Equilibrium deformation calculations of the ground state energies of 1p shell nuclei

[

I.D.I

I

Nuclear Physics 74 (1965) 33--58; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

EQUILIBRIUM OF THE GROUND

DEFORMATION

STATE ENERGIES

CALCULATIONS OF lp SHELL NUCLEI

A. B. VOLKOV * Department of Physics, McMaster University, Hamilton, Ontario Canada Received 13 April 1965 The equilibrium ground state energies of the I p shell nuclei are calculated by diagonalizing the full many-particle Hamiltonian with saturating forces in an M, rather than J, Slater determinant representation. The single-particle wave functions have variable oscillator constants which preserve cylindrical symmetry. The ground state energy is minimized with respect to the different oscillator constants. This is equivalent to minimizing the ground state energy with respect to nuclear volume, nuclear deformation, and, in a limited sense, to the form of the singleparticle radial functions. In the zero-deformation limit, the diagonalization and minimization with respect to nuclear size is equivalent to finding the best intermediate coupling result for the ground state energy consistent with the force being used. Most lp shell nuclei are found to be deformed with the deformation being a sensitive function of the amount of Majorana exchange in the force mixture. For light lp shell nuclei the lp orbitals are considerably larger relative to the I s orbitals than predicted by the usual simple harmonic single-particle potential.

Abstract:

1. Introduction

T h e l p shell nuclei, because o f their relative simplicity, have been investigated in c o n s i d e r a b l e detail b y means o f v a r i o u s m o d e l s such as the c o n v e n t i o n a l shell m o d e l i - 3 ) , the Nilsson m o d e l 4 ) , the cluster model, etc. These a p p r o a c h e s have h a d v a r y i n g degrees o f success b u t a n u m b e r o f p r o b l e m s still exist such as the " m y s t e r i o u s " 0 ÷ excited levels ( m o s t n o t a b l y in 12C a n d 160) o f B o h r a n d M o t t e l s o n 5) a n d the difficulties associated with fitting the light l p shell nuclei a n d the heavier l p shell nuclei with a consistent force o r a consistent set o f m a t r i x elements 6). I t a p p e a r s t h a t these difficulties m a y be associated with the d y n a m i c s o f nuclear d e f o r m a t i o n a n d / o r the i n a d e q u a c y o f assuming t h a t a single h a r m o n i c oscillator (either spherically or cylindrically s y m m e t r i c ) p o t e n t i a l is a g o o d r e p r e s e n t a t i o n o f the H a r t r e e - F o c k field o f nuclei 7). T h e m a n y successes o f the shell m o d e l a n d o f the Nilsson m o d e l d e m o n s t r a t e t h a t the n u c l e a r force does l e a d to a m e a n i n g f u l H a r t r e e - F o c k i n d e p e n d e n t particle p o t e n t i a l as a first a p p r o x i m a t i o n to the nuclear p r o b l e m . Several t w o - b o d y nuclear forces have been p o s t u l a t e d which explain m o s t existing scattering d a t a a n d which should be a d e q u a t e for use as realistic forces in low-energy nuclear calculations. However, the a p p a r e n t existence o f a very s t r o n g s h o r t range repulsive p o t e n t i a l t Work partially supported by the National Research Council of Canada. 33

34

A . B . VOLKOV

makes these forces unusable in conventional shell model calculations limited to configurations of very low-lying states. A short-range repulsive hard core would lead to extensive mixing of very high-energy states in order to create the appropriate shortrenge correlations. The usual answer to this objection is to accept the existence of an average Hartree-Fock field and postulate phenomenological residual forces to explain configuration mixing leading to observed spectra. The characteristics of the HartreeFock field are assumed (usually a hal monic oscillator potential) and fixed to some extent by choosing single-particle energies by some sort of experimental criteria. The difficulty of this approach is in its non-self-consistent nature. In general, the selfconsistent Hartree-Fock field can be expected to be both non-spherical and nonharmonic. Furthermore, the use of purely phenomenological forces and a fixed singleparticle potential could easily lead to the ignoring of important dynamical consequences, such as the possible change of the Hartree-Fock field in particle-hole excitations, in nuclear deformations, etc. A proper approach to the problem would be to do a Hartree-Fock calculation using Brueckner-Bethe techniques and realistic forces. However, this program is formidable even for the simplest nuclei. Another approach would be to do a HartreeFock calculation with phenomenological saturating forces of a "realistic" nature, i.e. with a "soft" repulsive core (so as to minimize the need for higher configurations) and with the proper low-energy scatteling and binding energy properties (the binding energy and size of 4He requires the existence of at least a soft repulsive core). A soft repulsive core effective force has been suggested as the type of force to be expected from the appropriate many-body treatment of realistic forces in a finite nucleus 8). Recent calculations in 160 have indeed led to effective forces which resemble the forces used in this paper 9). Even a Hartree-Fock calculation involving reasonable effective forces is non-trivial and consequently a limited form has been used to obtain the results of this paper. It is well known that a Hartree-Fock calculation is equivalent to finding the minimum of the expectation value of the Hamiltonian of the system with respect to a general variation of the single-particle wave functions of a single determinant. A limited HartreeFock calculation can then be obtained by a limited variation (i.e., by allowing only parameter variations in a fixed functional form) of the single-particle functions. However, in order to make the results comparable to ordinary intermediate coupling calculations tlae variation is not made with respect to a single determinant, but with respect to a complete basis (within the lp shell) of Slater determinants, i.e. the variation includes all possible configurations in the lp shell compatible with the given nucleus being considered. More strictly the functions are cylindrically symmetric harmonic oscillator functions and only in the case of no deformation are the functions true Ip functions. In practice a complete Slater determinant basis is generated for any given nucleus, the appropriate energy matrix is computed, and .then the matrix is diagonalized. The resulting ground state is minimized with respect to the singleparticle function parameters which includes nuclear size (or the separate orbital sizes

l p SHELL NUCLEI

35

in the more complete calculation) and nuclear deformation. The variation with respect to size requires that the effective force be saturating. The use of Slater determinants is necessary in the deformed region where angular momentum is no longer conserved and simplification resulting from the irreducible representations of rotation are no longer helpful. The use of Slater determinants is convenient since it is possible to calculate the matrix elements rapidly by means of a digital computer (the use of cylindrically symmetric harmonic oscillator wave functions and a potential which is the sum of an attractive and a repulsive gaussian is important to insure rapid computation). The resulting matrices are larger than usual in intermediate coupling calculations (the worst case is 84 x 84) but the diagonalization can be performed fairly rapidly with available computers. An important feature of the computing program is that force parameters can be varied easily to determine which factors produce or inhibit nuclear deformation. The calculations show that most nuclei in the lp shell are unstable at zero deformation for a suitable mixture of Majorana and Wigner exchange in the nuclear force. For the lighter nuclei (the beginning of the shell) maximum energy gain is obtained for prolate deformations while for heavier nuclei (middle and end of the shell) the maximum energy gain is obtained for oblate deformations. The energy gain from deformation decreases, but the overall binding energy increases if the Is orbitals and lp orbitals are allowed to have different oscillator constants. The nuclei 7Li, SBe and ~2C show the most significant energy gains with respect to deformation. In these nuclei, there may be states which must be considered in terms of more complicated representations than given by the traditional shell model wave function. 2. The Effective Nuclear Force

