Two-quasiparticle states in 164Ho

i 2.A.l: I 1.E.1 [

Nuclear Physics A150 (1970) 497--519; @ North-HollandPublishin9 Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

T W O - Q U A S I P A R T I C L E STATES IN 164H0 H. D. JONES+ and R. K. SHELINE The Florida State University, Tallahassee, Florida +t

Received 13 April 1970

Abstract: Thirty-one levels in 164Ho have been observed up to an excitation energy of 994 keV utilizing 12 MeV deuterons and the reaction 16Silo(d, t)164Ho. The ground state Q-value was determined to be --1730-4-15 keV. The spectrum has been interpreted in terms of the coupling of the [523~] Nilsson proton orbital with the neutron orbitals prominent in the (d, t) spectrum of 163Dy. Based on this interpretation, the singlet-triplet splitting energies for the [523~], [642~], [521~'], [4023¢] and [400]'] neutron orbitals coupled to the [523t] proton orbital were determined to be --144, +67, +171, --85 and -[-102 keV, respectively. Theoretical calculations of these energies made for a zero-range spin-dependent central potential gave values of --248, +74, +160, --82 and +73 keV, respectively. E [

I

NUCLEAR REACTIONS 16SHo(d,t), E = I 2 MeV; measured a(Et, O),Q. 164H0 deduced levels, J, n, K.

1. Introduction D o u b l y o d d d e f o r m e d nuclei can be a p p r o x i m a t e l y described as consisting o f a n inert tightly b o u n d core plus an extra p r o t o n a n d a n extra n e u t r o n which are relatively free c o m p a r e d to the particles in the core. The fact t h a t the r o t a t i o n a l m o t i o n o f these nuclei is well u n d e r s t o o d m a k e s it possible to experimentally d e t e r m i n e the energy resulting f r o m the residual interaction between the o d d p r o t o n a n d n e u t r o n when the N i l s s o n single-particle orbits o f these particles are known. The e x p e r i m e n t a l value o f this i n t e r a c t i o n energy can be used to evaluate calculations m a d e for a p a r ticular choice for the f o r m o f the t w o - b o d y potential. In this p a p e r we r e p o r t the results o f calculations m a d e for a zero-range spind e p e n d e n t central p o t e n t i a l for the nucleus 164Ho" I n o r d e r to evaluate these calculations, the energy levels o f 164Ho were studied in detail by m e a n s o f the r e a c t i o n 165Ho(d ' t ) l 6,~Ho" Since the possible means o f studying these levels do n o t directly p r o v i d e i n f o r m a t i o n on the spins a n d parities o f m o s t o f these states, our interpretation m u s t be based on the established systematic b e h a v i o r o f Nilsson states in def o r m e d nuclei. By relating the H a m i l t o n i a n o f 164Ho with that o f 163Dy ' the energy o f each b a n d h e a d in ~64Ho was estimated f r o m the k n o w n energy o f the related state in ~63Dy. T h e p o r t i o n o f the s p e c t r u m o f ~64Ho near this energy was then searched * Present address, Nuclear Effects Laboratory, Edgewood, Arsenal, Maryland. tt This research was supported in part by the US Atomic Energy Commission, by the Nuclear Program of the State of Florida, and by the US Air Force. 497

498

H. D. JONES AND R. K. SHELINE

for a well-characterized rotational band with the appropriate intensity pattern of a "fingerprint", which is characteristic of the angular momenta involved in the reaction and the Nilsson state of the transferred neutron. By requiring the relative energies to fit the rotational formula and the relative intensities to fit the fingerprint, almost all assignments of spins and parities can be made unambiguously. 27

Ho 165 (d,t) Ho 164 0:75 o

~

~z

,oo

~

Ed = 12 MeV

i

29

Exposure : 2 0 , 0 0 0 ~ C

i...J!~ ~.s.~" ~'., ~ , ' . ~

~./.~

L-

,, ~., .:~

"

"

•

~

~ 27

Q:: I-(.0 150~ E E

~\i ~';/~'.~.,.j~.~ ;.4+..,:: :.',:,':.~-,,."..

25

6=85*

12

I~: lOO-

n,. I.-

21

Exposure: 17,000# C

2;'

'ili, c'i,ii

29

28 }i

31

;'~ 3o ~

• .

z 0

rr" I-"

CI5

0=95 o

300-

Exposure: 4 0 , 0 0 0 , , , C 27

25

250200-

?;

150

I1 ~.

I00 50-

'

2

-3"~

9'0'~

i! 23 r~

,~

'~p., t~.1,

~ ' . : l~''~l'~"

i[ l/22F~24 Ti ~I 1/

~ J~'~

~o

*

~"

|~'l "~ j

rli !i,.

ii

LJ ~ ~ . ~ . . . . z

%.~..~.

6oo EXCITATION

ENERGY

(keY)

Fig. 1. Spectra obtained from the 165Ho(d, t)16*Ho reaction at 75, 85 and 95 °. 2. Experimental method and results Holmium targets with thicknesses of approximately 200 pg/cm 2 were bombarded with 12 MeV deuterons with the Florida State University Tandem van de Graaff accelerator. Tritons from the 165Ho(d ' t)164H0 reaction were recorded by Eastman Kodak nuclear emulsion plates which were spring-fitted to the focal plane of a broad

16~Ho

499

range magnetic spectrograph, which is a scaled-up (6 : 5) copy of one built by Browne and Buechner 1). The results of three exposures are displayed in fig. 1. These plots are of the usual type made for magnetic spectrograph data (counts per ½ m m strip versus plate disTABLE 1 Relative energies and intensities observed in the 165Ho(d, t)t64Ho reaction Peak

Energy

Error

number

(keV)

(keV)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

0 37 94 139 166 191 204 234 262 275 294 318 343 367 394 421 454 486 499 558 583 620 650 670 691 733 777 833 863 925 967 994

1 ! 1 1 2 2 3 3 2 2 2 1 4 4 1 1 2 2 1 2 1 3 2 4 1 2 1 4 1 5 4

Angle 75 o 7 14 a) 14 11 10 25 19 7 6 10 13 12 129 7 31 55 77 66 64 42 42 112 a) 78 51 134 50 153 29 84 24 28

85 ° 8 8 10 9 9 28 28 6 9 9 15 14 121 12 31 64 82 75 54 50 35 105 a) 56 86 54 159 48 172 34 96 23 38

95 ° 7 9 13 8 10 24 22 7 6 10 17 19 111 13 24 51 76 79 54 44 47 131 59 88 54 186 46 167 46 107 30 43

a) Obscured or partly obscured by 13C(d, t)12C impurity peak or plate edge.

tance) except that the horizontal scales have been adjusted so that triton peaks corresponding to the same Q-value are aligned. The horizontal scale is thus used to indicate the excitation energy relative to the ground state, whose Q-value was determined to be - 1730-+- 15 keV. Table 1 lists the observed relative intensities and average excitation energies with errors estimated from the spread in values obtained in the different exposures. The

500

H. D . JONES A N D R. K . S H E L I N E

experimental intensities are normalized to the total theoretical cross section in pb/sr of all observed levels resulting from the coupling of the [523T] proton orbital with the [523~] neutron orbital. The accuracy of the relative values of these intensities is believed to vary from about + 15 % for the strongest levels to perhaps a factor of two for those near the lower limit of observation.

