- Email: [email protected]
g-factors for 2+ states of doubly even nuclei (Ge, Se, Mo, Ru, Pd, Cd and Te)
1.E.3
J
Nuclear Physics A133 (1969) 310--320;(~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
o-FACTORS FOR 2 + STATES OF DOUBLY EVEN NUCLEI (Ge, Se, Mo, Ru, Pd, Cd AND Te) G. M. HEESTAND, R. R. BORCHERS and B. H E R S K I N D + Unirersity of Wisconsin, Madison, 14"isconsin tt and L. G R O D Z I N S and R. KALISH Laboratory for Nuclear Science, Physics Department Massachusetts Institate of Technology, Cambridge, Agassachusetts and D. E. M U R N I C K Massachusetts Institute of Technology, Cambridge, Massachusetts and Bell Telephone Laboratories, Inc., Murray Hill, New Jersey Received 1 April 1969
Abstract: The gyromagnetic ratio, .q, for the first excited 2 + states of 25 even-even isotopes of Ge, Se, Mo, Cd, Pd, Ru and Te were obtained using the ion implantation perturbed angular correlation technique. The .q factors were extracted from the measured precession angles using known lifetimes, equilibrium hyperfine magnetic field values and transient hyperfine magnetic field data for fast ions in ferromagnetic lattices. The results indicate that the ,q factors vary little for the states studied, and are close to the collective value Z/A. E
N U C L E A R MOMENTS (Static) 70, 72, 7~, V6Ge' 76, 78 80, 82Se ' 98, 100Mo ' 9s, lO~Ru ' 106, t08, UOpd ' ~0,~12, 1 1 4 , 1 1 6 C d ' and ~zo, 122,124., 126, 12s, t30Te; measured 77(0 ' A); deduced ~, for first 2 + states.
1.
Introduction
E x c i t e d s t a t e p r o p e r t i e s o f d o u b l y e v e n n u c l e i i n t h e r e g i o n 70 =< A =< 130 h a v e b e e n t h e t o p i c o f a n u m b e r o f r e c e n t e x p e r i m e n t a l a n d t h e o r e t i c a l ~' 2) i n v e s t i g a t i o n s . Specifically, c a l c u l a t i o n s u t i l i z i n g p a i r i n g p l u s q u a d r u p o l e f o r c e s h a v e b e e n s u c c e s s f u l in p r e d i c t i n g t r a n s i t i o n p r o b a b i l i t i e s f o r t h e s e " o n e - p h o n o n " The recent measurements
of non-zero electric quadrupole
first e x c i t e d s t a t e s 1). moments
for several
first 2 + s t a t e s 3), w h i c h t h e a b o v e t h e o r i e s p r e d i c t t o b e z e r o , h a s s t i m u l a t e d f u r t h e r theoretical investigations.
These
same calculations have also predicted
magnetic
m o m e n t s /~ = :tNg~I b u t u n t i l n o w , e x p e r i m e n t a l l y m e a s u r e d d i p o l e m o m e n t s b e e n r e l a t i v e l y f e w in n u m b e r . Excited state dipole moments
have
are usually measured either by perturbed angular
c o r r e l a t i o n s u s i n g e x t e r n a l o r i n t e r n a l fields o r b y t h e M 6 s s b a u e r effect. D o u b l y e v e n n u c l e i b e t w e e n A = 70 a n d A = 130 h a v e s h o r t l i v e d 2 + s t a t e s ( r ~ 10 p s e c ) o f f a i r l y t Present address: Niels Bohr Institute, Copenhagen, Denmark. tt Work supported in part by the U.S. Atomic Energy Commission. 310
g-FACTORS
311
high energy (E ,,~ 500 keV). Consequently, magnetic perturbations, even in internal fields of 500 kOe, are small with typical precession angles on the order of 15 mrad. The high gamma energies yield an almost zero recoiless fraction making M6ssbauer measurements impossible. What few data are available come from radioactive nuclei metallurgically doped into polarized Fe. Here the gamma-gamma correlation of the daughter nucleus is precessed by the large hyperfine magnetic field acting on dilute impurities in Fe. Measuring the precession allows the determination of the magnetic moments of the intermediate level in the gamma-gamma cascade 4). If Coulomb excitation is used to produce the states of interest and the decay gamma rays are detected in coincidence with backscattered ions, then a number of improvements in the sensitivity of the perturbed angular correlation technique can be realized. The angular distribution becomes very anisotropic while the recoil nucleus can be driven into a ferromagnetic backing by its recoil energy. In addition, there is no dependence on radioactive cascades to populate the states, so systematic investigations become easier. Although experimental evidence indicates that the recoil ions experience substantial perturbations while slowing down 5), it now seems that the origin of these perturbations is understood as the interaction of fast recoiling ions with polarized electrons in the ferromagnetic backing. The perturbations can be simulated by a large field ( ~ 10 MOe) acting on the nucleus during its slowing down time ( ~ lpsec). This field seems to vary slowly with Z and be the same for all isotopes of a given element. These empirical facts about the perturbation can be included in the data analysis to obtain the dipole moments of the excited states. This paper presents such measurements for 25 doubly even nuclei. The g-factors determined are compared with various theories but are best described by a simple hydrodynamical model of the nucleus 6).
2. Experimental The ion implantation perturbed angular correlation technique, IMPAC, has been described in detail in previous publications 7-1o). As is indicated in fig. 1, a high-energy ( 3 3 - 3 8 MeV) oxygen beam is used to Coulomb excite the target nuclei and concurrently implant them, by recoil into a suitable host lattice. Using NaI(T1) crystals, the decay 7-rays are counted in coincidence with oxygen ions backscattered into an annular detector. The coincidence requirement insures that only transitions from the m = 0 substate are observed, giving rise to the highly anisotropic E2(m = 0 ~ m = 0) radiation pattern. The forward momentum transferred to the excited nuclei by the backscattered particles is sufficient for implantation into the ferromagnetic backing. All targets used in these experiments were separated isotopes electroplated or evaporated onto 25 pm thick iron foils. Target thicknesses varied from 200/~g/cm 2 to 400/~g/cm 2. The recoil ranges and slowing down times calculated from the theory
312
o.M. HEESTAND e l at.
of Lindhard, Scharff and Schiott 11) indicate an average implantation depth greater than 500 pg/cm z, and a stopping time of about 1 psec. An external magnetic field of about 1 kOe perpendicular to the detection plane was used to magnetically saturate the iron foil and hence align the hyperfine fields acting on the impurity nuclei. This effective magnetic field (Herr) interacts with the magnetic moment of the excited state and causes it to precess with the Larmor frequency co(t) = -gpNH(t)/h. The angular distribution of the decay gamma rays at a time t after the excitation is given by: W(O, t) = W[O-.~co(t')dt']. For an integral
. ,moveable,~_
/
\
\ \ ?',3 Y2
/
]
Fig. 1. Schematic o f experimental apparatus, top view. The polarizing electromagnet is not shown.
precession measurement, in which the coincidence resolving time is much longer than the nuclear lifetime z, the integrated angular distribution weighted by the exponential decay of the excited state is observed. In this case:
w(o) = 2 foe-',w lo- flco(,')dC at.