The usual phenomenological forces used in shell model calculations are not properly saturating and would lead to over bound nuclei and/or to nuclei which are too small. The usual solution to this difficulty is to fix the size of the nucleus by choosing an appropriate oscillator constant (nuclear size) for the harmonic oscillator well. However, this constraint makes it impossible to perform an equilibrium calculation to test the stability of a given nucleus. Thus, it is necessary to use a force which will saturate in the proper manner. For the sake of this investigation we desire a force which should be applicable to the whole lp shell and yet be moderately "realistic" since we do not assume any particular single-particle Hartree-Fock potential, but instead minimize a Hamiltonian which includes all two particle interactions, i.e. our Hamiltonian is A

A

A

H = ~. T,-T¢.m.+ ~ V(ru)(l-m+mP~)+C~l,'s, 1=1

t
(2.1)

i=l

where the summations are over all particles in the nucleus. For the sake of simplicity only Majorana exchange P~j has been considered in the study of the ground state

36

A.B.

VOLKOV

equilibria. A later paper will discuss other exchange mixtures and the equilibrium spectra of the lp shell nuclei. The kinetic energy of the centre of mass Tc.m.must be subtracted in order to obtain the proper internal energy of the system. The potential V(rli ) is of the form

V(r) = - V, exp( -

2) + v,

e x p ( - (rip)2),

(2.2)

where Va, V,, ~ and p are parameters which must be fixed by presumably reasonable criteria. However, since this investigation is essentially exploratory in nature and since this potential is supposed to represent the result of a Brueckner-Bethe calculation using more realistic potentials, it did not seem reasonable to be too exact in the determination of these parameters. The following criteria were used to obtain possible values for the force parameters: (i) the s-wave scattering length should not be too far from the singlet and triplet scattering lengths; (ii) the effective range should not be too different from the triplet and singlet effective ranges; (iii) the binding energy and size of 4He should be given in an equilibrium calculation with the appropriate shell-model Slater determinant. The last criterion is the essential one and the first two were used primarily to limit the magnitude of the parameter search and to give a reasonable starting point. The last criterion leads to forces which are somewhat too strong in the relative s-state, but more conventional forces would lead to unacceptable binding energy and size values for 4He using simple shell-model wave functions. Since the purpose of the calculation is to investigate the equilibrium behaviour of systems described by Slater determinants, it was felt that the force must of necessity give at least the correct binding and size for 4He as a starting point. The choice of parameters is by no means unique in terms of the approximate nature of the criteria. An additional criterion is used to fix the amount of Majorana exchange m, and to discriminate between some of the choices allowed above. This criterion is: (iv) The binding energy and size of 160 should be given in an equilibrium calculation with the appropriate dosed shell Slater determinant. Since Coulomb folces have been neglected and since ground state correlations can be expected to make a fairly substantial contribution to the binding energy of 160, the last criteria was only used as a guide in choosing between possible forces and m = 0.6 was chosen as a reasonable Majorana exchange for the purpose of investigating equilibria behaviour. The effect of using different parameters is discussed at a later point. Several possible potentials with resulting scattering parameters, binding energies and sizes, are given in table 1. The equilibrium binding energy of 160 is quite sensitive to the amount of Majorana exchange and the value o f m in (2.1) must be in the vicinity o f m = 0.6 as can be seen in table 2. Force 1 has been used in almost all the calculations discussed in this paper.

lp SHELL NUCLEI

37

Other force choices lead to essentially identical results with respect to equilibrium deformations, spectra, equilibrium sizes, etc. However, recent investigations lo) have shown that different shaped forces (especially with respect to the size of the repulsive core) can have important results with respect to the spectra of 2s-ld shell nuclei. This point will be investigated in more detail when the present ealculational program is extended to the 2s-ld shell. TABLE 1 Nuclear potential parameters

Force 1 2 3 4 5 6 7 8

Vs

*t

--83.34 --60.65 --106.67 --76.69 --70.64 --73.23 --54.30 --53.98

1.60 1.80 1.50 1.50 1.70 1.70 !.80 1.80

p

f

ro

re

E(tHe) -- 28.20

a(4He) 1.32

E(ttO) -- 127.16

a(tsO) 1.72

0.82 1.01 1.05 0.45 1.01 1.01 0.81 0.71

+10.06 + 9.69 +!2.12 +10.77 +10.78 + 10.20 +10.53 +10.13

2.43 2.54 2.35 2.20 2.50 2.51 2.53 2.51

0.71 0.11 0.01 0.61 0.31 0.41 0.51 0.61

--28.01 --28.88 --28.55 --28.28 --27.52 --28.23 --27.45 --27.89

1.37 1.37 1.28 1.27 1.36 i.37 1.37 1.37

--144.47 --156.93 --147.31 --139.86 --145.85 --150.31 --146.39 --148.44

1.47 1.44 1.36 1.39 1.44 1.45 !.45 1.46

Vr 144.86 61.14 106.67 408.27 75.01 81.33 74.49 100.67

T h e parameters Vs, ct, Vr and p define the potential eq. (2.2). f a n d r0 are the scattering length and effective range o f the potential, r e is the radius at which the potential is zero due to an exact cancellation o f the repulsive and attractive parts. E and a are the binding energy and the oscillator constant obtained by the minimization of the g r o u n d state energy at zero deformation. The experimental values o f E and a are given below the headings. All energies are in MeV and distances in fm units. A Majorana exchange m = 0.6 is assumed in all calculations.

TABLE 2 Binding energy and oscillator constant a of aeO as a function of m

m• 0.45 0.50 0.55 0.60 0.65 0.70

I

--293.67 1.19 --234.77 1.27 --185.03 1.36 --144.47 1.47 --112.40 1.60 87.53 -- 1.73

2

--314.89 ! .17 --254.00 1.24 --201.14 1.33 --156.93 1.44 --121.42 1.57 --95.43 - - 1.72

3

--311.22 1.09 --246.88 i. ! 7 --192.08 1.26 --147.31 1.36 --i12.13 1.49 --85.27 1.63

4

--293.42 1.12 --232.12 1.20 --181.09 1.29 --139.86 1.39 --107.40 !.50 --82.29 1.62

5

--299.03 1.16 --239.43 1.24 --188.19 1.33 --145.85 1.44 --112.20 1.57 --86.23 1.72

6

--305.19 1.17 --244.99 1.25 --193.19 1.34 --150.31 1.45 --116.17 1.58 --89.76 1.72

7

--297.77 1.17 --238.97 i.25 --188.37 134. --146.39 1.45 --112.82 1.58 --86.76 1.72

8

--299.87 1.18 --240.97 1.26 --190.33 1.36 --148.44 1.46 --i14.76 1.61 --88.71 1.73

