Magnetic moments of neutron resonances in rare-earth nuclei

Nuclear Physics A267 (1976) 172-180; (~) North-Holland Publishing Co., Amsterdam N o t to be reproduced by photoprint or microfilm without written permission from the publisher

MAGNETIC MOMENTS OF NEUTRON RESONANCES IN RARE-EARTH NUCLEI V. P. A L F I M E N K O V , L. LASON t, Yu. D. MAREEV, O. N. O V C H I N N I K O V , L. B. P I K E L N E R and E. I. S H A R A P O V

Joint Institute for Nuclear Research, Laboratory of Neutron Physics, Dubna, USSR Received 16 March 1976 Abstract: The magnetic dipole moments #j of compound states (neutron resonances) in holmium and

terbium have been obtained by the neutron resonance energy shift method. This shift is due to the polarization of nuclei at ultra-low temperature in h.f. magnetic fields inside ferromagnetic samples. Shifts of resonances in neutron transmission were measured by the time-of-flight method in a booster mode of operation of the pulsed reactor IBR-30 with the 40 MeV Linac. The pj results are - 1.8 + 0.7 and 3.9 + 1.9 n.m. for the 3.93 and 12.7 eV resonances in t65Ho and - 0 . 2 + 1.0, 4.3 + 3.7 and - 1.7+4.4 n.m. for the 3.35, 4.99 and 11.1 eV resonances in 159Tb. The g-factors deduced from these data and those reported earlier were analysed. The mean value is 0 = 0.34+0.22. The intrinsic dispersion Ag = 0.51 + 0.20 was obtained from the statistical treatment of data. Both 0 and Ag values are compared with theoretical predictions. The results point out to the distinct fluctuations of magnetic moments of neutron resonances. .

E

i

N U C L E A R REACTIONS 165H0, 159Tb(n), E = 3-13 eV; measured resonance energy shifts. 166Ho, 16°Tb compound states deduced ~, g. Natural targets, 3He/4He refrigerator.

1. Introduction The magnetic dipole moment is a most important parameter of the nucleus. It has been measured for a great number of nuclei not only in ground states but also in excited states. All these data, however, have been obtained for excitation energies far below the neutron binding energy. There exists a region of nuclear states near the neutron binding energy (neutron resonances) in which the methods of neutron spectroscopy yield rich information on level positions and their widths and spins, whereas no data about their magnetic moments have been obtained as yet. Such widespread methods as M6ssbauer spectroscopy or perturbed angular correlations could not be applied for the high energy nuclear states with a lifetime of the order of 10-15 sec. In view of the complex nature of compound states, the investigation of magnetic moments of high energy states proves to be a valuable source of information about nuclear structure.

* On leave from the University of Lodz, Poland. 172

MAGNETIC MOMENTS

173

The possibility of measuring magnetic moments of neutron resonances was indicated for the first time by Shapiro 1, 2). He proposed to measure the shift of resonance energies due to the hyperfine interaction of the nuclear magnetic moment with the internal magnetic field. The shift AE for polarized neutron transmission through the unpolarized is, as compared to unpolarized neutrons, given by AE = ~ - ½ f . H { [ l + 2/(2I + l)]gs-I~t}, [--½f.H[(1 + 1/I)#,-- p j],

J = I+½ J = I-½,

(1)

while for unpolarized neutron transmission through the polarized target we get, as compared to unpolarized target nuclei, ~--fNH{[1--1/(2I+I)(I+I)]pj--#,}, AE = ( - fNH(l~s--I~t),

J = I+½ J = I-½.

