Cluster model wave function of 6Li

1.D.3

I

Nuclear Physics A97 (1967) 449--457; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

C L U S T E R M O D E L WAVE F U N C T I O N OF 6Li M. A. K. L O D H I

Department of Physics, Texas Technological College, Lubbock, Texas Received 6 February 1967

Abstract: The wave function of eLi has been constructed in the cluster model by considering an alpha and deuteron cluster configuration. This wave function is obtainable on proper antisymmetrization from a two-oscillator, parametric, single-particle wave function in the shell model. The separation parameter is a function of the parameters belonging to the alpha and deuteron clusters. This wave function has been used to calculate the r.m.s, radius of 6Li.

1. Introduction

F r o m the cluster model standpoint, the wave function of this nucleus has been presented with different assumptions by several workers 1) yielding good fits for certain observables but no general wave function is available hitherto. The r.m.s, radius which gives an indication of the size of this nucleus and the charge distribution is not well determined. This quantity has been widely measured and theoretical fits have been made for other p-shell nucleides quite successfully. The electron scattering data from Orsay 2) provides a value of ( r 2 ) ½ as 1.41 fm using a single harmonic oscillator in the shell model. This value differs considerably from values reported earlier 3). However, with this wave function the electron scattering cannot be fitted at large m o m e n t u m transfer. Wildermuth et al. i) constructed a wave function which gives a good agreement for energies of the various levels of 6Li, but the r.m.s, radius so obtained is too small to accept. In their wave function, 6Li is considered to have an alpha and a deuteron configuration. It further implies that the alpha cluster is larger than the deuteron :[uster. The width parameter for the alpha cluster is so chosen that the alpha particle binding energy and it~ r.m.s, radius are given correctly (although the alpha cluster in that mode[ is larger than the free alpha particle). The width parameter of the deuteron duster and the separation parameter of the alpha-deuteron clusters are their optimizing values giving the energy of 6Li. The separation parameter bears no correlation to the sizes of the clusters. The entire wave function is constructed in such a way that it reduces to the usual shell-model form with the oscillator functions in the lowest configuration if the three parameters are equal. In this paper a cluster model wave function has been derived starting from the two oscillator, shell-model wave function in such a way that the various sets of parameters, restricted within a reasonable range, always give the correct r.m.s, radius. 449

450

M.A.K.

LODHI

2. The wave function It seems quite proper to choose an alpha-deuteron description for this nucleus (Austern and Wackman 1), 1962, favour three-body alpha, p, n). It is assumed that the alpha cluster is larger than the free alpha particle but smaller than the deuteron cluster. It is also presupposed that the separation parameter is not arbitrary but depends upon the sizes of the two clusters. The two parameters belonging to the alpha and deuteron clusters, may however, be varied to a certain extent. Let the symbolic form of the wave function be 1) (Wildermuth, 1961) 7j = d{cbj(~)cl, k ( d ) z ( R , - R d ) ,

(1)

where #j and #k describe alpha and deuteron clusters, respectively, and z ( R , - R d ) refers to the relative motion between the two clusters. The operator d antisymmetrizes the wave function completely with respect to the exchange of all pairs of particles. The nucleons belonging to the alpha and the deuteron clusters may be assumed in ls and lp shells, respectively, from the shell model standpoint. If the s- and p-nucleons are considered moving in different potential wells with length parameters as and ap, respectively 4), the L S coupled shell model wave function of 6Li can be written in the lowest configuration with usual notations in the ground state as 7 / - [(ls)'(lp)2]Z=0,

(2) 4

6

1 d { U ~ e x p [ - ½ a 2 X r~J(Y°)4N~pexp [ - ½ a2 Z r2] ~/ =

~-~!

i=1

j=5

x r 5 r 6 ~ ( 1 1 # - p / O O ) Y ~ ( 5 ) Y ~ ( 6 ) ~ ( 1 2 3 4 ; 56), (3) #

where 4(1234; 56) is the spin and isospin function. Using the addition theorem for Y~(5) and Y~'u(6), the relation (2) can be written T -

4

6

1 ~/3 N2N2p~(exp [_½(a 2 ~ r i2 - a p 2 ~ r2)]r5 • r6~(1234; 56). ~/~.. 64" 3 i=1 j=s

(4)

