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Level densities of medium mass nuclei
Nuclear Physics 52 (1964) 630~640; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced b y photoprint or microfilm without written permission from the publisher
LEVEL D E N S I T I E S OF M E D I U M M A S S N U C L E I A. A S P I N A L L t, G. B R O W N t and J. G. B. H A I G H tt
Bradford Institute of Technology, Bradford
Received 26 October 1963 Abstract: An investigation is made of the nuclear level densities obtained from experiments with magnetic spectrographs and on slow neutron resonances. The data are analysed to give values of the singie-particle level densities; the values obtained from the two types of data are not consistent in magnitude or in the manner of their variation with nucleon numbers near closed shells. This analysis emphasizes the need for further investigations.
1 . ~ u c ~ n
In recent years many attempts have been made to find theoretical expressions for the level densities of excited nuclei x-4). Most of the theoretical investigations have been based on the Fermi gas model of the nucleus with a pairing interaction and lead to expressions for the density of nuclear states in terms of the density g of singleparticle states at the Fermi surface. The nuclear level density is then derived by consideration of the distribution of levels over different total angular momenta. This introduces a second parameter, to describe the dispersion of the angular momentum, and, because of its dimensions, this parameter is commonly called the nuclear m o m e n t of inertia. The theoretical expressions have been fitted to experimental data by adjusting both the value of the moment of inertia and O; in some cases these are treated as independent variables 5,6), but in other cases a definite relationship is established 1-3). Newton 1) a n d , C a m e r o n 2) analysed data on level spacings at neutron binding energy for m a n y nuclei, mainly of mass greater than 70; they observed marked decreases in the value of 9 in the mass regions corresponding to the closure of the nuclear shells at numbers 50, 82 and 126. The values of moment of inertia employed by Newton were approximately 0.13 times the classical rigid body value corresponding to a nuclear radius of 1.2 A ~ fm; those used by Cameron were 0.03 times the rigid body value 7). These data were also analysed by Lang s), who used a moment of inertia approximately equal to this rigid body value; he obtained values of 9 approximately twice those of Newton, but showing the same variation with nuclear mass. t Department of Physics. tt Department of Mathematics. 630
LEVELI)ENSlrmS
631
Ericson 5 ) h a s suggested a simple method for comparing data from magnetic spectrographs' with those at neutron binding energy; by inferring that the nuclear temperature is independent of excitation energy, he was able to use the data at neutron binding energy to derive a value of the nuclear moment of inertia. Both Ericson 5) and Bowman 6) have used this technique, and the values of moment of inertia which they find lie in the range 0.6 to 9.00 times the rigid body value 9). The disparity between the moments of inertia obtained from this and other methods 1,2,9-ts) suggests that a purely exponential variation of level-density with excitation energy is an oversimplification when extended to neutron binding energy. With the exception of the method outlined above, energy levels obtained from magnetic spectrographs have not been used for estimating the statistical parameters of nuclei. It has been asserted that the number of levels obtained from spectrographic measurements is insufficient to give a statistical estimate of the level density, and that the data are unreliable as there is a fair chance of levels being undetected. But the spectrographic data 16) on the iron isotopes clearly show the effect of pairing of nucleons, and a systematic decrease in level density as the neutron shell at N = 28 is approached. The magnitude of these changes is not consistent with that predicted using Lang's interpretation of the formulae suggested by Newton. In the present paper, the large quantity of available spectrographic data 17) for the mass range A = 40 to 70 has been analysed and values of g obtained for many nuclei in this range. These have been compared with values calculated from the results of experiments on the resonance capture of slow neutrons.
2. An Assessment of the Experimental Data It is important to discuss the reliability for level density calculations of the data obtained with different experimental techniques. Measurement of neutron resonances generally provides information on odd and odd-mass nuclei and most of the data currently available is for A > 100. The energy interval over which s-wave resonance assignments 13)have been made is less than 700 keV, the number of resonances observed varies from 2 to 21 and the variations in the quoted level spacing from 2 to 150 keV. At the lowest spacings the effects of resolution become important and at the highest spacings the application of the data to a statistical analysis is not acceptable. From such data the observed variation in level density of nuclei will be more qualitative than quantitative. Lang has analysed experimental data on nuclear temperature to obtain values of the single-particle level density. The experimental uncertainties lead to probable errors of approximately 20 ~ in g and only an approximate theoretical interpretation is available. It is doubtful whether such data can give more than superficial information on changes in the level density parameters with individual nuclear properties. Investigations using magnetic spectrographs are not restricted to nuclei of one type; even, odd-mass and odd nuclei can be studied by a suitable choice of reaction.