T h e first n u m b e r o f each entry is the m i n i m u m value o f the 160 binding energy. The second n u m b e r is the corresponding oscillator constant a. All calculations are performed at zero deformation. Experimental values are E = --127.16 MeV and a = i.72 f m - L

38

A.n. VOLKOV 3. Factors Leading to Nuclear Deformation

The single-particle representation used is very similar to that used by Nilsson in his calculations. For lp shell nuclei the following single-particle space states are required: ~, = Cs e x p ( - ½a(x 2 +y2)_½bz2), (3. In) o = Co(bz)exp(- ½a(x 2 + y 2 ) - ½bz2),

(3.1 b)

~b±1 = C± l(ax +__iay)exp(- ½a(x z + y 2 ) - ½bz2),

(3.1 c)

where Cs, Co and C± t are the appropriate normalization constants. This representation iqaplicitly assumes that the appropriate Hartree-Fock field has cylindrical symmetry and all the deformation calculations here have been subjected to this constraint. When a = b these functions reduce to simple spherical harmonic oscillator functions. From a shell model point of view these states represent a certain mixture of an infinite number of shell model states since e x p ( - ½a(x z + y2) _ ½bz2) = exp( - ½ar2) E (½(a -- b)z2)" n

n!

A deformation ~ is defined by the relation

a/b = (1 +½e)/(1-~e),

(3.2)

which is the same deformation parameter defined in appendix A of Nilsson's paper. This particular definition wffs used by Nilsson in order to diagonalize his deformation Hamiltonian with respect to the major shell quantum number. The use of this parameter in this work is for comparison purposes. It should be noted that e > 0 represents a prolate deformation while e < 0 represents an oblate deformation. In the course of the calculation, matrix elements of the Hamiltonian must be taken with respect to Slater determinants formed from spaee-spin-isospin single-particle product functions. These matrix elements involve the single-particle kinetic energies:

KE, = ( q~,lp2 /2m[ ~,),

(3.3a)

KEpo = (~bolp2/2mldPo),

(3.3b)

KEpt = (~b±t[p2/2m[t]~±1),

(3.3c)

and interaction matrix elements which are typically: •

V~p = (~(l)t~p(2)[V(r12)[~(1)~p(2)),

(3.4a)

V,px = (~b,(1)~p(2)[ V(r~2)l~bp(1)~b,(2)),

(3.4b)

where (3.4a) represents a direct matrix element and (3.4b) represents an exchange matrix element. The relative role of direct matrix elements and exchange matrix elements is determined by the amount of Majorana exchange, i.e. the size of m. It is

SHELL NUCLEI

lp

39

interesting, therefore, to examine the behaviour of the kinetic energies and the various matrix elements as a function of the deformation of the single-particle functions. One of the assumptions of the Nilsson model is that the nuclear volume, or equivalently a2b, remains constant throughout the deformation. In an equilibrium calI

I

I

I

I

i

KEpo

\

KEpt

/

\

/

\

/

\

"23

!

/

22

o

ne z

w a.

ul 20 ~

o2

19

Epo 12+

t

t

-0.6

t

t

l

-0.4

i

l

i

t

t

i

l

-0.2 0 0.2 DEFORMATION

l

l

18

t

0.4

0.6

Fig. 1. Kinetic energies as a function of nuclear deformation for the states ~., ~ and ~ t . The nuclear volume is constant with a~b = 0.064. culation the volume changes slightly as a function of deformation, but the constant volume assumption can be used for investigating the behaviour of the various matrix elements. Certain features o f the one particle kinetic energy matrix elements can be understood in terms of the space-momentum uncertainty relations. Thus, the state ~bo represents a particle with predominant motion in the z-direction (with only zero point motion in the x and y directions). A prolate deformation which extends space in the z direction initially decreases the ~bo kinetic energy. For constant volume con-

-5

;

:

:

;

i

I

I

I

f jVoo

Vlo

-6

I Vlo

VH ~ _ V o o ~ ~ •VH

VaVl

s

o

-7 >: 0 n,' l,d

z.8

V s o ~

- I° I vss~

ves

"ll÷t , i -0.8 -0.6

-01.4

, 0

-(~2

~2

DEFORMATION

~4

0'.6

0.8

C

FIE,. 2. The direct matrix elements V , = <~,(l)~j(Z)lF(r.)l~,(l)~j(2)>

as a function deformation. The nuclear volume is constant with ash = 0.064. O" i

i

I

i

i

I

i

i

of nuclear

'- i

Vlox~ Viox

- I

=:

~

vllx

V.x_11 ~ -2

Vsox

W

Vs0x

-3 Velx----~ V

s

-o'.8 -o'~ £4

~

-0'.2

o

~

x

0'.2 o'4

DEFORMATION ~"

=

016 ~.8

Fig. 3. The exchange matrix elements V(j= (~,(I)(~j(2)[ Y ( r l l ) l ~ j ( l ) ~ ( 2 ) > as a function of nuclear deformation. The nuclear volume is constant with a)b = 0.064.

lp SHELL NUCLEI

41

tinued increasing of the prolate deformation will eventually cause the tko kinetic energy to increase again because of the zero point x and y components of the kinetic energy. Similarly, the predominant x y plane of motion of the q~± 1 states lead to an initial decreasing of the kinetic energy for oblate deformations. Either deformation leads to increased kinetic energy for the Os state in which each direction is on equal footing. The kinetic energies per particle for these states are shown as a function of the deformation e in fig. 1. It should be noted that a decrease in kinetic energy of several MeV per particle is possible at the volume chosen (approximately the equilibrium size for 160), by allowing the appropriate deformation. The direct matrix elements of the interaction can be considered qualitatively as two-particle interaction "bonds" holding the system together very much like the bonds in a fluid. Furthermore, the strength of the bond is determined by the amount of overlap of the wave functions with the interaction. The direct interactions would develop a "surface tension" in the fluid model and as is well known a surface tension always opposes deformation since it attempts to minimize the surface of the system. The direct matrix element overlaps are generally minimized by the deformation which decreases the kinetic energy. Thus, the direct matrix elements oppose the deformation favoured by the kinetic energy. The behaviour of the direct matrix elements is shown in fig. 2. There does not seem to be a simple qualitative picture for the behaviour of the exchange matrix elements with respect to deformation, but explicit calculation shows that the exchange matrix elements increase for the same deformation that decreases the corresponding kinetic energies, i.e. the exchange matrix elements favour deformation. The behaviour of the exchange matrix elements is shown in fig. 3. It should be noted that while the exchange matrix elements are considerably smaller than the direct matrix elements, the changes with respect to deformation are of the same order of magnitude although opposite in sign. If the Majorana exchange mixture parameter m is small in the Hamiltonian, then not only will ~60 be overbound, but nuclear deformation becomes unlikely, because of the predominance of the direct matrix elements. However, as m increases the role of the exchange matrix elements also increases and nuclear deformation becomes more likely. For m somewhat greater than 0.5, i.e. for a force with somewhat more exchange than the Serber force, the total of all the potential interaction terms should be relatively constant for moderate deformations, the opposing tendency of the direct matrix elements being compensated by the deforming tendency of the exchange matrix elements. When this is the case, the equilibrium deformation of any given nucleus will be essentially determined by the kinetic energies of the particles of the filled orbitals. A continued increase of the value of m leads to decreasing binding for lp shell nuclei, but to larger deformations and larger deformation energy gains. Thus, the deformation of nuclei depends rather sensitively on the value of the exchange components of the nuclear force.

