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On the fine structure in delayed-proton spectra
2.D
1
Nuclear Physics/1,206 (1973) 583--592; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON THE FINE S T R U C T U R E IN DELAYED-PROTON SPECTRA V. A. K A R N A U K H O V , D. D. BOGDANOV and L. A. PETROV
Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, USSR Received 7 February 1973 Abstract: The fine structure in delayed-proton spectra is interpreted on the basis of statistical
fluctuations associated with the variation of/~-decay matrix elements and the relative proton widths o f the excited states. Formulae are derived which give the variance of relative proton intensity in terms of the average characteristics of the delayed proton emission and the level density. The experimental data on the variance of the relative proton intensity for 1~~Te and the calculations using these formulae are in satisfactory agreement. The fluctuations predicted for l°9Te considerably exceed the measured ones.
E
RADIOACTIVITY 111Te [from 1°2pd(12C, 3n)], J°9Te [from 96Ru (160, 3n)] measured fl-delayed protons; deduced variance of relative proton intensity tr2flr, (E)/(1, (E)), level density p(E). 111. logsb" Enriched target.
[
I
1. Introduction
Observed 1-3) delayed-proton spectra show a pronounced single-peaked gross structure, i.e., the average intensity of protons increases initially with energy and then decreases. In certain cases one observes some local resonances with a half-width of 0.5 to 1.0 MeV, which seem to be associated with resonances in the ]/-decay strength function. For instance, in the case of 109Te [ref. 4)] such a resonance is ascribed to the effect o f three-quasiparticle states pg~ lng~nd~ being populated strongly in /~-transitions. However, high-energy-resolution measurements can reveal the fine structure in delayed-proton spectra. For nuclei with Z < 30, the proton transitions between certain states are resolved. As Z increases, the level density increases and hence individual transitions become unresolved; however, intensity fluctuations remain. The magnitude of these fluctuations is dependent on the variations in the strength of individual transitions and on the detector energy resolution relative to the level spacing. Therefore, from the study of the fine structure, it may be possible to obtain some information about the level density in the excitation energy range of 3 to 7 MeV, for nuclei far from the fl-stability line. In an earlier work 2), a rather crude attempt was made to statistically analyse intensity fluctuations observed in the 11iTe delayed proton spectrum. The distribution of the relative intensities of certain proton peaks in the proton spectrum was compared with the g 2 distribution to determine the effective number o f degrees of freedom related to the average level density. 583
584
V.A. KARNAUKHOV
et al,
Hansen 5) suggests a more fruitful method of analysing the fluctuations, which is based upon the approach developed by Egelstaff 6) for total and partial neutron cross sections. A formula was derived s) to express the variance of proton intensity as a function of energy in terms of the level density and the average characteristics of the fl+ transition and proton emission. In the present paper we also take advantage of Egelstaff's concepts 6). We have derived an expression for the variation of relative proton intensity (in the proton spectrum) which is similar to the formula of ref. 5). However, there is a distinction which results in a factor of 2 difference in that part of the spectrum where the average proton width ( F p ) is much less than the total width ( F ) , The calculated variation of the relative proton intensity and that observed for a a aTe and 1o 9Te are compared. 2. Theory 2.1. RESOLVED PROTON LINES The condition F < 6E << D applies to this case, where 6E is the energy resolution of the detector and D is the average level spacing. The intensity of the delayed protons associated with the decay of the vth state with spin i to the final state f a t an energy E I is equal to =
eo-
A-1Epv-Es')#'" \F /'S,"
Here o~(Z, Q) is the known statistical rate function for the sum of positon decay and electron capture, #2.. is the square of the fl-decay matrix element, Qo is the JlV total K-capture energy, Bp is the proton binding energy in the daughter nucleus, and Ep is the kinetic energy of the proton. Differences in the structure of the states involved lead to the dispersion of the quantities Pff~vand (Fp/F)t$v with respect to their average values, even with a slight change in excitation energy. Consequently, the quantity Y~I~ fluctuates with the changing value of v. As is commonly done, we take the variances of the quantity Yo'~/(Y~Jr) as a measure of the fluctuations ~ou-: =
=
(2)
We assume that both the fl-decay and proton emission are statistically independent processes t. If we then employ the expression for the dispersion of the product of statistically independent quantities, we obtain
o,Aep )
_
+
+
a [(rdr),A
(3)
Here v is omitted for convenience. The denominators of all terms in eq. (3) are the squares of the mathematically t The problem of correlation between/z2 and Fp/F needs a special study.
