Measurements of M1 and E2 transition probabilities in Te125, I127, Xe129, Cs133, La139 and Pr141

Nuclear Physics 68 (1965) 352--368; (~) North-Holland Publishing Co., Amsterdam Not to

be reproduced

by photoprint or microfilm without written permissio n from the publisher

M E A S U R E M E N T S O F M1 AND E2 T R A N S I T I O N PROBABILITIES

in Te 12s, 1127, Xe 129, Cs 133, La 139 and Pr t4t J. S. GEIGER, R. L. G R A H A M , I. B E R G S T R O M * and F. BROWN General Physics and Research Chemistry Branches, Chalk River Nuclear Laboratories, Atomic Energy of Canada Limited Received 14 September 1964 The mean lives of the first excited states of Xe ts~, 1lz7, Cs lss and La is* have been measured electronically using a time-to-amplitude converter. The percentage E2 admixture in the dominantly M1 radiations de-exciting these states have been determined from measurements of the L subshell conversion line intensity ratios for the transitions. The percentage E2 admixture in the 145.43 keV y-transition in Pr m , the 165.84 keV transition in La x89 and the 35.48 keV transition in Te ~5 were determined in the same way. Summaries have been prepared of the presently available information on the M1 and E2 transition rate for the L forbidden M1 transitions of the classes represented by the above transitions (dt +_~ s t and g; ~_~ all; Z ~ 50).

Abstract:

I RADIOACTIVITY x27,x~gXe, measured X K ce-delay, co; lssXe, measured X K ce-delay; xs°Ce, measured flXtg-delay, cc; lZSXe, ~4xCe measured cc. E x~I, x28Xe, XS'La deduced T4z, t~(E2/M1); ~3sCs deduced Tt; 12sI, mPr, deduced 6(E2/M1). Sources from (n,y) on natural targets.

1. Introduction The advent of fast coincidence techniques little more than a decade ago 1) made it possible for the first time to measure, directly, the lifetimes of low energy E2 and M1 transitions 2, 3). The first survey of M1 transition probabilities 4) revealed that most are much slower than the predictions of the single-particle estimate 2) (typical hindrance factor ~ 102) and that most of these transitions could be classed as forbidden because of a change of AI = 2 in orbital angular momentum l. Further experiments 6) seemed to confirm this apparent regularity in the transition rates of l forbidden MI transitions. In the early experiments little was known about the E2 admixtures in these M1 transitions except that they were small. The advent of theoretical L subshell conversion coefficients 7, s) and higher resolution fl-spectrometers made it possible to deduce accurately the degree of E2 admixtures in dominantly M1 transitions from measured L subshell conversion line intensity ratios. The combination of lifetime measurements and M1/E2 mixing ratios yields both E2 and M1 transition rates which can be compared with theory. The acquisition of more experimental information in recent years makes it possible to study the trends of particular classes of M1 and E2 transition rates as functions of Z or N and it is of interest to see how well current theoretical predictions 9,17,1 s) * On leave of absence from the Department of Physics, Royal Institute of Technology, Stockholm 70, and Nobel Institute of Physics, Stockholm 50. 352

353

M 1 A N D Eg. T R A N S I T I O N P R O B A B I L I T I E S

can account for the new experimental results. In this paper we shall be concerned with l forbidden M1 transitions which have or appear to have the shell model description d~ ~ s~ (63 < N < 77) and g~ ~ - d~ (51 < Z < 63). New experimental information is presented for transitions in Te 125, 1127, Xe 129, Cs 133, La 139 and Pr 141 and a summary is given of other pertinent experimental data. 2. Experimental Method The mean lives of the first excited states in X e 129, 1127 and La 139 were determined from measurements of time correlations between conversion electrons and KX-rays. The experimental arrangement used for these measurements is shown schematically F,..~stON ELECT/ 80

