Pramana) Vol, 12, No. 2, February 1979, pp, 179-201, ~ printed in India
The collective bands of positive parity states in odd-A (fp) shell nuclei
D P A H A L P A R A
Physical Research Laboratory, Ahmcdabad 380 009 MS received 20 November 1978
Abstract. The low-lying collective bands of positive parity states in (fp) shell nuclei are described in the deformed Hartrce-Fock method by projecting states of definite angular momenta from ' the lowest energy intrinsic states in (sd) -1 (fp)n+l configura- tions. The modified Kuo-Brown effective interaction for (fp) shell and modified surface delta interaction (MSDI) for a hole in (sd) shell with a particle in (fp) shell have been used.
The collective bands of states are in general well reproduced by the effective inter- actions. The excitation energies of the band head states are however off by about one MeV. The calculated magnetic moments of the band head j = 3 ] 2 + states are in reasonable agreement with experiment. Using effective charges e p : l - 3 3 e and en=0"64 e we get fairly good agreement for E(2) transitions. The hindered M(1) transition strengths are reproduced to the correct order however they are slightly higher compared to experiment.
Keywords. Collective band of states; positive parity od| -1 and l s t - 1 states.
1. Introduction
In the low lying positive parity spectra of m a n y of the odd-A (fp) shell nuclei col- lective bands of positive parity states axe seen (StyoT_~R et al 1976 and references therein). Those states are strongly excited in the neutron and proton pick up rc- actions (p, d) and (d, H e s) respectively having large I=2 and I=0 transitions and hence have been traditionally regarded asod| -I or Isi-1 hole states (Maurenzig 1971).
In many of the nuclei these positive parity states form into well-developed rotational bands following the J(Jq-I) energy sequence to a very good accuracy.
There have been mainly four different kinds of theoretical attempts to describe the energy systematics of the positive parity states, namely (1) the shell model (Bansal and French 1964), (2) the phonon particle coupling model (Mitting et al 1974; G o o d e and Zamick 1969), (3) the rotation particle coupling model (Styczen et al 1976) of B6hr and Mottelson 0953) and (4) the deformed Haxtre¢ Fock (HF) model ( J o h n s t o n e 1968).
B a n s a l a n d F r e n c h (1964) m o d e l was t h e first s u c c e s s f u l a t t e m p t t o s t u d y t h e e n e r g y s y s t e m a t i c s o f t h e l o w e s t J = ~ + h o l e s t a t e s i n f t s h e l l n u c l e i . T h e m o d e l w a s l a t e r u s e d b y v a r i o u s a u t h o r s ( Z a m i c k 1965; S h e r r e t a l 1974; B e r n s t e i n 1972;
S h e r r a n d B e r t s c h 1975). T h e m o d e l h o w e v e r t r e a t s t h e l o w e s t J---~+ s t a t e s a s single o d i - 1 h o l e s t a t e s w h e r e a s t h e r e c e n t e x p e r i m e n t a l d a t a i n d i c a t e t h a t t h e y a r e n o t single h o l e s t a t e s b u t a r e b a n d h e a d s t a t e s o f c o l l e c t i v e b a n d s o f s t a t e s . B a n s a l 179
and Ashwanikumar (1977) have later done shell model calculation for positive parity states in 41Ca using the 2 particle-1 hole configuration (od] -- ls½) -1 (f{)z.
Using empirical effective interactions they obtain good agreement for the energies as well as electromagnetic properties of the hole states.
The phonon particle coupling model was used by Goode and Zamick (1969) and Mittig et al (1974) to study the positive parity states in 41Ca and alSo respectively.
They considered a weak coupling of a particle in (f-p) shell to the lowest one phonon odd parity states J = 3 - , 5-, 4- and 2- in 4°Ca.
The rotation particle coupling (RPC) model has been extensively applied to the description o f positive parity states in (fp) shell nuclei (Styczen et al 1976). See also (Tabor et al 1975a) and (Malik and Scholz 1967). A more satisfying microscopic calculation was done by Johnstone (1968) for the positive parity states in ~8Sc using deformed H F model. Using Kallio-Koltveit effective interaction they associate the positive parity states with the states of definite angular momenta projected from the lowest energy deformed H F intrinsic state in (s--d) -1 (fp)~ configuration.
Their calculation is quite successful in reproducing energies as well as electromagnetic transitions o f hole states in nOSe.
In an earlier paper (Ahalpara 1978) we have done deformed HF calculations in the (dl--f~)configuration space to study the energy systematics of hole states in (f~) shell nuclei. The emphasis was to examine the consistency between the empi- rical effective interactions defined in the (dl--f~)z configuration and the basic as- sumptions of the Bonsai-French model. For a detailed study of structure of hole states in (fp)n nuclei it is more appropriate to use a bigger model space including the single particle orbits in (fp) shell and in (sd) shell.
In the present paper we have studied the properties o f hole states in 41,4aCa, 4a,4z,47Sc, 45Ti, ~7,4s,49V and 51Mn nuclei using the deformed H F scheme in (sd)-(fp) configuration space. It is to be noted here that the extensive deformed H F calcula- tions by Dhar and Bhatt (1977a, b and references therein) for (fp) shell nuclei in (fp)n configuration space have been very successful in describing the properties of normal parity states in ~fp) shell nuclei. We therefore hope that the properties o f positive parity states in odd-A (fp)" nuclei would be well reproduced in the (sd) -1 (fp),+l configuration. For the effective two-body interactions in (fp) shell we use the Kuo-Brown effective interaction as modified by McGrory et al (1970). The modi- fied surface delta interaction of Glaudemans et al (1967) was used to generate the particle-hole effective two-body interaction.
We also study the electromagnetic static moments and transition rates for the positive parity states. For the E(2) transitions we have used the effective charges as given by Dhar and Bhatt (1977a): % = 1.33 e and e~---0-64 e. They have obtained these effective charges by fitting various experimental BE(2) values for the transitions amongst normal parity low-lying levels of 0')~) shell nuclei using deformed H F wave functions.
In §2 the present model is discussed. The results of energy spectra for hole states as well as their electromagnetic static moments and transition rates are com- pared with experiment in §3. Section 4 deals with a brief discussion about the results.