DELAYED-PROTON SPECTRA
585
expected values of the corresponding quantities for excitation energy E = Bp+ ( A / ( A - 1 ) ) E p + E f . One may expect that the squares of fl-decay matrix elements, like the partial widths for particle emission, obey the Porter-Thomas distribution 7). Then the first term in eq. (3) is equal to 2, and consequently
qhf = 2 + 3 a2[(Fp/F)iY]
(4)
"
Some experimental evidence in support of this assumption is provided by fig. 1 which displays the distribution function for the squares of the matrix elements for the known allowed ]/-transitions [the data from ref. s) are used]. The experimental data for fl-
~o
0
t0
20
30
,~0
xo (/-,~/:> ) ~
50
Fig. 1. The distribution o f the quantity (#2/(~2))~., where t~ is a fl-decay matrix element. Histogram 1 includes all allowed fl-transitions k n o w n before the appearance o f ref. s) (487 events) and histogram 2 corresponds to fl÷ transitions (139 events). Smooth curves show the distributions with the number o f degrees o f freedom n = 1 and 2. The inserted plot shows the initial distributions o f log ft.
and fl+ decay are close to the X2 distribution with the number of degrees of freedom equal to unity. Of course, the matrix elements should be expected to show a systematic behaviour with variation of A, Z and Q. It means that one should analyse the data over a rather narrow range of A, Z and Q to make a correct conclusion about the E-decays element distribution. However, this is impracticable because of poor statistics. In addition to fig. 1, we have considered the data for 70 < A < 110; the distribution of the matrix elements squared has also turned out to be close to the Porter-Thomas one. It is reasonable to assume that the proton width F v, like the neutron width F,, obeys the Porter-Thomas distribution. The distribution of the relative proton widths (Fp/F)~f depends on the value of (Fp)/(F~,). In the case where (F~)>> ( F p ) , the total width actually coincides with Ft. This means that the distribution of (l-'p:r)i $ is practically the same as that of ( / " p ) i f , since the dispersion of Fr/(Fr) is small. In
586
V.A. KARNAUKHOV
et al.
this case the second term in (4) is equal to 6. Now we consider the case where (Fy) << ( F v ) , and where the total width is approximately equal to the sum of partial proton widths: F i ~ ~¢(Fp)o,. If the transition to the ground state is predominant, the second term in (4) will be small. If the different final states are populated in p-decay with comparable probabilities, the situation becomes more complicated. In the transitions to the collective excited states one can expect a strong correlation between the partial proton widths. This will result in a small difference between the value of ( / ' P ) i f / 2 f ( F P ) i f and its average, and therefore the second term in (4) will be close to zero. However, if the correlation between the proton partial widths is weak, the second term in (4) may also appear to be rather large in the case of ( F p ) >> (F~). Thus the value of ~Oiy decreases monotonically (from a maximum of 8 to 2) with increasing proton energy. 2.0
I
-..
i
!
L ~ [ :' iil
:
i i ii i Fig. 2. The relative variance of 119/1' as a function of ( / ' p ) / ( / ' 7 ) . distribution was used; F~, ,~ ( F T ) .
For /'p the Porter-Thomas
The difference between the formula for the fluctuation (3) and the analogous one in ref. 5) consists in the presence in the former of the third cross term equal to the sum of the two first terms if ( F p ) << ( F r ) . Fig. 2 shows the dependence of a2(Fp/F)/(Fp/F)2 upon (Fp)/(Fr) for the case where only one partial proton width is predominant. Such a situation exists for the delayed proton emitters xx~Te and lO9Te" At least 90 % of the spectra are associated with the p-decay into the ground state of the final nucleus. The curve in fig. 2 is calculated with the assumption of a Porter-Thomas distribution for Fp and of a Gaussian one with negligible dispersion for F 7. 2.2. F L U C T U A T I O N I N P R O T O N I N T E N S I T Y S U M M E D O V E R A N E N E R G Y I N T E R V A L d E p >> D, 6E
This case is entirely analogous to that considered by Egelstaff 6) in an analysis of the average values of neutron cross sections. The proton intensity in the energy interval from Ep to Ep÷AEpis equal to
I(E.)AE. = Z E if
v
= E if
Z ( v
2r.lr),fv] •
(5)
With changes in energy, the value of I(Ep) will fluctuate around its average value
DELAYED-PROTON SPECTRA
587