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354

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in fig. 1. The transition of interest was selected by a beta-spectrometer which focussed K or L conversion electrons onto a 1 mm thick stilbene scintillator optically coupled to an RCA-7265 photomultiplier. The KX-rays from the source were detected in a 38 mm diam. by 25 mm high cylinder of NE-102 plastic scintillator which was optically coupled to a 56 AVP photomultiplier. A 6 mm thick polythene absorber between the source and the X-ray detector prevented conversion electrons from reaching the detector and also served as a vacuum seal, isolating this detector from the spectrometer vacuum. For the measurement of the mean life of the 80.99 keV state in Cs 133 this arrangement was replaced by one in which the photomultiplier made the vacuum seal, the beta-rays from the Xe 133 decay being detected in a thin (0.3 cm thick) plastic scintillator. Each photomultiplier fed a dual output transistorized limiter circuit, one output of which drove the time-to-pulse-height converter (lower clipping stub) while the other drove the supervisory fast coincidence circuit (upper clipping stub). The limiter circuits were modified versions of those developed by Dr. J. S. Fraser and Mr. R. B. Tomlinson of this laboratory 10). The use of the diode current switch as the time-to-pulse-height conversion element was suggested to us by Professor R. E. Bell and was found to give a more linear relationship between amplitude and time than an earlier version of this timesorter which employed a 6BN6 tube as the conversion element. The timesorter was calibrated in terms of pulse transit time in an air-cored variable delay line. During calibration, the pulse limiters were driven by fast-rising pulses from a mercury relay pulse generator as well as by the photomultiplier pulses. This calibration technique has been described in detail in an earlier publication 11). The spread in length of the electron orbits in the 100 cm radius Chalk River spectrometer 12) was a liability in these short lifetime measurements. The central orbit transit time as a function of energy is shown in the upper right-hand corner of fig. 1. For the best measurement conditions the prompt resolution curve should be as narrow as possible and in most cases the main contribution to that width arises from the spread in lengths of the electron orbits. In order to minimize this width contribution we have used a rather narrow transmission aperture ( ~ 0 . 0 7 ~o resolution setting) which reduced the electron detection efticiency. The relative L subshell conversion line intensities were also measured with the Chalk River ~ / 2 beta-spectrometer. For the Xe 129, 1127 and 112s measurements, a thin-window proportional counter 12) having a plateau of slope ~< 2 ~o per I00 V was used as the electron detector. For the La 139 and Pr 141 studies a 1 mg/cm 2 counter window was used. The Xe sources used in this work were prepared using the Chalk River electromagnetic mass separator of Scandinavian design 13), A1 foil was used as the target (source backing material). The X e 129 and X e 127 s o u r c e s used for the relative L subshell line intensity measurements were prepared from a natural Xe gas sample which had been irradiated in the N R U reactor at a flux of ~2.4 x 1014 n/cm 2. see for a sixweek period. Focussed 40 keV beams of Xe 127 and Xe 129 ions were collected on AI

M1 AND E2 TRANSITION PROBABILITIES

355

target foils of ~ 20 mg/cm 2 which served as the source backing. Ions of this energy penetrate into the backing to a mean depth of ~ 10 #g/cm 2 and a very small source thickness results. The total beam current at the target was ~ 10 #A and the run time 18 h. The Xe 12s source used for the L conversion line intensity measurements in 112s was prepared from Xe which had been irradiated for a much shorter period. The Ce 139 and Ce 141 sources were prepared by the vacuum sublimation method. A1 backings of surface density ,~ 800 #g/era 2 were used.

3. Results of the Lifetime Studies The features of the 1127 level scheme pertinent to the study are given in the inset of fig. 2. The 374.94 and 202.83 keV levels are populated in the K capture decay of Xe 127. In this investigation we measured the time correlations between the 1127

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M1 AND E2 TRANSITION PROBABILITIES

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x 10 - 1 ° sec. The K 1 4 5 . 2 2 - K X-ray time correlation is shown in fig. 2. This time correlation shows, in addition to a p r o m p t component, a component of mean life T = 2.68+0.16 nsec. In transferring the K 2 0 2 . 8 4 - K X-ray " p r o m p t " correlation curve, shown in fig. 3, to fig. 2 (dotted curve) a shift in abscissa was made corresponding to the calculated difference in transit times of the K 202.84 and K 145.22 electrons through the spectrometer (see fig. 1). The 2.68 nsee mean life is associated with radiations which follow the 145.22 keV transition. The L 5 7 . 6 0 - K X-ray time correlation was also measured; it showed the same mean life of 2.68 nsec with somewhat poorer counting statistics than those of the K 145.22- K X-ray correlation shown here and as expected the L 5 7 . 6 0 - K X-ray correlation does not have a p r o m p t component. These measurements confirm that the 57.60 keV transition populates the I ~27 ground state and yield a mean life for the 57.60 keV level of z = 2.68+0.16 nsec. Thieberger 14) has measured the life-time of this level using an oscilloscopic fast-pulsesampling technique and obtained the value • = 2.9+0.3 nsec, in good agreement with the result reported here. Jha and Leonard 1s) report a significantly shorter value for this life-time z = 1.8_+0.3 nsec. The K 1 7 2 . 1 0 - K X-ray time correlation was measured under somewhat poorer experimental conditions. The sides of the observed time correlation on a semilogarithmic plot (not shown) showed no departure from linearity over the two decades covered and their slope indicates that the mean life of the 374.94 keV level is < 6.5x 10 -1° sec. The decay scheme of the 8d isomeric state of Xe 129 is shown in the inset of fig. 4. The time correlation between the L 196.56 conversion electrons and the K X-rays given in the figure indicates a mean life of 1.46+0.06 nsec for the 39.58 keV state. This same mean life was also observed in a measurement of the L 3 9 . 5 8 - e - time correlation in which K and L electrons of the 196.56 keV transition were detected in a plastic scintillator placed behind the source. The present lifetime value is to be compared with the value 1.0+0.4 nsec obtained by Alv~iger, lohansson and Zuk 16) in earlier work. In the course of this investigation we have measured the fl-K 80.99 time correlation in the decay of Xe 133. The mean life obtained for the 80.99 keV state in Cs 13a is = 9.1 + 0 . 4 nsec. We have also measured the K 1 6 5 . 8 4 - K X-ray time correlation for the Ce 139 decay and obtained a mean life • for the 165.84 keV state in La 139 of 2.12_+0.09 nsec.

