Structure of the 14N nucleus using deformed orbitals

Nuclear Physics All6 (1968) 43--48; ~ ) North-Holland Publishing Co., Amsterdam N o t to be r e p r o d u c e d b y p h o t o p r i n t or microfilm without written permission f r o m the publisher

S T R U C T U R E OF T H E 14N N U C L E U S U S I N G D E F O R M E D ORBITALS S. RADHAKANT and NAZAKAT ULLAH Tata Institute o]" Fundamental Research, Bombay-5, India

Received 19 October 1967 Abstract: Axially deformed single-nucleon Hartree-Fock orbitals are obtained for the 14N nucleus

without assuming any degeneracy between the single-nucleon and its time-reversed state. Lowlying even-parity states are calculated by taking into account the band mixing. The groundstate magnetic dipole moment and the electric quadrupole moment are in excellent agreement with their experimental values. The M1 transition strengths are also calculated and compared with the earlier intermediate-coupling calculations. The single-particle Hartree-Fock Hamiltonian is used to construct the lowest unoccupied even-parity orbital. This deformed orbital is used to calculate the low-lying odd-parity states of the a4N nucleus.

1. Introduction

In the theoretical study of the properties of the ground state and the various lowlying excited states, the configuration mixing plays a dominant role. An elegant technique to take into account the mixing of the important low-lying configurations in the true many-particle wave function is to use a many-particle deformed intrinsic state and project out the physical states having good total angular momentum J and the parity n from it i). The intrinsic state is constructed from a set of deformed single-nucleon wave functions, which are obtained by carrying out a deformed Hartree-Fock calculation 2). In these calculations one usually assumes certain symmetries for the deformed single-nucleon Hartree-Fock orbitals, e.g. one common assumption so far has been that ~b_k is related to ~bk by the time-reversal operator. However, this is not true for nuclei, which have an odd number of neutrons and protons, i.e. the 14N nucleus. In the nucleus 14N, a further complication arises because the energies of the intrinsically deformed wave functions having total K = 1 and K = 0 are quite close. This means that the low-lying states of good J~ will have strong band mixing. The purpose of the present paper is threefold. (i) An axially deformed Hartree-Fock solution will be obtained for the 14N nucleus without assuming that the orbitals q5k and qS_k are degenerate. (ii) Low-lying even-parity states of the 14N nucleus will be calculated by properly taking into account the band mixing. (iii) Using the deformed single-particle Hartree-Fock Hamiltonian, unoccupied deformed single-nucleon orbitals will be calculated, which shall be used to construct low-lying odd-parity states of the 14N nucleus. The calculated quantities are compared with their experimental values and with the results obtained in the earlier calculations. 43

44

S. RADHAKANT AND N. ULLAH

2. Deformed orbitals for the 14N nucleus The intrinsically deformed ground state of the 14N nucleus is approximated by a single Slater determinant Z lo

Z = 1-I b~[0>,

(l)

),=1

where ]0> is the (ls) 4 spherical wave function, and 2 stands for k, m r The expansion of the operators bk+ is written as

b~ = ~ Cjka~+k.

(2)

TABLE 1 T h e spherical single-particle states j a n d their energies e~ Statej

Energy ej (MeV)

p~, p~ s t_ d~_ d~

-- 10.95 -- 4.95 -- 1.86 -- 1.10 3.39

TABLE 2 Single-particle d e f o r m e d self-consistent orbitals

k

~F

c~

c~

c~

0.999 0.040

--0.040 0.999

c~_~

c~_~

--0.974

--0.227

--0.227

0,974

(MeV) ,~ ½ ½'

--30.40 --30.27 --23.71

1.000

q_~ --½ --~ --½'

--29.20 --25.92 --22.83

1.000

ekHF are the eigenvalues o f the single-particle H a m i l t o n i a n h a n d Cs~ the c o r r e s p o n d i n g eigenvector components. The coefficients Cik are obtained by diagonalizing the matrix of the single-particle Hamiltonian It, the matrix elements of which are given by 10


(3)

,~=I

The single-particle energies ej are taken from the 13C levels as shown in table 1. The quantity VA denotes the antisymmetrized two-particle matrix elements of the residual

14N ORBITALS

45

nucleon-nucleon interaction, which we take to be the Rosenfeld mixture

V(r)

= ½1/o(~1 • xz)(0.3 + 0.7~1 ' a2) exp ( - # r ) , (4) #r with V = 50.0 MeV and p-1 = 1.37 fm. The harmonic oscillator parameter v, which enters into the radial wave functions exp(-r2/2v 2) is taken to be v = 1.65 fm. The values of the single-particle energies and the coefficients Cjk obtained by carrying out the deformed Hartree-Fock calculation are given in table 2. The t4N nucleus has an equal number of neutrons and protons, and the nucleon-nucleon interaction is isoscalar, therefore, the coefficients Cjk for neutrons and protons are the same. A prime on the value of k denotes that there is more than one orbital having the same value of k. As can be easily checked, the orbitals having the same value of k are mutually orthogonal.