(2)

Here H is the hyperfine magnetic field,f, andfN the neutron and nuclear polarization, respectively, I and J the spins of the target and compound nuclei, respectively, and #i and #j the magnetic moments of these states. The main difficulty in performing the experiment is due to the small value of the experimental effect. Indeed, if one assumes the maximum possible polarization fN = 1 and the maximum hyperfine magnetic field H = 10 7 0 e (which is true for several rare-earth nuclei), then, with P J - P l = 1 n.m. a shift of only 3 × 10 -5 eV is expected. This shift is to be compared with the total width of the resonance which is about 0.1 eV. The experiments were started several years ago in the Joint Institute for Nuclear Research. From the two possible ways indicated above we have chosen the unpolarized neutron transmission through the polarized nuclear target. The polarized neutron transmission method employed in Brookhaven 4) has revealed some side effects and probably requires improvement. The first experimental values were obtained for several 167Er [ref. 3)] and 161,163Dy [ref. s)] resonances. In this paper we present the magnetic moments for 159Tb and 165H0 resonances along with the statistical analysis of the y-factor data and a comparison with theoretical predictions.

2. Experiment The measurements were performed by the time-of-flight method at the pulse reactor IBR-30 with the 40 MeV Linac-injector, the pulse width being 4 #sec and flight path 58.5 m. The experimental set-up is shown in fig. 1. Sheets of metalic Tb and Ho with a thickness of 1.8 x 1021 nucl/cm 2 and 0.6 × 1021 nucl/cm 2, respectively were used as targets, ferromagnetic at temperatures below 219 K and 20 K. Both these nuclei exhibit low energy resonances and large internal magnetic fields (H = 3.1 x 106 Oe for Tb and 7.3 x 106 Oe for Ho). The latter feature is important not only in view of the proportionality of d E to H, but it also permits high polarization of nuclei to be achieved in the domains by cooling. In the absence of an external

174

V.P. ALFIMENKOV et al.

n

Fig. 1. Experimental lay-out: (1) vacuum neutron guides, (2) collimators, (3) the Te and Sb targets, (4) cryostat, (5) the Tb and Ho targets and (6) neutron detector. N~ /4o

7"6

S6 6.24

7-6 ,a.~

/.Co 3,0"2

"l-b 3,a5

/.5

~,0

O,5

Fig. 2. Part of experimental spectrum obtained after 12 h of measurements: t is the channel number, channel width being 2.5 ~sec and N is the number of counts per channel in 106 units. m a g n e t i c field t h e m e a n p o l a r i z a t i o n o f t h e t a r g e t is zero w h i c h simplifies the o b s e r v a t i o n o f r e s o n a n c e shifts. A s t h e i n v e s t i g a t e d r e s o n a n c e s o f t e r b i u m a n d h o l m i u m w h o s e p a r a m e t e r s are g i v e n in t a b l e 1 d o n o t o v e r l a p , it w a s p o s s i b l e to m e a s u r e simultaneously the transmission of n e u t r o n s through both targets placed TABLE 1

The parameters of investigated terbium and holmium resonances Target nucleus

1S9Tb

159Tb

~59Tb

165H0

165H0

Energy E 0 (eV) F, (meV) F~.(meV) J

3.35 0.35 80 2

4.99 0.081

11.1 7.8 87 2

3.93 2.1 85 4

12.7 10.5

1

4

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MOMENTS

175

in the cryostat one behind the other. A 3He/4He refrigerator 7) kept the temperature of 0.04 K at which the nuclear polarization in the domains fN > 0.98 for both targets. To destroy the polarization the temperature was increased to 1.5 K, the residual polarization being 0.08 for Tb and 0.29 for Ho. As it took an hour to pass from one temperature to another, the measurements with and without polarization alternated every 12 h. In these conditions we took great care of the apparatus time stability. Besides, it was necessary to be able to estimate the actual accuracy of the measured shift AE. A check upon the analyser time scale was carried out with the help of Te and Sb targets permanently placed in the neutron beam (fig. 1). Neutrons transmitted through the targets were registered with the aid of a liquid scintillator detector, while the time spectra were stored in the time analyzing system TPA-1001 computer 9). The whole system operating at 3 × 105 p/sec enabled us to achieve high statistical accuracy. At every 12 h period the spectrum was tape recorded for further processing in a BESM-4 computer. Fig. 2 presents part of such a spectrum. The two spectra obtained at different temperatures made a pair for data processing. Altogether we got 31 pairs of spectra.