Collective bands o f positive parity states in shell nuclei 181 2. Model
2.1. Model space and Hamiltonian
We consider the model space consisting of (s--d) and ( f - - p ) shell orbits. It is con- venient to work in particle-hole space. For an (fp) shell nucleus the ground state configuration would correspond to n nucleons filling the (fp) shell orbits with a com- pletely closed (sd) shell. The hole state configuration would then correspond to ( n + l ) nucleons in the (fp) shell orbits and a hole in the (sd) shell orbits.
The Hamiltonian in this model space is completely defined by single particle energies of (fp) shell orbits, single hole energies of (sd) shell orbits, the two-body matrix elements <(fp)2[V[ (fp)2> J T and the two-body particle hole matrix elements of the type <(sd) -1 (fp)l ] V[ (sd) -1 (jig)l> JT. Since we describe the hole states only by one hole configurations and do not mix them with three hole configurations, the matrix elements of the type <(sd)-al V I (fp)~) J T are not required.
The siogle partielz and single hole energies are taken from the spectra o f 41Ca and SgCa respectively. The single particle energy gap o f 7-3 MeV between the f ] and the d i orbits was taken from empirical binding energy differences.
For the two-body effective matrix-elements <(fp)~[ V I (fp)~) J T we take the Kuo- Brown (1968) matrix elements as modified by McGrory et al (1970). Recently Dhar and Bhatt (1977a & b and references tnerein) have done extensive deformed configuration mixing calculations for (f--p) shell nuclei using this interaction and have found that it gives fairly good description of the spectral properties o f low-lying normal parity states in ( f - - p ) shell nuclei.
For the two-body particle-hole matrix elements o f the type ((sd) -1 (fp)t[ V[
(sd)-i ( f p ) l ) J T we choose the modified surface delta interaction (MSDI) o f Glaudemans et al (1967).
Vii : - - 4rrAr ~ ( ~ 0 ) ~(ri - - R) 8(rj - - R) 8(r, - - R) + B r.
Dieperink et al (1968) have obtained a set of parametels A r and B r which fits the low-lying odd-parity states in 4°Ca in a one particle-one hole configuration. The values of the parameters are A0=0.7 MeV, A1=0-42 MeV, B o = - - l ' 3 MeV and
B1 =0.7 MeV.
2.2. Intrinsic states
The low-lying non-normal parity states of an (fp)n nucleus are assumed to arise from a few lowest energy deformed H F states with one hole configurations in which (n-t-l) particles move in the (f--p) shell orbits and a hole is restricted to the (s--d) shell orbits. We obtain such intrinsic states by carrying out axially symmetric deformed H F calculations.
For an odd-Z nucleus we find that the one-proton hole H F states are lower in energy than one neutron hole states. However, for an even-Z nucleus the one hole H F state resulting from neutron excitation are lower in energy. It is expected therefore that the hole states in odd-Z nuclei may be projected from a single-proton hole HF intrinsic state but for an even-Z nucleus it may also be necessary to consider H F intrinsic states involving neutron excitation. In contrast to the proton excited
intrinsic H F states the neutron excited intrinsic H F states do not have a definite isospin. In this later case, a linear combination o f proton and neutron excited intrinsic states is necessary to form good isospin intrinsic states. We illustrate this by showing typical one-hole intrinsic states o f odd-Z and even Z nuclei.
2.2a. Odd Z nuclei: In figure 1 is shown the proton excited intlinsic states of 4SSc.
The (f_p)e part o f the state has isospin T0=l and the total isospin is T=T0-+-~= ½ and s
(b) Even Z nuclei: In figure 1 is also shown the neutron excited intrinsic states of ~STi. None of the three intrinsic states has a definite isospin but they can com- bine to form (J'--p)6r o configuration with T0=0 and 1 leading to T=T0+½ and states. The defoxmed intrinsic H F states do not have a definite angular momenta.
We project out states with definite angular momenta from the intrinsic H F states following the standard projection formalism (Warke and Gunye 1967; Ripka 1968).
In the odd Z nucleus, the observed positive parity states are associated with states projected from single proton hole intrinsic state. For an even Z nucleus, good angular momentum states are projected from each intrinsic states. The Hamilto- nian is then constructed using these projected states and is then diagonalised to take care o f the non-orthogonality o f projected states flora different intrinsic states.
3. R e s u l t s
3.1. Ca isotopes
The low-lying positive parity states in odd Ca isotopes 4XCa and 4sCa have been studied by various reactions (Tabor et al 1975; Nann et al 1975 and 1976; Bohne et al 1975; Brown et al 1974; Seth et al 1973, Gorodetzky et al 1973; Olness et al
( f - p ) k
7/2
5 / 2
3/2 :: ::
(s - d )
3/z -~
1 / 2 p
45Sc
N
(f--p}
I s - d ) k 7 / 2 , 5 / 2 , I/2
3/2-¢
I/2 '
r ~ r ~
&
45Ti
F i g ~ e 1, Th© proton and neutron excited intrinsic states o f ~ and Wl'i.
Collective bands o f positive parity states in shell nuclei 183 1975; Martin et al 1972; Smith et al 1968; Poletti et al 1976). In 4zCa the neutron pickup reaction ~42a (p, d) 4zC.a by Martin (1972) has shown that the }+ (2-01 MeV) and ½+ (2.68 MeV) states are largely dl-X and s~ -z hole states respectively. The d| -1 hole state is less fragmented as compared to s½ -z hole state. Nann et al (1975) have studied the reaction UK (~, d)eC.a and have assigned a major dj -x x (]~y~
configuration to the high spin states 11/2 + (3.37 MeV), 13/2 + (3.92 MeV), 15/2 + (3-83 MeV) and 17/2 + (5-22 MeV).
Sartoris and Zamiok (1967) have done a conventional shell model calculation for the positive parity states in ~XCa. Using Kuo-Brown effective interaction they con- sidered the ls½ -z X (f_p)= and 0~I-1 X (.f_p)Z configurations and allowed m i x i n g
between the two. The model does not provide a satisfactory description o f low-lying positive parity states. The predicted excitation energies o f the lowest }+, ½+ and }*
states were 4.5, 6-05 and 6.62 MoV respectively. The first ½+ state was predicted to have mainly d | -z x (fi)yz, = 2 configuration whereas the pick up data o f 4=Ca 07, d) e C a by Martin (1972) and Smith et a! (1968) indicate that this state is largely an xl -z hole state.
