Nuclear deformations in the pairing-plus-quadrupole model

l 1.D.3 ] [

Nuclear Physics A122 (1968) 241--272; (~) North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission f r o m the publisher

NUCLEAR DEFORMATIONS

IN THE P A I R I N G - P L U S - Q U A D R U P O L E MODEL

(IV). Theory of collective motion * MICHEL BARANGER

Carnegie-Mellon University, Pittsburgh, Pennsylvania and KRISHNA KUMAR tt

Michigan State University, East Lansing, Michigan Received 26 June 1968

Abstract: A theory of collective motion based on the time-dependent Hartree-Bogolyubov approximation is shown to be possible for the pairing-plus-quadrupole model. It is a no,n-linear adiabatic theory, which makes it suitable for soft nuclei. The results are very close to those of the cranking theory, but they are not subject to the objections usually made against the latter. Explicit formulae are given for the potential energy, the three moments of inertia and the three vibrational inertial parameters entering Bohr's collective Hamiltonian for arbitrary quadrupole deformations and for the gyromagnetic ratios. A simple extension of the theory is found able to deal with core polarization. The theory has no obvious generalization to two-body forces other than pairing-plus-quadrupole.

1. Introduction I n previous papers of this series ~- 3), we discussed mostly the static aspects of nuclear deformations. We calculated the d e f o r m a t i o n energy of various nuclei, i.e. their energy as a f u n c t i o n of the parameters fi a n d ~ describing a static deformation, a n d we a s s u m e d that the stable nuclear shape was that for which the d e f o r m a t i o n energy is m i n i m u m . This is a suitable procedure for nuclei of fairly rigid shape, either well-deformed nuclei with p r o m i n e n t r o t a t i o n a l spectra or spherical nuclei close to magic n u m b e r s . But for most other nuclei, including the m a j o r i t y of those that do n o t show r o t a t i o n a l features, q u a d r u p o l e collective m o t i o n playes a very i m p o r t a n t role in the low-energy structure. T h a t such collective m o t i o n is very n o n - l i n e a r has long been evident f r o m the rather p o o r agreement between observed v i b r a t i o n a l spectra a n d the predictions of the h a r m o n i c oscillator model, b u t it has become particularly obvious since the m e a s u r e m e n t s of excited state q u a d r u p o l e m o m e n t s by the reo r i e n t a t i o n effect 4). O u r p u r p o s e here is to outline the theory of q u a d r u p o l e collective m o t i o n which we have used in c o n j u n c t i o n with the p a i r i n g - p l u s - q u a d r u p o l e residual i n t e r a c t i o n in * Work sponsored jointly by the U.S. Office of Naval Research under contract Nonr. 760(15) NR 024-439, and by the U.S. Atomic Energy Commission under contract AT(ll-1)-1051. *t Present affiliation: The Niels Bohr Institute, Copenhagen, Denmark. 241 December 1968

242

M. BARANGER AND K. KUMAR

all of our work beginning with Kumar's thesis 5-7). Actually the theory consists of two parts. The first part, which is given here, deals with the derivation from the pairing plus-quadrupole model of the seven functions of fl and 7 entering Bohr's collective Hamiltonian

~Bohr ~--- ~ / ' ~ - ~ ' r o t ~ -

(1)

~-~vib,

"~" = potential energy = ~V'(fl, 7), 3

~--rot = rotational kinetic energy = ~

½Ji([3,

(2) 7)~o~2

(3)

i=1

"Y--vlb = vibrational kinetic energy =

½B##(fl, 7)/~2 + Bpr([3, Y)l~':/+½Brr([3, 7)~/2.

(4)

The second part, which has already been published 8), deals with the properties and solutions of this Hamiltonian. We must apologize for the confusion caused by publishing the second part ahead of the first. Our only excuse is that we were so absorbed in the consequences of the model that we kept delaying the publication of its foundations. A preliminary report of the applications to the W-Os-Pt region has appeared 7), and more details will be given in paper V of the series. Applications to other parts of the periodic table are also in progress. We wish to make it clear at the outset that we do not believe that the theory of collective motion discussed here is entirely satisfactory. As far as we know, there exists at the moment no satisfactory microscopic theory of nuclear collective motion, if by "satisfactory" we mean that the theory is non-linear, applies to vibrations as well as to rotations, is simple enough for possible calculations throughout the periodic table and applies to an arbitrary two-nucleon interaction. Our theory satisfies the first three criteria but not the last one; it applies only to the pairing-plus-quadrupole model, and there does not seem to be any way to generalize it to an arbitrary two-nucleon interaction except by approximating the latter in ways that are, in effect, equivalent to going back to P + Q. This is why it is important to know how good the pairing-plusquadrupole model really is, and the answer 2, 9) seems to be: surprisingly good, as long as we restrict the comparison to phenomena involving only pairing and quadrupole degrees of freedom. The choice of a theory of collective motion is discussed in sect. 2. The theory that we adopt is an adiabatic one based on the time-dependent Hartree-Fock approximation. It resembles the cranking theory 10) but is distinctly different in some theoretical respects and is, we feel, more firmly based. In order to bring out these differences and to make clear the essentials of the theory, we discuss a simple model in sect. 3. The rather complicated application to the P + Q model is in sect. 4. In sect. 5, we include the core of the nucleus in the collective motion. The cranking theory is that which has been most often used ix, x2) in connection with the P + Q model. Our results are very similar to those of cranking, the differences being of the order of the errors caused by the adiabatic approximation. Hence the main point of this paper could be said to be that of justifying the cranking theory for

NUCLEAR DEFORMATIONS (IV)

243

P + Q. We emphasize once again that the theory works only for P + Q, therefore the preceding should in no case be interpreted as a justification of cranking for an arbitrary two-nucleon interaction. There is also some similarity between our discussion of sect. 3 and some work of Moszkowski 13). Finally, the strongest overlap is with the work of Belyaev 14), which is also based on the adiabatic approximation and the timedependent-self-consistent-field approach, although we differ from him in several respects. 2. The choice of an approximation

2.1. THEORIES OF COLLECTIVE MOTION There has been much discussion of microscopic theories of collective rotations. The theory that is needed here, however, must be capable of describing vibrations as well; in fact, vibrations are more important, since vibrational nuclei are less well understood at the moment. The first possibility that comes to mind is the random phase approximation [see for instance ref. 15) and other cited there; see ref. 16) for a critical discussion]. But this has to be rejected outright because it is a linear theory. The non-linear theories that are found in the literature fall into two categories. In the first category, are categories that may be rated "excellent" from the conceptual point of view, but whose application is beyond the reach of the present generation of computers. These include the approach of Peierls and Thouless 17) and the very beautiful theories developed recently by Klein and collaborators is). These authors are obviously "doing things right", but they are forced by practical considerations to limit themselves to models that are much simpler than the full quadrupole motion of a heavy nucleus. In any case, it is not clear that a crude model like P + Q deserves to be handled by such a good theory. The other category consists of theories using the adiabatic approximation. This assumes that the collective motion is slow compared to other motion in the nucleus, but it need not be linear; thus, it is well adapted to the treatment of transition nuclei, in which non-linearities are strong but collective levels are low. The traditional way of using the adiabatic approximation is known as the cranking model 10). For a development of the cranking model in the pairing-plus-quadrupole context, see ref. 12). However, the cranking model is subject to certain objections which we shall now review. 2.2. CRITIQUE OF THE CRANKING MODEL The calculation of the collective Hamiltonian by the cranking model involves the following steps. First, a collective coordinate, say D, is chosen. For each value of D, the true maoy-body Hamiltonian is approximated by some simpler H'(D), which depends explicitly on D. This may be for instance an independent-particle Hamiltonian in which the single-particle wave functions depend on D or an improvement of it along BCS lines 19). Let El(D) be an eigenvalue and [i(D)) the corresponding eigenstate of H'(D), the ground state being given the subscript i = 0. The next step

244

M. BARANGER AND K. KUMAR

is to assume that D is a slow function of time D(t) and to find a time-dependent solution of the Schr6dinger equation for H ' [D(t)]. This is performed by adiabatic perturbation theory 20), and the desired solution is that which reduces to the ground state in the limit of fixed D. There remains to take the expectation value of H'[D(t)] with this particular wave function. The result is the collective energy given by the wellknown formula .... k =

Eo(D)~-IB(D)b 2 ,

(5)

with

B = 2 ~ [(i]~n'/ODlO)lZ

,~0

(6)

