Nuclear Physics NorthHolland
A509 (1990)
MOMENTUM
117140
DISTRIBUTIONS IN AXIALLY DEFORMED NUCLEI: The Nilsson model
J.A. CABALLERO
and E. MOYA
SYMMETRIC
DE GUERRA
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 singleparticle 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 singleparticle and overall momentum distributions averaged over angle. We find that the dependence on the deformation of the mean field is much stronger for the singleparticle 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 shortrange 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
their
properties.
In ref. ‘) a thorough
analysis
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 pspace determined by the more developed mean field theories, such as densitydependent HartreeFock (DDHF), are quite general and can be reproduced with much simpler mean field calculations, such as those based on phenomenological WoodsSaxon 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/$03.50 (NorthHolland)
@ Elsevier
Science
Publishers
B.V.
t1s
.l.A. Cabaiferq
E. Mqm de Guerrn / Momentum
distributions
In a previous paper we used Nilsson model wave functions to study effects of nuclear deformation in quasielastic electron scattering 6)_We found that coincidence
(e, e’p) measurements target nucleus.
may show a strong
dependence
Fig. 1 shows the magnitude
cm the deformation
of the effect found
,I
.
28
\
,;’
of the
for 28Si. In this figure
Si (c i e*pJz7AI
',I,$ 10.54 E Q 12.0MeV \
P(MeV/c)
Fig. I. Effects of nuclear deformation in quasielastic electron scattering. Comparison of the deformed model results for 6 = 0.1 (dashedcurve) and d’= 0.07 (doubledotdashed curve) IU the spherical model (dotdashed curve) and to the experimental data and fit (fullcurve) of ref. ‘).
J.A. Caballero
experimental
, E. Moya de Guerra / Momentum distributions
data of the proton
momentum
distribution
in the missing
10.5 G E < 12.0 MeV for 28Si(e, e’p)27A1 are compared (dotdashed results
curve) and deformed
correspond
(dashed
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
119
parameter
model
6 = 0.10; a similar
(doubledotdashed
curve),
fit to
but with other
6values one cannot reproduce the lowp data. It may seem paradoxical that data on quasielastic 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 singleparticle 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 singleparticle 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 quasielastic electron scattering experiments presently being carried out at NIKHEFK. The paper is organized as follows. In sect. 2 we discuss the condition of isotropy in pspace at the equilibrium shape. In sect. 3 we analyse deformation of the isotropic parts of global and singleparticle 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 pspace
velocity
distributions
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 pspace 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 pspace 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 spinorbit interaction. In this particular case one can show on general grounds that isotropy in pspace follows from the selfconsistency condition “)_ When more realistic phenomenological potentials
120
J.A. Caballero,
E. Maya de Germ
/ Momentum
are used one has to resort to specific numerical in pspace
at the equilibrium
In what follows we discuss
results
calculations
calculated
of calculations
the case of the anisotropic
equilibrium
2.1. THE
we present
deformation
distributions
to check the isotropy
by the standard
for the Nilsson
HO to illustrate
procedures.
model,
the equivalence
but first
of different
conditions.
ANISOTROPIC
Consider
HARMONIC
the anisotropic
OSCILLATOR
harmonic
(
IT=;
oscillator
2 c
&++f
w’,(x,)’
.
a=x,y,z
>
The selfconsistency 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 rspace is characterof the potential, a, = (mw,)‘, ized by the mean square
values
=;, (nf +$/(Mw,) Rt=(j,(xb12) Then the selfconsistency
condition R a ..Er~RS a,,
which,
in terms of the a,
imposes w,~
potential
coefficients
condition
and the occupation
that at equilibrium
for any LY,LX’,
