ 0J 1共 ␬ 0 兲 ,
J 0共 ␬ 0 兲 ⫽
1
冑c
.
共43兲
Diffusion: (k 0 ⫽i ␩ 0 ⫽ ␻ ␬ 0 ). The question of diffusion and diffusion constant may be viewed
in two ways. One can note that far from the source the density follows n xx ⫹k 20 n⫽0, an equation
suggesting steady-state diffusion with diffusion coefficient
Dxx ⫽
共 1⫺c 兲
␩ 20
.
More generally,
Dxx ⫽Dy y ⫽
共 1⫺c 兲
␩ 20
,
Dyx ⫽⫺Dxy ⫽
␻ ⳵
D共 k, ␻ 兲 兩 k 0 ,
2k 0 ⳵ k
共44兲
the tensor components depending upon field strength and capture. This view is surely more helpful
to the experimenter than is the traditional attitude that ‘‘Fick’s law’’ holds throughout, that there is
everywhere a proportionality between density gradient and current. In fact, the law holds only
when spatial variations are so gentle 共‘‘long-ranged’’兲 that only the lowest powers in an expansion
in k need be retained. Then, Eq. 共16兲 leads to
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
冉 冊
冉 冊
1
1
1
Jx
1
⫽⫺ 共 ik 兲
,
Jy
2
1⫺c 1⫹ ␻ 2 ⫺ ␻
4071
共45兲
describing ‘‘classical’’ anisotropic diffusion, with diffusion tensor
Di j ⫽
冉
␻
1
1 1
2 1⫹ ␻ 2 ⫺ ␻
1
冊
共46兲
.
This simple picture, found in most textbooks, displays a ‘‘normal’’ diffusion, inhibited by the
magnetic field, and transverse diffusion, induced by the field. The tensor multiplication may also
be written
D•ⵜn⫽
1 1
关 1⫺ ␻ ẑ⫻ 兴 ⵜn.
2 1⫹ ␻ 2
The two treatments differ little when capture is almost negligible and k 0 is small. The case of large
field 共‘‘high frequency’’兲 is accessible through Eqs. 共16兲 and 共31兲 when Eq. 共44兲 yields
Dyx ⫽⫺Dxy ⫽⫺
J 1共 ␬ 0 兲
.
␻ 冑c ␬ 0
1
Thus, for fixed capture, the transverse diffusion is—again—inhibited by the magnetic field
when the field is large. D yx rises, proportionally to ␻ when ␻ is small, reaches some peak value,
then decreases, as 1/␻.
D. The distribution in angle
The angular distribution in the dominant, asymptotic mode is of particular interest. In the
notation of Eq. 共44兲 that quantity, F ⬁ (x, ␾ ), is
F ⬁ 共 x, ␾ 兲 ⫽
i ik x
e 0 ⌽ 0共 ␾ 兲
D0
with
⌽ 0共 ␾ 兲 ⫽
1
2␲
冕
⬁
0
冉冉 冊
dt e ⫺t exp i
k0
关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴
␻
冊
共47兲
for unit source. The transient modes have a similar appearance. One of the many possible rearrangements brings us to the convenient form,
⌽ 0共 ␾ 兲 ⫽
1
1
e 共 1/␻ 兲 F共 ␾ 兲
2 ␲␻ 1⫺e ⫺2 ␲ / ␻
冕
␾ ⫹2 ␲
␾
d␣ e ⫺ 共 1/␻ 兲 F共 ␣ 兲
共48兲
with F( ␾ )⫽ ␾ ⫺ ␩ 0 sin ␾ 关k0⫽i␩0(␻,c)兴. And there is always the differential equation
⫺␻
⳵
1
z 共 ␾ 兲 ⫹ 共 1⫺ ␩ 0 cos ␾ 兲 z 共 ␾ 兲 ⫽
⳵␾
2␲
which, when solved 共numerically兲 under the condition that z( ␾ )⫽z( ␾ ⫹2 ␲ ), yields a function
proportional to ⌽ 0 ( ␾ ).