42

^. B. VOLKOV 4. Calculat/on of Equilibrium

Energies as

a Function of Deformation

A conventional intermediate coupling calculation 1- 3, 4) makes use of the rotational invariance of the Hamiltonian to construct a convenient basis for each possible angular momentum and then diagonalizes the corresponding Hamiltonian matrix in order to obtain the eigenvalues and eigenvectors of the system. By the use of conventional shell model techniques the matrix sizes are kept as small as possible for calculational convenience. These techniques are no longer valid if a non-spherical basis is used. However, for comparison purposes, it is desirable to use a basis in the deformation calculation which will give the conventional intermediate coupling results in the limit of zero deformation. This can be done by using a Slater determinant representation characterized by the total M value (z component of angular momentum) of the system. As long as the system is assumed to have cylindrical symmetry the Hamiltonian matrix is reducible with respect to M and the Hamiltonian can be diagonalized separately for each M basis. These matrices are in general much larger than the J (total angular momentum) matrices used conventionally, but in the zero deformation limit the results are identical and the appropriate J for any given level can generally be determined by a simple counting of the number of degenerate states for that particular level. The M representation is necessary for calculations of deformed nuclei, but this necessity has certain compensating features as long as the system does not requite too many states as is the case for the natural parity states of the lp shell nuclei. First, it is fairly easy to generate the M representation automatically. Then it is fairly easy to construct automatically the corresponding Hamiltonian matrix, since the calculation of the matrix elements of Slater determinant states is quite straightforward. The energy eigenvalues and eigenvectors can then be determined by a machine diagonalization of the Hamiltonian matrix. The following procedure is used to calculate the equilibrium deformation energy curves. The basis is determined automatically for any choice of M, N v (the number of lp protons), and N. (the number of lp neutrons). For a given deformation e the value of b is determined by a choice of the parameter a. The Hami!tonian matrix is then constructed and diagonalized in order to find the energy of the lowest eigenvalue Eo(e, a) for the nucleus being considered. The process is repeated for a+Aa and a+2Aa to obtain the energies Eo(e, a+Aa) and Eo(e, a+2Aa). Investigation has shown that Eo "-. ( a - a(e)) 2 in the vicinity of a(e) the equilibrium value of a for the deformation being considered; a(e) is determined by making a parabolic fit to Eo(e, a), Eo(e, a+ da), and Eo(e, a+2Aa). The equilibrium value of a(e) is then used to calculate and record all the desired eigenvalues and eigenvectors. The whole procedure is then repeated for a new value of e. In this way an equilibrium energy curve is obtained and the equilibrium value of e can be determined from the curve. The equilibrium ground states of the even and odd-proton 1p shell nuclei are shown in figs. 4 and 5. The third component of the total isobaric spin Tz always has the small-

lp SELL

43

NUCLEI

est possible magnitude and T, and -7", are degenerate because of the charge independence of the assumed force. The force used to calculate the curves in figs. 4 and 5 is force 1 of table 1 with C = - 2 . 5 6 MeV for the spin-orbit force. All curves are calculated for a Majorana exchange parameter m = 0.6. It should be noted that the deformation curves do not represent "potential" surfaces in the usual sense. Volume .

.

.

.

.

.

.

.

.

.

-20t

...... J

~

A-6 > -22

-24

tJJ - 2 6 O z z

~

-28

A

-30

09 -38

(5 - 4 0

A-8

-42

-44

-%.6

*

i

-o4

i

-ds

i

;

DEFORMATION

i

o'2

i

;4

i

o~

~"

Fig. 4. T h e g r o u n d state binding energies o f o d d - p r o t o n a n d even nuclei f r o m A = 5 to A = 9 as a function o f nuclear deformation. T h e g r o u n d state represents the best possible a d m i x t u r e o f all possible natural parity states for a given nucleus. T h e energy is m i n i m i z e d with respect to nuclear size at each value o f the deformation.

is not necessarily conserved but the "best" possible volume is obtained by energy minimization. The deformation curves only indicate the best variational parameters for a set of particular states. Thus, the deformation curve measures the adequacy of the representation rather than a constrained physical system, which is the usual condition required to generate potential curves. However, the deformation curves do indicate preferred shapes of the lp shell nuclei and in this sense, but not in a dynamical one, do give an indication of the nature of the appropriate potential curves. In this connection Nilsson equilibrium energy curves represent potential surfaces only if one truly believes that nuclear volume does not change during the deformation.

44

A.B.

VOLKOV

T h e first p o i n t o f interest in these curves is t h a t m o s t o f the l p shell nuclei are unstable at zero d e f o r m a t i o n . Since the zero d e f o r m a t i o n calculation represents the best possible c o n v e n t i o n a l intermediate coupling calculation (for the force being considered), this lack o f stability indicates possible sources o f e r r o r in c o n v e n t i o n a l shell

_0o

/

>~ -62

g

n,W ~ -8(3

~

•

~ -82 ~

~ -84, I-.o

z

-86

o -96 -98

-116t

-"81 -0.6

, -d.4

'

-d.2

'

6

'

0'.2

'

I

0.4

0.6

DEFORMATION C

Fig. 5. The ground state binding energies of odd-even and even-even nuclei from A = 10 to A = 15 as a function of nuclear deformation. The ground state represents the best possible admixture of all possible natural parity states for a given nucleus. The energy is minimized with respect to nuclear size at each value of the deformation.

m o d e l calculations. A -- 7 (TLi, 7Be), A -- 8 (SBe), a n d A -- 12 (12C) s h o w the largest energy gains with respect to d e f o r m a t i o n . One reason for these nuclei having especially large d e f o r m a t i o n s is due to the large M a j o r a n a exchange c o m p o n e n t o f the force. I f the force d i d n o t c o n t a i n any spin-orbit term, then the H a m i l t o n i a n w o u l d

