Momentum distributions in axially symmetric deformed nuclei: The Nilsson model

Momentum distributions in axially symmetric deformed nuclei: The Nilsson model

Nuclear Physics North-Holland A509 (1990) MOMENTUM 117-140 DISTRIBUTIONS IN AXIALLY DEFORMED NUCLEI: The Nilsson model J.A. CABALLERO and E. MOY...

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Nuclear Physics North-Holland

A509 (1990)





and E. MOYA



Insiituto de Estructura de la Materia, CSIC, Serrano 119, 28006 Madrid, Spain Received 16 May 1989 (Revised 8 September 1989) Abstract:

We study velocity distributions in deformed nuclei within the Nilsson model. We analyse the condition of isotropy in momentum space as an equilibrium condition. We show results for Ne and Nd isotopes and discuss properties of overall and single-particle momentum distributions in these nuclei. We find that at the equilibrium deformation obtained with the Strutinsky method the overall momentum distribution is isotropic, or nearly isotropic, provided that AN # 0 admixtures are taken into account in the diagonalization of the Nilsson hamiltonian. We also analyse the dependence on deformation of the single-particle and overall momentum distributions averaged over angle. We find that the dependence on the deformation of the mean field is much stronger for the single-particle momentum distributions than for the global momentum distributions. The latter are shown to be similar for spherical and deformed nuclei at equilibrium.

1. Introduction The question of momentum distributions in nuclei, at both nucleon and quarkgluon levels, is gaining increasing interest with present (and the advent of future) experimental capabilities I). Although short-range corrections - not included in mean field approximations - are known to play a role ‘) in nucleon momentum distributions, the independent particle shell model is the natural starting point to study



In ref. ‘) a thorough


was made

of momentum

distributions of spherical nuclei in the mean field approximation. In that reference it was found that the momentum distributions in finite nuclei differ substantially from the nuclear matter momentum distribution. This has been experimentally confirmed by the (e, e’) data analysis of Day et al. “). In ref. ‘) it was also found that the bulk properties of the nuclear density in p-space determined by the more developed mean field theories, such as density-dependent Hartree-Fock (DDHF), are quite general and can be reproduced with much simpler mean field calculations, such as those based on phenomenological Woods-Saxon or even harmonic oscillator (HO) potentials. In this paper we analyse momentum distributions of deformed nuclei using as a first approximation the phenomenological Nilsson potential. DDHF calculations of momentum distributions in deformed nuclei are now under way and the results will be discussed elsewhere ‘). 03759474/90/$03.50 (North-Holland)

@ Elsevier





.l.A. Cabaiferq

E. Mqm de Guerrn / Momentum


In a previous paper we used Nilsson model wave functions to study effects of nuclear deformation in quasi-elastic electron scattering 6)_We found that coincidence

(e, e’p) measurements target nucleus.

may show a strong


Fig. 1 shows the magnitude

cm the deformation

of the effect found






of the

for 28Si. In this figure

Si (c i e*pJz7AI

',I,$ 10.54 E Q 12.0MeV \


Fig. I. Effects of nuclear deformation in quasi-elastic electron scattering. Comparison of the deformed model results for 6 = 0.1 (dashed-curve) and d’= -0.07 (double-dot-dashed curve) IU the spherical model (dot-dashed curve) and to the experimental data and fit (full-curve) of ref. ‘).

J.A. Caballero


, E. Moya de Guerra / Momentum distributions

data of the proton



in the missing

10.5 G E < 12.0 MeV for 28Si(e, e’p)27A1 are compared (dot-dashed results

curve) and deformed



for 6 = -0.07,

to results

energy range

of the spherical

curve) models 6X7).The deformed

to a value of the deformation

the data can be obtained




6 = 0.10; a similar



fit to

but with other

6-values one cannot reproduce the low-p data. It may seem paradoxical that data on quasi-elastic electron scattering are sensitive to the nuclear shape. Since the momentum distribution is expected to be isotropic at the equilibrium shape one would expect to find similar momentum distributions in spherical and deformed nuclei. However, the (e, e’p) data are sensitive to the momentum distributions of selected single-particle orbitals, which in turn may be sensitive to the deformation of the mean field. The Nilsson model seems particularly suitable to study the dependence on deformation of the single-particle momentum distributions. It offers the advantage over DDHF calculations that the deformation parameter can be changed at will, and it is more transparent and simpler to handle. On the other hand one may question the significance of these Nilsson model results, i.e. one may question whether the momentum distributions obtained from a selfconsistent mean field description of the nuclear ground state may exhibit any dependence on deformation, and whether the Nilsson hamiltonian is suitable for that description. In this paper we address these questions analysing the properties of single particle and overall momentum distributions in Ne and Nd isotopes, obtained by diagonalization of the Nilsson hamiltonian in a large basis containing eight major shells. The choice of nuclei under study has been motivated in part by quasi-elastic electron scattering experiments presently being carried out at NIKHEF-K. The paper is organized as follows. In sect. 2 we discuss the condition of isotropy in p-space at the equilibrium shape. In sect. 3 we analyse deformation of the isotropic parts of global and single-particle tions. Sect. 4 contains concluding remarks.