(1)
The angular correlation, W(O), is usually written
W(O) =
Z
A.P. (cosO),
n=O, 2, 4
where the coefficients A, are determined by the initial and final state spin and modified by the finite solid angles of the counters. The P,, are the Legendre polynomials. The typical angular distributions measured with polarizing fields " u p " and " d o w n " are shown in fig. 2; the example is that of the 434 keV state of 1ospd" The solid and
y-FACTORS
313
dashed lines are the best fits found by varying A2, A4, and the angular shift. The coefficients A 2 and ,44 are found to be in agreement with the angular correlation coefficients calculated from Coulomb excitation theory; there is no evidence for nonmagnetic perturbations. Small shifts in the angular correlation pattern are measured by monitoring the count rate at angles of maximum slope with frequent reversals of the magnetic field. ,40 is determined from the ratio: R = W(Ot)W(Ot)+ ......... ~- ....
"/
W(O~) W(O+)'
(+)
1°epd
(-)
.~.. ,,~
1500
,
~
/
',
on Fe backing
best fit: Az = 0.615 A4 : - [ 210
,,
oJr
= 0.014
~
1000
500
I
I
-120 -I00
I
-80
I
-60
I
-40
I
I
I
-20 0 20 8 (degrees)
I
I
/
I
40
60
80
I00
120
Fig. 2. Angular correlation for 1°SPd implanted into iron. The angular shift is used to determine the 2 + state magnetic moment. (+) and (--) refer to the external field direction. where the arrows refer to the external field directions. Each angle is treated independently, with theoretical A2 and A4 values used to determine AO. Four counters are placed symmetrically (usually __+22½° and ___112k °) so that two sets of equal and opposite effects are measured simultaneously, cancelling out asymmetries. Precession angles must be corrected for beam bending by the fringing field of the polarizing magnet which shows up as an apparent precession of the correlation. Such effects were measured using copper backed targets and also calculated from the measured fringing field. The correlations ranged from 2.0 to 2.5 mrad, depending on whether the incoming oxygen beam was in a 5 + and 6 + charge state. Precession angles corrected for beam bending are shown in table 1. Systematics of the lzresent data indicate that the hyperfine field seen by the implanted ion is considerably larger during slow down than after the ion has come to rest. The field occurring during slow down is referred to as the transient field and has
G.M. HEESTANDeta[.
314
TABLE 1 Summary of e×perimental data for states studied, including radioactivity data where available Isotope
Energy of 2 + (keV)
Meanlife (psec)
"°Ge "2Ge WGe ~°Ge v~'Se VSSe S°Se SaSe "aMo ~°°Mo 9SRu l°°Ru 1°2Rt.l t°4Ru ~°~Pd ~°6Pd ~°SPd ~°Pd J~°Cd ~t-'Cd ll~Cd ~°Cd tZ°Te ~22Te
1040 885 596 563 559 614 666 655 787 536 654 540 473 358 554 512 434 374 658 617 559 513 560 564
1.92--0.20 4.54~0.50 17.2 :1.4 25.2 : 2.6 16.0 ~ 1.5 12.4 ~:l.2 11.6 = 1.2 16.3 =1.6 5.1 ~ 0.5 14.9 +:1.5 8.5 " 0 . 6 ~) 17.2 ~1.2~) 25.4 _:1.8 ~) 83.5 :=6.0 ") 14.0 :=1.0 18.4 ~: 1.3 34.4 _:~2.4 66.0 =_4.0 7.2 :_0.761 8.9 :!-0.7 b) 13.0 : 1.1 b) 19.8 =:1.5 ~) 13.4 ~_2.6 II.0 q l.l
~2'*Te
603
9.5
126Te 128Tc t3°Te
667 743 840
H=
(o)z)¢xp
(kG)
(mrad)
+70~3
glMPAC
--11.2--1.5 --9.9±1.7 --11.3=1.3 --9.9~_1.5 --27.1~1.5 --24.0~ 1.6 --23.9~1.l --29.5--0.8 --12.1±3.0 --8.0-1.0 5.0" 4.0 3.3~3.2 2.8--4.4 47.8±4.4 --0.8~1,4 4.5--2.6 17.5:= 1,6 35.0-L2.3 --10.7~3.8 --7.3--0.9 5.2--2.5 --4.0--l.9 --15.673.6 - 15.8:!:1.2
0.29±0.17 0.30~0.04 0.25-3_0.03 0.39-~0.15 0.30~0.06 0.32~0.13 0.7l~0.38 0.21~-0.06 0.24-~0.06
=0.5~)
- 1 2 . 9 " 0.9
0.21±0.05
6.37~0.7 a) 4 . 5 9 / 0 . 6 d) 2.9 -~-0.3
- 13.l ±2.5 --10.1 ~2.0 -- 10.6:±2.2
0.25±0.07 0.21 ~0.06 0.25±0.07
+650±150
--256±5 -~505±15
--595±12
-348±10
i 620±20
0.59±0.29 0.50i0.25 0.46:{:0.23 0.37:t:0.18 0.404-0.11 0.4l:::0.11 0.42±0.12 0.43 ~:0.12 0.34=_0.18 0.34-~0.18 0.30:'0.17
gRAD
0.40==0.12 ¢)
0.42~0,03r) 0.34~-0,06 f) 0.29:L0.04 0.34:!-0.01 r)
0.27~ 0.09 g) 0.44j 0.06hi
0.395:0.03 ~1 0.46_'0.05 ~) 0.34±0.07 ~) 0.31 =_0.03 ~) 0.35~-0.05~1 0.21 ~_0.05 ~) 0.275 0.13 ")
a) b) ~) d) e) g)
E. K. McGowan, R. L. Robinson, P. H. Stelson, and W. T. Milner, Nucl. Phys. to be published. E. K. McGowan, R. L. Robinson, P. H. Stelson, and J. L. C. Ford, Nucl. Phys., 66 (1965) 97. M. Schumacher, Phys. Rev. 171 (1968) 1279. R. G. Stokstad and I. Hall, Nucl. Phys. A99 (1967) 507. See ref. is). f) See ref. 24). L. Keszthelyi, I. Demeter, I. Dezsi, and L. Varga in Hypcrfine structure and nuclear radiation, ibM., p. 155. h) See ref. 22). i) See ref. ~s). J) See ref. 1c'). k) See ref. 17). I1 See ref. 19). m) See ref. 2o). b e e n d i s c u s s e d in a p r e v i o u s p u b l i c a t i o n 5). L i n d h a r d a n d W i n t h e r 12) a t t r i b u t e t h e t r a n s i e n t field t o a n a m p l i f i c a t i o n o f t h e p o l a r i z e d e l e c t r o n d e n s i t y a t t h e n u c l e u s b y electron scattering d u r i n g the slowing d o w n o f the impurity ion. The time d e p e n d -
g-FACTORS
315
ence of the transient field is a function of the slowing down process. The slowing down times however are short compared to the lifetimes of interest allowing eq. (1) to be integrated to first order with the result that: or = o)0T+om,
(3)
o% is the Larmor frequency associated with the static field i.e., the field the impurity sees after it comes to rest. o91r~ represents the impulse precession due to the transient field. 3. Analysis Fig. 3 shows a plot of the experimentally determined shifts (o2r)exp versus the meanlife of the first 2 + state; values for a given element being grouped together. The line shown through the data is a least squares fit of eq. (3). The slope of the line depends on the average 9-factor of the states and the static internal field. The intercepts of the line depend only on the transient field. There are / Pd in Fe
40
Ge rrl Fe z
50
,o
/
/
20
;j -20 10
©
30
Te in Fe
-I0
5C
Ru in Fe
2C
//p/
/
o
i
l
Cd in Fe
5
0
20
20
0
5
rO
15
~ 20 5 •r (psec)-- --
I0
15
Mo in Fe
IO
L 15
20
Fig. 3. S u m m a r y of experimental precession angles for 2 + states of doubly even nuclei in Fe (and Co and N i for Se) plotted as a function of nuclear lifetime, 7.