632
A. ASPINALLet aL
A particular experiment yields a large number of levels but, as Ericson 5) has pointed out, many levels may be missed because of the limited resolution. It is worthwhile, therefore, to examine the features of a spectrographic experiment which contribute to the limited resolution. In a magnetic spectrograph charged particles of a particular energy E arising from a line source on the target enter a region of magnetic flux through a defining slit. In the magnetic field the particles follow a circular path and are brought to a focus on a photographic plate to give an image whose width is determined by (a) the size of the source, (b) the inherent aberrations of the spectrograph, (c) non-linear effects in the magnetic field, and (d) energy spread from the target source. It is convenient to express the position of a point P on the plate in terms of the effective radius of curvature p of a particle focussed at this point. If effects (c) and (d) are neglected, then the size, Ap, of the image at P depends solely on the geometry of the instrument, and hence on p alone. The radius of curvature is given approximately by E = kB2p 2, (1) where B is the magnetic flux density and k is a constant. Therefore, the energy width A E is given by
ae = 2e P
(2)
Hence the energy width is directly proportional to the particle energy for a given radius of curvature. If in two experiments, groups corresponding to particles of different energy fall at the same point on the plate, the energy spread will be greater for that group with the greater energy. Experiments involving (d, p) reactions with positive Q values have been carried out with deuteron energies which were similar to those of incident protons in (p, p') reactions. In these circumstances the energy resolution at a given point on the plate, corresponding to a given value of p, will be (eq. (2)) poorer for the (d, p) than for the (p, p') reaction. Owing to the limited resolution of magnetic spectrographs, there will always be some energy levels that are not observed and this is particularly evident where there is a high density of levels s). If results of different experiments are to be compared, it is desirable that the experiments should be performed with similar resolution. It appears that, in experiments reported to data, no account has been taken of the variation of resolution with energy, nor has any experiment been performed to show that, over a wide range o f excitation energy, the two types of reaction lead to population o f the same states. It would be possible to use data from different spectrographs, if corrections could be made for differences in resolution and this could be achieved if the distribution of level spacings were known. As yet no experiments have been
LEVEL DENSITIES
633
carried out with a resolution good enough to estimate the distribution of level spacings accurately. Although estimates of this distribution have been made from the results of neutron resonance experiments is), it is unlikely that it is the same as that pertaining to the results of spectrographic experiments, which yield levels of many angular momenta. In spite of the above limitations we feel that the data can be used profitably to obtain information on nuclear level density. 3. Analysis of Observations
3.1. METHOD OF ANALYSIS There have been several theoretical studies of the density of nuclear levels, each of which is based on the Fermi gas model, but which lead to different expressions for the relationship between excitation energy and level densities 1-3). However, these expressions do not differ greatly in their form, and it is not the aim of this paper to prove the validity of any one particular study but rather to give guidance to future work. We have chosen to use those derived by Lang and Le Couteur a); accordingly the density of states P ( U ) for an even nucleus at an excitation energy U is given by exp 1 7
--~Tz2#ae-"# + 1}
P(U) =
,
(3)
12(m)~'(~g) ~ U = {;#n2t 2{¼+ ¼(1 + a / t ) e - " / ' } - t,
(4)
a = 0.437A,
(5)
where t is a nuclear temperature, # the single-particle level density and A the pairing energy. The density of levels of spin J and both parities is p(U,J)-
( 2 J + l ) exp { 7r*(2c't) ~
2 ~ / e(u), (J+½)~t
(6)
where c' is the moment of inertia for spherical nuclei and according to the model varies with the excitation energy as c't = cte -"It + em 2,
(7)
where m is the magnetic quantum number, ~ the number of unpaired particles in the nucleus and c the rigid body moment of inertia given by c = 0.146A~#.
(8)
For levels of all spins and both parities the level density becomes 1
p ( U ) -- - P(U). (2nc't) ~
(9)
A. ASPINALL et al.
634
The level densities of odd-mass and odd nuclei are obtained b y correcting the observed excitation energy for pairing interaction. Thus one calculates the level density at an excitation energy U by adjusting this to a value U' such that
U ' = U+½ed.
(10)
In all analyses at least two variable parameters must be considered namely the single-particle level density and the pairing energy. Thus an analysis can be carried out in two slightly different ways; (a) a relationship between # and A is assumed and individual variations of level densities are accounted for by adjustment of the pairing energy t9,20), or (b) a relationship between A and A is assumed and individual variations 9) accounted for by adjustment of #. We have chosen the latter alternative and used for the pairing energy the expression of Stolovy and Harvey 21)
=
(1-
)•
(11)
3.2. A N A L Y S I S O F S P E C T R O G R A P H I C D A T A
At high excitation energies, owing to limited resolution, a considerable number of levels are not observed. Thus accurate analysis should be confined to a region where this is not important but where a statistical approach is reasonable; this limits the analysis to values of U" < 6 MeV and a number of levels N, such that 10 < N < 35. In our previous publication it was shown that there are marked changes in the level densities of the iron isotopes produced by the closure of the shell at a neutron loE 5£
10
2
/J
/I 1[ 3
4
,
,
5
6
,
?