2. Equilibrium In this section

we discuss

shape and isotropy

the isotropy

of overall

the dependence on momentum distribu-

in p-space



in stable

deformed nuclei. Intuitively it is easy to understand that the momentum distribution must be spherically symmetric in the nuclear ground state, for anisotropies in p-space lead to instabilities in the system. Yet, when using phenomenological deformed potentials there is no general proof that guarantees isotropy in momentum space at the calculated equilibrium deformation, and isotropy in p-space should be imposed as a consistency condition. The only case where such a proof can be given is that of the anisotropic harmonic oscillator without spin-orbit interaction. In this particular case one can show on general grounds that isotropy in p-space follows from the self-consistency condition “)_ When more realistic phenomenological potentials


J.A. Caballero,

E. Maya de Germ

/ Momentum

are used one has to resort to specific numerical in p-space

at the equilibrium

In what follows we discuss




of calculations

the case of the anisotropic


2.1. THE

we present



to check the isotropy

by the standard

for the Nilsson

HO to illustrate



the equivalence

but first

of different





the anisotropic






2 c






The self-consistency condition of Bohr and Mottelson “) states that for a given configuration {nb} the equilibrium shape can be obtained from the condition that, at equilibrium, the shapes of the density and the potential be equal. The shape of the potential is characterized by the lengths of the principal axes and the shape of the density in r-space is characterof the potential, a, = (mw,)-‘, ized by the mean square


=;, (nf +$/(Mw,) Rt=(j,(xb12) Then the self-consistency

condition R a .-.Er~RS a,,


in terms of the a,

imposes w,~




and the occupation

that at equilibrium

for any LY,LX’,


w, defined

ff&Ja = o;&,c this is a self-consistency

= hua/(Mw,) .

for any (Y,ff’,



in eq. (2) reads

the frequencies

(4) of the anisotropic

of the given configuration



That this self-consistency condition is an equilibrium condition can be shown by calculating the shape of the momentum distribution. The latter is characterized by the mean square values [email protected], From this expression

bl)‘)=Mho, 1i (n; +$) = Mho,u, one sees that the condition P’,= P;=

is identical

to the self-consistency

of isotropy

. in momentum



given in eq. (4). Furthermore

(5) space (6) the total

J.A. Caballero


, E. Maya de Guerra ,J ~omenfu~



of the system, (H)=(T)+;(V)=+C

is also seen to be minimum For instance


when condition

for a configuration



(3) is satisfied

({~}~i~ = 3P&/4M).

such that a, = aY = I&r,, with K a c-number,


we parametrize the HO frequencies as w, = w,$, , ct.+,= w,&, w, = w&3,/3J’ so that the volume is constant, the minimization ofthe energy in eq. (7) gives p1 = p2 = K-“3; or w, = oY = w,/K. Hence, minimizing the energy (for fixed configuration and volume) we get the same result for the equilibrium shapes of the potential (a, = a, = Ku,) and density (RX = R, = KR,) as using either of the conditions (3) or (6). In summary, within the anisotropic HO the three methods - minimization of the total energy with respect to deformation, isotropy in p-space and self-consistency condition - are equivalent and lead to the same equilibrium shape. However with the Nilsson model hamiltonian 9,‘“), h =$.1,(6)(-~~+r~-





eqs. (2), (5) and (7) do not hold any longer because of the spin-orbit and “squarewell” terms, and the simultaneous fulfilment of the various conditions is hard to impose in the numerical calculations. We can then at best expect that the numerical results be consistent.







That dN# 0 admixtures had to be taken into account to find the equilibrium deformation was early recognized by Nilsson “). However the problem of having to diagonalize the hamiltonian (8) in a large basis containing a given number of N shells was avoided by transforming to stretched coordinates “). In the stretched coordinate system the kinetic energy operator is non-isotropic and the term proportional to Aw,(S) in eq. (8) is diagonal in N. In addition the AN = 2 admixtures introduced by the spin-orbit and l2 terms are smaller than those introduced by the quadrupole term in ordinary coordinates “)_ The first microscopic calculations of equilibrium deformations were performed by Mottelson and Nilsson ‘I) using this transformation and redefining the spin-orbit and l2 terms to be N-diagonal in the stretched coordinates. The prescription used in ref. “) to find the equilib~um shape is to look for the e-value (deformation parameter of the stretched coordinate system) that minimizes the total energy, calculated as the sum of single-particle energies of the occupied levels, subject to the constraint that the volume contained in the equipotential surfaces be independent of E. These calculations were latter improved by Bes and Szymanski 12) to include pairing and Coulomb energies. Results of many calculations performed along these lines are available in the literature, which, on the whole, reproduce the results


J.A. Caballero,



forces at equilibrium Although correct,

E. Moya de Guerra / Momentum

in ref. I*), b ecause of the counterbalance

the method

of pairing

and Coulomb


the equilibrium

was recognized



failed to reproduce

13) that the Strutinsky

obtained absolute

in this way were approximately values

14) procedure

of binding



a more

and it


treatment. The Strutinsky procedure combines the global properties of nuclei, described by the liquid drop model (LDM), with the shell effects described by the Nilsson model (or any other phenomenological potential). It has been shown Is) that this method reproduces accurately results obtained with the much more involved Hartree-Fock calculations and is the optimal way to get microscopic results using phenomenological models. In what follows we show results for “Ne and Nd isotopes obtained with both prescriptions and analyse the shape of the density in p-space as a function of deformation. All the results presented correspond to single-particle (s.p.) energies E, and wave functions (P~ obtained by diagonalization of the Nilsson hamiltonian in a spherical basis IIV(ifl) of 8 major shells. For 20Ne the results do not change appreciably when a smaller basis of 6 major shells is used. The (real) coefficients in the expansion 14=;jc:illJWJ=






are obtained functions


the following $,~~(r)