l p SHELL NUCLEI

45

be invariant with respect to space permutations and the Hamiltonian matrix could be reduced to the irreducible representations of this group, i.e. the basis would be composed of functions of a definite space symmetry. When the Majorana component of the force is large then the states with maximum space symmetry form the lowest energy levels. In the cases being considered the states of maximum symmetry can be classified as [4, 3] (for A = 7), [4, 4] (for A = 8) and [4, 4, 4] (for A = 12). The first 4 in each case represents the four ~b, states of the "~-particle core", i.e. the first filled shell. A number of states can have the [4, 4] symmetry (there are three possible M = 0 states) one of which is the state in which every lp particle is in the ~bo state. This is the state that gains energy for a prolate deformation (the kinetic energy decreases and the V,ox matrix elements increase in magnitude which for a sufficiently large value of m compensates the loss of the Vss, Vso and Voo matrix elements since the relative contribution of the exchange matrix elements increase with increasing m). I f the correlation energy (the energy due to the best mixture of the different [4, 4] states at zero deformation) is not too great, then the system will deform prolately in order to benefit from the decreased kinetic energy of the ~bo state (this being the most important factor). Energy can also be gained by an oblate deformation, but the energy gained is much smaller mainly because less energy is gained by the kinetic energy decrease for oblate defol mation. Similar arguments hold for A = 6 and A = 7 and an overall prolate deformation is favoured since the states of maximum space symmetry include one state in which every lp particle is in the ~bo state. The situation for A = 12 (12C) is different since there is no possible state in which all the lp particles are in the ~o state. In fact at least half of the lp particles must be in the ~b± 1 states. There is one state of maximum space symmetry in which all the lp particles are in the states q~± 1. For this reason the maximum energy gain is for oblate deformations in this case. For A > 12 the nuclear systems have essentially spherical equilibrium since there are no states in which there are only q~± I states or ~bo states and thus while the kinetic energy of some particles decrease with deformation, the kinetic energy of other particles must increase. It is seen from fig. 1 that the rate of increase of kinetic energy is generally higher than the rate of decrease. The situation for 8 < A < 12 is intermediate to the clear cut prolate deformation of 8Be and the clear cut oblate deformation of 12C. The case o f A = 9 in fig. 4 is especially interesting since the addition of only one particle as compared to 8Be changes the equilibrium deformation from prolate to oblate. At zero deformation A = 9 is only slightly bound compared to A = 8 which is the experimentally observed case. However, A = 9 is not bound with respect to A = 8 at the equilibrium deformations unless the projection of the J = ½ state out of the A = 9 intrinsic deformed function results in considerably more energy gain than the corresponding energy gain resulting from the projection of the J = 0 state out of the A = 8 intrinsic deformed state. The problem of projection with respect to intrinsic states in a deformed representation appears to be quite difficult and has not been attempted in this study. There are two reasons for the A = 9 preference for oblate deformation instead of

46

A.B. VOLKOV

the prolate deformation that one might expect from a study of Nilsson diagrams or from the behaviour of the A -- 8 and A = l0 deformation curves. The first reason is the fact that there can be no state composed exclusively of 0 o orbitals while this is not the case for the 0 ~ 1 orbitals. Second, there are many more states in which three or more particles are in ~ ± 1 states compared to states in which there are three or more particles in the ~o states. This leads to the possibility o f important correlation effects -37

,

i

J

,

I

I

-38 -39

Be 6

-40 -41.

\\\

-42' z LU 0_Z -43' Q z

\\\\\ \\\

-44"

I.~

\\\\\\ -45"

Q

z

-46-

\

k

-47,

\

\ \x

-48-

\

x ~- ~NILSSON ~ _ _ ~

-49"

o

d,

0'.2

0'3 0'.4 DEFORMATION ~"

o's

o'.o

0.7

F i g . 6 . eBe deformation energy gain for prolate deformations. Curve I shows the minimized energy for the single Slater determinant having four particles in the ~ state. Curve II shows minimized energy for the best admixture of the five Slater determinants having at least three particles in the 4 0 state. Curve III shows the minimized energy for the best admixture of the twenty-three Slater determinants having at least two particles in the ~ state. Curve IV includes all possible ( f i f t y - o n e states) Slater determinants. The Nilsson equilibrium energy is shown for comparison.

which work against prolate deformation which is favoured by relatively fewer states than the oblate deformation. This is shown by the loss o f energy for small prolate deformations for A = 9. In almost every other case when zero deformation is unstable it is unstable with respect to either type of deformation. After 12C the lp shell nuclei are essentially spherical since the very small deformation energy gain in A = 13 can probably be ignored in favour of zero-deformation wave functions which have the virtue of being eigenfunctions o f 3.

l p SHELL NUCLEI

47

5. Importance of Correlations and Comparison with the Nllsson Model In fig. 6 the ground state of aBe is shown as a function of prolate deformation for several approximate calculations and compared with the exact (full representation) solution. Three of the approximations are typical deformation calculations of the type already discussed but with a limited number of basis states. Thus, curve I represents an equilibrium calculation with only one basis state, the state with maxiI

i

l

I

i

I

-75

-76 C 12

/

~" -78

~ -82 / "\ -83

/ \

/ ~" ~

,~NILSSON

/

/

/

-84

-o.7

-o'.6

-<~.5

-o'.,l

-c;~

DEFORMATION

-o'.2

-d.,

o

E

Fig. 7. ~=C deformation energy gain for oblate deformations. Curve I shows the minimized energy for the single Slater determinant having four particles in the ~ l states. Curve II shows the minimized energy for the best admixture of the five Slater determinants having at least three particles in the ~±x states. Curve III shows the minimized energy for the best admixture of the twenty-three Slater determinants having at least two particles in the ff±l states. Curve IV includes all possible (fifty-one states) determinants. The Nilsson equilibrium energy is shown for comparison.

m u m space symmetry with all four "1 p " particles in the ~0 state. Curve II represents a basis of the 5 Slater determinant states in which three or more particles are in ~bo states. Curve I I I represents a basis of 23 Slater determinant states with two or more particles in (k0 states. The last approximation is just the Nilsson equilibrium energy calculation with the nuclear volume chosen to be the same as the zero deformation nuclear volume for the exact solution (51 states) given by curve IV. The Nilsson calculation does not give an absolute energy and consequently the zero-deformation energy was chosen arbitrarily to be the same as that of the exact solution for convenience in plotting.

48

VOLKOV

^. B.

Fig. 6 shows that all the equilibrium calculations lead to essentially the same value of e at minimum energy while the Nilsson equilibrium leads to a substantially larger equilibrium deformation. A value of 8 = 0.5 represents an ellipsoid whose average major axis squared divided by the average minor axis squared (/) is 1.75 while e = 0.6 represents the ratio 2.00. Furthermore, it is seen that the Nilsson model i

'\ 05

i

!

)

i

i

i

)

t

i

\ \ "\

.

0.0

-0.5

\

> -I0 z

< (.9 )0

-]5

~'\

ki~ Be7

n-

'~\

uJ z -20

'!~\,\

o_

~ ~

-2.5

'~

-5.0 .....

-3.5

I11 •

/

./ \-J"

0.7

/

,.

;

Ill • 0 . 6

.....