3. Data processing and results Each pair was examined separately. It allowed us to reveal the possible systematic errors and to obtain an objective criterion for estimating the degree of accuracy. The shift in the position of the resonance in the spectrum of a polarized target with respect to that of an unpolarized one (i.e. a target with small residual polarization) was obtained in the following way. Let F~ a n d / ~ be the numbers of counts per ith channel of the time analyzer for pairs under consideration. The difference in spectra may not only be due to the shift under investigation, but may be caused also by other effects. These are a change in the time characteristics of the apparatus causing a displacement along the channel axis, a change in the reactor power together with that in the detector efficiency and a background causing a shift along the number-ofcounts axis. Let us form a new spectrum as follows:

TAt

k+~b;

(3)

and find a minimum of the functional

Z2 = ~ (F ° - rr,)~wi,

(4)

i

by varying the parameters At, k and qS. The summation is made with appropriate statistical weights W~ over all channels covering the resonance under consideration. The value of At found is the time shift in the position of the resonance in one

176

V . P . A L F I M E N K O V et al.

a

6"

e

,/

71 /

,

t

Y /I

-40 -20

0

20

40

sr

-40 -20

0

20

t"

-45

0

45

¢

Fig. 3. Examples of the experimental time shift distributions for resonances of (a) Tb (3.35 eV), (b) Ho (3.93 eV) and (c) Sb (6.24 eV). Solid lines are the normal distributions around ( z ) with dispersion caused by the count rate statistics.

spectrum with respect to the other. All investigated and reference resonances (Te and Sb) were treated by this least-squares fitting. The mean weighted value (At) taken over reference resonances is the time shift due to the apparatus. The differences zj = A tj - ( A t) are the level shifts for Tb and Ho caused by the nuclear polarization and hyperfine interaction. Here j is the number of the resonance. The above procedure was applied for every pair obtained. Since each pair is quite independent of the others, the z i values should be normally distributed around the value (z~) averaged over all the pairs, the width of the distribution being an objective estimate of accuracy. Fig. 3 presents histograms of the resonances Tb (3.35 eV), Ho (3.93 eV) and one of the reference Sb resonances (6.24 eV). A solid curve shows a normal distribution around the mean value (z j) with a dispersion calculated by least squares fitting taking into account only the count rate statistics. It is clearly seen that the histogram width and the expected width of the normal distribution are in good agreement, which indicates the absence of systematic errors. TABLE 2 Experimental data on shifts of reference resonances

Target nucleus Energy E o (eV) ( z ) (nsec)

123Te

121 Sb

121Sb

123Sb

123Te

2.33 - 1.5 +4.5

6.24 - 1.7 + 1.8

15.4 0.2_+ 1.0

21.6 - 1.8 _+ 1.2

24.1 3.1 + 1.6

TABLE 3 Experimental data on shifts and magnetic moments of terbium and holmium resonances

Target nucleus E o (eV) (z)(nsec) (AE'?(/~eV) /~j(n.m.)

~59Tb

159Tb

159Tb

165Ho

165Ho

3.35 -6.7+3.1 19 + 9 -0.2+1.0

4.99 3.8+ 6.2 -20 +33 4 . 3 + 3.7

11.1 - 1 . 8 + 2.2 31 + 3 9 - 1 . 7 + 4.4

3.93 - 9 . 7 + 3.2 36 +__12 1.8+ 0.7

12.7 - 0 . 2 + 1.4 4 +30 3.9+ 1.9

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177

Tables 2 and 3 give the experimental data on shifts o f reference resonances and those of Tb and Ho. The error of the mean value (3) was estimated from the variance of experimental z-values. The relation d E = - 2 E o z / t , where E o and t are the resonance energy and the neutron time of flight was used to transform the time shift to the shift of the resonance energy. In order to find magnetic moments of the compound states the experimental values A E and the known constants H and #t [for Tb,/~t = 2.0 n.m. and for Ho,/~1 - 4.0 n.m., ref. 1o)] were substituted into eq. (2). Instead of the nuclear polarizationfh the difference in polarizations calculated at 0.04 and 1.5 K was used. The values obtained for the magnetic moments are summarized in table 3.

