HYDRODYNAMIC APPROXIMATIONS TO
TIME-DEPENDENT HARTREE -
FOCK
by
Steven E. Koonin
California Institute of Technology (B.S.)
(1972)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF
PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 1975
Signature of Author
Signature redacted
Signature redacted
- -- - - -
Certified by . . . . .-
- - -
-
Department of Physics, May 9, 1975
Thesis Supervisor
Accepted by
Signature redacted
Chairman, Departmental Cobittee
on Graduate Students
ARCH VES
JUN 9 1975
2
HYDRODYNAMIC APPROXIMATIONS TO
TIME-DEPENDENT HARTREE - FOCK
by
Steven E. Koonin
Submitted to the Department of Physics
on May 9, 1975 in partial fulfillment of the
requirements for the degree of Doctor of Philosophy
ABSTRACT
By means of a Wigner transformation of the onebody density matrix, the Time Dependent HartreeFock equations are expressed as a quantal Vlasovlike equation describing the dynamics of a phasespace distribution function. Moments of this
equation result in an infinite hierarchy of nonlocal equations which yield to a hydrodynamic
interpretation.
The assumptions of an effective
two-body interaction of the Skyrme type and of
certain semi-classical properties of the distribution function allow a closed set of (almost)
local equations for the density, velocity, and
pressure fields.
These equations are applied to
small oscillations about the static Hartree-Fock
solution (RPA) for both infinite nuclear matter
and finite nuclei.
The kinematics of the nonlinear solutions corresponding to shock waves in
nuclear matter is also discussed.
Thesis Supervisor:
Title:
Arthur Kerman
Professor of Physics
3
Acknowledgement
Several people deserve thanks for help and
support during the cause of this work.
Primary among
these is Arthur Kerman, to whom I am grateful for exposing
me to a most enjoyable style of doing physics and for many
hours of enlightening conversation.
Economic support has been provided in part by the
National Science Foundation, the balance, and a good deal
of moral support being provided by my parents.
Finally, I want to thank my fiance, Laurie Card for
understanding and tolerating the needs of an often demanding
physicist.
4.
ABSTRACT------------------------------------------------
2
ACKNOWLEDGEMENTS----------------------------------------
3
I.
II.
III.
IV.
V.
VI.
INTRODUCTION------------------------------------THE TIME DEPENDENT HARTREE-FOCK EQUATIONS--------
11
TIME DEPENDENT HARTREE-FOCK IN THE WIGNER
REPRESENTATION-----------------------------------
32
TIME DEPENDENT HARTREE-FOCK WITH THE
SKYRME FORCE-------------------------------------
53
TRUNCATION
77
OF THE HYDRODYNAMIC
HIERARCHY-----------
A. Thomas-Fermi Approximation--------------------
78
B. The Classical Approximation------------------
81
C. The Isotropic Approximation------------------
83
D. The Irrotational Approximation----------------
84
ENERGY CONSERVATION------------------------------
91
Thomas-Fermi Approximation--------------------
96
B. The Isotropic Approximation------------------
99
A.
C. The Classical and Irrotational Approximations------------------------------VII.
6
SOUND
IN NUCLEAR MATTER--------------------------
100
103
A. Thomas-Fermi Approximation-----------------
106
B.
Isotropic Approximation--------------------
112
C. The Classical Approximation----------------
114
D. The Irrotational Approximation--------------- 118
VIII.
SIMPLE MODES
OF A FINITE NUCLEUS-------------------
123
5
IX.
ISO-SCALAR SHOCKS IN NUCLEAR MATTER--------------148
A. The Thomas-Fermi Approximation-------------154
B. The Isotropic Approximation----------------156
C. The Classical Approximation----------------161
D. The Irrotational Approximation-------------164
X.
SUMMARY------------------------------------------170
REFERENCES----------------------------------------------173
FIGURES-------------------------------------------------176
BIOGRAPHICAL
NOTE---------------------------------------
6
Chapter 1
INTRODUCTION
From almost the beginnings of our understanding
of the nucleus, a fundamental contradiction has existed. 1
As nuclei are composed of individual nucleons, they naturally
show many properties characteristic of this particulate
nature, such as behavior in accordance with the predictions
of the shell model.
However, certain aspects of nuclear
structure, and more recently many features of heavy ion
reactions, require descriptions in terms of a nuclear fluid
whose properties are characterized by such classical notions
as surface tension, pressure, and viscosity.
While it is
possible for calculations treating individual nucleons to
exhibit
"collective" behavior,2 the connection between the
"collectivity" expressed in the appropriate quantum mechanical
variable, the wave function, and that implicit in a hydrodynamic description in terms of density and velocity fields
remains obscure at best.
In this work, we seek to bridge the
gap between these two very different ways of describing
nuclear physics.
The reduction of an "exact" quantum mechanical manybody theory to a hydrodynamical description is desirable for
several reasons.
complexity.
Most important among these is a question of
The nuclear wave function contains far more
7
information than we may ever hope, or need, to use, and
indeed most of the interesting questions a physicist might
ask about a nucleus can be answered in terms of the one or
two body density matrices.
Hence, a theory dealing directly
with such physical observables as the density and velocity
fields is a substantial simplification and results in a
tremendous computational savings.
For example, even with a
particularly simple (and possibly inadequate) approximation
to the quantum theory, such as the Hartree-Fock approximation,3 a calculation describing the head-on collision of
as light a system as
Ca +
Ar would require tracing the
time evolution of approximately 30 complex fields
(the single
particle wave functions) defined over all space, whereas the
corresponding macroscopic calculation might involve as few
as 8 real fields
(the density, two velocity components, and
the pressure for both protons and neutrons).
Of course, the
savings become even more substantial for heavier systems.
Thus, while the microscopic calculation strains at the limits
of the present computing capabilities and has yet to be
carried out in a realistic situation, the hydrodynamic calculation may offer the same physics at a fraction of the
effort.
Apart from the prosaic questions of computational
tractability, there is a more fundamental need for establish-
8
ing the macroscopic-microscopic connection.
Current hydro-
dynamical calculations of nuclear dynamics are entirely
classical in nature.
They generally treat the dynamics of
a sharp surface, viscous liquid drop moving under the
influence of surface tension and Coulomb forces in accordance
with the classical equations of fluid mechanics.
quantum effects are absent.
Thus,
However, it may be possible to
improve these calculations by establishing the connection
with quantum mechanics and realizing quantal corrections to
the classical equations.
In addition, the properties of the
nuclear fluid which are input for these classical calculations
are not derived from first principles, but rather are
phenomenologically determined by nuclear properties over a
wide mass range.5
The establishment of the quantum connection
will elucidate the relationship between these properties and
the underlying inter-nucleon interaction.
A final motivation for understanding nuclear hydrodynamics in quantum mechanical terms rests upon the notunlikely situation of the tail wagging the dog, in the following sense:
It may be that the approximations used to solve
the quantum many-body theory are inadequate to describe the
actual physical situation (whether this is or is not the
case must be decided by actual computation, of course).
The
hydrodynamics derived from this approximation will then, of
course, be correspondingly inadequate.
However, by classical-
9
ly motivated modifications of the macroscopic equations, we
might obtain a closer description of the physics, and even
see our way clear to working backwards, making analogous
corrections to the approximate quantum theory.
This work is an investigation of the hydrodynamic
transcription of an approximation to the quantum many-body
theory.
Throughout, it is assumed that the time-dependent
Hartree-Fock approximation is a valid description of nuclear
motion, though as discussed above, such an assumption need
not necessarily be satisfied in order to arrive at a viable
hydrodynamic theory.
Chapter II presents an exposition of
the TDHF formalism, and establishes notation for succeeding
chapters.
Chapter III recasts TDHF in the language of the
Wigner function, resulting in a quantal Vlasow equation for
the motion of a distribution function in phase space, the
moments of which result in the hydrodynamic equations.
Chapter IV utilizes the phenomenological Skyrme interaction
to reduce these equations to an almost local form, though
also briefly indicates how the transcription may be extended
to arbitrary two-body forces.
Chapter V treats the approxi-
mation necessary to truncate the quantum mechanical hierarchy
of hydrodynamic equations, presenting four possible alternative prescriptions for specifying higher moments of the Wigner
function in terms of the lower ones, while Chapter VI treats
the conservation of energy in each of these approximations.
10
The remaining Chapters, VII-IX, deal with applications of
the hydrodynamic transcription.
the normal modes
(sound waves)
Chapter VII discusses
in nuclear matter.
Chapter VIII
treats the normal modes of 160, deriving expressions for the
isoscalar 2+ and giant dipole excitation energies.
Finally,
Chapter IX treats non-linear isoscalar hydrodynamics in one
dimension, discussing the kinematics of shock waves.
11
Chapter 2
THE TIME DEPENDENT HARTREE-FOCK EQUATIONS
In this chapter, we derive the Time Dependent
Hartree-Fock (TDHF) equations for an arbitrary two-body
interaction, and establish notation for succeeding chapters.
A brief derivation of the Random Phase Approximation
(RPA)
equations is also included, in anticipation of a linearization of the hydrodynamic equations.
The TDHF approximation furnishes a computationally
manageable scheme for treating a system of interacting
fermions, reducing the many-body problem to a set of coupled
one-body problems.
TDHF may be "derived" by several
different methods, such as finding the time-dependent Slater
determinant which minimizes the classical action, 6 or in the
limit of small amplitude motion, by means of a boson treatment of particle-hole excitations7
The treatment here
follows Reference 8.
To treat the quantum mechanics of a many-body
systems, it is convenient to use the techniques of second
quantization.3
a
and a
We define annihilation and creation operators
which destroy or create nucleons in the particular
quantum states denoted by a or 6
(these letters may label
such quantum numbers as the spatial, spin, and iso-spin wave
12
functions).
The a and a+,
which are adjoints of one another,
satisfy the anti-commutation relations
e&
(2.1)
States of the many-body system are defined with respect to a
particle vacuum, 10>, which has the property that it is annihilated by all the a :
Qj
=0
Io)
i.e. it contains no particles.
(2.2)
In particular, a state of
the nucleus may be represented as a linear combination of
Slater determinant kets
of the form:
tcttlo
where there occur
(2.3)
A different creation operators, A being
the number of nucleons in the system.
It is also useful to
define the number operator
(2.4)
whose eigenvalues are the possible numbers of particles in
the system.
13
To begin the discussion of the dynamics of the
system, we hypothesize a many-body hamiltonian of the form:
H= T+V
2 .5
)
c a.
The one body operator T
2(
is the kinetic energy, given by
--
(Note:
(2.6)
We henceforth work in units with h = m = 1, where m
is the nucleon mass, unless these quantities are explicitly
included in the formulae. For numerical evaluation, we take
h 2 /m = 41.57 MeV-fm 2 , and hc = 197 MeV-fm, so that Mc 2
934 MeV), while the two-body interaction is defined as
(2.7)
with V the two-body potential.
a> and
In these expressions,
|aS> denote the corresponding one and two-body
states, i.e.
ja> =a jo>
A complete treatment of the dynamics of the manybody system would solve for the state of the nucleus,
14
satisfying the Schroedinger equation
a generally impossible task.
However, the full many-body
wave contains more information than is useful.
We are
usually interested in the time dependent expectation
value of one-body operators, such as the density and velocity.
Such expectation values are conveniently given in
terms of the one-body density matrix, p .9
any one-body operator.
O
Let 0
denote
Then, in second quantized notation,
Qrj
(2.9)
*(A3CkwUf
with
O
Z<CwJOI
>
(2.10)
Then the expectation value of 0 is given by
)
0,3
+r (Qy)1b
(2.11a)
15
where p, which may be considered a matrix, is defined as
=
<1()a1 3 ( Ea I I W(2.12)
and equation (2.11b) is written as the trace of a matrix
product.
From the above discussion, it seems desirable to
avoid treating the full dynamics embodied in
IP(t)>,
and in-
stead attempt to write an equation for the time evolution of
p.
Indeed, from equation (2.12)
(2.13)
where the dot denotes the time derivative.
the Schroedinger equation,
and realizing that H
(2.8)
,
Utilizing
and its adjoint,
is hermitian, Equation (2.13) becomes
H]
)(2.14)
Using the relations
[+
]
(2.15a)
16
[ct~ct~)c4 dc~
g
a 01c4ac~~
(2.15b)
aga~a~cL)~+ ~a;a~a~a,~which may be simply derived from the anti-commutation relations for the a's,
Equation (2.5)
(2.1), Equation (2.14), together with
becomes
0
- F(
)
+(
(2.16)
where
Ckt4 GC-fS CLCr Cj A f
FA)
is the two-body density matrix.
(2.17)
In deriving Equation (2.16),
we have used the symmetry and hermiticity of the two-body
-
Vol
'a.
P'M(
'
interaction
6
iV'
OPV
W(2.18)
ScIP
Thus, as is evident from Equation (2.16), it is not possible,
in general, to obtain an equation involving the one-body
17
density only, for the two-body force couples the time
evolution of p to that of p
write an equation for p
2
.
If an attempt is made to
by using the Schroedinger equation
in a manner analogous to Equations (2.13-2.14), the threebody density must be introduced.
An infinite hierarchy of
dynamical equations results (analogous to the Martin-Schwinger
hierarchy in the many-body Green's function formalism,0 or
to the BBKGY hierarchy in statistical mechanics
1
,and so a
meaningful treatment is impossible.
The TDHF approximation assumes that the following
approximation to p
(2)is
valid
(2.19)
-
(Note:
The actual approximation necessary is that
where W is given by Equation (2.22).
This is a weaker con-
dition than (2.19), requiring the two-body density matrix
factorize into the product of two one-body density matrices
only when taking the expectation value of the potential.
discussed below, (2.19) implies that p
As
describes a Slater
determinant, while the weaker condition does not.
When only
this weaker condition is satisfied, interpretation of the
TDHF equations becomes difficult, as the derivation based upon
18
finding a Slater determinant which makes the action an
extremum is no longer valid. 6
Nonetheless, such a situation
may arise in the final state of an actual TDHF solution 12).
This approximation implies several properties of the system
which p
describes, which we reserve for discussion below.
However, if
(2.19) is valid,
(2.16) becomes
*
(2.20)
with the Hartree-Fock hamiltonian, h , given by
(2.21)
where the Hartree-Fock potential, W, is
o
(2.22)
(\4pa jvatp )-
Note that the effective Hartree-Fock hamiltonian governing
the time evolution of the one-body density matrix is a onebody operator.
Equation
(2.20), the TDHF equation, is thus
first-order in time, though non-linear in
p.
There are three quantities conserved by Equation
(2.20), and hence three corresponding constants of motion.
These are:
1) Particle number:
By taking the trace of (2.12),
19
we find
&r
pA
where we have used the definition of the number operator,
Equation (2.4).
The trace of Equation (2.20) yields
L r =
r E-,)f
(2.23)
0
where we have used the cyclic property of the trace to set
the trace of the commutator equal to zero.
do'
=
Thus,
0
and the number of nucleons in the system is conserved.
2) Energy conservation:
With the hamiltonian (2.5)
and the factorization (2.14), the total energy of the system
is
E=
I)
~.
p+Y
(2.24a)
Py{t~z I
Z236
r -Y*Y2. rf- W (2.24c)
The time derivative of this equation is
k~ ~3
(2.24b)
20
(2.25)
But from the definition of W, Equation (2.22), and the
symmetry properties of V, Equation (2.18),
so that
(2.25)
becomes
V
Inserting Equation (2.20) for
r(2.26)
p,
(2.27)
=
Hence, the total energy of the system is conserved.
3) Conservation of "Slater determinant-ness":
If at time t=0 the density matrix p
describes a Slater
determinant state of the many-body system, and if
p
is
evolved in time according to Equation (2.20), then at all
times later,
p
will describe a Slater determinant.
To
see this property, we must utilize the property that p
possesses if
property is
IT> is a determinant.
As discussed below, this
21
(2.28)
i.e. -
the density matrix is a projector.
From Equation (2.20),
(yC9Lp-cPZJtpLf
Hence
(2.29)
---
Ot
Thus, if p 2 - p= 0 is inserted as an initial condition into
(2.29), p2- p will be zero at all succeeding times.
While Equation (2.20) succinctly expresses the
TDHF equations, it is often more convenient to consider the
time evolution of the Slater determinant which p describes.
To this end, we consider the relationship between p
and the
single particle orbitals which make up jI(t)>.