The corresponding cluster model picture of this system would be regarded as follows: the four ls shell and two lp shell nucleons are moving about the centres of mass of their respective clusters in different potential wells. Let these relative coordinates be p~ and pj in the ~ and deuteron clusters, respectively, so that Pl = r i - R ~

and

pj = r j - - R d ,

(5)

where 4

R~ = ¼ ~ e, = R+½R,

(6)

i=1 6

gd = ½ 2 rj = j=5

(7)

WAVE FUNCTION OF eLi

451

clearly R = R ~ , - R d and R is the centre of mass of the entire system composed of these two clusters (see fig. 1). In order to antisymmetrize (4), only the part r 5 • !"6~(1234; 56) needs antisymmetrization as the exponential part is symmetrical under the exchange of nucleon coordinates. We have d r s • r6~(1234; 56) = ½ d ( r s +r6)2¢(1234; 56). (8) This follows from the fact that dr2~(1234; 56) = 0

and

~¢1"2~(1234; 56) = 0

~R 3

d

' Ps

Fig.

I. Vector representation

of alpha and deuteron

clusters in eLi.

due to the Pauli exclusion principle. Hence, d r 5 "r6 4(1234; 56) = 2dRd~(1234; 56)

(9) (10)

= 2 d ( R 2-~R`-" R+4R2)¢(1234; 56) = ~¢R2¢(1234; 56), since dR((1234; 56) = 0

and

(11)

~;'R. R~(1234; 56).

Further, 4-

4-

Z q = g p~+4(~+~R)', i=1

i=1

6

6

(12)

Y ~2 = E pj2 + 2 (-R- - 2z g )2, j=5

(13)

j=5

so that 4

6

.a~Z ri2 +• 12ap2 Zr,2

& 2

i=1

j=s

4-

6

2 a2 = ½a2 E e, + ½ ~ X e12

~=1

~=s

_{_~ . 1

2

2 2 2--2 4 2 +(2as2 +ap)R +~-(a~-a2)R 3-(as + 2ap)R

R.

(14)

452

M.A.K. LODHI

Omitting the last two terms in (14) as the c.m. motion cannot be separated out in this manner 5) and setting a = as,2

2 ~ = ap,

fl = ½(a2+2a2),

(15)

eq. (4) can be written with (10) and (14) as k~ = d exp

4

6

i=1

j=6

[-½ct ~ p2_½~ ~ p2_~flR2-]R2~(1234;56).

(16)

This wave function is similar in form to that of Wildermuth ~). It differs however, in choice of parameters and consequently, in normalization factor. When the exchange of the nucleons between the clusters is allowed, the normalization requires ~) N 2 = 6! f (~o-2~k~ +

~)*¢o dz

= 6!(a o - 2 A , +A2),

(17) (18)

where ~o = ~k(1234; 56)

no exchange

~'1 = ~,(5234; 16)

one-particle exchange

~k2 = ~,(5634; 12)

two-particle exchange.

3. The root-mean-~uare radius The r.m.s, radius of 6Li is given by 1), ~ = { ~ 2 f ( ~ k ° - 2 ~ l + ~ k 2 ) * ~ (~ pE+4R2)~k°dz} ' I = 1 Using wave function (16), this leads to

~ =

}'

( ~ o - 2 ~ + ~2) ,

(19)

(20)

where N, defined by (18) is given in terms of ~, x --- fl/~ and z = ~/0~ as h 0 = 1.5 X 10 - 2 × 2 9 Z ~ x - T z - ~ r c t - ~ ,

( q2] A~ = 10.1xE9zc~(4+6z) -~ p[5 2F 1 ~,~; 3; 4p2] ,

(21) (22)

~with 3 Pl =

ql=-~

11z+3z2 + x ) a, 6+9z 2 ( 6 + 4 z +6z 2]

~

/a,

(

A2 = 3.6x29zc~(l+z)P2 s zF1 ~, ~; 3; 4p2]

(23)

WAVE FUNCTION OF 6Li

453

with p2

=

+ 8

q2 =

~-~,

2F1 being the hypergeometric series and "-q~o= 7.6xlO_3x29nYx_7x_~ (~ +9 +~) ~ _z, ~-,

(24)

~1 = 8.5×29n¥(a+6z)-~P; 5 ( 1 - q2~-~ 4p] ] x [~(1+456z)(1

- q 2 ~ ( l + 2 q2~ 4p 2] 3 4p 2]