Recently Bansal and Ashwanikumar (1977) have also done shell model calcula- tion for the positive parity states using Od| -z x (.f{)= and lsi -x x (f{)= confi- gurations. They have used empirical effective interaction for (.fi)s J T states and Kuo-Brown interaction for ( 0 d | - l f i ) J T particle-hole states. In this truncated space they get improved results as compared to the calculations by Sartoris-Zamick (1967) in the full (sd) -z (fp)s configuration space. However, the excitation energy
4 - -
T_3__
3 ~ 3-80
3
>
0
17 3 2 1
T=3_
3 2
17
3 17
_ 13 11
15, 1. 0 11 .. . . 9
g t 5
11 1 - 3 6 7
9 1-19 9 ' 7
- - 5 5
7 0 . 8 7 7 -
7 5 '
5 .... 0-61
5
- - 3 "2"O1' 3 3 . 0 4 3 [ 2 . 0 1 ] 3 3 . 0 7 :3 ,3.19
2 J e x p HF RPC Phono.n porticle Shell
¢ou piing model
4 1 C Q Figure 2.
P.--6
of first J = ~ + state is still higher by more than 1 MeV compared to the experimental value. We show their calculated spectrum in figure 2.
Goode and Zamick (1969) have ped'ormed phonon particle coupling model cal- culation in which an f ; particle was coupled to the negative parity 3% 4% 2- and 5- states o f 4°Ca. The negative parity states o f ~°Ca were calculated microscopically in R P A calculation and were not phenomenological surface vibrations. The model is somewhat more successful. The ~+ and ½+ states are still high in energy but their separation is reduced to 0.33 MeV as compared to the experimental value of 0.66 MeV. Their calculated spectrum is compared with experiment in figure 2.
Recently Tabor et al (1975a) have used the rotation particle coupling (RPC) model to study the positive states in ~tCa. They employed lowest deformed intrin- sic states o f 2 particle---1 hole configurations with the two particles in the (fp) shell coupled to K = 0 , T = I and one hole in the (sd) shell. A linear combination o f these intrinsic states was taken to ensure good isospin. The low-lying part of the experi- mental spectrum is well reproduced by the R P C model. However as a consc~luence o f the assumed rotational Hamiltonian, the high spin states are predicted too high in energy (figure 2).
The present model contains some aspects o f both the above models. As in the Sartoris and Zamick shell model we have also used the (sd) -t (fp)~ configuration for 41Ca but the effective interaction used is different. The low-lying positive parity states o f aCa~ are assumed to correspond to the states o f definite angular momenta projected from the lowest three I-IF intrinsic states obtained b y the excitation o f a neutron and a proton from the (sd) to the (fp) shell. These intrinsic states are similar in structure to the ones used byTabor et al (1975a) in their RPC model. How- ever, instead of making a linear combination to form good isospin intrinsic states, we first project good angular momentum states from these intrinsic states and then allow mixing among these states. We find that although the mixing between Odj-l and ls½ -1 bands was allowed in the calculation, the effective interaction gives a very weak mixing between them. In the experimental spectrum it has been possible to identify the Odi-1 band o f states (Nann et al 1975 and Martin 1972) whereas the evidence for ls½ -1 structure o f states is available only for the first ½+ state at 2.67 MoV. Following Tabor et al (1975a) we associate second lowest ~+ (3.05 MoV)
>
O
Figure 3.
_
1 - -
O - - !
9 7
mC o 5
5 0-83 3
5
3 O-38
3
2.457 4 . 1 4 2 . 6 7
t 0 , 1
2 J exlD. H F A P C
Collective bands of positive parity states in shell nuclei 185 and 6 + (3-5 MeV) states to belong to the ls½ -1 hole b a n d with K = ½ +. In figures 2 and 3 we c o m p a r e the calculated spectra for d] -1 and s½ -1 b a n d s respectively with experiment as well as with the results o f R P C model, the phonon-partiele m o d e l and the shell m o d e l b y Bansal and A s h w a n i k u m a r (1977). T h e spectra are d r a w n relative to the energies o f b a n d head states J = ] + and J=½+. T h e excitation energies o f these b a n d head states relative to the ground states o f the nuclei are also s h o w n in the figures. W e find that in the present model the all-1 b a n d is fairly well r e p r o - dueed. Using the intrinsic states similar to the ones used in R P C model the effec- tive interaction in the plesent model does not give rise to r o t a t i o n a l sequence o f states and leads to b e t t e r agreement f o r the energies o f the high spin states as c o m - pared to R P C m o d e l results. The shell model results o f Bansal a n d A s h w a n i k u m a r (1977) in a t r u n c a t e d m o d e l space show almost similar features to the ones o b t a i n e d in the present m o d e l calculations. As shown in figures 2 a n d 3, the excitation energies o f b a n d h e a d states J = ] + a n d J=½+ are predicted higher b y m o r e t h a n 1 MeV in all the models except in R P C model where the energies o f these states were fitted to experimental values. The electromagnetic transitions f o r the positive p a r i t y states in ~ C a have been experimentally studied by E n d t a n d V a n d e r Leun (1967) and by G o r o d e t z k y et al (1973). Bansal and A s h w a n i k u m a r (1977) have recently calculated the electromagnetic transition rates for the positive parity states u s i n g their shell m o d e l wave functions in the Odl-X× (f~)~ a n d Is½ -1 × (~)~ configura- tions. T h e i r effective charges were e p = l . 5 e and en=0"5 e.
We c o m p a r e o u r calculated electromagnetic transition rates with experimental ones and with the results o f Bansal and Ashwanikumar (1977) in table 1.
Recently Y o u n g et al (1975) have measured the magnetic m o m e n t o f the J = 1 5 / 2 + state at 3.83 MeV. T h e y o b t a i n / z ( 1 5 / 2 + ) = 2 . 1 8 ± 0 . 1 5 n.m. Engeland et al (1976) have d o n e shell m o d e l calculation for 2 particle-1 hole configuration in (sd)--(fp) space using renormalised partiele-hole m.e. o f K u o - B r o w n a n d have f o u n d t h a t t h e J = 1 5 / 2 + state is 99~o m a d e up o f just two c o m p o n e n t s I [da-/~ × (fv2)~ 0] J with J 0 = 6 and 7. T h e i r wave function gives /~ (15/2+)=2.4 n.m. which is in g o o d agreement with experiment. O u r calculated value i s / z (15/2+)----2.62 n.m.