due to fluctuations in the intensities of individual lines and in the level density. It is shown by Egelstaff 6) that the contribution of the level density to the variances in average neutron cross sections is rather small. A similar situation characterizes the delayed proton emission. Indeed, the level spacing d obeys the Wigner distribution and consequently tr2(d)/ 2 = 0.27. This quantity is substantially less than (&s, i.e., the intensity fluctuations in the proton spectrum are mainly due to the variances of p21 and (Fp/F)t :, as has already been noted in ref. 5). If we make use of the known rules for the variance of the linear combination of statistically independent quantities and neglect level spacing dispersion, we obtain
dp = a 2 ( I(Ep) ~ = E2 Dig" \! i: ~ ~prP':"
(6)
Here/ = (oif (Ep) is essentially the weight factor for the contribution to the proton spectrum from the i ---}f transition; d Ep/Di: is equal to the average number of i ---}f transitions falling into the energy interval AEp. 2.3 UNRESOLVED PROTON LINES (6E >> D >>/')
This particular condition would apply to proton emitters in the region of Z > 50. In this case the proton spectrum is formed by summing the contributions from separate lines into channels (with a width of dE), the weight factor being determined by the detector response function which is assumed to be a Gaussian distribution with the half-width fiE. Thus in eq. (6) there appears the weight factor taking account of the contribution from the vth proton line, and the summation over v is made over the entire spectrum. If in finding the variance of I(Ep)dEp one goes from the summation over v to integration, one obtains (dEp << 6E): (~(Ep) =
-2 ,Z. : 2 Dif
(7)
The structure of this formula is the same as (12) of ref. 5), but the factors 9i: are different. The value of 1.53E/DI: is equal to the effective number of the i - ~ f t r a n sitions contributing to the spectrum interval of Ep to Ep + AEp. Eq. (7) is substantially simplified when in the proton decay one state of the final nucleus is predominantly populated. This situation occurs for the delayed proton emitters a o 9Te and 1t 1Te" In this case we can restrict ourselves to the consideration, with good accuracy, of the proton transition to the ground state of the l°SSn and 1lOSn nuclei, and hence eq. (7) is rewritten as follows: ~b(Ep)
= (l.56E)-l(•gjlSi)-2 i
_ L~ g j i2 S2D i icpi.
(8)
i
In deriving (8) it has been assumed that the strength functions associated with the
588
V.A. KARNAUKHOV et al.
allowed fl-transitions from the j-spin state to the different final/-spin states are distinguished only by the statistical weight factor gi~" In the summation, i takes the values ½, ~r and ~, since the spin of to9, lllTe is taken to be equal to ~+ [refs. 1, 4)]:
s,
3. C o m p a r i s o n o f calculations and e x p e r i m e n t a l data for lO9, l t l T e
To perform calculations using formula (8), one should know ##, t, Si, Di and ~pi. Following ref. 3) we take g~, i = (2i+ 1)/18. In order to obtain the Si values, we have used the proton and 7-ray widths calculated earlier 4). It should be noted that S~ does not exceed 0.015, i.e., the ~+ levels do not substantially change the picture of
/ /04-
,m
---
~
~s/~
,101
//./
"-., /¢/
L ///_
-
m. i
-
//I
2.0
3.0 ~0
5.0
~.0
7.o
80
9.0 E ~eV
Fig. 3. The density of the 3 + and [[+ levels for l°9Sb and IHSb calculated in accordance with ref. 9). fluctuations. The average level spacing was calculated following Gilbert and Cameron [ref. 9)] (fig. 3). The proton binding energies for 1°9Sb and 111Sb were taken to be equal to 1.3 MeV and 1.8 MeV, respectively 2, 4). For determining the experimental value of the variances of relative intensities in the proton spectrum, one should first find ( I ( E v ) } . It seems unreasonable to use for this function the spectrum calculated in the framework of the statistical model of delayed proton emission because of the possible presence of intermediate structures in the fl-strength function which makes this calculation somewhat ambiguous. To
DELAYED-PROTON SPECTRA
589
find (/(Ep)), we smoothed the experimental spectrum using the Gaussian weight function _ 1 [ . - ( x - x')2.7
x/~ A~
~, I(x')exp L 2A~
/ ,
(9)
where ~: represents the channel number. The choice of A~ is somewhat arbitrary. As a starting point in our calculations, we assume that the half-width of the weight function (2.36 A~) should considerably exceed the energy resolution but not the halfwidth of the possible local resonances of the fl-strength function and of the proton strength function 4). The value of A~ has been chosen to be equal to 127 keV (which corresponds to a half-width of 300 keV). 250-
Ip ,700
150
~0o
50.
151]
200
20
250
?5
Joo 30
35o 3~
4oo ~o
,~o
sbo N ~.5 Ep M,v
Fig. 4. The I°gTe delayed-proton spectrum. The smooth curve is obtained from smoothing by a Gaussian function with a hail-width of 300 keV.
