Symmetries and Semiclassical Features of Nuclear Dynamics: Invited Lectures of the 1986 International Summer School, Held at Poiana Brasov, Romania, September 1–13, 1986

458 Pages·1987·16.96 MB·English

by  Abraham Klein (auth.), A. A. Raduta (eds.)

A. SEMICLASSlCAL FEATURES OF NUCLEAR MOTION
QUANTUM FOUNDATIONS FOR A THEORY OF COLLECTIVE MOTION.
S EMICLASSICAL APPROXIMATIONS AND THE THEORY OF LARGE AMPLITUDE COLLECTIVE MOTION.
Abraham Klein
Department of Physics, University of Pennsylvania
Philadelphia, Pennsylvania 19104 USA
I. INTRODUCTION
In 1962 A. Kerman and the writer applied Heisenberg's equations of motion (matrix
mechanics) in a new way to the study of nuclear collective motion. We showed how to
preserve the symmetry properties of the underlying Hamiltonian in a fully quantum al-
beit approximate form of dynamics 1'2) The main consequences of this early work were
that all mean field approximations such as Hartree-Fock, Hartree-Fock-Bogolyubov, BCS,
RPA and self-consistent cranking were seen to be contained within the new framework.
This original formulation, variously termed the generalized Hartree-Fock approximation
or self-conslstent core-particle coupling method, can be characterized as a generaliz-
ed quasi-particle theory which requires a simultaneous self-consistent determination
of the properties of states in both even and neighboring odd nuclei. In the long run,
this most ambitious form of the theory has been found wanting both theoretically and
practically (.see ref. 12, 18). It has been revived, however, in a linearized form re-
quiring externally supplied models for the even nuclei 3'4). - In this form, it is still
the most complete and fundamental semi-phenomenological core-partlcle coupling method
extant.
Beginning in 19685) , it has become increasingly clear that the problems associat-
ed with the study of collective motion in even nuclei could be attacked on their own
merits, without dipping into the underlying single fermion structure, by utilizing
the Lie algebras of Fermion pair and multipole operators 6-II). Both the single-parti-
cle and Lie algebraic approaches have been thoroughly reviewed recently 12) . Shorter
recent reviews 13'14) may also be of interest.
Of the algebraic method, it may be said that it has proved to be a useful approach
to numerous models with collective properties and at the same time that its potential
has hardly been tapped. Until recently all applications have been to two types of
models. In the first, the Lie algebra is sufficiently simple that group representa-
tion theory may be applied without approximation. (We shall illustrate this type of
model below). In general such models are not very realistic (toy models). In the
second class of models, we deal with realistic shell structure where the Lie algebra
is already too complicated to be dealt with routinely but simplification is achieved
by using separable interactions such as pairing and quadrupole forces ("specificity"
forces), Under these circumstances it is also not difficult to derive a closed compu-
tational scheme which characterizes a set of states (band) called the collective sub-
space,
In the toy models the collective subspace is synonymous with the states of an
irreducible representation (irrep) of the Lie Algebra. In the realistic models the
collective suhspace is identified by reference to experiment. In an idealization it
may further be identified with a Hilbert space of small dimension ("geometric" model)
or with a finite dimensional space ("algebraic" model).
The main goal of the current lectures is to extend the algebraic method for even
nuclei to interactions which are not separable and to propose several new applications.
It goes without saying that many older applications can be "revisited", but we have
chosen not to look backwards because some of these subjects are out of fashion.
After these preliminary remarks, we turn to the substance of this section which
is to smooth the path from the toy models to the fully realistic ones by discussion
of an example. Consider the problem of N identical fermions in two N-fold degenerate
levels separated by the single-particle energy E. Let @t(po) create a fermion in the
magnetic sublevel p of the level ~ = ±i (for upper and lower respectively) and let
~(po) be the corresponding annihilation operator. A general shell-model Hamilton±an
for this system is
i
H = ~8 r O @+(pU)$(p~)
i + t
+ ~V (Pl~l,P2~2;P3o3,P4~4)@ (Pl~l)~ (P202)$(P4O4)@(P3~3) , (i.i)
where the matrix elements V are antisymmetrized.