Once again, the question of ␩ 0 ⬍1 or ␩ 0 ⬎1 enters. In the former case F共␾兲 is positive and
increasing, in the latter, not so, and one encounters more dramatic behavior. Analytical information
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4072
J. Math. Phys., Vol. 44, No. 9, September 2003
Noel Corngold
FIG. 3. Asymptotic angular distribution 共2D兲 for ␻⫽0.5, c⫽0.86. Arbitrary normalization. Capture moderate, field strong.
is available when ␻ is small, when familiar ideas from ‘‘asymptotics’’ enter.12 For example, as
␻→0 the dominant contribution from the integral in Eq. 共48兲 comes from the interval where F共␣兲
is minimum. In the case ␩ 0 ⬍1 the minimum is at ␣⫽␾ and expansion about that 共edge-兲 value
yields
⌽ 0共 ␾ 兲 ⫽
冋
册
1
␩ 0 sin ␾
1
1⫺ ␻
⫹¯ ,
2 ␲ 共 1⫺ ␩ 0 cos ␾ 兲
共 1⫺ ␩ 0 cos ␾ 兲 2
precisely the ‘‘outer expansion’’ of the solution to the differential equation. Since the outer solution satisfies the boundary condition, there is no need for a boundary layer. This simple expression
displays an interesting feature of the angular distribution—a small peak centered at a small angle.
The disturbance, vanishing with ␻, generates the transverse current.
When ␩ 0 ⬎1 the situation is different. F共␣兲 is oscillatory, and has a single minimum, at 0
⭐ ␾ ⬍ ␲ /2 where cos ␾ ⫽(1/␩ 0 ). In a subinterval of 共⫺␲⬍␾⭐␲兲, namely, ( ␾ ⬍0⬍ ␾
*
*
**
⬍ ␾ ), where F( ␾ )⫽F( ␾ ), the minimum lies inside the integration of Eq. 共48兲 and produces
*
**
*
boundary layer behavior. ⌽共␾兲 rises rapidly, proportional to exp关(1/␻ )(F( ␾ )⫺F( ␾ )) 兴 then falls
*
and passes to the nonexponential ‘‘outer’’-behavior for the remainder of the interval. These features are displayed in Figs. 3 and 4. Clearly, these angular distributions are not represented well by
an expansion-in-angle that is near-isotropic. The distributions associated with ␩ 0 ⬎1 become quite
singular in the limit of vanishing field—the continuum limit.
E. Transport in three dimensions „3D…
Since the key features are captured in the 2D case, we treat 3D briefly. We remain with plane
symmetry. Then, the modifications are relatively minor. It is convenient to use two sets of angle
variables. In one, the x axis is the polar axis and the polar and azimuthal angles are denoted 共␪,␺兲;
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4073
FIG. 4. Asymptotic angular distribution 共2D兲 for ␻⫽0.25, c⫽0.1. Arbitrary normalization. Capture very strong, field
moderate.
in the other, the z axis, to which the field is parallel, is the polar axis and the angles are 共␹,␾兲. The
streaming portion of the transport equation is altered only in that ‘‘␮⫽cos ␾’’ is replaced by
␮⫽sin ␹ cos ␾. Since ␹ enters the equation only ‘‘parametrically,’’ it may be absorbed through
much of the subsequent calculation by simply replacing the Fourier transform variable ‘‘k’’ by
k⬜ ⫽k sin ␹. The density is now a function of two angles, 共␹,␾兲. The in-scattering term is altered
through the replacement,
1
2␲
冕
2␲
0
d␾ →
冕
1
4␲
d⍀⫽
冕
1
4␲
2␲
0
d␾
冕
␲
0
d␹ sin ␹ .
The vector space is now a space of functions defined on the unit sphere, and we may use 共␪,␺兲
or 共␹,␾兲 in place of the variable, ␾. The inner product is now
具g兩 f 典⬅
冕
d⍀ g * 共 ⍀ 兲 f 共 ⍀ 兲 ⫽ 具 f 兩 g 典 *
and the ubiquitous
具␾兩a典⫽
1
,
冑2␲
具a兩 f 典⫽
冕
具a兩 f 典⫽
冕
2␲
0
d␾
1
f 共␾兲
冑2␲
become
具⍀兩a典⫽
1
冑4␲
,
d⍀
1
冑4␲
f 共 ⍀ 兲.
With these reinterpretations, most of the equations of the 2D case may be carried over, easily, to
3D, turning Bessel functions into spherical Bessel functions. We begin with the kernel,
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4074
J. Math. Phys., Vol. 44, No. 9, September 2003
Noel Corngold
FIG. 5. K3 , the 3D kernel 共Peierls兲 for the integral equation for density, n(x), with ␻⫽0.5, shown as a function of its
argument, y⫽ ␻ 兩 x 兩 /2. The kernel is zero when y⭓1.