4. Determination of the Transition Multipolarities

The L conversion lines of the 57.60 keV transition in 1127 are shown in the upper part of fig. 5. The intensities of these L lines are in the ratio ~/Lu/I.qii = 1/(0.118___ 0.004)/(0.0680.004). The measured LI/Lm and ~ / L n line intensity ratios are compared in the inset with those deduced f r o m the theoretical L shell conversion coefficients of Sliv and Band s) for small E2 admixtures in dominantly M1 radiation.

358

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These comparisons indicate that the multipolarity of this transition is (99.3 + 0 . 1 ) % M1 + ( 0 . 7 + 0 . 1 ) ~ o E2. The observed intensity ratios exclude all other multipolarity assignments. 5000

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The L conversion fines of the 39.58 keV transition in Xe lz9 are shown in the lower part of fig. 5. In taking these Xe t29 data, a 1 mm thick copper mask was used to restrict the source to an effective area of 5 mm × 3 cm. The L conversion line intensities

M1 A N D E2 TRANSITION PROBABILITIES

359

found from these data and f r o m data taken with an unmasked source are in the ratio L l / L n / L n ! = 1/(0.100+0.004)/(0.031+__0.003). The measured LI/LIn and L1/Lu line intensity ratios are compared in the inset with those deduced f r o m the theoretical L shell conversion coefficients of Sliv and Band s) for small E2 admixtures in dominantly M1 radiation. These comparisons indicate that the multipolarity of this transition is (99.925+0.025)~o M1 + ( 0 . 0 7 5 + 0 . 0 2 5 ) ~ E2. The observed line intensity ratios again exclude all other multipolarity assignments. The L conversion lines of the 35.48 keV transition in Te 12s were studied at a momentum resolution of ~ 0.2 ~o. The parent 18 h Xe 12s activity in the source had decayed through 30 half-lives at the time of the measurement. The relative L conversion line intensities of the 35.48 keV Te 12s transition were found to be LI/Ln/Lm = 1/0.089_+0.004/0.024+0.002. Comparison of these relative L line intensities with ratios of the theoretical L shell conversion coefficients of Sliv and Band leads to a transition multipolarity of (99.965 + 0.020) ~ M1 + (0.035 +- 0.020) ~ E2. The L conversion lines of the 165.84+__0.03 keV transition in La 139 were studied at a m o m e n t u m resolution of 0.06 ~o. Their intensities were found to be in the ratio LdLH/Lm = 1/0.072-+0.003/0.016+0.001. F r o m comparison of these ratios with ratios of the theoretical L shell conversion coefficients of Sliv and Band the E2 admixture in this dominantly M1 transition is found to be n. . .~+0.3 . 0.2 ~o. The K / ~ L line intensity ratio for this transition was found to be 7.4+0.2 and the K / ~ M + N + O line intensity ratio 28-t-1. The L conversion lines of the 145.43 +__0.02 keV transition in Pr 1.~ were studied at a m o m e n t u m resolution of 0.1 ~o. The relative L conversion line intensities were found to be L x / L l l / L l l I = 1.0/0.081+0.004/0.0172+__0.0025 and indicate an E2 admixture in this dominantly M1 transition of 0.4+0.3 ~o. The relative K/~,L/~,M/ ~ N + O intensities were found to be 1/0.136+_0.005/0.0294-+0.0015/0.0078___0.0006. 5. Discussion

The present investigations deal with the properties of ground state M1 transitions of the l forbidden type in spherical nuclei (Z ~ 50). Such transitions occur between shell model states differing by two units in their orbital angular m o m e n t u m and the transitions violate the orbital selection rule for M1 radiation, AI -- 0. The M1 ytransition probabilities for these transitions are < 0.01 times the single-particle Weisskopf estimate ,. In order to correlate the transition probabilities presented here with similar results in neighbouring nuclei we include summaries of d~ ~ g~ transition properties in tables 1 and 3 and of d~ ~ s~ transition properties in tables 2 and 4. In those cases in which the level lifetime has been measured by more than one group we have arbitrarily used the value with the smallest quoted error in deducing the 7 quantum emission rates 27, listed in the final columns of tables 1 and 2. The total t For the calculations in this work we have used the expressions of Wapstra et al. s), which correspond to a nuclear radius of 1.2 A~ fro.

360

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TAn~ 1 Summary o f the measured mean lives of ¢1~ ~ g~ odd-proton transitions in odd-mass nuclei having Z = 51-63 Nucleus

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rate has been corrected for internal conversion assuming pure M1 character for the transition and using the theoretical K and L shell conversion coefficients of Sliv and Band and an uM+ uN + ~O/C~L ratio of 0.264. The latter is the value measured for the 165.84 keV La x39 transition and has been assumed to apply to all the other transitions. This procedure for deducing 2 r is thought to be accurate to ~< 10 ~o and References for Table