Imagine A single particle wave functions {Ji(r)>,
i=l,... ,A}, with r a generalized particle coordinate
including spin and isospin labels).
$ i>
(i.e.
Furthermore, let the
be orthonormal
(2.30)
22
Then a Slater determinant formed from these wave-functions
has the form
A
)O> CLr(2.31)f
L=i
where a+
is the creation operator for a particle in state i,
given by
With the form (2.31) for the many-body wave-function, it is
simply shown that
Z<cc{>
1
9
> (2.32)
IKpl
We are now in a position to prove (2.28) is a
necessary and sufficient condition for p to describe a Slater
determinant.
To show that it is necessary, from (2.32) we
have
(2.33)
23
where we have used the completeness of the states y, and the
orthonormality of the |i > 's, Equation (2.30).
Equation (2.28) sufficient,
To show
realize that Equation (2.12)
implies that p is a hermitian matrix.
As such, it is diagon-
alizable, with real eigenvalues and orthogonal eigenvectors.
Furthermore, (2.28) requires that all eigenvalues of p are
either 0 or 1, and in fact, there are A eigenvalues which
are 1, as can be seen from Equation (2.21).
Thus, if we
identify the eigenvectors having eigenvalue 1 with the A
occupied states
I*i>,
Equation (2.32) is manifestly satisfied.
We may now prove that the TDHF approximation,
Equation (2.19), implies p is a Slater determinant.
the definition of p
(2),
Take
Equation (2.17), set 6 equal to a,
and sum over all states a.
Then
.
t
XJT (2. 34)
where we have used the definition of the number operator, (2.4).
Using the anti-commutation relations for the a+ and a,
Equation (2.1), it can be shown that
[~
(2.35a)
(2.35b)
24
Thus, Equation (2.34) may be written as
W~(.2.36a)
=
(2.36b)
where we have used the fact that
number of nucleons, A.
Equation
IT> contains a definite
Evaluating this same sum from
(2.19),
--
....
where we have used (2.21) for trp.
(2.36b) and
that
)(2.37a)
4(2.37b)
Comparing Equations
(2.37b), we see that (2.19) implies p 2 = p, or
IT> is a Slater determinant.
The TDHF equations may also be written for the
single-particle wave-functions
.>.
If p has the form
specified by Equation (2.32), it is easily seen that the TDHF
equation,
(2.20), may be satisfied if each
time according to
A(2.38)
IiP>
evolves in
25
Thus, each wave-function satisfies a time-dependent
Schroedinger equation involving the
dependent
lip
HF
hamiltonian, h.
(non-local) time
Because h is hermitian, the
> remain orthonormal at all times.
Equation (2.20), the TDHF approximation, is what
we shall be concerned with in the following.
This dynamical
equation is non-linear in the density matrix p (recall h is
proportional to p through Equations (2.21) and (2.22)), and,
when written in terms of the single particle wave functions
jip >, Equation (2.38), is also non-linear in these variables.
To date, the precise behavior of the solutions to
the TDHF equations for a given initial Slater determinant
is unknown.
With a little thought,
can be raised.
several serious questions
For example, how does TDHF "decide" upon a
final channel in a reaction situation.
It is, after all,
a deterministic theory in the sense that a given initial
condition gives rise to a specific final state.
On the other
hand, the probabilistic interpretation of quantum mechanics
assures that, in the actual situation, the final state is
spread over many different reaction channels.
Does TDHF then
select the "most probable" channel, or will the solution
spread out over all space in an effort to approximate a
coherent superposition of many reaction channels by one Slater
determinant?
If the latter is true, how are the reaction
amplitudes to be projected out?
A whole host of questions may
26
also be raised concerning the compound nucleus.
Does TDHF
contain enough flexibility to describe a compound nucleus?
If so, how will it decay, i.e. is neutron emission accounted
for? the possibility of fission?
While the speculations outlined above will ultimately be answered only by actual calculations, they do point
out the difficulty of understanding and interpreting the TDHF
solutions.
Hence, even a small amount of insight is welcome,
such as can be obtained by examining the linearized version
of the TDHF equations, the so-called Random Phase Approximation
.
(RPA)
Let us imagine that the static Hartree-Fock problem
has been solved.
We then have a density matrix p (0)
and
attendant Hartree-Fock hamiltonian h (0) satisfying
(o)
(
where h(u) is defined by Equations
p=p
(0)
.
(2.21) and
(2.39)
(2.22) with
and p c
commute, they may be simultaneous-
Letting i,j
denote occupied orbitals of p
Since h(O)
ly diagonalized.
-- 0
mn unoccupied orbitals, h( 0 ) assumes the form
(2.40a)
'
11h
A(2.40b)
27
(o)
(o)
(2.40c)
where the real E are the single particle energies, while p(O)
is given by
(2.40d)
(2.40 e)
(2.40f)
We now seek a solution to Equation
(2.20) by
linearizing in small excursions of p about the equilibrium
point p ().
Thus, we put
(o)
with the
Lw'.t
(
e LW
(2.41)
(complex) frequency 6 and the small transition
densities 6 p
to be determined as an eigenvalue problem.
Note that the condition that p be hermitian at all times requires that
t
(2.42)
while the condition p2 =p to lowest order in small quantities
requires that
28
(2.43)
Due to the unique diagonal structure of p (Q)
zeros),Equation (2.43) requires 6p
(all l's or
to have only particle-
hole matrix elements, i.e.
(2.44a)
Q
( 9P
(2.44b)
Inserting Equation (2.41) into Equation (2.20), linearizing
on 6p and extracting that component with an eiot time dependence, we obtain
4-
S
and the adjoint
9
L+
equation for 6p~.
J(2.45)
Taking the (i,n) element
of this equation, and using the fact that p (O) is diagonal and
vanishes for any subscript an unoccupied orbital, we obtain
A,
s t
(
A similar treatment of the (ni)element of
+
(2.45) yields
(2.46a)
29
&P-c
(xi ( S ')
+
(C
-E, )
-
(2.46b)
Using Equation (2.21-2.22) to evaluate
,
we obtain the
RPA equations
---!UJ
<Zllh
',1'60.
4
Lv
~v
v .X*
f
(2.47)
YC
elvi
with
x.
i'll
P4.
Y;
and the anti-symmetrized potential, V,given by
v~el
= <a6VIcCE >- <a.vlc)
(2.48)
-- v
CLbd
60
=V.
=-\/
b0. C
Equations (2.47) may be succinctly written in matrix form
by defining
A
t-A
=(
;)&S
+v
(.T1
(2.49a)
30
.
(2.49b)
.
in which case we have, in matrix notation,
X
A
X
\
(2.50)
Equations (2.50) provide an eigenvalue problem for the
normal modes, 6p, and normal frequencies, w, characterizing
the motion of the Slater determinant about the stationary
point p 0 .
If all eigenvalues w are real, the determinant
is stable with respect to small perturbations about this
point, as there is then no term in Equation (2.41) which
grows exponentially in time. 1 7
The solutions to Equation (2.50) also possess a
number of well-known but interesting properties.
First,
if (w,X,Y) is a solution to Equation (2.50), the transformation
*
31
+
Y
Y
+
Z
*
X
also results in a solution.
There are, then, pairs of
time reversed solutions, which, for real w, simply correspond
to the positive and negative frequency solutions.
Second,
because of the non-vanishing of B (the presence of ground
state correlations), the eigenvalues are non-linear in the
two-body force.
Finally, as Equation (2.50) is linear, any
(small) amplitude for the vector (X,Y) furnishes a solution,
so that the (conserved) energy of the time-dependent Slater
determinant of Equation (2.41) may be made arbitrarily close
to that of p (0)
The previous paragraph suggests the interpretation
that the diagonalization in Equation (2.50) is equivalent to
solving the problem of coupled oscillators to find the
normal modes in the space of particle-hole excitations.
Only
when these modes are quantized does the RPA furnish the excitation energies and wave functions for the excited states
built upon p(0)
Several models have been proposed18 to describe
low lying collective states in terms of shape oscillations
of the ground state density, though with the RPA equations
cast in the abstract basis of
the two is obscure.
ing chapter.
(2.50), the connection between
We furnish this connection in the follow-
32
Chapter 3
TIME DEPENDENT HARTREE FOCK IN THE WIGNER REPRESENTATION
In this chapter, we explore a special representation for the TDHF equations, i.e., that corresponding to a
Wigner transformation of the coordinate space one-body
density matrix.
We show that in this representation, the
TDHF equations reduce to a form easily recognizable as a
quantal version of the Vlasov equation, which approaches
the expected classical result in the limit that h
+
0.
In
the following, we restrict ourselves to a special class of
nuclei:
those that are spin saturated even-even nuclei.
also neglect the spin-orbit force.
density matrix p
We
For such a system, the
is diagonal in the spin projections as-
sociated with the labels a and S.
The generalization of the
following discussion to account for the presence of the spin
orbit force is straightforward, though adds nothing to the
concepts to be presented.
As far as a quantitative cal-
culation is concerned, it has been found that this particular
feature of the interaction may be safely neglected in certain
types of dynamical calculations.13
In addition to spin
saturation, we assume the wavefunction to represent a state
of definite
(even) number of both protons and neutrons, so
that the density matrix is also diagonal in the iso-spin
33
labels associated with a and F.
The density matrix then
assumes the form:
(3.1)
-T)t
or in terms of the single particle wavefunctions:
(3.2)
-- &
where the labels a and T denote the spin and iso-spin coordinates respectively.
Note that the assumption of spin-
saturation and the neglect of the spin-orbit force implies
that
(3.3)
C)--P
In 1930, Wigner suggested the following transformation of p in order to provide a semi-classical interpretation for its
dynamics.
He defined a "phase space
distribution function" f(R,k;a,T) given by
where the "center of mass" and "relative" coordinates are
defined by
34
R
=
/(r
I r
(3-5a)
(3. 5b)
respectively.
Equation (3.4) simply defines f(R,k) as the
kth component of the fourier transform of p(r,r') with re-
spect to the relative variable s at position R.
The density
matrix is, of course, given in terms of the phase-space
distribution function by inverting (3.4)
'-__-
(3.6)
While the uncertainty prohibits the simultaneous,
precise specification of a particle's position and momentum, 5
the Wigner function has many of the properties one would
expect of a classical phase space distribution function,16
which gives the probability density for finding a nucleon
at position R, with momentum k and spin-isospin labels aT.
The function f is real, due to the hermiticity of
the density matrix, as can be seen from the conjugate of
Equation (3.4)
(we henceforth drop the spin-isospin labels
whenever notationally convenient):
Y1
(3.7)
35
where the third line follows from a change of variable
5+
-s.
Note that though f is real, it is not necessarily
positive definite.
Various one-body observables are also given in
terms of f by their expected classical forms.
For a one
body operator 0 given by
0 (,)r)rO >
(3.8)
we have, from Equation (2.11)
<ljlC
E>
{(
z5d"3c rO Y )ar
(3.9a)
(3.9b)
d
R AO( R-
R+
)e{(4)
b
(3.9c)
where the operator expressed in the Wigner representation is
36
Thus, from Equation (3.9c), we see that O(R,k) provides
the appropriate weighting for the distribution function in
phase space needed to compute the required expectation
value.
The operator corresponding to the total nuclear
density at the point
0
S is given by
( -- S)
s,)s W )(3.11)
>
O'ii
(-R,"&)=
which, when expressed in the Wigner representation is
O(R)A)=
Thus,
(3.12)
from (3.10)
CO
3
_X
(.3.13)
%-4
16
which is the classical result,
i.e.
-
the total density
at any point in coordinate space is the integral of the
distribution function over all momenta.
Similarly, the
operator corresponding to the quantum-mechanical current at
1 15
the point S
th e is
37
-
(3.14)
which, when expressed in the Wigner representation is
O(R
(3.15)
so that the q uantum-mechanical current becomes
(3.16)
again, the expected classical result.
The Wigner function also has the expected form for
simple wave-functions.
momentum q,
For example, for a plane wave of
p is given by
--
(3.17)
so that f is
Tr
independent of R.
(3.'18)
Again, an expected classical result, is
obtained, i.e. that f is non-vanishing only at k=q, and is
independent of R.
38
Encouraged by the classical analogies presented
above, we may ask how the various features of p
The most elementary is the
total number of particles in the system:
-
which becomes
(cf.
(Equation (2.23)
)
themselves as properties of f.
express
A(3.19)
Equation (3.13))
3R S(i
; -1
(3.20)
The condition that p represent a determinant is a
bit more complicated.
In coordinate space
isospin labels) , this condition is
3
(dropping spin-
(cf. Equatin(2.28))
f(3.21)
or, in terms of f:
(3.22a)
/f
39
3
-3
d3*
-LA-
1
-- /
-
//
/2+
ki
Jdor"4
-
(3.22b)
9/
012-Y
fee)3
e
e
3.22c)
where the second line follows from Equation (3.4), and in
the third, we have introduced the spatial and momentum shift
operators
(3.23a)
* 1t
-ii
(3.23b)
In (3.22c), the superscripts
(1) and (2)
indicate which dis-
tribution function the shift operators apply to.
All in-
40
tegrals in (3.22c) are exponentials and so may be conveniently done with 6-functions to yield:
iD
D.
)
(3.24)
Upon taking the real part we have
4(~>&))
c6L(
4.
. )- (3.25)
where the cosine of the operator is to be interpreted as its
power series expansion.
Thus, when expressed in terms of f,
the condition for Slater determinantness, p2 =p, is a nonlocal, global restriction on f relating f to all derivatives
of itself at a given point.
It is interesting to see how
Equation (3.25) is fulfilled in nuclear matter.
system, as can be seen by summing Eqn.
such that
iqt-
In this
(3.18) over all q
kf , where kf is the fermi momentum
RR,
(3.26)
Thus, all spatial derivatives of f are zero, so that (3.25)
becomes
41
(3.27)
a condition manifestly satisfied by (3.26).
We now recast the TDHF equation in coordinate
space
in the Wigner representation.
In coordinate space,
Equation (3.20) becomes (we again drop the spin-isospin
labels for notional convenience)
(3.28)
Introducing f for the p's via Equation (3.6),
and defining,
in analogy with Equation (3.10)
(3.29)
we have
(3.30a)
42
C
Ot Oct
3 if
r
3
e
dp
gg
L10
eCtr~
7'~~/z4
4#rR,
r'
cg~
-
2..&
(3.30b)
with the shift operators defined by Equations
(3.23),
we
have
-t
Is
d3~
.- 6)
.-
d3
cI#r
4
C
e
M4fZDI~
er
L
1:.4
Lteiuiz,- i')e e
A
I
-Zb
(A-*
4
-34-
-A
)
LJ
1
eLDR-LW..
.A
j
-b
% qr,j#;
43
The integrals are all elementary exponentials, and may be
done with the aid of 6-functions to obtain
L
DbLO
(D)
(3.31)
R"A
ir,
.14)
taking the imaginary part,
(R )
_
_
_
_
_
_
_
(
4)
(3 .32)
For the hamiltonian (2.5), h is given by Equation (2..21)
as
(3.33)
or, using (2.6) for the kinetic energy operator, we have
YZ -A2 + W -ARI -AA
(3.34)
)
.,A ( R)"I AA
so that (3.32) becomes (only the first term in the expansion
44
)+0j)
+
for the sin works on the kinetic energy operator)
(3.35)
Note the similarity between this expression and the collisionless Boltzmann equation for a system in an external
potential U(R)6
(3.36)
__)k
Indeed, if W is a local potential (e.g. - if we had only
made the Hartree approximation, dropping the second term on
the right side of Equation (2.19) ),
(3.37)
")USC
( r-%- "-
)
VV-hi.'&
so that
and so
(3.38)
the expansion of the sin in Equation
(3.35) yields
45
V)
which, within terms of O(t),.is Equation (3.36).
(3.39)
-
Once again,
we find f is ripe for interpretation as a classical distribution function, in that it satisfies, in the classical
limit (Mi+), the equation governing the motion of that
classical distribution function.
For completeness in this chapter, we give the
Wigner representation expression for the total energy of
the system.
From Equation (2.24c) and Equation
(3.9)
(3.40)
I YZ
Thus, W may be thought of as functioning as a type of generalized potential, which, due to it's non-locality, depends
not only upon the position, but also the momentum of the
particle.