3. Theory 3.1. THE HAMILTONIAN Our interpretation of the spectrum of 164H0 is based on the assumption that the odd proton and odd neutron of this deformed nucleus occupy single-particle orbits described by the Nilsson model 2). We are primarily concerned with the low-energy portion of the spectrum where core excitations and collective vibrations can be ignored. The Hamiltonian is thus assumed to be H = Tro, + Hs.p.(p ) --}-H s.p.(n) + Hint,

(1)

where Trot is the rotational kinetic energy, H~.p.(p) and H~.v.(n ) are the Nilsson singleparticle Hamiltonians for the proton and neutron, respectively, and Hint is the residual interaction between the odd particles. Levels in the low-energy spectrum correspond to excitation of one or both of the odd particles and collective rotations of the nucleus as a whole. However, the spectrum we observe in the (d, t) reaction is simplified since excited proton states cannot be populated in first order and neutron particle states are only weakly populated. But even with these simplifications of the spectrum the level density is high and very few states are completely resolved. The resolution obtained in these experiments varied from 11 to 18 keV (FWHM). The expectation value of any residual interaction Hint can be written in the form 3) (2)

Hin t = A + ( - 1 ) I B b K o ,

where all of the/-dependence is written explicitly. This form results from the symmetry of the doubly odd wave function. The parameters A and B can be calculated from the particular type of interaction chosen and the Nilsson single-particle wave functions of the odd proton and odd neutron. The wave functions used in this work were calculated from the modified Nilsson model of Rost 4) which uses a more realistic Woods-Saxon potential rather than a harmonic oscillator potential. Terms to fourth order in the deformation parameter were included and matrix elements with A N = 2 were included in the Hamiltonian matrix to be diagonalized. The calculations were done with the code C H U L Y which was written by Rost. It can be shown 5) that the expression for the total energy is h2 E = _ _2[~I ( I + I ) _ K

2 _ ~ 2 - an2 + ( - 1)`+ ~A. An,~,,o ~ ,

~]

+Eprot+Enrot+Es.p.(p)+Es.p.(n)+A+(-1)'Bfro,

(3)

16 4-Ho

501

where h2

Ep rot = ~

(4)

E [C~[2Jp(Jp + 1) dr,

and

hz En to, = ~ 2 IC~2[2J~(J~+ 1).

(5)

For the analysis of data, a more convenient expression for the total energy is

E = Eo+ ~

{I(I + 1) - K(K + 1) + Ap A n 6/~o 3 tool, ~[1 - ( - 1)']} + [( - 1 ) ' - 1]BcSso,

(6) where Eo is the energy of the level in a rotational band which has I = K (the band head). In this form the rotational and single-particle energies of the proton and neutron as well as part o f the residual interaction energy are all absorbed by the single parameter Eo. For nonzero K, this is just the familiar rotational formula, but, for K = 0, there is always an odd-even shift due to the residual interaction and, if [f2p] = ½ and K = 0, there is also an odd-even shift due to a term in Trot [ref. 3)].

M

~

",

z Fig. 2. Angular momentum coupling scheme for doubly odd deformed nuclei. The symmetry axis is labelled z'. The other axis shown is the space-fixed z-axis.

502

H . D . JONES A N D R. K. S H E L I N E

The angular momentum coupling scheme for doubly odd deformed nuclei is shown in fig. 2. For axially symmetric nuclei the rotational angular momentum R is perpendicular to the symmetry axis so that the projection of the total angular momentum I on the symmetry axis is given by K = If2p+K2,l.

(7)

Since both the projections f2p and f2, can be either positive or negative, two states occur, one with a K-value of Ka = Ilf2pl- Ir~ll

(8)

K2 = IK2p[+ 1~2,1.

(9)

and the other with a K-value of

In the band with K = K1, f2p and [2, have opposite signs. In the band with K = /(2, g2p and f2n have the same sign. If we call the member of the band with I = K the band head then, except possibly for K = 0, the band head will have the lowest energy. Using the expression for the total energy, we see that the energy of one band head relative to the other is, for nonzero Kt hz E ( K , ) - E(Kz) = ~ ( K , - K2) + A ( K , ) - A(K2). (10)

For K1 = 0, the odd-even shift also contributes to E ( K 1 ) - E ( K 2 ) . Thus the rotational term tends to cause the state with the smaller value of K to lie lower in energy. However, the term arising from the residual interaction is usually larger and thus determines which state will have the lower energy. This can be predicted with remarkable accuracy using the Gallagher-Moszkowski coupling rule 6). If we define the total spin projection by Z = IZp+Z.[,

(11)

where Zp and Z, are the properly signed projections of the spins of the proton and neutron, respectively, then this rule simply states that the lower energy state is the one which has Z = 1. We call this the triplet state. The other member of the GallagherMoszkowski pair is then called the singlet state since it has Z = 0. The value of A ( K 1 ) - A(K2) in eq. (10) can be determined accurately from a knowledge of just the excitation energies in the doubly odd nucleus. Ifi this paper we refer to this value as the singlet-triplet splitting energy. The theoretical values of the singlettriplet splitting energies for two-quasiparticle states in 16'*Ho have been calculated for a zero-range spin-dependent central potential and compared to the observed energies. The calculations were performed using the formulae given by Pyatov 7).

164Ho

503

The form chosen for the residual neutron-proton interaction is Hint = V ( l r p - r , I ) [ 1 - = + ~ % " a , ]

(12)

and for simplicity we take

V(Irp- r.I) = -

4~gf(rp-

v.),

(13)

where 9 is the interaction parameter, e determines the amount of spin-spin force in Hi,t, and ~rp and e . are the proton and neutron spin operators. The resulting expression for the splitting energy is AE = A(K1)-

A(K2) = - 2~(Apn + Bpn)

(14)

and the odd-even shift strength as defined by eq. (2) is given by B = - 2~Apn Z~pZrn,

(15)

where np and ~zn are the parities of the proton and neutron states, respectively. The general expressions for Apn and Bpn are given by Pyatov. We see that, for the form we have chosen, only the spin-spin part of the residual interaction contributes to the splitting energy and the odd-even shift. Our calculations show that Ap. is always negative and Bpn is always positive. Thus the direction of the odd-even shift should depend only on the total nuclear parity, at least for a spin-spin force. The parameter 9 does not have the dimensions of energy and it is customary to consider instead of the parameter

where v is the quantity which appears in the expressions for the radial wave functions. Both Apn and Bp, are proportional to W. Since the energies we have calculated depend only on the product of a and W, we have only one parameter to choose. Our studies of 164Ho and other nuclei suggest a "best" value of aW = 0.85 MeV. The corresponding value originally considered by Pyatov was 0.24 MeV. By comparing the expression for the total energy of a doubly odd nucleus with proton number Z to that of its odd-A isotone with proton number Z - 1, it can be shown 5) that the band head excitation energies in these nuclei are approximately related by h2