(E;-~b)3

It is to be noted that the entire calculation is with the approximate Hamiltonian H ' ; the real H never enters. Though the cranking model manages to give good answers some of the time, it has been pointed out that it is conceptually incorrect 21). For instance, let us consider the case where H'(D) is obtained by some sort of self-consistent approximation similar to Hartree-Fock for each value of D. Self-consistence holds when D is constant; but once D is allowed to change with time, the approximate many-body wave function changes. The single-particle potential that it generates also changes, i.e. the singleparticle Hamiltonian is no longer H'(D) or, in other words, H'(D) is no longer selfconsistent. By neglecting the changes induced in H'(D) by the motion itself, the cranking model neglects terms which (in principle, at least) are of the same order of magnitude as the terms it attempts to calculate. 2.3. T I M E - D E P E N D E N T SELF-CONSISTENT A P P R O X I M A T I O N

To restore self-consistency, Thouless and Valatin 22) gave an improved version of the cranking model for rotations based on time-dependent Hartree-Fock theory. Here, we propose something similar for the pairing-plus-quadrupole model, but our problem is really quite different from theirs. Theirs was a linear problem (the differential equation for rotation of a nucleus with azimuthal angle q~ is (b = 0), and their solution resembled very much the random phase approximation (RPA). Our problem is non-linear as has already been pointed out, and the RPA is irrelevant. Instead, we shall use adiabatic perturbation theory, which assumes that the motion is slow, an assumption which is not implicit in the RPA. In order to see clearly the physical steps involved in this new approximation, it is useful to treat a simple example; in sect. 3, we consider a single quadrupole component and no pairing at all. The approximations used are time-dependent Hartree-Fock ( T D H F ) followed by adiabatic perturbation theory. In sect. 4, the full pairing-plusquadrupole model is treated, T D H F is replaced by time-dependent Hartree-Bogolyubov (TDHB). There are no new ideas, but the calculations are considerably more complicated. For a review of TDHF and HB, see for instance ref. 2 3).

NUCLEAR DEFORMATIONS (IV)

245

3. A simple example 3.1. THE MODEL The many-body Hamiltonian for n-particles is taken to be (in first-quantized language) n

n

H = Hs-½~ ~, Q(s) ~ Q(t), s=l

(7)

t=l

where H s is a spherical single-particle Hamiltonian and Q(s) the only non-vanishing component of the quadrupole moment of particle s, which, to be consistent with our previous notations, we should really have called Qo(s). This model does not contain pairing effects, the complications due to rotational degeneracy or the possibility of non-axial deformations. It might be viewed as a possible description of fl-vibrations of axially symmetric nuclei when pairing effects are unimportant, but our real purpose in considering it is just to give a feeling for the approximations. 3.2. STATIC HARTREE-FOCK THEORY As always with the pairing-plus-quadrupole model, we neglect the exchange term arising from the quadrupole force. In other words, it is really the Hartree approximation that we are using rather than Hartree-Fock, though we shall continue using the latter name. With this neglect, the one-body Hartree Hamiltonian is obviously of the form

HD = H s - D Q ,

(8)

D being a parameter which we call the deformation. The eigenstates and eigenvalues of H D are defined by the equations

Ho[A) = r/AIA),

for a filled level,

(9a)

HD[a) = ~/ola),

for an empty level.

(9b)

They all depend on D. The value of D that provides self-consistency is D = zq,

(10)

q = ~ (A[Q[A). a

(11)

with

To solve the static H F problem, one must solve eq. (10) for D. It is well known that the same result can be obtained by minimizing the expectation value of H with respect to D for the Slater determinant which constitutes the H F wave function. Since exchange is neglected, this expectation value is

Ho(D) = Z -½zq 2 a = ~ (AIHDIA) +Dq-½zq 2

(12a) (12b)

A

= X rlA+Oq--½xq 2. A

(12c)

246

~4. B A R A N G E R A N D K . K U M A R

The derivative with respect to D is dH 0/dD = ~ dqA/dD + q + D d q / d D - z q d q / d D . a

(13)

But, by first-order perturbation theory dqa/dD = - (A[QIA),

(14)

d H o / d D = ( D - xq )dq/dD.

(15)

hence

This vanishes when eq. (10) is satisfied. It is convenient for the following to introduce the one-body density matrix, whose general definition in second-quantized notation is (16)

Pr~, = (c~cr). J

The H F density matrix satisfies the supplementary conditions p2 ~- p,

(17a)

T r p = n,

(17b)

and its knowledge is completely equivalent to that of the Slater determinant. In the representation defined by eqs. (9), it is given by

(18)

DAB -~ (~AB, P,~b = P.a = PA,, = O,

therefore p and H D commute (19)

p H D - - H D p = O.

Some of the previous equations written using p are Eq. (11) Eq. (12a) Eq. (12b, c)

q = TrpQ,

(20)

H o ( D ) = Tr p H s - ½ Z q 2

= Tr p H o + D q - ½ z q

(21a) 2.

(21b)

3.3. TIME-DEPENDENT HARTREE-FOCK THEORY To treat the time-dependent problem, it is now assumed that the wave function is a Slater determinant which depend s on time, i.e. it is characterized by a density matrix p ( t ) obeying supplementary conditions (17). The Hartree potential generated by this wave function through the quadrupole force is of course - z q Q , q being given by eq. (20), and it also depends on time. We must now write that this Hartree potential generated by the wave function is just the potential that must be used in the Schr6dinger equation of which the wave function is a solution. In other words, self-consistency must be enforced at all times.

NUCLEAR DEFORMATIONS (IV)

247

If we write the usual one-body Schr~Sdinger equation for the density matrix as it5 = H D p - - p H o ,

(22)

then HD must be a one-body deformed Hamiltonian HD = H s - D ( t ) Q ,

(23)

in which self-consistency requires that the deformation D(t) be given by D(t) = zq(t) = g Tr [p(t)Q].

(24)

This self-consistency at all times is the basic ingredient that was missing from the cranking model. In the present treatment, there is no need for a distinction between potential deformation and wave function deformation since the two are always proportional. We shall usually consider D as our collective variable, though q could serve just as well. Of course, this treatment is only approximate. First, the true wave function is not really a Slater determinant. Moreover, even if it were a Slater determinant at a certain time, it would not remain so at later times. Finally, there is also a semi-classical approximation hidden in the theory. It enters because we are assuming D(t) to be a classical function, i.e. both the deformation and its time-derivative take definite values at time t in contradiction with the uncertainty principle. The slowness of the collective motion is the justification for this assumption. It will have to be remedied later by quantization of the collective Hamiltonian 8). The problem now is to solve the system of equations (22)-(24) connecting p(t) and D(t). In the RPA, the problem is linearized about the static equilibrium deformation. Instead, we shall use adiabatic perturbation theory. 3.4. ADIABATIC PERTURBATION THEORY Our adiabatic perturbation theory is identical to Schiff's z 0), but the use of density matrices makes the development much simpler. Let p0(D ) be the density matrix representing the Slater determinant built on the n lowest levels of H D, i.e. the matrix given by eq. (18). This would be an adequate description of the system, if D were independent of time. Since D is not, the correct p is different from this and-can be written p(t) = Po [D(t)] +p'(t).

(25)

Since the motion is slow, the correction p' is small. To find it, one substitutes expression (25) in eq. (22) satisfied by p. Since P0 commutes with liD, the result is i(tSo +t5') = H o p ' - p ' liD.

(26)

The "slowness" parameter which makes p' small may be taken to be D, the time

248

M. BARANGER AND K. KUMAR

derivative of D. Let p' be expanded in powers of this parameter P' = Pi +P2

"l-P3 @ ....

(27)

All time derivatives are of order unity in slowness. Then, the first-order part of eq. (26) is i~o = H D p l - p l H o . (28) Solving this equation will yield PlThe solution is easily carried out in the representation in which H o is diagonal. The matrix elements of/~o can be calculated as follows:

(a(t)llJolA(t)) = d ( a l P o l A ) / d t - ( a l P o l d A / d t ) - (da/dtlPolA~.

(29)

The first term on the right-hand side vanishes because (a]po[A) vanishes; the second term vanishes because (alpo vanishes; the third term equals - ( d a / d t [ A ) , which by first-order perturbation theory gives

- (da/dtlA) = - (a[~Hv/OtlA)(t h - r/A)-~ = +D(alQlA)(q~-r/a) -x

(30)

The other matrix elements of/~ o are calculated in a similar manner and one finds

(a(t)l~olb(t)) = 0,

(31a)

(A(t)llSolB(t)) = 0,

(31b)

( a( t)llJolA( t) ) = f) ( alQlA ) ( r / , - r/a)- ~,

(31c)

(A(t)llJola(t)) = b(AlQla)(rl~-r/a) -1

(31d)

Substituting in eq. (28) and taking matrix elements yields

(alp~lb) = (A[plIB) = 0,

(32a)

(alP~lA) = iD(alQlA)(~la-rla) -2,

(32b)

( AIp~ la) = - iD( AlQla)(r/a- r/a) -2.