4. Discussion of results The experimental data of the present work and those reported in refs. 3, 5) are given in table 4 together with the o-factor values (g = PJ/J) for all the resonances studied. As seen from this table the accuracy of measurement is not very high, the minimum error being 0.4 n.m. Further improvement in the accuracy can hardly be expected, since the results obtained required shift registration with the accuracy of (0.5-1) × 10-s eV for nuclei with the largest hyperfine field. Nuclei suitable for such measurements must have a hyperfine field of about 10 6 0 e , and besides their choice is confined only to those having low energy resonances, since the energetic shift error increases rapidly with an increase of energy. This leads us to the conclusion that one cannot expect an essential increase o f the number of investigated nuclei in the near future. But even the present status of information on magnetic momenta of compound states permits us to make a comparison with theoretical predictions and to draw certain conclusions. TABLE 4 Experimental data on magnetic m o m e n t s and y-factors of neutron resonances of nuclei

Product nucleus l°°Tb 16°Tb 16°Tb 162Dy 162Dy ! 62Dy 164Dy 166Ho 166Ho 16sEr 168Er

Eo

(eV)

3.35 4.99 11.1 2.72 3.69 4.35 1.71 3.93 12.7 0.46 0.58

l~J (n.m.)

y

-0.2+_1,0 4.3_+3.7 -1.7_+4.4 --0.4_+0.7 - 1.8+0.9 0.5 ± 1.2 2.8 _-+-0.5 1.8_+0.7 3.9-+ 1.9 0.9 -+0.4 1.8+0.9

-0.1 +__0.5 4.3 _+3.7 - 0 . 8 _+2.2 --0.13+0.23 -0.90+0.45 0.25 ___+0.60 1.40 + 0.25 0.45-+0.17 0.98+0.47 0,22 + 0.10 0.6 _+0.3

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In refs. 11,12) the magnetic momenta of the excited states of deformed nuclei are calculated within the frame of the thermodynamical approach giving the mean values of the g-factor as well as its fluctuation. For the g-factors of nuclei in the rare-earth region the two theoretical results are identical, ~ = Z/A ~ 0.4, but they are different as far as its fluctuation is concerned, Ag ~ 0.50 [ref. tl)] and Ag ~ 0.25 [ref. 12)]. In these calculations the results obtained for neighbouring nuclei are not distinguishable, therefore, all the experimental data may be considered as a single statistical ensemble.

4

rSJT6 8¢ Dy ~61Dy lSSHo 16ZEr 4.99 2.72 4.35 393 0.46 ~59T6 159T6 16¢Dy ~63D9 mSHo mZEr 335 11.11, 3.69 1.71 12.7 0.58

3 2 1 0

Fig. 4. The g-factor values for all the investigated resonances.

F r o m the data of table 4 and fig. 4 one may conclude that the difference between the g-values is noticeably greater than the experimental errors. It indicates that the g-value fluctuates distinctly from one resonance to another. Consequently, we shall suppose that the observed variance of g-factors Dobs consists of two independent components, namely, the true variance Dtr due to the nature of magnetic momenta and variance Der caused by the measurement errors: Dobs -----Dtr + Def.

(5)

Taking into account the different accuracy of individual g-values the following weights have been used in calculating the mean value ~: wi ~ (Dtr + Ag2)- l,

(6)

where Ag~ is the measured error of g~. Since the Dtr value is not known beforehand, we used successive approximations. As the first approximation to g we took the mean value over nine resonances: g(1) = 0.31. Two terbium resonances at 4.99 and 11.1 eV are excluded since their experimental errors are much greater than those of other resonances and cover the

MAGNETIC

MOMENTS

179

whole range of 0-value fluctuations. With this g(1) value we calculated the observed dispersion for the same resonances: 9

Oobs(1) = ~

[9,--,~7(1)] 2 = 0.45. i=1

It appeared to be much greater than that caused by experimental errors for the same resonances: 9

Dot =

(Agy

= 0.12.

