Unified theory of nuclear rotations

Unified theory of nuclear rotations

ANNALS OF PHYSICS: 56, 224258 Unified FELIX Laboratory Massachusetts (1970) Theory M. of Nuclear Rotations* H. VJLLARS AND G. COOPER+ for Nuc...

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56, 224258

Unified FELIX Laboratory Massachusetts


Theory M.

of Nuclear



for Nuclear Science and Physics Department, Institute of Technology, Cambridge, Massachusetts

The nuclear Hamiltonian is expressed in terms of the total angular momentum variables, and of intrinsic particle coordinates by means of a contact transformation. The intrinsic structure is approximated by a Hartree-Fock solution, and the Coriolis coupling diagonalized by admitting particle-hole excitations. It is shown that in this approximation, the rotational energy takes the conventional form, as given by the “unified” model, the moment of inertia being determined by a selfconsistent “cranking” formula. This, and corresponding results for matrix elements of multipole operators substantiate the validity of the more intuitive procedures based on the unified model and on the use of single particle orbitals in deformed potential wells.



This publication attempts to tie together and to bring to a conclusion a long series of attempts, by many authors, to present a consistent picture of nuclear rotation and intrinsic motion, and to offer a practical schemefor a really unified treatment of the dynamics of nuclei with rotational states. The method presented here is based on an explicit introduction of collective (angular and angular momentum) variables, and of a (redundant) set of intrinsic variables which remain in one-to-one correspondence with particle coordinates. This last feature is essentialfor any treatment of intrinsic structure in terms of independent-particle or quasi-particle methods (Hartree-Fock or B.C.S.) The introduction of collective rotational coordinates requires that a “body-fixed” coordinate system be defined. The three orthogonal unit vectors e”, (a = 1, 2, 3) which define this system are vector operators, and it follows from the scalar nature of the Hamiltonian H that it can be expressedin terms of powers of the scalar operators J, = j * &, (.? being the total angular momentum), and in terms of * This work has been supported in part by the Atomic Energy Commission under Contract No. AT(30-1)2098. + Present Address: Lawrence Radiation Laboratory Livermore, California.








additional scalar variables (1). The intrinsic coordinate axes may be defined in a variety of ways, as first pointed out by Eckart in 1935 (2). In the context of nuclear physics, explicit expressions of H in terms of the J, and intrinsic variables were first given by Lipkin, DeShalit and Talmi (3), Tamura (4), Nataf (5) and Villars (6). In Refs. 3 and 5, the method of redundant variables is used, which makes it possible to use the (scalar) body fixed particle coordinates & = (x’< . 2,) as independent variables. This method will be followed here, since it is conceptually the most straightforward. Not much use appears to have been made of these early results. The main reason for this curious fact seems to have been a. lack of confidence that these expressions for H in new coordinates would lend themselves to a realistic treatment of the problem. The standard device (followed by all above mentioned authors) of defining the body fixed coordinate axes by the principal axes of the quadrupole mass tensor leads to an expression, in which the rotational energy is not diagonalized exactly, and which in zero order approximation (neglect of all non diagonal terms), gives the “hydrodynamic” value for the rotational moment of inertia, familiar from the early Bohr theory of collective rotation, (7, S), (and in disagreement with all experimental evidence). The subsequent investigation with time dependent Hartree-Fock or B.C.S. theories (9) (or their corresponding R.P.A. versions) (IO), have convincingly shown that the moment of inertia, in any theory of independent particles or quasiparticles, is given by the “self-consistent cranking” formula of Thouless and Valatin (II). This is also in reasonable agreement with experimental data (12, 23). Despite these gratifying successesof particle and quasiparticle theories, a growing literature on the subject of rotational (and other collective) states in springing up (14-22). Two reasons may account for this: First, the treatment of rotation in the HF or BCS theories is restricted in scope, since it is embedded in a theory of small amplitude oscillations which in fact excludes finite rotations. Subtle arguments are then needed (23), to calculate matrix-elements of operators which engage the collective degrees of freedom. It appears therefore desirable to embed these results in a framework which would be more general and allow a straight-forward handling of transition amplitudes between collective states, such as y-transition, or collective excitations in nuclear collisions, for instance. Such a formalism would also, hopefully, justify the more intuitive-but so successful-prescriptions based on the “unified model” (24, 25). This last statement brings up the second point: It has in fact newer been clearly established that the method of coordinate transformation (with introduction of collective and intrinsic variables), does in fact lead to results which are in basic agreement with the prescriptions based on H.F. or B.C.S. theory. The purpose of the present paper is to supply at least the basic elements of this connection. 595/56/1-15





As a first step, we reproduce briefly the derivation of the transformed Hamiltonian (Section II). This follows pretty closely the lucid derivation given in Ref. 3; we add, however, a discussion of spin (Section III), and supplement it by a brief presentation of the symmetry properties of H, and of the symmetry conditions on the wave function (Section IV). Since it is the case of axial symmetry which is of greatest practical interest (and also the simplest case), we use it to illustrate a variational approximation to the eigenvalue problem for the transformed Hamiltonian, (Section V). It is then shown that in the framework of a Hartree-Fock approximation to the intrinsic problem (easily generalized to a B.C.S. method), the eigenvalue problem for the rotational energy leads to a relation very close to the self-consistent cranking formula for the moment of inertia. For the important K = l/2 case (Section VI) it is shown that the “decoupling” term is approximately given by the expression postulated by the “unified model”. In fact, the final discussion of matrix-elements of tensor-operators shows, that the present approach in a large measure substantiates the generally followed, more heuristic procedures, in which the general “structure” is supplied by the “unified model”, and intrinsic matrix-elements evaluated by independent particle or quasi-particle techniques, using suitably deformed single particle potentials.







