The nucleide 50Sc

I

1.D.1

[ I

Nuclear Physics All6 (1968) 33---42; ~ ) North-Holland Publishing Co., Amsterdam N o t to be r e p r o d u c e d by p h o t o p r i n t or microfilm without written permission f r o m the publisher

T H E N U C L E I D E S°Sc T. A. HUGHES t I B M Scientific Center, Houston, Texas and M. SOGA tt Bartol Research Foundation of the Franklin Institute, Swarthmore, Pennsylvania Received 1 April 1968 Abstract: The energy levels and wave functions of 5°Sc have been calculated within the framework

of the nuclear shell model using two-body matrix elements of both Soper and Rosenfeld mixture. The log f t for the fl-decay of both 5°Ca and 5°Sc have been computed. The agreement with experiment appears good. 1. Introduction

The purpose o f this paper is an investigation of the properties of the nucleus 5°Sc within the framework o f the nuclear shell model. This includes the spectrum o f 5°Sc as well as the fl-decay of 5°Ca to 5°Sc, its subsequent fl-decay to 5°Ti and the electromagnetic transition rates of nuclear states of 50Sc" The experimentally k n o w n data and inferred properties o f the mass-50 nuclei are given in fig. 1. The g r o u n d state of 5°Sc is k n o w n 1) to be 5 +, and the first excited state at 259 keV is suspected to be 2 + with a lifetime 3) of 0 . 3 5 _ 0 . 0 5 sec. Shida et al. 3) have formed 5°Ca by means of the 48Ca(t, p)S°Ca reaction and have observed the 7-ray spectrum following the fldecay to 5°Sc. A 72 keV line, attributed to the transition 331 -~ 259 keV and a 259 keV line attributed to 259 keV ~ g r o u n d were observed with a half-life of 9___2 sec. F r o m these data, the 331 keV level was assigned J~ = 1 + and was assumed to be populated by a Gamow-Teller type fl-transition f r o m the ground state of 5OCa" The energy levels o f 50Sc shown in fig. 1 are the results o f a study by O h n u m a et al. 4), who analysed the protons f r o m the reaction 48Ca(3He, p)5°Sc with a broad-range magnetic spectrograph. Previous theoretical w o r k on 5°Sc has been carried out by Vervier 5) and O h n u m a and Sasaki 6). Vervier calculated the levels o f 5°Sc using matrix elements derived from the experimental spectrum of 56Co and the (particleparticle)-(hole-particle) matrix element relationship. Only the g r o u n d state configuration f÷p~ was considered. O h n u m a and Sasaki extended the configuration space to include the p~ and f~ neutron orbits and fit the experimental data of 56Co and 5°Sc to the parameters of a phenomenological potential. t Part of this work was performed while T. A. H. was a post-doctoral fellow at the Bartol Research Foundation and supported by the U. S. Office of Naval Research. tt Work supported by the U. S. Atomic Energy Commission. 33

34

T . A . HUGHES AND M. SOGA

If one takes the view that the low-lying levels of 50Sc are determined by the coupling of an f~ proton and p~ neutron, there will be four levels 7) with J~ = 5 +, 4 +, 3 ÷, 2 +. As found in other doubly odd nuclei in which the neutron and proton are in different orbits, the energy splitting between states of different J~ should be small. Hence one would expect four low-lying closely spaced levels associated with the ground state configuration. If the 1 + assignment for the 331 keV level is correct, it requires a strong

-I0

0-~ 50 2 0C° 3 0 (:3 (J oo

,¢

-15

O (D >

54-

°s°29 \

Q) Q:~

MANY "~\

LEVELS

> q3~

-_P.Q_ 2+

O; 50 22Tt 28

Fig. 1. The A = 50 isobars.

depression of 1 + configuration from its zero-order position and a splitting between the 5 + ground state and the 3 + or 4 + state of the same configuration to be 1.8 MeV or greater. In hope of clarifying this situation, we have calculated the spectrum o f 5°Sc, a s well as the l o g f t for both fl-decays, the M3 lifetime of the 2 + isomeric state and the M1 and E2 lifetimes of the 1 + state.