(32c)

This completes the calculation ofp~. It can be checked that supplementary conditions (17) are satisfied to first-order. We shall also need the diagonal elements of P2, but a direct solution of equation of motion (26) is not necessary for them; they are obtained from the supplementary conditions, whose second-order part is

PoP2+P2Po+P~ = P2,

(33a)

Tr P2 = 0.

(33b)

The ( a l . . . ]a) matrix element of eq. (33a) gives

(alP21a) = E (aIPllAXAIP~Ia) --- 1)2 ~ i(alQlA)12(r/a_r/A)-,, A

A

(34a)

NUCLEAR DEFORMATIONS (IV)

249

while the ( A I . . . ]A) matrix element gives

(AIp2[A> = - ~ (,AIP~la)(alP,lA) = - D 2 ~ I(alQlA)lZ(rl.-rla) -4. d

(34b)

a

It is seen that eq. (33b) is also satisfied. 3.5. THE COLLECTIVE ENERGY We are now in a position to derive an equation of motion for D alone. There are two ways to do this. The more straightforward way is to carry the solution of eq. (26) to second order, and then to substitute the p(t) so obtained into the self-consistency condition (24) (there are.no first-order terms in this condition). One obtains a secondorder differential equation [eq. (40)] for D. However, the same result can be obtained in an easier way by first deriving art expression for the collective energy as function of D and /5 and then using Lagrange's equation to obtain the equation of motion. We follow this second course. The collective energy ~ is simply the expectation value of the many-body Hamiltonian H for our approximate time-dependent wave function, i.e. it is given by eq. (21b) carried to second order. However, since D -- xq to all orders, we have = Tr (Po + P t + Pz)HD + ½oZ/z.

(35)

The trace is easily calculated from the results of subsect. 3.4 Tr poHD = Z qa(D) ,

(36a)

A

Tr Pl HD = 0,

(36b)

Tr pzHo = /52 ~ i(alQ[A)lZ(q _na)-3.

(36c)

aA

Hence, the collective energy can be written

,Of = ¢"(D)+½B(D)D 2,

(37)

¢/'(D) = E t/a(O) + ½D2/z ,

(38)

with A

B(D) = 2 ~ I(aIQIA)IZ(tI.-qA) -3.

(39)

aA

The equation of motion derived by either of the two methods mentioned earlier is

B b + ½(OB/OD)/5 2 + D/Z - q o(D) = 0,

(40)

qo(D) = T r P o Q.

(41)

with

3.6. C O M P A R I S O N

WITH

THE

CRANKING

MODEL

The result of the cranking model for the collective energy is

~'~cr = Ho(D) + ½B/52.

(42)

250

M. B A R A N G E R A N D K . K U M A R

The collective potential energy Ho(D ) is the static H F calculated energy by eq. (21b) using only the Po part of the density matrix, i.e.

Ho(D) = Z qa(D) + Dqo(D)- ½zq~(D)

(43a)

A

= ~ (A(D)]Hs[A(D))-X2zq2(D).

(43b)

A

The first expression differs from the result just found with the T D H F method by

$~'(D)-Ho(D ) = 7(D , - )~qo)'/Z.

(44)

The two potential energies are the same at equilibrium (D = Zqo), which is as it should be since no objection has been raised against the static H F method. Away from equilibrium, the T D H F potential energy is always larger than Ho(D ). The difference between the two potential energies is of fourth order in the slowness parameter because ql = Tr pi Q vanishes, hence it can be argued that the replacment of H 0 by ~/F is not significant. Be that as it may, the fact remains that the present formulation is free of the objections against the cranking model that were reviewed in subsect. 2.2. Therefore Y/~(D), Ho(D ) or their analogs from sect. 4 can be used with confidence in calculations of actual nuclei. Actually, ~ ( D ) is the easier function to calculate since it depends only on r/a, while Ho(D) also requires the knowledge of q0- The close similarity between $~'(D) and the function minimized by Mottelson and Nilsson was discussed in ref. 2), subsect. 7.1. Lest it be thought that the present theory is merely a minor variation of the cranking theory, we shall emphasize again the conceptual differences between them. In the cranking the)ry, it is customary to distinguish the "potential deformation" D from the "wave deformation" Xqo. The two are equal only at equilibrium; in other words, only at equilibrium does one achieve self-consistency. In our formulation, the selfconsistency between the two deformations is enforced at all times. This is because the words "wave function" have for us a different meaning; they refer to the density matrix p, while the cranking approach considers Po as the density matrix defining the wave function. It is still true in our formulation that D and ;(qo differ everywhere except at equilibrium. The new ingredient, which is the T D H F ingredient, is that by improving the definition of the wave function, taking into account the effect upon it of the time-variation of the potential, we are justified in demanding self-consistency at all times. Turning now to a comparison of the kinetic energies from the two models, we see that they are identical. It may seem strange that an improvement which was originally designed to correct the time-dependent aspects of the cranking model has ended as a modification of the potential energy but not of the kinetic energy. Obviously the answer must be that the meaning of the deformation is not quite the same in both cases. One may well ask, where is the additional term in the inertial parameters which is said by Thouless 2~) to be required for self-consistency? It is there! The function of this

NUCLEAR DEFORMATIONS (IV)

251

term is to take into account the change in the potential induced by first-order changes in the wave function. In our case, this is taken care of by the definition of D as being self-consistent at all times. Perhaps it should be said that the surprising simplicity of these results is due entirely to the separable nature of the two-body interaction and to the fact that exchange terms are always omitted. This is why the theory cannot be extended beyond the pairing-plus-quadrupole model. The no-exchange assumption must be considered an integral part of the model; without it the P + Q interaction becomes just as complicated as any other interaction and loses its main virtue, which is its simplicity. One of the many unsatisfactory aspects of the theory is that it needs to be quantized. There is no unambiguous way of performing this 8). If the usual 24) quantization procedure is applied to Hamiltonian (37), the resulting ground state energy is obviously higher than the minimum of ~F'(D) or H o ( D ). But we know (subsect. 3.2) that this minimum can be obtained by the Rayleigh-Ritz variational principle, hence it should be an upper bound for the true ground state energy. This contradiction shows that the present method cannot be relied upon to calculate absolute energies. The relative energies are hopefully better.

4. The full pairing-plus-quadrupole model 4.1. H A R T R E E - B O G O L Y U B O V

THEORY

Consider a general many-body problem with Hamiltonian =

V~,p~c~,cpcac~, ~7

(45)

~tflT6

in which the interaction includes the exchange term, so that =

-

vp

,o

=

-

=

=

(46)

The Hartree-Bogolyubov (HB) approximation 2 3 , 2 5 , 26) is designed to treat field effects and pairing effects self-consistently and on the same footing. We shall only quote the results that we need. The proofs are in the references. The theory can be made very similar to Hartree-Fock theory by doubling the dimensionality of the representation, in particular, the Hartree-Fock one-body density p is replaced by the matrix ~c* l - p *

'

(47)

in which p is still given by eq. (16), and the pairing density ~c is defined by ~:p~, = (cac~,).

(48)

252

M. BARANGER AND K. KUMAR

The matrix ~ satisfies three supplementary conditions (49a) Tr~

= m,

(49b)

f . ~ f = 1 - ..~*,

(49c)

where m is the dimensionality of the original representation a n d f the matrix

((o) J =

(50)

(o)f

The HB ground state wave function does not conserve the number of particles as is evident from eq. (48) for x, therefore it is necessary to introduce a chemical potential and instead of H, to work with the Hamiltonian H' = H-2

• c~c~,.

(51)

In the time-dependent problem, it will be necessary to let 2 be a function of time determined by the requirement that the average number of particles be correct at all times Tr p = n.

(52)

The HB potential generated by density matrix ~ , i.e. the analogue of the Hartree potential, is the 2m-dimensional matrix

[T-X+r = \ -A*

A ) -T*+i.-F*

(53)

'

in which the Hartree-potential F and the pairing potential A arise from p and x according to the formulae F ~ = ~ V~p~p~p, #a

(54a)

A~p = ½ E V~araxar.

(54b)

~6

Self-consistency requires that this potential built upon ~ also generate wave function in return. In the static case, this is expressed by ~"-~g/'~

= o,

(55)

which is analogous to eq. (19). The role of the special representation of eqs. (9) is now played by the eigenvectors of YV'. The latter occur in pairs with opposite eigenvalues according to the Hartree-Bogolyubov equations ¢¢/'1d,) = E , [ d , ) ,

~/']M,) = - E i [ ~ , ) .

i = 1 to

m.