The nucleus, a system of A identical particles, will be described by means of A position vectors Zi , and suitable spin and isospin variab1es.r The -it, have space fixed, Cartesian components xiN(a = 1, 2, 3). The introduction of angular variables describing rotation of the total system requires the introduction of a “body-fixed” coordinate system. This system will be expressed by means of three orthogonal unit vectors C,, (a = 1, 2, 3): (@a. 4) = &xb

(e^, x &) = P, ,

(1) (abc) being a cyclic permutation of 123. The Cartesian components ear,of the vectors 6, define a rotation matrix, which is expressible in terms of suitable Euler angle variables, which we shall call 0, (s = 123). (Say 8, = y, 8, = 8, 8, = $, to tie it in with conventional rotation; See Appendix E.) The position vectors x’i have body fixed components xi, : x;, = (J;i * f?,), (2) and hence I . xia! = eaaxia (3) 1 These latter will be ignored in this section, but brought in Section III.






Notice that by (3), the xi, are scalar operators, and commute with the total angular momentum. The body fixed system may now be defined by imposing a collective constraint on the xi, . Sufficient, and convenient, is the condition that the quadrupole tensor

should be diagonal in the body-fixed system: QAb = C x:,xib = Q; tiab .


Let us make clear what is achieved by (4): First, it provides a definition of the Euler angles 8, (and hence of the vectors C,), in terms of the particle coordinates xia : Indeed, for (4) and (3), one has 2 .Yioxigemae8b= 0 (a # b) (5) holds identically in the xi, , and a straightforward shows that the %,/ax, satisfy the relation: e


EC( ax,






(5) argument






The Euler angles 0,(x,,) will then be part of the set of new variables, in terms of which we formulate the problem. It would be convenient to use also the xi, as new variables, were it not for their lack of independence. In fact, the arguments for having at hand variables which are in one to one correspondence with particle position are over-riding. We will therefore want to retain the xia . To give them the status of independent variables, we adjoin to the initial set xi, three spurious variables 5, , in such a way that 5, = 0 expresses the condition of constraint (4), on the xi, . The unconstrained xi,, then generate finite, non zero values of 5. This may be achieved in a variety of ways, all equivalent. Two transformations are particularly convenient: Transformation I:

Equation (7a, b) defines a transformation: xia(x;, , 0) and {(xi,) and its inverse: x&+ , [), 0(x:, , 5). Clearly 5 = 0 expresses the condition of constraint (4).





Transformation II: Here we introduce the principal moments Qp(x,,) of the mass tensor Qa, , but do not identify them with the Qa’ of Eq. (4). Rather we put:

In (8b), the R,, are rotation matrices, and the three redundant variables are the Euler angles E, . If they vanish, then QA, is diagonal, which again expresses condition (4) (QA, will also be diagonal if the E, are suitable multiples of 7r/2, in which case the axes abc are a permutation of the axes p. See also Section IV.). The redundant variables 5, (or es) will also appear in the wave-function of the system, Y(xi, , CC),with the understanding that Y(xia , 5 = 0) defines the physical probability amplitude. The transformations (7) or (8) generate a wave function in the new variables, defined by: wi,

3 5) = w;,

7 0)


The physical range of 9 is on a hypersurface, defined by QL, = 0 (b f will thus be normalized according to the prescription

For the transformation

c). It

(7), the Jacobian J is given by

J =

(notice that dQ, = dq d cos 19d+); for the transformation (8) J is unity, assuming de = de, d cos cp de, for the volume element of the three spurious Euler angles. For the transformation

(7) then, the normalization

of 9 is

One point must be clearly understood at this occasion: The purpose of the whole transaction lies in the expectation that q(xi, , 0,) will have a simple, approximate representation of the form







with the +K being determined by a variatimal principle. Now any trial function of type (12), with only approximate functions & , will invade the space of the spurious variables. If expanded in terms of the exact eigen-functions YJxiJ of H, (12) will give

Clearly the desirable goal to be achieved is not that p should be altogether independent of 5, but that in the series (13) only one term should be strong. If this is achieved, we don’t really care what function C,(l) is generated by ‘r/. In actual practice there will always be many terms in (13), and some information on the functions C,(l) will be useful. The trial function (12), or its equivalent (13), leads to a mean energy: (the proper normalization of p is understood) E=

j dx. za j 4 WJ(Y*Hy/)

= j dQo j d&x 6 ( pa,fbeb,

) (qf=/>

= cn EnI Cd5= W”.