2. Calculation We assume a 48Ca core and that the low-lying T = 4 levels of 5°Sc arise from the following j j-coupling states: ~.(f~,

J2)

7 • + bj:m~[co + r e), = ~ (~rml J2m21dM)a~m, ml, m2

~b(Jx,J2)-

1

/20 + 6 j J

~ .....

(j~m~j2mElJM){a+m~bj++aj+b+,.~}lcore),

35

5°Sc NUCLEIDE

~c(Jl , J2)

=

4 V~

V1

1

/20 +aj,j2) 1 41+fij,j2

ed(j2)

=

~// 2

(jlmlJ2m2lJM)(aj,m, bj2m=-

~ ....

"

2

+

+

aj2m2

+

bj,m,}lcore) +

+ bj=m2 + -~ 1 ~, a~=3 + b~m3lcore), T, (i1 ml J= m2lJM)bj,m

m,,m2

ma

Z ,--, { "~Jl-m*h+ ~.i,,.,~j,-.,, h +

2j1+1 m,>o × ~ (-)~-~(~m2~-

m31JM)a~m2b ~ l c o r e ) .

m2ra3

Here a + and b + are proton and neutron creation operators, respectively, their annihilation counter parts being a and b. The quantum numbers (Jl,J2) denote any pair of the set (2p~, 2p¢, lf~). The wave function kV~ includes all the configurations arising from the coupling of an f~ proton to any of the single-neutron states. The T = 0 coupling of the neutron and proton in single-particle states (J~,J2) is given by ~Pb with the restriction that for j l = J2 only odd J is allowed. The wave function ~ is made up of two components the first being the T = 1 coupling of the particles in orbits (Jl,J2) and the second the analog of the core coupled to the two-neutron configuration (J~,J2). In this case f o r j I = J2, ~c is limited to even J. The configuration ~d

×

j

T=O

(],,]2)

~,

T=I

:~ (J,,J=)

x x (J/2)

- V"/

;7=0

Fig. 2. Pictorial representation o f the wave function used in the calculation.

36

T.A.

HUGHES AND M. SOGA

corresponds to two neutrons with angular momentum 0 ÷ coupled to the low-lying T = 3 states of 4SSc. This configuration enters for J = 1 + ~ 7 +. All these wave functions correspond to T = 4 states and are illustrated in fig. 2. The Hamiltonian is taken to be H = H(single-particle)+ V(two-body), and with ~ , , kub, kv~ and 71a forming a basis was calculated and diagonalized for each total angular momentum J. The matrix elements are listed in table I. The expressions for the matrix elements in TABLE 1

Matrix elements for the states 7/., 7Jb, k~ and ~//a (~[ta, HTt,,) =

5(Z')l-Ep(f~) +En(j)] + (f~(p)j(n)[ Vlf~(p)j'(n)) s

(7~,, HTb) = x/~(f~jlVljl J2)s, r=o (Tt., HTJ~) = `/~(f~ JlVljl J2)s. T=O ( g t , HkUd) =

V

5(jj,)(ffl(p)j(n)[ V[ff l(p)f~(n)) s

~

=

(Jt J;) (J2 A)½[ p(Jd + E.(j,) + e.(j2) + E.(h)]

+(J~ Jz] glj; Jl)s, r=o =

(7~b' HkUd) =

o

Vv ~j

,/2(1+

16(jl J2)) {6(jja)(j~ l(p)j(n)[V[f~ 1 (p)f~(n))s + 6(jjl)( -)J' +J~-s(j; l(p)j(n)[ Vile- 1(p)f~(n))s}

.t 5 H~tc) = (~(Jl J 1)(~(J2 J2)[8-(Ep(J 1) + Up(J2)) '[- 3(En(Jl) -b En(J2)) ] + (J~ Jzl vI j'~ J'2)s, r =t 1V~ 1 (7~, H~'a) = ~. ~ x/2(1 + 6(j~ J2)) {5(jjz)(j-i l(p)j(n)[ V[L- l(p)f~(n))s