Ez = E z _ I + ~ { K _ K

o - [ a . 6KO -- a.o

6KOO]ap6l.Op],

- I~,,I + l Q ,ol + a , , ~ l ~ n i , ~ - - a . 6o1 ~ o i , ~ } + A - - A ° + B 6 K o - - B ° 6 K o o ,

(17)

where zero superscripts refer to the lowest neutron orbital in the doubly odd nucleus and K ° is the projection of the total angular momentum in the doubly odd nucleus

504

H. D. JONES AND R. K. SHELINE

for the ground state rotational band. This formula relates the excitation energy of a one-quasiparticle neutron state in the odd-A nucleus with the energy of each of the two-quasiparticle states formed from this neutron and the odd proton in the doubly odd nucleus. This form is obtained by assuming that the single-particle energies and moments of inertia are the same for the two nuclei. Using the known odd-A excitation energy in this way we usually obtain, in effect, a fairly reliable estimate of Enrot and Es.p.(n) for the various neutron orbitals. However, if the intrinsic states in the odd-A nucleus are considerably mixed with vibrational states, the energy estimates may not be very good. In the analysis, we have first used eq. (17) to estimate the energy range in which states in 164Ho should occur. Once each state was identified, this same equation was used to determine the value of A - A °, the relative residual interaction energy. Potential types other than that considered in eq. (12) can contribute to this energy. Because of this, the calculation of these energies is rather involved and depends on several parameters. These calculations are beyond the scope of this paper so only our experimental results are presented. 3.2. C R O S S

SECTIONS

The differential cross section for the population of pure two-quasiparticle Nilsson states in the (d, t) reaction may be written in the form 8) do

--(0)

d£2

=

V2<~brlq~>2 Z [Csgl ,]2(°t(O)[C(IiJnIf; Kf-(2., (2n, Ke)] z,

(18)

l, Jn

where Ii, If and J , are the angular momenta of the target, residual nucleus, and transferred neutron, respectively. The orbital angular momentum I of the transferred neutron is, in general, not unique, so that several of the single-particle cross sections ~b~(0) may contribute to the cross section of a given level. Kf, ~2n, and Ki are the projections on the symmetry axis of If, J,, and Ii, respectively. The Clebsch-Gordan coefficient is a projection factor representing the probability of finding the rotating target and incident deuteron in correct relative alignment for the neutron to be picked up 8). The vibrational overlap (~elqSi) and occupation probability V 2 a r e the same for all members of a given rotational band and thus do not affect the relative intensity pattern of a band. This "fingerprint" thus depends only upon geometrical factors, the expansion coefficients C~2, z, and the single-particle cross sections. The single-particle cross sections were calculated in the distorted-wave Born approximation 9,to) (DWBA) using the code D W U C K on the Florida State University CDC 6400 computer. The intrinsic cross section ~Pt(0) differs from the o"l~s(O) of refs. 9,1 o) by a normalization factor N. The normalization used was that suggested by Bassel 11) and was N = 3.33. The optical potential parameters used in the calculation were those which best fit the measured angular distribution of tritons from the 16°Gd(d, t) 159Gd reaction. The parameters 12) are given in table 2. The subscript

164Ho

505

TABLe 2

Optical-potential parameters used in the D W B A calculations Particle

Vs

ros

as

roi

al

ro

(MeV)

(fro)

(fm)

(MeV)

(fm)

(fm)

(fm)

86

1.15

0.87

48

1.37

0.70

1.3

154

1.10

0.75

48

1.40

0.65

1.4

deuteron triton

4Wa

2000

I000

I

~

\..

['1

i,o

-I=1 I=0

~t,2

--t,I

1"2

Z

_o

t-O

1"3

t-2

(..) LJ

1,2 !,4

t-3

O3

l-4

0 (_} IJJ _J 1,4

p(Y

t,4 1,5 | ,6

hi /

! ,5

1,6

Z i ,6

Q= -I.730

MeV

,oo

,so

/

/

~o o

5o

o

~o

~

~o

o

50

~

zo

LAB ANGLE (degrees) F i g . 3. I n t r i n s i c s i n g l e - p a r t i c l e c r o s s s e c t i o n s c a l c u l a t e d f o r 1 6 4 H 0 . T h e d a s h e d lines c o r r e s p o n d t o c r o s s s e c t i o n s c a l c u l a t e d f o r N - - 4 ( e v e n /). T h e s o l i d l i n e s c o r r e s p o n d t o c r o s s s e c t i o n s c a l c u l a t e d for N = 5 (odd-/) and N = 6 (even-/).

506

H . D . J O N E S A N D R. K . S H E L I N E

s refers to the volume potential and the subscript d refers to the surface potential. Spin-orbit and volume imaginary potentials were not included. The bound state wave function was calculated in a spherical Woods-Saxon well with its depth varied to bind the transferred neutron by an amount B.E. = - Q ( d , t)+6.257 MeV.

(19)

Other parameters used in the bound state potential were ros = 1.25 fm and as = 0.65 fm. The bound state wave functions and single-particle cross sections were calculated for Q-values corresponding to excitation energies in 164Ho of 0, 500, and 1000 keV. The values for the appropriate Q-values were interpolated assuming that the Qdependence of the single-particle cross section was of the form qS,(Q, 0) = exp (a + bQ + cQ2),

(20)

where a, b and c are functions of the angle only. Fig. 3 shows the angle and Q-dependence of the intrinsic single-particle cross sections calculated for the 165Ho(d ' 0164Ho reaction. This plot is made in such a way that values for the cross sections at any angle and any intermediate Q-value can be estimated from the graph. On this type of plot, the curve through the three points representing the cross sections for the three Qvalues at any given angle is closely approximated by the parabola of eq. (20).