(3)
w, defined
ff&Ja = o;&,c this is a selfconsistency
= hua/(Mw,) .
for any (Y,ff’,
relating
numbers
in eq. (2) reads
the frequencies
(4) of the anisotropic
of the given configuration
HO
{n,}.
That this selfconsistency condition is an equilibrium condition can be shown by calculating the shape of the momentum distribution. The latter is characterized by the mean square values
[email protected], From this expression
bl)‘)=Mho, 1i (n; +$) = Mho,u, one sees that the condition P’,= P;=
is identical
to the selfconsistency
of isotropy
. in momentum
P+gP&,
condition
given in eq. (4). Furthermore
(5) space (6) the total
J.A. Caballero
energy
, E. Maya de Guerra ,J ~omenfu~
diszr~butjo~~
121
of the system, (H)=(T)+;(V)=+C
is also seen to be minimum For instance
a
when condition
for a configuration
hw,a,,
(7)
(3) is satisfied
({~}~i~ = 3P&/4M).
such that a, = aY = I&r,, with K a cnumber,
if
we parametrize the HO frequencies as w, = w,$, , ct.+,= w,&, w, = w&3,/3J’ so that the volume is constant, the minimization ofthe energy in eq. (7) gives p1 = p2 = K“3; or w, = oY = w,/K. Hence, minimizing the energy (for fixed configuration and volume) we get the same result for the equilibrium shapes of the potential (a, = a, = Ku,) and density (RX = R, = KR,) as using either of the conditions (3) or (6). In summary, within the anisotropic HO the three methods  minimization of the total energy with respect to deformation, isotropy in pspace and selfconsistency condition  are equivalent and lead to the same equilibrium shape. However with the Nilsson model hamiltonian 9,‘“), h =$.1,(6)(~~+r~
8+r2f$(Cos
8))Kh&,(2!*
s+/.L(~~(~*)~)),
(8)
eqs. (2), (5) and (7) do not hold any longer because of the spinorbit 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.
2.2. EQUILIBRIUM
DEFORMATION
WITH
THE
NILSSON
HAMILTONIAN
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 nonisotropic and the term proportional to Aw,(S) in eq. (8) is diagonal in N. In addition the AN = 2 admixtures introduced by the spinorbit 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 spinorbit and l2 terms to be Ndiagonal in the stretched coordinates. The prescription used in ref. “) to find the equilib~um shape is to look for the evalue (deformation parameter of the stretched coordinate system) that minimizes the total energy, calculated as the sum of singleparticle 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
122
J.A. Caballero,
previously
obtained
forces at equilibrium Although correct,
E. Moya de Guerra / Momentum
in ref. I*), b ecause of the counterbalance
the method
of pairing
and Coulomb
13).
the equilibrium
was recognized
distributions
deformations
failed to reproduce
13) that the Strutinsky
obtained absolute
in this way were approximately values
14) procedure
of binding
provides
energies
a more
and it
satisfactory
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 HartreeFock 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 pspace as a function of deformation. All the results presented correspond to singleparticle (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=
c
N/A
a”,,,INlW,
(9)
with
are obtained functions
using
the following $,~~(r)
representation
= i’Rnl(r) Y’,(G) ,
in rspace
of the spherical
n=;(Nl),
basis (10)
with Y:, and R,, the standard spherical harmonics and radial functions of the isotropic harmonic oscillator. The representation in pspace of the spherical basis functions is then 1 4NI.4(P) z ~(2n)3,2 with r in units
dr$,,,(r)
eP’““=(l)“R,,(p)Yi($),
I
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 singleparticle 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 Abody system in the intrinsic frame have the form
where the different
density
multipoles
in r and pspace
are
p,,(r) = C ph”‘N”‘i’“R,,(r)R,,~,(~~
(I41
,
Nf N’I’
n,(p)
with common
=
123
(15)
C p,“L”“‘(l)“+“‘R,,(~)~~,,,(~), NI N’I’
coefficients
NI N’I’
Ph ’ where
n, are occupation
numbers
and the factor
2 accounts
for the degeneracy
of
time reverse states (Ia) and 1~7)). The shape of the density in r or pspaces 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