Should we decide to drop the constraint S(5) in this expression, (and adjust the normalization of 9 correspondingly) we get instead

a result which is not necessarily much worse; it could even be equal to (14) if the C,(s) were all of the form &f(l). This, of course, will at best be only approximately the case. But one will well want to balance the discomfort of having to use a projection against the benefits derived from it. We now proceed to calculate the kinetic energy CT) = /n&a

J” 4 S(5) V’*TY)

-!2i?Z jdQs jnd&d( In calculating


a?P* a!?






recall that by (7a) or @a):

and hence a9




eaa (a, * sj

2 x! a9 m j lb ax;,

ae - aP

+ e 2

aa axior




Now the term i??/aO, can be expressed in terms of the total angular momentum,2 projected on the body-fixed axes: (17)3 This relation is: (See Appendix B):

In this equation,

and in what follows, we write the angular momentum


boldface type whenever it is to be viewed as an operator on the collective variables

OS(For the explicit form of these operators, see Appendix E, Eq. (15).) Inserting this in (16), we notice again the appearence of the expression

The antisymmetry

of Mia,bc in bc allows the last term in (16) to be written as

We shall call L’ the intrinsic angular momentum. It operates on the scalar variables xta , and commutes with all components of the total angular momentum L, which operates on 8:

2We recall again that by refs. (7a, 8a), the eaa are vector operators, and the xi,, scalars; hence [L:s I xj,] = 0. a Properly speaking, we should define all particle coordinates relative to the center of mass of the nucleus. This introduces no complications (See Refs. 3 and 6). We avoid this here to keep all expressions as simple as possible.







Using (18) and (19) we can finally write

The difference (Lbc - L&) = B’,,c is often called the “rotational” angular momentum. We must keep in mind however, that for a function P(x~, , 6,). L operates on 0, L’ on the xi,, and the angular momentum eigenstates of the system are characterized by the eigenvalues of L, not by those of 9’. To get from (18) to the expression for (T), Eq. (13), we need the sum

which is easily found, using Eq. (6), and the fact that in the physical range (5 = 0 or E = 0), Qbb is diagonal: c (Mia,ac)a = 2’ (Pa iu C in




+ Qb’ -


Cc f




and hence

In (22), an additional

operator: (23)

has made its appearence. The result (22) has been known for a long time. (See Refs. 3-6) No efforts, however, seem to have gone into attempts to make use of this expression. Eq. (22) is an exact result, and all approximations will reside in the choice of the trial function P = xKfK(0) &(x~,), and in the neglect, if desired, of the constraint S(Qib/Q,’ - Qb’) in evaluating matrix-elements.





An important question is now: Does the form (22) allow us to define a Hamiltonian in the new variables 0, and xi, ? Or, technically, can we integrate by parts? The answer is simple, and as follows: (a) The operators pka , Lh, , NL,, , taken separately, are not hermitian operators; they do not commute with the constraints S(Ql,). (b) It is however possible to integrate by parts all derivatives on p* simultaneously, without generating any derivatives of the Q,‘, and without engaging the constraints S(Qi,). (This latter fact is to be expected, since I&?= Ql, is a redundant variable). This proof will be detailed in Appendix C. It demonstrates that we may indeed consistently define a Hamiltonian I? in the new variables 8,) x6 , which may be written as

(Lb- Gb)2 b%dL,b









where the following abbreviations and definitions are used: pia = -i L;b



a/ax;, (x6P;b






z N;b







,tQa’ m(Qa’ -


(25) +



This expression for fi must be supplemented by the prescription: of fi are defined as


If this prescription is used, then it follows: (a) fi is hermitian in the following sense: All derivatives may be shifted from p* on q or vice versa, as if the constraint 8(Qhb) did not exist, and as if the jab and D,, were constant coefficients. (b) The statements made for fi in (b) do not hold separately for the operators in that sense. Pia , Lkb or Nib , which are not hermitian







(c) The operator I?, Eq. (24) commutes with all the operators the sense that the commutator V%

Qi,(a f

b), in

Q:,l - Q:,

and hence all its matrix-elements vanish by the condition of constraint. transform of the statement that [H, lab] = 0 in the original variables.

This is the

Although we have made clear (hopefully) that there is no subsidiary condition Z@/a<, = 0, it is nevertheless interesting to know what #//a[ looks like in terms of q and the new variables:


from (7a) or (8a) it follows

In conjunction


with (18), this gives (27)

This shows clearly what the effect of a subsidiary condition a?P/a[, = 0 would be: The integrand in (22) is reduced to its first term; the BSbecome redundant variables, by means of which we can integrate out the ?3(Qi,/Qa’ - Qb’), and we are back where we started! The relation (27) acquires an even more transparent form in term of the alternative redundant variables, the Euler angles E, , defined by (8b). One may then define an angular momentum operator 9&, acting on function of the Euler angles E, . In analogy to (18), this is related to a/&, by (28) and a simple bit of algebra (Appendix -4O,b(E)


, E)iQ;,-O

D) shows that =





3 es).






So the rotational angular momentum W in our transformed Hamiltonian is spurious. But this must not be misinterpreted. The total fi is nonspurious, but in (22) we have just achieved a particular decomposition, in which the separate pieces are partly or totally spurious.





In the case of particles with spin, the amplitude Y is a function of the xia and of the spin variables nri , which are the eigenvalues of Siz : Y(xi, , m,). In order to generalize the previous separation of 9 into a function f of the collective angular variables, and a function c$(&) of scalar intrinsic variables, we must replace the mi by scalar spin variables, too. Such variables are provided by the eigenvalues pi = &l/2 of the scalar operators sj3 = (Sj * q.