,
8
U' gVlev)
Fig. l. The total n u m b e r o f levels N ( U ' ) up to an excitation energy U' for the isotopes of potassium.
number of 28. In figs. 1 and 2 the variation of log N(U') with U' for the data reported by Bueehner and co-workers 17) for the isotopes of potassium and calcium is shown. These data show effects similar to those of iron.
635
LEVEL DENSITIES
The calculated values of the single-particle level density are given in table 1; an analysis of the effect of uncertainties in the knowledge of the pairing energy and level density gives an uncertainty in g of approximately _+0.5 MeV-1. Any discrepancies 100
50i
43
I
2
Fig.
2.
3
The total n u m b e r o f levels
4
5
N(U') up
6 U' (MeV)
1
I
7
8
to an excitation energy U" for the isotopes o f calcium.
B O
B × 0
0
0
~7 0
K /x Ca • Sc + V o l',,In v Fe o Cox N/ • Cu#
0
i
I
28
i
Neutron
Fig.
3.
Jo
i
J2
nurr~3er
The variation w i t h neutron n u m b e r o f the single-particle level density g, obtained f r o m spectrographic data.
between the values in table 1 and those quoted previously 16) arise from the different method of analysis but it should be noted that in all cases they are within the quoted errors. It can be seen from table 1 that the data for potassium, calcium and iron
636
A. AsPnqALL e t al. TABLE 1 The values o f the single particle level density obtained f r o m the spectrographic data U' (MeV)
Nucleus
K 8' K 4° Ka K 42 Ca 4z Ca 's Ca ~ Sc ~ V 51 V 5= Mn 66 Mn 6e Fe ~4 Fe 55
g (MeV_I) Nucleus
6.2 5.9 4.3 5.1 5.0 4.1 4.4 3.7 4.9 5.4 4.3 4.6 4.8 4.5
4.4 5.4 7.3 6.2 5.9 8.7 6.9 10.0 7.4 5.8 8.6 7.8 6.3 7.1
Fe 56 Fe s~ Fe ss Fe s9 Co 5' Co 6° Ni 5' Ni 6° Ni 61 Ni s6 Cu sa Cu e4 Cu 65 Cu 66
U' g (MeV) (MeV -1)
4.3 4.1 4.2 4.1 4.2 4.6 4.0 3.7 3.9 4.8 4.0 4.4 4.0 4.6
6.8 8.4 7.9 8.4 8.8 7.6 8.2 8.9 8.8 6.7 9.7 8.1 8.8 7.4
The data for Fe 64 a n d Fe 66 are taken f r o m Aspinall e t al. xs); the remainder f r o m the publications o f Bueclmer and his co-workers x~).
TABLE 2 Evidence for p r o t o n shell effects Neutron number
z
g
z
g
21 23
19 19
5.4 6.2
20 20
5.9 8.7
no shell effect
28 29
23 23
7.4 5.8
26 26
6.3 7.1
conflicting evidence
30 31
25 25
8.6 7.8
26 ~26 28
6.8 8.4 8.2
conflicting evidence
32
26
7.9
f27 28
8.8 8.9
Comment
no shell
33
26
8.4
~27 28
7.6 8.8
effect
37
28
6.7
29
7.4
possible shell effect
LEVEL DENSITIES
637
isotopes show the effect of the closed neutron shells at N = 20 and N -- 28. This is also shown in fig. 3 in which the values of g are plotted as a function of neutron number; two maxima, for neutron numbers of 24 and 34, will be noted in this figure. It is of interest to search for similar effects due to the closure of proton shells; the relevant data from table 1 are collected together in table 2 and show little evidence for such effects. 3.3. A N A L Y S I S
OF THE NEUTRON
RESONANCE
DATA
Level spacings obtained from neutron resonance experiments have been taken mainly from the compilation of Bowman et al. 6). The analysis, which is summarized in table 3 is confined to nuclei where at least five resonances are known; the probable errors in the quoted values of g are between 0.2 and 0.3 M e V - 1. In fig. 4, the values of # obtained from neutron resonance data are plotted against proton number and against neutron number. The overall change with either proton or neutron number gives no indication of shell effects. The data for constant neutron numbers of 31 and TABLE 3 Values of # from neutron resonance data
Nucleus
U~
( M e #V _ l )
Nucleus
U'
( M e #V _ t )
K 4° K 4z C a 41 C a 45 Sc 48 T i ~7 T i t. V 5~ C r 51
10.8 10.5 9.9 8.9 1 1.6 10.4 9.6 10.3 10.7
5.0 5.2 4.7 5.3 5.9 5.0 5.8 6.8 4.9
C r 63 M n so F e s5 F e 5~ C o e° N i 59 Ni ~ C u ~4
9.4 10.1 10.7 9.1 10.3 10.4 9.2 10.4
5.2 6.8 5.1 6.2 7.0 5.2 6.2 7.6
T h e l e v e l s p a c i n g f o r , C u e4 is o b t a i n e d f r o m H u g h e s e t al. 23); d a t a for the o t h e r n u c l e i a r e f r o m B o w m a n e t al. *).
