Center-of-mass motion on shell-model calculations with arbitrary central potential

I

1.D.1

i !

Nuclear Physics A124 (1969) 609--623; (~) North-HollandPublishiny Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

C E N T E R - O F - M A S S M O T I O N IN S H E L L - M O D E L C A L C U L A T I O N S W I T H ARBITRARY C E N T R A L P O T E N T I A L H. N I S S I M O V a n d I. U N N A

Department of Theoretical Physics, The Hebrew University, Jerusalem, Israel Received 8 July 1968 In the f r a m e w o r k o f the r a n d o m - p h a s e a p p r o x i m a t i o n it is possible to give the explicit dependence o f the shell m o d e l H a m i l t o n i a n o n the c.m. degrees o f freedom. T h e smallness o f the t e r m s in this H a m i l t o n i a n which couple the c.m. to the internal degrees o f f r e e d o m can be e x a m i n e d in every case. It was found, t h a t for c o m m o n l y used potential wells the coupling is rather small a n d an a p p r o x i m a t e separation is possible. T h e m e t h o d described here can be generalized a n d used for the separation o f o t h e r collective degrees o f freedom.

Abstract:

1. introduction A well-known problem in usual shell-model calculations is the appearance of spurious excitations. They arise from the fact that the shell-model Hamiltonian is not translationally invariant but rather consists of a central potential fixed in space. The center of mass of the whole nucleus is therefore described by the shell model as if moving in a potential well. Thus, certain states, known as spurious states, correspond physically to excitations of the center of mass of the nucleus rather than to internal excitations. The latter are, of course, the only states of interest for the nuclear physicist. For example, the ground state of a certain nucleus, say 4He, is described by the shell-model configuration ls~. An excited state of this nucleus could belong to the configuration ls31p~. However, such an excited state corresponds only partially to internal excitation of the nucleus. It describes in part a spurious excitation corresponding to an excited motion of the center of mass. The problem of separating out the spurious part of such a state can be solved exactly if one assumes that the nucleons are moving in a harmonic-oscillator potential well. That is, if the single-particle state (nlj) is given by a harmonic-oscillator wave function. The procedure for carrying out the separation in this special case is described in several papers ~-3). Methods have also been developed for the general case, in which the form of the central potetatial is not limited to that of a harmonic oscillator 4-6)° However, all these methods suffer from ambiguities and the approximations involved in using them are not clear 5). Recently, it has been shown that certain collective degrees of freedom can be separated out in the framework of the random-phase approximation (RPA), known 609

610

rI. N I S S I M O V A N D I . U N N A

also as the quasi-boson approximation 7-9). In this paper we shall describe how this method can be utilized for the present problem. We shall be able to give a straightforward prescription for separation of the center-of-mass motion out of a general shell-model Hamiltonian whenever this is at all possible. We point out in a clear manner what approximations are made and how good these approximations are. Moreover, explicit expressions, as well as tables, are given for parameters which measure the goodness of the approximations. The method used in this paper can be generalized for the treatment of other collective degrees of freedom. This paper will therefore serve as an illustration for the general method here applied. 2. The quasi-boson approximation

We are given a shell model Hamiltonian of the general form H = Hs.p.-4-1-/....

(1)

where Hs.p. = E e~a+a~ , Hres = E V~#r~a ~+ a ,+ a~a 7 .

The term Hres is an effective two-body interaction which may be added in order to remove some of the degeneracies of the extreme single particle Hamiltonian, Hs.p.. The indices ~, fl etc. represent all the quantum numbers of the single-particle state like (n~lj~m~). It is convenient to transform this Hamiltonian into the quasi-particle language. That is, to take the ground state of the single-particle part plus the pairinginteraction part of the Hamiltonian to be the Bardeen-Cooper-Schrieffer ground state corresponding to the quasi-particle vacuum in the case of doubly even nuclei. In the special case of closed shell nuclei, which we shall consider here in more detail as an example, the quasi-particles reduce to simple particles above the closed shells or holes in the closed shells. Carrying out the transformation to the quasi-particle language in the latter example amounts to the replacement of the particle operators by quasi-particle operators in the following way anum = ~nljm for states (nlj) above closed shells, antjm = ( - - 1 ) J - m o ~ + j _ m for states (nlj) in the closed shells.