216
G. M. HEESTAND e l ,:ll.
two cases where the transient field effects can be isolated: (i) If the precession of the same state has been measured by radioactivity methods, which are essentially recoilless, then ( ( . O ~ ' ) e x p - - ( ( D r ) r a d = ( D 1 T 1 . (ii) I f (cor)exp = 0 then ~Ooz = - c o l t 1 or Hot = - H ~ z ~ . Both kinds of data have been used in the analysis, The method of data analysis makes no other assumptions about the transient field except that it is of an impulsive nature. The slope of the best fit straight line is o) o = - g # y H / h . By measuring the slope and inserting the static internal field Ho into the equation for coo it is possible to extract a g-factor which is an average of all g-factors of the isotopes under study. If an experimental point for an isotope falls off the fitted line, then either its g-factor may differ from the average g-factor (g,,,) or the lifetime may be in error. The field H o should be the same for all the isotopes of a particular element. It is assumed that A g / g , ~ = A(coz)/(cor)i, where A g is the deviation of g from g,v and A(coz) is the deviation of (coz),~p from (coT)r, the value of the angular shift given by the straight line fit at z equal to the lifetime of the excited state of the isotope in question. The corrected g-factor for each isotope of a specific element is given by: g = g . ~ + A g = g.~
l+ (corkl"
¢4)
This correction assumes no lifetime error. Typical errors quoted for lifetimes are about 10 % which, along with the error in A(c0z) and g,v are incorporated into the total error of g for each isotope.
4. Results
The extracted g-factors together with radioactivity g-factors, where available, are shown in table 1. Also shown are energies and lifetimes of the first 2 + states taken from the latest information available. In most cases these correspond to tabulations in ref. 13). The g-factor calculations and subsequent error analysis are discussed below for each element studied. T e l l u r i u m . The static field (Ho) for Te in Fe has been determined by the M6ssbauer effect in 125Te [ref. 14)]. The quoted errors in Te g-factors arise mainly from the error in the slope of the best fit straight line through the data points. The balance of the error comes from the uncertainties in z, Ho, and (co~)exp- The several radioactivity experiments in 122Te [refs. 15-18)] and 124Te [refs. ~7,~9,20)] exhibit a wider scatter than would be expected from the quoted errors. Consequently, radioactivity results were not incorporated into the g-factor analysis. In an earlier publication 21) most of these Te data were analysed without including transient field effects, although they were considered. The g-factors quoted here are significantly lower because of the transient effects. The Te g-factors are, on the average, the lowest of those measured for the spherical nuclei. The nucleus Te is also closest to the "transi-
O-FACTORS
3l 7
tional" region and the only element studied having protons and neutrons in the same major shell.
Cadmium. The four Cd isotopes studied, ii°Cd, H2Cd, llgCd and li6Cd have lifetimes which are short enough so that rotations due to the transient field outweigh the rotations due to the static field. The radioactivity point on ll4Cd [ref. 22)] was incorporated into the g-factor analysis. The large errors quoted are primarily due to the large uncertainty in the slope of the best fit straight line. The equilibrium internal field is known from a radioactivity measurement 23). Palladium. Significant g-factors could be extracted from data on 106pd ' ~ospd ' and l lopd" The experimental shift for 104pd is too close to zero to yield a meaningful g-factor since the transient field, which is positive, nearly cancels the effect of the negative static field. The best fit straight line incorporates the radioactivity measurements on l°6Pd [ref. 24]). The internal field for Pd in Fe is known from an N M R measurement 25). Ruthenium. Four Ru isotopes were studied. Only the 98Ru and a°*Ru g-factors could be extracted since (co~)expfor l ° ° R u and a°2Ru are close to zero. The error in the slope, which incorporated radioactivity data 24), is small because of the wide span in lifetimes of the Ru isotopes. The large error in the 98Ru g-factor is due to the fact that (O)~)exp for 98Ru is small. The internal field is known from a radioactivity measurement 26). Molybdenum. Only two Mo isotopes were studied so that the analysis can yield only an average g-factor. The internal field is known from an N M R measurement 25). Selenium. The field H o for Se in Fe is not known so a value was extrapolated from the systematics of known hyperfine fields of chemically similar elements. The value used in this analysis is H o = 650+ 150 [ref. is)] kOe, where the large error is due to the uncertainty in the extrapolation. This value of Ho yields g,v = 0.41_+0.12 where the error includes both the error in the slope and the error in H o. Data for Se in Co were also used. The experimental angular shifts for Se in Co are normalized to the Se in Fe data by subtracting the impulse shift due to Se in Co from the experimental Se in Co points, multiplying the results by the ratio of the Fe slope to the Co slope and then adding the impulse precession due to Se in Fe. This is necessary since the static field of Se in Co is also not known. The transient field in ferromagnetic metals should be proportional to the moment of the host material. The normalized Co points, averaged with the Fe points are displayed in table 1 under (o)z)exp. By combining Fe and Co data, the error in (6o'[)exp was reduced to under 6 % for each isotope. The radioactivity point on 76Se [ref. is)] was used in conjunction with the IMPAC points to determine the best fit line. Germanium. A recent experiment has yielded an internal magnetic field for Ge in Fe of +70_+3 kG [ref. 27)]. However, the measured slope of the best fit straight
318
G.M. HEESTANDe t
al.
line to the I M P A C data is small but positive. This could indicate that the sign o f the field is wrong or that there is a y-factor variation. The latter was assumed and a straight line corresponding to + 7 0 kg was least squares fitted in the following way. Eq. (3) can be rewritten as:
(~)oxp
=
-
yt, NHo~/h - gt, NH, ~,1~.
( S)
F o r Se H l z l = 3.8_+1.5" M O e ' p s e c . According to the theory o f Lindhard and Winter lz), the transient field varies smoothly with the atomic n u m b e r of the solute so the same H l z l can be assumed for Ge in Fe since the atomic numbers of Ge and Se differ only by 2. As a result H o and HjZl are fixed in eq. (5). The least squares fitting procedure consisted of adjusting g in eq. (5) to give the best fit. The result yielded Ya~ = 0.46_+0.23; the large errors being due to the large uncertainties in Ha~j. As in the case of Se, Ge in Co data were used along with Ge in 1% data. ,5 I~ ~ P A C
~ RADIOACTMTY KISSLINGER .......B S O R E N S E N .....G R E I N E R -- Z I A LOMBARD
i L i i
!
I
... ...'" ""... ,.
iLL I l T\.
I i
, ..........
",
1
/
T...............................
_
.5
gl. 0
i
/ i
! Ge • 70
74
Se 76
80
- Mo 92
96
Ru I0098
I02
Pd I04
I08
Cd IIO
II4
Te 120
124
128
Fig. 4. Experimental 2 + state .q-factors and various theoretical predictions.