3. Projection of even-parity states The intrinsic wave functions of even parity, having K = 1 and K = 0, can be constructed using the deformed orbitals of table 2. They can be written as ¢'K=1 = b - ¥ ~ b - ¥ ~ l ~ o ) ,

(5)

~r=o -- b ¥ ~ b - ¥ , l ~ o ) ,

(6)

where #o is the spherical wave function (1 s)*(lp) 12, and v and z~denote a neutron and proton, respectively. The expansion of the wave function #K in terms of the wave functions l JTKT=) can be easily obtained. It is given by r 2 1 , , ~K=I = (C~_~) [1010>a- ~-~ C~_~ C½_½[x/312010>c-11010>,] + ~55 (Ci-~)2[x/313010>b-- x/211010>b],

(7)

1 ~K = 0 = ~ C;_ ~ C;~[11000)a + 10100),] - ½C; _ ~ C;~ [12000>o+11100)o+11000)o-12100)=] - ~2 C~--~r ' r, "~k~r [[2000)c+[1100)=--[1000)o+[2100)o] ~C'-e= ' ~_~c~' [ . ~-~ 1 [1000)b+ [0100)b + [2100)b + ~3 13000)b 1 ,

(8)

where the subscripts a, b and c denote that the wave function belongs to the configura-

46

S. RADHAKANT AND N. ULLAH

tions (p½)-2, (p~)-2 a n d ( p C ) - 1 ( p ¢ ) - 1 , respectively. These wave functions have been written d o w n earlier 3). It is n o w an easy m a t t e r to project out the even-parity m a n y - n u c l e o n wave functions possessing a good a n g u l a r m o m e n t u m J a n d satisfying the usual time-reversal property 0 ~sm = ( - l ) S - m Us_m ' where 0 is the time-reversal operator, from the intrinsic wave functions given by expressions (7) a n d (8). These projected wave functions are used in the following to calculate the low-lying states, static m o m e n t s for the g r o u n d - s t a t e of the 14N nucleus a n d the M 1 t r a n s i t i o n strengths. TABLE 3

Low-lying even-parity states of the 14N nucleus d+, T

1+, 0 0+, 1 1+, 0 2+, 0 2+, 1 1+, 1 3+, 0

Energy (MeV)

Eigenvector components

exp.

calc.

xa

xb

xe

g.s. 2.312 3.945 7.030 9.170

0 2.816 8.022 5.736 9.519 12.242 12.977

0.956 0.999 0.292 0.000 0.000 0.000 0.000

--0.047 --0.013 0.084 0.000 0.034 0.000 1.000

--0.289 0.000 0.951 1.000 --0.999 1.000 0.000

The coefficients xa, xb and x c denote the admixtures of the configurations (p{)-2, (p~)-2 and (p~)-i (p~r)-1 in the various J+, T states. TABLE 4 M1 transition strengths M1 transition

experimentalenergies (MeV)

Transition strength A (M1)

(-/1,/'1) --~ (J~, Tf)

Et

Er

exp.

(0+,1)--~(1 +, 0) (2+,1)-+(2 +, 0) (2+, 1)---~(1+, 0) (2+, 0)--~(1+,0)

2.31 9.16 9.16 7.03

0.00 7.03 0.00 0.00

0.1-3.0 12.0±5.0 4.1

A

B

C

0.01 17.06 11.84 0.12

0.46 20.7 12.5 0.125

0.70 19.3 12.6 0.11

Column A gives the results of the present calculation and columns B and C the values obtained using the wave functions of Elliott and of Vischer and Ferrell, respectively. Columns B and C are taken from the table given by Warburton and Pinkston. We see from expansions (7) a n d (8) that the even-parity states having T = 1 arise from the K = 0 b a n d only a n d therefore have n o b a n d mixing. This p o i n t has also been n o t e d by K u r a t h a n d Picman4). The T = 0 states can be projected b o t h from K = 0 a n d K = 1 bands. These states are calculated by diagonalizing the residual i n t e r a c t i o n between the projected states. Table 3 gives the calculated even-parity states, which are f o u n d to be in better agreement with their experimental values t h a n those values calculated using only a radial H a r t r e e - F o c k calculation 3). W e see from table 2 that the excited state 1 +, T = 0 c a n n o t arise from the p-configurations alone b u t needs admixtures of 2s a n d l d configurations. This was f o u n d in earlier calculations 5) also.

47

14N ORBITALS

The c a l c u l a t e d m a g n e t i c d i p o l e m o m e n t a n d the electric q u a d r u p o l e m o m e n t in the g r o u n d state o f the ~4N nucleus t u r n o u t to be 0.325 n.m. a n d 0.794 e. fm z, respectively. T h e y are in r e m a r k a b l e a g r e e m e n t with their e x p e r i m e n t a l values o f 0.40 n.m. a n d 0.71 e . f m z. W e have also c a l c u l a t e d a few o f the M1 t r a n s i t i o n strengths using the wave functions o f table 3. T h e y are shown in table 4 with their e x p e r i m e n t a l values a n d the results o f earlier calculations 6). O u r values o f the M1 t r a n s i t i o n strengths seem to be better t h a n the earlier values o b t a i n e d using the i n t e r m e d i a t e - c o u p l i n g model.