(2)

A further transformation of the Hamiltonian can be made, for doubly even nuclei, by rewriting it in terms of quasi-particlepair operators. The pair operators are defined (in the case of spherical symmetry) by the relations t AjM(v, t~) = ~ (jvm~j,,m~lJM)cxso,, ~ s. . . . m~,tn~

BjM(~' ~)

=

• + (JvmvJa--D'ltrlJM)~j,,mC(j~mv.

E ( - 1 ) j~--m,r " mvma.

t Usually we shall leave out non-relevant quantum numbers, like nl, n'l" in the present case.

(3)

611

C.M. SEPARATION

From now on it will be sufficient for our purposes to consider only the extreme single particle part, Hs.p., of the shell-model Hamiltonian. The residual effective interactions between the nucleons depend only on relative coordinates and, therefore, have no influence on the spurious c.m. motion. It is true, that the single-particle part of the Hamiltonian is already in diagonal form. Talking of approximation methods, like the RPA, to diagonalize it seems therefore irrelevant. Still, we shall need the quasi-boson approximation in writing down approximate expressions for the c.m. coordinates and momenta. In terms of the pair operators, Hs.p. takes the fo~m

Hs.p. =

Z

e,(2jv + 1)~Boo(V, v)

-

y'

~v(2j~ + 1)~Boo(V, v).

(4)

v o c c u p i e d s . p . states

v empty s.p. states

A constant term has been omitted from this equation. Most convenient versions of introducing the quasi-boson approximation were given by Beliaev and Zelevinsky 10) and Marumori et al. 11). According to their methods, the various pair operators, AsM(v, (7), Bsu(V, (7) and their Hermitian conjugates, can be expanded into power series in terms of exact Boson operators AsM(v , a), AsM(V, (7), satisfying exactly the commutation relations

[AjM(V, a), As,M,(v, + t (7')] =

Jv-Jv,+J

-

(5)

The expansion for As~t(v , (7) turns out to have the form

AjM(v, tr) = AjM(v, a)+ terms to be neglected.

(6)

For Boo(V , v) one obtains

Boo(V, v) = (2jr+ 1) -~ Z ArM(v, (7)AjM(V, a),

(7)

JM

being an exact relation. The neglected terms in AjM(V , (7) are non-linear in the boson operators and supposedly of higher order in the smallness parameter f2 -~, where f2 denotes the number of available shell-model states. The operators BjM(v, (7) with J ~ 0 are neglected altogether in the framework of the quasi-boson approximation. The leading terms in their expansion are bilinear in the boson operators and, again, supposedly of higher order than the leading terms in any of the physical operators to be considered. The operators Boo(VV) appear in the Hamiltonian multiplied by the factors (2j~ + 1)3 and have therefore to be kept in the course of the calculation, being of the leading order. The shell-model (single-particle) Hamiltonian now takes the form

Hs.p. =

Z v e m p t y s. p . s t a t e s

(7)Aj.(v, (7) --

E

~vAj+M(v,(7)AjM(V, tr). (8)

~r JM v occupied s.p. states

612

H. NISSIMOV AND I. UNNA

3. Separation of the center-of-mass degrees of freedom

Let us express the center-of-mass coordinates and momenta in terms of the boson operators AjM(v, a), ArM(v, a). We proceed in the same way as we did for the Hamiltonian. In terms of the particle-creation and -annihilation operators they can be written as R. = 1 + (vm~lr~lamo)a~,.ao,.~, (9) vrnv ¢rnl~

P~

+ . = ~ (vm~lp~larn~)a~,.a~,,~

(10)

vmv ~mex

Here, r u and p~ are the # components of the single-particle coordinate and momentum vectors, respectively. Also, here, v and a stand for the quantum number (nvl~j~,), (n,,l,j,). The number of nucleons in the nucleus ~ is denoted by N. We now replace the particle operators by the corresponding quasi-particle operators (eq. (2) in our examples of closed-shell nuclei). The resulting expression can, in turn, be expressed in terms of the pair operators AsM(v, a), BjM(v, a) [eq. (3)] and their Hermitean conjugates. As can be easily seen, only pair operators with J = 1 and negative parity will appear. Finally, we apply the quasi-boson approximation replacing Alu(v, a), A-~u(v, ~) by the boson opeLators a lu(v, a), A+,(v, a), respectively, and omitting the Btu(v, a), B+u(v, a) terms. Thus, the center-of-mass coordinates and momenta take the form