5. Comparison between experiment and theory In fig. 4 are shown the 2 + g-factors for all of the isotopes studied. The gross features o f fig. 4 indicate that the y-factors are fairly constant with a slight d o w n w a r d trend from Ge to Te. However, nearly all of the g-factors are equal within errors. The dotted line in fig. 4 refers to the predictions of Kisslinger and Sorensen 1, 28). Only particles outside o f closed shells were considered where pairing plus quadrupole
#-fACTORS
319
residual interactions are assumed. The n-n and p-p pairing strengths are assumed equal while the n-p pairing strength is taken as zero. The long range quadrupole interactions are of equal strength between neutron-neutron pairs, proton-proton pairs, and neutron-proton pairs. This formalism allows low-lying states of spherical nuclei to be treated in terms of quasiparticles or phonons. A key assumption in the theory is the quasiparticle random phase approximation which linearized the equations of motion of quasiparticle pairs by dropping terms in the Hamiltonian whose effect is spread out over many pair states of different angular momenta. The magnetic moments predicted by the random phase approximation have substantial fluctuations near closed shells. Pd and Cd have g ~ 1 because their wave functions involve 9~ protons to a large extent. As can be seen, rapid variations in one phonon 9-factors have not been observed. Similar calculations have been made by Lombard and Campi-Benet 29). Their calculations involve a large number of configurations, e.g. core excitations of the hole-particle type are included. Their pairing constants are obtained from odd-even mass differences while their quadrupole and octupole forces are chosen to fit the Ez ÷ and E 3- energies of certain nuclei. Their predictions are shown by the dashed line. To date we have no #-factor for 92Mo. Such a number would be very interesting for Lombard predicts g = 1.27 while Kisslinger and Sorensen predict a large negative #-factor. Because of the shortness of the lifetime, z = 0.4 psec and the high energy of the first excited state, E2+ = 1.53 MeV, the experiment would be very difficult to do. The solid lines in fig. 4 are the predictions of the hydrodynamical model, # = Z/A. The agreement is good for Ge and Se but Z/A gives values which are consistently too high for the other isotopes. The hydrodynamical model does of course agree with the lack of large #-factor variations. The best agreement is with a model proposed by Greiner 2). Greiner uses the hydrodynamical model together with the fact that the pairing force between protons is stronger than that between neutrons. This enables the neutron distribution to become more deformed than the proton distribution and hence, participate more in collective motion. Greiner's g-factors for vibrational nuclei are:
#~AZ (1-4f)
where
f~
N (fl°(n) - 1 ) A \rio(P)
(6)
rio(n) is the neutron deformation parameter and rio(P) is the proton deformation parameter. The #-factors in eq. (6) can be evaluated by taking flo(n)/flo(p)= ,/Go/Gn where Gp and Gn are the neutron and proton pairing strengths respectively and Gp and Gn are found from odd-even mass differences. Nilsson and Prior 30) give G. = 18 MeV/A and Go = 25 MeV/A, while Marchalek and Rasmussen 31) use G, = 20 MeV/A and Gp = 30 MeV/A. The #-factors calculated from eq. (6) using Marshalek's pairing strengths are shown in fig. 4 and reproduce the data quite well.
320
G.M. HEESTANDet al.
W e w o u l d like t o t h a n k D r . J. D . B r o n s o n , P. R y g e , R. N o r d , G . C o h n a n d W . R o n e y f o r t h e i r h e l p o n t h e e x p e r i m e n t a l w o r k . C o n v e r s a t i o n s w i t h P r o f e s s o r s J. L i n d h a r d a n d A. W i n t h e r a b o u t t h e t r a n s i e n t field w e r e v e r y h e l p f u l .
References 1) L. Kisslinger and R. Sorensen, Rev. Mod. Phys. 35 (1963) 853 2) W. Greiner, Nucl. Phys. 80 (1966) 417 3) J. de Boer and J. Eichler, Advances in nuclear physics, ed. M. Baranger and E. Vogt (Plenum Press, New York, 1968) 4) Perturbed angular correlations, ed. E. Karlsson, E. Matthias and K. Siegbahn (North-Holland Publ. Co., Amsterdam, 1964) 5) R. R. Botchers, B. Herskind, J. D. Bronson, L. Grodzins, R. Kalish and D. E. Murnick, Phys. Rev. Lett. 20 (1968) 424 6) A. Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 (1953) no. 16 7) D. E. Murnick, ttyperfine interactions, ed. A. Freeman and R. Frankel (Academic Press, New York, 1967) p. 637 8) D. E. Murnick, L. Grodzins, J. D. Bronson, B. Herskind and R. R. Borchers, Phys. Rev. 163 (1967) 254 9) L. Grodzins, Hyperfine structure and nuclear radiation, ed. E. Matthias and D. A. Shirley, (North-Holland Publ. Co., Amsterdam, 1968) p. 607 10) B. Herskind, R. R. Borchers, J. D. Bronson, D. E. Murnick, L. Grodzins and R. Kalish, Hyperfine structure and nuclear radiation ibid p. 735 11) J. Lindhard, M. Scharff and H. Schiott, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1963) no. 14 12) J. Lindhard and A. Winther, private communication 13) Nuclear data (Academic Press) ed. K. Way 14) R. B. Frankel, J. Huntzicker, E. Matthias, S. S. Rosenblum, D. A. Shirley and N. J. Stone, Phys. Lett. 15 (1965) t63 15) K. Johansson, E. Karlsson and R. W. Sommerfeldt, Phys. Lett. 22 (1966) 297 16) K. Auerbach, B. Hams, K. Siepe, G. Wittkemper and H. J. K6rner, Phys. Lett. 22 (19661 299 17) S. K. Bhattacherjee, J. D. Bowman and E. N. Kaufmann, Phys. Lett. 24B (1967) 651 18) J. Murray, T. A. McMath, and J. A. Cameron, Can. J. Phys. 45 (1967) 1821 19) J. Murray, T. A. McMath, W. H. Brooker and J. A. Cameron, Can. J. Phys. 45 (1967t 1600 20) E. Bozek, R. Broda, J. Golezewski, A. Z. Hrynkiewicz, R. Kulessa, S. Ogaza, M. Rybicka and W. Walus, Hyperfine structure and nuclear radiation ibid. p. 158 21) R. R. Borchers, J. D. Bronson, D. E. Murnick and L. Grodzins, Phys. Rev. Lett. 21 (1966) 1099 22) S. K. Bhattacherjee, J. D. Bowman and E. N. Kaufinann, Phys. Rev. Lett. 18 (1967) 223 23) E. Matthias, S. S. Rosenblum and D. A. Shirley, Phys. Rev. Lett. 14 (1965) 46 24) K. Auerbach, K. Siepe, J. Wittkemper and H. J. K6rner, Phys. Lett. 23 (1966) 367 25) M. Kontani and K. Itoh, J. Phys. Soc. (Japan) 22 (1967) 345 26) O. C. Kistner, Phys. Rev. 144 (1966) 1022 27) E. Gerdau and J. Braunsforth, to be published 28) R. Sorensen, Hyperfine structure and nuclear radiation ibid. p. 13 29) R. J. Lombard and X. Campi-Benet, to be published 30) S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32 (1961) no. 16 31) E. R. Marshalek and J. O. Rasmussen, Nucl. Phys. 43 (1963) 438
J
Nuclear Physics A133 (1969) 310--320;(~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
o-FACTORS FOR 2 + STATES OF DOUBLY EVEN NUCLEI (Ge, Se, Mo, Ru, Pd, Cd AND Te) G. M. HEESTAND, R. R. BORCHERS and B. H E R S K I N D + Unirersity of Wisconsin, Madison, 14"isconsin tt and L. G R O D Z I N S and R. KALISH Laboratory for Nuclear Science, Physics Department Massachusetts Institate of Technology, Cambridge, Agassachusetts and D. E. M U R N I C K Massachusetts Institute of Technology, Cambridge, Massachusetts and Bell Telephone Laboratories, Inc., Murray Hill, New Jersey Received 1 April 1969
Abstract: The gyromagnetic ratio, .q, for the first excited 2 + states of 25 even-even isotopes of Ge, Se, Mo, Cd, Pd, Ru and Te were obtained using the ion implantation perturbed angular correlation technique. The .q factors were extracted from the measured precession angles using known lifetimes, equilibrium hyperfine magnetic field values and transient hyperfine magnetic field data for fast ions in ferromagnetic lattices. The results indicate that the ,q factors vary little for the states studied, and are close to the collective value Z/A. E
N U C L E A R MOMENTS (Static) 70, 72, 7~, V6Ge' 76, 78 80, 82Se ' 98, 100Mo ' 9s, lO~Ru ' 106, t08, UOpd ' ~0,~12, 1 1 4 , 1 1 6 C d ' and ~zo, 122,124., 126, 12s, t30Te; measured 77(0 ' A); deduced ~, for first 2 + states.