Let us first consider the widely known "Lipkin" model 15), which we regain when
the interaction takes the especially simple form
V(PlOl,P2~2;P3O3,P404) = V[~(p2,P4)~(o2,-~4)~(pl,P3)~(~l,-~3)
- (i+-+2)]~(01,o2)~(o3,o4) (1.2)
In this case (i.i) reduces to
H = gJ + i 2 2)
o ~V(J+ + J_ , (1.3)
where
j+ = (j_)t = I St(p+)$(p_) ,
(i.4)
P
i
l~[ ~#(p+)~(p+)-~#(p-)@(p-) ] (1.5)
Jo = ~p
are generators of SU(2). Many years ago, we showed 6) how the algebraic method could
be applied to this model. The equations which are utilized are the commutation rela-
tions of SU(2)
[ J+,J_ ] = 2J ° , (1.6)
[Jo,i± ] = ±i± , (1.7)
the "equations of motion"
[J±,H ]= • ~J± - VJ~ + 2VJoJ $ (1.8)
[Jo,H] , _ V(j+2 _ j_2) , (1.9)
and the equation which identifies the irrep of interest, [N] , giving the value of
the Caslmir invariant,
i i 2 1 1
J+J- + ~ J-J+ + Jo = ~(~N + i) (i.i0)
We do not wish to review this old work in detail, but only to retrieve a few
salient features. From the equations (1.6)-(1.10) (or from a subset of them) we ob-
tain a set of non-linear algebraic equations which characterize a collective subspace
of states. These statesconstitute a subspace of the irrep [~] which has N+I states
i
(IJo I ~ ~ N) in all. We choose this subspace as follows: (i) it consists of eigen-
states of the full Hamiltonian, which we label In>, n = 0,1,2,... in order of ascend-
ing energy. (ii) Matrix elements of products are evaluated by sum rules, e.g.
<n]JoJ+In'> = <nlJoln"><n"IJ+In'> (I.ii)
Thus matrix elements of (1.6)-(1.9) yield, by use of equations such as (i.ii), non-
linear algebraic equations where the variables are matrix elements of J+ and J and
- o
energy differences (En-En,). (iii) An essential element is that viable and accurate
approximation schemes may be obtained without having to include the entire space of
[N] because of the fact that in general, that is f_or all strengths of q0pp!ing, the
matrix elements <nlJ±In'> and <nlJoln'> decrease rapidly with In-n'[. Thus we may
start from the ground state and formulate schemes involving successively higher states.
(iv) Of course details may differ sharply according to the value of V. We even go
through a "phase transition" in this model: For small V the model is equivalent to
a one-dlmensional anharmonle oscillator, but for sufficiently large V we go over into
a double well oscillator with its characteristic almost degenerate doublet structure.
The number of levels utilized and the choice of suitable input for self-consistent
algebralc solution must reflect some knowledge of the regime in which one finds one-
self. (v) In general the information contained in the totality of commutation rela-
tions and equations of motion is redundant. For example, notice that
2 IJo'H] " [[J+'J-]'H]
= - [r~,J+],J ] - [[J_,H],J~ (1.12)
in consequence of the Jacobi identity, so that we do not utilize the equations of mo-
tion (1.9). But the scheme must include the information contained in Eq. (i.i0) be-
cause it expresses the Pauli principle for this problem.
For further details of this calculation and related topics - for example construc-
tion of a boson representation of the results - we refer to the previous literature
6,16) What we wish to extract most particularly from the above is the feature which
permits the formulation of a collective scheme within a relatively restricted subspace
of states. At first sight one is tempted to say that this is the group theoretical
structure of the model which restricts one to a single irrep. But even in this most
trivial of models, this is only a partial answer. A more profound reason is dynamical.