R, = - - -

1

(vllrlla)[A~.(v,

~,

Nx/3 voccupiedstates

a)+(-1)"Al_,(v, tr)],

(11)

a e m p t y states

Pu -

1

x/3

~

(vllplla)[AL(v, a ) - ( - 1 ) " A l _ , ( v , a)],

states

(12)

v occupied tr e m p t y states

where (vllrlla) and (vllplla) are the reduced matrix elements of the single-particle coordinate and m o m e n t u m operators. It is here assumed that the radial parts of the nucleon wave functions are real. The c o m m u t a t o r of the quasi-boson approximations to R and P [eqs. (11) and (12)], JR, P] is still exactly equal to i (in units of h). In order to terminate the usually infinite sum in eqs. (11) and (12) we make a further approximation. We keep only a finite number of terms in such a way that the c o m m u t a t o r -i[R, P] takes a value very near to 1 (0.98, say). This is easily done since the sum which makes up the c o m m u t a t o r consists of only positive terms as can be easily verified. Moreover, in all practical cases, a very small number of terms turns out to be sufficient. In fact, matrix elements between single-particle states which are far removed f r o m the Fermi level, one above and the other below it, cannot differ appreciably from zero. In general, the isospin quantum numbers should be included, However, in our examples we shall treat only idealized cases of identical nucleons. Generalization to actual problems is straightforward.

613

C.M. SEPARATION

F o r the following discussion it will be convenient to introduce some simplified / notations 1

A~-~)( i) -- V/--2 [A L(v, a) + ( - 1)tth 1 -u(V, A?p(i)

=

{X~1) ~

/3}1) =

1

O')],

(13)

[A+u(v, a ) - - ( - - 1)UAl_u(v, a)],

(14)

4 2

(15)

N~/3

(vllrllog,

!~32(vllplla).

(16)

The a r g u m e n t (index) i stands for the set * of q u a n t u m numbers v, a. Note, that A (-} and A (+) satisfy the simple c o m m u t a t i o n relations

[A('tM)(i)' A(s+M'(i')] = (-- 1)VtfjS'6M-M'fU'' [A(s~](i), A(7)M,(i')] = [AsM~I (+"" ), aj,n,,(1 (+) ., )] = 0.

(17)

The finite sums in eqs. (11) and (12) for center-of-mass coordinate and m o m e n t u m can now be rewritten in the new notation Ru

~ "v( 1)~( --)(;~ i=l

Pu = i ~ Yn(1),l(+)~ i "Zlg \'1"

(18)

(19)

i=1

The n u m b e r r of terms included in these sums is taken so that the relation r

~ (t) /~(l } - 1,

(20)

i=1

holds as nearly as it is practical and desired. It is clear f r o m the discussion in sect. 2 that, restricting ourselves to doubly even nuclear systems, any o p e r a t o r can be expressed in terms of the operators A~s+)(i), A~s~)(i) where J, M, i run over all the possibilities. O f course, any set of operators obtained f r o m these operators by a canonical t r a n s f o r m a t i o n is equally suitable as a basic set of operators by which any o p e r a t o r can be expressed. We can n o w choose a specific canonical t r a n s f o r m a t i o n which leads to pure centerof-mass operators, namely R and P, on one hand, and a set of purely internal operators, c o m m u t i n g with R and P, on the other hand. This is simply carried out by the use of eqs. (18) and (19), where the sums are actually limited to a small (finite) n u m b e r of terms, in accord with our previous discussion. The other, internal, operators, °(k), ru rC(uk) are now defined so that they c o m m u t e exactly with R and P. t In the following v corresponds to single-particle states below the Fermi level and a to states above the Fermi level.

614

H . N I S S I M O V A N D I. U N N A

Let us write

p(k) = ~ a(k)A(-)tDi

lu ~

J,

(21)

i=1 r

7t~k)

i ~" t~(k)A(+)~D .~ P'i 1/t \ },

k = 2,3,

" " ",

r.

(22)

i=1

Then the coefficients %(k), fl~k) are determined in such a manner that: (a) the internal operators commute with the center-of-mass operators and (b) canonical conjugate pairs of operators satisfy the correct commutation relations. P#(k),

R,.]

=

~ L P .(k)p,.] ,

=

L,,

--

[x~k), R,.] ---- [n~k), Pu']

r^(k) . e~(k')l = rL ~ ,(k). ,' J

[p~k), n~,k')] = i( -

~(,~.')] =

(23)

O,

6kk,.