4. Odd-parity states T h e o d d - p a r i t y states o f the 14N nucleus will be calculated by exciting a nucleon f r o m the l p c o n f i g u r a t i o n to the 2s, l d configurations. This will be d o n e by first constructing a d e f o r m e d single-nucleon o r b i t a l f r o m the set o f wave functions d~k, S½k, d~k. U s i n g this set o f wave functions a n d the values o f the single-particle energies in table 1, we d i a g o n a l i z e the m a t r i x o f the single-particle H a m i l t o n i a n given b y eq. (3), in which the o c c u p i e d o r b i t a l s q5a are a l r e a d y k n o w n by o u r H a r t r e e - F o c k calculation o f sect. 2. This gives us the lowest u n o c c u p i e d single-nucleon o r b i t a l o f even parity. I t is given by 0.327as~r~ (9) b+,, = 0.942ad~4-+ + + 0.073aa}~. + T h e o r b i t a l k = - 3 l l ! turns o u t to be a l m o s t degenerate with k = ½ a n d therefore we have t a k e n the coefficients which enter into its e x p a n s i o n as ( - 1 ) J - 4 o f the corr e s p o n d i n g coefficients in expression (9). T h e intrinsic wave functions o f o d d p a r i t y having K = 1 a n d K = 0 can be written as (10) ~ K -_ 1 = b¥,,~b_¥~b_¥~b¥,dCPo), + 45~=o = b+_¥,,~b_½,,~b_½,~b¥,~]q~o).

(11)

TABLE 5 Low-lying odd-parity states of the 14N nucleus

J-, T 0-, 0 2-, 0 1-, 0 3-, 0 2-, 0 1-, 1 0-, 1 3-, 1 2-, 1

Energy(MeV) Admixture of particle-hole configurations exp. calc. (s~r)(p~_)-a (d.t)(p~_)-3 (d})(p½) -~ (s~)(p})-l(p½)-2 (d~_)(p})-l(p½)-~ (d~)(p~)-l(p~)-~ 4.91 5.10 5.69 5.83 7.96 8.06 8.71 8.90 9.50

4.91 5.28 6.44 5.34 8.50 7.35 9.17 8.53 9.93

0.966 0.000 0.922 0.000 0.000 0.972 0.967 0.000 0.000

0.000 0.918 0.000 0.954 --0.280 0.000 0.000 0.983 0.982

0.000 0.028 0.125 0.000 0.397 0.010 0.000 0.000 --0.078

--0.076 0.258 --0.221 0.050 --0.613 --0.207 --0.076 0.051 0.094

Admixtures of various particle-hole configurations are also tabulated.

--0.240 0.297 0.289 --0.219 --0.620 0.109 --0.241 0.178 --0.148

--0.037 --0.042 0.046 --0.198 --0.069 0.026 --0.034 0.019 --0.015

48

$. RADHAKANT AND N. ULLAH

Using the Cjk coefficients given in table 2 and expression (9), we can expand ¢)~= 1, ~ = o in terms of the wave function ]JTKTz) as we had done earlier in the case of even-parity levels. The wave functions ]JTKTz) are constructed by first coupling the three holes in the p-configuration to a definite value of total angular momentum and isospin and then coupling it to a particle in 2s, ld configurations. A 3600 CDC computer program is used to construct these wave functions and diagonalize the residual interaction between the projected wave functions. Results are shown in table 5. The calculated energies of the various J - , T states listed in table 5 are relative to the lowest odd-parity state 0 - , 0. Since the coefficient of the component wave function which comes from three holes in the p+ configuration is much smaller than other configurations, it is not included in table 5. Our calculated energies of the odd-parity states are in better agreement with their experimental values than the results obtained for the even-parity states. This shows that the low-lying odd-parity states of the 1+N nucleus can be fairly accurately projected from the two intrinsic wave functions ~ = 1, ~ = o . References 1) M. Redlich, Phys. Rev. 110 (1958) 468; J. P. Elliott, Proc. Roy. Soc. A245 (1958) 128 2) I. Kelson, Phys. Rev. 132 (1963) 2189; I. Kelson and C. A. Levinson, Phys. Rev. 134 (1964) B269 3) Nazakat Ullah and R. K. Nesbet, Phys. Rev. 134 (1964) B308, erratum, 139 (1965) AB2 4) D. Kurath and L. Picman, Nucl. Phys. 10 (1959) 313 5) I. Talmi and I. Unna, Phys. Rev. 112 (1958) 452; E. K. Warburton and W. T. Pinkston, Phys. Rev. 118 (1960) 733; W. W. True, Phys. Rev. 130B (1963) 1530 6) J. P. Elliott, Phil. Mag. 1 (1956) 503; W. M. Visscher and R. A. Ferrell, Phys. Rev. 107 (1957) 781