1.
Introduction
E x c i t e d s t a t e p r o p e r t i e s o f d o u b l y e v e n n u c l e i i n t h e r e g i o n 70 =< A =< 130 h a v e b e e n t h e t o p i c o f a n u m b e r o f r e c e n t e x p e r i m e n t a l a n d t h e o r e t i c a l ~' 2) i n v e s t i g a t i o n s . Specifically, c a l c u l a t i o n s u t i l i z i n g p a i r i n g p l u s q u a d r u p o l e f o r c e s h a v e b e e n s u c c e s s f u l in p r e d i c t i n g t r a n s i t i o n p r o b a b i l i t i e s f o r t h e s e " o n e - p h o n o n " The recent measurements
of non-zero electric quadrupole
first e x c i t e d s t a t e s 1). moments
for several
first 2 + s t a t e s 3), w h i c h t h e a b o v e t h e o r i e s p r e d i c t t o b e z e r o , h a s s t i m u l a t e d f u r t h e r theoretical investigations.
These
same calculations have also predicted
magnetic
m o m e n t s /~ = :tNg~I b u t u n t i l n o w , e x p e r i m e n t a l l y m e a s u r e d d i p o l e m o m e n t s b e e n r e l a t i v e l y f e w in n u m b e r . Excited state dipole moments
have
are usually measured either by perturbed angular
c o r r e l a t i o n s u s i n g e x t e r n a l o r i n t e r n a l fields o r b y t h e M 6 s s b a u e r effect. D o u b l y e v e n n u c l e i b e t w e e n A = 70 a n d A = 130 h a v e s h o r t l i v e d 2 + s t a t e s ( r ~ 10 p s e c ) o f f a i r l y t Present address: Niels Bohr Institute, Copenhagen, Denmark. tt Work supported in part by the U.S. Atomic Energy Commission. 310
g-FACTORS
311
high energy (E ,,~ 500 keV). Consequently, magnetic perturbations, even in internal fields of 500 kOe, are small with typical precession angles on the order of 15 mrad. The high gamma energies yield an almost zero recoiless fraction making M6ssbauer measurements impossible. What few data are available come from radioactive nuclei metallurgically doped into polarized Fe. Here the gamma-gamma correlation of the daughter nucleus is precessed by the large hyperfine magnetic field acting on dilute impurities in Fe. Measuring the precession allows the determination of the magnetic moments of the intermediate level in the gamma-gamma cascade 4). If Coulomb excitation is used to produce the states of interest and the decay gamma rays are detected in coincidence with backscattered ions, then a number of improvements in the sensitivity of the perturbed angular correlation technique can be realized. The angular distribution becomes very anisotropic while the recoil nucleus can be driven into a ferromagnetic backing by its recoil energy. In addition, there is no dependence on radioactive cascades to populate the states, so systematic investigations become easier. Although experimental evidence indicates that the recoil ions experience substantial perturbations while slowing down 5), it now seems that the origin of these perturbations is understood as the interaction of fast recoiling ions with polarized electrons in the ferromagnetic backing. The perturbations can be simulated by a large field ( ~ 10 MOe) acting on the nucleus during its slowing down time ( ~ lpsec). This field seems to vary slowly with Z and be the same for all isotopes of a given element. These empirical facts about the perturbation can be included in the data analysis to obtain the dipole moments of the excited states. This paper presents such measurements for 25 doubly even nuclei. The g-factors determined are compared with various theories but are best described by a simple hydrodynamical model of the nucleus 6).
2. Experimental The ion implantation perturbed angular correlation technique, IMPAC, has been described in detail in previous publications 7-1o). As is indicated in fig. 1, a high-energy ( 3 3 - 3 8 MeV) oxygen beam is used to Coulomb excite the target nuclei and concurrently implant them, by recoil into a suitable host lattice. Using NaI(T1) crystals, the decay 7-rays are counted in coincidence with oxygen ions backscattered into an annular detector. The coincidence requirement insures that only transitions from the m = 0 substate are observed, giving rise to the highly anisotropic E2(m = 0 ~ m = 0) radiation pattern. The forward momentum transferred to the excited nuclei by the backscattered particles is sufficient for implantation into the ferromagnetic backing. All targets used in these experiments were separated isotopes electroplated or evaporated onto 25 pm thick iron foils. Target thicknesses varied from 200/~g/cm 2 to 400/~g/cm 2. The recoil ranges and slowing down times calculated from the theory
312
o.M. HEESTAND e l at.
of Lindhard, Scharff and Schiott 11) indicate an average implantation depth greater than 500 pg/cm z, and a stopping time of about 1 psec. An external magnetic field of about 1 kOe perpendicular to the detection plane was used to magnetically saturate the iron foil and hence align the hyperfine fields acting on the impurity nuclei. This effective magnetic field (Herr) interacts with the magnetic moment of the excited state and causes it to precess with the Larmor frequency co(t) = -gpNH(t)/h. The angular distribution of the decay gamma rays at a time t after the excitation is given by: W(O, t) = W[O-.~co(t')dt']. For an integral
. ,moveable,~_
/
\
\ \ ?',3 Y2
/
]
Fig. 1. Schematic o f experimental apparatus, top view. The polarizing electromagnet is not shown.
precession measurement, in which the coincidence resolving time is much longer than the nuclear lifetime z, the integrated angular distribution weighted by the exponential decay of the excited state is observed. In this case:
w(o) = 2 foe-',w lo- flco(,')dC at.