(56a) (56b)

NUCLEAR DEFORMATIONS (IV)

253

The quasi-particle energy E i is always positive. The paired eigenvectors are related by the equation 15~,) = ( ) I d , ) ) * ,

(57)

which follows from the symmetry relation

(58)

f ~ / ' f = - ~K" *

satisfied by ~-U. Eq. (55) is satisfied by making 1 ~ ) and IMi) eigenvectors of 6~ also. There is no sharp distinction between filled and empty levels. Instead, the correct relations are ~ldi)=0, ~[~i) = I~,),

i= 1

to

m.

(59a) (59b)

The breakdown of eigenvectors I d l ) and I,~i) (assumed to be normalized) into two sets of m-components is according to the notation

~)~i = B,*,

~2

= A~*,

(60b)

where we have used a superscript 1 or 2 to distinguish between the first m and the last m of the 2m components of I d i ) and IMi). The coefficients A and B are those that occur in the Bogolyubov transformation relating the quasi-particle creation operator a~ to the particle creation and annihilation operators a~ = E (A~,,c*~ + B~,c,,).

(61)

Finally, the expression of ~ in terms of ~¢ and ~ implied by eqs. (59) can also be written P~ = E B~i* B~,i,

(62a)

i

tq, p = E A~,IBp*"

(62b)

i

4.2. HARTREE-BOGOLYUBOV TREATMENT OF THE PAIRING-PLUS-QUADRUPOLE MODEL For simplicity, we shall discuss the theory as if only particles of a single charge were present. The extension to the physical case with both neutrons and protons is straightforward and has been discussed in ref. 2), sect. 2. All sums must now be extended over both neutron states and proton states. There are two pairing constants, g. and gp but a single quadrupole constant X. The different oscillator lengths for neutrons and protons are taken into account by replacing the quadrupole operator Q by P = ~2Q, where a2 is a correction factor slightly larger than unity for neutrons and slightly smaller for protons.

254

M. BARANGER AND K. KUMAR

In the pairing-plus-quadrupole model, the single-particle energy of Hamiltonian (45) has the form T,¢ = e,6,r,

(63)

and the interaction is

-z ~ (--)M(<~IQ~I~> - <~]QMI6>.

(64)

M

The notations are the same as in refs. "z). For convenience, we recall s~ = ( - ) J " - " ' . With the usual approximations [ref. 1), subsect. 2.1], formulae (54)for the Hartree and pairing potentials take a very simple form. The Hartree potential is F,r = - Z (-)~to~(c~lQMly),

(65)

M

with DM = ;~q~t,

(66)

qM = ~ (fllQMlfi)p~.

(67)

The pairing potential t is A~# = - s, 6pi A,

(68)

A

(69)

= +½9 Z s~x~.

One must keep in mind that is real and that its dependence on magnetic quantum numbers is given by the Wigner-Eckart theorem; various simple relations follow from this, for instance <=IQMI7> = <71Q*MI=> = <7I(--)MQ~Ic*>,

(70a)

q~t = ( _ ) M q ~ ,

(70b)

D* = (--)MD~,

(70c)

<=IQMIT> = s~sr<9]QMla>, ,

(70d)

F~ = s~s~F~.

With the special form for the pairing potential, the HB equations (56) are

E,A~. =

( e . - 2 ) A . , + Z C,.A~,-As~,B~,, , EiB~i = + As~A~i--(G-A)B~i-- ~ F~rBr~*'

(71)

7

With the definition B;, = - s , Bi,

(72)

* T h e sign c o n v e n t i o n s for some of the q u a n t i t i e s connec t e d with p a i r i n g are u n f o r t u n a t e l y different from those o f refs. 23, ~6). The new signs are m u c h m o r e co nve ni e nt especially w i t h re ga rd to eq. (88).

NUCLEAR DEFORMATIONS (IV)

255

and the use of eq. (70d), they can be transformed to E,A:i = ( e : - 2 ) A : i + Z F:+A~+ AB'~,, .g

E,B', = A A : i - ( e : - 2 ) B : , -t

I ~. F:,B~,.

(73)

vl C~i,

(74)

The solution is of the form A=i = u~ C=i,

B'i =

where C=i is a solution of the m-dimensional eigenvalue problem rhC,i = e~,C~,i+ ~ P, TC~i,

(75)

7

and ui and v i are solutions of the two-dimensional eigenvalue problem E,u, = ( r h - 2 ) u , + av,, Eivl = A u i - ( r h - 2 ) v i .

(76)

Explicit expressions for the eigenvalue (assumed positive) and the eigenvector of the latter problem are E, = [(q,-,~)2+A2]~, u i = [ ( E , + q i - 2 ) / Z E i ] ~,

v i = [(Ei-rh+2)/ZE[[ ~.

(77)

In other words, the simple form assumed by the pairing potential enables one to split the solution of the HB equations into two distinct parts; first, the diagonalization of the deformed one-body Hamiltonian Hs+F,

H D =

(78)

where we have used the notation H s instead of T to denote the spherical, singleparticle Hamiltonian as in ref. 1) and second, the standard BCS solution 19) associated with the eigenstates and eigenvalues of H o. The density matrices corresponding to this solution are given by eqs. (62) and are easily found to be p ~ = s~s~ Z v2, (79a) i t<~p =

St3 Z U i V i < ~ z [ i > < i ] f l > ' i

(79b)

with the notation : C=i.

(80)

Then, the potentials generated by these densities are given by eqs. (65)-(69). Upon substitution of the above expressions for p and K, the expressions for qM, eq. (67), and A, eq. (69), become qM = • v2(i]Qu[i>, (81) i

a = ½g Z u,v,. i

(82)

256

M. BARANGER AND K. KUMAR

Use has been made of eq. (70c) in the calculation of qM. The static solution of the HB problem is obtained from two self-consistency conditions, saying that the iterated potentials must equal the original ones. In the case of the pairing potential, this gives the very simple relation

1 = kg E E?'.

(83)

It is well-known that the solution of the self-consistency conditions is equivalent to the minimization of the average total energy with respect to variations of D M and A. With the usual approximations, this energy is

Ho(DM , A) = Tr ( H s p ) - ¼ g ~', s~K*~E srtc~

-½Z ~ (_)M Z P~,Z p~p.

(84)

Upon substitution of expressions (79) for p and ~:, it becomes

Ho(DM, A) = Tr ( H s p ) - l g ( Z u,v~)z - x Z ~', [qMIz" i

(85)

M

The minimization must be with respect to the original D u and A, those with which eqs. (75) and (76) were solved; self-consistency conditions (66) and (82) must not be used. The representation of the states [i> or Nilsson representation should be a convenient one to use in our further work. With this representation, we expect that the generalized Bogolyubov transformation (61) should reduce to the usual BogolyubovValatin transformation 27, 15) a~ = ui c~ - s, 01ci.

(86)

To check this, substitute (72) and (74) in (61) which becomes

a~ = u, Z c~<~]i>+ vi Z s,c,
(87)

The first term on the right-hand side of eq. (87) agrees with the corresponding term of eq. (86), given the customary transformation law for creation operators. To give agreement between the second terms also, the state li> must be defined by (s~ is assumed real) s, li> = - ~ sJ~> (88a) ct

= + Z s,l~)
(88b)

It can be seen easily with the help of eq. (70d) that sl]i) so defined is an eigenstate of the deformed one-body Hamiltonian eq. (75) or (78) for the same eigenvalue ~/i as ]i>; it is just the time reversed of 1i>. We have not defined sl; this is not necessary as the combination s~]i> is the only one that enters. However, it is convenient to allow for an arbitrary definition of ]i> subject to the condition that s~ be real, and s i is then

NUCLEAR DEFORMATIONS(IV)

257

determined by comparing this definition with definition (88) of sili). For instance, li) may be defined in such a way that, in the limit of zero deformation, the Ii), If) representation becomes identical with le), I~). Then, eq. (88b) shows that s t becomes identical with s~. Just as in the case of Ic~), we define [i) ~ [i). There follows, again just as in the case of leO, that si = - s v To see this, write eq. (88a) twice, once for i and once for i, in the form si(ct[~) = - s~( il~),

(89a)

si(fll i ) = - s~( ilfl ).

(89b)

Take the complex conjugate of eq. (89b) letting fl become ~, (90)

(ctl~) = s~s~(i]~).

Substitution in eq. (89a) gives s~s-~

=

(91)

-1.

We recall that all sums ~ in the present work include summation over the timereversed state i as well as i. There remains to write matrices ~ and Y#" of subsect. 4.1 in the/-representation. This is easily performed and the result is

\ "6i~i Sl Us Vl \ fiijSi A

'Sijs~u~v~'~ ~ijU 2 ] '

(92)

-fiijsiA

(93)

- ~j(n~-

'~ .