4. Analysis 4.1. EXPECTED SPECTRUM With sixty-seven protons and an approximate deformation 13) of/~ = 0.3, the Nilsson diagram z) indicates that the odd proton in 164Ho is expected to populate the [523]'] orbital. This is in agreement with the known ground state spins of the neighbouring odd-A isotopes of holmium, which are both 14) 7 - . Since the (d, t) reaction does not populate excited proton orbitals in first order, the proton assignment is the same for all the two-quasiparticle states which were observed. The nucleus 163Dy, appropriate for the application of eq. (17), has been studied extensively by Schult et al. 15) and detailed interpretation has been made in terms of the Nilsson model. Table 3 gives Schult's interpretation of the energy levels of 163Dy" Only assigned band heads and possible band-heads are listed. The K = ' + band head at 737.641 keV has a high (d, t) reaction cross section which probably indicates a sizable admixture of the [400]'] intrinsic state in that band. The unassigned level at 1058 keV has an extremely high (d, t) reaction cross section. ~lhe only intrinsic state which is expected to produce such a high cross section is the [4001"] orbital. The obvious conclusion then is that the 1058 keV state is the other part of the mixed state at 737.641 keV. Since the 1058 keV state has the much larger (d, t) cross section, it contains most of the [400I"] strength. This conclusion is also supported by the fact that the 1058 keV level and the [402+] state at 859.26 keV have about equal (d, t)

164Ho

507

TABLE 3 Interpretation a) o f intrinsic states in 16aDy K ~r

B a n d head energy (keV)

25-

Character

0

Configuration

g r o u n d state

5234 642~

~+

250.880

hole

½-

351.147

particle

521

3-

421.834

hole

521~

730

particle

737.641

complex

K - - 2 { 642~', ~-vib + . . . K - - 2 { 5214, ~,-vib + , ..

½+ 3+ ½+

512~"

820.782

complex

859.26

complex

884.285

complex

1058

4024 -1-651 ~ + . . . p411 ~' --p523 t +n523~, d- • ..

hole

unassigned

a) T h e interpretation is t h a t o f ref. 21). TABLE 4 Theoretical cross sections i n / t b / s r of levels in 164Ho at 75 degrees a) Nilsson neutron orbital 523~ 523~ 642~ 642~ 521~ 521~ 521~ 521~ 402~ 402+ 651~ 651~ 660~ 660~ 505~ 505~ 400~ 400~ 512~ 512~ 530~ 530~ 5324 5324

K zr

Eo (keV)

Total a n g u l a r m o m e n t u m in units o f 0

1+ 6+ 615+ 2+ 3+ 4+ 2525439+ 2+ 436+ 1÷ 4÷ 3÷ 5÷ 2+

0 191 139 160 343 486 421 499 620 733 700 700 700 700 500 500 833 925 900 900 900 900 900 900

1

2

3

4

5

6

7

17

22

21

17

10

4 51 10 7 78 9 15 28 1 11 7 13 14 11

1 23 20 4 35 2 3 7

1

4

8

11

91

71 244

174

85

53 71 265 25

2

5

9

4

5 9

10 186 29 34 45 4 255 10 6 12 13

5 402 67

2 31 10

103

32 125 69

30

20

13 64 26 58 9

13

10 318

14

40

30

48

1 5 1 84 3 16 9 30 3

1 3 12 9 6

8

9 15 1 2

9

10

5

1

1 6 4 2

2 1 1 32

72 5 2 7 1

a) Cross sections do n o t include the o c c u p a t i o n probability. States with cross sections less t h a n 1 /tb/sr are omitted. T h e b a n d head energy Eo is defined in eq. (6).

508

H. D. JONES AND R. K. SHELINE

intensities in 1 6 3 D y (where vibrational mixing reduces the cross section of the 1058 keV state), whereas in 157Gd the [400~] band head is over twice as strong as the [402+] band head in the (d, t) reaction 16). In 157Gd, the [400T] orbital has no opportunity to mix with a vibrational state. The points to be made are that the main part of the [400T] intensity appears about 200 keV higher than the [402J,] orbital, and that the rest of the [400~] intensity appears about 300 keV lower than the main part of the [4001"] intensity. Neutron orbitals which give rise to levels strongly populated in the (d, t) reaction are: [5211`], [402,~], [400T] and [503T]. It is expected that band heads in 164Ho resulting from the coupling of the [5231"] proton with these and other orbitals will occur within a couple of hundred keV of the energies observed in 1 6 3 D y . Table 4 gives the theoretical cross sections of levels resulting from all neutron orbitals which may be important in the low-energy spectrum of 1 6 4 H o " The cross sections of levels which were identified were calculated at the observed Q-values, and those of unobserved levels were calculated at Q-values corresponding to the estimated excitation energies. The cross sections in table 4 do not include the occupation probability. The pairs of Nilsson orbitals [4001"], [6601"] and [4025], [6511"] are expected to be mixed because of the nonzero matrix elements of the Nilsson Hamiltonian with AN = 2, where N is the principal oscillator quantum number 17). The Nilsson coefficients were calculated using the code C H U L Y of Rost 4), which includes these matrix elements of the Hamiltonian. The cross sections calculated using these coefficients were closer to the observed values than the cross sections calculated with the usual Nilsson coefficients. Agreement with experiment was still not good, however. This is probably because this mixing is sensitive to the relative energies of the affected states. We have made no attempt to describe the AN = 2 mixing quantitatively since the [660]'] and [651+] states were not observed. 4.2. D I S C U S S I O N O F T H E L E V E L S C H E M E

Levels resultin9 from the [523+] orbital. The ground state of the 1 6 4 H O is the triplet state resulting from the coupling of the [523T ] proton with the [523+ ] neutron, and the first three members of this rotational band have been observed in the decay [refs. 18,19)] of the 6- isomer at 139 keV. "Ihe K = 1 band head is the ground state and the 2 + member of the band is at 37 keV. From the energies of the first two members of this K --- 1 band and the rotational formula, the moment of inertia parameter is calculated to be 9.25+0.25 keV. Using this value, the 3 ÷, 1 member of this band is expected to be at an energy of 93___3 keV. This agrees well with the value of 9 4 _ 1 keV which we have measured. According to ref. ~9), an energy of 93 keV for the 3 ÷, 1 state is implied by the energies of the y-rays observed following the decay of the 6isomer at 139 keV. Ref. 2o), however, gives the energy of this state as 87 keV. The discrepancy lies in the energy of the y-ray transition between the 3 +, 1 and 2 +, 1 states. The energies of other 3,-rays observed agree within 1 keV between the two references. Our results indicate that the energies of ref. 19) are correct.

