Configuration mixing in mass 14 nuclei

Nuclear Physics 4 2 (1963) 4 5 4 ~ 4 6 1 ; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

C O N F I G U R A T I O N MIXING IN MASS 14 NUCLEI M. A. N A G A R A J A N

Case Institute of Technology, Cleveland, Ohio Received 26 October 1962 A b s t r a c t : T h e g r o u n d state configuration o f C 14 a n d N 14 have been determined using a zero-range

force. T w o m o d e l s have been used, one with O t* as the core a n d the o t h e r with C t ' as the core. F o r the sake o f simplicity, C 1' is treated as a closed shell o f l p ! nucleons, a n d l d ' ! a n d 2s*~ configurations are mixed into the g r o u n d states o f C '4 a n d N u with a spin-dependent zero-range potential. It is suggested t h a t a realistic m o d e l would be o n e wherein a n intermediate coupling configuration mixing calculation is made. T h i s is necessary in order to explain the stripping a n d pick-up data on m a s s 14 nuclei.

1. Introduction

The low excited states of C ~4 and N ~4 have been studied by many authors. Intermediate coupling calculations have been made by Inglis ') and Kurath 2), wherein good agreement has been obtained between theory and experiment in the prediction of the energy level spectrum. It has been pointed out by Inglis that the observed long lifetime of C t4 fl-decay cannot be explained on the basis of an intermediate coupling calculation. Visscher and Ferrell 3) were able to explain the anomalously long lifetime of C 14 by invoking a tensor force. The tensor force provided a coupling between the aS~ and 3 D t s t a t e s of N ~4 and its strength was sufficient to change the relative phase of the 3S x and IP 1 amplitudes in the ground state wave function of N ~4 and make way for a large cancellation in the C 14 ~ N ~4 fl-decay matrix element. Recent experiments 4) on C14(d, t)C ~a seem to show that there is evidence for configuration mixing in the ground state of C ~4. It is pointed out by French 5) that the wave functions obtained by Visscher and Ferrell do not predict the reduced widths extracted from stripping and pick-up reactions on C x4. An attempt has been made here to determine the wave functions from two different approaches. In the first approach, the ground state configurations of C t4 and N 14 are represented as two holes outside of the 016 core, and an intermediate coupling calculation is made using a zero-range force for the residual two-body force. The model being a crude one, quantitative agreement is not expected with experiment. In the second model, C 12 is assumed to be the core and the configurations assumed include ld~ and 2s~. The motive of the work is two-fold. Firstly to determine the effect of a pairing and quadrupole force in the j 2 configurations. (The effect of the quadrupole force in deforming the nucleus is not considered, but a direct diagonalization of the pairing + quadrupole force in the assumed j2 configurations is made.) 454

CONFIGURATION M I X I N G IN M A ~ 14 NUCLEI

455

Secondly, the evidence from pick-up reactions on C t4 suggests that the ground state of C t4 has a large overlap with the ground state of O ~6 as well as that of C 12. A realistic calculation would therefore be an intermediate coupling-configuration mixing calculation as has been done in the case of O t a by Redlich 6). The amount of overlap of the C t4 wave function with those of 016 and C t2 could possibly be studied from two-particle pick-up and stripping experiments on O ~6 and C t2 respectively.

2. Pairing Force Calculation During recent years the superconductivity model has been extended to the case of nuclei 7). There have been two approaches, one being an adapption of Bogoliubov's method, namely a transformation into a quasi-particle system and considering the ground state of an even nucleus as the part which corresponds to a quasi-particle vacuum. In the above method, the constraint is in the conservation of the average

number of particles, and the wave function obtained is a linear combination of wave functions of neighbouring even nuclei. The alternative approach of Kerman s) makes an extension of Racah's seniority concept, and the number of particles is conserved; as such the latter method is likely to be more convenient for nuclei containing a small number of particles or holes outside of a closed shell. In our calculation, the pairing force is a zero-range force, but is different from the seniority force because we allow a mixing of configurations. The residual two-body force is of the form

v = - Vo~ ( , , - ,j).

(1)

The following matrix elements are relevant to our calculation: ((n'l~]')2JM =

o[vI(nlj)~JM =

o) = G Y. ( 2 j + 1)(2j'+ 1)(2s+ 1)(2l+ l) LS

×(2,'+1)

½ jfi Ll ½ S

(llOOlLO)(l'l'OOlLO) '

(2)

S

where G =

V° R"t; .'t"

4,r 2 2 2 . R,I; fl'l' = _I u,,(r)u,,r(r)r dr,

u,~(r) is the radial part of the single particle wave function, ~ s

(3)

W,,nor

symbol and (II001LO) is a vector-addition coefficient 9). The long range part of the two-body force is chosen to be a quadrupole force, and is of the form 7) 5 Vlo., = -- - - x ½x ~, r~r2p2(cos w,j). (4) 4~t

~, j

456

M.A.