If there is collective motion, then there is at least one collective operator (in this
case a generator as well) which prefers or enforces transitions dominantly between
neighboring levels. In the Lipkin model, for small V, it Is J+, since Jo is almost
a good quantum number, but for large V, it turns out to be J = (J+ + J ).
x
To move closer tc the study of a realistic model, let us still consider the
Hamiltonian (1.3), but study the equation of motion for a general density fluctuation
operator, for example
[~%(pl-)~(p2+),H] - e ~%(pl-)~(p2 +)
+ ½V {[**(Pl-l*(P2-)- *%(pl+)*(P2+l],J+} (i.13)
Even though the operator ~%~ can connect different representations of SU(2), because
H has eigenstates within a given irrep, the equation of motion can once again be con-
fined to a given irrep. This is because a sum rule such as (i.ii) requires only one
of the two factors to be a generator. In fact, as long as we restrict ourselves to
matrix elements within a given irrep, we can replace (1.13) by the previous equations
of motion.
Finally, let us investigate the most general case represented by the Hamiltonlan
of Eq. (i.i). We introduce a condensed notation (
that there is a way of combining the two requSKements (i) and (ii) above. It is to
assume a generalized factorization of matrix elements of the two body operators that
occur in (1.15):
- (c~B) - (6+-~y) + (~+8,~+-~y)} (1.17)
This approximate ($enerallzed Hartree-Fock) factorization has a number of attractive
properties which will be discussed more fully in the next section. Here we remark
that for an interaction of the form (1.2), it becomes exact when averaged with the
interaction in forming either a matrix element of the Hamiltonian or a contribution to
the equations of motion. Thus it selects the part of the Hamiltonian effective in the
collective subspace. The factorization (1.17) will play a fundamental role in the
further development of the concepts in these lectures.
II. REVISED GENERALIZED DENSIR~I ~PclXMETHOD
A. Derivation. We study next a general non-relativlstic Hamiltonian of the
form
i
H = hab ~a*~b + ~ Vabcd ~a*~b*~d~c , (2.1)
where we use summation convention whenever possible, and the indices a, b,... on the
nucleon creation (~T) and annihilation (~) operators may refer, according to the
application intended, either to space, spin, and Isospin, a = (~,O,T) or to the quantum
numbers of a slngle-partlcle orbit, and satisfy the usual Fermion anticommutation rela-
tions. We take h and V to be Hermitian matrices, hab = hba , Vabcd = Vcdab and V to
describe antisymmetrized matrix elements, Vabcd - - Vbacd = - Vabdc. We suppose fur-
thermore, if H is a "realistic" Hamiltonian with h the kinetic energy operator, that
V is invarlant under translations, rotations and Galilean transformations.
Our initial aim is to derive from (2.1) an approximately closed set of equations
of motion for matrix elements of the one-body density operator, equations which charac-
terize a collective subspace. In the example discussed in the introductory section,
the collective subspace, here labelled as IA>, IB>, IC> ..... had a group theoretical
significance which made its choice more or less obvious. This simplification was, in
l)
fact, absent from our original conception , which was based on the idea that the re-
lationships among the members of the collective subspace was enforced by coherence
properties of sums of products of matrix elements of the two body interaction with
matrix elements of the density (or pairing) operator. (This conception also entered
the considerations of See. I.) It is to this initial conception that we shall return
here in order to effect an improvement in its implementation.
We study the equation of motion for the density operator
Oab= *b**a ' (2.2)
namely (Cf. (1.14))
i pa b = [0ab,H] = hacPcd - 0achcb
i i
+ 2 Vacde ~b~ct~e~d - 2 Vcdbe ~ct~dt~e~a (2.3)
By taking matrix elements of (2.3) between states of the collective subspace, we are
led to the study of the generalized density matrix (GDM) elements of the one and two
particle density operators,
p(aAlbB ) = <BI~bT~aIA> (2.4)
o(abAlcdB ) : <Bl~cT0d#0b~alA>
= 0(cdBlabA)* , (2.5)
where the latter also have certain obvious antisymmetry properties. Thus a matrix
element of (2.3), (EA .... are eigenvalues of H)
(EA-EB)P(aAIDB) = haC0(cAlbB) - o(aAlcB)hcb
1 0(deAlbcB) - ½ Vcdbe p(aeAIcdB) , (2.6)
+ ~ Vacde
can lead to closed equations only if the elements (2.5) can be expressed in terms of
the simpler objects (2.4). AS is well-known, this can never be the case if pairing
correlations are significant. We exclude such correlations from the present study,
since we have enough substance to communicate which does not depend in an essential
way on these currelations.
