Deformed binding fields

1.D.3

[

Nuclear Physics AI04 (1967) 683--691; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprmt or microfilm without written permission from the publisher

DEFORMED

BINDING

FIELDS

(lI). Deformed wave functions in 6Li arising from the tensor force N. FREED

Department of Physics, The Pennsylvania State Unwersity, University Park, Pennsylvania and PAUL G O L D H A M M E R t

Department of Physics, Universtty oJ Kansas, Lawrence, Kansas Received 24 April 1967 Abstract: A Hill-Wheeler representation is used to provide variational wave functions m a calculation of the T =: 0 levels of eLi. The model Hamiltoman contains a harmomc central force as well as a tensor interaction. It is found that the lowest levels with J = 1, 2 and 3 may be generated from a single intrinsic function with approximately the same stationary values of the variational parameters for each level and thus form a kind of collective band. The level scheme indicates that the vector effects of the tensor force have been taken into account. It appears that the 0~-partlcle core is soft towards deformation caused by the lp nucleons.

1. Introduction I n a p r e v i o u s p a p e r 1) we d e v e l o p e d a m e t h o d for u s i n g t h e H i l l - W h e e l e r

2-4)

integrals ~'MK(r') --

2./+lF j. , ~ z J D K M ( 0 a ) qbLsJ'K(ra)dOa

(1.1)

as trial w a v e f u n c t i o n s in a c a l c u l a t i o n o f t h e effects o f t e n s o r forces in t w o - b o d y p r o b l e m s . T h e intrinsic f u n c t i o n s q)'LSj.K(r,) are f o r m e d by c o n s t r u c t i n g c o n v e n t i o n a l s h e l l - m o d e l w a v e f u n c t i o n s in L S - c o u p l i n g (q~LSJ'K) w i t h o s c i l l a t o r o r b i t a l s a n d t h e n d e f o r m i n g t h e single-particle o r b i t a l s a c c o r d i n g to t h e p r e s c r i p t i o n exp ( - ½ 2 r ~ )

,, 2 exp (--2/~ra--6z~).

(1.2)

A l t h o u g h the p h y s i c a l significance o f L a n d 2 ' is lost after t h e d e f o r m a t i o n , we retain t h e m in t h e n o t a t i o n as a r e m i n d e r o f h o w o u r basic states w e r e c o n s t r u c t e d . A n i m p o r t a n t p r o p e r t y o f the H i l l - W h e e l e r i n t e g r a l is t h a t it p r o j e c t s a g o o d v a l u e o f J ( w i t h g o o d c o m p o n e n t M o n t h e lab z-axis) o u t o f 4 ' . W e d e m o n s t r a t e this by e x p a n d i n g the ~J.SJ'K in t e r m s o f f u n c t i o n s w i t h g o o d a n g u l a r m o m e n t u m '

=

(1.3)

L'S'M'J" t Work supported by the U.S. Atomm Energy Commission. 683

684

N. FREED AND P. GOLDHAMMER

We define our D-functions in such a way that the transformation from the body system a to the lab system l is given by

cPLSSr(r,) = ~, DSrM(O,)CPLSSM(rt).

(1.4)

M

Substitution of eqs. (1.3) and (1.4) into eq. (1.1) and use of the orthogonality relation

f DrM(O,,)Dr..~r(O,~)dO,, j. j, = . .8rt . . . c 6jj,6MM'6KK" 2.1+1

(1.5)

yields

=

CL'S',M tbz's'sM(r,).

(1.6)

L'S"

From the shell-model, point of view, the Hill-Wheeler integrals used here represent a means of generating a certain linear combination of higher configurations. When a tensor interaction $12

=

(al • rlz)(~r2

• r 1 2 ) - - - 3 - '0 " 1 • 0" 2

(1.7)

is incorporated into the Hamiltonian, such a generating procedure is absolutely essential since, in highly excited configurations 5-s), important effects have been shown to arise fi'om tensor interactions. In most earlier calculations the generating procedure has been effected through use of some form of perturbation theory. On the other hand, application of the Hill-Wheeler method to the deuteron 1) yielded a better energy eigenvalue than that obtained by third-order, Brillouin-Wigner, perturbation theory. In this paper, we demonstrate that the Hill-Wheeler method is easily applied in the case of several degenerate shell-model functions competing in zero order, in which case application of customary perturbation methods proves extremely difficult.