2)1

In each case, i is the line index and j the column index of an m x m submatrix. As for the eigenvectors (60) of ~¢/~,they become in the/-representation ~ l j = (~fjUi '

~ 2 ij = 6~jsivi,

(94a)

'¢~b = ~,isivi,

~

(94b)

= 6,jui.

The second subscript labels the eigenvector, while the superscript and the first subscript label the particular component of that eigenvector. 4.3, TIME-DEPENDENT ItARTREE-BOGOLYUBOV APPROXIMATION Now that the static Hartree-Bogolyubov theory has been developed in detail, the HB generalization of subsects. 3.3 and 3.4 is straightforward. The time-dependent HB wave function is completely characterized by a time-dependent density ~ sarisfying supplementary conditions (49). The motion of this density in potential ~ is given by the Schr/Sdinger equation i~ = :¢'~-

~3q:.

(95)

M. BARANGER AND K. KUMAR

258

Self-consistency must be enforced at all times, i.e. the potential W" must also be that which is generated by ~ according to eqs. (53) and (54). But the motion is assumed to be slow, thus ~ can be written ~' = d o + ~ ' ,

(96)

where ~ o is the static density defined in eqs. (59) by its effect on the eigenvectors of ~¢/', and ~ ' is a small correction which can be expanded in powers of the slowness parameter ~ t = "~1 + ~ 2 + ~ 3 At- . . . . (97) The first-order part of eq. (97) is i ~ o = ~ g ' ~ l - ~ 1 ~¢/'.

(98)

The solution of this equation in the representation in which ~/" is diagonal is carried out exactly as in subsect. 3.4, and the result is (d~[~lldj}

= (~'i[~l[~j)

= 0,

(99a)

<~t~i[ '"~1 I ~ j >

=

--

i(E, + Ej)- 2,

(99b)

(~jl~lld,)

=

+i(~jIYP'Id,)(E,+Ej) -2.

(99c)

It can be checked that supplementary conditions (49) are satisfied to first order. As in subsect. 3.4, we also need the diagonal elements of ~ 2 and, once again, they come out of the supplementary conditions in second order. They are = ~ J

II2(E,+Ej)-',

<~',1~'21~,> = - Z J

t<'~,IYP'Isdj>I2(E,+Ej)-'*.

(100a) (100b)

4.4. THE COLLECTIVE ENERGY As in subsect. 3.5, the quickest procedure at this point is to take the expectation value of the many-body Hamiltonian H for our approximate time-dependent HB wave function. In the pairing-plus-quadrupole model, this expectation value is given by eq. (84). In the present case, however, self-consistency conditions (66) and (69) are satisfied at all times, therefore the collective energy becomes 5¢{ = Tr

(Hsp)-g-lA 2-½Z-'

~ IDMI2.

(101)

M

The problem is to calculate Tr(Hsp ) to second order in the slowness parameter; eq. (79a) is not sufficient as it gives us Po but not Pl and P2. In analogy with the simple model of sect. 3 [see transition from eq. (21a) to eq. (21b)], we might suspect that Tr(Hsp ) is simply connected to T r ( # ~ ) ; this is indeed the case. Recalling expressions (53) and (47) for ~ and ~ and definition (78)

NUCLEAR DEFORMATIONS (IV)

259

of HD, one finds Tr ($¢r~) = 2 Tr (HDp)--22 Tr p - T r ( H D - 2 ) + T r

(A~c*+A*t¢).

(102)

The self-consistency conditions enable one to write Tr (HDp) = Tr ( U s p ) - Z -1 ~ iDM[2,

(103)

M

Tr (Ax* + A*~c) = - 4 g -

1A2.

(104)

Upon solving for Tr(Hsp) in terms of Tr(¢t/'~) and substituting in eq. (101), one gets = ½ Tr ( $ / / ' ~ ) + ( n - l m ) 2 +½ Tr H s + g - l d 2 + ½ Z - 1 ~, [DM[2. M

(105)

The time-dependent ~ was calculated in subsect. 4.3, and Tr($//'~2) has the following three terms: Tr (~'~2o) = - ~

E,,

(106a)

i

Tr ($/:~1) = 0, Tr ( ~ 2 )

(106b)

= Y, I<~¢,I¢:I~j>I2(E, + E j)- ~.

(106c)

ij

Finally, the collective energy consists of the sum of the potential energy $/" = ½ Tr

Hs+(n-½m)2-½ ~ EI+g-IA2+½Z -1 ~ i

IDol 2,

(107)

M

and the kinetic energy J

(108)

= ½ ~ I ( d , lC~l~'j)12(E,+ Ej) -3 ij

In view of the remark following eq. (91), the kinetic energy can just as well be written in the form ~-- = 1 E I(dzl:~l~'))12(Ez+EJ) -3,

(109)

ij

which turns out to be more convenient. There remains to write the kinetic energy in terms of the time derivatives of the collective parameters, i.e. DM, A and 4. An expression for ~ cannot be obtained directly from eq. (93) since the latter is written in the Nilsson representation which is time dependent. One must start from an expression valid in a fixed representation, such as eq. (53), take the time derivative, and then go over to the Nilsson representation. The result is

:" = \

6,)s,/i

~':2 + ~Mb~(i

QMIJ)*

"

(110)

Expression (110) can now be combined with eqs. (94) in order to obtain (d,lY]fi#2j).

260

M. BARANGER AND K. KUMAR

A little care is needed in handling the intermediate sums, and one finds

-

= 6,j sj[(u~ - v~)A + 2u~ v, 4]

+ u~sjvj ~ b* + s~viuj ~ bM*. M

(111)

M

It is easy to show using the explicit expression (88) for s,li) and relation (70c) for QM that sisj(ilQMlj)*

= (-)~(ilQ~lj).

(112)

With the help of eq. (70b), the matrix element entering in the kinetic energy (109) takes the form -

<~e',l¢~l~j> = sj(u, vj + v, u~) Z D* M

+ sj 6,j[(u 2 _ v2)fl + 2u i vi ~].

(113)

4.5. CONSERVATION OF THE NUMBER OF PARTICLES It has already been mentioned in subsect. 4.1 that 2(0 is not an independent para,meter, but that it must be chosen so as to give the correct number of particles at all times according to eq. (52). However, the time-dependent density entering the latter equation is not identical with the static density P0 whose trace is easily written Tr Po = ~ v2,

(114)

i

but differs from it by terms of first and second order in the slowness parameter, therefore the correct way to determine 2 is through the equation L v 2 + T r Pl + T r P2 = n.

(115)

i

We have examined these correction terms to Trp and found them to be extremely small in most cases. It seems pointless to further complicate the theory by trying to use eq. (115), when the BCS approximation which we are using already treats the problem of the number of particles in a very rough way in any case. Therefore, in all our calculations, we have determined 2 by the simple condition Z v~ = n.

(116)

i

4.6. T H E T I M E V A R I A T I O N

OF THE PAIRING

POTENTIAL

Thus far, D M and A have been considered as six independent collective variables. Collective motion in the D M directions constitutes the familiar rotations and quadrupole oscillations. Collective motion in the A direction represents the pairing vibrations which have recently received much attention 2a). These pairing fluctuations can lead to 0 ÷ excited states in certain nuclei. There have been some apparently suc-

NUCLEAR DEFORMATIONS ( I V )

261

cessful calculations of such 0 ÷ states using the RPA or the two-quasi-particle approximation and conventional two-body forces 29). However, we prefer not to include pairing vibrations together with collective quadrupole motion in our theory. We do this for simplicity, and also because the two phenomena usually manifest themselves in different nuclei and at different energies. After this decision, we do not consider A as an independent variable, and connect the value of A to that of DM. It turns out that this can be done without changing the formalism already developed by a simple re-interpretation of the theory. The only reasonable way to specify A is to restrict it to being a solution of the static self-consistency condition (83) for each deformation D M. In other words, instead of regarding any choice of D M and A as a possible trial potential, we allow only those choices satisfying eq. (83). The corrections to density ~ due to the motion are still given by eq. (95). Already in the work until now, the use of eq. (95) involved an approximation, namely the projection of the true wave function on the subspace of HB wave functions. Now, an additional projection is performed, namely projection on the subspace of HB wave functions derived from a A satisfying eq. (83). However, the approximate solution of eq. (95) is not changed, and the same expression is obtained for the collective energy (subsect. 4.4), except for the fact that A, like 2, is now a function o f DM. Therefore, the time derivatives A and J. appearing in eq. (113) are given in terms of DM by the expressions A = E (eZ/aOM)bM,

(l17a)

M

}" = Z (~32/ODM)bM,

(117b)

M

and the kinetic energy takes the general form ~-'- ~

1 -2BMN D" *M D" N .