164Ho

509

Higher members of this band have been observed for the first time in this work. The calculated energy of the 4 +, 1 level is 167-t-5 keV, and its intensity should be about the same as that of the ground state. Since all the levels in the K = 1 band are very weakly populated in the (d, t) reaction, there are large uncertainties in the measured intensities. The intensity of the level observed at 166 keV is within experimental error of the intensity expected for the 4 +, 1 state. Since the 166 keV level is the only one within 24 keV of the expected energy, it is assigned as the 4 +, 1 state. The calculated energy of the 5 +, 1 level is 259 + 8 keV, and its intensity should be a little more than half that of the ground state. The weak triton group centered at about 265 keV looks like an unresolved doublet. A least-squares fit yields peaks with average energies of 262 and 275 keV and roughly equal intensities. The intensities are consistent with that expected for the 5 +, 1 state. Since the 262 keV level has the closer energy, it is tentatively assigned as the 5 +, 1 state. Higher members of this band are not expected to be seen, since their cross sections are small and they are expected near other levels which are very strong. Since both the rotational Hamiltonian and the residual interaction are expected to cause the K = 1 [523+] band to have a lower energy than the K = 6 [523+] band, then the energy separation between the K = 1 and K = 6 band heads is expected to be large. The 139 keV level has been assigned in previous investigations 18,19) as arising from another configuration. The observed (d, t) intensity is consistent with this interpretation. The level we observe at 139 keV is much too weak to be the K = 6 [523~] band head. The first unassigned triton group is centered near 200 keV and is an obvious doublet. A least-squares fit yields levels with average energies of 191 and 204 keV with the 191 keV level having a slightly greater intensity. These intensities are near that expected from the K = 6 band head, and there are no other states within 100 keV which are strong enough to be the K = 6 band head. Consequently, one of the states in this doublet is probably the 6 +, 6 state. At this point in the discussion, it is not clear which of the two levels is the 6 +, 6 state. As will be seen later, however, the other member of the doublet must be the 7 - member of the K = 6 [642]'] band which starts at 139 keV. It will be seen that both the energies and intensities favour the assignment of the 191 keV level as the 6 +, 6 state and the 204 keV level as the 7 - , 6 state. Using the moment of inertia parameter found for the ground state band, the second member of the K = 6 [523~.] band is expected near 320 keV with an intensity which is about half that of the 6 +, 6 state. The intensity of the experimental level at 318 keV is about half that of the 191 keV level, and it is very near the expected energy. Although there is another level at 294 keV with a reasonable intensity, the energy of this level would require an apparent moment of inertia parameter of 7.4 keV. Coriolis coupling calculations show that the amount of mixing expected for the K = 6 band would only reduce the apparent moment of inertia parameter by about 0.2 keV from the unperturbed value, which should be close to that found for the ground state rotational band. Therefore, the level at 294 keV is assigned as the 7 +, 6 state. From the

510

H.D.

J O N E S A N D R. K. S H E L I N E

energies of the first two members of this band, the moment of inertia parameter is determined to be 9.07+0.20 keV. Using this value, the 8 +, 6 member of the band is expected at an energy of 4 6 3 _ 6 keV with an intensity about y1 that of the band head. Although there seems to be a b u m p on the tail of the strong peak at 454 keV, evidence for the 8+, 6 state is not conclusive. Higher members of this band have very small theoretical cross sections and are not expected to be observed. Levels resulting from the [642]'] orbital. The 139 keV level has already been assigned as the K = 6 band head resulting from the [642]'] orbital 18,19). In 163Dy and other odd-A nuclei, this orbital has been observed to be strongly affected by Coriolis coupling is). The mixing coefficients connecting this orbital with the [633]'] and [651]'] orbitals are both very large. In 163Dy, this mixing causes an unusually small moment of inertia parameter (3.8796 keV) to be required to fit the levels to the rotational formula. This is not too surprising since the [633]'] orbital is a particle state and is expected to be close in energy to the [642]' ] orbital, which is a hole state. In nuclei where those orbitals are both hole states or both particle states, the energy separation is much greater and the mixing is less. The relative energy of the [651]'] orbital is uncertain, but it should be farther away than the [633]'] orbital. Since the mixing coefficients connecting these orbitals are essentially the same in 164Ho as in ~63Dy and the energy separations are similar, we can expect strong mixing effects in 164Ho as well. In particular, the apparent moment of inertia of the K = 6 [642T ] band should be similar to the one observed in a63Dy for this orbital. However, the K -- 1 [642]' ] band mixes with a K = 0 band (which has an odd-even shift) so the apparent moment of inertia parameter of the K = 1 [642]' ] band may be different for the odd and even members of the band. This means that the energies and the rotational formula will be almost useless as means of assigning levels to the K = 1 band. However, the intensities should still be useful. A rough mixing calculation shows that the mixing is such that strength always goes from the [633]' ] and [651]' ] levels to the [642]'] levels. Since the [633]'] and [651T] orbitals are very intrinsically weak and in this case the [633]'] orbital is a particle state, the [633T] and [651T] orbitals have very little strength in the (d, t) reaction to contribute to the [642]'] levels. Consequently, the intensities observed for the [642T ] levels should be close to the theoretical unmixed intensities. The 204 keV level is probably the second member of the K = 6 [642]'] band. It is the only unassigned level within 100 keV which has an intensity large enough to be the 7 - , 6 state. The energy of this level implies a moment of inertia parameter of 4.64+_0.16 keV. This is fairly close to the value observed in 163Dy, and is entirely reasonable. The intensity of this level relative to that of the 6 - , 6 state is about 2, which is close to the expected intensity. The 204 keV level is thus assigned as the 7 - , 6 state. The calculated energy of the 8 - , 6 level is 278_+_5 keV, and its intensity should be about 1.5 times that of the 6 - , 6 state. The experimental level at 275 keV is very near the expected energy and has an intensity about equal to that of the 6 - , 6 state. It is

164H0

511

thus the most likely choice as the 8 - , 6 state. Although the 294 keV level has a somewhat better intensity, it is a little too far away in energy. The 275 keV level is therefore assigned as the 8 - , 6 state. The calculated energy of the 9 - , 6 level is 362__ 8 keV, and its intensity is expected to be about half that of the 6 - , 6 state. The experimental level at 367 keV is very near the expected energy and has an intensity relative to that of the 6 - , 6 state of 0.6 at 75 °, 1.3 at 85 °, and 1.6 at 95 °. The uncertainty in the intensity of this state is large since it is not resolved from the much stronger peak at 343 keV. The 367 keV level is thus tentatively assigned as the 9 - , 6 level. The expected intensities of higher members of this band are too small for them to be observed. Since the K = 1 [642T] band is the singlet state, it is expected to be at a higher energy than the K = 6 [642T] band. However, the rotational energy of the K = 1 band head is about 50 keV less than that of the K = 6 band head, and the K = 1 band head is pushed down by the Coriolis interaction, but the K = 6 band head is not. The two band heads may, therefore, be fairly close together. The only unassigned states near the K -- 6 [642T] band head are at 234 and 294 keV. These levels have intensities consistent with those expected for the K = 1 [642]'] band. As mentioned previously, the moment of inertia parameter for the even members of this band may be different from that of the odd members. Therefore, the 234 and 294 keV levels will have to be assigned on the basis of intensities alone. The most probable assignments are 3 - , 1 for the 234 keV level and 4 - , 1 for the 294 keV level. These assignments must be considered highly tentative. Levels resultin9 from the [521T] orbital. From the band head energy of the [521T] orbital in 163Dy, the K--- 5 and K = 2 band heads resulting from this orbital in 164Ho are expected between 200 and 600 keV. Since this orbital is the lowest intrinsically strong hole state in 163Dy' the lowest strong levels in 164Ho probably result from this orbital. The K" = 5 band is the triplet state and is thus expected to be at a lower energy than the K = 2 band. As table 4 shows, the band head of the K = 5 band is by far the strongest level expected in either of the two bands. The lowest as yet unassigned level is at 343 keV, and it has a higher intensity than any other levels below 600 keV. It is therefore confidently assigned as the 5 +, 5 state. The observed intensity is about three-quarters of the theoretical intensity. The mixing coefficient connecting the [5211"] orbital with the nearby [512T ] particle state is large so that the intensities will be affected. Levels resulting from the [512T ] orbital are expected ~ 300 keV higher than the [521T] levels. For this separation, a calculation shows that the mixing increases the cross sections of the 6 +, 5 and 7 +, 5 states, but leaves the cross sections of other members of the K = 5 band essentially unchanged. Using the moment of inertia found for the ground state band, the second member of the K = 5 band is expected to have an energy of about 454 keV, and its unmixed intensity relative to that of the 5 +, 5 state is about 0.4. As previously mentioned, the observed intensity should be a little greater than this because of the Coriolis coupling. There are as yet unassigned levels at 421,454 and 486 keV with intensities relative to