NAGARAJAbl

The matrix elements o f l"~,.s can be obtained f r o m the relation ~o)

= ( - 1)t('' +,2+r, +,'2)+L 5 ×

[-(211 + 1)(212+ 1)(2/'1 + 1)(21~ + 1)~ ~

L

J

5 ( 2 L + 1)

x V(l t lz 1, lz, L 2 ) ( / t It 0ol20>, l

t

.

p

l

(5)

where U(abcd; ef ) is related to Racah's W-function t h r o u g h the relation

U(abcd; ef) = 1"(2e+ 1)(2f+ 1)]tW(abcd; ef).

(6)

Since the t w o - b o d y force we have chosen is independent o f isobaric spin, we shall omit the mention o f this variable. The energy matrix for the J = 0 state is given in table 1. T o obtain it, a value o f 1.95 was chosen for G (see eq. (2)), and the energy TABLE 1

Energy matrix for the Ipt -2 lp| -9

J = 0

-- 1.95

state -- 2.76 --8.76

The Hamiltonian assumed is H = 2ej--Vor(rl-ra) difference between the states o f the ~P! hole and Xp~ hole was taken as 6.33 MeV, as obtained f r o m the N 15 energy spectrum. The energy difference between the two J = 0 states was 11.9 MeV. Diagonalizing this energy matrix, we obtain for the g r o u n d state o f C 14 ~'(J = 0) = 0.970(lp~) -2 +0.248(1p~) -2 or, alternatively, ~P(J = 0) = 0.762 I S o + 0 . 6 4 8 3Po.

(7)

The agreement between our wave function and that o f Visscher and Ferrell is not surprising because tensor force did not play a strong role in the C ~4 wave function. TABLE 2 E n e r g y m a t r i x for the J =

lp~ -2 l p t -2 {Ipt -1, ip~ -~}

-- 1.95

1.23 10.14

1 state

0 --2.46 2.34

The energy matrix for the J = 1 state is shown in table 2. Fig. I shows the spectrum o f the J = 1 states. The first J = 1 state appears at an energy o f 3.60 MeV above the

CONFIGURATION MIXING IN MASS 14 NUCLEI

457

g r o u n d state, as c o m p a r e d to the experimental value o f 3.95 MeV. The g r o u n d state wave function of N 14 is ~P(J = 1) = 0 . 9 9 8 ( l p i ) - 2 + 0 . 0 3 3 ( l p , ~ ) - 2 + 0 . 0 2 5 ( l p ~-1 lp~-1)

(8)

= --0.153 aS 1 + 0 . 4 8 4 1P t + 0 . 8 6 8 aD 1. It was found that the inclusion of a q u a d r u p o l e force did n o t make much difference in the N 14 wave function. It could not change the relative phase of the aS 1 and IP t wave functions. W i t h i n the framework of the ~p-shell, a tensor force is necessary to explain the long lifetime of C 14 fl-decay.

360

I+IT= 0

J~= i'~ T,O

Fig. I. The J = 1 levels of mass 14 nuclei using only the lp} -s and lpi-~ configurations.

3. Configuration Mixing Calculation The configurations t chosen were lp~,

ld[,

2s~.

Here it is implied that C ~2 in its g r o u n d state is chosen as an inert core. A value o f G = 2 was chosen. The energy matrix for the J = 0 a n d J = 1 states is shown in tables 3 a n d 4. In o b t a i n i n g it, the l d t - lp½ a n d 2 s ½ - l p ½ energy differences were TABLE 3 E n e r g y m a t r i x for the J = 0

lp½2 ld| 2 2s½2

--2

state

--2(3)4 i.70

--2 --2(3)4 4.18

TABLE 4

Energy matrix for the J = 1 state lP½2 ld| 2 2s½2

-- 2

--0.84 4.44

4.38 1.37 4.18

taken from the C ~3 spectrum. T h e level spectra o f the J = 0 a n d J = 1 states are shown in figs. 2 a n d 3. There are three J = 0 states, the second a n d third appearing t As the effect of quadrupole force is small compared to the central zero-range force, it was not considered in the configuration mixing calculations.