具a兩
1
1
兩a典→
O
4␲
⫽
冕
⬁
冕
⬁
dt e ⫺t
0
dt e ⫺t
0
冕
1
2
冕
␲
0
d⍀ e i ␬⬜ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴
冉
d␹ sin ␹ J 0 2 ␬ sin
冊
␻t
sin ␹ .
2
共49兲
Here, and in the discussion of currents, the relation
冕
␲ /2
0
d␹ sinn⫹1 ␹ J n 共 z sin ␹ 兲 ⫽ j n 共 z 兲 ,
between ordinary and spherical Bessel functions is useful. Then,
K3 共 k, ␻ 兲 ⬅ 具 a 兩
1
兩a典⫽
O
冕
⬁
0
冉
dt e ⫺t j 0 2 ␬ sin
冊
␻t
2
⫽
2
␻
1
sinh
冉冊
␲
␻
冕
␲ /2
0
d␾ cosh
冉 冊冉
冊
2␾
k
j 0 2 cos ␾ ,
␻
␻
共50兲
with j 0 (z)⫽sin z/z.
Upon expansion to produce a power series in k, one finds
冋
K3 共 k, ␻ 兲 ⫽ 1⫹
⬁
兺
m⫽1
m
册
1
共 ⫺k 2 兲 m
.
2m⫹1 r⫽1 1⫹r 2 ␻ 2
兿
共51兲
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4075
Setting ␻⫽0 gives the familiar logarithm of the field-free case, and expanding for small ␻ in
the manner given earlier 关e.g., Eqs. 共27兲 and 共37兲兴 yields the singular and asymptotic sequence
K3 共 k, ␻ 兲 ⫽
1
k2
1⫹ik 1 2
⫹¯⫹exponentially small terms.
log
⫹ ␻ 2
2ik
1⫺ik 3 共 k ⫹1 兲 3
共52兲
The regime ␻Ⰷ1 is dealt with best via the series, Eq. 共49兲, giving
K3 共 k, ␻ 兲 ␻
2k
冕
2k/ ␻
0
共53兲
dz J 0 共 z 兲 .
These expressions are helpful in determining the zeros of D3 (k, ␻ ,c), which, like its predecessor,
is entire-in-k. The behavior of these zeros is quite similar to their 2D counterparts. Turning to the
picture in x coordinates, we note that the Peierls kernel may be obtained by Fourier-inverting Eq.
共49兲 to get a pretty, ‘‘3-Sine’’ formula for the even function,
K3 共 x⫺x ⬘ , ␻ 兲 ⫽
冕
1
1⫺e
⫺4 ␲ / ␻
2␲
0
d␪
冕
d⍀ e ⫺2 ␪ / ␻ ␦
冉
冊
␻
共 x⫺x ⬘ 兲 ⫺sin ␪ sin ␾ sin ␹ .
2
共54兲
Comments made earlier about its compact support continue to hold. On the other hand, we may
Fourier-invert Eq. 共50兲 to get quite a different compact form,
K3 共 x, ␻ 兲 ⫽
1
冉冊
␲
2 sinh
␻
冕
␾ 共␻x兲
*
0
d␾
2␾
cosh
cos ␾
␻
共55兲
(cos ␾ ⫽(␻/2)x, and 兩 ( ␻ /2)x 兩 ⬍1). In fact, K3 may be shown to vanish with vertical tangent.
*
共See Fig. 5.兲 That this rather peculiar expression does become the familiar exponential integral
when ␻→0 may be seen by setting
␾⫽
冉 冊
␲
⫺␺ ,
2
expanding the cosh, and passing to the limit.
Currents. We have, generally,
Jx,y,z 共 k 兲 ⬅J共 k 兲 ⫽ 具 j 兩
1
1
1
c
兩Q典⫹
具 j 兩 兩 a 典具 a 兩 兩 Q 典 .
O
D共 k,c 兲
O
O
共56兲
When the source is isotropic, and normalized to 冑4␲,
J共 k 兲 ⫽
1
1
具 j兩 兩a典,
D共 k,c 兲
O
共57兲
and we discuss the numerator of this expression,
1
冑4 ␲
具 j x,y 兩
1
1
兩a典⫽
O
4␲
冕
⬁
0
dt e ⫺t
冕
d⍀ sin ␹ 关 cos ␾ ,sin ␾ 兴 e i ␬⬜ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴 ,
共58兲
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4076
J. Math. Phys., Vol. 44, No. 9, September 2003
1
冑4 ␲
具 j z兩
冕
1
1
兩a典⫽
O
4␲
⫽
冕
⬁
⬁
冕
dt e ⫺t
0
dt e ⫺t
0
1
2
冕
␲
0
Noel Corngold
d⍀ cos ␹ e i ␬⬜ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴
冉
d␹ sin ␹ cos ␹ J 0 2 ␬ sin ␹ sin
冊
␻t
⫽0,
2
by symmetry.