1

L) u~, ~Lx, ~L~x and ~Lxn for pure M1 radiation taken from Sliv and Band. It is further assumed that U(M+N+O) = 0.264 U~L as found experimentally for La x89. b) 2~ = I/~(~T+I ). No allowance is made for any small E2 admixture which may be present. uz is assumed to be exact. c) T. Alvager, B. Johansson and W. Zuk, Ark. Fys. 14 (1958) 373 d) H. Beekhais, Physica 28 (1962) 1199 e) E. Berlovich, Izv. Akad. Nauk SSSK (ser.fiz.) 20 (1956) 1438 t) E. Ye. Berlovich et aL, Nuclear Physics 37 (1962) 469 g) E. Ye. Berlovich et al., Phys. Lett. 2 (1962) 344 h) E. Bodenstedt, H. J. Korner and E. Matthias, Nuclear Physics 11 (1959) 584 l) E. Bodenstedt et al., Z. Phys. 160 (1960) 33 J) E. Bozck et al., Phys. Lett. 6 (1963) 89 k) A. R. Brosi and B. H. KeteUo, Phys. Key. 103 (1956) 917 l) A. R. Brosi ,B. H. Ketelle, H. C. Thomas and R. J. Kerr, Phys. Rev. 113 (1959) 239 m) N. A. Burgov, A. V. Davydov and G. R. Kartashov, JETP 36 (1959) 1946 n) y . y . Chu et al., Phys. Rev. 133 (1964) B1361 o) H. de Waard and T. R. Gerholm, Nuclear Physics 1 (1956) 281 P) H. de Waard, M. H. Garrell and D. Hafemeister, Phys. Lett. 3 (1962) 59 q) M. S. El-Nest and E. Bashandy, Phys. Lett. 2 (1962) 287 r) T. C. Engelder, Phys. Rev. 90 (1953) 259 s) G. T. Ewan (1954), quoted in Nuclear Data Sheets ~) S. Gorodetzky, R. Manquenoville, R. Ricbert and A. C. Knipper in Electromagnetic lifetimes and properties of nuclear states, Publication 974 (National Academy of Sciences-National Research Council, Washington, D.C., 1962) u) I. M. Govil, C. S. Khurana and H. S. Hans, Nuclear Physics 45 (1963) 60 v) R. L. G r a h a m and R. E. Bell, Can. J. Phys. 31 (1953) 377 w) R. L. Graham, J. M. Hollander and P. Kleinheinz, Nuclear Physics 49 (1963) 641 x) K. J. Gromov et al., submission to the 14th annual symposium on nuclear spectroscopy held at Tbilisi, USSR (Feb. 14-22, 1964) Y) R. E. Holland, F. J. Lynch and E. N. Shipley, A.N.L. 6376, (June, 1961) z) D. J. Horen, J. M. Hollander and R. L. Graham, Phys. Rev. 135 (1964) B301 ss) j. Jastrzebski, J. Phys. Rad. 21 (1960) 12 bb) S. Jha and R. Leonard, Phys. Rev. 136 (1964) B1585 oc) N. Kaplan, S. Ofer and H. Zmora, Nuclear Physics 52 (1964) 249 dd) W. H. Kelly, G. B. Beard, W. B. Chaffer and J. M. Gonser, Nuclear Physics 19 (1960) 79 ee) G. Lang and R. Leonard, Bull. Am. Phys. Soc. 8 (1963) 86 tt) p. Lchmann and J. Miller, Compt. Rend. 240 (1955) 1525 gg) F. R. Metzger, J. Franklin Inst. 261 (1956) 219 hh) T. D. Nainan, Phys. Rev. 123 (1961) 1751 it) S. L. Ruby, Y. Hazoni and M. Pasternak, Phys. Rev. 129 (1963) 826 ~J) M. Schmorak, A. C. Li and A. Schwarzschild, Phys. Rev. 130 (1963) 727 kk) V. S. Shirley, W. G. Smith and J. O. Rasmussen, Nuclear Physics 4 (1957) 395 11) A. A. Sorokin, Izv. Akad. Nauk SSSR (ser. fiz.) 23 (1959) 1445; Columbia Techn. Transl. (1960) p. 1434 m~) D. Stromingcr, UCRL-3374 (1956) nn) p. Thieberger, Ark. Fys. 22 (1962) 127

362

J. $. OEmER et al.

TABI~ 2 Summary of the measured mean lives of d t ~ st odd-neutron transitions in odd-mass nuclei having N = 63-77, Z = 48-54

Nucleus

Er (keY)

,sCd~ 11

342

~ --> ½

< 29 0.039 e)

g) 5)

0.018 e)

2.5 x 10+x°

4aCd.X~a

300

~t "-->½

0.046 e)

1)

0.026 o)

2.1 × 10 +1°

6oSn~'

161

m)

0.15

[1.9 -4-0.2] × 10 +9

k)

4.10

[7.4 =1=0.4]× 10 +s

soSn~ °

23.83

Spin sequence

~ --. ½ t --~ ½

Mean life (nsec)

0.45-4-0.04 26.7 + 1 . 4

R.ef.