46
To summarize what has been accomplished to this
point, we have succeeded in obtaining an exact equation of
motion (i.e. exact within the TDHF formalism)
space distribution function.
for a phase
This distribution function
possesses many of the properties of a classical distribution
function, and it's equation of motion, which involves a
momentum dependent potential, reduces to the collision-less
Vlasov equation in the classical limit.
We now consider more explicitly the relationship
between the non-local HF potential W, defined by Equation
(2.22) and the fundamental two-body interaction.
We shall
show that for a general two-body force, the resultant
dynamical equation for the distribution function f, Equation (3.35) is non-local in both the spatial and momentum
variables, leading to similar difficulties of physical
interpretation as those encountered in the abstract representation of the TDHF Equations (2.20).
However,
it
will be
seen that the introduction of an effective, short range
expansion of the two-body force of the Skyrme form 21
reduces the Wigner representation of TDHF to an almost
local form, amenable to treatment and approximations in the
classical hydrodynamic spirit.
We begin by expressing Bkguation (2.22) in coordinate
space.
We again restrict the discussion to spin saturated
systems with central forces, as in the previous chapter.
47
In addition, we also momentarily assume the two body force
to be spin independent, as well as iso-spin independent.
In
this situation, the HF potential is diagonal in both spin
and isospin labels, and independent of the former.
The co-
ordinate version of (2.22) then reads
'e
-AX
V('r'r
(3.41)
with the anti-symmetrized matrix element of V defined by Equation (2.48), and in an obvious notation for the spin-isospin labels.
The symbol a denotes an arbitrary spin index.
The spin-iso-spin independence of V and Gallilean invariance
imply that V must have the following form:
"C
T
(3.42)
~r
where v is the (non-local) two-body potential, and the overall 6-function insures that the total momentum of the
interacting nucleus is conserved.
(3.41) becomes
With Equation (3.42),
48
(3.43)
A
7S~~3..r
~
.(%? L
pJh
i-Crx- rjagfrrI
-
where we have used
S ("X/Z)
=:Sg9(9)
(3.44)
and have defined
JP-C
(3.45)
T- T.
rM
,
Introducing the Wigner transformation as in Equation (3.29)
and defining
Eqn.
(3.46)
.
0o-%
(3.43) becomes
-- :V
cpr
(1770-)
49
~+
e.
J
--
O(Q
S
(-r~
_
a
*K
.C
-x
r
_
e4/Z
C
-%
+ _41
04 t
.
.-
C
t
(3.47)
With the use of the spatial shift operator, DR , we can
write
t
L/7t
+
i)
(3.48)
R
so that all integrals in (3.47) become elementary exponentials.
The result for W is then found to be
fd4
9tWi
.Mftf
t
(3.49)
t-A)
-~
-~
L,~)
50
While the form of (3.49) is, in general, nontransparent, several important points can be gained by
examining its
force.
behavior in the case of a local two-body
If the two-body force is of the local form
)
(3.50)
then, from (3.46)
--
51)
3(3.
so that, from Equation (3.49)
The first integral (direct) may be reduced, by means of the
convolution theorem for Fourier transforms,20 to
c
3
r'
-LrG')
(3.53)
Thus, the direct term depends only upon the density, the
zeroth moment of f in momentuM, as per Equation (3.13), at
all points in coordinate space, so that all properties of f
need not be known in order to compute this integral.
How-
51
ever, the second (exchange) integral in (3.52) may be
expressed by expanding the potential in a Taylor series as
(3.54)
mom
t3
where v(n) (k) is a schematic representation for the expansion
coefficients.
The exchange integral then involves all
moments of f in momentum space (i.e. current, density, etc.),
and all powers of k appear in W.
From (3.52-3.54), it is apparent that a solution to
the dynamical problem expressed in terms of the Wigner
representation involves as much complexity, and therefore as
much information, as does a solution in the more abstract
representation, (2.20).
However, in an actual calculation,
the quantities of direct physical interest are generally the
few lowest moments of the Wigner function in momentum space,
i.e. the density and current fields. Hence it is reasonable
to attempt to generate a dynamical equation involving the
few lowest moments.
In the classical regime, these equations,
-
formed by taking moments of the Vlasov equation, (3.36), in
16
momentum space, yield classical hydrodynamics,
i.e.
equations for the conservation of mass
(the continuity
equation), the conservation of momentum (Euler's equation),
52
and the conservation of energy.
Certain, physically
plausible assumptions concerning the form of the distribution function (e.g. a local Maxwellian distribution
characterized by a temperature field varying on a spatial
scale slow compared to the mean force path of the particles
involved)
close this set of classical equations, and
directly provide for the dynamical treatment of the fields
of interest.
As can be seen from the form of Equation
(3.52),
any
long-ranged two-body force involving many powers of Q or ZL
will negate an attempt to deal only with the lowest moments
of f, due to the non-locality of the Hartree-Fock potential
caused by the exchange term.
There is, however, a particular
form for the two-body force which handles this problem,and
is discussed in the following chapter.
53
Chapter 4
TIME DEPENDENT HARTREE FOCK WITH THE SKYRME FORCE
The Skyrme f orce
proposed in 1956,21 is specifical-
ly designed for self-consistent field calculations in finite
nuclei.
This force consists of two terms:
three-body force.
The
a two-body and a
two-body force is taken to be of the
form 2 2
o ((:r~
+Q
(4.lb)
4In this expression, x , t , ti, and t2 are constants, while
P
is the nucleon spin-exchange operator.23
Consistent
with our previous restriction to spin-saturated systems with
central forces, we have omitted the usual spin-orbit term.
Using the definition
(4.1)
(3.46), a fourier transform of
reveals that, when expressed in co-ordinate space, the
Skyrme force consists of 6-functions and derivatives of 6functions in the relative nucleon co-ordinate.
The t
(6-function) acts in relative S-waves, the only partial
piece
54
wave which does not vanish at the origin, of the relative
coordinate, with effective strength t (1-x ) in spin singlet
states and 't (l+x ) in spin triplet states.
The tj piece
acts in relative P waves only (the only partial wave whose
derivative does not vanishes at the origin), while the t2
piece acts in both relative S and D waves, where the second
derivative does not vanish.
Note that for a local Skyrme
force (independent of 2), t1 =-t 2 .
When first proposed, the
form (4.1) was believed to represent in lowest order terms
of an expansion of the nucleon-nucleon force in momentum
space.
However, recent work
has shown that, actually
it represents, in an average sense, the folding of the
relative wave-function with a more realistic two-body
interaction (see the discussion at the end of this chapter).
The three body term in the Skyrme interaction is
taken to be of the form
(4.2)
t
-~
where t 3 is yet another constant.
r.
Note that this three-
body force is equivalent to a two-body force whose strength
is linearly dependent upon the density.
As such, it is
providing for the density dependence of the effective twonucleon interaction well known from microscopic calculations
with realistic forces. 2 4
55
With the force (4.1) and (4.2), it is straightforward to express the Hartree-Fock potential, W(R,k)
(3.49) in terms of the two-body force.
using
Some extra care must
be taken in handling the spin exchange force and the threebody force, but the final expression is obtained by a simple
extension of the arguments used for the two body force (the
detailed treatment for static systems has been worked out
in Ref.
25).
The Hartree-Fock potential for the force of
Equations
(4.1-4.2) may be expressed as
tR)(4.3)
4Q
where
2
(4.4b)
+
-
(4.4b)
In these equations, T' denotes the iso-spin index not equal
to T.
The moments of the distribution function,
T are defined as
p, J, and
56
3
0(3t
,(ZTT)5
.. "
-A
'7t ( R)
IO
5
T4 (1
Tr
.
0-4 -4)
1R)-t
~
(9)~
L
P2
Pi
tj
(density)
(4.5a)
(current)
(4.5b)
(kinetic energy
"-4y1
(4.5c)
tensor)
T"C
In terms of the single particle wave functions, these
quantities may be written as
4L
(i~t)
.C
T.
T
(4.6a)
Y-~'(t~LI
(4.6b)
(
-L
L
-
-J
(v V)
Re
As promised,
aL
(F C
(4.6c)
-
%
t l -.
L
iLV~'(0I;C)
V0
only the three lowest moments of f appear in
57
Equation (4.4).
Furthermore, the quantity C (R) may be
viewed as being related to an effective mass, since the
coefficient of k.
in h,(R,k) is given by (see Equation
(3.34) and Equation (4.3)
+
(4.7)
This leads to the identification of the effective mass as
Y-c
if we write the k
~
2 L*C~(
contribution to W as k /2m*.
(4.8)
From
Equation (4.4c), we see that the inverse of the effective
mass is linearly related to the proton and neutron densities.
For completeness here, we also give several other
expressions relevant for use with the Skyrme parameters.
The Schroedinger equation for the single particle wavefunctions, Equation (2.38), is
with
RC
'C(2-
J3(4.10)
7T)
58
obtained by inverting Equation (3.10). With Equation
(4.3)
for
W, we have
(4.11)
so that (4.9) becomes
0a
(4.12)
Note that
despite the appearance of the i in Equation
(4.12), the Hartree-Fock hamiltonian is still hermitian, as
(4.13)
13+ B- P +1.
59
where the hermitian momentum operator P
-
is, as usual,
..
(4.14)
The total energy of the system is also simply
expressed for the Skyrme force.
Equation
In particular, from
(3.40) and Equations (4.3-4.34), we find
(Note that the three-body tone requires a factor of 1/3!
rather than the 1/2 in front of W in
=1/6
(3.40) ):
(4.15)
where the energy density I(r)
(
)
'/2
Tr
given by
/t
71f a[0+ XO/ ) P2 (X.
4
1~P~iz
is
4.
(4.16)
60
where the moments subscripted n and p denote isospin labels
neutron and proton respectively, while those without any
subscript denote total values, i.e.
Th
(4.17)
For isoscalar systems (protons and neutrons identical),
Equation (4.16) simplifies to
I
Ork
=V
Tri
+JP7r
3 /8+
(4.18a)
- /9V)+
p3
/16
t
2
-T
with
9'=
(3-,sY/
(4.18b)
(S0I
9bxV 6
(4.18c)
(4.18d)
61
Having seen that the Skyrme form of the internucleon interaction leads to a particularly simple form
for the Hartree-Fock potential in phase space, we are now
in a position to recast TDHF in terms of an infinite hierarchy of hydrodynamic-like equations by the process of taking moments on the momentum.
The zeroth moment of the Wigner function in the
momentum variable (i.e. - the integral over all momenta)
yields the density (cf.
Equation (3.13)).
The time evolu-
tion of this field may be found by integrating Equation (3.35)
over all momenta and using the definitions
(4.3):
(4.19)
4 Len
(a
IV (74)} (AtA -.&+t Y
The first two terms integrate directly to yield
ve
J."c(4.20)
The sin term, however, required a bit more work.
Consider,
first, the lowest order term in the power series expansion
of the sin:
62
~~
d
fj-8t
2Cei -VAi(Ac 4a
'4) -
J~2 T)3Lvft~2Ct~Rt(4.21)
=.
Integrating the last 3 terms by parts, we obtain
9
+2 c.-
(v#,)+ If,
r
V6) +2.T-VC,
T
or
B. (+-2 C
)
(4.22)
It is easily verified that all higher order terms in the
expansion of the sin give no contribution, so that the
combination of Equations (4.20) and (4.22) yields the final
continuity equation
where we have used Equation (4 . 7 ) f or the ef f ective mass, 1/m*
.
63
Equation (4.23) expresses a generalized form of
current conservation.
For a system in which the HF poten-
tial is local, i.e. if B = C = 0, so that W(Rk)
is
dependent only upon R, then (4.23) would read
-0- Q
which is what is expected classically.
(4.24)
However, the failure
of local current conservation indicated by (4.23) is not
surprising, as it is well known that current is not conserved locally for a non-local potential.26
The total
number of nucleons of a given iso-spin type is, of course,
conservedas seen by integrating over all space and dropping
surface terms
~
CLe
Q~C
(4.25)
which is the analogue of Equation (2.23).
Equation (4.23) also possesses the surprising
property of conserving the total (iso-scalar) nucleon
density locally.
Indeed, by reference to the definitions
(4.4b-c), the continuity equation may be written as
64
+ V-
-
(4.26)
The sum of Equation (4.26) for both charge types results in
i"+OW
)
(4 .2 7
Thus, the non-locality of the current conservation is
directly related to the non-isoscalar behavior of the
system.
The equation resulting in conservation of momentum
is found by taking the first moment of
by k
(3.35).
Multiplying
and integrating over all momenta, we find only the two
lowest terms in the expansion of the sin contribute, so
that
(4.28)
After several integrations by parts, and a judicious rearrangement of terms, we obtain, in component notation
65
+rTjA
4
Equation (4.29) is written with the summation convention
for repeated indices.
There is no distinction between super-
and sub-scripted indices, which we used only for notational
convenience.
+
In a tensor notation, Eqn.
~~
4T.
(4.29) becomes
?cVAt+ Y2.Tr(4irJ)'7(_;)
(4.30)
Equation (4.30) reveals that nucleons move under
the influence of two types of forces:
a "pressure" associated
with the divergence of the kinetic energy tensor, and interparticle forces of both the static (VA) and velocity dependent type.
Note also the term V 2 (pVc) arising from the
higher order terms in the expansion of the sin, which is
66
explicitly of a quantal nature.
Lastly, note that the
integral of (4.29) over all space does not yield zero, as
the momenta of protons and neutrons are not separately conserved.
However, when (4.29) is summed on charge states
and integrated, the result is, of course
(4.31)
3rJ(
so that the total momentum is conserved, as expected (see
Chapter 9 below for an explicit demonstration of this in a
special case).
The equation describing the dynamics of the third
moment of the distribution function, the kinetic energy
tensor, is obtained by multiplying (3.35) by k k] and
integrating over, all momenta.
Again, only terms to third
order (second term) in the expansion of the sin contribute,
and the resultant equation is
AA
67
()
-L
1y-~
4V/ ~
Ak
4 ~ xZ4 ~
0
41
-tK
A ~xJ (I
)
2
-T,
,
3x
L-'~) ;x&
z
e
)
'WI
-4.
;jX4k D~
m
x
Ilk
C
r
(I.
)
I
;'
+
&
+1
24.
7A o
the heat flow tensor
~~aA
,
In Equation (4.32) we have introduced the third moment of f
J3'A
FT-r)3
,
(4.32)
defined as
(4.33)
68
In tensor notation,
+(v
(4.32) becomes
-t +
o vd)>Tr
Q,
(+ 34 +
)
t {vT-cl
(4.34)
4~~7~?(B)
(V0~
)v
v~
where the symmetric tensor product of two vectors is
defined
as
A) 3
= ABJ 4
(4.35)
the symmetric tensor product of two tensors is
SoT
-A T Aj. Ti A
(4.36)
( Trt
(Try)
'
and the trace of the third-rank tensor Q is
(4.37)
(4.37)
69
Equations
(4.34) and (4.32) are written in a form so that
the quantal correction terms arising from the expansion
of the sin past the lowest order are enclosed in the final
set of large square brackets.
In contrast to the two lower moments, the integral
Equation (4.34)
over all
space does not yield a conservation
law, even when summed over iso-spin labels and traced on
spatial indices.
This is because kinetic energy alone is
not conserved, but rather the total energy kinetic plus
potential, is.
Thus, to obtain the third conservation law,
(4.34) must be used to examine the energy balance in
greater detail, which will be done in Chapter 6.
The three Equations (4.23, 4.30, and 4.34)
can be
expressed in more useful and transparent form by defining
velocity and pressure tensor fields in the following manner
(4.38a)
and the reduced third moment, Q-, as
÷u-k
(4.39)
70
The velocity field U is analogous to the classical
definition, while the fields
T
and
a
are the kinetic
energy tensor and heat flow tensor as seen in a frame moving
locally with the fluid, i.e.
U C)
(4.40)
and
-CU
(4.41)
It is also useful to define a convective velocity field,
7 , given by
U
Mm
Li(4.42)
as will be seen shortly, U is the velocity field appearing
in the convection derivatives of the hydrodynamic equations.
Note that if the dynamics are iso-scalar in the sense that
Up = Un=U, then UT
finitions
is
i,
as is easily seen from the de-
(4.4).
With these definitions of the velocity and pressure
fields, the continuity equation,
(4.23) becomes
71
Ut
U t.-F
while the momentum conservation equation,
It
at
LAt
(4.30), becomes
7r
R
vj~.)