/=1

Using eq. (5) we estimated the true dispersion Dt~(1) = Dobs(1)-De~ = 0.33. Introducing it into eq. (6) one finds the weights w; and the mean weighted value ~(2) = 0:33. The result does not change when two omitted Tb resonances are included because o f their small statistical weights. The calculation of dispersion Dobs (2) using weights, 9

9

Dob~(2) = Z (g,-.q)2wj ~ w,, /=1

/=1

gives Dob~(2) = 0.38 for nine resonances. The next approximation gave practically the same values, i.e. ~ = 0.34 and Dob~ = 0.38 which may be regarded as final. An ordinary estimation of statistical error from the observed dispersion gives ff = 0.34+0.22. In order to estimate errors AD of the observed variance D ob~ let us employ the following relation

ADobs =

Dob s.

(7)

It holds for a r a n d o m sample of n r a n d o m variables distributed according to the normal law with a variance Dobs [see for example ref. 13)]. In our case the above requirements are met since the experimental errors are smaller than or equal to the true fluctuations. F o r m u l a (7) gives ADobs = 0.19. The same error may be ascribed to the value Dtr which we are interested in, as the Der is much smaller. Finally, for the true dispersion we have D t r = 0.26+0.19. F r o m here we obtained the mean square deviation

A9 = 0.51 ___0.20. The error A 9 is obtained from the error interval o f D t r a s a median value.

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V.P. ALFIMENKOV et al.

Let us now compare the experimental results and theoretical estimates. The mean g-value obtained in the experiment is consistent with the theoretically calculated = 0.4 and no comment is necessary. The experimental value of Ag is consistent with 0.5 estimated in ref. t 1)] and is somewhat greater than 0.25 predicted in ref. 12). It should be noted, however, that the author of ref. 12) regarded the value 0.25 as a lower limit and pointed to possible ways of increasing it. Thus, the experimental data on magnetic moments of nuclear compound states are consistent with theoretical estimates for the mean value of g-factors. They point to comparatively large fluctuations of magnetic moments, though the statistical plausibility of this conclusion should be improved. We are pleased to express our gratitude to Prof. I. M. Frank for his interest in the work, to G. G. Bunatian for useful discussions, to T. S. Afanasieva, B. A. Rodionov, V. A. Vagov and S. Salai for help in preparing the experimental equipment and for assistance during the measurements, and to T. F. Dmitrieva for the translation. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

F. L. Shapiro, Research applications of nuclear pulsed systems (Vienna, IAEA, 1967) p. 176 F. L. Shapiro, Polarized targets and ion sources (Saclay, CEA, 1967) p. 339 V. P. Alfimenkov et al., Yad. Fiz. 17 (1973) 13 K. H. Beckurts and G. Brunhart, Phys. Rev. C1 (1970) 726 V. P. Alfimenkov et al., Phys. Lett. 53B (1975) 429 Neutron cross sections BNL, 3rd ed., 1 (1973) V. P. Alfimenkov and O. N. Ovchinnikov, JINR Report P8-9168 (1975) H. Malecki et al., JINR Report 13-6609 (1972) V. A. Vagov, V. N. Zamrij and S. Salai, Proc. 7th Int. Syrup. on nuclear electronics, Budapest, 1973 (JINR Report D13-7616, 1974) p. 358 G. Goldring and R. Kalish, Hyperfine interactions in excited nuclei (New York, 1971) p. 1255 R. N. Kuklin, Yad. Fiz. 6 (1967) 969 G. G. Bunatian, JINR Report P4-8889 (1975) D. J. Hudson, Lectures on elementary statistics and probability (Geneva, 1964)