共59兲
Thus, quite generally,
Jz 共 k 兲 ⫽ 具 j z 兩
1
兩Q典,
O
an expression which is, at first sight, puzzling, for the current appears to be independent of
capture. A moment’s thought convinces one of its correctness, and that J z is, in any case, trivial.
We merely note that with isotropic scattering, a beam becomes distributed isotropically at the first
collision. After that collision, only a density gradient will drive a current. But in our problem, there
are no gradients in the z direction; J z exists only in the interval from birth to first collision when
the value of ‘‘c’’ is irrelevant. This argument holds for particles immersed in an arbitrary external
field that is independent of z. The z current is carried only by uncollided particles. Adding a bit of
anisotropy to the scattering changes the result significantly.
The current components, Jx,y (k), are simply the weighted ␹ average of their counterparts in
two dimensions, Eq. 共39兲. The connection between ordinary and spherical Bessel functions then
gives the concise result,
iJx 共 k 兲 ⫽
1
D3 共 k,c 兲
冕
1
⫺iJy 共 k 兲 ⫽
D3 共 k,c 兲
⬁
dt e ⫺t cos
0
冕
⬁
dt e
⫺t
0
冉
冊
␻t
␻t
j 1 2 ␬ sin
,
2
2
冉
冊
␻t
␻t
sin
,
j 1 2 ␬ sin
2
2
共60兲
for currents in three dimensions.
Angular distributions. Expressions for the angular distribution associated with dominant and
transient modes may be obtained easily. One simply replaces k 0 with k 0 sin ␹ in Eq. 共47兲. The
distributions are then symmetric with respect to the (x – y) plane and fixing a value of ␹ is
equivalent to selecting one of the 2D distributions we have described earlier. As ␹ decreases from
its in-plane value of ␲/2, the effective ␩ 0 diminishes and the corresponding distribution is
smoother. Overall, angular distributions in 3D appear to be smoother than those in 2D.
One can proceed further with details of the 3D case, in a manner similar to that of 2D but it
is clear that the 2D case displays almost everything that is interesting about the problem. The next
step should be an attack upon the time-dependent problem, which is simple enough after Laplacetransform, but whose inversion is a complicated matter.
ACKNOWLEDGMENTS
The author is grateful for a friendly and helpful correspondence with Professor R. E Robson
and Professor K. Kumar, during the early phase of this research.
APPENDIX
An alternative, easier way to evaluate the key quantity,
具a兩
1
1
兩a典⫽
O
2␲
冕
⬁
0
dt e ⫺t
冕
2␲
0
d␾ e i ␬ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴
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J. Math. Phys., Vol. 44, No. 9, September 2003
Transport in magnetic field
4077
is to notice that 关 sin ␾⫺sin(␾⫹␻t)兴⫽⫺2 sin(␻t/2)cos(␾⫹␻t/2), and that the replacement of ( ␾
⫹ ␻ t/2) by ␾ does not alter the value of the integral. The expression
具a兩
1
兩a典⫽
O
冕
⬁
0
冉
dt e ⫺t J 0 2 ␬ sin
␻t
2
冊
follows at once. In fact, evaluation of the matrix element in the expression for the current induced
by an isotropic source
J x,y 共 k 兲 ⫽
1
冑2 ␲
具 j x,y 兩
1
1
兩a典⫽
O
2␲
冕
⬁
0
1
1
具 j 兩 兩a典,
D共 k 兲 x,y O
dt e ⫺t
冕
2␲
0
d␾ 关 cos ␾ ,sin ␾ 兴 e i ␬ 关 sin ␾ ⫺sin共 ␾ ⫹ ␻ t 兲兴
is made easy by the same approach. The change of variable ( ␾ ⫹ ␻ t/2)→ ␾ and integration by
parts produces the results noted earlier.
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2
E. A. Mason and E. W. McDaniel, Transport Properties of Ions in Gases 共Wiley, New York, 1988兲.
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See, for example, F. F. Chen, Introduction to Plasma Physics 共Plenum, New York, 1974兲.
9
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11
R. E. Peierls, Proc. Cambridge Philos. Soc. 35, 610 共1939兲.
12
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York, 1978兲.
1
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