(ec,

• t

•

•

-

6(Jl J ) ( - ) J ' + h - J ( j f l(p)j(n)[ Vlffa (p)f~(n))s}

"" r [j. tj. t )s =o, r =t (TJa, H~d) = 6(jj'){2E.(j)+ ( f f ~(n)fk(p) [V[f~-~(n)f~_(p))j} + (jj[

table 1 are straightforward except for Ep(j), En(j) and/~,(j); Ep(j) and En(j) represent respectively the single-particle energy of a proton and neutron in the state j relative to a 4SCa core. The neutron energy is taken directly from the experimental 49Ca spectrum. The Ep(f~) relative to *SCa can be obtained from the mass table of Mattauch et al. 8). The Ep(j) f o r j = p~, p~ and f~ have to be calculated from the experimentally observed T = -~ spectrum of 49Sc and the symmetry energy for the appropriate state [ref. 9)]. Relative to the "9Sc ground state, the proton single-particle energy lies above

5°Sc NUCLEIDE

37

the T = ~ position by an amount equal to UJ2T¢, where Uj is the symmetry energy for the state j and T¢ the isospin of the 4SCa core. The Ep(j) and E.(j) calculated with the above prescription are given in table 2. TABLE 2

The proton and neutron single-particle energies with a 4sCa core j

Ep (j) (MeV)

f~ p~ p+ f~

--9.623 --5.703 --4.589 --3.841

En (j) (MeV)

--5.144 --3.116 --1.185

The expression in the bracket for the matrix element ( T d, HTa) has also been estimated from experimental data;/~n(J) is given by the expression

~.(j) = E.(j)+ AE(j), where 1

AE(j) - 8(2j + 1) ~ ( 2 J ' + 1)[(f~(p)j(n)] V[f~(p)j(n)) s, +

(fk-'(n)j(n)[V [ f ( ' (n)j(n))s, ].

Since/~n(J) is independent of the total angular momentum J of Ta, we estimate it for J = 0. In this case - A E ( j ) equals the difference between the energy of the analog of a single neutron in the statej coupled to the core and the energy of the state in which a neutron in the statej is coupled to the analog of the core. This difference is Uj/2Tc. Hence /~'n(J) = E . ( j ) - Uj2Tc. The two-body interaction between an f~(p) and f¢(n) hole has been calculated from the 4SSc spectrum 10,11). The two-body nuclear interaction was taken to be central and both Soper and Rosenfeld mixtures were used. The strength of the interaction Vo was assumed to be - 4 5 MeV. Harmonic oscillator radial functions with an oscillator parameter v = 0.27 were used. The force-range parameter was taken to be 1.8 fro. 3. Results

The experimental and calculated spectra are given in fig. 3. The spectra B and D were calculated without the configurations T a for Soper and Rosenfeld mixture, respectively. Spectra A and C include the T a configurations, while in spectra E and F the ground state eigenvalues have been subtracted from the calculated energies so that

38

T. A. HUGHES AND M. SOGA

the calculated ground state is normalized to the experimental energy. The fit to the experimental data is good especially for the low-lying states. For both mixtures, the 331 keV level is calculated to have J~ = 3 +, and this result differs from the 1 + assignment of Shida et al. 3). The spin assignments for the four lowest levels based on the calculation are ground state - 5 +, 259 keV - 2 +, 331 keV - 3 + and 761 keV - 4 +. The calculation indicates that the most promising candidate for the 1 + assignment is the 1857 keV level. The 1 + level with Soper mixture is at 1.76 MeV and with Rosenfeld 3

.

0 --

/]/

3

I - -

6,4

/

- -

_5

~

6

zj-- 13

- - I

-~-5 6 - -

/

3

~

g

4_

4_

//

//

- -

3 _5

3 5 §

- - 3 --

~.