Fig. ,4 shows the 109Te delayed proton spectrum measured in the experiments described in ref. 4) and smoothed by formula (9). In order to separate protons from a-particles, use has been made of a proportional-counter-semiconducting-detector telescope. As a result, the energy resolution was moderate, ranging from 85 to 60 keV with increasing proton energy in the region of interest. Nevertheless the spectrum exhibits a pronounced structure which disappears after the smoothing procedure. However, a broad peak remains in the region of 3.5 MeV, which is associated with the resonance in the//-strength function 4). In the determination of the variance of the relative intensity I(Er,)/(I(Ep) ) the spectrum was divided into intervals of 250 keV each. For each of these intervals the estimate of the variance was made 1o) corresponding to the average energy of the interval. The experimental variance consists of two components: the first one is as-
590
V . A . K A R N A U K H O V et aL
sociated with the fluctuations in the strength of the different partial transitions contributing to the intensity, while the second is due to the counting rate statistics. Here we are interested in the first component. In order to determine this, we have subtracted the variance due to the statistical error of the measurement from the measured values of the variances of the relative intensity. In obtaining the experimental value of ~(Ep) it is assumed that the values of the relative intensities in the different channels are statistically independent. However, if the width of a channel is less than the energy resolution, one cannot avoid the correlation of intensities in neighbouring channels. This may affect the experimental value of the variances and thus result in some deviation from that predicted by formula (8).
i
'
\xx ~
i
'
~%
J
÷l
EpHeV Fig. 5. The variance of relative intensity in the 11XTe proton spectrum with the resolution of 27 keV (dots) and 55 keV (open circles). Curves correspond to the calculation by formula (8).
Fig. 5 shows a comparison of the experimental and calculated values of the relative intensity variances for 111Te. The 1~1Te delayed proton spectra measured with resolutions of 25-30 keV and 50-60 keV are taken from ref. 1); the calculations are performed for fiE of 27 and 55 keV. We note that the values calculated by formula (8) for b E = 27 keV are close to the experimental ones for Ep > 2.7 MeV. At lower energies the intensity fluctuations are smaller than expected. This may be due to the contribution of fl-background to this part of the spectrum; the fl-background increases the average intensity without changing the fluctuating part. The spectrum measured with a resolution of 50-60 keV exhibits fluctuations that are by 2.7 ___0.8 times weaker than those of the first spectrum. According to formula (8) with increasing 6 E from 27 to 55 MeV, one might expect a two-fold decrease in the relative variation. A further possible suppression of fluctuations is due to the effect of the correlation between the intensities in neighbouring channels which increases with increasing fiE. Although the accuracy in the estimation of this effect is not high, this effect does not essentially influence the results obtained at the resolution of 6 E = 27 keV.
DELAYED-PROTON SPECTRA
591
After considering the data presented in fig. 5 one can draw the following general conclusion. The calculation describes satisfactorily the fluctuations in the delayedproton spectrum for 11'Te. The level density for l'~Sb differs from that calculated following Gilbert and Cameron by no more than a factor of 1.5. This deviation is less than that corresponding to the uncertainty in the level density due to the error in the estimation of Bp for ~l~Sb [ref. 2)]. In fact, the proton binding energy for ~llSb has been estimated to an accuracy of about 200 keV. Consequently, this is the uncertainty in knowledge of the excitation energy of the state which emits protons. According to fig. 3 a change of 250 keV in energy results in a variation of the level density by a factor of 1.5. f0*
o_ ,.._.
20
30
~o
EpMeV
Fig. 6. The variance ofrelativeintensityin the l°9Te proton spectrum. Fig. 6 presents the results for l°9Te. The calculations using formula (8) with the level density from ref. 9) predict considerably larger fluctuations than those observed experimentally. To explain the measured magnitude of the fluctuations, one can assume that either the level density of 1o9Sb exceeds by 4-5 times the calculated one, or the proton binding energy in 1°98b is 2.5 MeV, and not 1.3 MeV. The latter assumption seems less probable 4). The data obtained for ~o 9Te should be regarded as preliminary. We are unable to eliminate completely the systematic errors due to the insufficient energy resolution, which damp the observed intensity fluctuations. It is desirable to repeat an analysis of the fine structure in the proton spectrum of 1o 9Te after measurements with a higher resolution. We would like to stress the importance of performing experiments which would permit the determination of Bp, such as the measurements of the quantities Qo and Qo-Bp for delayed-proton emitters. The problem of determining Qo for isotopes with low yield is not easy. However, it is worth tackling since a precise knowledge of Bp will enable us to obtain more accurate information from the delayed-proton spectrum studies. The authors are thankful to Academician G. N. Flerov for his interest, and to S. Batselev and S. N. Bogdanova for assistance in performing numerical calculations.