(III. 1)

We shall therefore write the trial function Y/(x,, , mi) first in the form yCxiav





but then express 4(x:= , mi) in terms of amplitudes relating the two amplitudes is:




C&X;, , pi). The transformation

the 9(e) being the Wigner rotation matrices, in the notation of Brink & Satchler, and Rose (26) (See Appendix E). They are unitary matrices, satisfying in particular the relation (111.3) It should be stated perhaps that we make no attempt to absorb the factor L%(e) in f(0). The reason of not doing this will be clear presently: Going back to Eq. (11.20), it is evident that with the trial function (111.2), new terms will arise, containing

4 The arguments presented in the first part of this section are well known; See Refs. 27, 28.







In our notations, the LB%,(e) are angular momentum follows (See Appendix E), that


and it

where (,u’ 1sub 1p) are the S = l/2 spin-matrices (s12 = 40~ , etc.) As a result of (111.4), the relation (11.20) is modified to read

This makes it at once clear that we can eliminate the 9(e) from the expression for (T) by means of Eq. (ITI.3). Formally, the net effect will be the replacement of #J(x;= , mi) by C&X;, , pi) and the replacement of L;b








In substance, however, there is an additional change, in the fact that, for the trial function (IIL2a,b) the term L&f(8) now expresses the total (not just the total orbital) angular momentum of the system. Since this is an important point to appreciate, let us verify this directly. Define the total angular momentum, projected on the body fixed axis, as

Operating on the trial function (IK2a,b)

one has

-f(e) c (... pi ... / s,, ; ... pi’ ...> J(.&, &,I u’



In the last term, one can



(m j s&, j m’) into the p, p’ representation:


It is then seen that the two last terms cancel out, so that indeed

It will be useful to express this in a notational change, and henceforth write Jab for Lab . It is singly the angular momentum operator, acting on a function of the Euler angles. After elimination of the 9~~(8) from (T), we deal effectively with a new representation of the state, in which the whole angular dependence is contained infV>; B(& , pi> 1s . now a function of scalar variables only, and Labf = Jabf, necessarily expresses the total angular momentum. In terms of 9 = C f(e,) && , pLi), the expression for the mean kinetic energy is now given by


The arguments leading to a definition of a Hamiltonian in new variables are not affected by the spin variable. We can define an effective Hamiltonian A, whose structure is 17 = HN + Hmt + f-f, (7)

5 dr stands

for II dx:,

df& 6(Qha/Q.’









The statements made in the previous section concerning the Hermiticity of B, the definitions of its matrix elements etc., hold separately for the three terms of (7).





The Hamiltonian (11.24) or (111.7,8) commutes with all space fixed components of the total angular momentum, JaB . The functionsf(0) in pare therefore Wigner D-functions (see Appendix E), for which h3*JMK = J(J+

1) S3zK (1)

From Eq. (11.29) one would conclude that A should also commute with the rotation operator gab = Jab - Jib , since H clearly commutes with L?&(C). We must realize, however, that in view of the definitions of matrix elements of Z? through Eq. (11.26) ( an d corresponding definitions for matrix-elements in general), the relevant invariance group is that of

rather than that of A alone. This group is that of the 24 discrete rotations generated by products of ei(n’2)ge~~,and reflects the 24 ways of labelling the intrinsic axes. Notice that these symmetry operations leave the particle coordinates xiLvinvariant, but in yl(xi, , 5,) permute the spurious coordinates <, (or, if the Y(xi, , EJ are used, the E, are changed by multiples of n/2). But the important point is that, whatever the transformation of Y(xi, , 5,) under ei(““@~b may be, the reduction to the physical domain Y(x, , iJ + Y(xi, , [ = 0) must lead to a unique answer; in other words, Y/(x,, , [ = 0) must be invariant under these operations,6 so as to define a unique amplitude in the physical variables xior (29). Of particular interest is the subgroup of the above mentioned octahedral group, formed by the rotations by 7r about the intrinsic axes: eein*ab. Since eVzni@ag= 1, and e-inJtLe-if1492 = e-i”“?e-i”“l = e-i”” 3, this is an abelian subgroup of 4 elements, 6 These symmetry conditions need not be invoked if the three principal moments are distinct, and their vibration amplitudes small enough. In this case one may assign QI’ > Q2’ > Q,’ for instance, and ignore the quantum mechanical tunneling into a different configuration. If these premises do not apply, then the symmetry considerations become crucial. See Refs. 7 and 29.





with 4 representations or rank I, which define the well known 4 symmetry classes of the asymmetric rotator (30). In our case, only the symmetry class A, for which e-in%by



all (ab),


will be allowed. Let us now consider the important special case of axial symmetry. This situation arises whenever two of the principal moments of the mass tensor are sufficiently close to each other, so that all matrix elements containing a positive power of their difference, (say (Q,’ - Q2’)n, n > 0) may be neglected, or treated as a small perturbation. There is then a favoured direction, which we will identify with the f3 - axis. With this assignment fixed, we have ei@3 as a symmetry operation, for any es . It is seen from (11.25) that the rotational energy associated with R3 becomes infinite in the axially symmetric limit; on the other hand, it is now possible and consistent to impose on 9 the subsidiary condition w,!P = 0,


which eliminates this term, which is totally spurious. In this case, one of the three conditions of constraint becomes redundant. In fact, the value of Q&Q,’ - Qz’ becomes unspecified; its value represents the tangent of a spurious angle variable Ed, which together with 19~= fi defines the position of the l-axis in the 1 - 2plane. This angle does not enter into any matrix-element calculated with wave functions satisfying Eq. (3), and we simply drop S(Q;,/Ql’ - Qz’), (This of course calls for a corresponding adjustment in the definition of the normalization of p). Thus, in the axially symmetric case, we deal with a Hamiltonian: f? = &

cp;u” + V(x,‘,) +2+{(J,

- &{N&(J,

- jr’)’ + (Jz - J9’)‘)

- Jr’) + N;,(J, - Jz’) + h.c.),


where d = dQ1’ - Q~‘MQI’ + Qs’) z.z m(Q,’ - Qa’MQz’ + Qs’)


D = m(Q1' - Q3')z


m(Qz' - Q;).