~ r s)

2y b) (see =1)

s2Te~ 1

212.2

(~t --~ ½)

< 7 0.089+0.022

f) m)

0.08

[1.044-0.25] X 10 +1°

e,Te:: 3

159

~z --~ ½

< 1.4 0.274-0.04 0.274-0.03

t) h) m)

0.19

3.2 4-0.3 X 10+9

< 3

5,Te~] 5

35.48

~ --.'- ½

6.Xe~ 9

39.58

~ --> ½

54Xe~ 1

80.164

½~ ~

l)

2.28_L.0.22

h)

13.2

[3.1 4-0.31× 10 +7

1.0 -4-0.4 1.46-+-0.06

a) pres. work

11.8

[5.4 4-0.2] × 10 +T

< 6

0.694-0.29 0.424-0.09

n)

e)

l)

1.57

[9.3 4-2.0] × 10 +a

a) ~K, 0eLI, 0~LIxand 0~Lm for pure MI radiation taken from Sliv and Band. It is further assumed that ~(M+N+O)= 0.264 0Cy~Las found experimentally for LalSL b) ~t~ = 1/T(~rq- 1). No allowance is made for any small E2 admixture which may be present ~T is assumed to be exact. e) Mean life deduced from measured B(E2) and mixing ratio. ~ r value used by McGowan et al. in their analysis. a) T. Alvager, B. Johansson and W. Zuk, Ark. Fys. 14 (1958) 373 e) R . E . Bell, R. L. Graham and H. E. Petch, Can. J. Phys. 30 (1952) 35 t) M. Deutsch and W. E. Wright, Phys. Rev. 77 (1950) 139 s) D. Engelkemeir, Phys. Rev. 82 (1951) 552 h) R . L . G r a h a m and R. E. Bell, Can. J. Phys. 31 (1953) 377 l) F . K . McGowan, Phys. Rev. 85 (1952) 142 1) F . K . McGowan and P. H. Stelson, Phys. Rev. 109 (1958) 901 k) j . L . Olsen, L. G. Mann and M. Lindner, Phys. Kev. 106 (1957) 985 1) J. Samueli and A. Sarazin, J. Phys. Rad. 21 (1960) 390 m) M. Schmorak, A. C. Li and A. Schwarzschild, Phys. Rev. 130 (1963) 727 n) W. E. Wright and M. Deutsch, Phys. Rev. 82 (1951) 277

p r e f e r a b l e t o t h e u s e o f e x p e r i m e n t a l CtK v a l u e s , f e w o f w h i c h a r e m e a s u r e d w i t h t h i s a c c u r a c y . T h e E 2 t r a n s i t i o n p r o p e r t i e s a r e s u m m a r i z e d i n t a b l e s 3 a n d 4. T o s i m p l i f y comparison of the E2 transition rates with Sorensen's theoretical predictions we have t a b u l a t e d v a l u e s o f B ( E 2 ) / ( 2 I e + 1), w h e r e If is t h e f i n a l s p i n , r a t h e r t h a n v a l u e s o f the quantum

e m i s s i o n r a t e 2~(E2). T h e e x p r e s s i o n r e l a t i n g t h e s e q u a n t i t i e s is t h e

following: 2~(E2) = 1.21 x 1 0 + 6 1 E 5 B ( E 2 ) ,

bll AND I/9.TRANSITION PROBABILITIES TABLI3 3 Comparison of deduced and theoretical B(E2) values for d i ~ gl odd-proton transitions in odd having Z ~ 51-63. Nucleus

51Sb~[a

E¢

Spin

E2 admixture

(keV)

sequence

(~)

160

~ --->~

Coulomb ex. +1.5 ad

Ref.

l) ~)

--0.8

Coulomb ex.

o)

Expt.

Sorensen ~)

0.08t0.02 0 12 -kO'17 " --0.10 0.057

0.146

1 ~: 1 1.2 ~=0.1

0.156

6aIl~7

57.60

½ "-> i

0.6-4-0.6 0•7=[=0.1

ac Lss

~Cs]~ I

78.69

~ --->~

0.5-4-0.4

Lss

P)

0.06-4-0.05

0.174

6~Cs]ss3

80.99

~ --->~

Coulomb ex. 2.5-4-0.3 ac 1.9-4-0.2 ac 2.64-0.2 Lss 3.5-4-0.5 ac 2.6-4-0.2 Lss

1) g) 1) h) e) u)

0.4 =1=0.1 0.31 ~=0.04 0.23 =k:O.02 0.32±0.02 0.43±0.06 0.32±0.02

0.097

6~Lasl~°

165.84

~ --->~

Coulomb ex. ~4 adt

n) b)

6.Pr~ 1 )

145.43

~- --> ~

0 4 ~-0"4 " --0.3

adt

0 2 +0.3 • --0.2

Lss

Coulomb ex. Coulomb ex. Coulomb ex. 0 6+0.4 "

q) present work

B(E2) (2It-b 1-------~in units o f 10-6° e:

r) present work n) c) t)

adt

~)

--0.2

0.4±0.3

Lss

present work

6sPry,.8

57.37

~ --> ~

~0.3

Lss

m)

+~PmsX~7

91.06

½ --> ~

3 -{-6 --3 6 ±2 ~3 0.8±0.2 1.64-0.4

adt

f)

ac adt Lss adt

~0.2

Lss

+aEua~s6~

21.7

~ --* j

<0.17 ~0.12

0.0014

0 012 "{-0"012 " --0.009 0 006 +0.009 " --0.006 <0.13 0.045 <0.055

0.0008

0 033 +0.022 " --0.011 0.022-{-0.017 ~0.11

0.0098 0.090

s) b) k) v)

0.6 -{-1.2 --0.6 1.2 ± 0 . 4 ~0.6 0.15-4-0.04 0.31 ± 0 . 0 8

a)

~4.4

a d - angular distribution following Coulomb excitation. ac - Y-7 angular correlation• Lss - L subshell line intensity ratios. adt - angular distributions f r o m aligned nuclei. t j. F. Schooloy, D. D. Hoppes and A. T. Hirshfeld, J. o f Research N.B.S. 66A (1962) 317 : admixture o f (0.46J:0.11) %