= i’Rnl(r) Y’,(G) ,

in r-space

of the spherical


basis (10)

with Y:, and R,, the standard spherical harmonics and radial functions of the isotropic harmonic oscillator. The representation in p-space of the spherical basis functions is then 1 4NI.4(P) z ~(2n)3,2 with r in units




of b =~‘h/(Mo&,)

and p in units

of bp’. The values

(11) used for the

parameters K and /1 are those given in ref. lo). Alternatively one could use a cylindrical basis characterized by the quantum numbers INn,AE) (commonly used in deformed HF calculations) or the stretched basis IN,Z,A,E) defined by Nilsson “). However for further calculations it is more convenient to use the spherical basis in ordinary coordinates. This allows one to analyse in a direct way the content of different spherical orbitals in any given deformed single-particle orbital (Y.This is particularly useful with regards to future comparison with experimental data.

3.A. Caballero

, E. A4oya de Guerra / ~omenfum d~strjbutio~~

For any given value of the deformation parameter 8 the densities of the A-body system in the intrinsic frame have the form

where the different



in r- and p-space


p,,(r) = C ph”‘N”‘i’-“R,,(r)R,,~,(~~



Nf N’I’


with common




C p,“L”“‘(-l)“+“‘R,,(~)~~,,,(~), NI N’I’



Ph ’ where

n, are occupation


and the factor

2 accounts

for the degeneracy


time reverse states (Ia) and 1~7)). The shape of the density in r- or p-spaces can be conveniently discussed in terms of these density multipoles. Within the Nilsson model, hexadecapole and higher order terms in the multipole expansions (12), (13) are at least of order ?j2 and to investigate the isotropy of the momentum distribution at equilibrium we may restrict our attention to the quadrupole densities. In analogy to the intrinsic quadrupole moment, QO, in coordinate space QoQ;=~~Ip(r)r”Y:(i)dr=1711p2(r)lld~, we define

an intrinsic



in momentum

(17) space

(18) and study its dependence on the deformation parameter 6. As said before one could use the condition of isotropy in momentum space (i.e. QOp= 0) as a condition to find the equilibrium deformation. However in the search for the equilibrium deformation we do not fix the configuration, but rather take the lowest energy configuration for each S-value. In particular for 6 = 0 we average over all degenerate configurations. This means that we may find more than one 6 value at which Qg = 0. In fact, with this procedure we find Q{ = 0 at 8 = 0 for any nucleus, but this solution is only meaningful for nuclei that are known to be spherically symmetric. As we shall see in what follows, for the spherical nucleus 14’Nd, Q$ has a zero only at S = 0, while for the deformed nuclei considered, QOp

J.A. Caballero, E. Moya de Guerra / Momentum distributions


has also a zero at approximately and the quadrupole experimental



test in determining

the same S-value in coordinate


at which the energy is minimum


is approximately

This is what one should

the equilibrium



to the

as a consistency


That Q{ can be zero for 6-values

at which

Qh is large can be easily understood

by inspection of eqs. (14) to (18). Using these equations units of b2 and bm2, respectively, Q; = $7~ 1 /$:r”‘INLN,,,

we find for Q& and Q{ in





;, p~:;“‘ZN,,NzIz(-l)(N-N’)/Z,


N'I' where the coefficients

IN,,N,ls are the radial I N,,N’,‘=

These integrals



~,,(X)~,~,~(X)X4 dx.


from zero for 1’ = I,1 f 2 and N’ = N, N * 2 (see

are only different

table 1) and are the same as the ones that enter in the diagonalization of the Nilsson hamiltonian. Then if we split the contributions to Q& and QOpinto a contribution (Qo)AN=o, coming from the terms with N = N’ in eqs. (19), (20) and a contribution from the terms with N’ = N * 2, we can write (QcI)LIN=~, coming Qh=(Qo)a~=o+(Qo)~~=z,




For 6 = 0, ( Qo)nN=2 = 0 and, for the reasons above mentioned, (Qo)aN=o is also zero. For 6 # 0 both contributions are proportional to 6, and as 6 approaches the equilibrium deformation, the two contributions in eq. (22) tend to be equal so that, as QT, increases

Q{ goes to zero.


Values of the radial





N+2 N+2






in eq. (21)

J.A. Caballero

This behaviour moments

is indeed

for protons

, E. Moya de Guerra / Momentum distributions seen in fig. 2 for *“Ne, where the intrinsic

in r- and p-space

are shown

one sees that at S = 0.35 Q{ is zero. At this 6 value which gives for the p value defined

as functions



of S. In this figure

Qh = 16.92 b* and (T*) = 2.75 b*,

by “)



5 .Z(r’) ’


p = 0.49, in excellent agreement with the p value determined from the experimental data on the charge r.m.s. radius 16) (r = 3.02 fm) and the quadrupole moment “) ( Q0 = 58.5 fm*). This P-value is also in agreement with that obtained from HF calculations 18) and is somewhat larger than the p-value obtained from relativistic Hartree calculations 19). From eqs. (22), (23) it is apparent that when major shell admixtures are neglected one has QR = Q&= ( QJdNZO and Qt grows always with deformation. Hence, taking into account AN = 2 admixtures is essential to satisfy the condition of isotropy in p-space for deformed nuclear ground states. This condition imposes that at the equilibrium deformation

(Q~A~=~= (Qo)-rN=2 or

0; = ~(Q~A~=o.