(118)

MN

Now that A satisfies eq. (83), it is possible to transform expression (107) for the potential energy into the perhaps more familiar expression 3~(DM) = E v2~li=o-lAZ + l z - ' E IDM]2" i

(119)

M

Another possible refinement in the theory would consist in allowing for the decrease of A at large rotational quantum numbers 30). We can ignore this complication since we do not intend to use this work for such large quantum numbers, most of our interest being concentrated ia the low-lying ( ~ 1 MeV or less) states of transiti 3n nuclei. This effect would be of fourth-order in the slowness parameter, which is why it does not appear in the present treatment. 4.7.

COMPARISON

WITH

THE

CRANKING

MODEL

The comparison with the cranking model leads to the same conclusions as in subsect. 3.6. The kinetic energy [eq. (109) completed by eq. (113)] agrees formally with

M. BARANGERAND K. KUMAR

262

that of the cranking model, though the meaning of the variables is different. For instance, an expression equivalent to ours is given in the appendix of the paper by B~s 11). On the other hand, the potential energies are different. That of the cranking model is the deformation energy of ref. ~) and is given by

Ho(DM) = ~ v2(ilnsl i) -g-1az-½• ~ lqoMI2, i

(120)

M

where qoM is the average quadrupole moment calculated with the static density Po, i.e. qoM is given by eq. (81). Our potential energy is given by eq. (119). In order to compare them, one can for instance replace H s by HD-F in eq. (120), and one finds

"¢/'(Du)- Ho(DM) = ½Z-1

~ M

I D a - Zqo~12.

(121)

The two are the same at the equilibrium deformation, and elsewhere ~ is always larger. See subject. 3.6 for further discussion. 4.8. REDUCTION TO INTRINSIC AXES In ref. 1), subsect. 2.2, we introduced an alternative choice for the five components of a quadrupole tensor. In this notation, the components are real and are called 0, 2', xy, xz, yz, respectively, or # for a variable component. This choice is convenient because, when the quadrupole tensor is reduced to its intrinsic axes, only components 0 and 2' differ from zero. In the case of the deformation tensor, we shall adopt the following connection 1) between Do, D 2 , , and the usual deformation parameters fl, ~: Do = D cos 7,

(122a)

D sin 7,

(122b)

O = mo92fl.

(122c)

D 2, =

Here coo is the harmonic oscillator frequency of the single-particle well and m the mass of the nucleon. In the intrinsic frame of D the solutions of the Nilsson equation (75) mix only states whose magnetic quantum numbers differ by 0 or 2. It is convenient to refer to states with m = ½, - 1 , ~, - ~ . . . . as being of the "direct" kind, while those containing - ½, z-, 2 --~, ½. . . . are said to be of the "reverse" kind. The time-reversed [as defined by eqs. (88)] of a state of the direct kind is a state of the reverse kind and vice versa. In the intrinsic frame, the potential energy (119) obviously depends only on D O and D2,. Now, we consider the vibrational kinetic energy (4). It is that part of the kinetic energy which remains, when the intrinsic axes are held fixed and only D O and D 2, are changed. If it is written as

~-vib

= 12Boo

"2 + Do

B o 2 ' D"o D"2 '

+½B2,2,/)22,,

(123)

NUCLEAR DEFORMATIONS (IV)

263

then eqs. (109), (113) and (117) give for the inertial coefficients (#, v = 0, 2')

Bj,,,(Do , D2, ) = 2 (ui vj + v i u j ) 2 ( ilOul J)(JlQ,,I i)(Ei + E j)- 3 i-¢ j

+ E [2u,v,((ilQ~,li)+ 02[OO.) + (u~ - v~)Oh/OD~,] i

x [2u,v,((ilQvli) + O2/ODv) + (u~ - v~)OA/OD~](2E,)- a.

(124)

Expressions for O2/OD~, and OA/OD~, at the equilibrium deformation were given by B6s 11) and in ref. 1). However, close examination of the derivation reveals that the expressions are actually valid for any deformation. For convenience, we repeat these results here O2[OD, = - (aG, + bd~,)/(a 2 + b2), (125a)

OA/OD~, = (ad~,- bG,)/(a 2 + b2),

(125b)

with the definitions

a = ~_, A/E 3,

(126a)

i

b = ~. ( q , - 2)/E~,

(126b)

i

c~, = ~., A(,ilQuli)/E~,

(126c)

i

d,, = ~_, (rh- ).)( ilQ.,,li) /EP. i

The other part of the kinetic energy is the rotational kinetic energy. write it in form (3), we must place ourselves in the intrinsic system and infinitesimal rotation around one of the axes. In a rotation of the wave infinitesimal angle ~ around axis k, the tensor operator TLM transforms

TIM = (1 + iSJk)ZLM(1 -- i~Jk) ~, TLM -t- i~(J k T L M - TLMJk).

(126d) In order to perform an function by into (127)

The commutator is given by (ref. 31), p. 53)

Jk TLU-- TLMJk = ~', T L N ( L N I J R I L M ) . N

(128)

The deformation D u equals )~ times the expectation value of the second-rank tensor Qu, hence its transformation in this rotation is given by

D'M = DM + is ~ O~(2NlJd2M). N

(129)

The matrix elements of Jk are well known. Now this expression must be written in terms of the real component of D. The results are: (i) for an infinitesimal rotation around intrinsic axis 3, D~ = Do, D~. D'xy = 2eDz,, t D'z = Drz = 0;

=

D2, ,

(130)

264

M. B A R A N G E R A N D K . K U M A R

(ii) for an infinitesimal rotation around intrinsic axis 1,

D;

=

DO

D 2, =

DE, ,

D;z = e(D 2, +x/3Do), ' = D'xz = 0; Oxy

(131)

(iii) for an infinitesimal rotation around intrinsic axis 2, D~ = Do,

DE, = DE,,

x/300),

D;z = t

t

D~y = Dr z = 0.

(132)

Thus, for an infinitesimal rotation around axis 3, for instance, the only component of D which changes is D~y, and its time derivative is bxr = 2D2,o93, where e~3 is the angular velocity. Hence term ~~- ~ 3 ( - 0 23 in eq. (3) must be identified with term 1Bxy' xfl)Ey in kinetic energy (118), therefore the third moment of inertia is given by ~3

=

2 4DE'Bxy, xy = 4D 2 sin 2 ~Bxy, xy

(133a)

calculated in the intrinsic axes. Similarly the other moments are J a = (DE' + ~3Do)EBy~, r~ = 4D2 sinE(7-120°)Byz, yz,

(133b)

J 2 = (DE,-~/3Do)EB . . . . . = 4D E sin E (7+120°)B . . . . . .

(133c)

Explicit expressions for the B-coefficients are

B:,r, ~y = ~, (uivj + v~u~)2l< ilQ,¢rlJ>lE(E~ + Ey) - 3,

(134)

ij

and similarly for B:,.... and Brz ' rz; there are no derivatives of 2 and A as in eq. (124), because 2 and A do not change in a rotation. The expressions just found for the moments of inertia involve matrix elements of the quadrupole operator, but they can be transformed so as to involve matrix elements of the angular m o m e n t u m , in which case they are recognized once again as the usual cranking-model formulae. To see this in the case of J 3 , consider the c o m m u t a t o r of J3 and HD,

J3 UP-- HDJ3 = - Z O~t(Ja Q m - QMJa).

(135)

M

The c o m m u t a t o r of J3 and QM can be calculated by eq. (128), and the result valid only in the intrinsic system is

J3 H o - HD J 3

=

-- 2 i D E '

Q~y.

(136)

Substitution in the above expression for ~'3 yields

J a = ~.. (uivj+viuj)21(ilJaHD--HoJalj)]2(g,+

Ej) -a

IJ

-- ~ (u~ vj + v, u j) 2(q, - q~)21(ilJa IJ)l JJ

2(E, + E j)- 3.

(137)

NUCLEAR DEFORMATIONS(IV)

265

By virtue of the identity = Ii bl)j - - 13i u j ,

(138)

J 3 = ~ (ui vj - viuj)2l(ilJ3l j)IZ(E, + E j)- 1.