Table 1. M1 strengths and E2 strengths for positive parity states in 4×Ca (in W.U-)
M1 strengths E2 strengths
Jl - - ~ Jy Exp. HF Sh.Modta)
5/2 3/2 - - 0.24 - -
7/2 5]2 ~ 0.33
9/2 7/2 - - 0-18
1112 9/2 0-144-0"01 0"68 7.23 1312 11/2 < 1"0 0"47 0.04 15/2 13/2 - - 0.52 - -
( a ) Bansal and Ashwanikurmtr (1977)
Jl ~ Jf Exp. HF Sh.Modta)
5/2 3/2 - - 2.53 - -
7/2 5/2 - - 0.79 - -
3/2 _--_ 0 1-96 0-71 9/2 7/2 - - 0.52 - -
5 / 2 - - 1 "73 - -
1 1 / 2 9/2 - - 0 " 5 4 0"23
7/2 - - 1-37 0"65 13/2 11/2 - - 0"51 0"10 9/2 - - 0"72 0"02 1512 13/2 - - 0"86 - - 11/2 1"024-0"11 1-13 0"53
The positive parity states in UCa have been studied by Poletti et al (1976); Martin (1972); Brown et al (1974) and Nann et al (1976). A band o f positive parity states starting from 3 = ~ + (0.99 MeV) upto J = 1 5 / 2 + (3-94 MeV) belonging to the 4 par- tiele-1 hole eontiguration Odi-Xx (fp)~ have been identified (Nann et a11976). The pick up reaction ~ C a (p, d) ~ C a by Martin (1972) have strongly populated the band head state 3 = ] + (0.99 MeV) and also have assigned a major Odi-1 component to this state.
In the present model we consider the lowest few deformed H F intrinsic states with 4 particle-1 hole configurations. Five such intrinsic states involving both proton and neutron excitation are needed to form states with definite isospin. The states with definite angular momenta were projected from these five intrinsic states.
The Hamiltonian was then constructed with these projected states and diagonalized taking care o f the nonorthogonality of the projected states. In figure 4 we com- pare the calculated spectrum with experiment. The agreement between the two spectra is fair. The excitation energy o f the ]+ band head state is also in good agree- ment with experiment.
The electromagnetic transitions amongst positive parity states in ~ C a have not been experimentally measured so far. We give in table 2 the E(2) and M(1) transi- tions as predicted by our model.
...,.
>
2 -
1 -
0 -
1 5 2 . 9 , 5
1 3 '" 2 . 3 8
11 " , " " ' , , , - - - 1, 9 6
9 1 - 4 1
t 3
11 9
7 , 0 - 9 1 7
,5
5 ~ 0 . 4 1
o - g g 3 - - ' - - - . . - - 0 2,]
exp
43Ca
0 - 8 6 H F
1figure 4.
Collective bands o f positive parity states in shell nuclei
Table 2. E(2) and M(1) transitions in 'sCa
E(2) strength M(1) strength dt ""-'* JS in (e ~ fm 4) in 10 -2 x (n.m.) I
HF HF
5/2 312 54 2"1
712 5/2 34 10-5
3/2 26 --
9/2 7/2 14 1-4
5/2 32 --
1112 9/2 11 8"0
7]2 38 - -
1312 11/2 4 1-9
9/2 35 - -
15/2 13/2 2 3-9
1112 32 - -
187
3.2. Sc isotopes
The odd Se isotopes have shown up very low-lying bands o f positive parity states.
The lowest ~+ and ½+ states, reached with l = 2 and l = 0 transfers and large spectroscopic strengths in pick-up reactions have been identified as having large Odi-X and ls½ -1 components respectively (Yntema and Satchler 1964). In 4SSc the positive parity states have been experimentally studied by many workers (Poletti et al 1976;
Afford et al 1971; Ball et al 1970, 1971; Forster 1970; Phillips et al 1967;
Manthurithil et al 1970; Endt and Van der Leun 1967).
Malik and Scholz (1967) and recently Styczen et al (1976) have studied the positive parity states in 4~Sc in the strong coupling model (R.PC) o f B6hr and Mottelson (1953). Their calculated spectra for the band of states starting with band head states J = ] + and J=]r + are in fairly good agreement with experiment. These spectra are shown in figures 5 and 6.
A more satisfying microscopic calculations was done by Johnstone (1968). In the (sd)-i (fp)4 four particle-one hole configuration they performed I-IF calculations.
They used three different effective interactions namely Kallio Kollveit, Serber and Rosenfeld. All the three effective interactions give very similar energy spectra and agree fairly well with experiment. In figures 5 and 6 we show their calculated spectra.
The present model is very similar to that used by Johnstone (1968) except for a different effective interaction. We associate the low-lying positive parity states in 43Sc to the states projected from the lowest energy proton hole intrinsic H F state in the (sd) -1 (fp)~ configuration. The calculated band o f states with large di-1 and s½ -1 components are shown compared with the experimental ones in figures 5 and 6 respectively. We find that for the d,, -x band of states we get improved agreement with experiment as compared to previous calculations. The excitation energy o f the band head state J = ~ + is however higher by about 2 MeV.
P.--7
We also study the deetromagnetic properties o f the positive parity states in 4sSe. The calculated magnetic moment of ~+ state at 0.15 MeV is 0.21 n.m. which is in better agreement with the experimental value o f 0"348-t-0.012 n.m. (Mitchell et al 1977) as compared to the RPC model prediction o f 0-11 n.m. The electro- magnetic transition strengths have been experimentally studied by Styczen et al (1976). In table 3 we compare the calculated electromagnetic transition strengths with experiment. The E(2) strengths calculated in present model and by Johnstone (1968) are in better agreement with experiment as compared to RPC results. For
>
Q,) 4 n
15 ' 3 . 6 0
3 - - 1 3 , 2.-99
11 , - , - , , , - - 2 . 4 0 2 - -
g ,1.78
7 1.1g
1 - -
5 ~ 0.73
0 - - 3 0.15 0 2 J exp.
]L?Ig~re 5.
1.5
13
11
11' 9---'-" g'
7 m
7
5 - 5 '
2.41 3
HF P~e~en|
cole,
3 ,0"16' ,, HF Johnsfone
43sc
15
1 3 - - - ~
9
7
7 " ' " 5 - - ~ "
5
3 3
RPC RPC
Styc zen Molik & Scholz et ol
2
>
0
5 ' 0 . 8
3 ' 0 . 3
- 1 o e 6 , o 2 J exp.