This also explains why when AN = 2 admixtures are neglected in the diagonalization of the Nilsson hamiltonian, the 6 value required to get the experimental p value is substantially larger (6 -0.5 for “Ne) than the one obtained here. In figs. 3 and 4 we show the variation with 6 of the Strutinsky energy ( ELDM+ SE), relative to the spherical liquid drop model energy, and of the total Nilsson energy (E =$C,_ eras,), in units of h;,. Fig. 3 is for *‘Ne and fig. 4 is for 142~‘46~‘50Nd.In

Fig. 2. Mass quadrupole


in r-space

(0;) (dashed curve) and in p-space QOp(full curve) for *‘Ne.


J.A. Caballero,

E. Moya de Guerra / Momentum

Fig. 3. Dependence

on the deformation

parameter 6 of the Strutinsky *‘Ne (see text).


(a) and Nilsson


~ A=142 _ _ _ A=146 A=150

Fig. 4. Same as fig. 3 for ‘42.‘46.‘50Nd nuclei.

(b) energies


, E. Maya de Guerra / Momentum

LA. Caballero

the calculations BCS equations

for Nd isotopes with a constant

of the Strutinsky energy


(6E, including

have been calculated


the oscillating in a consistent


has been taken into account

gap parameter

the LDM


by solving


(A, = A, = 1 MeV). In the calculations



part of the pairing way as described

and the shell correction energy

for Nd isotopes)

in ref. 14) (see also refs. 107”)).

In particular, the smoothed level density (g(e)) and occupation numbers (;ii) required for the computation of 6E were calculated using the averaging function f(( E = Ed),/y) proposed in ref. lo),

with M = 3 and y = 1.3 hui,, for “Ne,

y = R& for the Nd isotopes

(hr.& = 41 A-“3).

In fig. 3 for *‘Ne one can see that both, Strutinsky (fig. 3(a)) and Nilsson (fig. 3b) energies are minimum at 6 = 0.35, which is the same &value at which QOp= 0 and Q; = Qr”“. Hence, within the Nilsson model, 6 = 0.35 is a well-defined value of the equilibrium deformation for “Ne, since the criteria of minimum energy and isotropy in p-space are both satisfied. Figs. 4 and 5 for Nd isotopes show that only in the case of the spherical isotope, 14’Nd, is the situation as clear as in the case of “Ne discussed above. For “‘Nd and, specially, for 146Nd the situation is less clear. In fig. 4a one can see that the Strutinsky energy has a deep minimum at 6 = 0 for 14*Nd and at 6 = 0.22 for “‘Nd, while the corresponding curve for ‘46Nd shows a minimum at 6 = 0.12 which is very shallow to the left. These &values are very close to the equilibrium deformations obtained from HF calculations with the Skyrme III force “). The Nilsson energies




Fig. 5. Intrinsic

mass quadrupole

moments in p-space (Qg) for ‘42,‘46S’S0Nd nuclei as functions deformation parameter 6.

of the

J.A. Caballero, E. Moya de Guerra / Momentum distributions


TABLE 2 Experimental

and theoretical experimental

charge r.m.s. radii, quadrupole moments and p values from eq. (24). The rC and Q. values are from ref. 16) and ref. “), respectively


rc Nuclei


‘46Nd lsoNd










3.02 4.97 5.0

1.66 2.10 2.13

58.5 276.0 525.8

16.92 53.68 94.94



0.51 0.15 0.27

0.49 0.16 0.28

S (the.)

0.35 0.12 0.22

for these isotopes (see fig. 4b) have much more shallow minima with locations at somewhat lower 6-values, except for the spherical 14’Nd isotope. The intrinsic mass quadrupole moments in p-space for ‘42,146Y150Ndare plotted in fig. 5 as functions of 6. As already mentioned, for 142Nd QOpis only zero at 6 = 0, and at that point the slope of the curve is large. In the case of “‘Nd one also finds Q[=O at 6 =0.15 which is close to but somewhat smaller than the equilibrium deformation 6 = 0.22 obtained with the Strutinsky method. It is also interesting to note that for this nucleus the slope of the Q,” curve at 6 = 0 is of opposite sign to (and smaller than) that at 6 = 0.15, indicating that S = 0 is a local maximum of the energy as seen in fig. 4. For ‘46Nd QR is only zero at 6 = 0 but at this point the slope is very tiny. As seen in fig. 5 the curve is very flat in the range -0.07 c 6 G 0.1, and IQOplincreases rapidly with 161 at these limiting values. This agrees with the behaviour observed in fig. 4a for the Strutinsky energy, which suggests that ‘46Nd has not a well defined shape and is very soft in the range of 6 values -0.1 s 6 G 0.15. In summary from the analysis of figs. 4 and 5 we may then conclude that unlike 14*Nd shape oblate dipole