1)~'6._u,

(24)

Due to eqs. (17) all these commutation relations reduce to the following relations for the al k) and fl}k) coefficients r

(1)fli(k) = 2 O~i i=1

~ ~i . (k)o(k') t~i

~k)flll)

=

(25)

0,

i=1

=

fikk" ,

k, k'

=

2, 3,.

..,

r.

(26)

i=1

These equations can be easily solved for ~k) and fl}k) (k = 2 . . . . , r). Since the number of coefficients exceeds the number of equations, there is still freedom left in choosing the solution. This freedom can be utilized in order to cast the internal part of the Hamiltonian into normal form. An explicit procedure for the calculation of the coefficients o~k), fl~k) (k = 2 . . . . . r) is outlined in appendix A. Thus, we are able to define a set of r - 1 momentum vector operators and r - 1 coordinate vector operators which correspond to purely internal degrees of freedom. It must be remembered that there exist many operators which have not been considered so far because they trivially commute with the center-of-mass ones [see a), AjM(v, a) with J ~ 1. Second, all those eq. (5)]. First, all the operators Asu(v, + operators A~-~) (i), At +) (i) which were not included in the sums (18) and (19). We are now able to rewrite the single-particle shell-model Hamiltonian [eq. (8)] in a form in which the center-of-mass degrees of freedom are almost separated from the internal degrees of freedom. By "almost" we mean that the separation, though usually incomplete, can be shown in many cases to be very good. In any case, it is the best separation to be achieved in the framework of the random-phase approximation. After rewriting the Hamiltonian one will be able to tell immediately whether there is good separation between the center-of-mass and internal degrees of freedom, and to what extent it is not complete.

C.M. SEPARATION

615

Let us divide the Hamiltonian [eq. (8)] into two parts,

Hs.p. = H i + H 2 .

(27)

The first part, H1, includes all those degrees of freedom which are relevant to our discussion, since they partially describe center-of-mass motion r

H1 = Z Ai A ~,(i)A lu(i),

(28)

i=1 /t

where A~ is the difference between the single-particle energies Ai = e¢-e~,

(29)

(a corresponds to empty states and v to occupied ones). The sum in eq. (28) extends over the same terms i which appear in the sums of eqs. (18) and (19). The second part of the Hamiltonian, H2, consists of all the remainder and pertains to trivially internal degrees of freedom. There is no coupling between the degrees of freedom appearing in H 2 and those included in H1. Therefore, we shall consider hereafter only H 1. The operators A+u (i), Alu (i) can be easily expressed in terms of the operators R,, Pu, ptk), rctk). This is done by inverting eqs. (18), (19), (21) and (22) with the use of the relations (20), (25) and (26) for the coefficients ct~k) a n d fl~k) (k = 1, 2, . . . . r). We obtain

A+(i)

=

~ {[fl}l)Ru_i~}1)p~]+k=2 [/3i(k) p~(k) - t ~- i( k ) ( nk ).- l )j ; ,

, % ( 0 - ( -4)2 1

{[fl}X)R-~'+io~}l)e-S]+k=2~[fl[k)o~k)u+io~k'z~)U]}"

(30)

Here, the separation into center-of-mass and internal degrees of freedom is clearly evident. Note, that the coefficients ct}1), fl}l) are given by eqs. (15) and (16) and el k), fl}k) (k = 2 ..... r) are obtained by simple calculations based on eqs. (25) and (26) (see appendix A). Substitution of eqs. (30) into the expression (28) for H 1 yields the desired form for the Hamiltonian r

H~ = ½ E A , [ 0f(1)2 , P 2 +fli(1) 2R 2 ] i=1 r

+½

Z &k,k'=2 i ~~k)~(~')_(k)

i=l

"~

Al

i=l

{~i~(1)~(k)D~i ~ " J~ _ ( k ), . " T 'ap(i1 ) Po (i k ) a~ D . k=2

p(k)] _~ E Ai" i=1

(30

616

H. NISSIMOV AND I. UNNA

In this form of H1, we have revealed the pure center-of-mass part, the pure internal part and the inevitable term which couples center of mass with internal motion. A suitable choice of the coefficients ~i~(k),fll k) (k ----2, . . . , r) reduces the internal part of H1 to normal form (see appendix A). With this special choice we have for H1

H, = ½(9(I')P2 +

h(ll)R2)

+ ½ ~ [g(kk)7~(k)2+ h(kk)p(k)2] k=2

+ ~ [ g " k T • g(~)+

k=2

h(lk)R

•

p~)],

(32)

where the constant term is omitted. The coefficients g(kk'), h(kk') are defined as follows

g(kk')

~ A ~(k)~(k') i=1

h (kk') = ~ A. R!k)[~(.k') (33)

i=1

With the special choice employed in eq. (32) one has

g(kk') -~ h(kk') ~. O,

k • k';

k, k' > 2.