43

~ ~

-14

< Z o

5 - - . 4

4

/- - I - - I

o

I

/ ' f

/

I >

m. G')

-15 4 - -

4 - -

- - 4

03 E3

- - 4 3 2

Z J -

- - 5 3 _ _

3

5 - -

2

2 - - 5

- - 5

0

- - 5

-16

5 _ _

A

- - ' 5

B

C

D

EXP

E

F

Fig. 3. Calculated and experimental spectra of 5°Sc. E X P is the experimental spectrum. A B E / C D F are spectra calculated with Soper/Rosenfeld mixture. A and C include k~a, while B and D do not. E --- A and F =- C but normalized to the experimental ground state.

mixture it is 1.2 MeV. The depression of the 1 + with Rosenfeld mixture is due to a strong off-diagonal matrix element between ~b( 3, 3) and qj (32). The states which have an appreciable component of ~rj ( j 2 ) , i.e. greater than 0.5 in amplitude, have underlined J-values. These states start at about 2.5-3.0 MeV. Although the effect of the 7~a states in the 1 + energy matrix is to depress the lowest state, it does not appear to be sufficient ta make it lie at 331 keV excitation. The levels at 2227 keV and 2334 keV seem to correspond to J= = 4 + and 3 +, but the splitting is very small and there is the possibility of their being inverted. At higher energies there are many closely spaced calculated levels making a one-to-one corre-

S°Sc NUCLEIDE

39

spondence impossible. The ~Pa and ~Po components of the wave functions of the lowest five levels in terms of their amplitudes are given in table 3. The kuc and ~d amplitudes are small for these lowest levels and were disregarded in the calculation of the decay properties except for the fl-decay to the 1 + level. In the case of the 1 + level, the ~d component is 0.17 for Soper mixture and 0.32 for Rosenfeld mixture. TABLE 3 Configuration amplitudes o f the five lowest states in s°Sc

J

f~Pk

f~zfl-

1 a)

2

3

4

5

f~P~

P~-Pk

P~f~r

PkP~

f~f~

f~P~

P~P~

mixtureb)

--0.4947

--0.6243

--0.0278

--0.5515

+0.0601

+0.0999

S

--0.3063

--0.6295

+0.0089

--0.5707

+0.0465

+0.1417

R

--0.9571

+0.1356

+0.1394

40.1174

+0.1075

S

--0.9243

+0.1484

+0.1914

+0.1377

+0.1694

R

--0.9145

--0.0746

--0.3168

--0.2074

--0.0469

+0.0226

+0.0439

S

--0.8984--0.0823

--0.2909

--0.2536

--0.0665

+0.0263

+0.0594

R

--0.9779

--0.0767

+0.1427

+0.0972

S

--0.9673

+0.1072

+0.1375

+0.1575

R

--0.9963

--0.0488

+0.0040

S

--0.9965

--0.0461

+0.0030

R

a) The ~gd c o m p o n e n t o f this state is 0.17 (S) and 0.32 (R). b) S -- Soper mixture, R -- Rosenfeld mixture.

The M3 lifetime of the 2 + state has been calculated for both Soper and Rosenfeld mixture using an oscillator parameter of 0.27 and taking the magnetic moment of the nucleons to be 2.79 n.m. for the proton and - 1.91 n.m. for the neutron. The results are given in table 4 with the experimental value of Karras and Kantele. The agreement between experiment and the values calculated with the model wave functions is good. TABLE 4 The M3 lifetime (in sec) o f the 2 + isomeric state in 5°Sc Transition

2 + --~ 5 +

Mixture Soper

Rosenfeld

Experiment

0.358

0.404

0.35:~0.05

40

T.A.

H U G H E S A N D M. SOGA

Although no experimental data exist on the 3,-decay of the 1 + level to the 2 + and 3 + levels, the theoretical values may be of use in an experiment, such as 48Ca(aHe, p3,) coincidence measurement *. The transition rates for the E2 and M1 7-decay of the 1 + to the 2 + level and for the E2 3'-decay to the 3 + level are given in table 5. This calculation indicates that the branching is approximately 55 ~ (E2) to the 3 + level, 40 ~ (E2) to the 2 + level and 5 ~ (M1) to the 2 + level. TABLE 5 Transition rates (in sec -1) for the y-decay o f the 1+ state o f 5°Sc