(spherical) and “‘Nd (deformed) ‘46Nd has not a well-defined equilibrium and may in principle be described as a mixture of spherical, prolate and shapes. This is in qualitative agreement with experimental data on the giant resonance in these isotopes 20). In fig. 27 of ref. 20) one can see that the peak

of the El-strength


in r4*Nd gets much broader in ‘46Nd and splits up in two peaks in 15’Nd, revealing that 146Nd is much softer than 142Nd and that “‘Nd has a stable deformed shape. However the experimental /3 values for ‘46Nd and “‘Nd are in agreement with the /3 values obtained in our calculations for 6 = 0.12 and 0.22, respectively [see eq. (24) and table 21. Therefore we take in the next section 6 = 0.12 and 0.22 as the equilibrium deformation parameters for ‘46Nd and r5’Nd, respectively. 3. Average momentum distributions In sect. 2 we showed that the momentum of the Nilsson hamiltonian are isotropic

distributions obtained (or nearly isotropic)

by diagonalization at the equilibrium

J.A. Caballero



, 17.Mop de Guerra / Momentum

by the Strutinsky

dence on the deformation carry out this analysis


In this section

6 of the average

for both global

For a given single-particle




and single particle


we study the dependistributions.




state ICX)we define the average momentum



with the normalization The average




j dp ii,(p) distribution

= 1. is defined

to be the isotropic

term in

eq. (13) (27) which is normalized


to the total number


of particles

(j dp ii(p) = A).


The isotropic part of the overall proton momentum distribution in Ne and Nd isotopes is shown in figs. 6 and 7, respectively, for different values of the deformation parameter 6. As seen in these figures the average momentum distributions fi( p) are not very sensitive to the deformation parameter 6. The main effect of the deformation on g(p) is to produce an increase of the central density in momentum space due to the admixtures of higher shells. However these differences in momentum distributions, caused by the deformation, are of the same order as the differences found in




Fig. 6. Average




in Ne for different


(see text).


J.A. Caballero,


E. Moyn de Guerra

/ Momentum







Fig. 7. Same as fig. 6 for Nd nuclei

ref. ‘) for momentum distributions of different spherical nuclei. This is best seen in fig. 8 where we compare total momentum distributions in 20Ne, ‘423’46S150Nd,at their equilibrium deformations (fig. 8a), with total momentum distributions of several spherical nuclei (fig. 8b). In fig. 8 fi(p)/A is in units of fm3 and p is in units of fm-’ and the b-values have been adjusted to give the experimental charge r.m.s. radius for each nucleus 16). In this figure it is clearly seen that, apart from shell fluctuations, the general trend of overall momentum distributions in deformed nuclei is analogous to that observed in spherical nuclei and their bulk properties can be discussed on the basis of the simplified models used in ref. ‘).


In this section particle



we discuss


the properties

and study their dependence

of momentum

on the deformation

average single-particle momentum distributions defined in terms of its spherical angular momentum components fiU(P) =; where the Zj components

no(p) =&e+

are explicitly




of single-


6. The

in eq. (26) can be written as, (28)


given by (see eqs. (9), (11))

1; Grj(-1)“~’



where p is in units of b-’ and nt( p) in units of b3. The percentage



of Zj contributions

J.A. Cabaflera

) E. Moya de Guerra / Momentum distributions


Fig. 8. Average total momentum distributions in nuclei at their equiiib~um deformations. (a) Ne and Nd isotopes: *‘Ne at 6 = 0.35 (full curve), 14*Nd at 8 = 0 (dotted curve), la6Nd at 6 = 0.12 (dashed curve) and “*Nd at 8 = 0.22 (dot-dashed curve). (b) spherical nuclei: “0 (full curve), 4”Ca (dotted curve), curve) and “‘Pb (double-dot-dashed curve). “Zr (dashed curve), ‘*“Sn (dot-dashed

to fi,( p) are then conveniently



dpnE(p)=C(C”,lj)z. N Expression (28) allows one to analyse a given deformed single-particle orbital.

the content of different U components in As remarked in sect. 2, this is of interest

J.A. CahaNero,


for future comparison in coincidence

E. Mop

de Guerra / Momenrum

with experimental

(e, e’p) experiments

data, since the spectral


out specific

particle momentum distributions 677). Figs. 9 and 10 show results for the least bound different

values of the deformation




Zj components


in “Ne


of the single-



6. Figs. 11-14 show results for several

proton orbitals around the Fermi level in Nd isotopes. in units of (b/v’%)3 and p is in units of bP’.

In all the graphs


Fig. 9. Momentum distribution of last bound orbital in “Ne. Results for S = 0.35 (full curve) are compared to, (a) results for 6 = 0 (dashed curve) and, (b) results for 6 = 0.1 (dotted curve), 6 = 0.2 (dot-dashed curve) and pure s-wave limit (dashed curve).