458

M.A.

NAGARAJAN

at 8.05 MeV and 12.05 MeV above the ground state. The 8.05 MeV level could possibly l~) correspond to the 6.89 MeV state o f C 14. 11.95 O~e~T• I

7,87

I+IT:O

I~Q5

5.20

I÷=T=O

O+=T=I

j~= o'l'l T, i

d"~" I+= T.O

Fig. 2. The J = 0 states of mass 14 nuclei with mixing of Idt= and 2s~2 configurations.

with mixing of Id| ~ and 2s~) configurations.

F i g . 3. T h e J =

1 states of mass

14 n u c l e i

The ground state wave function o f C 14 was obtained as ~u(J = 0) = 0.761(1p½)2 + 0.543(1d~) + 0.354(2s~) = 0.975 ISo+0.223

(9)

3P 0 .

F r o m (9) it is seen that there is a considerable mixing o f the l d | and 2s, configurations. This could be altered by changing the value o f G and the quantitative results o f this crude model cannot be taken seriously t There are three J = 1 states, the second and third being at 5.20 MeV and 7.87 MeV above the g r o u n d state. The ground state wave function is given by ~ ( J = 1 +, T = 0) = 0.987(lp~)+0.142(ld~)-0.078(2s~).

4. C a l c u l a t i o n s

with a Spin-Dependent

Force

In order to explain the (J = l + ) - ( J = 0 +) splitting in N 14, it is necessary to choose a spin-dependent force. It has been shown by De Shalit ~2) that the spectra o f odd nuclei could be reproduced by a zero-range force o f the type 1:12 = - V0[0 - ~) + ~ , "

~216(r, - '2),

with ~ > 0-10. Using the above force, it is seen that the ratio o f the spin singlet and triplet forces is (l-4a). We choose the ratio to be 0.25, which corresponds to the value ~ =

3 TZ-"

Treating the radial integrals to be constant, we obtain the energy spectrum shown in fig. 4. The strength o f the zero range force was adjusted to yield the first excited state in the correct position. The second J = 1 + state is obtained at 4.60 MeV, and it t It was found that by choosing the two-body force of the form V = --I,'0(0.85+0.15 ~t" o=) ~(rl--r2), it is possible to obtain the correct (J = 0 ) - - ( J = i) difference, and that the amount of mixing ofthe 1 dr= and 2s½=configurations is reduced and is within the limits prescribed by Baranger and Meshkov. ~). See fig. 4.

CONFIGURATION

MIXING IN MASS 14 NUCLEI

459

was tempting to make a correspondence with the 5.10 MeV level, which is speculated to be a 2 + level la). There is seen to be a good correspondence between the other J = 0 and J = 1 levels. 11.36

OtIT=l

902

I%T=O

8.15

O÷,T • I

4.60

I+tT = 0

2.32

O÷IT=I

I+~T=O

972

O~=T=I

8.62

~ I~),T=O

5,10

j'=r I+IT, 0 THEORY

O÷IT=I

i.t T,O EXPT.

Fig. 4. The energy spectrum o f J = 0 and J = I states of mass 14 nuclei treating all radial integrals as

equal.

The radial integrals that enter into the calculations are shown in table 5. It is seen that the value of J2,; 2, is very much larger than the rest. The exact value of the TABLE 5

Radial integrals appearing in the matrix elements Jnl; n'r = ~ Radial integrals J J J ,~ Jr J

lp; Ip lp; ld Ip; 2s

Id; ld Id; 2s 2s; 2s

Value

(b-Sl~/~z)

0.589 0.413 0.324 0.371 0.486 0.906

Rnt(r) R~'r(r)rZdr

460

M. A. N A G A R A J A N

integrals was n o w e m p l o y e d and tables 6 and 7 s h o w the energy matrices for J = 0 ÷ and J = 1 + states. The c o m p l e t e energy spectrum is shown in fig. 5. It is seen that the second excited (J = 1 +) state, which appeared at an excitation of 4.60 MeV has been TABLE 6 E n e r g y m a t r i x f o r J = 0 +, T = 1 s t a t e s w i t h t h e i n t e r a c t i o n o f t h e t y p e V = - - V0[(l - - ~ ) + c t o ' t • o'=} 6(rl-- r,) lp½ =