33 and for constant p r o t o n numbers of 26 and 28 suggest the effect of a closed shell at a nucleon number of 28, but other groups of nuclei do not substantiate this. I f shell effects are present they are certainty less marked than for the spectrographic data. 3.4. C O M P A R I S O N
OF THE SPECTROGRAPHIC
AND NEUTRON
RESONANCE
DATA
In columns 2 and 3 of table 4 we compare the values of g obtained for a particular nucleus through the two experiments; the neutron resonance data generally give lower values of g. It is possible that the magnitude of g does in fact decrease with increasing excitation energy, resulting from the excitation of nucleons from well
A. ASPINALL et al.
638
+
7
+
+
2> +
A
o
/
//
o
/M
• D
19,,
2O •
g;" 28 32
Neutron numbec
i
I
34
8
•
~>7f 6
"t" 5~,
3
•
',
•
.,,,,,,
o
•
210
O,
•
'
•
Q
212
2r4
N 21 25 27 29 31 33
hw
"O •
' 26 2 Proton number
• o • o •
3
Fig. 4. T h e v a r i a t i o n w i t h p r o t o n a n d n e u t r o n n u m b e r o f the single-particle level de ns i t y 9, o b t a i n e d f r o m the n e u t r o n r e s o n a n c e data.
TABLE 4 C o m p a r i s o n o f the values o f g f r o m spectrogra phi c a n d n e u t r o n re s ona nc e d a t a N e u t r o n re s ona nc e d a t a Nucleus
K 4° K 42 C a 4~ Ca 45 V~ Mn b8 Fe 55 Fe 5~ Ni 69 Ni ~ Cu ~
Spectrographic data
5.4 6.2 5.9 6.9 5.8 7.8 7.1 8.4 8.2 8.8 8.1
D i s p e r s i o n c't 5.0 5.2 4.7 5.3 6.8 6.8 5.1 6.2 5.2 6.2 7.6
D i s p e r s i o n ct 5.5 5.9 6.0 6.8 7.5 7.5 6.2 8.0 6.5 8.0 8.8
LEVEL DENSITIES
639
below the Fermi level. The concept of an average single-particle spacing does not readily lend itself to speculation on such a variation and it is worthwhile therefore to look for other ways of accounting for this discrepancy. The first possibility is, as suggested by Lang and Le Couteur 3), that the pairing energy decreases with increasing excitation energy; such a reduction follows naturally from the use of a superconductivity model 22). Whilst retaining this as a possibility, we feel that the type of calculation used in our analysis would become unduly complicated. As a second alternative it is possible to remove the discrepancy by allowing the moment of inertia to change more rapidly than given by the form of eq. (7). This is justifiable because knowledge of the form of the dispersion of total angular momenta is extremely sparse; little consistency is revealed among the many attempts to derive this parameter from experimental data lO-15). If, as shown in column 4 of table 4, the moment of inertia at neutron binding energies is assumed to be given by eq. (8) the values of g from these data are similar to those from the spectrographic data. This moment of inertia corresponds approximately to the classical rigid body value for a nuclear radius of 1.4 A ~r fm. These calculations, which are in agreement with similar ones made by MacDonald 9), indicate that the moment of inertia used to interpret neutron resonance data should be approximately 4 times that used for spectrographic data; a ratio of only 1.7 is obtained using the methods of Lang and Le Couteur. 4. Discussion
The following anomalies are revealed by the present analysis: (1) Neutron shell effects but no proton shell effects are observed with the spectrographic data. (2) The shell effects observed with spectrographic data are stronger than any that may be present with the neutron resonance data. (3) Using the model of Lang and Le Couteur, the values of g for the spectrographic and neutron resonance data are not consistent. Further investigations are required to clarify these points. At present, analyses of spectrographic data must be confined to low excitation energies so that the effects of lack of resolution are not important. It is desirable that the resolution be improved so that reliable data can be obtained at higher excitation energies. It is equally important to discover whether at reasonable excitation energies a (d, p) reaction and (p, p') reaction leading to the same nucleus populate the same levels; if the same levels are not populated then the scope of spectrographic work is restricted more than it appears at present. Experiments should be performed with magnetic spectrographs to provide data on the magnitude of the proton shell effects, the nuclei chosen being such that the neutron number is kept constant but the proton number varied such that it is at or near the