512

H . D . JONES A N D R. K. S H E L I N E

the 5 +, 5 state of about 0.4, 0.6, and 0.5, respectively. The K = 5 band moment of inertia parameters implied by these levels are 6.50, 9.25 and 11.92 keV, respectively. It is noted from the calculation that the mixing is such as to cause a reduction in the apparent moment of inertia parameter of approximately 2 keV. Thus the unperturbed moment of inertia parameter implied by the 486 keV level is much too large. Since the mixing increases the intensity of the 6 +, 5 state, the 421 keV level is not a likely choice for the 6 +, 5 state. The intensity and energy of the 454 keV level thus imply that this level is the 6 +, 5 state. The moment of inertia parameter for the K = 5 band is then 9.25-t-0.12 keV. Using this value, the third member of the K = 5 band is expected at 584-t-3 keV, and its unmixed intensity relative to the 5 +, 5 state should be about 0.2. The observed intensity should be somewhat larger than this as a result of mixing. The only experimental level close to that energy is at 583 keV and has an intensity relative to the 5 +, 5 state of about 0.3. Both the energy and intensity are close to the expected values. The 583 keV state is therefore assigned as the 7 +, 5 state. As table 4 shows, the theoretical intensities of higher members of this band are very small. Thus they are not expected to be observed. Since the K = 2 band is the singlet state, it is expected at a higher energy than the K = 5 band. Possible candidates for the K -- 2 band head are at 394, 421,486 and 499 keV. The intensities of these levels relative to that of the K = 5 band head are about 0.2, 0.4, 0.5 and 0.4, respectively. A mixing calculation shows that the cross sections of the levels in the K -- 2 band are only slightly increased as a result of the mixing. Since the intensity of the K = 2 band head should be close to half of the K = 5 band head intensity, the 486 keV level is the best choice for the K = 2 band head. It is found that this assignment also allows a more consistent interpretation of higher members of the K = 2 band than do other possible assignments. Therefore, the 486 keV level is tentatively assigned as the 2 +, 2 state. Using the moment of inertia parameter found for the K = 5 band, the second member of the K = 2 band is expected near 542 keV, and its intensity relative to that of the 2 +, 2 state should be about 0.8. The only unassigned level near that energy is at 558 keV with an intensity relative to that of the 2 +, 2 state of about 0.7. The 558 keV level is thus tentatively assigned as the 3 +, 2 state. The moment of inertia parameter is determined to be 12.00+__0.47 keV. Using this value, the 4 +, 2 member of the band is expected to be at an energy of 6 5 4 + 7 keV, and its intensity relative to that of the 2 +, 2 state should be about 0.6. There are as yet unassigned levels at 650 and 670 with intensities relative to that of the 2 +, 2 state of about 0.6 and 1.2, respectively. Since the energy and intensity of the 650 keV level are closer to expectation, it is tentatively assigned as the 4 +, 2 state. The calculated energy of the 5 +, 2 level is 774+ 11 keV, and its intensity relative to that of the 2 +, 2 state should be about 0.3. There are experimental levels at 733 and 777 keV with intensities relative to that of the 2 +, 2 state of about 2.1 and 0.6, respectively. The 733 keV level is definitely not the 5 +, 2 state, but because it is so strong it

164140

513

may be obscuring the 5 +, 2 state. The energy of the 777 keV state is very close to that expected for the 5 +, 2 state, and although its intensity is somewhat high, it is tentatively assigned as the 5+, 2 state. The calculated energy of the 6 +, 2 state is 918_+ 17 keV, but its intensity relative to that of the 2 +, 2 state is expected to be only about 0.1. Although there seems to be a bump on the tail of the strong level at 925 keV, evidence for the 6 +, 2 state is not conclusive. The expected intensities of higher members of the K = 2 band are much too small for them to be observable. Levels resulting from the [402+] and [400T] orbitals. These orbitals have been observed in several odd-A nuclei but usually with cross sections which are much less than would be expected for the pure [402~ ] and [400T ] states. This has been attributed to AN = 2 mixing between the [402+] and [651]'] orbitals and between the [400T] and [660T] orbitals 17). Part of the high (d, t) cross section of the [402~.] and [400]'] states is usually transferred to the [651T] and [660]'] states, respectively, so that four strong hole states are seen in the spectra of odd-A nuclei rather than just two. This has been experimentally observed in several odd-A nuclei with neutron numbers between 91 and 97. However, only two strong hole states are observed in 163Dy below 1.5 MeV [ref. a 5)]. Either the other parts of the mixed states are much weaker or they occur at energies higher than 1.5 MeV. In either case, this means that in 64Ho we expect to see band heads resulting from only one component of the mixed states. From the band head energies of the [402+] and [400T] orbitals observed in 163Dy and the above remarks, we expect to see four strong hole states between about 650 and 1250 keV. These correspond to the band heads resulting from the [402+] and [400]'] orbitals, as shown in table 4. There are indeed four strong hole states observed at 620, 733, 833 and 925 keV. In 163Dy, the [402+] orbital occurs approximately 200 keV below the [400T ] orbital. Since the residual interaction singlet-triplet splitting is usually less than 200 keV, the two lower energy levels are probably the two [4021 ] band heads and the other two are the two [400]' ] band heads. In a63Dy, the [400T] orbital appears to mix strongly with the K - 2 ])-vibration built on the [642]' ] orbital, thus reducing the intensity of the [400T ] state. In 164Ho ' both of the [400]'] bands have the opportunity to mix with ])-vibrations built on the [642]'] bands. This is apparently the case since the observed intensity of the 833 and 925 keV levels is about 30 ~ of the theoretical [400]'] intensity, whereas the intensity of the 620 and 733 keV levels is about 60 ~ of the theoretical [402,~] intensity. As table 4 indicates, the strongest [402+] band head should be the 5-, 5 state. Since the 733 keV level is about 1.5 times as strong as the 620 keV level, this would indicate the assignments 2-, 2 and 5-, 5 for the 620 and 733 keV levels, respectively. This is also consistent with the Gallagher-Moszkowski coupling rule since the 2 - , 2 band is the triplet state. There are unassigned levels at 670 and 691 keV which are candidates for the second member of the K = 2 band. Coriolis coupling is expected to decrease the apparent moment of inertia parameter of the K = 2 band. This makes the 670 keV