/

re=O,5

\\ \ \

///

-4.0 i

-0.4 '

'

- 8 .2

'

0'

'

DEFORMATION

c; .2

'

0.4 '

0.6

(¢

Fig. 8.7Li, VBe g r o u n d s t a t e d e f o r m a t i o n e n e r g y g a i n f o r d i f f e r e n t a m o u n t s o f M a j o r a n a e x c h a n g e in the n u c l e a r f o r c e . T h e m i n i m i z e d e n e r g y i n c l u d e s t h e best a d m i x t u r e o f t h e t w e n t y - o n e p o s s i b l e Slater d e t e r m i n a n t s t a t e s for t h e s e n u c l e i .

overemphasizes the energy gain due to deformation by approximately a factor o f two. These are characteristic results, i.e. the Nilsson model leads to too large deformations and excessive deformation energy gains. The effect o f correlations can also be seen in fig. 6. Thus at zero deformation the exact 51 state solution has 3.5 MeV more binding than the single Slater determinant solution. The increased binding, roughly I0 % of the total, is due to the correlations in the wave function required to obtain a definite angular momentum for the state. Curve I represents a limited Hartree-Fock solution of aBe (the solution is limited since

l p SHELL NUCLEI

49

the functional form of the single-particle functions was not allowed to be varied with complete generality). Curves II, I I I and IV are then ground states in which varying degrees of correlations have been allowed. At the equilibrium deformation, there is still about 1.5 MeV difference in energy between the limited Hartree-Fock solution and the best correlated state even though the determinant of curve I occurs with 93.6 ~o probability in the complete solution of curve IV. In fact, the 18 states having only two particles in q5o orbitals contribute approximately 0.75 MeV to the binding even though these states occur with very small probability. Thus it is seen that correlations still play a significant role even at large deformations and the "intrinsic" state obtained in these calculations is probably more complicated than the equivalent Nilsson state. Similar calculations for the oblate deformation of 12C are shown in fig. 7. The results are essentially equivalent to those of aBe. The complete correlated solution has about 3.5 MeV extra binding compared to the limited Hartree-Fock solution at zero deformation. At equilibrium deformation the energy gain is still slightly over 1 MeV. The Nilsson equilibrium energy calculation again over-estimates both the equilibrium deformation and the energy gain. The over-estimates of the Nilsson model are due to the neglect of the short-range characteristics of the nuclear force. It is well known that the inclusion of pairing force correlations always leads to smaller deformations. The effect of changing the component of Majorana exchange in the nuclear force is shown in fig. 8 for 7Li. It is seen that increasing m results in increased values for the equilibrium deformation and for the deformation energy gain. These results are quite general and hold for all the deformable lp shell nuclei. For purposes of comparison only the change from zero deformation energy has been plotted. However, an increase in rn also results in an overall decrease in the total binding energy. Table 2 shows the extreme sensitivity of the 160 binding energy as a function of rn and this binding energy effectively determines m once the radial part of the nuclear force is determined. However, the 160 binding energy is also a sensitive function of the Bartlett and Heisenberg exchange mixtures. Inclusion in these mixtures would result in the need for a different value of m, the Majorana exchange mixture. The investigations on the effect of these other exchange mixtures is now in progress and will be reported later.

6. Excited States The ground state and some of the first few excited states of aBe are shown as a function of deformation in fig. 9. At zero deformation the J = 0, 2, 4 states form the expected rotational spectrum with however, a spacing about two-thirds the experimentally observed spectrum. As can be seen considerable energy can be gained by deformation. However, the interpretation of the various minima is not clear cut. The ground state has two local minima, one at a prolate deformation and another at oblate deformation. It seems likely that the prolate deformation represents an in-

50

A.B.

VOLKOV

trinsic state out of which the J = 0 ground state could in principle be projected. The situation with respect to the oblate deformation is more obscure. First, there is the problem of whether the projected J = 0 state is really significantly different from the prolate state. As has been pointed out by Mottelson i z), the "prolate" single-particle state ze -°'2 is really not different from the "oblate" single-particle state (x+ iy)e -''~

-,oI ,,,,,, T

"\ \

....

I "%.

,, ?i,

\

\\ %

- s2 /

>

,,,

"'--

,'

",\

....

,

,

,/!

, /

/:

,// /

/

, - x ' " .-'"

//

/ '/

I

>.: - 5 8 Ld

I -44

...... -" - -

-"

M.O M-I M,2

\ Be 8

",.

M,3

-46

-o~

\

M,4

......

'

-d.4

\ \

-d2

b

DEFORMATION

d.a

o'.4

06

e"

Fig. 9. Some of the lowest energy levels of SBe as a function of deformation. The nuclear volume for all levels is determined by minimizing the ground state energy at each deformation.

(one is just the state obtained from the other by a rotation. There is no essential physical difference between the two states). The situation is, of course, very different for the complicated mixture of Slater determinants and the cylindrical single-particle functions considered here, but the ambiguity in interpretation still remains and one cannot be certain that the two minima are really independent. The situation with respect to the excited states is also difficult to interpret. The

lp SHELL NUCLEI

51

M = 2 first excited state is seen to have an oblate minimum. The lowest J state that could be projected out of this state is J = 2. Thus it is clear that the J = 2 state is at least as low as the first M = 2 minimum with an oblate deformation. In fact, it would be possible to carry out a minimization in the M = 2 representation in exactly the same way as performed for the ground state (all nuclear volumes in fig. 9 were obtained by the ground state minimization only). This has been done and while the energies of the M ~- 0 levels decrease slightly, the qualitative picture is the same as shown in fig. 9. This leads then to the behaviour of the first M = 1 state. This state has its minimum for prolate deformation as opposed to the M = 2 oblate deformation. What then is the correct deformation for the J = 2 state? There does not seem to be any simple answer to this question and it is possible that the best way to approach the problem of excited states is to ignore deformations when the energy gains are not too large, and hope that the errors thus made are more or less systematic from level to level so that the level spacings are not too badly represented. This point of view is more meaningful than might first be apparent, since, as will be shown in the next section, the energy gain with respect to deformation is being exaggerated because of the form of the representation being used. Needless to say, for heavier nuclei, those normally considered deformed, the deformation of the system must not be ignored and the same possibly holds true for excitation of particles from one major shell to another in light nuclei. Particle-hole excitations from the lp shell into the 1s-2d shell probably lead to important deformation effects and account for many of the peculiarities of the unnatural parity states in the lp shell nuclei and for the "mysterious" low 0 ÷ states.

7. Modifications of the "Hartree-Fock" Harmonic Oscillator Potential Implicit in all the calculations so far discussed is the assumption of the harmonic oscillator nature of the Hartree-Fock single-particle potential. The potential was allowed to deviate from spherical symmetry to cylindrical symmetry and the oscillator constant (the nuclear size) was determined by energy minimization in each calculation, but the essential harmonic oscillator character (in the cartesian sense) of the potential remained, i.e. a potential characterized by infinitely high walls. In most conventional spectroscopic calculations it has been found that the assumption of a harmonic oscillator single-particle potential is quite adequate, since only a slight modification of the effective residual interaction is necessary to duplicate results using more realistic wave functions, i.e. wave functions derived from a Woods-Saxon singleparticle potential. However, it is not apparent that this will be the case in an equilibrium calculation involving more "realistic" forces. In this connection a detailed investigation has been made to study the effects of possible deviations from a harmonic oscillator potential on the ground state equilibrium. An obvious point of departure would to be use wave functions characteristic of a Woods-Saxon potential. This was not done for computational reasons. Instead, the

52

A.B.