J.A. Cab&era

For the discussion the numbers 1,2,3,. Fermi level, number Nilsson




of the figures


to 20Ne we order the levels by

to the first level below the . . , so that number 1 corresponds 2 to the next one below and so on. In *‘Ne, characterizing the

in terms

of the asymptotic

levels for S ~0.4

1= [2,2,0,

+J, 4=[1,1,0,$],

Fig. 10. Momentum


, E. Moya de Guerra / Momentum disfributions




[TV, n,, rrrI, 01,


are, 2=[1,0,1,$,

3=[1,0,1,;], 5 = [O, 0, 0, 41)

of orbitals 2 (a) and 3 (b) in “Ne (dashed curve) (see text).

for S = 0.35 (full curve)

and 6 = 0


J.A. Caballero,

E. Maya de Guerra j’ ~ornenr~rn


Fig. 11. Momentum

distributions of orbital 1 in Nd isotopes for S = 0.12 (full curve), curve) and S = 0 (dot-dashed curve) (pure Id,,,) (see text).

6 = 0.22 (dashed

In fig. 9a we compare the momentum distribution of the last bound orbital in **Ne (orbital 1 = [Z, 2,0, $1) for 6 = 0.35 (equilibrium deformation parameter) with the momentum distribution obtained in the spherical limit corresponding to a Od,,, orbital (both cases normalized to 1). In this figure one can see the large effect of the deformation on the single-particle momentum distribution which is particularly strong for p - 0 and quite noticeable for all p-values below p - 1 fm-‘. These huge differences

at low p remain 0.18

when one neglects

AN = 2 admixtures

/,,,I,I~III/I,IJIJI,~~/~~~,~/,~~~~~~~~_ ,I’. I i


i !’










i i i i i i i i



\ i i


i f I i


'\ i


i i i i i i ! !

: ! _ .’ - L #’ L a 1 ‘>

0.00 0.00




P Fig. 12. Same as fig. 11 for orbital


and can be easily

J.A. Caballero


, E. Moya de Guerra f Momentum












‘\ \

I i i i I ! ! ! I




/ I




\ \ '\ \ \ \ \ '\ '\ '\




P Fig. 13. Momentum

distribution of the orbital 1 in Nd isotopes for S = 0.12 (full curve), curve) and 6 = 0 (dot-dashed curve) (pure Og,,,) (see text).

6 = 0.22 (dashed

understood analysing the Z-wave content of the Nilsson orbital at 6 = 0.35. From table 3 one can see that at this 6 value the Nilsson orbital is essentially -70% d-wave and -30% s-wave. Since, for this last bound orbital the admixture of s-wave increases with deformation the momentum distribution at p - 0 is roughly proportional to 6. This is best seen in part (b) of fig. 9 where we plot the momentum distribution for several values of the deformation parameter 6 as well as the momentum distribution for the pure OS,,~ wave (all normalized to 1).





P Fig. 14. Same as fig. 13 for orbital



LA. Caballero,

E. Moya de Guerra / Momentum


TABLE 3 ij contributions to the three last occupied levels in ‘*Ne for 6 = 0.35, as defined in eq. (30). Only nz values larger than 0.01 are quoted Level:

1 P, -7% fl



n:,,,,=0.265 n;5,2=0.627 n: 3,2 = 0.072 n: g/z = 0.03 1

n:,,,=0.144 $i,,=0.840 3 S,* = 0.012

In fig. 10 we show the momentum


n:,,,=0.982 n:,,,=0.014

for orbitals

2 (fig. 10(a)) and 3

(fig. 10(b)). We compare the results for S = 0.35 (full curve) and for 6 = 0 (dashed curve). In the spherical limit the level 2 is a Op,,, and the level 3 is a Op,,,. In the distribution has been normalized to 2, to latter case (fig. lob) the Op,,, momentum take into account that in the spherical case there are two pairs of protons in the Op,,, orbital while in the deformed case there is only one pair of protons in the [ 1, 0, 1, $1 orbital. Apart from this normalization effect, the momentum distributions for these odd-parity orbitals are not very sensitive to the deformation parameter S. This is because, as seen in table 3, these orbitals are dominated by the p-wave contribution even at rather large S-values. This is the only contribution that remains when AN # 0 admixtures are neglected. In the case of Nd isotopes pairing plays an important role and there are many levels that can be of interest for the analysis of experimental data 21). To simplify the discussion, in this paper we restrict our analyses only to the two first proton levels below and above the Fermi level. For the discussion of the figures and tables we label by 1 and 2 the first and second levels above the Fermi level, and by 1 and 2 the first and second below. In the range of 6 values 0 < 6 s 0.3 several positive and negative-parity levels cross each other, resulting in a different sequence of levels for different S values. As we said in sect. 2.2 we adopt here 6 =0.12 and 6 =0.22 as equilibrium S-values


the sequences


of proton

for ‘46Nd and “‘Nd,


For these

n, = 0.396, n, = 0.365, nr = 0.629,

a2 = 0.786,

levels are,



I = [4,1,3,3 for S = 0.12, with occupation and,


1 = [4, 1,3, $1) 1= [5,5,0,

$1 )

2 = [5,5,0,






for S = 0.22, with occupation numbers n, = 0.461, n, = 0.436, nl = 0.609, nz = 0.903. The Ij components of these orbitals (as defined in eq. (30)) are given in table 4. It should be pointed out that in these cases the neglect of dN Z 0 admixtures leads to

J.A. Caballero

, E. Moya de Guerra / Momentum distributions


TABLET Ijcontributions

to the two first proton states below (1,2) and above (1,2) the Fermi level in ‘4hNd for S =0.12, and in “‘Nd for S = 0.22

(a) ‘46Nd Level:

1[4, 1, 1, $1 n: n; n: n:

3,2 = 5/z = ,,* = 9/Z =

0.013 0.896 0.067 0.022

2 [5,5,0,


n: 7,2 = 0.037 ?I: ,,,Z = 0.941 n: ,5,2 = 0.020

1 [4,1,3,11 ?I: 5,2 = 0.028 n: ,,2 = 0.959

2 [4,&O, n; ,,* = nzz 5,2 = n:,,> = nj9,2 =


0.092 0.195 0.072 0.035

(b) “‘Nd Level:

level crossings


21X4, Ltl

1[5,5,0, t1

n: 5,2= 0.033 n: ,,* = 0.927 n: 9,z = 0.020 n; ,,,z = 0.019

n: ,,> = 0.087

nf ,,* = 0.118 n!,,,2=0.815 n$ ,5,2 = 0.060

1 [4, 193,

at different


The results

2 [4,2,0,


n$ ,,z = 0.146 nz,,, = 0.560 nj,,,=0.194 n:9,2 = 0.082

shown here are only those obtained

taking into account AN # 0 admixtures. Figs. 11-14 show the momentum distributions of levels 1,2,I, 2, respectively, for 6 = 0.12 (full curve) and 6 = 0.22 (dashed curve) normalized to 1. Also shown in these figures are the corresponding momentum distributions in the spherical limit (dot-dashed curve). Pairing effects can be easily taken into account multiplying the results shown by the corresponding occupation numbers. In all the figures fi, (p) is in units

of (b/G)’ and p is in units of b-‘. Because of the level crossings between 6 = 0.12 and 6 = 0.22 the first level below the Fermi level in ‘46Nd, [4, 1,3, $1, becomes the first level above Fermi level in “‘Nd. Inversely, the level [5,5,0, $1, which is the second level above Fermi level in ‘46Nd, becomes the first level below Fermi level in 15’Nd. Note also that the first level above Fermi level in ‘46Nd, [4,1, l,;], becomes the third level above Fermi level in “‘Nd, while the level [5,4, 1, $1 is the third and second level above Fermi level in 146Nd and I50Nd, respectively. This results in the quite different momentum distributions for ‘46Nd and I50Nd observed in the figures for levels 1 and 1. In fig. 11, for level 1, the momentum distribution for ‘46Nd (6 = 0.12) is essentially that of a d-wave and is very close to the momentum distribution of the last occupied level in the spherical limit (ld&, shown by the dash-dotted line. As seen in table 4 the l-wave content of level 1 in 146Nd is of the order of 90% d-wave and 9% g-wave, while in “‘Nd the I-components of level 1 are of the order of 95% g-wave, 3% d-wave and 2% i-wave. Contrary to this different structure of the level 1 observed in fig. 11, one can see in fig. 12 that level 2 has a rather similar structure in 146Nd and “‘Nd because this level is predominantly an h-wave level in both cases (see

J.A. Caballero, E. Maya de Guerra / Momenium distributions


table 4). Also shown for comparison in this figure is the momentum the last occupied orbital in ‘42Nd (Id,,, orbital). Finally the orbitals

in figs. 13 and

14 we compare

1 and 2, respectively,

in the spherical

the average

in ‘4hNd and “‘Nd

limit. The latter has been normalized


distribution distributions

of for

with that of the Og,,, orbital to 4 to take into account


in the spherical limit this level is occupied by 4 pairs of protons. The l-wave content of level 1 (see table 4) is essentially -96% g-wave and -3% d-wave, for S =0.12, and -82% h-wave, -6% j-wave and -12% f-wave for 6 = 0.22, resulting in the different behaviours of the momentum distributions for different deformations observed in fig. 13. On the other hand level 2 contains the same l-wave components for S = 0.12 and 6 = 0.22 (see table 4) but in different proportions: -80% d-wave, -10% g-wave and -9% s-wave, for 6=0.12; -56% d-wave, -30% g-wave and -15% s-wave, for 6 = 0.22. This is seen in fig. 14, where one can also appreciate major


from the spherical

Og7,2 limit in the entire p-range.

4. Concluding To summarize, we have analysed using a phenomenological one-body


momentum hamiltonian

distributions (the Nilsson

in deformed hamiltonian).

nuclei From

our calculations on Ne and Nd isotopes several general conclusions can be drawn. The calculated overall momentum distribution in the nuclear ground state has the desired property of being isotropic (or nearly isotropic) at the equilibrium deformation determined by the Strutinsky method. The equilibrium deformation parameters obtained by the Strutinsky method for 20Ne, r4’Nd, ‘46Nd and “‘Nd are 6 = 0.35, 0, 0.12, and 0.22, respectively. At these S-values the calculated p-values (defined by the ratio of the intrinsic agreement with experimental tum


are zero

quadrupole moment to the mean square radius) are in data and the intrinsic quadrupole moments in momen-

or very




the phenomenological


considered, as a basis for the discussion of properties of momentum distributions that can be experimentally observed. As shown in sect. 2, an important technical ingredient of our calculations is the inclusion of admixtures between different major shells that result from the diagonalization of the Nilsson hamiltonian in a large spherical basis. For deformed nuclei, isotropy in momentum space can be obtained only when these admixtures that the momentum are taken into account. Yet, for 146Nd and “‘Nd the requirement distribution is isotropic at the equilibrium deformation is not very well met. This may be an indication that higher multipoles should be included in the calculation of the equilibrium deformation or that a larger HO basis should be used. In our view the most important ingredient missing in the Nilsson model hamiltonian, in comparison to DDHF calculations, is the dependence on the neutron density of the mean field felt by the protons (and vice versa). Work is now in progress ‘) to elucidate the role of these different features.