-. 4
V)
(4.43)
'P-C
-j-
IV 'k-
+- I
)C.
(4.44)
Finally, the equation for the pressure field, using
(4.34), becomes
Q+t
t
V)
If
%I
".Oft
WOM
7T-c
Lit -2 /\
)
Equation
Z."
+
v4) xI-1
t
(4.45)
172
!V(I')
V Pr
'P I V (::' 'tt )) -U,I ) - '
)Try
'Ut
+ V Vil'. VIRC + V. (PA vvu97
+
(VW
YIA 'V
(V X)
72
where the symmetric tensor A
is given by
V2 (i
(L~1)L
(446
(4.46)
and the cross product of a vector and a tensor is defined as
(Vx T)~
(Tx V
E1V- T-
= T AVA T
( 4 . 4 7a)
(4.47b)
where cijk is the usual alternating symbol, 1(-l) if ijk is
an even
(odd)
p ermutation
of indices, and zero otherwise.
73
Equations
(4.43-4.45) are the set of hydrodynamic-
like equations we shall deal with in the following.
It is,
of course, possible to extend the moment treatment further,
and generate the whole hierarchy of equations for all
moments of the distribution function, but
as such a
procedure adds little in terms of insight or utility we
stop at three.
The static HF solutions passes several interesting
properties when considered in terms of the moments dealt
with in the hydrodynamic equations.
Equations
As is seen from
(4.6), successive moments of the distribution
function in the Wigner representation are equivalent to
derivatives of the density matrix in coordinate space,
evaluated along the diagonal (r=r').
Hence the restriction
of the hydrodynamic treatment to the lowest moments is an
expression of the belief that the dynamics of the density
matrix may be adequately described by it's near-diagonal
behavior.
All odd (i.e. - odd under time reversal) moments
of the distribution function, such as the current and heat
flow tensor, depend upon the non-vanishing of the imaginary
part of the single-particle wave-functions.
Hence, in a
static system, for which the wave functions may be chosen
to be real, such moments vanish, implying that the distribution function is invariant under the transformation
k-
-L
Equations
(4.43) and (4.47) then become automatic-
ally satisfied when all odd-moments are set to zero, while
74
(4.26) becomes
_0(4.49)
This is the quantal analogue of the condition ofr hydrostatic equilibrium, expressing a balancing of "pressure"
and two-body forces.27
However, in the usual classical
treatments (e.g. in the case of stellar equilibrium), H
is assumed to be diagonal (isotopic) and specified uniquely
in terms of the density through an equation of state so
that
7T
7TF (o)
(4.50)
Static HF solutions, on the other hand, satisfy Equation
(4.49) under the constraint p2 =p, which, as is seen from Equation (3.25), is not simply implemented in terms of
these low moments.
If an ansatz such as (4.50) is
inserted into (4.49), we immediately obtain a differential
equation describing the density distribution in a static
system.
The hydrodynamic equations also retain the property
of Gallilean invariance, as they ought to, since the original TDHF theory possessed this property.
pT (R), T (R)
If
is a solution to the static problem,satisfy-
75
ing Equation (4.49), then the transformation
-V+-(4.51a)
-*
RV
(4.51b)
which describes, classically, the translation of the
system at a uniform velocity V, leaves Equations
(4.34-4.35)
unchanged.
While the above features are certainly desirable
ones for a theory which purports to be a hydrodynamical
description of nulcei, there is still a problem in the
4ctual implementation of the theory.
Equations
In particular,
(4.43-4.45) are three equations dealing with
four moments, and are therefore not a closed set of equations.
Hence, some specification of one of these moments, most
naturally
a-,
in terms of the other three, is required.
Had the full hierarchy of moment equations been dealt with,
,
of course, would have been allowed to be an independent
variable, and such a specification would have been unnecessary.
Thus, the question we must address in the next
chapter is whether or not a meaningful closure of the hydrodynamic equations can be performed.
76
Before concluding this chapter, we briefly discuss the possibility of the extension of the hydrodynamic
transcription to forces more realistic than those of the
Skyrme type.
Equation
As was seen in the discussion following
(3.54), any finite range two body force foils an
attempt to restrict discussion to the lowest moments of
the Wigner function due to the non-locality of the HF
potential in co-ordinate space
momentum dependence).
(or equivalently, its
A way of by-passing this obstacle
would be to parameterize the k dependence of distribution
function, f(R,k"),in terms of a few spatially varying
parameters.
(A physically sensible parameterization of
this type, in which f may be given in terms of p and H has
been given for the static system28 under the assumption of
isotropy for H.
The extension to deal with time dependent
systems and the anisotropy of HI is straightforward.).
Under such a parameterization, all moments in k of f are
given in terms of these parameters, so that the non-locality
of W may be handled.
Such a procedure would be entirely
analogous to the classical assumption that the distribution
function is a local maxwellian, parameterized by a local
density, mean velocity, and temperature. 1 6
77
Chapter 5
TRUNCATION OF THE HYDRODYNAMIC HIERARCHY
In this chapter, we discuss several semi-classical
truncation prescriptions for the hierarchy of moment
equations derived in the previous chapter.
In particular,
four different prescriptions are discussed, the ThomasFermi, Classical, Isotropic, and Irrotational.
We discuss
the motivation for each type of truncation, as well as the
approximations to the many-body wave function inherent in
each.
As is the case in classical hydrodynamics, the
ultimate test of any type of truncation is the physical
plausibility of the solutions the resultant equations
generate, or a comparison with the exact TDHF solutions,
questions we address in the following chapters.
Hence, it
should be born in mind that the exact validity or applicability of these prescriptions, in so far as they yield
solutions which behave as "exact" solutions to the TDHF
equations, remains an open question in the absence of such
exact solutions.
Of course, as discussed in the intro-
duction, it may be the case that TDHF itself is not an
adequate description of collective nuclear dynamics, and
by the judicious choice of a truncation procedure, the
missing features necessary to reproduce realistic results
78
may be supplied.
A.
Thomas-Fermi Approximation
We begin with the simplest possible approximation,
which is within the spirit of the Thomas-Fermi approximation to the many-body system29 (more accurately, it is
closer to a Slater-type approximation, as is discussed
below).
In this approximation, we limit discussion to
only the continuity and momentum balance equations, and
attempt, as was hinted at in Equation (4.50), to specify
the pressure field in terms of the density.
This is ac-
complished by assuming that the distribution function looks
like that of nuclear matter, moving with the local velocity
U4R)
and having the local density pT(i_).
By referring to
Equation (3.26), it can be seen that this approximation
supposes the distribution function to be unity inside a
sphere in the momentum variable of radius k
(R), the local
Fermi momentum, and centered about U (R), the local velocity
field, while zero elsewhere.
The Fermi momentum is given
in terms of the density by
__0-_
(5.1)
where the extra factor of 2 arises from the spin degree of
freedom.
From the definition of the pressure field,
79
Equation
(4.40) of the previous chapter, it is clear R is
isotropic (proportional to the unit tensor) and given by
(5.2a)
(5.2b)
Equation (4.26) furnishes the "equation of state" necessary
to truncate the hydrodynamic hierarchy.
The nuclear fluid
behaves as a perfect gas under adiabatic compression or
expansion, the polytropic exponent,27 or ratio of specific
heats being 5/3.
The validity of the Thomas-Fermi approximation for
static systems has been discussed by several authors previously. 2 9
Briefly, it is expected to hold in situations
where the spatial scale of variation of the self-consistent
potential is large enough so that the single-particle wavefunctions in this well may adjust themselves so as to
locally mimic plane waves.
The characteristic length for
density (and hence potential, due to the short range nature
of the nuclear force) fluctuations is given by
IVpI/p
while the scale on which the wave-functions change is given
80
by kf.
Thus, from (5.1), the condition of validity for
the Thomas-Fermi approximation is
1/3
(5.3)
_____
This condition is manifestly not satisfied in realistic
nuclei, where the surface thickness is approximately 2.5 fm
and the fermi momentum -1.3 fmn.
Nonetheless, the hydro-
dynamic equations with this Thomas-Fermi truncation describe
the compressible hydrodynamics of the nuclear fluid, and,
as is the case classically, are expected to show a rich
variety of phenomena.
Hence, the Thomas-Fermi truncation
is not entirely trivial in its consequences, and, in some
situations may aid in understanding the behavior of the
We remark that for consistency in
hydrodynamic equations.
regard to the conservation of energy, the quantal correction
term in Equation (4.44) must be discarded when using the
Thomas-Fermi truncation, as will be discussed in the followBefore turning to more sophisticated ap-
ing chapter.
proximations, we remark upon a seeming ambiguity in the
specification of R in terms of p.
the Skyrme force,
22
In previous work with
the quantity H has been given in terms
of the "kinetic energy density", T, defined by
T
T
=T
T
+ 1/4VVp
T
81
or in terms of the single particle wave functions
S =120 V
n
,)
(,)Vin
While T possesses a particularly simple form in terms
of the single particle wave-functions, its usage has been
purely ad-hoc.
To the extent that we wish to make state-
ments concerning the one-body density matrix, it is T,
the second moment of this function, which should be approximated by its nuclear matter value, not
'",
as has been
done in a previous calculation. 3 0
B.
The Classical Approximation
We now turn to the classical method of truncating.
In this case, the essential approximation is to set the
reduced heat flow tensor,
Eqn.
a..,
to zero.
By reference to
(4.41) it can be seen that this implies that in a
frame moving locally with the fluid (i.e. with velocity
U
(R)) the distribution function is invariant under the
transformation k+-k.
This approximation is thus analogous
to the classical truncation prescription which describes
the distribution function in the momentum variable as a
maxwellion,
characterized by a local temperature.
However,
in contrast to the classical situation, we may still retain
the tensor character of H.
Thus, in the classical trunca-
82
tion of the TDHF equations, all terms in Q arise solely
from convection.
In contrast to the Thomas-Fermi approximation, the
classical truncation prescription permits the nuclear
fluid to get "hot" in the following sense:
In the Thomas-
Fermi truncation, the distribution function is "close packed" in that at any point, the lowest orbitals
be local plane waves)
(assumed to
are taken to be occupied.
Therefore,
R becomes a unique function of p, with the kinetic energy
density,
-Tr H , being the smallest value possible for a
given p, consistent with the Pauli principle requiring no
more than one nucleon in a given state.
The classical
prescription, however, turns the kinetic energy density
loose, so that it becomes independent of p and hence capable
of describing random (i.e. - non-organized) heat motion of
the nucleons.
In this approximation, Equations
(4.43-4.45)
describe the compressible, non-isothermal flow of the
nuclear fluid, or more specifically, flow which is adiabatic in the classical sense, i.e. no heat flow. 1 6
Note that the classical truncation procedure also
possess the property that the HF stationary solutions
remain stationary solutions, as discussed in the previous
chapter.
It may not be implausible, then, to utilize these
solutions as initial conditions for the truncated hydrodynamic equations, so that quantum mechanical effects like
83
shell fluctuations in the density are automatically included in the initial conditions.
It is important to re-
member that through this truncation procedure, all hope
of satisfying p2 =p in the time dependent solution has been
lost, though of course, it is not at all clear that this is
a physically important constraint on the wavefunction.
C.
The Isotropic Approximation
The isotropic approximation is a form of the
classical approximation in which the pressure tensor,
, is taken to be isotropic.
This is usually done
classically, on the grounds that intermolecular collisions
occur rapidly enough so that any anisotropies in this tensor
are smoothed out on a time scale much shorter than the
scale of collective motion.
Of course, this may or may
not be the case in a quantal system, though it should be
remarked that for a static HF solution for Pb, for example,
even in a region as anisotropic as the surface, the tensor
H computed from the single particle wave-functions is very
.
. 28
nearly isotropic.
As will be shown in the following chapter, the
classical, or even isotropic truncation respects energy
conservation, and soEquatiors(4.43-4.45)
cation except, of course, setting
require no modifi-
QL to zero.
84
D.
The Irrotational Approximation
We now turn to the final truncation prescription,
the irrotational approximation.
As a motivation, we first
consider the dynamics of a single particle under the
influence of a local hamiltonian.31
wave function $(r,t)
The time-dependent
satisfies the Schroedinger equation
~H
Y(5.4)
with the hamiltonian H given by
=~
(5.5)
~ ~/zV'V(re)
where V is a local (possibly time dependent) potential.
The wavefunction $ may always be written in the form
/,(
(5.6)
where the real functions p and S are the density and velocity
potential fields associated with
i.
From Equation (2.32)
(Y;
fop(5.7)
85
If the Schroedinger equation is now recast in the form
(cf. Equaticn2.20)
L d-
(5.8)
[ H?]j
a Wigner transformation in the manner of Chapter 3 yields
the following moment equations
(5.9a)
jc'
r
A-vt*T-
(+
V
T
(5.9b)
(5.9c)
where the velocity field U is given by (cf. Equations (4.6a)
and (4.38a))
(5.10)
H is defined by Equations (4.6b) and (4.38b) and A
(cf. Equation (4.46)
)
defined as
is
86
(5.11)
--
2
S plays the role of the velocity potential 3 2 for the irrotational velocity field U, i.e.
7KL .
(5.12)
As might be suspected, Equations
what redundant.
(5.9) are some-
The original Schroedinger equation (5.4)
was an equation for one complex (or two real)
(or Re* and Im$), while Equations
real fields.
(5.4)
fields
i
deal with the
In fact, by reference to Equations
(4.6b) and
(4.38b), it can be seen that the "equation of state"
relating the pressure tensor to the density is
ft-
-.).4L.
4-p
(5.13)
;Pc)X
so that only Equations
C3i
(5.9a,b), together with (5.13) are
all that is actually necessary to describe the dynamics.
In the case of single particle motion, it is also
possible to exactly express the reduced heat flow tensor,
87
in terms of the lower moments of the distribution
function.
shows that
Reference to Equation (4.33) and (4.39)
L is given by
2___A
(5.14)
Q.is)of course, symmetric in its indices, as
(5.10)
implies
(5.15)
--
which is manifestly symmetric.
With the realization that
may be exactly ex-
pressed in terms of the lower moments in the case of the
dynamics of a single particle, we now consider TDHF for
the dynamics of many particles.
the following situation to hold:
In particular, we assume
that the phases of the
single-particle wave functions for nucleons of a given isospin label are all describable by a common function, i.e.
E? L
(5.16)
where ST is the velocity potential for isospin T and is
assumed independent of the wavefunction label i.
If the
condition (5.16) holds, the density matrix is then given by
88
Equation
(2.32)
as
P ((5.17)
i.e. the phase assumes a
factorizable form.
The velocity
field is then found to be
QzR~?-&C~*
(5.18)
while, in analogy with the single-particle case, the reduced heat flow tensor is
A -+
(5.19)
t"
-x
Note that (5.18) implies, as previously, that the velocity
field U is irrotational.
The form (5.16) for the single-
particle wave-functions is certainly a sufficient condition
for irrotationality, but its necessity is an unproven conjecture.
We also note the connection between the form
(5.17) and the adiabatic TDHF theory of Baranger and
Veneroni,
in which the density matrix is written in the
form
X
(P)
-'x
(5.20)
89
with p
a time-even Slater determinant density matrix,
and X a one body operator having only particle-hole
matrix elements with respect to p
local,
.
For the operator X
(5.20) reduces to (5.17).
The irrotational assumption expressed by (5.16)
implies a different physical situation than does the
corresponding assumption in classical hydrodynamics.
In
that situation, Kelvin's theorem32 states that irrotation
flow remains irrotational, i.e.
(
c O GO7~E.) + V
VFW M)
0
(5.21)
which holds under the assumptions that
(vTT)o
V
x C V47.-
(5.22a)
(5.22b)
Equations (5.22) are manifestly satisfied for situations
in which R is isotropic and uniquely related to the density
e.g.
in the Thomas-Fermi truncation discussed above.
However, no such analogous theorem exists quantally, and,
as is seen from (5.16), vorticity in the velocity field is
generated by a loss of coherence in the individual velocity
fields of the nuclear fluid described by each single
90
particle wave function.
Nonetheless, as
(5.16) furnishes
the only quantaly rigorous truncation procedure available,
and as irrotation flow has long been used as a condition
in classical treatment of nuclear hydrodynamics,
the
insight offerred by this procedure into the connection
between the classical assumptions and corresponding statement concerning the single particle wave function of TDHF
is not inconsiderable.