VOLKOV

wave functions given by (3.1) were modified to be

4" --- c, exp(- ½a(x2 +y2)_ ½bz2),

(7.1a)

4'0 = C o b o z exp(-½ao(x 2 +y2)_½boz2),

(7.1b)

~ : l = C± 1al (x__+i y ) e x p ( - ½a~ (x 2 +y2) _ ½btz2),

(7.1c)

i.e. each orbital is characterized by its own set of oscillator constants. However, these oscillator constants are subjected to the restriction that a / b = a o / b o = a J b t,

(7.2)

i.e. it is assumed that the deformation e is the same for each orbital even though the orbital size can differ from that given by the harmonic oscillator potential where a = ao = al and b = bo = bl. The assumption contained in (7.2) implies that the single-particle potential has a definite deformation e since the probability density of each orbital shows the same deformation. Furthermore, although each orbital is of the harmonic oscillator form, each has its own characteristic size and in this sense these orbitals can approach the behaviour of Woods-Saxon type wave functions with the lp orbitals being either larger or smaller with respect to the ls orbital than would be the case for a simple harmonic oscillator single-particle potential. The calculation of the ground state equilibrium is the same as before except the ground state energy must in general now be minimized with respect to a, ao and a t rather than just a (at zero deformation only a minimization of a and ao is necessary since ao = a~ is required from symmetry considerations). In practice a minimization with respect to all parameters would be much too time consuming. Therefore, the following procedure has been used. The energy is minimized at zero deformation with respect to a, ao and al. In this process the parameter tr = a o / a = a~/a is determined at zero deformation and is used for all subsequent deformations to determine the value of ao for prolate deformations and to determine the value of al for oblate deformations. Thus, ao = tra for e > 0 and a~ = tra for e < 0. Once ao or at has been determined in this way, it is necessary to find the best corresponding value of a t or ao. For a non-zero deformation it has been found that the best value of a t for e > 0 and for ao for e < 0 is almost always that which makes the q~ t orbital kinetic energy equal to the ~b~ orbital kinetic energy. Thus if we define the parameter 3f = KEpl/KEvo, then 3f = I generally minimizes the energy for non-zero deformations. This requirement means that a~ < ao for e > 0 and a t > a o for e < 0. In allowing the variation of the different orbital oscillator constants, it is no longer possible to separate exactly the centre-of-mass motion from the internal motion of the nucleus. The argument of Elliot and Skyrme ~2) is still valid for the case of a deformed harmonic oscillator potential, but not exactly for the case of different oscillator constants. However, the argument is still qualitatively correct and there should be no "pure" states of centre-of-mass excitation as could be the case for particle-hole excited states. The small centre-of-mass excitation that occurs always occurs with a simul-

l p SHELL NUCLEI

53

taneous relative motion excitation and consequently there is no special favouring o f "spurious" states and these will not occur with any significant probability. However, in a previous paper 7) the centre o f mass was extracted from the energy in a way which, while correct for a cylindrically symmetric harmonic oscillator, over-estimated the centre-of-mass energy for the case o f different oscillator constants. An improved calculation shows that the change o f energy for SBe is no longer so large. See fig. 12 o f this paper. i

*-

t

i

!

i

i

i

i

i

i

i

i

i

-18

He

> -I£ ~r

6

i

w

\\\

z w -2¢ z z ~ -21

~

,' / ,/

Q z

/, /

~ -22 "IT

/

,

J

-23 -0.6

-0.4

-0.2

0

DEFORMATION

02

04

6

~"

F i g . 10. 6He g r o u n d state b i n d i n g e n e r g y as a f u n c t i o n o f d e f o r m a t i o n w i t h different o s c i l l a t o r c o n s t a n t s for the different orbitals. In c u r v e I ~ = 1 a n d Z = 1. I n c u r v e I I a = 0 . 4 7 5 a n d Z = 1, these v a l u e s are the best p o s s i b l e at z e r o d e f o r m a t i o n .

The effect of varying the individual oscillator constants is most profound for the lightest lp shell nuclei. Thus for 6He, fig. I0, we see that there is considerable energy gain at zero deformation as a changes from a = 1 to a = 0.475. This gain is over 3 MeV which is considerably more than the original gain from deformation. The equilibrium curve for a = 0.475 (curve II) shows essentially no energy gain with respect to deformation and the system can be considered spherically symmetric (though very soft with respect to moderate deformations). The case of 7Li is shown in fig. 11. Here the situation is not quite so dramatic as for 6He. Only about 1.5 MeV is gained at zero deformation for the best value of a = 0.665. It is seen that at higher prolate deformations, the proper minimization o f energy would cause a to increase to at least o" = 1.

A.B. VOLKOV

54

The m i n i m u m energy for curve II occurs at about e = 0.4 while the m i n i m u m for curve I occurs at 8 = 0.5. The true m i n i m u m would appear to fall at some intermediate value a r o u n d 8 = 0.45. The net effect o f the more complete variation is to reduce the deformation equilibrium and to reduce the energy gain due to deformation. The case o f SBe is shown in fig. 12. The best value o f a at zero deformation is ,r = 0.82 and the energy gain is small, about 0.5 MeV. It is seen that the equilibrium I

I

I

I

I

I

I

I

I

I

I

I

-25

I

I

I

Li 7

°7 -27

-29 t

-30

I

-d.6

'-S.4

'-d2

'

;

'

DEFORMATION

0.2

'

d4

'

0:6

'

E"

Fig. 11. ~Li, 7B¢ ground state binding energy as a function o f deformation with different oscillator constants for the different orbitals. In curve I a = 1, and Z = 1. In curve II ~r = 0.665 and % = 1, these values are the best possible at zero deformation.

m i n i m u m is determined by a = 1 at e = 0.5. The nucleus aBe represents a case where there is considerable deformation energy gain even with variation o f all parameters. By the time one reaches A = 11 the harmonic oscillator potential is a fairly g o o d representation o f the l p shell nuclei. After 12C the tendency is for a > 1, i.e. the l p orbitals try to minimize their size and maximize their overlap with the ls orbital. The effect and significance o f the variation o f a = ao/a = a l i a for zero deformation 2 i.e. the mean squared is shown in fig. 13. This plot shows the variation o f r,2 and rp, spread o f the ls and l p orbitals respectively, as a function o f the nuclear mass number A. Two cases are shown. The solid line is for the case when both oscillator con-

l p SHEI L NUCLEI

55

stants are equal, i.e. a = 1. The dashed line is for the best value of a, i.e. the result of the additional variation of the oscillator constants. The effect of adding two lp particles to an alpha-particle core in the first case is to drastically increase the size of r 2. This is necessary to prevent too large a kinetic energy for the lp particles. The kinetic energy is especially important in this case since I I

I

I

i

t

1

1

1

1

-40"

-41

>

~-42 tY LO 0

~X

z_ z° -43

(0

z°

Bem

-44

\\

-45

46

'-d6

'-d,

'-d2

'

6

'

o'.2

'

d.4

'

d.6

'

DEFORMATION ~"

Fig. 12. SBe g r o u n d s t a t e b i n d i n g e n e r g y as a f u n c t i o n o f d e f o r m a t i o n w i t h d i f f e r e n t o s c i l l a t o r c o n s t a n t s f o r the d i f f e r e n t o r b i t a l s . In c u r v e I a = 1 a n d Z = 1. I n C u r v e I I cr = 0.82 a n d Z = 1, t h e s e v a l u e s are the best p o s s i b l e a t z e r o d e f o r m a t i o n .

there are relatively so few interaction bonds. As additional lp particles are added there is a tendency for the values of r 2 and rp2 to become smaller as the number o f interaction bonds increase and the interaction energy plays a more important role than the kinetic energy, i.e. the tendency to increase the interaction overlap is more important than the increase in kinetic energy. It can be seen that aBe, 12C and 160 represent local minima with respect to orbital size. These are nuclei which can display the maximum space symmetries [4, 4], [4, 4, 4] and [4, 4, 4, 4] and a large Wigner and Majorana exchange component in the force would result in this sort of behaviour.