514

H . D . J O N E S A N D R. K . S H E L I N E

level the more likely candidate. Since its intensity is also closer to the expected value, it is assigned as the 3 - , 2 state. The strongest [400]'] band head should be the 4 - , 4 state. Since the 833 keV level is almost twice as strong as the 925 keV level, this would indicate the assignments 4 - , 4 and 3 - , 3 for the 833 and 925 states, respectively. This is also consistent with the Gallagher-Moszkowski coupling rule since the 4 - , 4 level is the triplet state. The unassigned level at 994 keV is near the energy expected for the second member of the K = 3 band and has a reasonable intensity. Consequently, the 994 keV level is tentatively assigned as the 4 - , 3 state. Because of the large Coriolis, vibrational and A N = 2 mixing effects which are important for these orbitals, the above assignments must be considered highly tentative. Since A N = 2 mixing is not well described theoretically even for odd-A nuclei, the effects in doubly odd nuclei are even more ditticult to predict. Other levels. Since the [521~] orbital has a high intrinsic strength, it is expected that it may be visible in the low-energy (d, t) spectrum of 164Ho, even though it is a particle state. The coupling of this orbital with the [523]' ] proton produces bands with K = 3 and K = 4. Both of these bands have been identified in the doubly odd nucleus of 166Ho where the two band heads are separated by about 180 keV with the K = 3 band at the lower energy 20). Since the configuration of these levels is the same in 164Ho as in 166Ho, we expect about the same spacing in ~64H0. F r o m the energy of the [521J, ] orbital observed in 163 Dy, the occupation probability for that state is about 0.15. This implies that the cross section of the K = 3 and K = 4 band heads should be about 40/~b/sr. In 163Dy ' the [521+] band head is observed at 351.147 keV. Thus the [521~] band heads should be between about 150 and 550 keV in 164Ho. The only unassigned levels in this energy range are at 421 and 499 keV with cross sections of about 50/tb/sr. This makes them possible candidates for the [521~] band heads. However, the separation of these levels is only 78 keV, much less than the 180 keV expected. If we assume that the lower energy level is the K = 3 band head and look about 180 keV higher for the other band head, we see that it could be obscured by the strong level at 620 keV. Thus a possible assignment for the 421 keV level is as the 3 +, 3 state. However, in view of the weakness of the experimental evidence no formal assignment is made. As mentioned previously, both of the [400]'] bands have the opportunity to mix with 3,-vibrations built on the [642]' ] bands. Since this type of mixing seems to be appreciable in 163Dy ' it could be appreciable in ~64Ho as well. In 163Dy, the weaker part of this mixed state occurs about 300 keV lower than the stronger part. In ~64Ho, the K -- 4 - [400]' ] band mixes with the K - 2 7-vibration built on the K = 6 - [642T] band, and the K = 3 - [400]'] band mixes with the K + 2 7-vibration built on the K = 1- [642]'] band. If we assume that A N = 2 mixing affects the [400]'] cross sections the same way it affects the [402].] cross sections, then about ½ of the K = 4 [400]'] strength and about ½ of the K = 3- [400T] strength is still unaccounted for. Presumably, this strength goes into the 7-vibrational states which should be about

164Ho

515

three hundred keV or so lower in energy than the strong parts of the [400T] band heads. Both of the above fractions correspond to cross sections of about 50/~b/sr. It is interesting to note that two of the unassigned levels below 800 keV occur 333 and 225 keV lower in energy than the corresponding [400T] band heads with cross sections of about 50/~b/sr. Thus the 499 keV level could be the K - 2 y-vibration built on the K = 6 - [642T] band, and the 691 keV level could be the K + 2 y-vibration built on the K = 1- [642T] band. Although this interpretation is certainly possible, the experimental evidence is not conclusive enough for formal assignments to be made. :4SO 1432 1356

1084

45+

777

4'

650

5-327~ -

583

6~"

454

54

343

833

4--

994

S-

925

E"S~-

967

[400t3

i-sfl.E, oo~

733

Emt-]+E4o2~

670 620

691

~ 4 9 9 ~ 9-

7* 6*

3~'

94

21 It

37 0

191

E"'fl'l~'O

4

2

1

~'7

318

394

UNASSIGNED 8-

275

7- •

204

~"

,s,

(4-1 (3-)

294 234

l-3~-i--~fj

Fig. 4. Level scheme for 164Ho. Highly tentative spin assignments are enclosed in parentheses. In the notation for the configurations, the first set of quantum numbers refers to the proton and the second set refers to the neutron. The algebraic signs shown are the signs of the projections of the particle angular momenta. 4.3. S U M M A R Y

Fig. 4 shows the proposed level scheme for 164Ho. Highly tentative spin assignments are enclosed in parentheses. Fig. 5 shows the relation between the excitation energies of band heads in 164Ho and in the odd-A isotone, ~63Dy. Dotted lines connect the band heads in 164Ho with their parent states in 163Dy. The lines in the center 6 4 H o scheme represent the doubly odd excitation energies calculated from eq. (17) assuming no residual interaction. The difference then between these levels and the corresponding observed levels is due to the residual interaction and second order mixing effects. The relative residual interaction energies are summarized in table 5.

516

H . D . JONES AND R. K. SHELINE

Fig. 6 shows the experimental and theoretical cross sections for all levels which have been assigned. The theoretical cross sections include the occupation probability I000

3-,3 900~

i

/

4- ,4

x:l

/ /

80O

x:o

2 '2

// /

/ 5-,5

i

/

/ /

i

/

/

/

/

700

/ /

~ 6oo

2%2

X=l

/

u.i

5 Z 0

500

2+,2

X=O

t..-

x

i

/

2 '2

/

4O0

/ 5¢,5

X=I

/ /

2,O0

/ / 200

6+,6

X=O X=O

I-,I G -,6

~=1

/

/ S

/'

5

/// /,I /

\ \ \

I00

\ \ \x

0

l+,l

~1

~,~K Ho164 EXPERIMENTAL

52 ' z5

----~ -~

Ho164 CALCULATED WITH NO RESIDUAL INTERACTION

Dy163

z,,K

EXPERIMENTAL

Fig. 5. Correspondence between one-quasiparticle states in 163Dy and two-quasiparticle states in 164Ho.

which was estimated from the observed excitation energies in 16aDy" The values used for V z were 0.5, 0.83, 0.88, 0.90, and 0.94 for the [523+], [642T], [521T], [4025], and [400T] orbitals, respectively.