quadrupole
moment
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 Svalue. 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
124
has also a zero at approximately and the quadrupole experimental
moment
quadrupole
test in determining
the same Svalue in coordinate
moment.
at which the energy is minimum
space
is approximately
This is what one should
the equilibrium
require
equal
to the
as a consistency
deformation.
That Q{ can be zero for 6values
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
,
(19)
NI N'I'
QOp=$r
;, p~:;“‘ZN,,NzIz(l)(NN’)/Z,
(20)
N'I' where the coefficients
IN,,N,ls are the radial I N,,N’,‘=
These integrals
i’”
integrals
~,,(X)~,~,~(X)X4 dx.
(21)
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,
(22)
QR=(Qo)AN=cI(Q~)AN=~.
(23)
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.
TABLET
Values of the radial
N
1
N
r+2
N+2 N+2
1+2
Ni2
r2
integrals
defined
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 pspace
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
125
quadrupole
of S. In this figure
Qh = 16.92 b* and (T*) = 2.75 b*,
by “)
p=
JToo
5 .Z(r’) ’
(24)
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 Pvalue is also in agreement with that obtained from HF calculations 18) and is somewhat larger than the pvalue 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 pspace for deformed nuclear ground states. This condition imposes that at the equilibrium deformation
(Q~A~=~= (Qo)rN=2 or
0; = ~(Q~A~=o.
(25)
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
moments
in rspace
(0;) (dashed curve) and in pspace QOp(full curve) for *‘Ne.
126
J.A. Caballero,
E. Moya de Guerra / Momentum
Fig. 3. Dependence
on the deformation
parameter 6 of the Strutinsky *‘Ne (see text).
distributions
(a) and Nilsson
1.20
~ A=142 _ _ _ A=146 A=150
Fig. 4. Same as fig. 3 for ‘42.‘46.‘50Nd nuclei.
(b) energies
for
, E. Maya de Guerra / Momentum
LA. Caballero
the calculations BCS equations
for Nd isotopes with a constant
of the Strutinsky energy
energy,
(6E, including
have been calculated
pairing
the oscillating in a consistent
127
has been taken into account
gap parameter
the LDM
distributions
by solving
the
(A, = A, = 1 MeV). In the calculations
deformation
energy
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 welldefined value of the equilibrium deformation for “Ne, since the criteria of minimum energy and isotropy in pspace 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
10.00
RQ Q
0.00
Fig. 5. Intrinsic
mass quadrupole
moments in pspace (Qg) for ‘42,‘46S’S0Nd nuclei as functions deformation parameter 6.
of the
J.A. Caballero, E. Moya de Guerra / Momentum distributions
128
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
QO
rc Nuclei
“Ne
‘46Nd lsoNd
P
exp.
the.
exp.
the.
lfml
lb1
WI
lb’1
3.02 4.97 5.0
1.66 2.10 2.13
58.5 276.0 525.8
16.92 53.68 94.94
exp.
the.
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 6values, except for the spherical 14’Nd isotope. The intrinsic mass quadrupole moments in pspace 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 welldefined 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 Elstrength
distribution
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
deformation
obtained
, 17.Mop de Guerra / Momentum
by the Strutinsky
dence on the deformation carry out this analysis
parameter
In this section
6 of the average
for both global
For a given singleparticle
method.
d~st~i~ution~
momentum
and single particle
129
we study the dependistributions.
momentum
We
distributions.
state ICX)we define the average momentum
distribution
as,
with the normalization The average
condition
global
momentum
j dp ii,(p) distribution
= 1. is defined
to be the isotropic
term in
eq. (13) (27) which is normalized
3.1. OVERALL
to the total number
MOMENTUM
of particles
(j dp ii(p) = A).
DISTRIBUTIONS
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
3.00
2.40
1.80
Fig. 6. Average
proton
momentum
distribution
in Ne for different
Svalues
(see text).
130
J.A. Caballero,
4.80
E. Moyn de Guerra
/ Momentum
distributions
Y
3.60