56

A.

,. VOLKOV

There even seems to be evidence of an odd-even effect with the even nuclei having relatively smaller orbitals than the neighbouring odd-proton nuclei. The extra degree of freedom gained by allowing a separate variation o f the ls and lp oscillator constants leads to a very dramatic increase o f the value rp2 for the lighter lp nuclei. At the same time the value of r,2 remains much closer to the 4He value. 6.8

I

I

I

I

P

t

l

I

I

I

6.4 ¸ 6.0' 5.6, 5.2' c

4.8" 4.4. 4.0' 3.6" "3.2"

,o

,_

6

_

_

8

I0 12 MASS NUMBER A

14

16

Fig. 13. O r b i t a l sizes, determined at zero deformation by e n c r ~ j minimization, as a function o f mass n u m b e r A . T h e solid curve is for the case of one set of oscillator constants for all orbitals. T h e dashed curve is for the case of the best set o f different oscillator constants for different orbitals at zero d¢-

formation.

The "alpha" particle core expands gradually with increasing A with very little apparent structure. The increase in the r2 average is a result of the increasing importance o f the interaction overlap between the ls and lp orbitals as more particles are added. The value of r~ decreases rapidly with increasing A and shows the same sort of structure as in the single harmonic oscillator case. Thus the symmetry structure is obviously related to l p particle orbitals. The symmetry structure can be related to the cluster model and/or to the alpha particle nature of light nuclei.

l p SHELL NUCLEI

57

It is reasonable to believe that the 2s-ld orbitals for the lightest 2s-ld shell nuclei may also be anomalously large, i.e. display a behaviour similar to that shown by A = 6 and 7. This would also have a profound effect on those excited states of l p shell nuclei which are due to particle-hole excitations in the lp shell, e.g. the 6.06 MeV 0 ÷ state of 160 and the unnatural parity states of the lp shell nuclei. The success of Boryzowicz 13,14) in explaining the 160 excited state spectra with the exception of a large gap between the ground state and the excited states can probably be explained because of the non-equilibrium nature of his calculation and the assumption of a single harmonic oscillator constant for all orbitals. The rapid variation of rp2 in fig. 13 also indicates the reason for the difficulty o~ explaining the spectra of light nuclei consistently with parameters fixed by the spectra of the heavier lp shell nuclei. Amit and Katz 6) made this attempt by fixing the value of the appropriate matrix elements by using selected data and then using these matrix elements to calculate spectra. If there is any radical change in the size of orbitals, one would have to expect corresponding changes in the matrix elements which would invalidate the Amit and Katz approach. Even with all the difficulties and ambiguities associated with nuclear forces, a nuclear force calculation has the virtue of being able to give more dynamical results and a more fundamental understanding of the equilibrium processes involved.

8. Conclusions For a given nuclear force which saturates in the lp shell, it appears that all the Ip shell nuclei up through A = 13 are not spherically stable. The lighter nuclei up to aBe display equilibrium in the prolate state while nuclei up to 13C display varying degrees of oblateness. From A = 13 to 160 the lp shell nuclei are spherical. A fairly large amount of Majorana exchange is necessary (at least as much as the Serber mixture) to lead to significant deformations and deformation energy gains. However, only 7Li, aBe and 12C show energy gains of over 2.5 MeV. If the singleparticle orbital size is varied separately for all orbitals, then it is found that the deformation energy gain is reduced. It may be quite reasonable to ignore deformation effects while calculating the natural parity states of lp shell nuclei, but a simple harmonic oscillator single-particle potential is probably not adequate for very light lp shell nuclei or particle-hole excitations into the 2s-ld shell. All the lp shell nuclei intermediate to 4He and 160 are underbound for central forces which bind 4He and 160 properly. This seems to be a characteristic of central forces. The error is not too large in terms of the overall interaction energy, but the lack of proper binding seems to indicate the need for some tensor force in the force mixture. The tensor force will not contribute to the binding of 4He and 160 and therefore will not alter the situation for these nuclei, but will provide the additional binding for the intermediate nuclei.

58

A.B.

VOLKOV

The d e f o r m a t i o n and binding is a sensitive function o f the v a r i o u s exchange mixtures. In o r d e r to o b t a i n g o o d spectra it a p p e a r s that a fairly sizable a m o u n t o f a t t r a c tive Bartlett exchange is necessary in the force mixture. This will increase the required a m o u n t o f M a j o r a n a exchange necessary to give the correct 160 binding. Preliminary calculations indicate that the overall effect o f Bartlett exchange will be to reduce still further the d e f o r m a t i o n o f l p shell nuclei. The a u t h o r w o u l d like to t h a n k Professor G. E. Brown for his interest a n d suggestions which inspired this work, Dr. John G u n n of N O R D I T A for his instruction a n d a i d in the art o f c o m p u t e r p r o g r a m m i n g and Mr. D a v i d J. Hughes for extensive c o m p u tations. The late Dr. N. G . Silvermaster was a source o f strength a n d e n c o u r a g e m e n t in the face o f m a n y difficulties. T h e a u t h o r w o u l d also like to t h a n k Professor A a g e B o h r for a F o r d F o u n d a t i o n G r a n t a n d for extending the hospitality o f the Institute for Theoretical Physics, University o f C o p e n h a g e n , where m o s t o f this w o r k was done.

References I) 2) 3) 4) 5) 6) 7) 8) 9) 10) I 1) 11) 13) 14)

D. Kurath, Phys. Rev. 101 (1956) 216 D. Kurath, Phys. Rev. 106 (1957) 975 A. N. Boyarkina, Izv. Akad. Nauk SSSR (ser. flz.) 28 (1964) 629 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 No. 16 (1955) A. Bohr and B. Mottelson, private communication D. Amit and A. Katz, Nuclear Physics 58 (1964) 388 A. B. Volkov, Phys. Lett. 12 (1964) 118 Baird H. Brandow, Congr~s Int. Physique Nucl6aire, Paris (1964) 2/C157 C. W. Wong, Congr~ Int. Physique Nucldaire, Paris (1964) 2/C133 S. P. Pandya and I. M. Green, Nuclear Physics 57 (1964) 658 B. Mottelson, private communication J. P. Elliott and T. H. R. Skyrme, Proc. Roy. Soc. A232 (1955) 561 J. Boryzowicz and R. K. Sheline, Phys. Lett., 12 (1964) 219 J. Boryzowicz, Nuclear Physics, to be submitted