592
V . A . K A R N A U K H O V et aL
References 1) D. D. Bogdanov, S. Daroci, V. A. Karnaukhov, L. A. Petrov and G. M. Ter-Akopyan, Yad. Fiz. 6 (1967) 893 2) V. A. Karnaukhov, Yad. Fiz. 10 (1969) 450 3) P. I-[ornshaj, K. Wilsky, P. G. Hansen, B. Jonson and O. B. Nielsen, Nucl. Phys. A187 (1972) 609 4) D. D. Bogdanov, V. A. Karnaukhov and L. A. Petrov, J I N R preprint P6-6861 5) P. G. I-Iansen, Nuclear structure lectures, Alushta, 1972, J I N R D-6465, 365 (Dubna, 1972) 6) P. A. Egelstaff, Proc. Phys. Soc. 71 (1958) 910 7) C. E. Porter and R. G. Thomas, Phys. Rev. 104 (1956) 483 8) C. Gleit, G. W. Tang and C. D. Coryell, Nucl. Data Sheets 5, $5 (1963) 109; C. S. I-Jan and N. B. Gove, ibid., p. t46 9) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 10) D. Hudson, Statistics for physicists (Mir, Moscow, 1967)
1
Nuclear Physics/1,206 (1973) 583--592; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
ON THE FINE S T R U C T U R E IN DELAYED-PROTON SPECTRA V. A. K A R N A U K H O V , D. D. BOGDANOV and L. A. PETROV
Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, USSR Received 7 February 1973 Abstract: The fine structure in delayed-proton spectra is interpreted on the basis of statistical
fluctuations associated with the variation of/~-decay matrix elements and the relative proton widths o f the excited states. Formulae are derived which give the variance of relative proton intensity in terms of the average characteristics of the delayed proton emission and the level density. The experimental data on the variance of the relative proton intensity for 1~~Te and the calculations using these formulae are in satisfactory agreement. The fluctuations predicted for l°9Te considerably exceed the measured ones.
E
RADIOACTIVITY 111Te [from 1°2pd(12C, 3n)], J°9Te [from 96Ru (160, 3n)] measured fl-delayed protons; deduced variance of relative proton intensity tr2flr, (E)/(1, (E)), level density p(E). 111. logsb" Enriched target.
[
I
1. Introduction
Observed 1-3) delayed-proton spectra show a pronounced single-peaked gross structure, i.e., the average intensity of protons increases initially with energy and then decreases. In certain cases one observes some local resonances with a half-width of 0.5 to 1.0 MeV, which seem to be associated with resonances in the ]/-decay strength function. For instance, in the case of 109Te [ref. 4)] such a resonance is ascribed to the effect o f three-quasiparticle states pg~ lng~nd~ being populated strongly in /~-transitions. However, high-energy-resolution measurements can reveal the fine structure in delayed-proton spectra. For nuclei with Z < 30, the proton transitions between certain states are resolved. As Z increases, the level density increases and hence individual transitions become unresolved; however, intensity fluctuations remain. The magnitude of these fluctuations is dependent on the variations in the strength of individual transitions and on the detector energy resolution relative to the level spacing. Therefore, from the study of the fine structure, it may be possible to obtain some information about the level density in the excitation energy range of 3 to 7 MeV, for nuclei far from the fl-stability line. In an earlier work 2), a rather crude attempt was made to statistically analyse intensity fluctuations observed in the 11iTe delayed proton spectrum. The distribution of the relative intensities of certain proton peaks in the proton spectrum was compared with the g 2 distribution to determine the effective number o f degrees of freedom related to the average level density. 583
584
V.A. KARNAUKHOV
et al,
Hansen 5) suggests a more fruitful method of analysing the fluctuations, which is based upon the approach developed by Egelstaff 6) for total and partial neutron cross sections. A formula was derived s) to express the variance of proton intensity as a function of energy in terms of the level density and the average characteristics of the fl+ transition and proton emission. In the present paper we also take advantage of Egelstaff's concepts 6). We have derived an expression for the variation of relative proton intensity (in the proton spectrum) which is similar to the formula of ref. 5). However, there is a distinction which results in a factor of 2 difference in that part of the spectrum where the average proton width ( F p ) is much less than the total width ( F ) , The calculated variation of the relative proton intensity and that observed for a a aTe and 1o 9Te are compared. 2. Theory 2.1. RESOLVED PROTON LINES The condition F < 6E << D applies to this case, where 6E is the energy resolution of the detector and D is the average level spacing. The intensity of the delayed protons associated with the decay of the vth state with spin i to the final state f a t an energy E I is equal to =
eo-
A-1Epv-Es')#'" \F /'S,"
Here o~(Z, Q) is the known statistical rate function for the sum of positon decay and electron capture, #2.. is the square of the fl-decay matrix element, Qo is the JlV total K-capture energy, Bp is the proton binding energy in the daughter nucleus, and Ep is the kinetic energy of the proton. Differences in the structure of the states involved lead to the dispersion of the quantities Pff~vand (Fp/F)t$v with respect to their average values, even with a slight change in excitation energy. Consequently, the quantity Y~I~ fluctuates with the changing value of v. As is commonly done, we take the variances of the quantity Yo'~/(Y~Jr) as a measure of the fluctuations ~ou-: =
=
(2)
We assume that both the fl-decay and proton emission are statistically independent processes t. If we then employ the expression for the dispersion of the product of statistically independent quantities, we obtain
o,Aep )
_
+
+
a [(rdr),A
(3)
Here v is omitted for convenience. The denominators of all terms in eq. (3) are the squares of the mathematically t The problem of correlation between/z2 and Fp/F needs a special study.