2.40

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 bvalues 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. ‘).
3.2. SINGLEPARTICLE
In this section particle
orbitals
MOMENTUM
we discuss
DISTRIBUTIONS
the properties
and study their dependence
of momentum
on the deformation
average singleparticle momentum distributions defined in terms of its spherical angular momentum components fiU(P) =; where the Zj components
no(p) =&e+
are explicitly
I
distributions
n;(P)
of single
parameter
6. The
in eq. (26) can be written as, (28)
3
given by (see eqs. (9), (11))
1; Grj(1)“~’
dz
L!?‘2(~2)
where p is in units of b’ and nt( p) in units of b3. The percentage
I23
(29)
of Zj contributions
J.A. Cabaflera
) E. Moya de Guerra / Momentum distributions
131
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 (dotdashed curve). (b) spherical nuclei: “0 (full curve), 4”Ca (dotted curve), curve) and “‘Pb (doubledotdashed curve). “Zr (dashed curve), ‘*“Sn (dotdashed
to fi,( p) are then conveniently
defined
as
dpnE(p)=C(C”,lj)z. N Expression (28) allows one to analyse a given deformed singleparticle orbital.
the content of different U components in As remarked in sect. 2, this is of interest
J.A. CahaNero,
132
for future comparison in coincidence
E. Mop
de Guerra / Momenrum
with experimental
(e, e’p) experiments
data, since the spectral
project
out specific
particle momentum distributions 677). Figs. 9 and 10 show results for the least bound different
values of the deformation
distributions
parameter
functions
Zj components
orbitals
in “Ne
measured
of the single
corresponding
to
6. Figs. 1114 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
i&(p)
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 (dotdashed curve) and pure swave limit (dashed curve).
is
J.A. Cab&era
For the discussion the numbers 1,2,3,. Fermi level, number Nilsson
levels
different
occupied
of the figures
corresponding
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,$],
Fig. 10. Momentum
133
, E. Moya de Guerra / Momentum disfributions
distributions
quantum
numbers
[TV, n,, rrrI, 01,
the
are, 2=[1,0,1,$,
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
134
J.A. Caballero,
E. Maya de Guerra j’ ~ornenr~rn
dist~butions
Fig. 11. Momentum
distributions of orbital 1 in Nd isotopes for S = 0.12 (full curve), curve) and S = 0 (dotdashed 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 singleparticle momentum distribution which is particularly strong for p  0 and quite noticeable for all pvalues 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
0.75
i !’

I
i
at2
0.09
0.06

0.03

i i i i i i i i
i

\ i i
i
i f I i

'\ i
i
i i i i i i ! !
: ! _ .’  L #’ L a 1 ‘>
0.00 0.00
1.00
8.00
2.00
P Fig. 12. Same as fig. 11 for orbital
2.
and can be easily
J.A. Caballero
0.30
, E. Moya de Guerra f Momentum
I
0.24