(34)

It is shown in appendix B that, as expected,

9(11) _ i M

(35)

where M is the total mass of the system.

4. Practical application of the separation method Let us denote the coupling term in H1 [eq. (32)] by H ' and the separated part by Ho H 1 = Ho+H'. (36) The eigenstates of H o can be characterized by quantum numbers nl, n2, n3 . . . . nr where n k stands for a set of three quantum numbers corresponding to the three Cartesian components. The set nl describes the degrees of excitation of the pure center of mass motion (which one might call the numbers of "spurions"). The remaining quantum numbers, n 2, n a , . . , nr correspond to excitations of the purely internal motions. The eigenstates of H o are [nl, n 2 . . . . n , ) corresponding to the eigenvalues Z ~k=100k(nkx + nky + nkz) where

ok = [O(kk)h(kk)]~,

k = 1, 2 . . . .

and the ground-state energy has been subtracted.

r,

(37)

C.M. SEPARATION

617

All the states with n 1 different from 0 are spurious states, in our approximation. If the coupling term H ' could be neglected, the physical states would be all the states 1 0 n2, n3, . . . , n r ) . However, the coupling H' cannot always be neglected. As we shall see from the following examples the amount of mixing between various eigenstates of H o depends on the choice of the shell-model potential. It will be of interest to check, for every given potential, to what extent the separated ground state [0) provides a good approximation to the exact H I ground state. This is best done by considering the smallness parameters Yk

( l ,lkln'[o) 7k =- - O) 1 -~" 0.) k

1 gOk)[hO 1)h(kk)/gO 1)g(kk)']~:__hOk)[gO 1)g(kk)/h(11)h(kk)]~ =-2

[g(li )h(l a)]½ + [g(kk)h(kk)]~-

'

k = 2 , 3 . . . . r, (38)

which are the amplitudes of the lowest-order perturbational corrections. It is preferable to work with a shell-model potential which leads to st-hall 7k values. Yet, another requirement has to be fulfilled by the potential. It should not lead to admixtures between excited eigenstates of Ho, like I1 x) and [lk) etc. This is necessary in order to obtain physically meaningful results. As a demonstration of the method we performed calculations for two model systems. For the sake of simplicity we assumed the systems to consist of only one kind of nucleons, eight identical nucleons in the first case and twenty in the second. Generalization to systems consisting of both protons and neutrons is straightforward. However, one has to remember that an additional collective degree of freedom appears in this case which corresponds to total center of mass at rest but relative motion between neutrons' center of mass and protons' center of mass. As is well known this degree of freedom corresponds to the collective giant dipole vibrations. We carried out the calculations for three commonly used central potential wells. (a) The infinite square-well potential. The results in this case are independent of its size. (b) The Woods-Saxon potential

V(r) = - Vol ( 1 + exp [ ( r - R)/al}.

(39)

We used the following values for the parameters t V0 = [53.2 MeV for the system of 8 nucleons, (54.0 MeV for the system of 20 nucleons, /3.15 fm for 8 nucleons, R 14.27 fm for 20 nucleons, a = 0.65 fm. /

(40)

(c) The harmonic-oscillator potential (again, the spring constant is irrelevant). The t The numerical values are those calculated to give the known experimental binding energies in ~60 and 40Ca, respectively.

618

H. NISSIMOVAND I. UNNA

latter case leads, of course, to exact separation and was merely used as a check on various stages of the calculation as well as computer programs. Self-evidently, addition of a spin-orbit coupling potential does not make any difference so long as it is independent of the radial variable r. In table 1 we list all those excited particle-hole configurations which describe at least to some extent excited center of mass motion, so that they were included in eqs. (18) and (19). TABLE 1 Configurations of one particle-one hole states and their overlap ~ with the one spurion center-of-mass excitation II1) N=8 (a)

N=20

(b)

(c)

(a)

config.