DELAYED-PROTON SPECTRA
585
expected values of the corresponding quantities for excitation energy E = Bp+ ( A / ( A - 1 ) ) E p + E f . One may expect that the squares of fl-decay matrix elements, like the partial widths for particle emission, obey the Porter-Thomas distribution 7). Then the first term in eq. (3) is equal to 2, and consequently
qhf = 2 + 3 a2[(Fp/F)iY]
(4)
"
Some experimental evidence in support of this assumption is provided by fig. 1 which displays the distribution function for the squares of the matrix elements for the known allowed ]/-transitions [the data from ref. s) are used]. The experimental data for fl-
~o
0
t0
20
30
,~0
xo (/-,~/:> ) ~
50
Fig. 1. The distribution o f the quantity (#2/(~2))~., where t~ is a fl-decay matrix element. Histogram 1 includes all allowed fl-transitions k n o w n before the appearance o f ref. s) (487 events) and histogram 2 corresponds to fl÷ transitions (139 events). Smooth curves show the distributions with the number o f degrees o f freedom n = 1 and 2. The inserted plot shows the initial distributions o f log ft.
and fl+ decay are close to the X2 distribution with the number of degrees of freedom equal to unity. Of course, the matrix elements should be expected to show a systematic behaviour with variation of A, Z and Q. It means that one should analyse the data over a rather narrow range of A, Z and Q to make a correct conclusion about the E-decays element distribution. However, this is impracticable because of poor statistics. In addition to fig. 1, we have considered the data for 70 < A < 110; the distribution of the matrix elements squared has also turned out to be close to the Porter-Thomas one. It is reasonable to assume that the proton width F v, like the neutron width F,, obeys the Porter-Thomas distribution. The distribution of the relative proton widths (Fp/F)~f depends on the value of (Fp)/(F~,). In the case where (F~)>> ( F p ) , the total width actually coincides with Ft. This means that the distribution of (l-'p:r)i $ is practically the same as that of ( / " p ) i f , since the dispersion of Fr/(Fr) is small. In
586
V.A. KARNAUKHOV
et al.
this case the second term in (4) is equal to 6. Now we consider the case where (Fy) << ( F v ) , and where the total width is approximately equal to the sum of partial proton widths: F i ~ ~¢(Fp)o,. If the transition to the ground state is predominant, the second term in (4) will be small. If the different final states are populated in p-decay with comparable probabilities, the situation becomes more complicated. In the transitions to the collective excited states one can expect a strong correlation between the partial proton widths. This will result in a small difference between the value of ( / ' P ) i f / 2 f ( F P ) i f and its average, and therefore the second term in (4) will be close to zero. However, if the correlation between the proton partial widths is weak, the second term in (4) may also appear to be rather large in the case of ( F p ) >> (F~). Thus the value of ~Oiy decreases monotonically (from a maximum of 8 to 2) with increasing proton energy. 2.0
I
-..
i
!
L ~ [ :' iil
:
i i ii i Fig. 2. The relative variance of 119/1' as a function of ( / ' p ) / ( / ' 7 ) . distribution was used; F~, ,~ ( F T ) .
For /'p the Porter-Thomas
The difference between the formula for the fluctuation (3) and the analogous one in ref. 5) consists in the presence in the former of the third cross term equal to the sum of the two first terms if ( F p ) << ( F r ) . Fig. 2 shows the dependence of a2(Fp/F)/(Fp/F)2 upon (Fp)/(Fr) for the case where only one partial proton width is predominant. Such a situation exists for the delayed proton emitters xx~Te and lO9Te" At least 90 % of the spectra are associated with the p-decay into the ground state of the final nucleus. The curve in fig. 2 is calculated with the assumption of a Porter-Thomas distribution for Fp and of a Gaussian one with negligible dispersion for F 7. 2.2. F L U C T U A T I O N I N P R O T O N I N T E N S I T Y S U M M E D O V E R A N E N E R G Y I N T E R V A L d E p >> D, 6E
This case is entirely analogous to that considered by Egelstaff 6) in an analysis of the average values of neutron cross sections. The proton intensity in the energy interval from Ep to Ep÷AEpis equal to
I(E.)AE. = Z E if
v
= E if
Z ( v
2r.lr),fv] •
(5)
With changes in energy, the value of I(Ep) will fluctuate around its average value
DELAYED-PROTON SPECTRA
587
dp = a 2 ( I(Ep) ~ = E
(6)
Here
This particular condition would apply to proton emitters in the region of Z > 50. In this case the proton spectrum is formed by summing the contributions from separate lines into channels (with a width of dE), the weight factor being determined by the detector response function which is assumed to be a Gaussian distribution with the half-width fiE. Thus in eq. (6) there appears the weight factor taking account of the contribution from the vth proton line, and the summation over v is made over the entire spectrum. If in finding the variance of I(Ep)dEp one goes from the summation over v to integration, one obtains (dEp << 6E): (~(Ep) =
(7)
The structure of this formula is the same as (12) of ref. 5), but the factors 9i: are different. The value of 1.53E/DI: is equal to the effective number of the i - ~ f t r a n sitions contributing to the spectrum interval of Ep to Ep + AEp. Eq. (7) is substantially simplified when in the proton decay one state of the final nucleus is predominantly populated. This situation occurs for the delayed proton emitters a o 9Te and 1t 1Te" In this case we can restrict ourselves to the consideration, with good accuracy, of the proton transition to the ground state of the l°SSn and 1lOSn nuclei, and hence eq. (7) is rewritten as follows: ~b(Ep)
= (l.56E)-l(•gjlSi)-2 i
_ L~ g j i2 S2D i icpi.