0.18

0.12

I
.I
./‘\
.\
‘\ \
I i i i I ! ! ! I
1.00
135
‘\
/ I
i
0.00
distributions
\ \ '\ \ \ \ \ '\ '\ '\
‘\
2.00
3.00
P Fig. 13. Momentum
distribution of the orbital 1 in Nd isotopes for S = 0.12 (full curve), curve) and 6 = 0 (dotdashed curve) (pure Og,,,) (see text).
6 = 0.22 (dashed
understood analysing the Zwave 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% dwave and 30% swave. Since, for this last bound orbital the admixture of swave 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).
0.00
1.00
3.00
2.00
P Fig. 14. Same as fig. 13 for orbital
2
136
LA. Caballero,
E. Moya de Guerra / Momentum
dis~ribufio~~
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
3LO,l,tl
2IL0,Ltl
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
distributions
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, $1 orbital. Apart from this normalization effect, the momentum distributions for these oddparity orbitals are not very sensitive to the deformation parameter S. This is because, as seen in table 3, these orbitals are dominated by the pwave contribution even at rather large Svalues. This is the only contribution that remains when AN # 0 admixtures are neglected. In the case of Nd isotopes pairing plays an important role and there are many levels that can be of interest for the analysis of experimental data 21). To simplify the discussion, in this paper we restrict our analyses only to the two first proton levels below and above the Fermi level. For the discussion of the figures and tables we label by 1 and 2 the first and second levels above the Fermi level, and by 1 and 2 the first and second below. In the range of 6 values 0 < 6 s 0.3 several positive and negativeparity levels cross each other, resulting in a different sequence of levels for different S values. As we said in sect. 2.2 we adopt here 6 =0.12 and 6 =0.22 as equilibrium Svalues
deformation
the sequences
parameters
of proton
for ‘46Nd and “‘Nd,
respectively.
For these
n, = 0.396, n, = 0.365, nr = 0.629,
a2 = 0.786,
levels are,
1=[4,1,1,$],
1
I = [4,1,3,3 for S = 0.12, with occupation and,
numbers
1 = [4, 1,3, $1) 1= [5,5,0,
$1 )
2 = [5,5,0,
$1,
2=[4,2,0,+],
2=[5,4,
l,$],
2=[4,2,0,$
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
137
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,
t1
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 =
$1
0.092 0.195 0.072 0.035
(b) “‘Nd Level:
level crossings
$1
21X4, Ltl
1[5,5,0, t1
n: 5,2= 0.033 n: ,,* = 0.927 n: 9,z = 0.020 n; ,,,z = 0.019
n: ,,> = 0.087
nf ,,* = 0.118 n!,,,2=0.815 n$ ,5,2 = 0.060
1 [4, 193,
at different
Svalues.
The results
2 [4,2,0,
$1
n$ ,,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. 1114 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 (dotdashed 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 dwave and is very close to the momentum distribution of the last occupied level in the spherical limit (ld&, shown by the dashdotted line. As seen in table 4 the lwave content of level 1 in 146Nd is of the order of 90% dwave and 9% gwave, while in “‘Nd the Icomponents of level 1 are of the order of 95% gwave, 3% dwave and 2% iwave. 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 hwave level in both cases (see
J.A. Caballero, E. Maya de Guerra / Momenium distributions
138
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
momentum
distribution distributions
of for
with that of the Og,,, orbital to 4 to take into account
that
in the spherical limit this level is occupied by 4 pairs of protons. The lwave content of level 1 (see table 4) is essentially 96% gwave and 3% dwave, for S =0.12, and 82% hwave, 6% jwave and 12% fwave 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 lwave components for S = 0.12 and 6 = 0.22 (see table 4) but in different proportions: 80% dwave, 10% gwave and 9% swave, for 6=0.12; 56% dwave, 30% gwave and 15% swave, for 6 = 0.22. This is seen in fig. 14, where one can also appreciate major
deviations
from the spherical
Og7,2 limit in the entire prange.
4. Concluding To summarize, we have analysed using a phenomenological onebody
remarks
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 Svalues the calculated pvalues (defined by the ratio of the intrinsic agreement with experimental tum
space
are zero
quadrupole moment to the mean square radius) are in data and the intrinsic quadrupole moments in momen
or very
close.
This
validates
the phenomenological
model
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
From
our analyses
of total densities
in sect. 3.2, we conclude peculiar feature relating deformed. exhibit
in pspace
at the equilibrium
139
deformations
that the overall momentum distributions do not show any to whether the nuclear density in rspace is spherical or
On the contrary,
in general
, E. Maya de Guerra / Momentum distributions
a strong
the momentum dependence
distributions
on the nuclear
of singleparticle deformation,
orbitals
as shown
in
sect. 3.3. In particular, this implies that in electron scattering at quasielastic kinematics strong effects of nuclear deformation can be observed in exclusive (e, e’p) experiments, which single out specific singleparticle orbitals, but not in inclusive (e, e’) experiments. This is so because in quasielastic electron scattering the main effect of the collective nuclear shape enters indirectly through the dependence of the singleparticle 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 singleparticle 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 singleparticle 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
Nucleon
momentum
Phys. Conf. Harrogate p. 615
and density
distributions
86 (IOP Pub. Bristol,
Nucl. Phys. A399 (1983) 51; A427 (1984) 473 J. Treiner, Nucl. Phys. A473 (1987) 429
in 1987)
140
J.A. Caballero,
E. Maya de Guerra / Momentum
distributions
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 manybody 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