6

config.

6

config.

6

config.

lp-lld lp-12s lp-12d ls-12p lp-13s

0.80 0.15 0.03 0.01 0.01

lp q d lp-12s

0.84 0.16

lp-lld lp-12s

0.83 0.17

ld-llf 2s-12p ld-12p ld-12f ld-13p lp-~2d 2s-13p lp-13s ls-12p

(b) 6

config.

0.68 0.15 0.11 0.02 0.01 0.01 0.01 0.004 0.003

ld-llf 2s-12p ld-12p ls-12p

(c) 6 0.71 0.16 0.13 0.001

config. ld-llf 2s-12p ld-12p

6 0.70 0.17 0.13

(a) Infinite square well. (b) Woods-Saxon potential well, eqs. (39) and (40). (c) Harmonic oscillator potential well.

In cases (a) and (b) the internal degrees of freedom are coupled to the center of mass degrees of freedom. As discussed above, the effect of this coupling on the separated ground state [0) is best measured by the quantities 7k, eq. (38). The TABLE 2 Values of the parameters •k [see eq. (38)] N=8

N=20

(a)

(b)

(a)

0.06 --0.10 0.05 0.06

0.01

0.05 --0.04 0.08 --0.04 --0.05 --0.02 0.06 --0.04

(b) 0.01 --0.02 --0.01

(a) Infinite square well. (b) Woods-Saxon potential well, eqs. (39) and (40).

C.M. SEPARATION

619

number of these coefficients in each case is ( r - 1 ) , where r is the number of configurations appearing in table 1. The values of Vk for the cases treated are given in table 2. They vanish, of course, exactly in the case of harmonic oscillator potential [case (c)]. It is seen that the separated ground state 10) is very near to the exact ground state also in cases (a) and (b). Comparison of these two cases shows that the Woods-Saxon potential [case (b)] is somewhat preferable over the infinite square well. The operators A+M (i) are eigenoperators of the complete shell-model Hamiltonian Hs.p. [eq. (8)]. We have now seen that the separated ground state, 10), is, to a good approximation, an eigenstate of Hsp. The states Aj+M (i)[0) can therefore also be considered as approximate eigenstates of the complete shell,model Hamiltonian H~.p.. If the separated excited states I l l ) I/k) (k = 2 , . . . , r) would be good eigenstates of H~.p. as well, then one of them should coincide with A;-u (i)10) and the others should be orthogonal to it (assuming that there is no degeneracy). In practice, the coupling term H ' may cause mixing between such excited states. A measure of how spurious a shell-model state, Aj+ (i)10), is, can be obtained by considering the quantities fl ~, = (lxlafM(i)[O).

(41)

The usual shell-model calculations are meaningful whenever these quantities have values of nearly 1 or nearly 0. In table 1 the overlap values, fig between the eigenstates, A ~+~(i) [0), and the spurious state, Il l ) , are tabulated. It should be emphasized that in cases (a) and (b) the quantity fl cannot be interpreted as the percentage of spuriousity in the state A~'~ (i)10). It enables one to decide whether this state is nearly internal (f~ small) or nearly spurious (f, near to 1). On the other hand, for case (c) it is indeed the percentage of spuriousity. This is so since in the case of a harmonic oscillator potential the various states A+(i)]0) appearing in the table are exactly degenerate and the states I11), I / k ) are exact eigenstates of the full Hamiltonian, 5. Discussion

We have arrived at a form for the Hamiltonian [eq. (32)] in which the center-ofmass motion and its coupling to the internal motion have been made explicit. We can, of course, speak of separating out the spurious center-of-mass motion only when the coupling vanishes. This happens exactly only when the nucleons move in a harmonic-oscillator potential. In that case it is easily seen (see appendix B) that the quantities g(lk), h(ik) (k = 2 . . . . , r) in eq. (32) vanish exactly. An approximate separation remains, however, possible whenever the coupling terms are small enough. To check the smallness of these coupling terms we consider them as a perturbation and investigate their effect on eigenstates and eigenvalues of the "unperturbed" part of the Hamiltonian which is exactly separated. We have carried out detailed calculations for two nuclear systems and two different