In summary then, we have presented four procedures
by which the moment expansion of the TDHF equations may be
truncated to result in a closed set of equations.
The
Thomas-Fermi procedure assumed the nuclear fluid to behave
locally as uniformly translating infinite nuclear matter,
while the classical and isotropic approximations were
classically motivated.
Finally, the irrotational approxi-
mation provides an exact truncation under the assumption
that the velocity fields for each single particle wave
function are coherent.
91
Chapter 6
ENERGY CONSERVATION
In this chapter, we discuss the conservation of
energy as embodied in each of the truncation prescriptions
presented in the previous chapter.
Rankine-Hugoniot
In preparation for the
relations needed in the treatment of
shock solutions of the hydrodynamic equations,
are derived for the energy flux vector.
34
expressions
For simplicity of
treatment, we restrict the discussion to one dimensional
iso-scalar dynamics, although the results are applicable to
more complex situations in a straightforward way.
fields
All
(density, velocity, and pressure tensor) then become
functions of one coordinate, z , only, and are.identical for
both protons and neutrons.
only a non-vanishing
Furthermore, the velocity has
z component, while only the diagonal
elements of the pressure tensor are assumed to be non-zero,
this following from the symmetry of the problem.
As can be seen from Equation (4.38a), for isoscalar
dynamics, the convective velocity field U is identical with
the real velocity field, so that the hydrodynamic Equations
(4.43-4.45) simplify to
P
f+f
(6.1)
92
U///
-(6.3)
-
TT
+ Lt 77w
Th
63
(6.4)
T
YrYY
where the dot and prime indicate time and spatial derivatives
respectively.
p and
V are
the total (neutron plus proton)
density and pressure tensor, while u and B are the components
of the respective vectors in the
direction. The component
zz is abbreviated by Q, while the functions A,B,
and m*
are given by (see Equations (4.4))
A3 o~/
3
3
/6f
'Kt tyr
(6.6a)
93
(6.6b)
'-
1I+2C.
(Note:
(6.6c)
In general, symmetry considerations do not rule out
a vanishing of
<zxx
or azyy
.
However, in anticipation of
the use of the irr6tational truncation procedure, for which
these quantities do vanish, we only retain the component
zz
The total energy of the system, which is conserved
in the TDHF approximation, is given by (cf.
Equations
(4.15-4.16)
J'TrT +3
A
2.
(6.7a)
16
JcLTr lT
+ypA
+
e+
-LC -,T~~3(6b
(6.7c)
Here fd 2 x denotes integration over the two non-participating
spatial directions.
As can be seen from (6.7c), the total
94
energy may be written as the volume integral of an energy
density I(z).
Note that (6.7b) is the explicit reali-
zation of Equation (3.40) in the case of a Skyrme force,
with a correction to effect proper treatment of the threebody force.
As the three-body force is concerned with the
interaction of triplets of particles, the 1/2 in front of
W(R,k)
in Equation (3.40), which avoids the over-counting
of pairs of particles for the two body force, is not
sufficient to avoid over-counting of these triplets.
In order to maintain energy conservation for a
given truncation procedure, the time derivative of the
energy density, I, must be of the form
(6.8)
where q is the energy flux.
If Equation (6.8) holds,
where we have assumed q vanishes at
=
<x>
in order to
neglect surface terms.
In order to evaluate I, it is not most convenient
to directly take the time derivative of Equation (6.7b).
Rather, using (2.26), we may obtain
95
T 4 A+Tf'
E~dkL TrS+(6.10)
(even in the presence of a three-body force!) so that,
comparing with the time derivative of Equation (6.7c),
-0
TrT
+ T&
(6.11)
From the definitions of H and U, we have
(6.12a)
TA
T
+ ('
TW
(6.12b)
so that (6.11) may be written as
-
Tr
(I- pp)/z +
( A +(A
(6.13a)
+p)
12
(6.13b)
Inserting p and ii from Equations (6.1) and (6.2), we have
96
I TTr
( pp)/2
~
(
4
3
u'+t
C
(Pu)'LA+
(lf,>p//
TTk/a
A'-
p
(Tr'1u
(puD'/ tJ3p/
or, upon rearranging terms
/
(6.14)
Equation (6.14) is the expression we shall use in
the subsequent analysis of the various truncation procedures.
A.
The Thomas-Fermi Approximation
We begin with the Thomas-Fermi approximation, for
which
97
TT XY , IT YY = -P-Z*
-
Tr f
(6.15a)
3P
(6.15b)
with, as per Equation (5.1)
V,/
P
T13
53
(6.16)
3oIP
fP
It then follows that
/
7r
P/A
e
Tr- lz
3,p
-S
/1 Z
,l
0)~~~~~~P/P=--j6(U
(6.17)
(6.18)
so that (6.14) becomes
Li-p
~
r j'),3P
,
(PtOI (1-9f) - U
U.
(6.19)
/
+ 3/a
or, upon rearranging terms
K
(Af
(1y3
I/
- -r(p4'
(6.20)
z_6.22)
3ifr
)P+
f~-
98
Equation (6.20) is not of the form (6.8), due to the
presence of the last term.
leading to
An aalysis of the manipulations
(6.20) reveals that this term is due to higher
order terms in the expansion of the sin of derivative
operators appearing in Equation
(3.35), and is concerned
solely with the spatial variation of the effective mass.
Energy conservation in the Thomas-Fermi approximation demands the neglect of this term in the treatment of the
dynamics, a step not completely unjustified.
The essence
of the Thomas-Fermi approximation is that such variations
may be neglected, so that the dropping of this term results
in a consistency of approach.
The energy flux then becomes
OU L- ( S'P/12*÷p./
(6.21)
or, explicitly in terms of the Skyrme parameters as
3
'I..
Thus, energy flows with the local fluid velocity u, with
an effective energy density composed of intrinsic kinetic
(through P), collective kinetic
(pu 2 ) and potential energies.
99
B.
The Isotropic Approximation
In the case of the isotropic truncation pro-
cedure, energy conservation must be explored using equation
giving the time variation of the pressure field, which is
assumed to be an independent dynamical quantity.
In this
approximation, we have
P
while, setting the reduced heat flow
summing equations (6.3) and
(6.23)
to zero, and
(6.5)
Tr TflS u TrW
NF-(Trfrt)
Inserting (6.16) into (6.14) and rearranging, we obtain
for the energy flux
+U
4
"/
(6.25)
100
Equation
(6.25) is similar to that obtained in the Thomas-
Fermi approximation, though with several crucial differences:
First, the pressure P now appears as an in-
dependent variable, not explicitly given in terms of p.
Secondly, additional terms, non-linear in the Skyrme
parameters, and involving high derivatives of the density
and velocity fields appear.
These terms are due to the
presence of an effective mass different from unity
(i.e. 1/m* = 1 if S=0).
of shocks in Chapter 9,
As will be seen in the discussion
such terms certainly affect the
detailed dynamics, but not overall conservation laws.
C.
The Classical and Irrotational Approximations
We now treat the classical and irrotational trunca-
tion prescriptions.
Equations (6.3-6.5) are still utilized
to compute Tr, but the possibility of anisotropy in H and
a non-vanishing
is included.
rI
+
Thus
1
(6.26)
r 7) T (77
/ (2
(6.27)
101
Inserting (6.27) for Tr
and (6.5) for H
into (6.14),
we obtain, after some rearrangement
i-=.[u (LW9 +37.
32!t
21*
y
"-
+
4-
(6.28)
In the classical case,
c~=0,
so that
W=U 3 T-t+271(
(6.29)
which is seen to be similar to (6.25), and identical if
Hzz _
xx _
For the irrotational case,
<.
is given by Equation
(5.19) as -pu" /4, so that (6.28) becomes
(6.30)
+ p LL~jp'-~V)/S
102
In summary, we have shown energy to be conserved
one-dimensional isoscalar flow by the hydrodynamic equations
in the classical, isotropic, and irrotational truncation
approximations,equations (6.25, 6.29, and 6.30) giving q,
so that (6.9) holds in each of these cases.
The Thomas-
Fermi approximation, however, required the neglect of
qnantal terms concerned with the spatial variation of the
effective mass in the Euler equation in order to conserve
energy, equation (6.22) giving the energy flux.
Thus, we
have shown that the proposed truncations of the moment
expansion of the TDHF equations do not destroy energy
conservation.
103
Chapter 7
SOUND IN NUCLEAR MATTER
We now begin a study of the solutions to the
equations derived and discussed in Chapter 4.
Of course,
becuase these equations resemble those of classical hydrodynamics, analytical solutions can be found only in situations
of very simple geometry and under often radical simplifying
assumptions.
Nonetheless, the results that can be obtained
in such situations indicate the implications of the various
truncations proposed in Chapter 5, and may be useful in
answering questions concerning the validity of the TDHF
approximation itself.
In this chapter, we treat the de-
scription of sound waves
(small amplitude oscillations)
in
nuclear matter.
Nuclear matter is a hypothetical system of equal
(infinite) numbers of protons and neutrons in equal spin
populations.
It fills all space with a uniform density of
nucleons, p 0 and a binding energy per nucleon, s 0 , given by
the volume term in the semi-empirical mass formula as about
16 MeV.
Many calculations of nuclear matter properties have
been performed in recent years,35 as this type of system
provides a laboratory for testing both the nucleon-nucleon
potential and many-body techniques.
104
The calculation of nuclear matter with a force of
the Skyrme type is particularly simple.25
In Equation (4.18a)
we take an infinite, uniform, static system with
pn =
p = p/2,
p
,
In
T
p = T/2, so that the energy density is
given by
..
T4_3
2-rT 4_17~
ttyd
(7.1)
The density is simply related to the Fermi momentum, kf , by
(recall that all single particle orbitals are plane waves,
and all spatial orbitals with
kII kf are occupied by 4 nuc-
leons, one in each spin-isospin state)
33
(7.2)
while the trace of the kinetic energy tensor is given by
(cf.
Eqn.
(4.5c)
s
t s ndt
so that as a function of the density only
T
3
+~
fT
1"
Z. +
(7.4)
11b
105
The relevant intensive property, the energy per nucleon,
which is minimized in equilibrium, is given by
(7.5)
A plot of e(p) is shown in Figure 1 for the SKM III parameters
listed in Table I.
The combination of the attractive 6-
force (t ) with the repulsive 3-body force (t3 ) and kinetic
energy with effective mass (t), combine to produce a
minimum in e(p) at the saturation density p0 .
This occurs at
(7.6)
Another physically important property of the curve, as will
be seen later on, is the adiabatic compressibility,
KAD'
directly related to the curvature of e(p) at p = p0 :
106
3,.
(7.7)
8
Note that this is actually 1/9 the compressibility, Kusually
defined in treatments of nuclear matter35
X 26
where we have used Eqn.
(7.8)
C
(7.2) to relate kf to p.
The state of nuclear matter described above, a
Slater determinant of plane waves, with orbitals having
kl
k
filled, is the HF ground state.
What we wish to
investigate now is the stability and oscillations of this
state about the equilibrium point, using Equations
(4.45) in the same spirit as the RPA equations,
A.
(4.43)-
(2.47).
The Thomas-Fermi Approximation
We begin with the simplest form of truncation,
Thomas-Fermi, and assume the following forms for the moments
of the density matrix, following Equation (2.41)
107
(7.9a)
'79b)
(7.9c)
-
Here, all quantities labeled with a 6 are position and time
independent and assumed small so as to be treated only in
first order.
k is an arbitrary vector in the direction k,
and we seek a dispersion relation between w and k.
Thomas-Fermi approximation,
Tr
Jr
from Equation
In the
(S. 2.)
(7.10)
9r-
Inserting forms (7.9) into the hydrodynamic equations
(4.43-4.45) and linearizing in small quantities we obtain
(
A/
-. '+a
c) O
(7.lla)
....
(73.11b)
108
where we have used Equatin (4.8) to identify the effective mass
and have eliminated the quantal correction term in Euler's
equation, consistent with the discussion of the previous
From Equation (4.4) we have
*N A
EA _C
t=r
Ar
?z
+ 3A-V
(7.12a)
p-C
pe
T
+ #,
gT.Tt
-aTrT
+
chapter.
- rTT-/
(7.12b)
(7.12c)
3
bEt
with
+
D2
DAt
-.1
"Now
. -t-
(7.13a)
INO-MMMO
)DT
4
(7.13b)
(7.13c)
/
(7.13d)
/
S7rTr/..0STr'T'
+M-- E)
(7.13e)
(7.13f)
109
-r/
3...:=
6
Xf
X
-t./g
t /.+ 00
(7-.13g)
(7 .13h)
Equations (7.11) may then be rearranged to yield
(7.14a)
H2
+
gT
a
r-MW
s -C T7r
+
+zo
Lot)
IMMMMMW
T
A
Equations
6P
,,
6J1 T
4.
4'A -'N
-rrT
r
Opr.
.r..
(
;14,
(7.14b)
ap DTrTr -f p
A
(7.14) furnish an eigenvalue condition for 6pT,
6JT,, with w being related to the eigenvalue.
Note that (7.14) is actually two sets of equations, one for
each species of charge.
110
By the symmetry of the problem, two solutions of
(7.14) exist with the following properties:
the isoscalar
solution has protons and neutrons oscillating in phase,
so that
/
gPI
S
isoscalar
'C
(7.15a)
while the isovector solution has the two charge types oscillating
7
out of phase
~4 5?j
~J~=
I
isovector
(7.15b)
Inserting the isoscalar condition (7.15a) into Equations (7.14), the eigenvalue condition is immediately obtained as
W 1-"
?
(7. 16a)
CVZ
77
(Off13
3
t3Uv2f2.f
( 1b
bI
ill
where c+ is the isoscalar speed of sound.
Using the condi-
tion (7.6), (7.16b) may be rewritten as
+X _Y b
yt~
(7.17)
Equation (7.17) has several interesting features.
From (7.16a), both +w and -w are solutions, as is expected
from more general considerations of the RPA discussed in
Chapter 2.
In this case, of course, these are solutions
corresponding to waves running in both directions along k.
Secondly, for k<<kf, c
value, K
,16
reduces to the expected classical
though for larger k, (7.17) becomes dispersive.
Lastly, note that for a non-interacting system (all t parameters zero, m*=l), Equations (7.16b) reduces to
132
(7.18)
a well known result for Fermi systems at zero temperature.3
Upon inserting the isovector condition, Equation (7.15b),
into Equations
(7.14) a similar eigenvalue condition
yields the isovector speed of sound, c_, as
112
t3 0, Pt+ z(7.19)
Note that in contrast to c2, C1
is non-linear in the inter-
action parameters, as is to be expected of a general solution
Equations (2.47).
c2 = CZ = k
Furthermore,
for a non-interacting system,
/ 3 , again as expected.
With the set of Skyrme III parameters listed in Table
I, Equations
(7.17) and
(7.19) may be evaluated numerical-
ly to yield the dispersion curves shown in Figures 2 and 3.
Note that the iso-scalar mode propagates at all wave numbers,
with a long wavelength speed smaller than that of the isovector.
The isovector mode cuts off at k ~ 4.4 kf and is much
less sensitive to variations in k.
B.
The Isotropic Approximation
As discussed in Chapter 5, the isotropic truncation,
assumes that
77c
(7.20)
113
Taking the trace of Equatimc(4.45)
in order to obtain an
equation for TrH ,we obtain, with a similar travelling
wave space-time variation
-- Tr 7T
Ur A TrTo_- :& D
2.
8
8.
(7.21)
while a variation of Equation (2.44) yields
3P
3%
.(7.22)
or, using Equations (7.12)
2:: .!
r
trt
4..4
aZi
trT.
c
z
(7.23)
2ftrTc 3P5-J
Equations (7.lla,7.21, 7.23) form a eigenvalue system similar
to Equations (7.14).
Upon inserting the iso-scalar condition,
Equation (7.15a), we obtain
C
/D
AD
g
_---+
(7.24)
(7
114
while the isovector condition results in
(7.25)
.1464
Note that Equations
(7.24-7.25) are identical with the
Thomas-Fermi results, except for the dispersive terms.
numerical evaluation of Btuiation (7.24)
and (7.25)
A
with the
SKM. III parameters results in the dispersion curves shown in
Figures 2 and 3.
C.
The Classical Approximation
We continue in the hierarchy of increasingly more
sophisticated truncations, now assuming the classical
truncation, i.e. Q= 0.