(139)

(U i Vj

"~/)i Uj)(?]i- ~/j)(Ei "{-E j ) - I

this reduces to the well-known formula 32) l.I

The argument is the same for the other two moments, and so is the final formula. We repeat that, though eq. (139) is the cranking formula, the theory by which it was derived is not the cranking theory; it is a time-dependent HB approach, free from the objections that are often raised against the cranking theory but valid only in connection with the P + Q model. 4.9. MAGNETIC EFFECTS In order to be able to calculate electromagnetic transition probabilities,it is desirable to know the electric and magnetic properties of the collective motion. They are given in terms of the intrinsic electric quadrupole and magnetic dipole moments which were defined in eqs. (133) and (143) of ref. 8). These are simply the moments expressed in the intrinsic frame for a given deformation. They are calculated as q = Tr(p Q) and p = Tr(pM), respectively, where p is the time-dependent density. In keeping with the assumption of small collective velocities, it should be sufficient to expand to first order in the slowness parameter. But T r ( p l Q ) vanishes, as one would expect from the evenness of Q under time reversal, and therefore the static quadrupole moment can be used. The situation is different for the magnetic moment since it is odd under time reserval. The Tr(P0M ) vanishes, but T r ( p l M ) does not; the magnetic moment is entirely due to the motion, obviously. To calculate Tr(paM), we rewrite it in the doubled representation as p = Tr (plM) = ½ Tr (~1 ~//), where

o)

~( =

- M*

"

(140)

(141)

The trace (140) is performed in the d , ~ representation Tr ( ~ 1 J / / / ) = 2 < d ~ l ~ l l ~ j > < ~ j l J / g t d , > ij

+ ~. (~,l~lI~¢j)(djV/gl~,). t3

(142)

The matrix elements of ~ 1 are given by eqs. (99) and (113). Those of .//g can be related to the matrix elements of M in the Nilsson representation with the help of eqs. (94), which give


(143)

266

M. BARANGER AND K. KUMAR

In this equation, M is the single-particle operator

M = gzL+gsS ,

(144)

with different gyromagnetic ratios g~ and gs for neutrons and for protons. The oddness of M under time-reversal is expressed by the equation

s, sj ( [lMlj> = -
(145)

which can be derived from eqs. (88) and from the properties of M in the spherical representation. This relation may be used to rewrite eq. (143) as s s ( ~ j [ J / / I d ~ ) = (u,v s - v~us)(jlMli ).

(146)

Finally, the intrinsic magnetic moment takes the form

P = ½~ <~l~ll~j>sjsj<~)lMIdt>

+ complex conjugate

(147)

iS

-- - i½ ~ ss(u,v s - v, uj)(E,+ Ei)- 2 + complex conjugate. (148) ij

At this point, we have to use the fact that we are in the intrinsic frame and look at various types of motion. For vibrational motion, only /)o and 1)2, are non-zero. Then eq. (113) shows that is real, and that i a n d j must be states of the same kind, from which it follows that ( j l M l i ) is also real. Hence expression (148) vanishes; there is no magnetic moment due to vibrations. Now look at rotations. Then A and ~, vanish. For a rotation around intrinsic axis 3, it was shown earlier that only Dxy changes, and that/)xy = 2D2,o93- Substituting this in eq. (113) and using eqs. (136) and (138), we obtain



- i ( ~ ¢ ' , [ ~ [ ~ j ) s j = (u,vj+ v, uj)(ilQxrlj>2iDz, 093

(149a)

= -o93(u~v.i+viuj)(i[J3Ho-HoJ3]j>

(149b)

= + o~3(uivj + v, uj)(rh- rlj)

(149c)

= w 3 ( E , + E j ) ( u , v j - v, uj).

(149d)

Similar results hold for rotations around the other two axes. They must now be inserted in eq. (148). It is found as expected that, for rotation around intrinsic axis k, only component k of M yields a non-zero result. This is a consequence of the facts that the 3-components of M and J always connect two states of the same kind (direct or reverse), while the 1 and 2-components connect states of opposite kinds, and that the matrix elements of the 1 - a n d 3-components are real, while those of the 2-components are imaginary. The final result for the k-component of the intrinsic magnetic moment is Ilk = W-OkE (Uil)j -- vlus)2 < ilJk]J>(Ei"b EJ) tJ

1.

(150)

But o9k = lk/Sk, where I k is the k-component of the collective angular momentum s)

NUCLEAR DEFORMATIONS (IV)

267

and J k the kth moment of inertia as given by eq. (139). Therefore

I~k = gklk,

051)

gk = jeffi ~ (uivj_viuj)Z(Ei_t_Ej)-l,

(152)

with t.I

which is the cranking result once again 33).

5. Core effects in the collective motion 5.1. STATEMENT OF THE PROBLEM The pairing-plus-quadrupole model is good only if the action of the forces is restricted to a rather small range of states near the Fermi surface. Reasons for this were given in ref. 2). It does not mean that the other states forming what may be called "the core" have no effect. The core manifests itself in a renormalization of the parameters of the theory. Sect. 6 of ref. 2) described a simple way of understanding how the core increases the quadrupole force constant and replaces the proton and neutron charges by certain effective values. Likewise, when the nucleus undergoes collective motion, the inertia of the core must be added to the kinetic energy somehow. In order to do this, we seek a time-dependent treatment of the core model of ref. 2), in which the core is represented by a harmonic oscillator coupled to the external nucleons by a quadrupole force. We shall perform it in detail for the simple case containing a single collective coordinate as developed in sect. 3 of the present paper. The extension to the full P + Q model is straightforward but more involved. The Hamiltonian of the system is taken to be

H = H s - l X ~ Q(s) ~ Q ( t ) - k R ~ Q(s)+K. S

t

(153)

$

The first two terms are the same as in eq. (7). The last one is the Hamiltonian of the free core K = - ½ b - 1~2/0R 2 + ½cRz, (154) i.e. an oscillator Hamiltonian for the coordinate R, the quadrupole m o m e n t of the core, with inertial parameter b and stiffness constant c. The remaining term of H is the quadrupole interaction of strength k between the core and the external nucleons. 5.3. STATIC SOLUTION The wave function of the whole system is taken to be a product 17"o) = I~o)lq~),

(155)

where Iq~) is an arbitrary wave function of the coordinate R and l~bi~) a Slater determinant of single-particle wave functions calculated in the one-body Hamiltonian of eq. (8), i.e. I~ko) is just the type of wave function considered in subsect. 3.2. The

268

M. BARANGER AND K. KUMAR

expectation value of H with the exchange term neglected is n o ( D ) = (WDIHI wD) = ~ tla + O q - ½zq z - kq(~olRko) + (~olKilo).

(156)

,A

The notations are the same as in subsect. 3.2. This expression has to be minimized with respect to both [~o) and D. The minimization with respect to k0) yields the ground state of the following oscillator: K - kqR = - ½b- ' ~2/~R2 + ½c(R - kq/c) 2 - ½k2q2/c, (l 57) whose origin has been displaced to R = kq/c and whose ground state energy is 71 w - ~ k1

2

q 2 /c,

(158)

co = (c/b) ~:.

(159)

with After this choice of [~p), the energy becomes H o ( P ) = ~ qa + D q - ½(g + k2/c)q 2 + !o9"

(160)

A

Henceforth, the situation is the same as in subsect. 3.2, except for the additional constant ½o9, the zero-point energy of the oscillator, and except for the replacement of the quadrupole force constant by an effective value Xeff =

g "]- k2/c.

(161)

Once again, it is convenient to introduce density matrices. In addition to p, which has the same meaning as before, there is the oscillator density matrix g. Since we have assumed that the oscillator had to be in a pure state, a obeys the supplementary conditions 6 2 ---- 6 , (162a) Tr 6 = 1. 5.3. T I M E - D E P E N D E N T

(162b)

SOLUTION

It is now assumed that both the Slater determinant and the oscillator wave function vary slowly with time. Self-consistency must hold at all times, and therefore the singleparticle Hamiltonian seen by the external particles is as in eq. (23) with D = zq+kr,

(163)

q = TrpQ

(164a)

r = Tr aR, while the effective oscillator Hamiltonian is K - k q R .

(164b) The Schr6dinger equations for

NUCLEAR DEFORMATIONS (IV)

269

the density matrices are eq. (22) and ib = ( K - k q R ) a - a ( K - k q R ) .

(165)

Solutions of these equations in the adiabatic approximation proceed as in subsect. 3.4. The solution for p is exactly the same as previously. The solution for a is written in the form a = ao(t)+al(t)+a2(t)+ .... (166) where ao(t ) is the density corresponding to the instantaneous ground state of the displaced oscillator, eq. (157), and al and a 2 are of first and second order, respectively, in the slowness parameter. Let the eigenvalues and eigenstates of the displaced oscillator be defined by the equations ( K - k q R ) l q ~ i ) = ei[¢Pi),

(167)

i = 0 denoting the ground state. Then, all matrix elements of a l are found to vanish except those connecting the ground state to an excited state (i # 0), which are QPilal[~po ) =

ik~(qgi[Rl~oo)(ei-eo) -2,

(q~ o ]al IcPl) = - ikdl(CPo[Rlcpl)(e i - e o ) - 2.