(

(4q12~ ~p12]]"] , (25)

--1.5 q, Q 4P 2q-1-1 25+10 4~12) +22.5 Pp, 1+ with p-

31+66z+63z 27(2+3z)2 '

Q-

2+60z+18z 2 27(2 + 3z) 2

~2 = 2 . 7 x 2 9 ~ z g ( l + z ) - 3 p 2 5 ( l -

q2~-~ 4p 2]

x [ ~ _ 4 ( 1 + l +4z ) ( 1 - q22~( 1 + 2 q2~ 4p 2] 3 4p 2] 4 ~ ) 2 qg ( 25+10 q2 4p2

+ - 7"5( 1+ 4 q 2 ~ q . p2 3 4p22].J

(26)

The r.m.s, radius has been computed by varying the parameters g and ~ within the range 0.1 < g < 1 and 0.1 < z < 6.5, respectively. The value of ½ is obtainable for several sets of parameters g and ~ (see fig. 2). The correct wave function with the suitable choice of these parameters must be decided by other physical considerations. It is worth noticing here that no reasonable value of ~ is obtained with parameters (15) derived from those of shell model wave function of 6Li used in electron scattering 4, 6 - 8 ) . The cluster model wave function (16) will correspond to the single oscillator shell model wave function in the lowest configuration when ~ = 8 = fl = y (say) are set in (16). This, however, gives the correct
454

•. A. K. LODHI

wave function (fitting the elastic electron result) for the cluster model wave function (16), it does not yield any acceptable (r2) {. This is what we expected as the wave function (16) was derived from a shell model standpoint in the lowest configuration (2). But in order to fit the elastic electron scattering results with a single oscillator parametric shell model wave function, it is important that the higher configuration mixing due to residual two-body interaction must be taken into account 8) in 6Li. Therefore, with e = ~ = /3 = 7 one must include the other configurations in (2) and reduce it to cluster model form if possible (see appendix).

2 0

1

2

3

z

F i g . 2. T h e r . m . s , r a d i u s o f 6Li ( f r o m t h e t o p ) . T h e s o l i d c u r v e s a r e f o r c~ = 0.3, 0 . 4 , 0.5, 0.6, 0 . 7 0 . 8 , 0.9, 1.0 f m -2 a n d d o t t e d c u r v e s a r e f o r ct = 0 . 3 3 8 , 0 . 4 4 3 a n d 0 . 5 8 f m -2.

4. Conclusion An attempt has been made to describe 6Li from the cluster model point of view. For its ground state an alpha cluster and a deuteron cluster representation have been chosen. The alpha cluster in this representation is expected to be larger than the free alpha particle so that the width parameter ~ (fm -2) of the alpha cluster, treated as one of the variational parameters, may be varied up to the width parameter of the free alpha particle (0.58 fm -2) as its upper limit. As the deuteron cluster is even less firmly bound, it is expected to be larger than the alpha cluster so that the ratio of the width parameter of the deuteron cluster to that of the alpha cluster, z should be less than unity. The separation parameter is not arbitrary but depends on the sizes of the two clusters given by (15).

WAVE FUNCTION OF 6Li

455

This wave function is equivalent to the single particle shell model wave function in which the s- and p- nucleons are allowed to move in different potential wells. Further, when ~ = ~ = fl -- y (say) the wave function is reduced to a single parametric shell model wave function. The wave function (16) thus constructed is employed for calculating the r.m.s. radius of 6Li. An acceptable value of(r2> ~ lying between 2.4 fm to 2.8 fm is obtained for the set of the parameters ~ and z such that: 0.58 fm -2 >- ct > 0.34 fin -4 and

1.0

\ \

0.~

o.L

0.2

0~4

"

0.6

0.8

F i g . 3. z as a f u n c t i o n o f ~ f o r ( f r o m t h e t o p ) ( r 2 > l - = 2 . 0 0 , 2 . 2 5 , 2 . 5 0 , 2 . 7 5 , 3 . 0 0 f m .

0.4 < z < 1. The r.m.s, radius is almost constant for large z. This trend (see figs. 2 and 3) favours our proposition for choosing the width parameter of the deuteron cluster. I am thankful to Professor L. R. B. Elton for many helpful discussions, and to Professor D. M. Brink for reading the manuscript and making useful suggestions.