364

J.s. GI~IGBRet al.

where 2 is in see - x i f E is expressed in M e V a n d B(E2) in units o f e2cm 4. A r i m a , H o r i e a n d S a n o 17) were the first to include configuration mixing in calculating the M1 t r a n s i t i o n p r o b a b i l i t i e s with the single-particle spherical shell m o d e l . T h e i r t r e a t m e n t s h o w e d c o n s i d e r a b l e success in a c c o u n t i n g for the o b s e r v e d M I t r a n s i t i o n p r o b a b i l i t i e s as well as for g r o u n d state m a g n e t i c a n d electric q u a d r u p o l e m o m e n t s . M o r e recently Berlovich a n d B u k a t 1 s) have carried t h r o u g h this t y p e o f c a l c u l a t i o n f o r n e a r l y all the cases o f interest to this p a p e r . T h e i r p r e d i c t i o n s a r e c o m p a r e d with the e x p e r i m e n t a l results in figs. 7 a n d 9 a n d show quite g o o d agreement. Sorensen has e v a l u a t e d b o t h the M1 a n d E2 t r a n s i t i o n rates 9) using the KisslingerSorensen wave functions calculated f o r a p a i r i n g - p l u s q u a d r u p o l e force. T h e E2 t r a n s i t i o n rates he o b t a i n s s h o w general a g r e e m e n t with experiment. The M1 t r a n sition rates, however s h o w m u c h p o o r e r a g r e e m e n t t h a n those calculated in the m a n n e r o f A r i m a e t al. T h e e x p e r i m e n t a l E2 e n h a n c e m e n t factors for the d÷ ~ - g t o d d - p r o t o n t r a n s i t i o n s are c o m p a r e d with Sorensen's p r e d i c t i o n s in fig. 6. It is a p p a r e n t f r o m this p l o t t h a t Sorensen's p r e d i c t i o n s r e p r o d u c e qualitatively the v a r i a t i o n in the e x p e r i m e n t a l E2 e n h a n c e m e n t factors with n e u t r o n n u m b e r . H o w e v e r , q u a n t i t a t i v e l y there are signi-

References for Table 3

4) D. G. Alkazov et aL, submission to the 14th annual symposium on nuclear spectroscopy held at Tbilisi, USSR (Feb. 14-22, 1964) b) E. Ambler, R. P. Hudson and G. M. Temmer, Phys. Rev. 97 (1955) 1212, 101 (1956) 196 e) D. S. Andreev, A. P. Grinberg, K. I. Erokhina and I. Kh. Lemberg, Izv. Akad. Nauk. SSSR, (ser. fiz.) 25 (1961) 70 d) N. M. Anton'eva, A. A. Bashilov, B. S. Dzhelepov and B. K. Pr¢obrazhenskii, Izv. Akad. Nauk SSSR (set. fiz.) 22 (1958) 135 e) A. P. Arya, Phys. Rev. 122 (1961) 549 ~) G. R. Bishop et aL, Phil. Mag. 2 (1957) 534 ~) E. Bodenstedt, H. J. Korner and E. Matthias, Nuclear Physics 11 (1959) 584 h) F. Brown, R. L. Graham, G. T. Ewan and J. Uhler, Can. J. Phys. 39 (1961) 779 l) C. F. M. Cache et aL, Phil. Mag. 46 (1955) 1287 J) F. M. Clikeman and M. G. Stewart, Phys. Rev. 117 (1960) 1052 k) G. T. Ewan and R. L. Graham, private communication; in Alpha, beta and gamma ray spectroscopy, ed. by K. Siegbahn (North-Holland Publ. Co., Amsterdam, 1965) 1) L. W. Fagg, Phys. Rev. 109 (1958) 100 in) R. L. Graham, J. M. Hollander and P. Kleinheinz, Nuclear Physics 49 (1963) 641 a) N. P. Heydenburg and G. M. Temmer, as quoted by Alder et aL, Revs. Mod. Phys. 28 (1956) 432 o) R. E. Holland, F. J. Lynch and E. N. Shipley, ANL 6376 (June 1961) P) D. J. Horen, J. M. Hollander and R. L. Graham, Phys. Rev. 135 (1964) B301 q) S. Jha and g. Leonard, Phys. gev. 136 (1964) B1585 r) A. Knippcr, Ann. de Phys. 6 (1961) 211 8) T. Lindqvist and E. Karlsson, Ark. Fys. 12 (1957) 519 t) R. C. Ritter, P. H. Stelson, F. K. McGowan and R. L. Robinson, Phys. Rev. 128 (1962) 2320 u) K. Siegbahn et aL, Nucl. Instr. 27 (1964) 173 v) G. A. Westenbargor and D. A. Shirley, Phys. gev. 123 (1961) 1812

Mt AND E2 TRANSITION PROBABILITIES

365

TABLE 4 Comparison of deduced and theoretical B(E2) values for d t ~ st odd neutron transitions in odd-mass nuclei having N = 63-77, Z = 48-54 Nucleus

Er (keV)

Spin sequence

E2 admixture (%)

B(E2) in units of 10 -+° e z cm 4 (2It+ 1) Expt. Sorensen ~) Single particle -

Ref.