(1)
The angular correlation, W(O), is usually written
W(O) =
Z
A.P. (cosO),
n=O, 2, 4
where the coefficients A, are determined by the initial and final state spin and modified by the finite solid angles of the counters. The P,, are the Legendre polynomials. The typical angular distributions measured with polarizing fields " u p " and " d o w n " are shown in fig. 2; the example is that of the 434 keV state of 1ospd" The solid and
y-FACTORS
313
dashed lines are the best fits found by varying A2, A4, and the angular shift. The coefficients A 2 and ,44 are found to be in agreement with the angular correlation coefficients calculated from Coulomb excitation theory; there is no evidence for nonmagnetic perturbations. Small shifts in the angular correlation pattern are measured by monitoring the count rate at angles of maximum slope with frequent reversals of the magnetic field. ,40 is determined from the ratio: R = W(Ot)W(Ot)+ ......... ~- ....
"/
W(O~) W(O+)'
(+)
1°epd
(-)
.~.. ,,~
1500
,
~
/
',
on Fe backing
best fit: Az = 0.615 A4 : - [ 210
,,
oJr
= 0.014
~
1000
500
I
I
-120 -I00
I
-80
I
-60
I
-40
I
I
I
-20 0 20 8 (degrees)
I
I
/
I
40
60
80
I00
120
Fig. 2. Angular correlation for 1°SPd implanted into iron. The angular shift is used to determine the 2 + state magnetic moment. (+) and (--) refer to the external field direction. where the arrows refer to the external field directions. Each angle is treated independently, with theoretical A2 and A4 values used to determine AO. Four counters are placed symmetrically (usually __+22½° and ___112k °) so that two sets of equal and opposite effects are measured simultaneously, cancelling out asymmetries. Precession angles must be corrected for beam bending by the fringing field of the polarizing magnet which shows up as an apparent precession of the correlation. Such effects were measured using copper backed targets and also calculated from the measured fringing field. The correlations ranged from 2.0 to 2.5 mrad, depending on whether the incoming oxygen beam was in a 5 + and 6 + charge state. Precession angles corrected for beam bending are shown in table 1. Systematics of the lzresent data indicate that the hyperfine field seen by the implanted ion is considerably larger during slow down than after the ion has come to rest. The field occurring during slow down is referred to as the transient field and has
G.M. HEESTANDeta[.
314
TABLE 1 Summary of e×perimental data for states studied, including radioactivity data where available Isotope
Energy of 2 + (keV)
Meanlife (psec)
"°Ge "2Ge WGe ~°Ge v~'Se VSSe S°Se SaSe "aMo ~°°Mo 9SRu l°°Ru 1°2Rt.l t°4Ru ~°~Pd ~°6Pd ~°SPd ~°Pd J~°Cd ~t-'Cd ll~Cd ~°Cd tZ°Te ~22Te
1040 885 596 563 559 614 666 655 787 536 654 540 473 358 554 512 434 374 658 617 559 513 560 564
1.92--0.20 4.54~0.50 17.2 :1.4 25.2 : 2.6 16.0 ~ 1.5 12.4 ~:l.2 11.6 = 1.2 16.3 =1.6 5.1 ~ 0.5 14.9 +:1.5 8.5 " 0 . 6 ~) 17.2 ~1.2~) 25.4 _:1.8 ~) 83.5 :=6.0 ") 14.0 :=1.0 18.4 ~: 1.3 34.4 _:~2.4 66.0 =_4.0 7.2 :_0.761 8.9 :!-0.7 b) 13.0 : 1.1 b) 19.8 =:1.5 ~) 13.4 ~_2.6 II.0 q l.l
~2'*Te
603
9.5
126Te 128Tc t3°Te
667 743 840
H=
(o)z)¢xp
(kG)
(mrad)
+70~3
glMPAC
--11.2--1.5 --9.9±1.7 --11.3=1.3 --9.9~_1.5 --27.1~1.5 --24.0~ 1.6 --23.9~1.l --29.5--0.8 --12.1±3.0 --8.0-1.0 5.0" 4.0 3.3~3.2 2.8--4.4 47.8±4.4 --0.8~1,4 4.5--2.6 17.5:= 1,6 35.0-L2.3 --10.7~3.8 --7.3--0.9 5.2--2.5 --4.0--l.9 --15.673.6 - 15.8:!:1.2
0.29±0.17 0.30~0.04 0.25-3_0.03 0.39-~0.15 0.30~0.06 0.32~0.13 0.7l~0.38 0.21~-0.06 0.24-~0.06
=0.5~)
- 1 2 . 9 " 0.9
0.21±0.05
6.37~0.7 a) 4 . 5 9 / 0 . 6 d) 2.9 -~-0.3
- 13.l ±2.5 --10.1 ~2.0 -- 10.6:±2.2
0.25±0.07 0.21 ~0.06 0.25±0.07
+650±150
--256±5 -~505±15
--595±12
-348±10
i 620±20
0.59±0.29 0.50i0.25 0.46:{:0.23 0.37:t:0.18 0.404-0.11 0.4l:::0.11 0.42±0.12 0.43 ~:0.12 0.34=_0.18 0.34-~0.18 0.30:'0.17
gRAD
0.40==0.12 ¢)
0.42~0,03r) 0.34~-0,06 f) 0.29:L0.04 0.34:!-0.01 r)
0.27~ 0.09 g) 0.44j 0.06hi
0.395:0.03 ~1 0.46_'0.05 ~) 0.34±0.07 ~) 0.31 =_0.03 ~) 0.35~-0.05~1 0.21 ~_0.05 ~) 0.275 0.13 ")
a) b) ~) d) e) g)
E. K. McGowan, R. L. Robinson, P. H. Stelson, and W. T. Milner, Nucl. Phys. to be published. E. K. McGowan, R. L. Robinson, P. H. Stelson, and J. L. C. Ford, Nucl. Phys., 66 (1965) 97. M. Schumacher, Phys. Rev. 171 (1968) 1279. R. G. Stokstad and I. Hall, Nucl. Phys. A99 (1967) 507. See ref. is). f) See ref. 24). L. Keszthelyi, I. Demeter, I. Dezsi, and L. Varga in Hypcrfine structure and nuclear radiation, ibM., p. 155. h) See ref. 22). i) See ref. ~s). J) See ref. 1c'). k) See ref. 17). I1 See ref. 19). m) See ref. 2o). b e e n d i s c u s s e d in a p r e v i o u s p u b l i c a t i o n 5). L i n d h a r d a n d W i n t h e r 12) a t t r i b u t e t h e t r a n s i e n t field t o a n a m p l i f i c a t i o n o f t h e p o l a r i z e d e l e c t r o n d e n s i t y a t t h e n u c l e u s b y electron scattering d u r i n g the slowing d o w n o f the impurity ion. The time d e p e n d -
g-FACTORS
315
ence of the transient field is a function of the slowing down process. The slowing down times however are short compared to the lifetimes of interest allowing eq. (1) to be integrated to first order with the result that: or = o)0T+om,
(3)
o% is the Larmor frequency associated with the static field i.e., the field the impurity sees after it comes to rest. o91r~ represents the impulse precession due to the transient field. 3. Analysis Fig. 3 shows a plot of the experimentally determined shifts (o2r)exp versus the meanlife of the first 2 + state; values for a given element being grouped together. The line shown through the data is a least squares fit of eq. (3). The slope of the line depends on the average 9-factor of the states and the static internal field. The intercepts of the line depend only on the transient field. There are / Pd in Fe
40
Ge rrl Fe z
50
,o
/
/
20
;j -20 10
©
30
Te in Fe
-I0
5C
Ru in Fe
2C
//p/
/
o
i
l
Cd in Fe
5
0
20
20
0
5
rO
15
~ 20 5 •r (psec)-- --
I0
15
Mo in Fe
IO
L 15
20
Fig. 3. S u m m a r y of experimental precession angles for 2 + states of doubly even nuclei in Fe (and Co and N i for Se) plotted as a function of nuclear lifetime, 7.