For such a case, the variation of
the pressure H zz differs from that of H xx and H
y, the
former being parallel to the direction of propagation, the
latter equal and two transverse to this direction.
A linear-
ization of the pressure equation in this approximation yields
TT.
6
XX
-
.
u
g fYY
I(7.26a)
Tr '
12
(7.26b)
115
while the momentum equation becomes
ru
.
t
.
rz
'P
T.
(7.27)
Tr T
.
-.
3&
-
S
where, in evaluating 6AT from Equation (7.12a) we must use
9\ r 71,.w
-f-
_._ PO A 3 Bt /0
Tr T.
as is found from Equations
(7.26).
tinuity equations, Equations
(7.28)
Together with the con-
(7.26-7.28) form an eigenvalue
condition for the frequency w.
For the isoscalar case, the
speed of sound is readily found to be
5-. "'.
A z "W"ft
z
-A
+
o+
3
- 3 (4g)
1
which, using (7.6-7.7), may be rewritten as
(7.29)
116
-1
-}P-
-
?%t
(7.30)
Similarly, for the isovector case, we obtain
z
z A" .1 k
(7.31)
A comparison of
(7.29-7.30)
with Bauations(7.17-7.19
and 7.24-7.25) which have been derived under less sophisticated truncation procedures indicates that the general
form of the dispersion relation remains, although the precise
value of the numerical coefficients changes.
for a non-interacting system,
In particular,
(7.29) gives for the isoscalar
mode
(7.32)
which is greater than the thermodynamic (Thomas-Fermi) value
given by a factor of 3/Y5-.
This difference is due to the
underlying difference in the assumptions of dynamics implicit in the two truncation methods.
The Thomas-Fermi
method assumes the pressure tensor isotropic at all times,
117
a condition expected to hold in a classical gas when intermolecular collisions occur at a rapid enough rate,(i.e. far
greater than the frequency) so as to keep the local isotropy
of the distribution function, and hence pressure tensor.
In
the TDHF theory, such collisions are absent, so that the
more sophisticated approximation is called for.
Indeed, for
a density disturbance dependent only upon distance along a
wave vector k, it is seen from Equations (4.3-4.4) that the
corresponding changes in the self-consistent (Hartree-Fock)
potential W are also functions of only this distance.
As a
result, the single particle wave functions, which are initially plane waves in the directions transverse to k remain
so at all times.
However, the wavefunctions in the direction
parallel to k*, also initially plane waves, must change in
time due to the changes in W.
Hence the tensor N is expected
to be invariant under rotations mixing directions transverse
to k, but anisotropic in the sense that k-H-k is different
from components of H transverse to k.
This statement is
also true for forces more realistic than those of the Skyrme
form, and is a special case of a more general property that
the TDHF equations possess, i.e. - that a Slater determinant
having a symmetry conserved by the many-body hamiltonian
(such as parity or isospin invariance) retains that symmetry
when evolved with the TDHF equations.
The symmetry in the
case of sound waves is that of rotations about k.
The dis-
tinction between the isotropy of the pressure tensor in the
118
Thomas-Fermi theory, and the anisotropy built into the more
exact approach (i.e. close to TDHF) is analogous to the distinction between zero and first sound found in liquid 3He
which has been discussed by many authors.3
The former is a
density wave dependent upon the many randomizing intermolecular collisions for its propagation, while the latter is an
oscillation sustained through the self-consistent potential,
and hence much closer to the solutions expected to arise out
of a TDHF treatment.
The numerical evaluation of (7.30) and (7.31) in
the case of the SKM III force is straightforward,
in the curves shown in Figures 2 and 3.
and results
As before,
(7.30)
and (7.31) are non-linear in the Skyrme parameters, as is
expected of the RPA solutions, but not of the more naive
semi-classical arguments. 36
D.
The Irrotational Approximation
As a final calculation in this chapter, we consider
the irrotational truncation defined by Equations (5.19)
By reference to Equation (4.45), this is seen to add a
factor of
(7.33)
tOnd
to the right side of Bguation (7. 26b)
and (7. 28) .
The disper-
119
sion relations are then simply found as before to be
+-
(7.34a)
and
21
Z.
34 b)
Pat+(0.
Thus, the only effect of the irrotational truncation
upon, the dispersion relation obtained in the classical approximation
is to modify the k 2 term.
Equations
(7.34) computed with the Skyrme III parameters are shown in
Figures 2 and 3.
Equations (7.34) admit to a physical interpretation
in the limits of very large and very small k,
frequencies).
For k small, Equations
(high and low
(7.34) describe a
zero-sound type of mode with constant speed,as have all the
other truncated versions of the linearized TDHF equations we
have been considering.
However, in the limit of very large
k, this isoscalar mode approaches
(7.35)
120
i.e. the frequency goes quadratically as the wave number.
This is expected to be a property of the exact RPA modes of
the nuclear matter system for the following reason.
first a non-interacting system.
Consider
In such a case, the normal
modes (excited states) of the system with wave number k may
be formed by promoting a particle from a plane wave of
momentum q
Ik"+q 4i>kf.
(IqI: +kf) to the plane wave orbital kfq, provided
In such a case, the excitation energy (frequency)
is given by the difference between the single particle energies
involved as
(7.36)
Thus, in the limit of very short wavelength excitations,
the normal modes of a non-interacting system have a singleparticle spectrum, i.e.
for a free particle.
(7.36) is just the energy expected
It is gratifying to see that (7.35)
also approaches this limit for the non-interacting system,
i.e. the "graininess" or particulate nature of the nuclear
fluid, becomes apparent at short wavelengths.
In the case of an interacting system, the failure
of
k2
(7.35) to approach the expected 2m*
,
in analogy to (7.36)
may be traced to the fundamental form of the Skyrme force.
As seen from (4.1) , at larger momenta, particles interact
121
more strongly with one another, so that there is no reason to
expect decoupling of the single particle modes as the wave
number increases.
For all physically plausible potentials
of finite range, the force falls off with increasing
momentum, so that an approach to (7.36) would be expected
with a more realistic interaction of this type.
In summary, this chapter has considered small
amplitude perturbations of infinite nuclear matter about
the Hartree-Fock ground state.
Four truncation procedures
have been considered (Thomas-Fermi, Isotropic, Classical,
and Irrotational).
All lead to a dispersion relation of the
form
)
W2 = k 2 (a, + alk 2
where a, and a 2 are constants non-linear in the Skyrme
parameters and dependent upon both the truncation procedure
and the mode of oscillation.
The Thomas-Fermi truncation
led to the classical expression relating the low frequency
isoscalar modes to the adiabatic compressibility.
The iso,
tropic truncation retained this property, but modified a 2
the dispersive term.
The classical truncation, which retains
an anisotropy in the pressure tensor, was argued to be closer
in form and spirit to the exact RPA solutions, and was
shown to lead to a substantially different expression and
interpretation for the isoscalar modes.
Finally, the ir-
rotational truncation was shown to lead to the correct
122
collective behavior at small wavenumber and single particle
behavior at high wavenumbers in the case of a non-interacting nuclear matter, but failed to do so in the case of the
Skyrme force due to the unphysical behavior of this force
at high momentum.
123
Chapter 8
SIMPLE MODES OF A FINITE NUCLEUS
In this chapter, we demonstrate the application of
the linearized version of the hydrodynamic equations to the
small amplitude motion of a finite nucleus.
Irrotational
and incompressible flow patterns are assumed, so that the
irrotational truncation procedure of Chapter 5
is utilized.
Formulas for the isoscalar quadrupole and
giant dipole oscillations of a spherical N=Z nucleus are
derived in terms of the ground state properties of the
system.
Numerical results are presented for 160, using harmon-
ic oscillator single-particle wave functions.
We begin by assuming knowledge of the HF solution
for an N=Z nucleus possessing spherical symmetry.
For such
a system (in the absence of Coulomb forces), the density and
pressure tensor a functions of the radius only
a
r)(8.1)
In general, radial symmetry implies that II
possesses only
the property that it must be invariant under rotations about
124
the radial direction, that is, that the two directions
perpendicular to the radius vector at any point
equivalent.
are
However, as is found to be very nearly the case
in actual Hartree-Fock calculations,24 and as will be seen
to be true in the harmonic oscillator model of 160 discussed
below, we assume H
to be isotropic
0(8.3)
where
1
is the unit tensor, given by
ij-
6
and P0 a function of
ij
Irl
only.
To treat the isoscalar quadrupole oscillations,
we assume that the protons and neutrons move together in an
irrotational, incompressible flow pattern given by
-6 )
(
-(Ae,(-The velocity potential, $,
=;
.64
V
%
(8.4)
is given by
(8.5)
and a is a small time dependent quantity, independent of
position, in which the hydrodynamic equations are to be
linearized.
While the validity of such an assumption may be
125
questioned, for it is known that nuclear inertias are not
given by their irrotational values,37 it has been shown
that irrotational flow provides an upper estimate of the
true RPA frequency. 1 3 ,38
Note that the velocity field
v(r) is given by
C?*) VO V
y)(8.6)
and is irrotational
(8.7)
and incompressible
(8.8)
implying
- Q
(8.9)
From Equation (4.42) for the convective velocity
field u (r), we have
-
U(8.10)
Following the form of the assumed wave function in the RPA
equations (2.47), we put
O (
/2
(8.11)
126
(6p is independent of T, as protons and neutrons move together) and linearize the continuity equation to obtain
-
.
0" 4--
(8.12)
which is satisfied by
n
)
A linearization of the momentum equation
a&r+
(8.13)
j
V. Vs_+
(4.44), results in
. v;A.,+
VA
(8.14)
4~Trh
77St V(t
rf
N
where, since protons and neutrons move together, the subscript
T
is superfluous, and will henceforth be dropped.
Taking the
inner product of (8.14) with the velocity field and integrating over all space, we obtain, after several integrations
by parts
(8.15)
2/Y (mTr A)g7
~r7C-)
rrTF7
Moo
127
where the tensor A is defined as
41
0
--
01(16
(8.16)
and the quadrupole mass, M2, is defined as
t~8 4 Trfl(8.17)
- A<rl>
The mean-square radius is defined as
fr, rL p (r)
and A
=
(8.18)
N+Z=2N is the number of nucleons.
From the definitions of A
p
(4
Gu
so that (8.15) becomes
and C
T
2i:
,
we have
p
(8.19)
(8.20)
128
-
dr
Tr SV -1
4t
+A /aP 1)
)
- f
Vg-2
)
'4if
--- - WcV-"O V ) Vp'f-'
--- 1
+
Maa
t
-a
. V )Tr
(8.21)
oet
or, after rearrangement,
Maz
)
4-
-. J
(8.22)
-
rvp~ir
SV
From (8.13) we have
V 2.( Iro Vro )
ubm
)
f OtIr
dw=W
Cal
3
100
while, using (8.6)
(V6rp
V1L(L
7) 4
.23)
(8
129
)Y p
(
'
(8.25)
where the prime denotes differentiation with respect to
rI, so that (8.23) becomes
(8.26)
Z
Vrr)
AL2
(8.22) becomes
d
+
-
r
+rpt
(8.27)
)
Using the fact that trA=O,
We now turn to evaluating the variation of the
pressure tensor.
As all second derivatives of v
tion (5,,i9), the irrotational truncation) demands
tion (4.45).
C> .
are zero,Equa=0 in Equa-
Furthermore, it is clear that 6H is linear in
Indeed, linearizing Equation (4.45) gives
IMMvP0 )I
SIT
4 Puw~
+
130
(8.28)
vz
VC))
( 22
-&
lv
(
40.4&
A
so that
. P,,,
4J
A
0
/
taw
as 4M
(8.29)
~1i%
Pv.
)
-.
(vp'+
From (4.4c)
(8.30)
so that
^2
mnow
fr
r.. T
r
41
%mom
t r ' -A
Cer P
q
(1
0
f
L&)( Vol/
qlp
)
.
(7A%(.8.31)
131
where we have integrated by parts (one or twice) all terms
in the square brackets, and have used
= 424
M
.0"
(8.32)
Simplifying (8.31), we obtain
~
rx1 mofm'own4
f
062
Gal
pv
(.8.33)
We now treat more explicitly the brackets in Equation (8.33).
The second integral is
p,
ak
1%
mom
rawY
ay )
roj
(
X
OMMOM
ZT
Jtr3r
(8.34)
132
But
(8.35)
so that (8.34) becomes
(8.36)
where we have defined the cosine of the polar angle
(8.37)
The angular integral is easily done, to give 167r/5, so that
(8.36) becomes
(.8.38)
or integrating by parts, we obtain
CO
CO
r)j
a
0
o
(8.39)
133
Similarly, the first integral may be done to give
(8.40)
f
Combining (8.40) with (8.34), (8.33) becomes
(8.41)
+V
Inserting (8.41) into (8.27), we obtain
co
(8.42)
Equation (8.42) describes the equation of motion for a simple
harmonic oscillator with coordinate a, and frequency of
quadrupole oscillation, W 2 , given by
<L 3 f
&0+rpip) O d'
r)p4(8.43)
0
0
0
134
The quadrupole frequency has thus been found in terms of the
properties of the ground state HF solution.
The terms in
(8.43) may naturally be associated with the pressure and
effective mass, ground state correlations, and a surface
term.
Before explicitly evaluating (8.43) for 160, we
treat the case of iso-vector dipole (giant dipole)
lations.
oscil-
In this mode, the protons are imagined to oscil-
late ff radians
out of phase with the neutrons, with a flow
pattern given by a dipole velocity field.
Thus, for spheri-
cal N=Z nucleus,
(r
t
4,-e
eVc ~(8.44)
where the velocity potential $ is given by
I Tr
(8.45)
and the velocity field is
(8.46)
The convective velocity field for protons is given by
(4.42)
as
(8 .47)
135
or using (4.46)
(8.48)
( +t".-'
Using the fact that v is incompressible and irrotational,
we have, from Equation (8.48)
V
VP (LAt
(8.49)
(8.50)
-
As in the quadrupole case, we now proceed to linearize the hydrodynamic equations.
The continuity equation
reads, for protons
~
(8.51)
or, when linearized, with (8.48) gives
((8.53)
The momentum equation may also be linearized to
give for protons
136
:P. V + v2
6A
AV
+ Le
2
7?
p
7AO
(8.54)
+/a Tr Th
Using
c,+(Tr
7T)V =po
(8.52) and the fact that
(8.55)
~77
we have, from the definitions of A
6
and Cp,
P
~(8.56)
(8.57)
Dotting the velocity field into (8.54), integrating over
all space, and integrating by parts, we obtain
M
[6-tAp
,Py -V4,(.5
)Ap
f
ONE y
CQ3Vsr7>/
(8.58)
( P)TrsY"7T
o
137
or, using (8.56)
;14 J43r
- tt
~v
+0x-VZ.)SP
Tr9Y Iy,
2.
O1-V
P
"9
Tr lT
)
M,
-Ulm= Vylvpo)
SP -. .6 - Pp
m
Simplifying, we obtain
j
Mr. VP.)
+
POA
)
(
+ fsir
(8.59)
,
C'V)/ 'P.
4.
Tr Si P(-p>
'P
O")
0
.
where
tie
fd3r 5t-
From (4.45), we have, upon linearizing
(8.60)
138
r 1,
2.
N
L
--ftC P.
4.~
( -"V)*r-Te
frTC
V-,Vcp)
g
4f
VA
(~)
a
+
4%WA
Vd, (f
0
v.)
(8.61)
72-(p
)
t
, *Vr
I
or
2..
9%
!..P
CO, vp ) ,p V
0
)
(
-oktr ) P
Using Equations (8.47-8.50),
-6r 8"
P@Y
+ Cl r1
.8.62)
v)(AV
)
(i00P
-6
(8.62) becomes
CA (j+-tPWZ)CiA-eo) -f
a t jv(- 11
(8.63)
( ID)
+.WON*
,
,1~
(itv)(fV)
(4
Inserting
(8.63) into (8.60) and using (8.53) we obtain
139
) 4tj eO/Z)
M 0
5
It
T
N
VP
+5a+
are
(8.64)
I
ScL'* 9P,
I
i-m/4 p V done ogP)
The angular integrals are simply done to give
ft
( (+ me+)) P"f
r
(v~a)7~
$t3+ Po p
1um
Equation
P.
Ljt C1 .