(168a) (168b)

Moreover, from well-known properties of the oscillator, matrix element (q~i[R[q~o) itself vanishes unless cp, is the first excited state (i = 1). Also needed are the matrix elements of a 2, which can be obtained from the supplementary conditions, as usual. The only non-vanishing ones are (qh[azlq~l) -- [(q~olal[cpl)[ 2 -- k2~Zl(qgolR[cpz)12(el-eo) -4, (~oolG21~Oo) =

-(~oxla2ltpx).

(169a) (169b)

Explicit expressions can be substituted for the oscillator matrix element and eigenvalues and then QPlla2lq~l> = -- (q~o[a21q~o) = k2q2/2c c03.

(170)

5.4. T H E C O L L E C T I V E E N E R G Y

The collective energy , ~ is obtained by taking the expectation value of the manybody Hamiltonian (153) with the result •~

= Tr p H s - ½xq 2 - kqr + Tr irK = Tr pHD + D q - ½xq 2 + Tr a ( K - kqR).

(171)

One must include terms up to second order in the slowness parameter. The first term Tr pHD is the same as in subsect. 3.5, namely Tr p H D = E tlA(D)+½B(D) bz" A

(172)

270

M. BARANGER AND K. KUMAR

The last term Tr ~r(K--kqR) breaks up into three contributions Tr g o ( K - kqR) = ½09- ½k2q2/c,

(173a)

Tr a l ( K - k q R ) = O,

(173b)

Yr t r 2 ( K - k q R ) = ½k2d12/c092.

(173c)

Altogether the collective energy ~

can be written as the sum of a potential energy

"If(D) = ½~o+ Z ~la(D) +Dq - ½;Gffq2,

(174)

A

gaf being defined in eq. (161), arid a kinetic energy

J - = ½B(D)D 2 +½kEd12/co92.

(175)

The term ½co in V is just a constant and can be omitted. However, this is not the end, because ~ still contains the two variables D and q; since q is a function of D, it must be eliminated. The relation between D and q is given by eq. (163), r being calculated as r = Tr a o R + T r ~ r l R + T r o'eR.

(176)

The first term Tr aoR = kq/c, while Tr a i R actually vanishes. Hence

r = kq/c+ second order term,

(177)

D = Zeff q + second order term.

(178)

or

Since the theory does not attempt to go beyond second order, it is permissible to write the kinetic energy as J - = ½[-B(D) +

k2/C(D2Z2ff]b2.

(179)

In the potential energy, one can write 1

2

1

2'

Dq--½Zeffq 2 -~ ~D / Z e f f - - ~ ( D - - z e f f q ) / Z c f f ,

(180)

the last term being of fourth order and therefore negligible. Hence, the potential energy may be written 3e~(O) = ~ rla(O ) + ½D2/x~ff. (181) A

The collective energy is now identical to that of sect. 3 except for the addition of a constant to the inertial parameter B(D) and the substitution of an effective value ~(eff for the quadrupole force strength Z. 5.5. E X T E N S I O N

TO THE

REALISTIC

CASE

The full pairing-plus-quadrupole model leads to the same conclusions. The interaction of the outside particles with the core has the effect of substituting Zeff for •.

NUCLEAR DEFORMATIONS(IV)

271

These two quantities were called Z and •, respectively, in ref. E); their actual values were discussed there and also the renormalization of the quadrupole moments. The core contribution to the kinetic energy has the form ½Bo E b ~,, E

(182)

/t

with B c = k2/coE)~eff "2

(183)

Because o) is not known, it does not seem possible to estimate B c reliably. It has to be kept as a parameter to be fitted to the data. The quantity B c must be added to the vibrational inertial coefficients Boo and B2, 2, Of eq. (123). It also yields a core contribution to each m o m e n t of inertia. By eqs. (133), this is J k c = 4D2 sine (Y--2~zk)Bc •

(184)

The rotation of the core also contributes to the collective magnetic moment the quantity Pk~ = ~ k J k , 9 ¢ , (185) which must be added to expression (150). Here 9o is the gyromagnetic ratio of the core, which may be taken equal t to Z/A in the usual units of e/2mc. Since now OOk = l k / ( J k + Jk~), the modification of eq. (152) is Ok = (~b~k"~-J k e ) - - l [ o e J k ¢ ; "~- ~ (u, vj--v, uj)2(ilJklJ)(jlMkli)(E,+ Ej)-a] • (186) ij

The sum in this expression extends over both neutron and proton states. We thank E. Marshalek, S. Moszkowski and M. V6n6roni for stimulating discussions. M. Baranger acknowledges the hospitality of the Aspen Center for Physics during the summer of 1963, when most of this theory was formulated. K. K u m a r thanks H. M c M a n u s for his encouragement and support during the years 1963-1966, while the theory was being put in calculable form. t See eq. (12) of ref. 3) for a better estimate.

References 1) 2) 3) 4) 5) 6) 7) 8)

M. Baranger and K. Kumar, Nucl. Phys. 62 (1965) 113 M. Baranger and K. Kumar, Nucl. Phys. A l l 0 (1968) 490 K. Kurnar and M. Baranger, Nucl. Phys. A l l 0 (1968) 529 J. de Boer and J. Eichler, in Advances in nuclear physics, Vol. I, ed. by M. Baranger and E. Vogt (Plenum Press, New York, 1968) K. Kumar, Ph.D. thesis, Carnegie Institute of Technology (1963) unpublished M. Baranger and K. Kumar, in Perspectives in modern physics, ed. by R. E. Marshak (J. Wiley, New York, 1966) K. Kumar and M. Baranger, Phys. Rev. Lett. 17 (1966) 1146 K. Kumar and M. Baranger, Nucl. Phys. A92 (1967) 608

272

M. B A R A N G E R A N D

K. KUMAR

9) D. R. B~s and R. A. Sorensen, in Advances in nuclear physics, Vol. 2, ed. by M. Baranger and E. Vogt (Plenum Press, New York, 1969) 10) D. R. Inglis, Phys. Rev. 96 (1954) 1059; 103 (1956) 1786 11) D. R. B~s, Mat. Fys. Medd. Dan. Vid. Selsk. 33, No. 2 (1961) 12) A. K. Kerman, Ann. of Phys. 12 (1961) 300 13) S. A. Moszkowski, Topics in nuclear models, unpublished lecture notes (1965) 14) S. T. Belyaev, Nucl. Phys. 64 (1965) 17 15) M. Baranger, Phys. Rev. 120 (1960) 957 16) D. J. Rowe, Nucl. Phys. 85 (1966) 365 17) R. E. Peierls and D. J. Thouless, Nucl. Phys. 38 (1962) 154 18) A. K. Kerman and A. Klein, Phys. Rev. 132 (1963) 1326, 138 (1965) B1323; A. Klein, L. Celenza and A. K. Kerman, Phys. Rev. 140 (1965) B245; G. Do Dang and A. Klein, Phys. Rev. 156 (1967) 1159 19) J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175 20) L. I. Schiff, Quantum mechanics, second ed. (McGraw-Hill, New York, 1955) p. 215 21) A. B. Migdal, Nucl. Phys. 13 (1959) 655; D. J. Thouless, Nucl. Phys. 21 (1960) 225 22) D. J. Thouless and J. G. Valatin, Nucl. Phys. 31 (1962) 211 23) M. Baranger, in 1962 Cargbse lectures in theoretical physics, ed. by M. L~vy (Benjamin, New York, 1963) 24) W. Pauli, in Handbuch der Physik, Vol. 24/1 (Springer, Berlin, 1933) p. 120; A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1952) 25) N. N. Bogolyubov, Usp. Fiz. Nauk 67 (1959) 549; J. G. Valatin, Phys. Rev. 122 (1961) 1012; M. Baranger, Phys. Rev. 130 (1963) 1244; F. Herbut and M. Vuii6i6, to be published 26) M. Baranger, Phys. Rev. 122 (1961) 992 27) N. N. Bogolyubov, Nuovo Cim. 7 (1958) 794; J. G. Valatin, Nuovo Cim. 7 (1958) 843 28) A. Bohr, in Congr. Int. de physique nucl6aire, Paris, ed. by P. Gugenberger (Editions du C.N.R.S., Paris, 1964) p. 488; D. R. B~s and R. A. Broglia, Nucl. Phys. 80 (1966) 290; R. A. Broglia and C. Riedel, Nucl. Phys. A92 (1967) 145; S. T. Belyaev, Sov. J. Nucl. Phys. 4 (1967) 671 29) R. Arvieu et al., Phys. Lett. 4 (1963) 119 30) B. R. Mottelson and J. G. Valatin, Phys. Rev. Lett. 5 (1960) 511 31) D. M. Brink and G. R. Satchler, Angular momentum (Oxford University Press, 1962) 32) S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 11 (1959) 33) S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 16 (1960)