2. The Hamiltonian and intrinsic functions We choose as our model Hamiltonian for 6Li 1

1

H = ~ p l 2 - A ( ~ p,)2+ ~ Z ( q , - - q j ) 2 + V O ~ , ( q , - - q j ) 2 S L j , t

i
(2.1)

i
where p = - i ~ / S q . We are therefore considering harmonic interactions between pairs of particles in addition to the tensor force. This particular choice was made so that there would be no tendency for the central force to distort the ~-coret. As a result we can concentrate our investigation on the effects of the tensor force in deforming the shell-model orbitals. The term ( ~ p,)2/A represents the kinetic energy of the centre of mass and is subtracted out so that we deal only with the relative kinetic energy of the nucleons. + Distortion

of the core due to central forces has been considered

by Sharon

a).

DEFORMED BINDING FIELDS

685

The choice of intrinsic wave function requires some elaboration. Since the levels we want to investigate have T = 0, the undistorted 3S~, IP 1 and 3DI,2, 3 shell-model states (generated from the (ls)4(lp) z configuration) are all candidates. Since various non-negative values of K are also allowable 1), we have nine negligible basis functions with which to calculate the J = 1, 2 and 3 levels. We shall restrict ourselves in the following to distorted functions constructed from the 3S1 states with K = 0 and 1. That is, we shall take

tPsu - 2J+l{bojDo,vtCbo,,8 2 4

J-~

t

od0+b~ f DJ~' q~ot 1~ d0} .

(2.2)

At first glance it would appear that this is a poor choice for the J = 2 and J = 3 levels since the 3Dj states are not included in ¢b~ ~K when 6 = 0. Such objection turns out to be unwarranted for the following reason. Although the depend on the coordinates of all six nucleons in the system, the Hamiltonian refers only to the intrinsic relative motion. Therefore, if we expand the ¢~' in powers of 6 [cf. eq. (1.3)], we find higher-order terms in which the relative motion is identical to that in the (I s)4(1p)2 configuration but with the centre of mass in an excited state. Some of these terms correspond to the 3D s states so that one can project out their energy eigenvalue even in the limit 6 ~ 0.

Cb'LSJK

3. Calculations Matrix elements of the Hamiltonian

(~.~MKIHI~.rMK'--) 2J8~2+1yO~*,K(O)~ D~,K(O)(~'Aj,,,II~II~'~..c,)dO

(3.1)

are calculated by the methods described in ref. l) (outlined for the present case in the appendix to this paper). We should remark at this point that the most unattractive feature of this method is often thought to be the appearance of overlap integrals between different single-particle orbitals in the Slater determinants. In the present calculation no such integrals arise. All "ls-orbitals" are orthogonal because no two have the same spin-isospin quantum numbers; the two "Ip-orbitals" are orthogonal in charge space; and the overlap integrals between the ls and lp orbitals vanish because the deformation leaves the resulting states with good parity. Our trial function Os~ contains several variational parameters. The K-dependence (K = 0, 1) of,;~ and 6 was found to be essential in the two-body problem and turns out to be just as important in 6Li. Furthermore, one should employ distinct parameters for the deformed s- and p-orbitals. Such a procedure would involve eight variational parameters in addition to the band-mixing parameter and is almost completely impractible because of the computational difficulties associated with our search routine. Furthermore, its use is unjustified because of the preliminary nature of our Hamiltonian.

686

N. FREED AND P. GOLDItAMMER

On the other hand, the manner in which the ls orbitals tend to follow the deformation in the lp orbitals constitutes one of the more interesting aspects of the investigation. In order to get some idea of this effect and yet not be faced with an intractable calculation we decided to consider the following two extreme approximations: (i) The Is and l p orbitals lie in the same deformed oscillator well " " ~ = )-~P = )-K, AK

(3.2a)

6~ s = ~3~Y-_- 6K.

(3.2b)

(ii) The ls orbitals are undeformed "" LK

= 2~ r' = /"' K

(3.3a)

=

(3.3b)

o, t

6~ p = 6K.