,sCd~ t

342

~ -+ ½

13.2

4-1.2

ad

g)

2.7 4-0.2

2.09

0.16

,aCd6X~a

300

~ --~ ½

7.7

4-0.5

ad

g)

2.7 4-0.2

2.02

0.16

5oSn~7

161

t --~ ½

0.15 +0.16 --0.10

ac

r)

0.134

0.17

b)

0.1 +0.1 --0.06 ~ 0.07

Lss

i)

<8

0.0001

0.17

ac Lss

e) c)

5.4 4-1.4 4.6 4-1.3

0.174

0.18

o) e) e) 4)

0.454-0.15 1.6 4-0.2 0.854-0.20 0 5 +0.7

0.86

0.18

ae LSS ad

Coulomb ex. 60Sn~9

23.83

~ --~ ½ < 0.2

6tTe~ t

212.2

({ ~ ½)

6~Te~ a

159.0

~~ ½

5.6 4.8

4-0.5 4-0.3

Coulomb ex. 1.3 4-0.1 0.67 4-0.11 0 4 +0.6 "

--0.3

"

--0.4

52Te~6

35.48

~t "-~ ½

0.0354-0.020

Lss

present work

56Xe~9 5~Xe~7sz

39.58 80.164

{r ~ ½ 0.0754-0.025 ½--* {t < 3

Lss Lss

present work 1.7 4-0.6 h) <18

0.8 4-0.5

1.4

0.19

1.87 1.58

0.20 0.20

ad - angular distribution following Coulomb excitation. ae - ~'-7 angular correlation. Lss - L subshell line intensity ratios. adt - angular distributions from aligned nuclei. ~) D. G. Alkazov et aL, submission to the 14th annual symposium on nuclear spectroscopy held at Tbilisi, USSR (Feb. 14-22, 1964) b) D. S. Andreev et al., Izv. Akad. Nauk. SSSR (ser.fiz.) 25 (1961) 832 c) y . y . C h u e t a L , Phys. Rev. 133 (1964) B1361 a) L. W. Fagg et aL, Phys. Rev. 100 (1955) 1299 e) N. Gold.berg and S. Frankel, Phys. Rev. 100 (1955) 1350 ~) R. K. Golden and S. Frankel, Phys. Rev. 102 (1956) 1053 8) F. K. McGowan and P. H. Stelson, Phys. Rev. 109 (1958) 901 n) j. W. Mihelich, Phys. Rev. 87 (1952) 646 l) j. L. Olsen, L. G. M a n n and M. Lindner, Phys. Rev. 106 (1957) 985

ficant discrepancies which are well outside the limits of experimental error, particu l a r l y f o r t h e 1127 a n d C s 133 t r a n s i t i o n s . T h e M 1 h i n d r a n c e f a c t o r s f o r t h e s e s a m e t r a n s i t i o n s a r e c o m p a r e d w i t h t h e p r e d i c t e d v a l u e s in fig. 7. I n t h i s c a s e t h e r e is o r d e r o f m a g n i t u d e a g r e e m e n t o n l y f o r n u c l e i h a v i n g N < ~ 82. I n fig. 8 t h e E 2 e n h a n c e m e n t

f a c t o r s f o r t h e d+ ~ - s½ o d d n e u t r o n t r a n s i t i o n s a r e

compared with Sorensen's calculations. These predictions indicate a marked reduction i n t h e e n h a n c e m e n t f a c t o r s f o r 5oSn 119 a n d 52Te t21. T h e 23.83 k e Y t r a n s i t i o n i n

366

J . s . G£IGER et aL i

i

I

I

i

1

i

l

i

l

127

531 u. I00 53C S 153

Z

S

Z W

b

ID 147 61' m

Pr+4+ I

~

+

I0

`cS

"T

+

Pr'

@

=SOR[NSEN PREDICTIONS

\

?

00 0.1

68

I 70

I 72

1 74

I 76

I 78

Blo

/~Pr

I 82

814

8

I6

I 8S

N

Fig. 6. Enhancement factors for the E2 components in g~ <~ d4 odd-proton transitions. The experimental points are taken from the data in table 3. Sorensen's predictions i ) a r e based on wave functions

derived from the pairing-plus-quadrupole force model for atomic nuclei.

"OFF SCALE " ~sCs

o

i

1000 --

°z

I00

;s

u.

St)

++. -'..

i,Li

,+, ."'+"+Sh S

Cs

.,..

~

~

/

/ 137 L ~.- t+T ~

I3S'~,133

~

<. . . .

.¢+-. /

~-:-'~9 / • ++

]

~x

~ ~

,,'+, ,z-. ,47 "-...

x~

,/.o / /

t.

-,- ; 7

x ~.

EXPERIMENT

I0 - ---×--

I

I 68

SORENSEN'$ PREDICTIONS, PAIRING ~ QUADRUPOLE FORCE SHELL MODEL WITH CONFIGURATION MIXING

I 70

I 72

I 74

I 76

I 78

[ 80

I 82

t 84

I 86

[ 88

N

Fig. 7. Hindrance factors for g~ ~ dt M1 transitions. The experimental data are taken from table 1. Sorensen's predictions 0) are given by the s o l d lines.