620

H . N I S S I M O V A N D I. U N N A

potential wells. The calculations were applied to systems of both 8 and 20 identical nucleons. The two potential wells treated were the infinite square well and the WoodsSaxon potentials. For the Woods-Saxon potential (eqs. (39) and (40)) it turns out that the ground state of the separated part of the Hamiltonian is in both systems almost unaffected by the coupling terms [table 2 case (b)]. The effect of the coupling terms for the infinite square potential is slightly bigger but still small. Corrections to separated ground state wave functions in the latter case are of the order of 2 ~o [see table 2 case (a)]. The coupling term H ' has bigger effects on excited states. This is already demonstrated by the values of the overlap integrals 6~ = (111A+(i)]O) given in table 1 [cases (a) and (b)]. The fact that these values differ appreciably from 1 or 0 indicates that the spurious state [11), which is an eigenstate of the separated part of the Hamiltonian, coincides or is orthogonal only partially with the state A +10) which is to a very good approximation an eigenstate of the full Hamiltonian t. It is advisable to prefer shell-model potentials for which the overlap integrals 6 take extreme values. One should remember, that in the case of the harmonic oscillator [case (c)] the overlap 5 has no such significance. In this case the various states A + (i) L0) are degenerate and it is therefore always possible to take the combination which is identical or exactly orthogonal with the separated state [11). In the cases where no exact separation exists it is still possible to identify, by means of the values 6~, those states which are nearly spurious, since these states have big overlap with the purely spurious state II1). Those states which have only small overlap with I11) can be considered as corresponding to almost purely internal excitations. It has already been mentioned that a generalization of the methods outlined here to cases where both protons and neutrons are considered is straightforward. However, in that case, the differences between the neutrons' center-of-mass and the protons' c.m. coordinates and momenta can be treated similarly to the total c.m. coordinates and momenta. These differences are genuinely internal degrees of freedom. Nevertheless, they are of highly collective nature and deserve separation similar to the separation of the spurious effects, discvssed in the present work. Indeed, they give rise to the highly collective giant dipole resonances. Other collective degrees of freedom can be treated on the same lines in the framework of the random phase approximation. F o r instance, the "collective" number deviations (lq-N), causes spurious effects in the BCS treatment of pairing correlations. In an earlier paper 8) our methods have been applied to the separation and elimination of these number spuriousities. Additional examples of collective degrees of freedom to be considered are quadrupole vibrations and rotations 9). t It is here assumed that there is no degeneracy between states A+(i)I0). In the case of a degeneracy combinations of the degenerate states have to be chosen which extermize the overlap with 111).

C.M. SEPARATION

621

Appendix A In this appendix we shall demonstrate how one can construct a set of ( r - 1) internal coordinate vectors and their canonical conjugates. The center-of-mass coordinates and m o m e n t a are given by eqs. (18) and (19). The 2r coefficients ~I '), fl}l) (i = 1, 2 . . . . . r), are given explicitly by eqs. (15) and (16) and satisfy the relation r /=1

..(1)a(1) ~i pi = 1.

(A.1)

Each term o f the sum in eq. (A.1) is positive. We look for the 2 r ( r - 1 ) numbers ctlk), fl~k), (i = 1, 2 . . . . . which satisfy the relations

r; k = 2, 3 . . . . . r)

r

Z e~k)fl~k') = 5kk,,

k, k' = 1, 2 . . . . .

r.

(A.2)

i=1

F u r t h e r m o r e , we look for the normal set of internal coordinates and m o m e n t a . Thus we require the submatrices g (kk'), h (kk') [see eqs. (33) and (29)] with k, k' = 2, 3, . . . . r to be diagonal. Let us first build a set of vectors e(k), fl(k) which satisfy the relations (A.2) without any further restrictions. This is easily done by taking, for k = 2 . . . . r •

=

.....

4'_)1,

if(k) =

(fl(ll)fl(21) . . . . .

fl(l) l,

o,...,

fl~u), 0 . . . . .

o), 0).