Transition

Mixture Soper

Rosenfeld

T(M1)

1.5 × 1013 2.2 x 1012

1.7 × 10 x3 0.6 x 1012

T(E2)

2.0 × 1013

2.2 × 1013

1+-->2 + T(E2)

1+-->3 +

In order to calculate the/?-decay of s OCa to 50Sc ' the wave function of the 0 + state of S°Ca has to be known as well as the 1 + state in 5°Sc. Vervier 12) has calculated the energy levels and wave functions of 50Ca and satisfactorily explains the ground state binding energy as well as most of the experimental levels up to 5 MeV. The wave functions were sufficiently accurate to account for the strong population of the ground state when compared to the excited 0 + levels in the (t, p) reaction on 48Ca. The energy matrix was formed using the matrix elements of Auerbach 13) which were derived from an analysis of the Ni isotopes. Using these matrix elements, we have calculated the wave function of the ground state of 50Ca and used it together with our calculated 1 + wave function to compute the l o g f t for the fl-process. T h e f t value was computed using the relation 14) 4487

it=

I<~)['

"

The l o g f t for the/?-decay to both calculated low-lying 1 + levels is given in table 6. In this calculation, the 7Jd(p~) state was included with ~ , and ~b configurations shown in table 3, since the (f~[lallf~) matrix element is quite large. The value of the l o g f t depends sensitively on the amplitude of 7Jd in the 1 + wave function. In the case of the Soper mixture, the effect is less than for that of the Rosenfeld mixture. The other important components of the 1 + state are ~b(P~, P~) and 7Jb(p~, p~). These components make up approximately 60 ~ of the wave function and are significant because the S°Ca ground state is approximately 90 ~ p~(n). The configuration amplitudes calculated for the 7Jd state result in a contribution to ( a ) which is almost twice as large and of opposite sign to that derived from 71 and 7Jb . In the limit that the qJn compo* This calculation was suggested by Dr. Ohnuma.

"5°ScNUCLEIDE

41

nent goes to zero (which would give a better fit o f the 1 + state to the 1857 keV level), the log f t value would be 4.0. TABLE 6

The log ft for the beta decay of s°Ca to 5°Sc Transition

Mixture

0 + --->11+ 0 + --+ 12+

Soper

Rosenfeld

4.46 4.86

5.86 4.16

As a further test o f our wave functions, we have calculated the beta decay o f the 5 + g r o u n d state of 5°Sc to the 4 + and 6 + levels in 5°Ti. The 4 + and 6 + states in 5°Ti were assumed 15) to be the pure (f~)2 configuration. The experimental data in the beta decays come f r o m the work of Chilosi et al. 1). The calculated and experimental values are given in table 7. The calculated values are in both cases too large indicating that the calculated matrix element is too small. However, the ratio l o g f t ( 5 + --+ 4+)/ l o g f t ( 5 + ~ 6+), which is also given in table 6 does agree with the experimental data. TABLE 7

The log jt for the beta decay of 5°Sc to 5°Ti Transition

5+ --> 4+ 5+ --->6+ 5 + -->

4+~

+ ~ ] 5+--->6

Mixture Soper

Rosenfeld

Experiment

6.99 5.75

7.04 5.80

6.2 5.2

1.21

1.21

1.19

This means that the c o m p o n e n t ~ua(f~, fl) in the 5°Sc ground state is too small. I f we take the configuration amplitude to be three times larger than the value listed in table 3, the absolute values of the l o g f t will agree with the experimental data. Such a change in the small c o m p o n e n t of the 5 + wave function could not appreciably affect the energy levels or other transition rates. F r o m the results o f the calculation o f the 50Sc energy spectrum and transition properties, it would seem that the shell model gives an adequate description of a n u m b e r o f the low-lying states and possibly some of the higher levels of 5°Sc. The experimental assignments o f spin and parity would be welcomed in order to evaluate the present model. A b e t a - g a m m a coincidence measurement and a 48Ca(3He, PT) experiment would, hopefully, resolve the uncertainty concerning the 1 + level. Discussions with Dr. S. Fallieros are greatly appreciated.