M1 A N D

E2 T R A N S I T I O N

367

PROBABILITIES

50Sn 119 should provide the most crucial test of this feature but as yet there is only an upper limit of 0.2 % on its E2 admixture based on the h / ( L n + h n ) > 4 intensity ratio measurement of Olsen et al. ~9). While one can hope for an order of magnitude l

I

I

I

1

I

I

I

o

~Xe

I00

Cd|,|

3°Sn119 T .~=Te |2(

Cd '13

~

,0

I

3ETe'23

,.ca ~

131

~(cl

~Xe'29

~Xe

~-'~--~×~

IL7

=:u;

3°Sn

3

7

?

1.0 i.a

N w

O'J i-

P

I 65

I 65

1 67

~

~ 69

I

~ [ 71

@ L SORENSEN'S " ° - L PSOEDICT|ONS

[ 75

I 75

I 77

79

N

Fig. 8. Enhancement factors for the E2 component in ¢1~ ~ s} odd-neutron transitions.

o "OFF SCALE"

I000

4e Cd , 3oSr~, 32Te | SORENSEN'S PREDICTIONS

$ z Z

×

,<

Cd . . . . . i 43 o 48cdll3 o

3~T~;2~-

,,,

Sn ~

~ Tel23 3°SxnH o ~Zx 0 121 3aTe

I

I Xe

×

.... x ~

x ""

¢

o 129 { ~Xe 131 34 xe

EXPER~MENT, CdIII ond Cd"s CORRECTEDFOR E2 AOMIXTURE X SHELL MODEL WITH CONFIGURATIONMIXING i / I I I 69 71 73 715 6'3 65 67 N

I

77

Fig. 9. Hindrance factors for d} ~ s½ MI transitions.

368

J. $. GEIGERe t al.

i m p r o v e m e n t in setting a limit on E2 a d m i x t u r e f r o m a m o r e a c c u r a t e m e a s u r e m e n t o f L subsheU intensity r a t i o s it seems unlikely t h a t this limit c o u l d be l o w e r e d b y the f a c t o r ~ 104 suggested b y the predictions. W e n o t e also t h a t the E2 e n h a n c e m e n t f a c t o r for the 212.2 k e V t r a n s i t i o n in 52Te T M is 30 times higher t h a n the p r e d i c t i o n a n d in fact is the largest in this class. This l a c k o f a g r e e m e n t between e x p e r i m e n t a n d p r e d i c t i o n is even m o r e p r o n o u n c e d for the M1 h i n d r a n c e factors s h o w n in fig. 9. T h e e x p e r i m e n t a l M1 h i n d r a n c e factors f o r this class are surprisingly c o n s t a n t , s h o w i n g only a slight decrease with increasing n e u t r o n n u m b e r . W e c o n c l u d e t h a t while Sorensen's t r e a t m e n t o f the g~ --, d~ a n d d~ ~ s~ t r a n s i t i o n rates using the wave functions o f the p a i r i n g - p l u s - q u a d r u p o l e force m o d e l p r o v i d e s a fair description o f the E2 t r a n s i t i o n rates it fails to p r o v i d e a n a d e q u a t e d e s c r i p t i o n o f the M1 t r a n s i t i o n rate for these classes o f transitions.

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

R. E. Bell, R. L. Graham and H. E. Petch, Can. J. Phys. 30 (1952) 35 R. E. Bell and R. L. Graham, Phys. Rev. 78 (1950) 490 R. L. Graham and R. E. Bell, Phys. Rev. 84 (1951) 380 R. L. Graham and R. E. Bell, Can. J. Phys. 31 (1953) 377 V. F. Weisskopf, Phys. Rev. 83 (1951) 1073; A. H. Wapstra, G. J. Nijgh and R. van Lieshout, Nuclear Spectroscopy tables (North-Holland Publ. Co., Amsterdam 1959) H. de Waard and T. R. Gerholm, Nuclear Physics 1 (1956) 281 M. E. Rose, Internal conversion coefficients (North-Holland Publ. Co., Amsterdam, 1958) L. A. Sliv and I. M. Band, Tables of internal conversion coefficients of gamma rays; I. K-shell, II. L-shell (Academy of Sciences Press, USSR, Moscow-l_.vningrad, 1956, 1958) R. A. Sorensen, Phys. Rev. 132 (1963) 2270, 133 (1964) B281 J. S. Fraser and R. B. Tomlinson, private communication R. L. Graham, J. S. Geiger, R. E. Bell and R. Barton, Nucl. Instr. 15 (1962) 40 R. L. Graham, G. T. Ewan andJ. S. Geiger, Nucl. Instr. 9 (1960) 245 T. Alv~ger and J. Uhler, Ark. Fys. 13 (1953) 145 P. Thieberger, Ark. Fys. 22 (1962) 127 T. Alv~ger, B. Johansson and W. Zuk, Ark. Fys. 14 (1958) 373 A. Arima, H. Horie and M. Sano, Progr. Theor. Phys. 17 (1957) 567 E. Ye. Berlovich and G. M. Bukat, Izv. Akad. Nauk SSSR 28 (1964) 214 J. L. Olsen, L. G. Mann and M. Lindner, Phys. Rev. 106 (1957) 985