(A.3)

The coefficients ,,(k) a(k) are determined by the conditions ~(k). fl(~) = 0 and ~(1). fl(k) = ~k , ek 0, respectively. It is easy to see that the vectors ~(k), fl(k) obtained in this way are never orthogonal. They can, therefore, be normalized to yield

~t(k). fl(k) ---- 1. In order to obtain the normal internal degrees of freedom we shall now m a k e the transformation

e ( k ) ~ ~ Vklo~(1), 1=2

fl(k)~ ~ Uktfl(,,,

k = 2, 3 . . . . . r,

(A.4)

/=2

in such a way that the sub-matrices g(kk'), h(kk') for k, k' = 2, 3 . . . . . r become diagonal. Moreover, we shall be able to require that g(kk') transforms into the unit matrix. We m a k e use of the facts that the submatrices g and h are symmetric, regular and have only positive eigenvalues. The matrices U, V (which are of order r - 1) have to satisfy

Vg ~" = I, Uhgr = A,

(A.5)

622

r l . N 1 S S I M O V A N D I, U N N A

where A is any diagonal matrix, as well as the relation VU = I.

(A.6)

Let us first find an orthogonal matrix A of order ( r - 1 ) which diagonalizes the matrix g AgJ~ = 2,

(A.7)

2 being a diagonal matrix with positive elements. Now, let us find an orthogonal matrix B which diagonalizes the symmetric (still regular and still with only positive eigenvalues) matrix 2~Ah~2 ~. We can write B2½Ah.42~B = a.

(A.8)

V = B2-~A, U = B2~A.

(A.9)

Now, put

These matrices have the required properties. Appendix B

In this appendix we shall prove the relation g(11)=. 1 , NM

(B.1)

where N is the number of nucleons in the system, M is the mass of the nucleon, and ½0(11) is the coefficient in the center-of-mass kinetic energy term of the Hamiltonian [see eqs. (32) and (33)]. We shall also prove that if there is no spin-orbit coupling g(lk) = 0,

k ~ 1.

(B.2)

This means that in H 1 [eq. (32)] there is no coupling between the center-of-mass momentum and the internal momenta. The Hamiltonian for a single particle is pZ H = -+ V ( r ) + ~ ( r ) ( / • s). 2M

(B.3)

The relation, [ [ H , z], z] =

-

1

--, M

(B.4)

is obvious. Taking a diagonal matrix element of both sides we obtain (e,,_8¢)l(vmlzlv,m,>l 2 = ~,m,

1 . 2M

(B.S)

C.M. SEPARATION

623

Let us now sum over all the occupied states [vm). We get t

p

( e v - - e v , ) l ( v m l z l v m )l

2

N

=

2M

vm o c c u p i e d states v'm" e m p t y states

(B.6)

In eq. (B.6) the sum is extended only over the unoccupied states ]v'm') since only these terms contribute to the sum. Use of eqs. (15), (29) and (33) together with the Wigner-Eckart theorem leads to the proof of eq. (B.1). To complete the proof one should remember that the difference between the infinite sum in eq. (B.6) and the finite sum in eq. (33) is assumed to be negligible. To prove eq. (B.2) we note that when there is no spin-orbit coupling we have

EH, z] = - i

(B.7)

M

This leads to the equation

(e,-ev,)(vmlzlv' m') =

i

!

t

M (vmlPzlv m ).

(B.8)

Using eqs. (15), (16) and (29) one gets

N M Ai Inserting this result into eq. (33) and applying eq. (A.2) proves eq. (B.2). Finally, it is worthwhile noting that, in the case of no spin-orbit coupling, use of eqs. (A.3) and subsequent discussion, leads immediately to a diagonal form of the whole matrix g~kk'). The procedure described in the second part of appendix A is in that case simpler. References 1) 2) 3) 4) 5) 6) 7) 8) 9)

J. P. Elliott and T. H. R. Skyrme, Proc. Roy. Soc. A232 (1955) 561 I. Unna and I. Talmi, Phys. Rev. 112 (1958) 452 E. Barranger and C. W. Lee, Nucl. Phys. 22 (1961) 157 S. Gartenhaus and C. Schwartz, Phys. Rev. 108 (1957) 482 H. J. Lipkin, Phys. Rev. 110 (1958) 1395 F. Palumbo, Nucl. Phys. A99 (1967) 100 D. J. Thouless and J. G. Valatin, Nucl. Phys. 31 (1962) 211 I. Unna and J. Weneser, Phys. Rev. 132 (1965) B1455 J. N. Ginocchio and J. Weneser, Phys. Rev. 170 (1968) 859; E. Marshalek and J. Weneser, private communication 10) S. T. Beliaev and V. G. Zelevinsky, Nucl. Phys. 39 (1962) 582 11) T. Marumori, M. Yamamura and A. Tokunaga, Progr. Theor. Phys. 31 (1964) 1009