(8)
i
In deriving (8) it has been assumed that the strength functions associated with the
588
V.A. KARNAUKHOV et al.
allowed fl-transitions from the j-spin state to the different final/-spin states are distinguished only by the statistical weight factor gi~" In the summation, i takes the values ½, ~r and ~, since the spin of to9, lllTe is taken to be equal to ~+ [refs. 1, 4)]:
s,
3. C o m p a r i s o n o f calculations and e x p e r i m e n t a l data for lO9, l t l T e
To perform calculations using formula (8), one should know ##, t, Si, Di and ~pi. Following ref. 3) we take g~, i = (2i+ 1)/18. In order to obtain the Si values, we have used the proton and 7-ray widths calculated earlier 4). It should be noted that S~ does not exceed 0.015, i.e., the ~+ levels do not substantially change the picture of
/ /04-
,m
---
~
~s/~
,101
//./
"-., /¢/
L ///_
-
m. i
-
//I
2.0
3.0 ~0
5.0
~.0
7.o
80
9.0 E ~eV
Fig. 3. The density of the 3 + and [[+ levels for l°9Sb and IHSb calculated in accordance with ref. 9). fluctuations. The average level spacing was calculated following Gilbert and Cameron [ref. 9)] (fig. 3). The proton binding energies for 1°9Sb and 111Sb were taken to be equal to 1.3 MeV and 1.8 MeV, respectively 2, 4). For determining the experimental value of the variances of relative intensities in the proton spectrum, one should first find ( I ( E v ) } . It seems unreasonable to use for this function the spectrum calculated in the framework of the statistical model of delayed proton emission because of the possible presence of intermediate structures in the fl-strength function which makes this calculation somewhat ambiguous. To
DELAYED-PROTON SPECTRA
589
find (/(Ep)), we smoothed the experimental spectrum using the Gaussian weight function _ 1 [ . - ( x - x')2.7
x/~ A~
~, I(x')exp L 2A~
/ ,
(9)
where ~: represents the channel number. The choice of A~ is somewhat arbitrary. As a starting point in our calculations, we assume that the half-width of the weight function (2.36 A~) should considerably exceed the energy resolution but not the halfwidth of the possible local resonances of the fl-strength function and of the proton strength function 4). The value of A~ has been chosen to be equal to 127 keV (which corresponds to a half-width of 300 keV). 250-
Ip ,700
150
~0o
50.
151]
200
20
250
?5
Joo 30
35o 3~
4oo ~o
,~o
sbo N ~.5 Ep M,v
Fig. 4. The I°gTe delayed-proton spectrum. The smooth curve is obtained from smoothing by a Gaussian function with a hail-width of 300 keV.
Fig. ,4 shows the 109Te delayed proton spectrum measured in the experiments described in ref. 4) and smoothed by formula (9). In order to separate protons from a-particles, use has been made of a proportional-counter-semiconducting-detector telescope. As a result, the energy resolution was moderate, ranging from 85 to 60 keV with increasing proton energy in the region of interest. Nevertheless the spectrum exhibits a pronounced structure which disappears after the smoothing procedure. However, a broad peak remains in the region of 3.5 MeV, which is associated with the resonance in the//-strength function 4). In the determination of the variance of the relative intensity I(Er,)/(I(Ep) ) the spectrum was divided into intervals of 250 keV each. For each of these intervals the estimate of the variance was made 1o) corresponding to the average energy of the interval. The experimental variance consists of two components: the first one is as-
590
V . A . K A R N A U K H O V et aL
sociated with the fluctuations in the strength of the different partial transitions contributing to the intensity, while the second is due to the counting rate statistics. Here we are interested in the first component. In order to determine this, we have subtracted the variance due to the statistical error of the measurement from the measured values of the variances of the relative intensity. In obtaining the experimental value of ~(Ep) it is assumed that the values of the relative intensities in the different channels are statistically independent. However, if the width of a channel is less than the energy resolution, one cannot avoid the correlation of intensities in neighbouring channels. This may affect the experimental value of the variances and thus result in some deviation from that predicted by formula (8).
i
'
\xx ~
i
'
~%
J
÷l
EpHeV Fig. 5. The variance of relative intensity in the 11XTe proton spectrum with the resolution of 27 keV (dots) and 55 keV (open circles). Curves correspond to the calculation by formula (8).