640
A. ASPlNALL et al.
closed shells for the numbers 20 to 28. These experiments would also give additional information on the values of # near the maxima and the minimum already observed. Additional direct comparisons of the spectrographic and neutron resonance data are dearly desirable. The method of comparison used here is inadequate because the energy interval between the two sets of data is too large. The experiments with spectrographs must be extended under conditions of constant resolution, to excitations approaching neutron binding energy for the nuclei to which the neutron resonance technique is applicable. If the spectrographic data are thus extended, a simple extrapolation for all isotopes to the neutron binding energy will enable reliable comparisons o f differences to be made. 5. Conclusions It has been demonstrated that the data now available from experiments with magnetic spectrographs can yield useful information on nuclear level densities and we suggest that much more can be readily obtained. It is desirable that additional experiments should be performed at least with a constant resolution but preferably with improved resolution so that the following information may be obtained: (i) the relative level density for isotopes and isotones, (ii) the distribution of level spacings so that it may be possible to correct for lack of resolution, (iii) the level density at excitation energies approaching neutron binding energy, so that changes in the single-partide level density and pairing energy may be investigated and comparisons made with the neutron resonance data.
We are grateful to Drs. N. MacDonald and R. Middleton of A.W.R.E., Aldermaston, for some interesting and stimulating discussions and to Dr. Smare of the Physics Department, Bradford, for his encouragement. References 1) T. D. Newton, Can. J. Phys. 34 (1956) 804 2) A. C. W. Cameron, Can. J. Phys. 36 (1958) 1040 3) D. W. Lang and K. J. Le Couteur, Nuclear Physics 14 (1959) 21 4) T. Ericson, Nuclear Physics 6 (1958) 62 5) T. Ericson, Nuclear Physics 11 (1958) 481 6) C. D. Bowman, E. G. Bilpuch and H. W. Newson, Ann. of Phys. 17 (1961) 319 7) A. C. Douglas and N. MacDonald, Nuclear Physics 13 (1959) 382 8) D. W. Lang, Nuclear Physics 26 (1965) 434 9) N. MacDonald, private communication 10) N. MacDonald, A. W. R. E. report N R / P -- 2/61 11) C. T. Hibdon, Phys. Rev. 114 (1959) 179, 124 (1961) 500 12) T. Ericson and V. Strutinski, Nuclear Physics 8 (1958) 284 13) J. H. Carver and G. A. Jones, Nuclear Physics 19 (1960) 184 14) J. R. Huizinga and R. Vandenboseh, Phys. Rev. 120 (1960) 1305, 1313 15) J. H. Carver, G. E. Coote and T. R. Sherwood, Nuclear Physics 37 (1962) 449 16) A. Aspinall, G. Brown and S. E. Warren, Nuclear Physics 46 (1963) 33 17) W. W. Bueehner, M.I.T.L.N.S. Progress Reports and appropriate publications 18) J. A. Harvey and D. J. Hughes, Phys. Rev. 109 (1958) 471 19) D. L. Allan, Nuclear Physics 24 (1961) 274 20) I. Dostrovsky, Z. Fraenkel and G. Friedlander, Phys. Rev. 116 (1959) 683 21) A. Stolovy and J. A. Harvey, Phys. Rev. 108 (1957) 353 22) D. W. Lang, Nuclear Physics 42 (1963) 353 23) D. J. Hughes, B. A. Magurne and M. K. Brussel, B.N.L. 325 second ed.
LEVEL D E N S I T I E S OF M E D I U M M A S S N U C L E I A. A S P I N A L L t, G. B R O W N t and J. G. B. H A I G H tt
Bradford Institute of Technology, Bradford
Received 26 October 1963 Abstract: An investigation is made of the nuclear level densities obtained from experiments with magnetic spectrographs and on slow neutron resonances. The data are analysed to give values of the singie-particle level densities; the values obtained from the two types of data are not consistent in magnitude or in the manner of their variation with nucleon numbers near closed shells. This analysis emphasizes the need for further investigations.