216
G. M. HEESTAND e l ,:ll.
two cases where the transient field effects can be isolated: (i) If the precession of the same state has been measured by radioactivity methods, which are essentially recoilless, then ( ( . O ~ ' ) e x p - - ( ( D r ) r a d = ( D 1 T 1 . (ii) I f (cor)exp = 0 then ~Ooz = - c o l t 1 or Hot = - H ~ z ~ . Both kinds of data have been used in the analysis, The method of data analysis makes no other assumptions about the transient field except that it is of an impulsive nature. The slope of the best fit straight line is o) o = - g # y H / h . By measuring the slope and inserting the static internal field Ho into the equation for coo it is possible to extract a g-factor which is an average of all g-factors of the isotopes under study. If an experimental point for an isotope falls off the fitted line, then either its g-factor may differ from the average g-factor (g,,,) or the lifetime may be in error. The field H o should be the same for all the isotopes of a particular element. It is assumed that A g / g , ~ = A(coz)/(cor)i, where A g is the deviation of g from g,v and A(coz) is the deviation of (coz),~p from (coT)r, the value of the angular shift given by the straight line fit at z equal to the lifetime of the excited state of the isotope in question. The corrected g-factor for each isotope of a specific element is given by: g = g . ~ + A g = g.~
l+ (corkl"
¢4)
This correction assumes no lifetime error. Typical errors quoted for lifetimes are about 10 % which, along with the error in A(c0z) and g,v are incorporated into the total error of g for each isotope.
4. Results
The extracted g-factors together with radioactivity g-factors, where available, are shown in table 1. Also shown are energies and lifetimes of the first 2 + states taken from the latest information available. In most cases these correspond to tabulations in ref. 13). The g-factor calculations and subsequent error analysis are discussed below for each element studied. T e l l u r i u m . The static field (Ho) for Te in Fe has been determined by the M6ssbauer effect in 125Te [ref. 14)]. The quoted errors in Te g-factors arise mainly from the error in the slope of the best fit straight line through the data points. The balance of the error comes from the uncertainties in z, Ho, and (co~)exp- The several radioactivity experiments in 122Te [refs. 15-18)] and 124Te [refs. ~7,~9,20)] exhibit a wider scatter than would be expected from the quoted errors. Consequently, radioactivity results were not incorporated into the g-factor analysis. In an earlier publication 21) most of these Te data were analysed without including transient field effects, although they were considered. The g-factors quoted here are significantly lower because of the transient effects. The Te g-factors are, on the average, the lowest of those measured for the spherical nuclei. The nucleus Te is also closest to the "transi-
O-FACTORS
3l 7
tional" region and the only element studied having protons and neutrons in the same major shell.
Cadmium. The four Cd isotopes studied, ii°Cd, H2Cd, llgCd and li6Cd have lifetimes which are short enough so that rotations due to the transient field outweigh the rotations due to the static field. The radioactivity point on ll4Cd [ref. 22)] was incorporated into the g-factor analysis. The large errors quoted are primarily due to the large uncertainty in the slope of the best fit straight line. The equilibrium internal field is known from a radioactivity measurement 23). Palladium. Significant g-factors could be extracted from data on 106pd ' ~ospd ' and l lopd" The experimental shift for 104pd is too close to zero to yield a meaningful g-factor since the transient field, which is positive, nearly cancels the effect of the negative static field. The best fit straight line incorporates the radioactivity measurements on l°6Pd [ref. 24]). The internal field for Pd in Fe is known from an N M R measurement 25). Ruthenium. Four Ru isotopes were studied. Only the 98Ru and a°*Ru g-factors could be extracted since (co~)expfor l ° ° R u and a°2Ru are close to zero. The error in the slope, which incorporated radioactivity data 24), is small because of the wide span in lifetimes of the Ru isotopes. The large error in the 98Ru g-factor is due to the fact that (O)~)exp for 98Ru is small. The internal field is known from a radioactivity measurement 26). Molybdenum. Only two Mo isotopes were studied so that the analysis can yield only an average g-factor. The internal field is known from an N M R measurement 25). Selenium. The field H o for Se in Fe is not known so a value was extrapolated from the systematics of known hyperfine fields of chemically similar elements. The value used in this analysis is H o = 650+ 150 [ref. is)] kOe, where the large error is due to the uncertainty in the extrapolation. This value of Ho yields g,v = 0.41_+0.12 where the error includes both the error in the slope and the error in H o. Data for Se in Co were also used. The experimental angular shifts for Se in Co are normalized to the Se in Fe data by subtracting the impulse shift due to Se in Co from the experimental Se in Co points, multiplying the results by the ratio of the Fe slope to the Co slope and then adding the impulse precession due to Se in Fe. This is necessary since the static field of Se in Co is also not known. The transient field in ferromagnetic metals should be proportional to the moment of the host material. The normalized Co points, averaged with the Fe points are displayed in table 1 under (o)z)exp. By combining Fe and Co data, the error in (6o'[)exp was reduced to under 6 % for each isotope. The radioactivity point on 76Se [ref. is)] was used in conjunction with the IMPAC points to determine the best fit line. Germanium. A recent experiment has yielded an internal magnetic field for Ge in Fe of +70_+3 kG [ref. 27)]. However, the measured slope of the best fit straight
318
G.M. HEESTANDe t
al.
line to the I M P A C data is small but positive. This could indicate that the sign o f the field is wrong or that there is a y-factor variation. The latter was assumed and a straight line corresponding to + 7 0 kg was least squares fitted in the following way. Eq. (3) can be rewritten as:
(~)oxp
=
-
yt, NHo~/h - gt, NH, ~,1~.
( S)
F o r Se H l z l = 3.8_+1.5" M O e ' p s e c . According to the theory o f Lindhard and Winter lz), the transient field varies smoothly with the atomic n u m b e r of the solute so the same H l z l can be assumed for Ge in Fe since the atomic numbers of Ge and Se differ only by 2. As a result H o and HjZl are fixed in eq. (5). The least squares fitting procedure consisted of adjusting g in eq. (5) to give the best fit. The result yielded Ya~ = 0.46_+0.23; the large errors being due to the large uncertainties in Ha~j. As in the case of Se, Ge in Co data were used along with Ge in 1% data. ,5 I~ ~ P A C
~ RADIOACTMTY KISSLINGER .......B S O R E N S E N .....G R E I N E R -- Z I A LOMBARD
i L i i
!
I
... ...'" ""... ,.
iLL I l T\.
I i
, ..........
",
1
/
T...............................
_
.5
gl. 0
i
/ i
! Ge • 70
74
Se 76
80
- Mo 92
96
Ru I0098
I02
Pd I04
I08
Cd IIO
II4
Te 120
124
128
Fig. 4. Experimental 2 + state .q-factors and various theoretical predictions.