Any trial function satisfying (3) may be written as (6)







with J3’$J7d&)

= wbK(&),


and n standing for all additional quantum numbers needed to specify $. With (3) satisfied, Eq. (2) leads to the additional condition e-in%Ip






In what follows, we choose C& to express this symmetry; clearly, “2” has no geometrical meaning beyond representing an axis perpendicular to 3; it establishes, however, a phase convention associated with the 2-component of angular momentum. To enforce condition (8), we observe that e-inJz 9’,(e)

= (- l)J+K 9zK(e)


and a p which conforms to (8) will have the structure:

For K > 0, we are free to define &,-K as

the “time reversed” partner to $n,K. For K = 0, we may have (12)

Both signs are possible, but the + sign generally occurs in ground states of even nuclei (See Nilsson, (25)). A final comment on the symmetry properties of 9 concerns the parity, which relates Y(xi, , 5) and Y(-xh , 5). Now the transformations (117, 8) for xia , { to xi, , 19,establish 8 as eaen function of the xiol , so that !P(S& ) 5) = 9(x:, , 0) entails Y(-xi-

, 5, = ?P-x:,

) B).

The parity of p is therefore a property of the intrinsic wave function $(xi,).










In the Hamiltonian fi, (IV. 43, the terms linear in J provide a residual coupling of the rotational to the intrinsic motion. The transformation leading to A was not designed to minimize the effects of this coupling. The decoupling of the rotational energy is a dynamic problem, and no simple transformation of variables can achieve it. In the attempt to diagonalize the Hamiltonian (IV.4,5)in an approximate fashion, several simplifying assumptions and approximations wilI be introduced: (a) We consider an even, axially symmetric nucleus. In this case, K is either 0 or > 1. To order J2, no dynamic effects are produced by the symmetrization of 9, Eq. (IV. 12), and we need not explicitely consider it in the calculation which follows. (The case K = l/2 will be treated separately later). (b) We will neglect the constraints 8(Q;,) 8(Qi3) in the matrix elements of A. The physical reason which makes this approximation possible is the assumed strong deformation of the nucleus in question. Quantitatively, strong deformation is defined by the property (for axial symmetry) (q)

= = 1 dT($&&)

> 1


(do stands for 17 d&). As a result, C& does not violate the conditions


- Qa’) = 0


Q;d(Qz’ - Qa’, = 0 very badly: In fact, from

i-h’, Q;sl = 4Qz’ - Q;>, it follows that or

M2> m (Q2’ - Qd2, (2)

The error committed in approximating body operators F by

matrix elements of intrinsic one- or two-

(along with aab = s dT(&,*&)), is of order 1/(5i2), as a more detailed analysis will show. The approximation is therefore invalid for weakly deformed nuclei, but very good in the strongly deformed Rare Earth and transuranic region.







(c) Another approximation, consistent with strong deformation only, is the replacement of the operators ,$ and D in Eq. IV.5 by their mean values:


We thus assume that the axially symmetric shape is stable, and neglect all effects of vibration of the principal moments. The ensuing expression for A will be simplified notationally be introducing J, & iJ, = J* ,

JI’ i iJi = J+‘,

N& f iN& = Nk’,



With these definitions, one has

H Rot



(J2 - Ja2)>

HN = T + Y + +(J+‘X-

- -&

(54 +

J-‘A’+ + X-J+’ + X+J-‘1

(J+‘J-’ + J-‘J,‘) 0

H, = -(J+X-

+ J-X,).


The HN commutes with J3’ and is, of course, time reversal invariant. There are therefore eigenfunctions of HN of type (bn,*K, degenerate with respect to the sign of K. The associated energies will depend on 1K /, however. One would therefore first think of a trial function

which (for K + l/2) would give a rotational Eiot



(J(J + 1) - P). 0


energy (7)





This result is, however, not even approximately correct, as we shall see. In fact the structure of F(O) as given by (6) ignores the important coupling effects of the term (59.

We attempt to solve the eigenvalue problem for I? variationally, in an approximation, in which a Hartree-Fock solution is used for the intrinsic wave function 9 n.iK * The necessary improvements of the form (6) for q become evident if we operate on it with If, , as given by (5~): Hc 9,$+,,

= - dJ(J + 1) - K(K - 1) 9ZK-r(X&)

- dJ(J + 1) - KW + 1) ~~,+,cx++~,,,>. Visuahzing I& as a Slater-determinant, satisfying J3’& = K#J~~, the effect of the one-particle operators X, on +nK will be to generate particle-hole excitations, with J3’ = K f 1. This suggests a more general expression for a trial function p, in which the variational principle still leads to a Hartree-Fock problem for a single Slater determinant & : We generate a trial function q by operating on ly/(O) with an operator ei9: (we omit in what follows the symmetrization in K, -K, and also the tilda over Y’ and H) Y = eiFYy(0) = ei” LB~(d)





where 9- = J,F-

-I- J-F,

and Fi are one particle operators on &,K. The Fi are vector components in the intrinsic variables, so that w, ,w

= 0,

and hence 9’,Y = @JJ(O) = 0. The determination of Fi will be part of the variational presently. To order J2, one has

principle, as will be seen

+ J+(i[H,

, F-1 - X-) + J-(i[H,

, F+] - X+)

+ i [J+ (+ tHN> F-1 - X-1, s] + i [J- ($- [HN , F+] - x,), T] 1 Y(O).