1 . ~ u c ~ n
In recent years many attempts have been made to find theoretical expressions for the level densities of excited nuclei x-4). Most of the theoretical investigations have been based on the Fermi gas model of the nucleus with a pairing interaction and lead to expressions for the density of nuclear states in terms of the density g of singleparticle states at the Fermi surface. The nuclear level density is then derived by consideration of the distribution of levels over different total angular momenta. This introduces a second parameter, to describe the dispersion of the angular momentum, and, because of its dimensions, this parameter is commonly called the nuclear m o m e n t of inertia. The theoretical expressions have been fitted to experimental data by adjusting both the value of the moment of inertia and O; in some cases these are treated as independent variables 5,6), but in other cases a definite relationship is established 1-3). Newton 1) a n d , C a m e r o n 2) analysed data on level spacings at neutron binding energy for m a n y nuclei, mainly of mass greater than 70; they observed marked decreases in the value of 9 in the mass regions corresponding to the closure of the nuclear shells at numbers 50, 82 and 126. The values of moment of inertia employed by Newton were approximately 0.13 times the classical rigid body value corresponding to a nuclear radius of 1.2 A ~ fm; those used by Cameron were 0.03 times the rigid body value 7). These data were also analysed by Lang s), who used a moment of inertia approximately equal to this rigid body value; he obtained values of 9 approximately twice those of Newton, but showing the same variation with nuclear mass. t Department of Physics. tt Department of Mathematics. 630
LEVELI)ENSlrmS
631
Ericson 5 ) h a s suggested a simple method for comparing data from magnetic spectrographs' with those at neutron binding energy; by inferring that the nuclear temperature is independent of excitation energy, he was able to use the data at neutron binding energy to derive a value of the nuclear moment of inertia. Both Ericson 5) and Bowman 6) have used this technique, and the values of moment of inertia which they find lie in the range 0.6 to 9.00 times the rigid body value 9). The disparity between the moments of inertia obtained from this and other methods 1,2,9-ts) suggests that a purely exponential variation of level-density with excitation energy is an oversimplification when extended to neutron binding energy. With the exception of the method outlined above, energy levels obtained from magnetic spectrographs have not been used for estimating the statistical parameters of nuclei. It has been asserted that the number of levels obtained from spectrographic measurements is insufficient to give a statistical estimate of the level density, and that the data are unreliable as there is a fair chance of levels being undetected. But the spectrographic data 16) on the iron isotopes clearly show the effect of pairing of nucleons, and a systematic decrease in level density as the neutron shell at N = 28 is approached. The magnitude of these changes is not consistent with that predicted using Lang's interpretation of the formulae suggested by Newton. In the present paper, the large quantity of available spectrographic data 17) for the mass range A = 40 to 70 has been analysed and values of g obtained for many nuclei in this range. These have been compared with values calculated from the results of experiments on the resonance capture of slow neutrons.
2. An Assessment of the Experimental Data It is important to discuss the reliability for level density calculations of the data obtained with different experimental techniques. Measurement of neutron resonances generally provides information on odd and odd-mass nuclei and most of the data currently available is for A > 100. The energy interval over which s-wave resonance assignments 13)have been made is less than 700 keV, the number of resonances observed varies from 2 to 21 and the variations in the quoted level spacing from 2 to 150 keV. At the lowest spacings the effects of resolution become important and at the highest spacings the application of the data to a statistical analysis is not acceptable. From such data the observed variation in level density of nuclei will be more qualitative than quantitative. Lang has analysed experimental data on nuclear temperature to obtain values of the single-particle level density. The experimental uncertainties lead to probable errors of approximately 20 ~ in g and only an approximate theoretical interpretation is available. It is doubtful whether such data can give more than superficial information on changes in the level density parameters with individual nuclear properties. Investigations using magnetic spectrographs are not restricted to nuclei of one type; even, odd-mass and odd nuclei can be studied by a suitable choice of reaction.