5 3
1 ,4.3 HI:
Present colc.
3 1 t-g2
H F Johnstone
4 3 S c
5
3 . . m . , . ~ 1 1:2
RPC Styczen et ol
3
RPC Molik 8, S r h o l z
FiguR 6.
Collective bands o f positive parity states in shell nuclei
Table 3. E(2) and M(1) transition strengths in °So.
HF HF RPC
Styczen Jt , Jj, Exp. Present Johnstone et al E2 strengths in e2frn"
179_n6 203 243 115
512 3/2 + 1 . o
214-x08 135 195 88
7/2 5/2 +x92
106-22 87 129 55
312 +22
9]2 712 42-28 +29 81 81 38
129_2. 122 149 73
5/2 +*'
11/2 9/2 280-176 +292 67 95 46
7/2 162 +3s_j. 145 196 102
13/2 1112 - - 40 - - 15
912 - - 143 - - 94
M1 strength in 10 -2 x (n. m.)*
512 312 2.04-0.9 5.1 3.8 5.2
7/2 5/2 5"4+_~'~ 7"0 7"2 7"9 9/2 712 2-5 4-0.9 7"5 5-0 5"0 1112 9/2 194-3.2 8"2 8-1 8-5
189
the M(1) transition strengths we find that all the calculated results are hindered as observed but are s o m e w h a t larger compared with the experimental values except f o r
~+ transition.
the ~.!+ __>
The positive p a r i l y states in 45Se have been investigated b y (Rust et al 1974;
Kownacki et al 1973; Blasi et al 1970; T o u l e m o n d e et al 1976; Sehulte et al 1974;
Y n t e m a and Satehler 1964). In 45Sc the excitation energy o f J--g--s÷ (0-012 M e V ) is the lowest a m o n g the positive parity states in all the odd-A (fp) shell nuclei. T h e p r o t o n pick-up recation 4eTi (d, H e 8) ~ S e by Y n t e m a a n d Satehler (1964) have since long indicated large d, -x hole c o m p o n e n t f o r the J = ] + state at 0.012 MeV.
I I
Styczen et al (1976) have done rotation-particle c o u p l i n g model (RPC) calcula- tions f o r the positive p a r i t y states in 45Sc. They o b t a i n a qualitative agreement f o r the b a u d s o f states with b a n d head states J = ~ + a n d J = ~ + . T h e i r calculated spectrum is s h o w n in figure 7.
In the present m o d e l we have considered lowest energy d e f o r m e d one hole H F intrinsic state corresponding to excitation o f a p r o t o n f r o m (sd) shell to (fp) shell.
The projection o f g o o d
angular
m o m e n t u m states leads to two bands o f states with large all-1 a n d s] -1 components. The calculated spectra are c o m p a r e d with experi- ment in figure 7. W e find that there is a qualitative agreement with experiment.T h e electromagnetic properties o f the positive p a r i t y states in ~ S e are experi- mentally studied b y T o u l e m o n d e et al (1976) and Styczen et al (1976), We eompaxe
>
ID
13 11
2 - -
9
1 - -
7 . . . .
5 O - - 3
2 , 3 e x p , 2 . 5 5 2.15
13----'--"
1 3 "
1.55 11
11
ii
0 . 0 1
9 ,,m-,-m- 9 ~
0 - 9 5
7 , 7
0 . 4 2 5 " 3
5
O 3 - 0 . 3 1 , 3 [O.013 I
HF RPC 2 J
Figure 7.
45Sc
Table 4. The E(2) and M(1) strengths in 45Sc 0 . 9 4
ex p.
0 " 8 6 0 . 3 6 0
Jt ~ J$ Exp. H F R P C
E(2) strengths in e2Jm 4
5/2 3/2 360+_~g 250 170
7/2 5/2 115_91 +2~1 166 122
3/2 119_+~ 106 78
9/2 7/2 57 +65 --$5 99 61
5/2 131-4-55 151 108
11/2 9/2 87+_~g 9 81 62
7/2 220_+~ 8 183 142
13/2 11/2 < 4,25 × 10 a 48 26
9/2 126_+~g 184 138
M(1) strengths in 10-~ × (n.m.) 2
5/2 3/2 2-3 -4- 1"0 3"7 5"7 7/2 5/2 2"7 -4- 0.8 5"1 8"2
9/2 7/2 1.4 -4- 0'7 5"8 6"5
11/2 9/2 6.0 +4.0 6"4 9-1
. 5 n l
HF 5 3 i
1 1.3
RPC
Collective bands q f positive parity states in shell nuclei 191 the calculated E(2) and M(1) strengths with experiment and with RPC results o f Styczen et al (1976) in table 4. The experimental uncertainties in the E(2) transi- tions are rather large. The calculated strengths in H F and R P C models agree fairly well with experiment. For the M(1) strengths we find that our model con- siderably improves upon the RPC results. Still the predicted strengths are large compared to experiment.
The magnetic moment o f the band head J = ~ + state at 0.01 MeV has not been measured so far. Our model predicts a value of 0-22 (n.m.) as compared to R.PC model result of 0.09 (n.m.).
The positive parity states in ~TSe have been investigated by a variety o f experi- ments (Toulemonde et al 1974; Ohnuma 1971, Baudinet-Robinet et al 1971; Lewis 1970; Fossart and Poletti 1968; Yntema and Satchler 1964; Delbrauck-Habaru et al 1971 and Vingiani et al 1971). The proton pick-up reactions (d, He 3) by Yntema and Satchler (1964) and Ohnuma (1971) and (t, d) by Baudinet-Robinet et al (1971) have shown that the J=~-+ (0.77 MeV) and J = ~ + (1.39 MeV) states are largely odi-1 and ls& -1 hole states.
In figure 8 we compare our calculated bands with the experimental ones. We find that the agreement for d~ -z band of states is fairly good. The excitation energies of the band head states are however off by more than 1 MeV.
The electromagnetic transitions of low lying positive parity states in ~7Sc have been observed by Toulemonde et al (1974) and recently by Styczen et al (1976).
Styczen et al (1976) have also calculated these transition strengths in the RPC model.
In table 5 we compare the E(2) and M(1) transition strengths with experiment and with R.PC model results. We find that the predicted strengths are within the ranges of uncertainties o f the experimental strengths. In M(1) transitions both the H F and RPC models give large values compared to experiment.