We now consider two types of admissible variations of Y(O), consistent (J, - J3’) Y(O) = 0, and assuming &to be a determinant with J3’ = K.

243 with


SYr(O) = QzK S& : p ar t’IC1e-h o 1e excitations preserving the value of J3’.



= S;fKil

S+K~l : p-/z excitations changing the value of J3’ by &I.

The only terms contributing to 6(H) in a variation of Type b are the terms linear in J, and J- in (8). To assure a stationary (H), one needs then

This condition determines the matrix elements of Fi , once & is known, that is, particle & hole states defined. This is done by considering variations of Type a, which give

which determines +K (assumed to be a determinant), as the Hartree-Fock solution for HN. With Y(O) so determined, we can now write down the actual mean value of H according to (8). The two last terms contribute to the rotational energy. Using (9), one has

where COP>, = j d7 WP&


There remains the determination of F by Eq. (9). Let us begin with some notational preliminaries: HN is the sum of one and two-body operators H,,, = c h(i) + i ,z w(i,,j). z 2il


Let +K be the determinant of single particle states y,(p = I,..., A); Call q0 the unoccupied s.p. states; call # a p - h excitation of +K with J3’ = K’.





For any one-body operator F = 2 f(i), 4 Ci+j g(i, j), define matrix-elements

and two-body

hu=j %*f% * dwi” =is%*u> dL %*(2)

Let E, be the Hartree-Fock





(144 - 9%(l) (1W %m).

single particle energies (CZ= CIor CL>

(1% It then follows, (31) for any one-body operator F, that

Equation (16), and its adjoint, then are the equations by means of which the matrixelements of Fh , Eq. (9), are determined. One might think of approximating the rhs of (16) by its first term: (E, - ~,)f,, . An approximation of this kind is often made in the time-dependent Hartree-Fock theory of rotational energies, (9) and leads then to the well known “Inglis”formula (32) for the moment of inertia. In the present case, this approximation is not allowed. The reason for this is that w contains products of one particle operators: See Eq. (5~). These give in part rise to factorable two-body matrix elements in (16) and produce “coherent” effects that cannot be ignored: From (5b), one has:

We see from Eq. (17) that the following “coherent” c Lrf+w 7”

- f+vL)

terms arise in (16).

= W-‘7 F+l>o = iq

(184 Wb)

and the complex conjugates

([J,‘, F-l),

= iT*, ([X+ , F-I),

= i.$*. We drop,



however, the incoherent contribution the equation forf,, reduces to:





of the j- and x-matrix element in (17). Then,

The equation for fIOU is identical, with 7 * and t* replacing q and .$. There exists, however, a representation of single particle states in which j*bll and v&, , etc. are real, hence the fO, imaginary, and t and 17real. We notice also that with Eqs. (18), the moment of inertia yerf , Eq. (12), is given by

~ = $ + (5+ t*) = f ipI 0

+ 25.



Remains for Eq. (19) to be solved. We can simplify this equation, by observing that the operators F are only determined up to a multiple of the redundant variables Q;, 9 Q;, , which both commute with HN . Constructing the operators

Z* = QhIt iQ&


we observe that Eq. (9) is satisfied for both F*


p+ = F+ + AZ*

since Z* and HN commute. But in the implementation of Eq. (9) through Eqs. (16)-(19), F and P do not appear equivalent. Because of the commutation relations [J,‘, Z-1 = -[.I-‘,

Z-k] = -2(Q,’

- Qs’)

[X+ , z-1 = [X- , Z,] = 0 the substitution F-+ P leaves < unchanged, but changes 7: 7 + 77+ 2h( Q,’ - Q3’)o. Hence the moment of inertia (Eq. 20), is not affected by this “gauge transformation” of F, but the Eq. (19) for F is. It is convenient to choose a gauge which will make r] = - 1. With this, (19) reduces to





and Eq. (18a) becomes i([L’,


= i&I+‘,


= 1.


Now, Eqs. (22) and (23) are exactly of theform of the Thouless-Valatin equations (11) for & , derived from time-dependent Hartree-Fock theory. It must be pointed out that they will not produce the exactly identical moment of inertia as the Th.V. prescription. This is due to the fact that in our case, the single particle basis of states is determined by the Hartree-Fock problem for HN , Eq. (5b), whereas the Th.V. formula employs H.F. states of H. From a recent investigation of the Kelson criterion (33) by Banerjee et al. (34), it appears that the extra terms in HN will not affect the numerical value of $,rr very drastically. They can be solved explicitely in the approximation that the coupling term to rlA in (22) are ignored. In this case (24) and by Eq. (23):

which is the well known Inglis cranking formula (32). With this, we have traced the shortest path which, in the framework of this general approach, leads to a familiar result. This path contains a fair number of approximations, which are not unconditionally valid. Most of them are not easily relinguished. But the main purpose of this section (and of the next) was to demonstrate the sucessful use of Hartree-Fock methods-generalizable to a BCS method-in the context of this general approach to collective rotation. VI. THE CASE K = f 4