164Ho

517

Hol64

95 ° 4*4

3-3

5-5 jc~:)

2-2

5+5

Zo F-

IOO

6+5 Z*2

-

3+2

•

4-3

tlA 6+6

4-1~6 3.1 ~

10-

i'l

5

3-1 t

tl..

I

J.O

I

i

i

1(30

=

=

=

2o0

3oO

I

400 500 600 EXCITATION ENERGY (keV)

700

l

J

800

I

900

1000

100<3,

Hol64

75 ° 4~4 3-3

5-5 Zo2

5~'5

Z 6~'5

•

2*2

3*2

"

4-3

? I

6+6

'7.-6

,'r" j ,¢ Z

8-~

IO

I

2~'1t 311

t~ m

1.0

I 7

I

i=~

=

I 100

i

I I

/ 4

'

i

=

700

800

I 9oo

i

|

to(3o

EXCITATION ENERGY (keV)

Fig. 6. C o m p a r i s o n o f theoretical and experimental cross sections for the Z65Ho(d,t)Z6~Ho reaction at 75 ° and 95 °. The solid lines represent theoretical cross sections and the large dots represent experimental cross sections. D a s h e d lines represent the intensities o f unassigned levels. The numbers above each level are the assigned values o f I~K.

518

H. D. JONESAND R. K. SHELINE TABLE 5 Experimental relative residual interaction energies for a64Ho Configuration proton

neutron

K~

2?

523~ 523~' 523]` 523]` 523~ 523~ 523]` 523~ 523]` 523]`

523+ 523~ 642~' 642~ 521]` 521~ 402~. 402.~ 400]' 400]`

1+ 6+ 615+ 2+ 2543-

1 0 l

Relative residual interaction energy (keV) 0 144 -- 158

0

-- 91 a)

1 0 1 0

--126 45 a) -- 259 b) -- 174 b)

1

-- 115 U)

0

-- 13 b)

a) This state is strongly affected by the Coriolis interaction and the band head is not directly observed. The energy, therefore, has a large uncertainty. b) These states are highly mixed. Energies and even the assignments are very uncertain.

5. R e s i d u a l i n t e r a c t i o n c a l c u l a t i o n s

B o t h the singlet and triplet states have been observed for five configurations in X64Ho. We thus have five splitting energies that we can compare to the values calculated f r o m eq. (14). The splitting energies for the [5235], [642T], [521T], [402+] and [400T ] neutron orbitals were measured to be - 144, + 67, + 171, - 8 5 and + 102 keV, respectively. F o r our choice of the parameter (ocW = 0.85 MeV), the values calculated f r o m eq. (14) for these neutron orbitals were - 2 4 8 , + 74, + 160, - 8 2 and + 73 keV, respectively. With the exception o f one o f these values, the agreement is remarkably good. This indicates that the spin-spin interaction can account for most of the singlet-triplet splitting energy. Similar calculations have been performed for several experimental splitting energies observed in other nuclei. The results o f these calculations which will appear in a forthcoming publication determine the value o f the parameter e W quoted above.

6. C o n c l u s i o n s

The energy levels o f 164H0 have been observed by means of the (d, t) reaction. Using data on the energy levels o f 163Dy and the expected systematic behavior o f energy levels in deformed nuclei, it has been possible to interpret the energy levels o f 164H0 below a b o u t 1 MeV as resulting f r o m the coupling of the [523T] p r o t o n Nilsson orbital with the neutron orbitals prominent in the low-energy (d, t) spectrum of 163Dy"

164Ho

519

This i n t e r p r e t a t i o n has resulted in the d e t e r m i n a t i o n o f relative energies due to the residual n e u t r o n - p r o t o n i n t e r a c t i o n for all o f the assigned t w o - q u a s i p a r t i c l e states. T h e five singlet-triplet splitting energies o b s e r v e d were c o m p a r e d to theoretical calculations for a simple z e r o - r a n g e s p i n - d e p e n d e n t central potential. These calculations p r e d i c t t h a t the triplet state has the lower residual i n t e r a c t i o n energy in each case, in a c c o r d a n c e with b o t h the e x p e r i m e n t a l results a n d the G a l l a g h e r - M o s z k o w s k i c o u p l i n g rule. F o u r o f the five theoretical splitting energies agree very well with the o b s e r v e d energies. It thus a p p e a r s that the spin-spin interaction can a c c o u n t for m o s t o f the singlet-triplet splitting energy. W e w o u l d like to t h a n k Drs. C. T. Hess a n d Y. Shida for helpful discussions o f v a r i o u s aspects o f this study a n d Dr. M. M. M i n o r , who p r o g r a m m e d p a r t o f the residual i n t e r a c t i o n calculations. T h a n k s also to M r . K e n n e t h Chellis for p r e p a r i n g the targets used in this research a n d to the plate c o u n t i n g staff o f Mrs. M a r y Jones, Mrs. Hazel Benton, Mrs. V e r o n i c a Lanier, a n d Mrs. Ella Jean W e h u n t . The help o f M r . R o b e r t M c D o l e in d r a f t i n g for this m a n u s c r i p t is gratefully a c k n o w l e d g e d .

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

C. P. Browne and W. W. Buechner, Rev. Sci. Instr. 27 (1956) 899 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29 (1955) No. 16 N. D. Newby, Jr., Phys. Rev. 125 (1962) 2063 E. Rost, Phys. Rev. 154 (1967) 994 H. D. Jones, Ph.D. dissertation, The Florida State University (1969) unpublished C. J. Gallagher and S. A. Moszkowski, Phys. Rev. 111 (1958) 1282 N. I. Pyatov, Izv. Akad. Nauk SSSR (ser. fiz.) 27 (1963) 1409 G. R. Satchler, Ann. of Phys. 3 (1958) 275 R. H. Bassel, R. M. Drisko and G. R. Satchler, Oak Ridge National Laboratory Report 3240, 1962 (unpublished) G. R. Satchler, Nucl. Phys. 55 (1964) 1 R. H. Bassel, Phys. Rev. 149 (1966) 791 M. Jaskola, K. Ngboe, P. O. Tjoem and B. Elbek, Nucl. Phys. A96 (1967) 52 K. Kumar and M. Baranger, Nucl. Phys. A l l 0 (1968) 529 C. M. Lederer and J. M. Hollander, Table of isotopes, 6th ed. (Wiley, New York, 1967) O. W. B. Schult et al., Phys. Rev. 154 (1967) 1146 Y. Shida, private communication (1968) R. K. Sheline, M. J. Bennett, J. W. Dawson and Y. Shida, Phys. Lett. 26B (1967) 14 B. Sethi and S. K, Mukherjee, Nucl. Phys. 85 (1966) 227 M. H. Jorgensen, O. B. Nielsen and O. Skilbreid, Nucl. Phys. 84 (1966) 569 H. T. Motz et al., Phys. Rev. 155 (1967) 1265