+m
/*,4)
OD
I'
r1k)v
o( 43t4p
(8.65)
)
410%,2t
OvY7
(8.65) describes the harmonic motion of the ampli-
tude a(t).
=
M Lid1
(Ot C I +>V 1 )
+tPO
)
The frequency of oscillation is given by
0
'~A PO'Ig '(qumim3t
TA
14T+P)POI
%
r
Muma
) I -t
it
I P. r 7- 4
)rI
R' (r/
(8.66)
140
The frequency of the dipole mode has thus been
reduced to an evaluation of the ground state properties of
the nucleus.
Note that W 2 is non-linear in the Skyrme
parameters, as is expected of a general solution to the
RPA equations.
We now turn to an evaluation of Equations (8.43) and
(8.66) in the case of 160.
As is known to be the case from
actual HF calculations,25 the single particle wave functions
in the interior of this closed shell nucleus are accurately
described by eigenfunctions of an isotropic harmonic oscillator, so that the single particle HF hainiltonian may
be taken to be
. -++r
(8.67)
2.
160 has 4 nucleons (2 protons and 2 neutrons)
in
the OS1/2 orbital, 4 in the OP
orbital, and 8 (4 protons
and 4 neutrons) in the OP 3 / 2 orbital.
The normalized
spatial wave-functions are given by 3 9
(1)
-
rz
(8.68a)
141
with the square of the oscillator length, y, given by
It
with m the nucleon mass.
/(8.69)
Note that there are, of course,
3 spatial wave functions for the p-shell (each containing
4 nucleons), denoted by P ,P
,Pz.
From the wave functions
(8.68), the ground state density may be constructed
)~
~
+I (/p(PIi 1Gpjn I~(~I
(I*%
~~~o
3/Trgr
1~
(8.70)
as may be the pressure tensor, HQ
tropic
which is found to be iso-
(consistent with the assumption at the beginning of
this chapter)
and given by
()
&
(8.71a)
We will choose the oscillator length, y, so as
to minimize the energy of
16
0. 21
The resultant
142
ground state must then be very nearly the HF solution.
From (8.70), the mean-square radius of 160 is given by
rl->
6
(8.72)
(8.73)
A minimization of the energy gives a value of y of
=~Oe332 S->
From Equations
(8.74)
(8.70-8.71), the integrals required
for the evaluation of W 2 from Equation (8.43) may now be
calculated
0D
(8.75a)
fdL,,gPoo
0
51 vNET
(V7
(8.75b)
143
(8.75c)
(8.75d)
so that (8.43) becomes
zs,z
-d
3*6
---f- 2-6
Tr
"-Now"
-v-Z
Zr
I.Ir ( A)[
VL-T
(8.76)
-M
+ to T,
z Vrz- 7
T
VS"
where we have used, from (8.17) and (8.70)
(8.77)
ML
Equation (8.76) may be simplified to yield
a
?.J[.f
77T,
Evaluating
(8.78) with the SKM III parameters listed in
(8.78)
144
Table I and using
Equation (8.74), for y, we have
(8.79)
An equation similar to (8.78) has been obtained
previously under the assumption that the quadrupole oscillations of the nucleus were due to a scale transformation
40
of the wave functions.
However, the term non-linear in
the Skyrme parameters (t2 ) was not obtained.
This term,
however, is a small shift in the frequency, and omitting it
yields Equaticr (8-79)
to 3 places.
The final result is
in
excellent agreement with generator coordinate calculations
and the experimental value.4 1
We turn now to the evaluation of the giant dipole
oscillations.
The integrals required for the evaluation of
(8.66) are
(IPpA
(8.80a)
(8.80b)
'PfJ
ap
04rJb
~~/
7g'03
(8.80c)
145
a*~
(UI0S
2 v
'?~' r~cb~
(8.80d)
,2"2 Tr
CO
(8.80e)
w
91?
(8.80f)
4S Tr
(7r(,)74
a vv
00
-wZ
s-z8r
,
(V-POL)v
( 8 . 8 0g)
(8.80h)
0l
~
so that (8.66) becomes
1331Tr
(4)75
(8.80i)
146
T
L'r.'
(T
4
(1
)
S~4)]
1,bVI
An evaluation of (8.81) with the SKM III parameters
of Table I yields
I4
2t
AeJ
(8.82)
This value is considerably larger than the accepted
.
experimental value of 23 MeV7
Equations (8.79) and (8.82) provide estimates for
the isoscalar quadrupole and giant dipole frequencies of
160.
They were derived under specific assumptions for the
nuclear velocity fields, and the extent to which they are a
valid approximation to the exact RPA solutions depends upon
the validity of those assumptions.
Of course, it is also
possible to directly seek solutions to the eigenvalue problem
obtained by linearizing Equations
(4.43-4.45) about the HF
ground state, thus obtaining a more precise solution to the
truncated problem.
It is encouraging to note that the giant
147
dipole frequency lies above the experimental result for, as
discussed above, the value (8.82) must be considered an upper
bound.
148
Cha pter 9
ISO-SCALAR SHOCKS IN NUCLEAR MATTER
The preceeding chapters have dealt with the solutions of the truncated TDHF equations in cases in which the
equations were linearized about a static HF solution.
However, for situations of interest in heavy ion reactions
and fission, large amplitude dynamics must be dealt with,
requiring the retention of the full non-linearity of the
equations.
This chapter presents an example of such a
treatment.
We consider the case of one-dimensional iso-
scalar hydrodynamics treated in Chapter 6, and so discuss
reactions between slabs of nuclear matter.
Such reactions
involve the formation of shock waves, whose kinematics we
treat analytically in each of the four truncation approximations discussed in Chapter 5.
While one-dimensional
hydrodynamics may seem somewhat removed from situations
actually encountered in heavy-ion physics, such a system
actually bears upon two questions of interest.
First, it
might be expected to approximate the dynamics in head-on
collisions of very large nuclei, indicating the maximum
densities attainable in such collisions, and so having
direct bearing upon the possibility of the formation of
abnormal nuclear matter. 42
Second, the TDHF equations have
149
recently been solved for Skyrme type forces in the case of
colliding slabs.12
A comparison of such solutions with the
hydrodynamic calculations presented here will demonstrate
the range of validity of the various truncation procedures
proposed and possibly indicate new ones.
The type of initial conditions we shall consider
are shown in Figure 4a.
Two very thick slabs of nuclear
matter, with the characteristic diffuse surface, initially
move toward one another with relative velocity 2U0, the left
hand side at velocity +U0, the right at -U0 . As the slabs
come into contact,
center.
matter begins to pile up in the
After a long time, the situation will be similar to
that shown in Figure 4b.
reached a density pc
The central region, will have
rpo
dependent upon U0 , and the fluid
in this region will then have zero velocity.
Larger values
of U0 will, of course, produce larger values of r.
Some
distance to the right (and left) of this central region will
be a transition region, in which the density falls from p
to P , and the velocity changes from U=O to U=-U.
0
0
velocity changes to +U
(The
at the left hand shock, of course.)
The exact density profile in this region, which translates
outward from the origin at shock speed s', depends, as will
be seen below, upon the nature of the two-body force.
classical calculation, such a profile is related to the
viscosity.34
Adjacent to the transition region is an
In a
150
asymptotic region, in which the fluid moves leftward, as
yet unperturbed.
The shock speed s' need not be equal to
the fluid speed U0 , but is instead related
to U0 and pc
by the need to conserve matter as it streams into the
central region.
In the subsequent treatment, it will be convenient
to work in a different frame of motion, i.e. that moving
with velocity U0 to the left.
The central region now moves
with velocity U0 to the right, while the shock speed
s=s'-U . The matter in the rightward asymptotic region now
appears to be at rest, as is shown in Figure 4c.
We begin our discussion of the kinematics with
the equations governing the motion of the density and
velocity fields.
From Equations (6.1 ) and (6.2)
we have
(9.1)
h
(9.2)
where the notation has been introduced in Chapter 6.
the special case being considered
For
151
=
yf
A
ta
(9.3)
-
Z
(9.5)
In addition, as discussed in Chapter 6, we have
7
(9.6a)
70
7(9.6b)
(9.6c)
so that (9.2) may be rearranged to give
(9.7)
Y
Noting that the left side is a perfect differential, we can
see that (9.7) simply expresses the conservation of momentum,
since, if the density and H vanish at z =
-o
(to allow
dropping surface terms), an integration of (9.7) over all
space gives
152
(9.8)
In order to proceed further, it is simplest to
In particular,
assume a specific form for the solution.
if a steady state has been reached, in the frame moving
leftward at speed u0 , the shock profile, velocity, and
pressure fields will be functions of a single variable
w = z-st, i.e. the shock translates rightward at speed s.
Hence, we have the transcriptions
(g)
(9.9a)
yaw
(9.9b)
so that Equations (9.1) and (9.7) may be readily integrated
to yield
+
41L
7T*
7T + My/
%p
?U
(9.10)
C
-t
pt
CLo
(9.11)
Here, CCp and Cu are integration constants, and as all fields
153
are now functions of w, primes denote differentiation with
respect to this variable.
To evaluate CP, we consider
W-+ +M,
.
where, in the frame we are working in, U=0,
p=p0
Thus, from (9.10)
so that
OA
~
(9.12)
Equation (9.12) gives the relation between s and u due to
mass conservation mentioned at the beginning of the chapter.
Cu may also be evaluated by considering w-++o.
Since in this
region we have unperturbed nuclear matter, the pressure
tensor is given by (cf.
1
Equation (7.3))
-I
(9.13)
so that for large w, where the derivatives of all fields
vanish, Equation (9.11) gives
which, by comparing with Equation (7.7) is found to be
154
For the case of colliding slabs of nuclear matter,
C u =0.
Substituting (9.12) into (9.11) and rearranging
gives
(9.15)
8
r2.
With a given truncation procedure, hence a specification of
fI,
(9.15) yields two pieces of information.
W+-o,
Evaluated at
(actually a value of w corresponding to the central
region) where p=pc, U=U , and all derivatives are zero, we
may solve for s as a function of pc and hence use (9.12)
to find UO as a function of pc.
known,
In addition, once s is
(9.15) may be used to furnish a differential equation
determining the shock profile p as a function of w.
We
first proceed through the four truncation procedures discussed in previous chapters, evaluating the shock parameters
not only for nuclear matter with the Skyrme force, but also
for the case of a non-interacting system.
A.
The Thomas-Fermi Approximation
We begin with the Thomas-Fermi approximation, for
which
155
so that
(9.15) becomes
(9.16)
where we have identified pz-s/Dp for nuclear matter from
Equation (7.7), and have dropped the last term involving
derivatives, consistent with the discussion of energy conservation in Chapter 6.
For the central region,
(9.16)
gives
4-
PrP,
(9.17)
PC
Note that for weak shocks, p c
-MAW"j)
P0 , and (9.17) becomes
Ab(9.18)
i.e. the shock speed approaches the long wavelength sound
speed computed in the same approximation (cf.
(7.17).
Equation
156
For the non-interacting system
so that (9.17) becomes
rr/
and from (9.12)
Figures 5 and 6 shows s and u as functions of r for the noninteracting system in the Thomas-Fermi approximation.
B.
The Isotropic Approximation
In the case of the isotropic truncation procedure,
we have
(9.18)
Tr3P
157
so that (9.15) becomes
f2f-
P
/>
fi)+
+2 *y
0
3((9.19)
-
&
+ P/S ( p')
+
(
--
In order to evaluate P, it is most convenient to use the
equation for energy conservation, Equation (6.Z)
S yAP +44
P/y)
(9.20)
D3
where
.
YaTTTVOjp)+tpUt
(9.21)
When (9.2U) is integrated with respect to w, we have
42
@/&
(,
L
+
U
P/
)+..
>A
+
Cf
(9.22)
158
C , the constant of integration,imay easily be evaluated by
taking w-*+
resulting in
-t
= ot
Inserting the explicit form of A,
(9.23)
(9.5), into (9.22), and
using (9.12) for U, we may solve for P, to obtain
LO
LID)
(p)+
D,JPa+ p3(ppo ',
4,
'
9
. 2 4)
with
(9.25a)
+tZP3 -3-01
101P)
/(0 (Z
(9.25b)
)
- I fO>lp
5,
1), fF)
A
(I -
ZP)z
4t?) Ch-
With
It
(9 .24 ), Equat ion
(9.25c)
(9.25d)
(9.25e)
)
P
I
tP P
PC
(
D3jy?)
1) LP~)
SC
f -Pr/I - Pur ))
(9.19) may be rearranged to give
159
,
3
9
(_
-E
+ ; WBr(I
+ 4Do (r)
Q - 54 f )
(9.26)
)
+g3
-0-IrPIP)
a
ic (P) I
+
P
In order to compute the shock speed, we evaluate (9.26) in
the central region to obtain
D0,9)
(9.27)
In contrast to the Thomas-Fermi approximation, the
variation of the pressure is independent of that of p, so
that it is useful to define a "heat" energy per nucleon,
6,
160
given by
43
3u(9.28)
i.e.
we have subtracted the "cold" energy from the total
energy.
As in the Thomas-Fermi case, it is helpful to
consider the non-interacting system.
From (9.25), we have
-c
(9.29a)
(9.25b)
((9.29c)
(6
0
(9.29d)
so that (9.27) becomes
4 $,.
For weak shocks, r+l and s+kf/vI
approximation.
(9.30)
4- r
,
as in the Thomas-Fermi
Thus, for weak shocks, at least, the iso-
tropic and Thomas-Fermi approximations agree.
interesting that at pc =
4p
0
It is also
, the shock speed becomes infin-
ite, as does the fluid velocity, given by Equation (9.12) as
161
-r
(9.31)
j
As is a well-known result in classical physics43 for the
adiabatic shock of a perfect gas, no matter how violent the
collision, the maximum density attainable is four times the
initial density.
(9.28),
Equations
The heating is also calculable from
(9.24), and (9.30),
and is given by
L-/3
(9.32)
This is positive definite for r1l and approaches w as r+4.
Figures 5-7 show s,u, and G as functions of r for the noninteracting system with the isotropic truncation.
C.
The Classical Approximation
We now turn to the classical truncation procedure.
Because this differs from the
isotropic one only in the
treatment of the pressure tensor, we expect D 1 and D
Equation (9.24)
of
(which now becomes an equation for H zz)
to be different, but D 2 'D3 'D 4 remain as given by Equations
(9.25c-e).
results in
A treatment similar to the isotropic case
162
IF
=(9.33)
+-D~ (p/]
with
I
AfF/2.
Qf)
D+
fj)
tT
(9.34a)
(.
/
(9.34b)
The transverse component of the pressure tensor, Hl" must
be evaluated from the equation governing its dynamics,
Equation (6. 3 ) ,which may be integrated in the present case
to yield
PO
_(9.35)
or, using (9.12)
(9.36)
Thus, in thisapproximation, the transverse pressure is tied
to the density, flowing along with it.
given by (9.27) with D
The shock speed is
and D1 replaced by D' and D
given by an equation similar to (9.27), and the heat energy
is replaced by
163
=JT4 +-TA
a.q-~
-oL
(_is 5f/
(9.37)
zP-'o
In the case of the non-interacting system, we have
(9 .38a)
(9.38b)
//
D? (PL)zf" -gPO/(r)2
(9.38c)
so that (9.27) gives
+ r
,-~-
(9.39)
2.+_
and so
(r
(A ,
7T~
1)
'h
2r-J
2~r
and
Vs-T_?U - 0"0'"1
(%C)
(9.40)
(9.41)
164
..oft0(
()
i
--al3
rr
(9.42)
Again, as can be seen from (9.39), as r-l, s approaches the
speed of sound,
/3/5
kf, computed in this approximation.
In
contrast to the isotropic case, however, the maximum density
attainable is now only twice the initial density.
D.
The Irrotational Approximation
The irrotational truncation is essentially the same
as the classical, though the coefficients D3 and D4 will
change due to the assumption that
As these coefficients are not necessary for the computation
of s and u as functions of r, the irrotational truncation
results in shock parameters identical to those of the
classical truncation, though due to a modification of the
derivative terms, the shock profile is expectedto be different.
As can be seen from Figures 5-6, for the noninteracting system, the various truncations of the TDHF
equations result in very different shock kinematics.
The
isotropic and Thomas-Fermi truncations give similar parameters for weak shocks, but for strong shocks, rt2, the
165
isotropic truncation pumps energy into heating the gas
than compressing it further.