In the case K = &l/2, already approximation (V6) to gives a correction to the rotational energy spectrum due to an interference effect between K = l/2 and K = -l/2, and involving H, (Eq. V.5c): LIE,“,, = -(-1)J+“2(J,

l/2 I J- I J, -l/2)

s dr +1,2X++--1,2







This approximation to dE,,,t is just as poor as Eq. (V.7) for the quadratic (J(J + 1)) term. The method for correcting the wave functions will be the same as in Section V. So Eqs. (V.8) remain valid. It is in considering Eqs. (V.9, 10) that some care must be exercised. We must assume that the Hartree-Fock problem for HN , which emerges from (V.lO), leads to a doubly degenerate ground state, with the wave functions $K and 4-K 9 which differ only by a single unpaired one particle state, which is cpU,in I&, and y,, = ein’*fpGo



Equation (V.9) is then generally assumed to hold, except for 8& that is, it holds for all excited particle hole states 6$,*t, . The corrected decoupling energy is then

Got = (6 l)‘+W



= &;

j” dT~1,dI’W~, F+l - X+) +-l/2 ,


and it follows from the assumptions on &,, and $-1,2 , and in analogy to Eq. (V.16) that s

dT A,,tH,- 3F+l d-m = C (~~&.Ci,,J+rv - s+v~~$,.v,& 7”

We concentrate again on the “coherent” of (3) is approximated by


terms, as in (V.17), in which case the rhs

and hence ~&ot 2% (-l)J+“2(J

+ l/2> 1-u

+ rl)(x+L,p, + &

- e) (j_)U&j

Again the “gauge” 7 = -1 is used, and by (V.20) we have

e3t E (-- 1Y’YJ + VII.



-& (j+)&&



It is fairly straightforward to apply the results so far obtained to the evaluation of matrix-elements of tensor operators. In the case that such operators are functions





of particle momenta, some care must be exercised, however, to get the proper approximation. The case of the magnetic moment is a good example, which will be used to illustrate this point. The magnetic moment operator z= C zi, in units of nuclear magnetons, is

(1) Using Eqs. (11.3) and (X20), the components of A4 may be written (in a tensornotation):


On the explicit assumption that mass and charge distribution the form (11.6) for i14ia,ed gives

are proportional,

where (5)

The magnetic g-factor of the nucleus in a state of angular-momentum

J is

It is in the evaluation of the mean value in Eq. (6) that we must use care. In the case of axial symmetry, the trial function Y(O) (Eq. V.6) applied to (6) gives gJ=x+


Z i gK--

K2 1 J(J+l)’



KgK= j dr &&+$K .








This result is not correct, and indeed, the correction of the wave function defined by Eq. (V.7) profoundly modifies this result: The correction terms linear in 3 add to the matrix element in (6) the term $(J(J + 1) - P) (i [A+) - $ J+‘, F-1 + i [A?‘-’ - $ J-‘, F+]) 0

(the notation of Section V is used.) Now by (V.18a) and (V.23), (i[J+‘, F-l),

= (i[J-‘,

F+]h-, = 1,

and hence, adding in the correction terms gives, in place of (7): K2 &=&+k-gg,)


where g, = $(i[A+‘,

F-1 + i[dfl-‘, F+]), .


This expression for g, is of course just the self-consistent cranking formula for the collective g-factor, and in the approximation (V.24) for F:

a well known result (33). The generalization of this result, taking into account pairing (13, 32), would of course also follow from a corresponding generalization of the method presented here. No corresponding subtleties arise if purely geometric quantities are, considered, like the electric quadrupole moment. Any spherical tensor Tmz(xJ which is a function of the particle coordinates xia is related to the corresponding function T,(&) of the xi, by the relation

In the case of the quadrupole tensor Qm2(x), only the term Q&L) gives a non zero mean value, and in lowest approximation (using the trial function ‘F$&), we have

There are no first order corrections due to .F in this case; they would bring in Q&(x’), but these tensor components zero under the assumed proportionality of mass and charge distribution.






OF EQ. (11.6)

From (11.5):

one has, after differentiation

with respect to xi,, :

Re-expressing this in terms of the xi, , and using Eq. (11.4) again gives $beva




(Qb’ Q;) -


. +)



from which (11.6) follows immediately.




OF EQ. (11.18)

Eq. (11.15) for Lab gives iLab

= c (x;,eRb - xibeBa) f$ $ . -18 s z

Now there is an orthogonality




(B.1) with the expression

(qt. Pbj= - (6, *z&j ) and using (B.2) one has: (B.3) which is the desired result.







Since the collective operators L,, are hermitian, the proposed property must hold separately for every power of L occurring in (T). For the terms quadratic in L, the proposition is trivial. The contributions to (T) from the terms linear in L contains the expression (see Eq. (24)).

which, upon integration

by parts gives

plus some commutators.

These are

(C.3) To see that these commutator-terms

are zero, we observe that





Q;bl = -i(Q,’





9 &bl





Hence, in (C.3), the term

V-h , Qhl = -iQh , etc.


[X2 , Pi31 = +iQk, , etc.