456

M. A, K. LODHI

Appendix As an example, a state at 2hco higher than the lowest one be mixed in (2) so that ~, = (as)4(lp) ~+ c(ls)4(lp2p)

N(Cl~1(0) _~_C2 ~(1>).

(A.1)

The function ~(o) has been shown earlier to reduce to cluster model in (16). We shall now show that when ~b(1) is mixed in, the total 71 can still be reduced to cluster model (ref. 9)). Let 0(1) =

1 1 ~/3 4 ~ / ~ x/2 64~3 N ~ N x p N 2 p exp [-½~=~,_.r~] x x]r,.

r615--~(r~+r~)]4.

(A.2)

The first term ~ t r s • v6 has been treated while antisymmetrizing ~o; we shall now take up d r s • v6(rZ5+ r62)4 = ½~¢[(r s + r6) z -- (r 2 + r62)](r~ + r62)4 2

2

2

2 2

= d [ 2 R a ( r 5 + r 6 ) - r s r614,

(A.3)

since d r y 4 = 0. Now d R ~ ( r 2 + r2)4 = d E ( R 2 - 4 ~ . R + ~R2)(p~ + p2 + 2R~)] 4. Expressions written in terms of R, which contain R as a factor, are spurious state. They will be omitted on antisymmetrization 5). Hence ddR~(r~ + r~)¢ = d [ - } R Z ( p ~ + P6) z +~-Y a2 R 2]4.

(A.4)

Next d r 2 r z 4 = - ½ d (r 2 - r2)2 ~

= -2d(S~.

r)2~

(where r = r s - r 6 ) =

= -8vdR2r2[1

+2Pz(cos 0)]¢

(0 being the angle between the vectors R and t) 2 --~ = - i v1. 6. ~ R ( p 2, + p26 ) [ l +

8T¢

(22#-/~/00) r~'(Y~)r~-u(co)l ~.

(A.S)

WAVE FUNCTIONOF eLi

457

Eq. (A.2) is thus written with the help of (A.3)-(A.5) as ~b(1) =

5/1 16x/5 (~) ~ 6 x/6! 9x/3 d exp [ - ½Yi=IZp2 _ 3~_~2 _ ~}yR2-]

x y R 2 E1-~-57

{~R2+(p2+p 2) ( 5 +

16rc~ ~ (22p-kt/00))

Y~'(f2)Y;U(~)}] ~,

(A.6)

where M is the normalization constant to allow for removal o f spurious state. Finally the wave function (A.1) is given by =

116 (~)~ 6 x/~." 9x/3 N d exp [-½~i=IZ p2 _]])R 2 _2y_~2-] x R 2 Ecl..~4-~Mc2 { 1 _ 1 , ( ~ R , . ( , ~ . , ~ ) ) ( ~ +

16~.)

.u Provided c = c2/c1 is small the wave function (A.7) can be written approximately in terms o f two cluster wave functions given by • ~ = N~ d exp

[ 2:' -½y

-½~(1 +

],]-5Mc) ~ p~. .]=5 945

4-

6

i=1

j=5

] 6

pj-7~'R -37R -] ~ p2 j=5

x ~ <22p-#/OO>Y¢(t2)y2"(co)}~. /t

References 1) K. Wildermuth et al., Phys. Rev. 123 (1961) 548; T. I. Kopaleishvili et al., JETP 11 (1961) 1268; N. Austern and P. H. Wackman, Nuclear Physics 30 (1962) 529; D. F. Jackson, Proc. Phys. Soc. 79 (1962) 1041 2) G. Bishop, quoted in Elton and Lodhi, Nuclear Physics 66 (1965) 209 3) A. Galonsky and M. T. McEllistrem, Phys. Rev. 98 (1955) 590 4) L. R. B. Elton, Nuclear Sizes (Oxford, 1961) p. 25 5) K. Wildermuth, CERN 59-23 (1959) 6) G. R. Burleson and R. Hofstadter, Phys. Rev. 112 (1958) 1282 7) H. Oberall, Phys. Rev. 116 (1959) 218 8) L. R. B. Elton and M. A. K. Lodhi, Nuclear Physics 66 (1965) 209; M. A. K. Lodhi, Nuclear Physics 80 (1966) 125 9) M. A. K. Lodhi, Ph.D. thesis, University of London, 1963

(A.9)