3
0
>
:E
15 13
11 9
9 1-64 9 7
7 1 . 0 9 7
5 0 . 6 4
7 5
3
0.7 7 -0.8 3 1- 3 9
- - 3 0 3 I
2 J exp. HF 2 j exp.
4 7 S c Figure 8.
1"14 5 0.6.1 3 0-41
0 . 3 4
0 I
HF
Table 5. E(2) and M ( 1 ) transitions in 4vSc
Jl -'~ J,t Exp HF RPC
E(2) strengths in e2fm 4
+ 5 7 0
5/2 3/2 432-293 192 102
7/2 5/2 < 4200 130 81
3/2 111+~ t . 86 51
9/2 7/2 - - 73 33
5/2 - - 121 66
11/2 9/2 - - 62 43
7/2 - - 147 94
M(1) strengths in 10 -~ × (n.m.) ~
5/2 3/2 9.A+2"4 4"5 4"1
7/2 5/2 < 6"0 7"5 6"4
9/2 7/2 9-8 3-4
11/2 9/2 12.I 6-7
The magnetic moment o f the band head state J = ~ + (0.77 MeV) has been observed experimentally by Fossan and Poletti (1968). The R P C model (Styczen et al 1976) gives a value /~=0.1 n.m. which is smaller compared to experimental value
~=0"35:k0"05 n.m. In the present model the calculated magnetic moment o f ~+
state is 0"35 n.m.
3.3. ~STi
The low lying positive parity states in ~STi have been studied in past by various reactions (Kownacki et al 1973; Blasi et al 1971; Haas et al 1975; Maurenzig 1971;
Lutz and Bohn I968; Rosner and Pullen 1967; Borlin and Braid 1967; Kashy and Conlon 1964). The band o f states ~-+ (0.33 MeV), ~+ (0-74 MeV), ~+ (l'23MeV),
~-+ (1-88 MeV), 3~a_+ (2.48 MeV) and 15/2 + (2.94 MeV) follow nicely the rotational J ( J + l ) rule.
We consider the lowest three intrinsic H F states with proton and neutron excita- tion from the K = ~ + and K----½ + orbits. The states o f good angular momenta pro- jeered from these states are then mixed taking care o f the orthogonality. This gives rise to K = ~ + and K=½ + bands of states corresponding to di-1 and s½ -1 hole states respectively. From the neutron pick-up reactions ~6Ti(p, d)45Ti (Kashy and Conlon 1964) and 46Ti (He 3, He 4) 4STi (Lutz and Bohn 1968; Rosner and Pullen 1967) it is known that the lowest J=~-+ (0.32 MeV) and J==½+ (1.58 MeV) states contain large components of oda-t and ls½ -1 respectively. In figure 9 we compare the energy spectra for the K = ~ + and K=½ + bands o f positive parity states with experiment. The two bands are fairly well reproduced. The excitation energies o f band head states are however off by about 2.0 MeV.
The electromagnetic transition rates for the positive parity states in 45Ti have
Collective bands of positive parity states in shell nuclei 193
4.
(11,13 ,
3 - -
11
9
1 - -
7
5 0 . 3 3
0 - - 3
2 J exp :E 2
3-61
15 13 2.15
1.,55 11 9 0 - 9 0
7 0 . 4 2 5 0 3 2.8
H F
9 iii
7 (3.5) 0 . 8 6
5 ' (I .3) 0.36 3
1.58 4.9
I 0 I
2 J exp HF
45Ti
Figure 9.
Table 6. Electromagnetic transitions in UTi
Yi "-~ Jy Exp H F
. . . . . . . ~ . . . . . L . ....
E(2) strengths in eSfm4
R P C
5/2 3/2 ~Lt.d+l"'~._,., 2"1 3"3
7/2 5/2 4 8-,.4 . +s.o 2-8 4"8
9/2 7/2 2.o+8"v ~ - - I . . S 2"9 3"5
11/2 9/2 - - 3.1 5.3
M ( I ) strengths in 10 -s × (n.m.) ~
5/2 3/2 ~'~-*u~'m+*47 248 217
7/2 5/2 233+~| * I57 132
3/2 l ~ + e o . . . . • o 103 92
9/2 7/2 104_,= +so* 100 85
5/2 StT+=** ~ v - - - - ? l 150 138
11/2 9/2 - - 73 57
7/2 - - 176 163
been studied b y Styczen et al (1976), Blasi et al (1971), a n d K o w n a k c i et al (1973).
I n table 6 we c o m p a r e the E(2) a n d M(1) transition strengths with experiment.
T h e E(2) strengths calculated in the present m o d e l are in b e t t e r a g r e e m e n t with ex- p e r i m e n t as c o m p a r e d to R P C results. T h e M(1) strengths agree with experiment w i t h i n the uncertainties.
2 - - 9
>
o I - -
7 5
0 . 2 6 0 - 3
2 J exp.
F i g u r e 10.
11 1.49
g I iii
0.88 7
0.4 5
3.73
0 3
HF
4 7 v
Table 7.
i
E(2) and M(1) transition strengths in ~TV Jl ~ J . t
E(2) strengths in e2fm ~
Exp HF RPC
5/2 3/2 < 12.5 × 108 403 243
7/2 5]2 120+~o'~ 264 171
226-117 170 111
3/2 +16~
9/2 7/2 < 4"49 × 108 160 88
5/2 343+~]~ 244 155
11/2 9/2 ~ 127 85
7/2 - - 295 202
M ( 1 ) strengths in 10 -2 × ( n . m . ) 2
5/2 3/2 < 14 4"6
7/2 5/2 4 8-,.a 6"3
9/2 7/2 - - 6.7
11/2 9/2 - - 7.5
6.2
8.9 6-8
9"9
Collective bands o f positive parity states in shell nuclei 195 The magnetic moment o f the J = ~ + (0.33 MeV) states was measured by Haas et al (1975) and is the first measurement of magnetic moment of di-1 hole state in f ] shell nuclei. The observed value is p=(0-98-}-0.24) n.m. In the present model the cal- culated magnetic moment o f ~-+ state is 0-74 n.m. and RPC model prediction is
1.2 n.m.