Also [Li, , S(Q;,)] = -iQi, s’(Qi,), but this is accompanied by a factor i3(Qk3) and hence vanishes again. Finally, all commutators of L' or N’ with (Qa’ & Qb’) vanish, for a similar reason: [Liz, Q,'] = 2iQiz,


all these commutators are multipled with a &function which makes them vanish. A similar, albeit somewhat more lengthy argument shows that the proposition concerning H made in the Section Ii is generally true.




BETWEEN 3fiaa AND pab

If the angles es are chosen as redundant variables, Eq. (11.27) remains valid, with a(?,/&, replacing 30,/a<, . Now, Eq. (11.8b) implies that

and hence, since xi, = xi,(xj, , EJ: P-2)

In conjuction with CD.31 one derives the result:

In the constrained range of variables, when QA, is diagonal, this reduces to p.4) So assuming (Pa’ - Qb’) f 0 and Qi, = 0, one has, from (D.4) and (11.27):

and hence, by (11.28):







This appendix will contain a brief review of topics which may help to clarify some of the manipulations of Sections II and III. Our definition of the Euler angles @I) is the sameas that of the corresponding angles ~$37in Ref. (26): X1 , Xz , X, indicate the axes of a (space fixed) frame of reference S. Vector components with respect to these axes will be labelled D, (or ,B,y); The 1, 2, 3 label for the axes of the rotated frame S; vector components with respect to the axes in S will be labelled 5, (or b, c).


e tx3 \ “I \-



‘p 0 +



1 Ic, I'

It will be good to pay attention to three distinct representations of total angular momentum: (i) The operator P

= c ((x’i x &) + Si) I


and its components

J,“” in S



J,““e,, = eyQJyO, in S.


(ii) The matrices J1 , J, , J3 in a standard representation


J1 :t iJ, Ijm) = Sm~,m*l l/j(j

+ 1) - mm’

(jm’l J3 ljm) = m6mm#


and generally labelled J, or J, , depending on the context. This also defines the matrices f(JN). (iii) The differential operators J, and i, = eyaJyon the Euler angles 9, 0, #. We consider first a case where the q+ 0, # are external parameters, that is, 110 identification of S as a “body-fixed” system is made yet.





In this case, a component D, (in S) of a vector operator fi, and the corresponding component Cfl(a = a) in S are related by a unitary transformation 92: 5, -- C z:e,, = &lc,W,



The inverse relations are

A basis eigenstates of a scalar Hamiltonian, / njnt), (n stands for the set of all non angular momentum quantum numbers) in S is related to the corresponding basis of states 1nTi> in 5 by

= c I$>.


In the last step we have used the fact that the kinematic relation expressed by the matrix-element of W can equally be expressed through the corresponding function of the matrices, whence (jpl R 1jm) = (jpl euJ3eieJzeioJ31jm).


The inverse to (E.5) is then the relation 1 Fp)

The conventional (jm IT)

= 92-l 1 rzjp) = c ( njm)( jm I R-l ) jp).



definition of the D-function is in terms of the matrix R-l: =


R-’ 1jp)

s S?&,(~&))

= e-i”m(jmi

em*” ljp) ewdurL,

and (E.7)



The role of .9&,, as the transformation matrix between the two representations of angular momentum is used in Eq. (III.6), for the casej = l/2

where (\p 1s,~ 1p’) is now a spin-matrix in standard representation. THE SYSTEM SAS BODY-FIXED


In this case, the angles I## are defined as functions of the particle coordinates xi, . The variables xin , ms, may now be replaced by the equivalent set q@, 6,) pFLi , the f, being scalar functions of the xia , and the pi eigen-values of (e, . Si) = fii3 . The wave function

(xi,, msi I nJW = (q-4& 4, ; pi I nJW, may be expressedin terms of amplitudes in 3 by using Eq. (E.5):

<@#; to; /JGI nJM) = C COO; to 3pi I ~~JK)(JKI R(@‘$llJM> K

This establishesthe structure of the wave function in the new variables, and identifies LSj&($$) as eigenfunctions of the total angular momentum. Indeed:

(xim7mSilJ,“” lnJM> = C ~,,C&, , pi> C 9;~
= c bdJW H(iW Ja IJW = c &KJ,

In this last step, we use the fact that the matrix R(@$) J, can be expressedin terms of derivatives of R(@#), and leads to the differential operators J, . For an explicit representation of these, see Refs. (1) and (37) (In this latter, a different definition of Euler angles is used!) In a similar way, the operators 3, for the body-fixed components are found. The crucial step here is the matrix-relation which corresponds to (E.3):

or (E. 10)





Hence, following the steps of (E.9),

This last expression defines both the differential operator 3, and a matrix-representation for it, by means of the eigenfunctions g+~:

The two different matrix-representations generated by J,“” and ji” (or respectively, by J, and 3,) make it clear that all J, commute with all 5, . Similarly, a comparison of (E.9) and (E.12) show how the different sign in the commutation relations arises: J,J, 9,

= ; g&
(E. 13a)

j,;i, 9j&

= 5 (JK I J,J, 1Ji?) S$& ,



which give at once

[J, , JoI = iJ,


In Eq. (111.4), the operation i,$AP of the matrix iJy that ~:,ww

[i, , ii,] = -iiC


occurs. It follows from (E.7) and the reality

= (--1)-u


and with (E. 12): ia .s&, = c (-l)“-“(j, P

- /.L/J, lj, - ,I+ LS;L,

= -~~,AjFl J, Up>.




We close this appendix with a tabulation representations:

of the J-operators and their matrix-

RECEIVED: May 6, 1969

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