LA. Caballero


our analyses

of total densities

in sect. 3.2, we conclude peculiar feature relating deformed. exhibit

in p-space

at the equilibrium



that the overall momentum distributions do not show any to whether the nuclear density in r-space is spherical or

On the contrary,

in general

, E. Maya de Guerra / Momentum distributions

a strong

the momentum dependence


on the nuclear

of single-particle deformation,


as shown


sect. 3.3. In particular, this implies that in electron scattering at quasi-elastic kinematics strong effects of nuclear deformation can be observed in exclusive (e, e’p) experiments, which single out specific single-particle orbitals, but not in inclusive (e, e’) experiments. This is so because in quasi-elastic electron scattering the main effect of the collective nuclear shape enters indirectly through the dependence of the single-particle orbital on the shape of the potential, and this dependence is large for specific orbitals but becomes small after averaging over all occupied orbitals. This is analogous to the case of magnetic elastic scattering 22), where the dependence on deformation enters mainly through the single-particle current, and is in contrast to the case of longitudinal inelastic scattering to the yrast 2+, 4+, __ . levels where one measures directly the overall nuclear shape. Comparison of our Nilsson model results to the experimental data on Nd isotopes recently obtained at NIKHEF requires a simultaneous study of the corresponding Pr isotopes spectra on the basis of the rotational model “). Work along these lines is now under way and will be presented elsewhere *‘). A general remark can however be made here concerning the occupation of nlj orbitals in the deformed case. Because the deformation of the mean field produces a splitting of single-particle levels with different absolute Jz values - as well as admixtures of I waves and N shells theoretically one expects to find in deformed nuclei a larger fragmentation of the nlj strengths. We wish to thank Dr. M. Casas, Dr. J. Martorell and Dr. P.K.A. de Witt Huberts for stimulating discussions, and Dr. D.W.L. Sprung for a careful reading of the manuscript. This work was supported in part by Direction General de Investigation Cientifica y Tecnica of Spain under Grant PB 87/0311. One of us (E.M.G.) also acknowledges support by the NATO International Collaborative Research Grant 0702187.

References 1) A.N. Antonov, P.E. Hodgson and I.Zh. Petkov, nuclei (Clarendom Press, Oxford, 1988); P.K.A. de Witt Huberts, in Proc. of the Int. Nucl. J.L. Durrell, J.M. Irvine and C.C. Morrison, ed. R.L. Jaffe, NucI. Phys. A478 (1988) 3c 2) S. Fantoni, B.L. Friman and V.R. Pandharipande, S. Fantoni and V.R. Pandharipande, Nucl. Phys. 3) M. Casas, J. Martoretl, E. Moya de Guerra and 4) D. Day et af., to be published



Phys. Conf. Harrogate p. 615

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in 1987)


J.A. Caballero,

E. Maya de Guerra / Momentum


5) D.W.L. Sprung and E. Moya de Guerra, in preparation. 6) E. Moya de Guerra, J.A. Caballero and P. Sarriguren, Nucl. Phys. A477 (1988) 445 7) J. Mougey et al., Nucl. Phys. A262 (1976) 461; S. Frullani and J. Mougey, Adv. Nucl. Phys. V14 (1985) 8) A. Bohr and B.R. Mottelson, Nuclear structure. Vol. 2 (Benjamin, New York, 1975). 9) SC. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955) 16 10) P. Ring and P. Schuck, The nuclear many-body problem. (Springer, Berlin, 1980); See also V.G. Soloviev, Theory of complex nuclei (Pergamon, Oxford, 1976) 11) B.R. Mottelson and S.G. Nilsson, Mat. Fys. Skr. 1 (1959) 8 12) D.R. B&s and Z. Szymanski, Nucl. Phys. 28 (1961) 42 13) S.G. Nilsson et al., Nucl. Phys. A131 (1969) 1 14) V.M. Strutinsky, Nucl. Phys. A95 (1967) 420; Al22 (1968) 1; A218 (1974) 169 15) M. Brack et al, Rev. of Mod. Phys. 44 2 (1972) 320; M. Brack and P. Quentin, Phys. Lett. B56 (1975) 421; P. Quentin and H. Flocard, Ann. Rev. Nucl. Part. Sci. 28 (1978) 523 16) C.W. de Jager, H. de Vries and C. de Vries, At. Data Nucl. Data Tables 14 (1974) 479; 36 (1987) 495 17) S. Raman, C.H. Malarky, W.T. Mimer, C.W. Nestor and P.H. Stelson, At. Data Nucl. Data. Tables 36 (1987) 1 18) W.H. Bassichis, A.K. Kerman and J.P. Svenne, Phys. Rev. 160 (1967) 746 19) C.E. Price and G.E. Walker, Phys. Rev. C36 (1987) 354 20) B.L. Berman and S.C. Fultz, Rev. Mod. Phys. 47 (1975) 734 21) J.A. Caballero et al., in preparation 22) E. Graca et aI., Nucl. Phys. A483 (1988) 77