3.4. l," isotopes
_ a + 5+ 5+ and ~+ have been observed by In 47V low-lying positive parity states J--~- , ~- ,
several reactions (Schulz and Toulemonde 1974; Blasi et al 1973: MuUigen et al 1975; Cujec and Szoghy 1969; Halbert 1977; Styczen et al 1976). These positive parity states form a nice rotational band.
In the present model we obtain the deformed intrinsic H F states with ground state and one hole configurations. The hole state configuration corresponds to the excita- tion of a proton from K=~- orbit o f (sd) shell to the lowest available orbit K = ] in (fp) shell. The states with definite angular momenta are projected from these in- trinsic states. In figure 10 we compare the spectrum of positive parity states with experiment. We find that the rotational type of sequence is obtained in the calcu- lated spectrum but the spectrum is slightly compressed. In table 7 we compare the electromagnetic transition strengths with experiment (Styczen et al 1976). The table also contains the results of RPC model calculation (Styczen et al 1976). The present model gives better results for both E(2) and M(1) transitions.
Experimental data regarding the magnetic moment of band head state J---~-+
(0.26 /vieV) are not available. The present model predicts a value /z=0.23 n.m.
while the R P C model gives/z=0-09 n.m.
Much experimental data exist on the positive parity states in 49V (Hass et al 1975;
Styczen et al 1976; Malan et al 1972; Legg et al 1969; Blasi et al 1967; Claude et al 1967; M e et al 1970).
4 w
5 - - 13
- - 1 1
9 1 - -
7 0 ` 8 6
,5 0 . 3 9 0 - - 3 0.75 0
2 J e x p
>
1 . ~ ii
1 5 ,
1 3 - -
" 2-59 13
F i g u r e 11.
1.99 11 1.43
9
7 m
5
1.24 3
HF
11
9 1 . 4 9
9 ~
7 1.17
7 5 , 0 . 7 4
5 3 0-35
,3 [0,7,5] 1 1.65 0
RPC 2 J exp
4 9 V
11
9 7
1 t I i
9 - - - - "
7 ...
5 1
5 3
3.57 3
1 1 1-95'
HF R PC
Table 8. Electromagnetic transitions in ~gV
Jl ~ JS Exp HF RPC
i t . . . . . .
E(2) strengths in e~fm 4
+ 3 7 6
5/2 3/2 246-xas 334 173
7/2 5/2 < 364 223 127
3/2 160 q-43 143 81
9/2 7/2 < 254 128 60
5/2 249+_I°~ 202 110
11/2 9/2 < 8"18×108 107 64
7/2 363-194 +~as 248 148
13/2 11/2 - - 59 24
9/2 - - 248 141
M(1) strengths in 10 -~ × (n.m.) ~
5/2 3/2 3.9+I:~ 6-2 5-1
7/2 5/2 4.3 4-1.2 8.8 7.5
9/2 7/2 5 0-3.x . +7.8 8"6 5.2
11/2 9/2 - - 10.4 8.2
Styezen et al (1976) have considered the R P C m o d e l to describe the positive parity states in 49V. T h e y considered the lowest p r o t o n hole intrinsic states in dl-X (fp)iO a n d s½ -1 (fp)10 configurations. In figure 11 we show their calculated spectrum.
I n the present m o d e l we have considered lowest energy intrinsic H P states with o n e hole configurations. We project good angular m o m e n t u m states f r o m the K = ~ + a n d K = ½ + p r o t o n holeintrinsic states which have pure di-1 (fp)lO and s½-X(fp) 1°
configurations. In figure 11 we also c o m p a r e the calculated s p e c t r u m with experi- ment. F o r b o t h the bands our calculated spectra have i m p r o v e d agreement with experiment over the results by R P C calculation.
T h e electromagnetic transition rates have been obtained by Haas et al (1975) a n d Styezen et al (1976). In table 8 we c o m p a r e the calculated E(2) and M(1) tran- sition rates with experiment as well as with R P C m o d e l results (Styczen et al 1976).
F o r the ~+-~ ~+ transition, the uncertainty in the experimental value is relatively small a n d the calculated strength o f 143 e z f m ~ is in m u c h better agreement with the experimental value o f 1604-43 e 2 f m 4 as c o m p a r e d to the R P C result o f 81 e2fmL
T h e calculated M(1) strengths f o r ~+ ~ ~ , -~ and ~ transitions are consistent with experimental values.
T h e static magnetic m o m e n t o f the band head state J----I+ (0.75 MeV) has n o t yet been experimentally determined. The present m o d e l predicts a v a l u e / ~ = 0 - 3 6 n.m.
The R P C m o d e l gives/~---0.11 n.m.
Recently two-odd-parity rotational like bands in 4sV built on the J = 1 - ( 0 . 5 1 9 MeV) and J = 4 -1 (1-099 MeV) states have been observed (Samuelson et al 1977; see also Hass and Tarns 1974; Tarns et al 1974; Rickel et al 1976; M a n t h u r i t h i l et al 1975 a n d Brown et al 1975). Haas and Taras (1974) have suggested (rrd|-a) -1 (fp)8
Collective bands o f positive parity states in shell nuclei 197 configuration for these two bands. The energies of the states within these bands follow the the rational J(Jq-1) rule to a very good accuracy.
We have considered two deformed intrinsic H F states with ~rdi-1 (fp)9 configura- tions in which a proton is promoted from K = ~ -+ orbit o f (sd) shell to K=~-- orbit in f p shell. The last odd proton in K = ~ -+ orbit and the last odd neutron in K = ~ orbit gives rise to total K = K v 4- K ~ = 4 - and 1-. We project good angular momentum states from these intrinsic states and compare their energies with experiment in figure 12. We find that there is an excellent agreement between calculated and experimental spectra for the two negative parity bands. The excitation energies o f the band head states are however off by about 2 MeV.
The electromagnetic transitions have been measured experimentally by Brown et al (1975) and Taras et al (1974). We compare the E(2) and M(1) strengths in table 9. There is a good agreement for the E(2) and M(1) transition strengths in the K = I - band o f states.