The goal of closing the set (2.6) is accomplished by means of the fundamental
assumption of our work, the generalized factorlzation hypothesis first encountered in
(I.17),
<Bl~c*~dt~b0a IA> i (<Bl,c*~alC><Ci,d**bl~ >
- (e+-+d) - (a+-+b) + (c+-+d,a+-+b)} (2.7)
This factorizatlon 17'18) which generalizes the proposal found in our earliest work has
the following desirable features: (1) It preserves the antlsymmetry and hermiticity
properties of the two-body GDM. (ll) The approximate equations of motion to which it
leads satisfy all the conservation laws inherent in the original equations of motion.
It is thus a conserving approximation. (lii) In the semi-classical limit it reduces
to tlme-dependent Hartree-Fock theory. (iv) Though the previous criteria provide ex-
cellent recommendations for the decomposition (2.7), to look more deeply into its
possible origin and significance, we must return to the physical arguments contained
in our original work. There it was reasoned that it was asking too much to expect
(2.7) to hold for an arbitrary choice of single particle indices. All that can Be
expected is that selected averages of (2.7) lead to coherent sums which favor transi-
tions within the collective subspace. Where these sums form generators of Lie algebras
as in the model studied in Sac. I, the matter is especially clear. But if we wish to
deal with realistic interactions, we cannot insist on such mathematical clarity and
must hope that physical intuition will pull us through. We view this mode of reason-
ing as the moxt extensive use of the random phase argument encountered in the theory
of collective motion: Certain coherent subsets of matrix elements favor transitions
within the collective subspace whereas the overwhelming majority of the remaining
elements cancel out because of "random phases".
If we accept (2.7) in the sense described, the equation of motion (2.6) can be
rewritten with the help of several convenient definitions. These include a collective
Hamiltonian in which energies are referred to the ground state energy,
~c(aAlbB) = ~ab6AB(EA-Eo) E ~ab~AB~A (2.8)
and a generalized Hartree-Fock Hamiltonian H,
4H(aAIbB) = hab ~AB + v(aAlbB) ' (2.9)
v(aAIbB) = Vacbd p(dAIcB) (2.10)
We then find that (2.6) may be written in operator form as
1 i
[~c '~ = ~ [~'~ ]+ ~'~]e (2.11)
and, e.g. (Hp) is a special matrix product
e
~P)e(aAlbB) =~(aC[cB)o(cA[bC)
= 0(cA[bC~H(aC[cB) . (2.12)
According to the order in (2.12), it is thus seen that the symbol e may refer either to
exchange of the collective coordinates or to the exchange of the single particle in-
dices.
If we replace the time-independent operators ~a by time-dependent operators,
~a(t) = exp(iHt)~a exp(-iHt), we may replace (2.11) by an equation which is properly
termed "quantized" TDHF, namely
dp i i
i ~ = ~ [~,p] + ~ [~,0] e (2.13)
From the form (2.13), which is that of an initial value problem, it is especially
clear that we need a kinematical constraint to set the scale of p. This can again be
derived from (2.7) by setting b=d, sunning over b and using number conservation in
the forms
N]A> = N]A>. N *alA> " (N-I),alA> , (2.14)
where
= E *a** (2 .15 )
a a
We find (replacing p2 . 0 of Hartree Fock theory)
1 2 i( 2.
P = ~ P + ~ P )e " (2.16)
Equations (2.11), (2.13), and (2.16) will provide the starting points for the re-
mainder of these lectures. Here Eq. (2.16) replaces the Casimir invariant of the SU(2)
model of Sec. I. Its derivation from (2.7) represents an extended application of that
factorization. To include the analogue of all the elements of the simple model we
should also take note of the algebra of the density operators. This we do only in
passing, since these relations will not play any direct role in the applications to
be discussed.