5. Comparison between experiment and theory In fig. 4 are shown the 2 + g-factors for all of the isotopes studied. The gross features o f fig. 4 indicate that the y-factors are fairly constant with a slight d o w n w a r d trend from Ge to Te. However, nearly all of the g-factors are equal within errors. The dotted line in fig. 4 refers to the predictions of Kisslinger and Sorensen 1, 28). Only particles outside o f closed shells were considered where pairing plus quadrupole
#-fACTORS
319
residual interactions are assumed. The n-n and p-p pairing strengths are assumed equal while the n-p pairing strength is taken as zero. The long range quadrupole interactions are of equal strength between neutron-neutron pairs, proton-proton pairs, and neutron-proton pairs. This formalism allows low-lying states of spherical nuclei to be treated in terms of quasiparticles or phonons. A key assumption in the theory is the quasiparticle random phase approximation which linearized the equations of motion of quasiparticle pairs by dropping terms in the Hamiltonian whose effect is spread out over many pair states of different angular momenta. The magnetic moments predicted by the random phase approximation have substantial fluctuations near closed shells. Pd and Cd have g ~ 1 because their wave functions involve 9~ protons to a large extent. As can be seen, rapid variations in one phonon 9-factors have not been observed. Similar calculations have been made by Lombard and Campi-Benet 29). Their calculations involve a large number of configurations, e.g. core excitations of the hole-particle type are included. Their pairing constants are obtained from odd-even mass differences while their quadrupole and octupole forces are chosen to fit the Ez ÷ and E 3- energies of certain nuclei. Their predictions are shown by the dashed line. To date we have no #-factor for 92Mo. Such a number would be very interesting for Lombard predicts g = 1.27 while Kisslinger and Sorensen predict a large negative #-factor. Because of the shortness of the lifetime, z = 0.4 psec and the high energy of the first excited state, E2+ = 1.53 MeV, the experiment would be very difficult to do. The solid lines in fig. 4 are the predictions of the hydrodynamical model, # = Z/A. The agreement is good for Ge and Se but Z/A gives values which are consistently too high for the other isotopes. The hydrodynamical model does of course agree with the lack of large #-factor variations. The best agreement is with a model proposed by Greiner 2). Greiner uses the hydrodynamical model together with the fact that the pairing force between protons is stronger than that between neutrons. This enables the neutron distribution to become more deformed than the proton distribution and hence, participate more in collective motion. Greiner's g-factors for vibrational nuclei are:
#~AZ (1-4f)
where
f~
N (fl°(n) - 1 ) A \rio(P)
(6)
rio(n) is the neutron deformation parameter and rio(P) is the proton deformation parameter. The #-factors in eq. (6) can be evaluated by taking flo(n)/flo(p)= ,/Go/Gn where Gp and Gn are the neutron and proton pairing strengths respectively and Gp and Gn are found from odd-even mass differences. Nilsson and Prior 30) give G. = 18 MeV/A and Go = 25 MeV/A, while Marchalek and Rasmussen 31) use G, = 20 MeV/A and Gp = 30 MeV/A. The #-factors calculated from eq. (6) using Marshalek's pairing strengths are shown in fig. 4 and reproduce the data quite well.
320
G.M. HEESTANDet al.
W e w o u l d like t o t h a n k D r . J. D . B r o n s o n , P. R y g e , R. N o r d , G . C o h n a n d W . R o n e y f o r t h e i r h e l p o n t h e e x p e r i m e n t a l w o r k . C o n v e r s a t i o n s w i t h P r o f e s s o r s J. L i n d h a r d a n d A. W i n t h e r a b o u t t h e t r a n s i e n t field w e r e v e r y h e l p f u l .
References 1) L. Kisslinger and R. Sorensen, Rev. Mod. Phys. 35 (1963) 853 2) W. Greiner, Nucl. Phys. 80 (1966) 417 3) J. de Boer and J. Eichler, Advances in nuclear physics, ed. M. Baranger and E. Vogt (Plenum Press, New York, 1968) 4) Perturbed angular correlations, ed. E. Karlsson, E. Matthias and K. Siegbahn (North-Holland Publ. Co., Amsterdam, 1964) 5) R. R. Botchers, B. Herskind, J. D. Bronson, L. Grodzins, R. Kalish and D. E. Murnick, Phys. Rev. Lett. 20 (1968) 424 6) A. Bohr and B. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27 (1953) no. 16 7) D. E. Murnick, ttyperfine interactions, ed. A. Freeman and R. Frankel (Academic Press, New York, 1967) p. 637 8) D. E. Murnick, L. Grodzins, J. D. Bronson, B. Herskind and R. R. Borchers, Phys. Rev. 163 (1967) 254 9) L. Grodzins, Hyperfine structure and nuclear radiation, ed. E. Matthias and D. A. Shirley, (North-Holland Publ. Co., Amsterdam, 1968) p. 607 10) B. Herskind, R. R. Borchers, J. D. Bronson, D. E. Murnick, L. Grodzins and R. Kalish, Hyperfine structure and nuclear radiation ibid p. 735 11) J. Lindhard, M. Scharff and H. Schiott, Mat. Fys. Medd. Dan. Vid. Selsk. 33 (1963) no. 14 12) J. Lindhard and A. Winther, private communication 13) Nuclear data (Academic Press) ed. K. Way 14) R. B. Frankel, J. Huntzicker, E. Matthias, S. S. Rosenblum, D. A. Shirley and N. J. Stone, Phys. Lett. 15 (1965) t63 15) K. Johansson, E. Karlsson and R. W. Sommerfeldt, Phys. Lett. 22 (1966) 297 16) K. Auerbach, B. Hams, K. Siepe, G. Wittkemper and H. J. K6rner, Phys. Lett. 22 (19661 299 17) S. K. Bhattacherjee, J. D. Bowman and E. N. Kaufmann, Phys. Lett. 24B (1967) 651 18) J. Murray, T. A. McMath, and J. A. Cameron, Can. J. Phys. 45 (1967) 1821 19) J. Murray, T. A. McMath, W. H. Brooker and J. A. Cameron, Can. J. Phys. 45 (1967t 1600 20) E. Bozek, R. Broda, J. Golezewski, A. Z. Hrynkiewicz, R. Kulessa, S. Ogaza, M. Rybicka and W. Walus, Hyperfine structure and nuclear radiation ibid. p. 158 21) R. R. Borchers, J. D. Bronson, D. E. Murnick and L. Grodzins, Phys. Rev. Lett. 21 (1966) 1099 22) S. K. Bhattacherjee, J. D. Bowman and E. N. Kaufinann, Phys. Rev. Lett. 18 (1967) 223 23) E. Matthias, S. S. Rosenblum and D. A. Shirley, Phys. Rev. Lett. 14 (1965) 46 24) K. Auerbach, K. Siepe, J. Wittkemper and H. J. K6rner, Phys. Lett. 23 (1966) 367 25) M. Kontani and K. Itoh, J. Phys. Soc. (Japan) 22 (1967) 345 26) O. C. Kistner, Phys. Rev. 144 (1966) 1022 27) E. Gerdau and J. Braunsforth, to be published 28) R. Sorensen, Hyperfine structure and nuclear radiation ibid. p. 13 29) R. J. Lombard and X. Campi-Benet, to be published 30) S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32 (1961) no. 16 31) E. R. Marshalek and J. O. Rasmussen, Nucl. Phys. 43 (1963) 438