632
A. ASPINALLet aL
A particular experiment yields a large number of levels but, as Ericson 5) has pointed out, many levels may be missed because of the limited resolution. It is worthwhile, therefore, to examine the features of a spectrographic experiment which contribute to the limited resolution. In a magnetic spectrograph charged particles of a particular energy E arising from a line source on the target enter a region of magnetic flux through a defining slit. In the magnetic field the particles follow a circular path and are brought to a focus on a photographic plate to give an image whose width is determined by (a) the size of the source, (b) the inherent aberrations of the spectrograph, (c) non-linear effects in the magnetic field, and (d) energy spread from the target source. It is convenient to express the position of a point P on the plate in terms of the effective radius of curvature p of a particle focussed at this point. If effects (c) and (d) are neglected, then the size, Ap, of the image at P depends solely on the geometry of the instrument, and hence on p alone. The radius of curvature is given approximately by E = kB2p 2, (1) where B is the magnetic flux density and k is a constant. Therefore, the energy width A E is given by
ae = 2e P
(2)
Hence the energy width is directly proportional to the particle energy for a given radius of curvature. If in two experiments, groups corresponding to particles of different energy fall at the same point on the plate, the energy spread will be greater for that group with the greater energy. Experiments involving (d, p) reactions with positive Q values have been carried out with deuteron energies which were similar to those of incident protons in (p, p') reactions. In these circumstances the energy resolution at a given point on the plate, corresponding to a given value of p, will be (eq. (2)) poorer for the (d, p) than for the (p, p') reaction. Owing to the limited resolution of magnetic spectrographs, there will always be some energy levels that are not observed and this is particularly evident where there is a high density of levels s). If results of different experiments are to be compared, it is desirable that the experiments should be performed with similar resolution. It appears that, in experiments reported to data, no account has been taken of the variation of resolution with energy, nor has any experiment been performed to show that, over a wide range o f excitation energy, the two types of reaction lead to population o f the same states. It would be possible to use data from different spectrographs, if corrections could be made for differences in resolution and this could be achieved if the distribution of level spacings were known. As yet no experiments have been
LEVEL DENSITIES
633
carried out with a resolution good enough to estimate the distribution of level spacings accurately. Although estimates of this distribution have been made from the results of neutron resonance experiments is), it is unlikely that it is the same as that pertaining to the results of spectrographic experiments, which yield levels of many angular momenta. In spite of the above limitations we feel that the data can be used profitably to obtain information on nuclear level density. 3. Analysis of Observations
3.1. METHOD OF ANALYSIS There have been several theoretical studies of the density of nuclear levels, each of which is based on the Fermi gas model, but which lead to different expressions for the relationship between excitation energy and level densities 1-3). However, these expressions do not differ greatly in their form, and it is not the aim of this paper to prove the validity of any one particular study but rather to give guidance to future work. We have chosen to use those derived by Lang and Le Couteur a); accordingly the density of states P ( U ) for an even nucleus at an excitation energy U is given by exp 1 7
--~Tz2#ae-"# + 1}
P(U) =
,
(3)
12(m)~'(~g) ~ U = {;#n2t 2{¼+ ¼(1 + a / t ) e - " / ' } - t,
(4)
a = 0.437A,
(5)
where t is a nuclear temperature, # the single-particle level density and A the pairing energy. The density of levels of spin J and both parities is p(U,J)-
( 2 J + l ) exp { 7r*(2c't) ~
2 ~ / e(u), (J+½)~t
(6)
where c' is the moment of inertia for spherical nuclei and according to the model varies with the excitation energy as c't = cte -"It + em 2,
(7)
where m is the magnetic quantum number, ~ the number of unpaired particles in the nucleus and c the rigid body moment of inertia given by c = 0.146A~#.
(8)
For levels of all spins and both parities the level density becomes 1
p ( U ) -- - P(U). (2nc't) ~
(9)
A. ASPINALL et al.
634
The level densities of odd-mass and odd nuclei are obtained b y correcting the observed excitation energy for pairing interaction. Thus one calculates the level density at an excitation energy U by adjusting this to a value U' such that
U ' = U+½ed.
(10)
In all analyses at least two variable parameters must be considered namely the single-particle level density and the pairing energy. Thus an analysis can be carried out in two slightly different ways; (a) a relationship between # and A is assumed and individual variations of level densities are accounted for by adjustment of the pairing energy t9,20), or (b) a relationship between A and A is assumed and individual variations 9) accounted for by adjustment of #. We have chosen the latter alternative and used for the pairing energy the expression of Stolovy and Harvey 21)
=
(1-
)•
(11)
3.2. A N A L Y S I S O F S P E C T R O G R A P H I C D A T A
At high excitation energies, owing to limited resolution, a considerable number of levels are not observed. Thus accurate analysis should be confined to a region where this is not important but where a statistical approach is reasonable; this limits the analysis to values of U" < 6 MeV and a number of levels N, such that 10 < N < 35. In our previous publication it was shown that there are marked changes in the level densities of the iron isotopes produced by the closure of the shell at a neutron loE 5£
10
2
/J
/I 1[ 3
4
,
,
5
6
,
?
,
8
U' gVlev)
Fig. l. The total n u m b e r o f levels N ( U ' ) up to an excitation energy U' for the isotopes of potassium.
number of 28. In figs. 1 and 2 the variation of log N(U') with U' for the data reported by Bueehner and co-workers 17) for the isotopes of potassium and calcium is shown. These data show effects similar to those of iron.