B. ConserVation Laws' and Sum Rules. Suppose that we have an exact conservation
law associated with the Hamiltonlan (2.1), of the form
d /dx ~(X) ~(x) = 0 (2.17)
dt
where 8 can, for example, be the single-particle linear or angular momentum operator.
The invar~ance properties of (2.1) which result in (2.17) can be w~Itten in the form
(Xl[e, Id IY) = exhxy " hxy0y = O (2.18)
and
(xy][ (81449 2) ,V] ]zW) = (Sx+Sy)Vxyzw - Vxyzw(SZ+Ow) = 0 (2 .19)
We now show that in consequence of (2.18) and (2.19) and the approximate equations of
motion (2.13), Eq. (2.17) is satisfied within the collective subspace. We calculate
Id/f /dx <A' l * * (x )0* (x ) IA> (2.20)
Since the contributions from the single-particle and two-particle parts of H vanish
separately, we consider only the interaction part, which is
!2/0x[ Vxz,~, P(wAIzS)0(YSIxA') - 0(xA]yS)VyzxwO(wSlzA,)
+ Vxzy,w 0(wSlzA')0(yAIxB) -p (x B I Y A ' ) V y z x w 0(wAlzS)
= 21--/ (xyI[01-~2,V] Izw)0(wAlzB)p(ySlxA') = 0 , ( 2 .21 )
requiring only a reshuffling of variables.
Next we show that our method satisfies both energy weighted and energy squared
weighted sum rules. Let F be an arbitrary one-partlcle ~ermitian operator
^ ^
= feb Pba = Tr f0 , (2.22)
and consider the ground state expectation values
^ k ^
Sk(F) ~ <O[F(H-E o) FIn> (2.23)
For k = I, two exact versions of Sl~f) are
I <0I [~, tH,~] ] i . > (2.24)
fOA " <O[FIA> (2.2S)
We prove that this sum rule continues to be satisfies in the GDM approximation, pro-
vided the left hand side is evaluated from the solution of our equations and provld=,/
FI0> is a state in the collective subspace. For then the right hand side of (2.24) is
10
equal to
1 i <O[EH,F] IA> (2.26)
foA <AI[H'FI lO> - 7 fAO
But according to the GDM approximation, we evaluate
i [0,~] (cOIdA)
<A[[H,F][O> = fdc {~
i *
+ 2 [0'~]e(COldA) = fdc[O'~4c ](cOIdA) = ~AfOA (2.27)
Combined with a similar evaluation for the second term, we easily reach the required
identity,
The method described works not only for the S I sum rule, but also the S 2 sum rule)
which takes the form
S2(F) ~ Z(~A)mlfoA 12 = <OI[F,H][H,F]I0> (2.28)
The proof is immediate if one follows the outline of the argument given above.
III. SEMI-CLASSICAL LIMITS
In this section we shall describe several different problems whose central ele-
ments can be understood by starting with the GDM equations and passing to the semi-
classical limit in which the theory reduces to a version of time-dependent Hartree-
Fock theory (TDHF).
A. Periodic Motion. 19)" For illustrative purposes only, we restrict our study
to a one-dlmensional collective mode, a breathing mode, for example, so that IA>+In>,
with n an integer, n = 0 referring to the ground state. We study the matrix element
0(xltn[x2tn') = <n'l*t(x2t)*(Xl t) In>
= exp [i(En,-Bn)t]<n'I~+(x2)~(xl)In > (3.1)
One standard method of approaching the semi-classlcal limit is as follows: Introduce
new variables
= ½(n+n'), ~ = n'-n , (3.2)
and approximate the energy difference
En,-E n ~ (dE/dn)~ E ~(n)9 , (3.3)
If the correspondence principle approximation (3.3) is Justified, then (3.1) becomes
. . i~(~)~t
0(xltnlx2tn' ) ~ p,,~XlX2)e , (3.4)
and the Fourier sum
O~(XlX21t ) =~O(xltnlx2tn' ) (3,51
defines a periodic function of t, period T(n) = 2z/~(n). From its definition, the
Fourier coefficients of 0~(xlx21t) are the various transition matrix elements of the
density matrix.
Provided 0~,v(XlX2) is a slowly varying function of n and a function peaked in V,