-- 1

- - 1.21

--0.55

5.81

- - 1.43

Idt= 2s~. =

4.64

TABLE 7 E n e r g y m a t r i x f o r J = 1 +, T = 0 s t a t e s w i t h t h e i n t e r a c t i o n o f t h e t h e V = - - Vo[(l - - c t ) + ~ t e ] • o=] 6 ( r l -- r=) lp~ 2

--3.63

I d t=

--0.107

0.380

5.14

2.06

2s½ =

0.60

10.51

0 + IT,I

9.11

14" IT, 0

7.59

0 '1" =T- I

3.8]

2.31

I ÷ (T=O ?}

9.72

I + .(T.O ? 0 'l" , T - I

8.98 8.62

I ÷ =T-O

I÷= T ' O

3.9:3

0'1" =T=I

O f iT "1

231

jW,l~" T, 0

j.w, i + .T,O EXPT.

THEORY

F i g . 5. T h e e n e r g y s p e c t r u m o f d = 0 a n d J = 1 s t a t e s o f m a s s 14 n u c l e i w i t h e x p l i c i t e v a l u a t i o n o f t h e radial integrals.

CONFIGURATIONMIXING IN MASS14 NUCLEI

461

depressed to c o r r e s p o n d to the 3.95 MeV level o f N 14. T h e g r o u n d state wave functions o f N t4 a n d C t4 a r e ~ ( J = 0 +) = 0 . 9 7 4 ( l p ~ ) + 0 . 1 9 1 ( l d ~ ) + 0 . 1 1 9 ( 2 s ~ ) , ~ ( J = 1 ÷) = 0 . 9 9 4 ( l p ~ ) + 0 . 0 3 4 ( l d ~ ) - 0 . 1 0 4 ( 2 s ~ ) .

5. Summary It is k n o w n t h a t the a s s u m p t i o n o f t r e a t i n g C 12 as a magic core is not realistic. The present a t t e m p t was m a d e to d e t e r m i n e w h e t h e r the energy levels o f mass 14 nuclei c o u l d be o b t a i n e d by a mixing o f higher configurations. It is seen that one could o b t a i n r e a s o n a b l y g o o d a g r e e m e n t with a s p i n - d e p e n d e n t zero range potential, a n d the a m o u n t o f a d m i x t u r e o f the ld~ a n d 2s~ configurations is in a g r e e m e n t with the prescriptions o f M e s h k o v a n d B a r a n g e r 4). T h e stripping experiments on N t4 leading to the states o f N t5 indicate, however, t h a t there should be a considerable a d m i x t u r e o f lp~ configuration also in the g r o u n d state o f N t4. In o r d e r to o b t a i n a fairly realistic wave function, o n e has to m a k e an i n t e r m e d i a t e - c o u p l i n g configuration mixing calculation. This w o u l d p r o b a b l y be able to explain the C ~4 ~ decay as well as the details o f the stripping a n d p i c k - u p experiments. T h e a u t h o r acknowledges his gratitude to Professor R. A. Ferrell for his interest in the p r o b l e m a n d to Professor L. L. F o l d y for helpful discussions. T h e a u t h o r also wishes to t h a n k Professor M. K. Banerjee a n d Dr. M. K. Pal for the c o m m u n i cation o f their results.

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

D. R. Inglis, Revs. Mod. Phys. 2.5 (1952) 390 D. Kurath, Phys. Rev. 101 (1956) 216 W. M. Visscher and R. A. Ferrell, Phys. Rev. 107 (1957) 781 W. E. Moore, J. N. McGruer and A. I. Hamburger, Phys. Rev. Lett. 1 (1958) 29; E. Baranger and S. Meshkov, Phys. Rev. Lett. 1 (1958) 30 M. H. Macfarlane and J. B. French, Revs. Mod. Phys. 32 (1960) 567 M. G. Redlich, Phys. Rev. 99 (1955) 1421 S. T. Belyaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 1i (1959) ; L. S. Kisslinger and R. A. Serenson, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 9 (1960) A. K. Kerman, Ann. of Phys. 12 (1961) 300; A. K. Kerman, R. D. Lawson and M. H. Macfarlane, Phys. Rev. 12.4 (1961) 162 E. U. Condon and G. H. Shortley, Theory of atomic spectra (Cambridge University Press, 1935) J. P. Elliott and A. M. Lane, Handbuch der Physik, vol. XXXIX (Springer-Verlag, Berlin, p. 404 A. M. Lane, Revs. Mod. Phys. 32 (1960) 519 A. De-Shalit, Phys. Rev. 91 (1953) 1479 E. K. Warburton and W. T. Pinkston, Phys. Rev. 118 (1960) 733