3.5. 51Mn
In 51Mn the unambiguous identification of positive parity states has become some- what difficult by the inaccessibility of this nucleus to single nucleon pick-up reactions which would selectively populate the hole states. However, from the data o f elec- tromagnetic transitions amongst various low lying levels, tentative assignments o f
_
>~ 7 m
m
3 - - 7 , 2-9~
6~ - 2.26 2 - -
5 1.54
5
1 1 4 ' ~ " " 1 . 0 4 4
3 0.54 3
2 0-52"~'0"23 2
0 - - 1 ~ - ~ o 1 2 . 4 6
2 J exp H F
48 V
8 " ' - - ' - - 2 . 8 8
7 " - - - 2 . 0 8
6 - , ~ , - ~ 1.30 9 " ' - "
8
7
5 0 , 5 9 5
4 , 1 ' 1 0 ~ 0 4 2.31
Z J exp .HF
Figure 12.
Table 9. Electromagnetic transitions in ~V
J l --'> .If EXp.
E (2) strengths in w. u.
HF
2 1 ~ 22.2 31-6
3 2 ~ 31.1 14.6
1 23 4-as.0 . +15-, 18-2
4 3 -- 8"1
2 ~ 11"1 25.0
5 4 -- 5"0
3 -- 28"2
M (1) strengths in w. u.
2 1 0.104-0.03 0.13
3 2 0.144-0-04 0.16
4 3 ;~ 0.036 0.18
5 4 - 0.18
positive parity states have been m a d e b y various a u t h o r s ( N o e et al 1977, F o r s b l o m et al 1972; R a p a p o r t et al 1967 and Cujee and Szoghy 1969).
N o e et al (1976) have recently observed two bands o f positive p a r i t y states starting with J = , ] + (1.82 MeV) and J=½+ (2.28 MeV) states. B o t h the b a n d s follow the J(J-t-1) r o t a t i o n a l sequence fairly well.
In the present model we obtain the lowest energy p r o t o n hole intrinsic states with K = ~ -+ and K = ½ + which have mainly dl-X (fp)12 and s½ -1 (fp)lZ configurations respec- tively. G o o d angular m o m e n t a are then projected f r o m these intrinsic states.
In figure 13 we c o m p a r e the positive parity spectrum with experiment. It is seen t h a t b o t h the bands are well reproduced by the present model.
T h e r e have been n o experimental data regarding electromagnetic transition strengths f o r the positive parity states. In table 10 we show t h e calculated E(2) and M(1) transition strengths for positive parity states.
4 . D i s c u s s i o n
Various models have been employed in the past to study the positive parity states in some o f the odd-A (fp) shell nuclei. N o attempt has so far been m a d e to simultane- ously describe the properties o f these states from a single p o i n t o f view. In the present p a p e r we have studied the positive parity states in the d e f o r m e d H F model b y using few lowest energy deformed intrinsic states with one hole configurations
(sa)-I (.fp).+~.
W e find that the bands o f the positive parity states are in general well reproduced b y the effective interactions in the chosen configuration space. T h e excitation
Collective bands of positive parity states in shell nuclei 199
>
1 - -
_
11
9 '"'
7 1-07 7
5 O 7 5
5 0 . 4 9 5 3
3 0 . 4 2
1-82 3.38 2-28 4 . 6 6
3 0 3 , 0 1 0 1
2J" exp HF 2J" exp HF
Figure 13.
51Mn
Figures 2-13. Comparison of~t:alculated and experimental energies of collective bands of dl-X and s½ -1 hole states. The spectra are drawn relative to the band head states.
The figures near these states indicate their excitation energies relative to ground states.
Table 10. E(2) and M(1) transition rates in 5XMn E(2) strengths
J~ ~ Js in e~fm 4 HF
5/2 3/2 274
7/2 5/2 186
3/2 118
9/2 712 101
5/2 162
M(1) strengths in 10 -~ × (n.m.) ~
HF 2"7 39"8
31"3
energies o f t h e b a n d h e a d states J=~-+ relative to their g r o u n d states are h o w e v e r off b y a b o u t 1 MeV. Recalling the success o f the schematic m o d e l by Bansal a n d French we feel t h a t this m a y be due to inability o f ( f p ) 2 effective interactions to fit t h e g r o u n d state b i n d i n g energies o f (fp)" nuclei exactly.
T h e calculated m a g n e t i c m o m e n t s o f the b a n d h e a d J = a÷ states in ~ S c , 47S¢ a n d 45Ti are in g o o d a g r e e m e n t with experiment. T h e m a g n e t i c m o m e n t s o f J = ] ÷ states in 4z,~7Sc, 47,49V and 51Mn have n o t been measured so far. I t would be g o o d to m e a s u r e these m a g n e t i c m o m e n t s to test the predictions o f the present m o d e l .
Using the effective charges e p = l . 3 3 e a n d e , : 0 . 6 4 e which are f o u n d a p p r o p r i a t e for the (fp) shell region ( D h a r a n d Bhatt 1977a) we get an overall g o o d a g r e e m e n t f o r the E(2) t r a n s i t i o n strengths. The calculated 3//(1) strengths are highly h i n d e r e d as o b s e r v e d b u t the actual values are s o m e w h a t larger t h a n the observed ones.
T h e l i m i t a t i o n o f t h e p r e s e n t m o d e l is t h a t we h a v e c o n s i d e r e d o n l y t h e o n e h o l e c o n f i g u r a t i o n s a n d h a v e n o t m i x e d t h e m w i t h m u l t i - h o l e c o n f i g u r a t i o n s ( f o r e x a m p l e t h r e e h o l e c o n f i g u r a t i o n s ) . I t is h o w e v e r difficult t o d e f i n e a p p r o p r i a t e effective i n t e r a c t i o n s f o r t h e s e h i g h e r c o n f i g u r a t i o n s a n d h e n c e it is n o t clear h o w t h e y will affect t h e p r e s e n t results i n w h i c h c o n f i g u r a t i o n s a r e r e s t r i c t e d t o o n e h o l e o n l y .
Acknowledgements
T h e a u t h o r wishes t o express his g r a t i t u d e t o D r K H B h a t t f o r g i v i n g v a l u a b l e g u i d a n c e t h r o u g h o u t t h e c o u r s e o f t h e p r e s e n t w o r k . H e a l s o t h a n k s h i m f o r care- f u l l y g o i n g t h r o u g h t h e m a n u s c r i p t a n d m a k i n g v a l u a b l e s u g g e s t i o n s . T h a n k s a r e also d u e t o D r A K D h a r f o r p r o v i d i n g t h e c o m p u t e r c o d e s o f H F a n d p r o j e c t i o n c a l c u l a t i o n s i n t h e ( f p ) shell c o n f i g u r a t i o n space.
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