Fig. 5 shows a comparison of the experimental and calculated values of the relative intensity variances for 111Te. The 1~1Te delayed proton spectra measured with resolutions of 25-30 keV and 50-60 keV are taken from ref. 1); the calculations are performed for fiE of 27 and 55 keV. We note that the values calculated by formula (8) for b E = 27 keV are close to the experimental ones for Ep > 2.7 MeV. At lower energies the intensity fluctuations are smaller than expected. This may be due to the contribution of fl-background to this part of the spectrum; the fl-background increases the average intensity without changing the fluctuating part. The spectrum measured with a resolution of 50-60 keV exhibits fluctuations that are by 2.7 ___0.8 times weaker than those of the first spectrum. According to formula (8) with increasing 6 E from 27 to 55 MeV, one might expect a two-fold decrease in the relative variation. A further possible suppression of fluctuations is due to the effect of the correlation between the intensities in neighbouring channels which increases with increasing fiE. Although the accuracy in the estimation of this effect is not high, this effect does not essentially influence the results obtained at the resolution of 6 E = 27 keV.
DELAYED-PROTON SPECTRA
591
After considering the data presented in fig. 5 one can draw the following general conclusion. The calculation describes satisfactorily the fluctuations in the delayedproton spectrum for 11'Te. The level density for l'~Sb differs from that calculated following Gilbert and Cameron by no more than a factor of 1.5. This deviation is less than that corresponding to the uncertainty in the level density due to the error in the estimation of Bp for ~l~Sb [ref. 2)]. In fact, the proton binding energy for ~llSb has been estimated to an accuracy of about 200 keV. Consequently, this is the uncertainty in knowledge of the excitation energy of the state which emits protons. According to fig. 3 a change of 250 keV in energy results in a variation of the level density by a factor of 1.5. f0*
o_ ,.._.
20
30
~o
EpMeV
Fig. 6. The variance ofrelativeintensityin the l°9Te proton spectrum. Fig. 6 presents the results for l°9Te. The calculations using formula (8) with the level density from ref. 9) predict considerably larger fluctuations than those observed experimentally. To explain the measured magnitude of the fluctuations, one can assume that either the level density of 1o9Sb exceeds by 4-5 times the calculated one, or the proton binding energy in 1°98b is 2.5 MeV, and not 1.3 MeV. The latter assumption seems less probable 4). The data obtained for ~o 9Te should be regarded as preliminary. We are unable to eliminate completely the systematic errors due to the insufficient energy resolution, which damp the observed intensity fluctuations. It is desirable to repeat an analysis of the fine structure in the proton spectrum of 1o 9Te after measurements with a higher resolution. We would like to stress the importance of performing experiments which would permit the determination of Bp, such as the measurements of the quantities Qo and Qo-Bp for delayed-proton emitters. The problem of determining Qo for isotopes with low yield is not easy. However, it is worth tackling since a precise knowledge of Bp will enable us to obtain more accurate information from the delayed-proton spectrum studies. The authors are thankful to Academician G. N. Flerov for his interest, and to S. Batselev and S. N. Bogdanova for assistance in performing numerical calculations.
592
V . A . K A R N A U K H O V et aL
References 1) D. D. Bogdanov, S. Daroci, V. A. Karnaukhov, L. A. Petrov and G. M. Ter-Akopyan, Yad. Fiz. 6 (1967) 893 2) V. A. Karnaukhov, Yad. Fiz. 10 (1969) 450 3) P. I-[ornshaj, K. Wilsky, P. G. Hansen, B. Jonson and O. B. Nielsen, Nucl. Phys. A187 (1972) 609 4) D. D. Bogdanov, V. A. Karnaukhov and L. A. Petrov, J I N R preprint P6-6861 5) P. G. I-Iansen, Nuclear structure lectures, Alushta, 1972, J I N R D-6465, 365 (Dubna, 1972) 6) P. A. Egelstaff, Proc. Phys. Soc. 71 (1958) 910 7) C. E. Porter and R. G. Thomas, Phys. Rev. 104 (1956) 483 8) C. Gleit, G. W. Tang and C. D. Coryell, Nucl. Data Sheets 5, $5 (1963) 109; C. S. I-Jan and N. B. Gove, ibid., p. t46 9) A. Gilbert and A. G. W. Cameron, Can. J. Phys. 43 (1965) 1446 10) D. Hudson, Statistics for physicists (Mir, Moscow, 1967)