(3.3c)

The correct result lies somewhere between (i) and (ii). Calculations which included both the K = 0 and K = 1 bands showed very slight improvement over those using one band alone. As a result, band mixing was ignored. Minimized energy eigenvalues and the associated optimum values of 2 and 6 are given in tables 1-4 as a function of tensor force strength. Perhaps the most striking feature of the calculation is that within a given band, the parameters )-K and 6 K which minimize the energy for a fixed Vo are approximately equal for the J = 1, 2 and 3 levels. For the strongest tensor force used (V o = - 1.0), the optimum value of 20 was the same for all three levels in approximation (ii) and varied by less than 2 ~ in approximation (i). Similarly, for the same tensor force, 6 o varied by 7 ~ [3 ~ ] for approximation (ii) [(i)]. This feature becomes less in evidence as the strength of the tensor force decreases. The reason for this lies in the presence of a number of near-lying relative minima which can, for certain values o f Vo, compete strongly with the "equal ).-equal 6" absolute minimum. F o r certain combinations of J and V o, one of these relative minima may actually be pulled down sufficiently far to give the o p t i m u m eigenvalue. On the other hand, it must be pointed out that the gain in energy is always extremely slight. F o r example, with Vo = - 0 . 8 in approximation (i) we find for the o p t i m u m values 21 = 1.35 (1.36) and 61 = - 0 . 4 1 ( - 0 . 4 1 ) f o r J = 1 (2) but find for J = 3, )-1 = 1.22 and 6| = - 0 . 3 0 . A n evaluation o f the J = 3 energy for 21 = 1.36 and 61 = - 0 . 4 0 leads to a result o f 18.667, 0.2 ~ higher than the true minimum of 18.631. As the tensor force becomes stronger and the deformation larger the tendency to constancy of the band parameters with J becomes more firmly established. It is apparent, therefore, that these levels can all be generated approximately from a single simple intrinsic function with constant parameters and in this sense form a kind of collective band. Furthermore the exact choice of intrinsic function is not crucial. Our calculation was based on the distorted 3S 1 state 4);11K. For some values

DEFORMED BINDING FIELDS

687

TABLE 1 Minimum energy eigenvalues In approximation 0) K~0 Vo

J:

0.0 .... 0.2 --0.4 -0.6 --0.8 --1.0

1

band

K ~ 1 band

J--2

19.00 18.85 18.52 18.08 17.55 16.96

19.00 18.89 18.59 18.19 17.70 17.12

J~3

J--

1

19.00 18.80 18.44 18.15 17.77 17.27

19.00 18.90 18.70 18.40 18.04 17.59

J=2

J=3

19.00 18.97 18.88 18.73 18.50 18.17

19.00 19.01 18.92 18.78 18 63 18.43

TABLE 2 Parameters yielding m i n i m u m energy eigenvalues in approximation (0

Vo

J~

0.0 --0.2 -0.4 - 0.6 --0.8 -- 1.0

1 J--2

1.00 0.87 0.80 0.74 0.69 0.63

J=3

1.00 0.86 0.79 0.74 0.69 0.64

J=l

J=2

0 O0 0 21 0 33 0.42 0.48 0.55

1.00 0.80 0.77 0.76 0.71 0.65

0.00 0.25 0.35 0.44 0.49 0.58

J:3 0.00 0.37 0.36 0.32 0.37 0.58

J=

1 J=-:2

1.00 1.16 1.31 1.33 1.35 1.39

J=3

1.00 1.14 1.21 1.31 1.36 1.39

J:l

1.00 1.22 1.19 1.27 1.22 1.39

J=2

0.00 0.22 0.35 0.38 0.41 0.47

0.00 0.17 0.27 0.36 0.41 0.46

J-

3

0.00 0.28 0.27 0 30 0.30 0.44

TABLE 3 M i m m u m energy eigenvalues in approximation (u) K = 0 band 11"o

J=

0.0 0.2 -0.4 0.6 -0.8 -1 0

1

19.00 18.80 18.42 17.97 17.46 16.91

K=

1 band

3~2

Y=3

J== 1

J::2

19.00 18.84 18.51 18.09 17.60 17.06

19.00 18.15 17.82 17.59 17.69 17.18

19.00 18.90 18.62 18.15 17 43 16.30

19.00 19.01 18 88 18.64 18.06 16.79

J--3 19.00 18.42 18.34 18.29 18.41 17.44

TABLE 4 Parameters yielding minimum energy eigenvalues in a p p r o x i m a h o n (ii) ,;.'o Vo 0.0 -0.2 -0.4 -0.6 -0.8 -1.0