Thus, for a given initial
velocity, the isotropic truncation has a lower central
density.
The classical truncation "heats" much more
rapidly than either of the others, and so attains a far
lower maximum density.
For the interacting system, the shock parameters
computed with the SKM II
force are shown in Figures 7-9.
The basic trends evident in the non-interacting system
remain, though there are, of course, quantitative differences.
The maximum density attainable in the isotropic
truncation is 1.92p
value is 1.59p
.
, while the classical or irrotational
Note also that at p
= 1.51p
:in the iso-
tropic and at pc = 1.41 in the classical and irrotational
(corresponding to bombarding energies of 10.4 and 14.4 MeV/
nucleon respectively in the center of mass system)
the
nuclei in the central region become unbound.
The classical and irrotational discussions given
above for the interacting system contain somewhat more
information than might be obvious at first glance, for the
following reason.
Note that as far as the shock kinematics
are concerned, the specification of the reduced heat flow
tensor, Q-, was irrelevant.
It might be argued that, as
discussed in Chapter 6, by allowing only
zzz to be non-
166
zxx and
zero, we have neglected the influence of
ing the results dependent upon the truncation.
zyy, mak-
However,
the inclusion of these moments does not affect the diszxx
zy
cussion above for, by symmetry, Q" and :)y
must vanish
in the central region, while they are certainly zero in the
asymptotic region.
As the relations above were derived by
a comparison between these two regions, they are exact.
Thus, provided the system being considered is large enough
to exhibit shock-like behavior, Equations(9.27, 9.34) are
exact expressions for the shock speed and central density
in terms of the initial velocity.
With the SKM III force,
will never give a central density larger than 1.59p
.
then, a 1-dimensional TDHF calculation of isoscalar nuclei
As a final topic in this chapter, we consider shock
profiles in the Thomas-Fermi approximation.
From Equation
(9.16) we have, with Cu=0
0
PF
p(9.44)
where s 2 is to be computed in terms of pc from Equation
(9.17).
Using the identity
w "000 a(9.45)
where p'
is
ce
is considered a function of p,
(9.44) may be
167
integrated on p to give
(9.46)
2.+-
where X is an integration constant.
To determine the
integration constant, consistent with the discussions in
the previous chapters we require the right hand side of
(9.46) to be zero (at least a single zero) at both p=p 0
and p=p C.
However, with a single parameter available, .,
this is impossible.
We may force (9.46) to vanish only at
p0 or pc, but not both.
(9.44), at both p=p
In addition, from (9.17) and
and p=pc we have
Hence, if p' = 0 at either of these points, the right hand
side of (9.46) will have a double root at the same point and
the density will approach that value as an exponential in w.
In summary then, we are forced to choose between a shock
profile exponentially approaching p0 as
, with a dis-
ing pc as w+-0- with a discontinuity in derivative at p=p
.
continuity in derivative at p=p c, or exponentially approach-
Neither choice is necessarily physical, and the failure to
obtain solutions continuous in both value and first derivative
may signal the fact that a stable shock profile does not
exist.44
Nonetheless, it is still interesting to examine
168
the form of the solutions.
To this end, we choose the
solution exponentially approaching p
so that A in (9.46)
is chosen to be
(9.47)
Equation (9.46) may then be integrated by standard techniques 45 to find p as a function of w (we choose p=p 0 at w=0).
The resultant shock profiles are shown in Figures 8 and 9.
Note that the shock becomes steeper as the strength becomes
larger, and that the 10%-90% rise distance is typically
1-2 fm.
In summary, we have investigated shock solutions to
the truncated TDHF equations in the case of one-dimensional
iso-scalar flow.
The kinematics were investigated in all
truncation procedures and found to yield markedly different
results.
The irrotational and classical approximations were
shown to yield exact (in the sense of reproducing the TDHF
results) relations for the shock speed and impact velocity
in terms of the central density.
The shock profile was
investigated in the Thomas-Fermi approximation and no continuous solution found, possibly indicating no stable shock profile exists. For the most physical discontinuous solution, the
shock front was found to be 1-2 fm in thickness, with the
169
thickness larger for weaker shocks.
170
Chapter 10
SUMMARY
This work has explored the meaning and interpretation of the Time-Dependent Hartree-Fock equations.
It was &OAn
that the Wigner function satisfies a non-local Vlasov-like
equation governing its motion in phase space.
For zero-
range two-body forces of the Skyrme form, the equation becomes local, and successive moments in the momentum variable
satisfy hydrodynamic-like equations.
The problem of
truncation the hydrodynamic hierarchy of equations was considered, and four prescriptions presented for specifying
higher moments of the Wigner function in terms of the lower
ones.
These prescriptions were shown to be consistent with
almost all quantities conserved by the original TDHF equations,
except that the Wigner function does not necessarily remain
one which describes a Slater determinant.
The various forms of truncation were then tested in
a number of physical situations utilizing the Skyrme III
force.
Expressions were derived for the dispersion relations
for both isoscalar and isovector sound waves in nuclear
matter, and interpretations given for the physical meaning of
the several truncations.
Both isoscalar quadrupole and giant
171
dipole resonances of spherical N=Z nuclei were treated
and expressions derived, under the assumption of incompressible irrotational flow, for the frequencies of these
modes in terms of the ground state nuclear properties.
These expressions were then applied to 160 using harmonic
oscillator single-particle wave functions.
Excellent re-
sults were obtained for the isoscalar quadrupole excited
frequency, and a result consistent with an upper bound for
the giant dipole frequency was obtained.
As a final application, the kinematics of shock
waves expected to occur in heavy ion reactions was discussed for colliding slabs of nuclear matter.
Expressions
were derived for the maximum density attainable and the
shock speed as a function of bombarding energy, and it was
shown that two of the truncations lead to an exact description of the expected TDHF results.
The matter of the shock
profile in the Thomas-Fermi approximation was also considered, and it was shown that no continuous profile existed.
For a physically reasonable discontinuous profile, it was
shown that the shock front extends over a region of 1-2 fermi,
the profile becoming steeper at higher bombarding energies.
As stated in the introduction, verification and
extension of the ideas presented in this work must come from
two sources.
lations.
The first, is, of course, actual TDHF calcu-
The formalism and insights presented here offer
172
the possibility of an interpretation of these calculations.
At the same time, it will be important to ascertain to
what extent the hydrodynamic description presented here is
compatible with the actual behavior of the TDHF solutions.
The second source will be the actual physical situations
amenable to a hydrodynamic interpretation, such as fission
and heavy ion reactions, and whether or not the equations
presented here offer the possibility of an accurate
description of such processes.
The few analytical cal-
culations presented here offer some hope that these
equations
describe
sensible physical phenomena, but the
final arbitermust, of course, be the experimental results.
173
References
1.
A. Bohr and B. Mottleson, The Many Facets of Nuclear
Structure in Annual Review of Nuclear Science, Vol. 23,
E. Segre, ed., (Annual Reviews, Inc., 1973)p. 363.
2.
G.E. Brown, Unified Theory of Nuclear Models and Forces,
(North-Holland, 1971), Chapter 4.
3.
A.L. Fetter and J.D. Walecka, Quantum Theory of Many
Particle Systems, (McGraw Hill, 1971).
4.
A.J. Sierk and J.R. Nix, in Proceedings of the Third
International Atomic Energy Agency Symposium on Physics
and Chemistry of Fission, Rochester, 1973 (International Atomic Energy Agency, Vienna, 1974), Vol. II, p. 273.
5.
W. Myers and W. Swiatecki, Nucl. Phys. 81
6.
F. Villars, MIT Preprint, CTP #459
7.
A. deShalit and H. Feshbach, Theoretical Nuclear Physics,
Vol. 1, (Wiley, 1974)p 541.
8.
F. Villars, Hartree-Fock Theory and Collective Motion
in Dynamic Structure of Nuclear States, D. Rowe,
L. Trainor, S. Wong, T. Donnelly, eds. (University of
Toronto Press, 1972) p. 3.
9.
Reference 3, p. 64.
(1966).
(1975).
10.
Abrikoson, Gorkar, and Dzyaloshinski, Methods of Quantum
Field Theory in Statistical Physics, (Prentice Hall,
1963), ch. 2.
11.
J.G. Kirkwood, Journal of Chemical Physics 14, (1946) 180.
N.N. Bogolubov in Studies in Statistical Mechanics.
J. deBoer and G. Uhlenbeck, eds., (North-Holland, 1962),
Vol. 1.
12.
P. Bonche, S. Koonin, J. Negele, to be published.
174
13.
G. Bertsch, Los Alamos preprint, 1975.
14.
E.P. Wigner, Phys. Rev. 40
15.
L. Schiff, Quantum Mechanics,
16.
K. Huang, Statistical Mechanics,
17.
D.J. Thouless, Nucl. Phys. 21
18.
M. Goldhaber and E. Teller, Phys. Rev. 74
19.
G. Bertsch and S. Tsai, Physics Reports, in press.
20.
P. Morse and H.
Physics
(1930) 749.
(McGraw-Hill, 1949).
(Wiley, 1963), Ch. 5.
(1960) 225.
(1948) 1046.
Feshbach, Methods of Theoretical
(McGraw-Hill, 1953) Vol. 1, p. 464.
21.
T.H.R. Skyrme, Phil. Mag. 1 (1956) 1043.
T.H.R. Skyrme, Nucl. Phys. 9 (1959) 615.
22.
M. Beiner, H. Flocard, Nguyen Van Giai, and P. Quentin,
Orsay preprint IPNO/TH 74-27 (1974).
23.
J.M. Blatt and V.F. Weisskopf, Theoretical Nuclear
Physics, (Wiley, 1952), p. 135.
24.
J.W. Negele and D. Vautherin, Phys. Rev. C5
25.
D. Vautherin and D.M. Brink, Phys. Rev. C5
26.
Reference 40, and
L. Kadanoff and G. Baryon, Quantum Statistical Mechanics
(Benjamin, 1962) p. 56.
27.
S. Chandrasekhar, An Introduction to the Study of Stellar
Structure, (Dover, 1957).
28.
Ref. 24 and J. Negele and D. Vautherin, MIT Preprint,
CTP #425 (1974).
29.
W. Meyers and W. Swiatecki, Ann. Phys. 55
30.
C.D. Bennett and D.G. Ravenhall, Phys. Rev. C10
2058.
31.
K.-K. Kan and J.J. Griffin, Phys. Lett. 50B (1974) 241.
K.-K. Kan, Univ. of Maryland Ph.D. Thesis (1975).
32.
R.P. Feynman, The Feynman Lectures on Physics,
Wesley, 1963), Vol.II, Ch. 40.
(1972) 1472.
(1972) 626.
(1969) 395.
(1974)
(Addison-
175
33.
M. Baranger, Journal de Physique, Supplement 33 (1972)
61.
M. Baranger and M. Veneroni, to be published.
34.
R.D. Richtmeyer and K.W. Morton, Difference Methods for
Initial Value Problems, 2nd ed. (Wiley, 1967).
35.
H.A. Bethe, Ann. Rev. Nucl. Sci. 21 (1971) 93.
36.
E. Galssgold, W. Heckrotte, and K. Watson, Ann. Phys. 5
(1959) 1.
37.
Reference 7, p. 415.
38.
F. Villars and E. Guerra, private communication.
39.
A. Messiah, Quantum Mechanics,
Vol. I, Ch. XII.
40.
D. Vautherin, Orsay preprint IPNO/TH75-2 (1975), submitted to Physics Letters.
41.
H. Flocard and D. Vautherin, Orsay Preprint IPNO/TH74-40
(1974).
42.
T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291.
43.
F. Harlow and A. Amsden, Fluid Dynamics, Los Alamos
Monograph LA4700 (1971).
44.
S. Koonin and P. Bonche, to be published.
45.
J.A. Zonnefeld, Automatic Numerical Integration (MCT-8),
Mathematisch Centrum, Amsterdam, p. 23.
(North-Holland, 1962),
176
FIGURE CAPTIONS
Figure 1
The energy per nucleon as a function of density, e(p), for
nuclear matter. The force used is SKM III as given in
Ref. 22. The saturation density is .145 nycleons/fm 3 corresponding to a fermi momentum of 1.29 fm~ . The binding
at saturation is 15.77 MeV, while the adiabatic compressibility, p 2D2/ap 2 , at saturation is 40 MeV.
proximations are coded:
; classical
Thamas-Fermi
.
.
;
; isotropic
irrotational . . . .
.
Figure 2
The isoscalar dispersion relations for sound in nuclear
matter calculated with the SKM III force. The various ap-
Figure 3
The isovector dispersion relations for sound in nuclear
matter with the SKM III force.
The curves are labelled as
in Figure 2.
Figure 4
.
a) The initial conditions in the center of mass frame corresponding to two very thick colliding slabs of nuclear
matter.
b) The expected configuration at times long after the collision has taken place.
c) Figure 4b as seen in a frame moving leftward
at speed U
Figure 5
The shock speed and fluid velocity for the non-interacting
system as a function of the density in the central region.
The labelling of the curves is as in Figure 2. The irrotational result is identical with the classical result.
Figure 6
The internal energy per nucleon and heating in the region
behind the shock for non-interacting nuclear matter as a
function of the central density. The curves labelled e
give the internal energy, with the various approximations
indicated as in Figure 2. Those labelled e give the heat
energy per nucleon in the central region. Energies are
measured in units of 6%, the internal energy per nucleon in
the region preceeding the shock.
177
Figure 7
The shock velocity, in units of the speed of light, as a
function of central density for nuclear matter with the
SKM III force.
The coding of the curves is as in Figure 2.
The irrotational approximation gives results identical to
those of the classical approximation.
The maximum central
density attainable in the isotropic approximation is 1.92PO,
while in the classical approximation, it is 1.59PO.
The
curves have a value at p=po of the isoscalar sound speed
computed in the corresponding approximation.
Figure 8
Same as Figure 7, but for the fluid velocity.
The righthand
ordinate indicates the center of mass kinetic energy per
nucleon corresponding to the fluid velocity.
Figure 9
Same as Figure 6, but for nuclear matter with the SKM III
force.
The curves labelled e now indicate binding energy
per nucleon.
Note that the isotropic approximation predicts
the nucleons behind the shock will become unbound at a
density of 1.51PO while the classical trunction gives 1.41PO.
Figure 10
Shock profiles for various shock strengths in'the ThomasFermi approximation for the SKM III force.
The curves are
labelled by r, the ratio of the central density to the
equilibrium density.
Figure 11
The 10%-90% rise distance of the shock profile as a function
of shock strength for the SKM III force.
178
Table I
SKM III FORCE PARAMETERS2 2
=
-1128.75 MeV -fm 3
x
=
.45
t
=
14000.0 MeV -fm 6
t
=
395.0 MeV -fm 5
t
=
-95.0 MeV -fm 5
t+
=
t 1 + t 2 = 300.0 MeV -fm 5
t
=
t.-.
t2 = 490.0 MeV -fm5
t
=
(3t1 + 5t 2 )/16 = 44.38 MeV -fm 5
t
=
(3t1 - 5t 2)/16 = 103.8 MeV -fm 5
t
0
179
-
15
0
(MeV)
-5-
-IC
-15 -~
.05
.10 r
F*M15
L
.15
.20.25.3
p (IWJueois/;.?)
180
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190
Biographical Note
The author was born on December 12, 1951 in
Brooklyn, New York.
He attended Stuyvesant High School in
New York City, graduating in June, 1968.
The following September, he entered Cal Tech where
he is undoubtably remembered by his peers mor for his performances as a rock organist than for his physics.
A memor-
able senior year included an experimental thesis under
Borje Persson on radiative electron capture, a theoretical
project in nuclear astrophysics with Tom Tombrello, teaching
a section of freshman physics, and the courting of his
future wife, Laurie Card.
He received the Bachelor of
Science degree in June, 1972, winning the George Green Memorial Award for creative scholarship.
In September, 1972, he entered the graduate school
at M.I.T. as a National Science Foundation Gratuate Fellow.
Interspersed with idyllic summers in the New Mexico mountains
of Los Alamos, his work here with Arthur Kerman has included
such topics as a formal description of the quasi-equilibrium
reaction mechanism and various formulations of dissipation
in nuclear collective motion.
He leaves M.I.T. expecting to divide the next two
years between Cal Tech and the Niels Bohr Institute.