J=

1 J~2

1.00 0.92 0.87 0.83 0.79 0.76

1.00 0.92 0.88 0.83 0.80 0.76

~5"o J=-3 1.00 0.89 0.87 0.86 0.80 0.76

J=

1 J-~2

0.00 0.32 0.54 0 68 0.80 0.89

0.00 0.38 0.54 0.76 0.87 1.00

)-'t J=3 0.00 0.40 0.40 0.44 0.94 1.04

J~-= 1 J = 2 1.00 1.07 1.12 1.16 1.18 1.18

1.00 1.10 1.10 1.18 1 22 1.23

6'~ J=3 1.00 1.06 1 10 1.16 1.19 1.26

J--

1 J=2

0 O0 -0.21 --0.32 --0.40 -0.46 --050

0.00 --0.23 -0.29 --0.39 --0.48 - 0.54

J--3 0.00 - 0 24 - 0.25 --0.29 -0.43 -0.55

688

N . F R E E D A N D P. G O L D H A M M E R

of 0-, 6) these functions strongly overlap the distorted 3Dj states qs~ t i r- If mixing of these states had been important, band mixing would have been significant [there is, in fact, a 94 % overlap between the K = 0 and K = 1 states for Vo = - 1.0, approximation (i)]. Another interesting feature of the calculation is the dependence of level order on Vo. Consider the K = 0 band eigenvalues in either approximation. For a weak tensor force the J = 3 level is the ground state. For a more realistic tensor force [realistic in the sense of giving reasonable aD 1 state admixture in the two-body calculations ~), the J = l level becomes the ground state with the J = 3 and J = 2 levels (in that order) excited states. This is in fact the observed level order in 6Li. For a stronger tensor force the order becomes J = l, 2 and 3. On the other hand the diagonal matrix elements of the tensor force in undeformed LS-coupled states (3Sl1S12135t) = O,

(3.4a)

( 3 D j I S l z I 3 D j ) = (S = 1,L =

211SlzllS

--- 1,L = 2)W(J212; 12).

(3.4b)

The 3Dj states are as a result split in the order J = 2, 3, 1 and the (3SllSlz1391) off-diagonal element is usually strong enough to pull the J = 1 level down far enough to be the ground state. But under no circumstances can the correct ordering in 6Li arise from the first-order shell model effects of the tensor force. The observed variation of level order as a function of Vo could result only from an effective vector force having diagonal matrix elements proportional to

( L S J M I L . SI LSJM) ,,., J(J+ I ) - L ( L + 1 ) - S ( S + 1).

(3.5)

This effective vector interaction is almost certainly related to that which arises in conventional tensor force shell-model calculations carried out in a spherical basis [refs. lo-12)].

4. Conclusions Tables 1 and 3 illustrate that approximation (ii) usually yields a lower (more accurate) energy eigenvalue than approximation (i). The difference between the two approximations is almost always rather slight. We can conclude that although the l s orbitals do not follow precisely the deformation caused by the I p orbitals the core is certainly soft against this type of deformation. Correct treatment of the tensor force by standard shell-model methods has hitherto required a great deal of effort, usually involving the application of second-order perturbation theory in addition to the shell-model calculation. Our prescription of using single-particle orbitals of mixed I [see also ref. 13)] appears at least as accurate as second-order perturbation theory and conceptually far simpler because of its intimate connection with the collective nuclear model.

DEFORMED BINDING FIELDS

689

The authors are grateful for the assistance of Mr. Robert Fornaro in the machine programming of the problem. They also wish to acknowledge the kind cooperation of The Pennsylvania State University Computation Centre in allowing us access to their computing facilities.

Appendix In this appendix we outline some of the more basic features of the calculation. Since the results are for the most part amplifications of the two-body development, we refer the reader to ref. 1) for background information. The minimization of ~aM [eq. (2.2)] with respect to the b K leads to the energy eigenvalue equations det IHrK,--IKK, Ei = 0.

(m.l)

Hr~,-~ (~JMrlHI~,MK),

(A.2a)

In this expression

IrK'

--

(OSMrlOaMr').

(A.2b)

A generalization of eq. (A.6) in ref. 1) leads after integrating over space and spin to the result (¢'J~K[(bJ,w) = ~as4- 2ap- 3lpp(1) '

(A.3)

with

Ipp(l)

=

J~ = ).~+2~(5r~ +6r,~)+).~ar.~ar, ~ sin z fl,

(A.4a)

Jp = ;tap+ 2zp(arp+ar,p)+ ).pa~par,p sin2fl,

(A.4b)

I --3 {3% " 4 +42~(6rp+ar,p) aJp

+ 2;~(a,,. + a,,,./'- + 2;~.(a,,. + a,,,.)a,,.a,,., sin ~/~ + aL a~,,. sin"/~},

(A.4c)

2s = ½()-rs + 2r,~),

(A.4d)

2p = ~(2rp +)-r,p).

(A.4e)

The s and p label the orbitals and K and K' the bands (K, K' = 0, 1). The overlap integrals Irr, are then found by integrating (A.3) times the appropriate Dfunctions over the second Euler angle ft. The integrations can be carried out analytically by utilizing the explicit r-dependence of the D-functions. The final results, as well as those for the [IRK,, are not reproduced here. The matrix elements tiff, are somewhat more complicated. For this reason we merely tabulate those basic matrix elements for the intrinsic motion which are later

690

N . F R E E D A N D P. G O L D H A M M E R

combined and integrated over the collective variables for the final Ifrw. We find


(A.5a)

( Z z~z) = ~ J : Zlpp(zz) +-~6lpp(1)Is,(Z2)J; ~-,

(A .5b)

( E p2) = "~'Js 2/pp(1)[53~. + 2t)p + 62~ + 43~] i

(A.5c)

2

z l z:~,j 2]] Lr,2 '~,'l,','(r 2)+46v(2p+fv)lpp(zZ)_46plpp (,,~,

'-?.s;-%~(1)[;.~ M, -~)+ 4a~(,~+ a.OMz ~)], i
( Z q," q~> = --1Rj-~I, ", ",'z"

(A.5e)

t
In the following call S

=

2 Ei
(1 ISl I ) = ]J~ 2[tv,(z2)--~-/pp(r2)],

(m.5 D

(llSl0) = 3 16 \ / 2 j~_2[lvp(xz)_ ilpp(yz)],

(A.5g)

(IIS[ - 1) = ~J~- 2[l~p(X_ iy)2], <0ISI1 > = ~lv6/~J; 2[Ipp(xz) + ilpp(yz)], <01Sl0>


8 - 2 lpp(-sr 2 2 - 2 z 2 ), ~.J~

=

1) =

! 6/ j~_ 2[Ipp(xz)_ ilvp(yz)], 3,v2 •

(A.5h) (A.5i) (A.5j) (g.5k)

where

l ss(f) ----f f exp { -- ¼(2,s+ 2,p + 22s + 22 p)(r z + r 2) -a,(z'~

" ~ +z~)}drxdr2 + z z" )--a2(z,

-= f f e x p ~drldrz, lpp(f) =

A -

(g.6a)

ff( , •

(A.6b)

exp adr~ dr2,

(A.6c)

f(r • r3)2e-gdrtdr2dr3,

(A.6d)

B ~ f z 1Z3Pt " rae-°d,t dr2dr 3 ,

(A.6e)

DI-FORMED B I N D I N G FIELDS

69t

with o -

2

~

;t~+½(<~+;%)<

2

~ 2 2 +½(,~,~+~Or~+6~pZ~ r 2 2 2 t 2 t 2 Ol-(~lp(Z3) "3[-(~2SZ2DV(~2pZl 3Vals(Zl) -~-alp(Z2) .

T h e terms I,,.,2 and

lz,~2 are

(A.6f)

exchange integrals o f a rather c o m p l i c a t e d f o r m an d

are n o t r e p r o d u c e d here. All o f the a b o v e integrals can be evaluated analytically and inserted into expressions (A.5). With these results one can calculate the intrinsic elements (~"IH[4~') a nd then insert these into eq. (3. I). T h e angular i n t eg r at i o n can then be carried o u t (also analytically) and the results for Hrr, and IKK' inserted into eq. ( A . I ) for the final m i n i m i z a t i o n process which is carried out on a c o m p u t e r .

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

N. Freed and P. Goldhammer, Nuclear Physics 63 (1965) 323 D. L. Hdl and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 M. G. Redhck and E. P. Wlgner, Phys. Rcv. 95 (1954) 127 R. E. Pcmrls and J. Yoccoz, Proc. Roy. Soc. A70 0957) 381 P. Goldharnmer, Phys. Rev. 116 (1959) 676 T. T. S. Kuo and G. E. Brown, Phys. Lett. 18 (1955) 54 T.T.S. Kuo and G. E. Brown, Nuclear Physics 85 (1966) 40 P. Goldhammer, Revs. Mod. Phys. 35 (1963) 40 Y. Y. Sharon, thesis, Princeton University (1965) A. M. Feingold, Phys. Rev. 101 (1956) 450 A. Arima and T. Teresawa, Progr. Thcor. Phys. 23 (1960) 115 P. Goldhamrner, Phys. Rev. 125 (1962) 660 W. H. Bassichis, A. K. Kerman and J. P. Svenne, Bull. Am. Phys. Soc. 11 0966) 304