635
LEVEL DENSITIES
The calculated values of the single-particle level density are given in table 1; an analysis of the effect of uncertainties in the knowledge of the pairing energy and level density gives an uncertainty in g of approximately _+0.5 MeV-1. Any discrepancies 100
50i
43
I
2
Fig.
2.
3
The total n u m b e r o f levels
4
5
N(U') up
6 U' (MeV)
1
I
7
8
to an excitation energy U" for the isotopes o f calcium.
B O
B × 0
0
0
~7 0
K /x Ca • Sc + V o l',,In v Fe o Cox N/ • Cu#
0
i
I
28
i
Neutron
Fig.
3.
Jo
i
J2
nurr~3er
The variation w i t h neutron n u m b e r o f the single-particle level density g, obtained f r o m spectrographic data.
between the values in table 1 and those quoted previously 16) arise from the different method of analysis but it should be noted that in all cases they are within the quoted errors. It can be seen from table 1 that the data for potassium, calcium and iron
636
A. AsPnqALL e t al. TABLE 1 The values o f the single particle level density obtained f r o m the spectrographic data U' (MeV)
Nucleus
K 8' K 4° Ka K 42 Ca 4z Ca 's Ca ~ Sc ~ V 51 V 5= Mn 66 Mn 6e Fe ~4 Fe 55
g (MeV_I) Nucleus
6.2 5.9 4.3 5.1 5.0 4.1 4.4 3.7 4.9 5.4 4.3 4.6 4.8 4.5
4.4 5.4 7.3 6.2 5.9 8.7 6.9 10.0 7.4 5.8 8.6 7.8 6.3 7.1
Fe 56 Fe s~ Fe ss Fe s9 Co 5' Co 6° Ni 5' Ni 6° Ni 61 Ni s6 Cu sa Cu e4 Cu 65 Cu 66
U' g (MeV) (MeV -1)
4.3 4.1 4.2 4.1 4.2 4.6 4.0 3.7 3.9 4.8 4.0 4.4 4.0 4.6
6.8 8.4 7.9 8.4 8.8 7.6 8.2 8.9 8.8 6.7 9.7 8.1 8.8 7.4
The data for Fe 64 a n d Fe 66 are taken f r o m Aspinall e t al. xs); the remainder f r o m the publications o f Bueclmer and his co-workers x~).
TABLE 2 Evidence for p r o t o n shell effects Neutron number
z
g
z
g
21 23
19 19
5.4 6.2
20 20
5.9 8.7
no shell effect
28 29
23 23
7.4 5.8
26 26
6.3 7.1
conflicting evidence
30 31
25 25
8.6 7.8
26 ~26 28
6.8 8.4 8.2
conflicting evidence
32
26
7.9
f27 28
8.8 8.9
Comment
no shell
33
26
8.4
~27 28
7.6 8.8
effect
37
28
6.7
29
7.4
possible shell effect
LEVEL DENSITIES
637
isotopes show the effect of the closed neutron shells at N = 20 and N -- 28. This is also shown in fig. 3 in which the values of g are plotted as a function of neutron number; two maxima, for neutron numbers of 24 and 34, will be noted in this figure. It is of interest to search for similar effects due to the closure of proton shells; the relevant data from table 1 are collected together in table 2 and show little evidence for such effects. 3.3. A N A L Y S I S
OF THE NEUTRON
RESONANCE
DATA
Level spacings obtained from neutron resonance experiments have been taken mainly from the compilation of Bowman et al. 6). The analysis, which is summarized in table 3 is confined to nuclei where at least five resonances are known; the probable errors in the quoted values of g are between 0.2 and 0.3 M e V - 1. In fig. 4, the values of # obtained from neutron resonance data are plotted against proton number and against neutron number. The overall change with either proton or neutron number gives no indication of shell effects. The data for constant neutron numbers of 31 and TABLE 3 Values of # from neutron resonance data
Nucleus
U~
( M e #V _ l )
Nucleus
U'
( M e #V _ t )
K 4° K 4z C a 41 C a 45 Sc 48 T i ~7 T i t. V 5~ C r 51
10.8 10.5 9.9 8.9 1 1.6 10.4 9.6 10.3 10.7
5.0 5.2 4.7 5.3 5.9 5.0 5.8 6.8 4.9
C r 63 M n so F e s5 F e 5~ C o e° N i 59 Ni ~ C u ~4
9.4 10.1 10.7 9.1 10.3 10.4 9.2 10.4
5.2 6.8 5.1 6.2 7.0 5.2 6.2 7.6
T h e l e v e l s p a c i n g f o r , C u e4 is o b t a i n e d f r o m H u g h e s e t al. 23); d a t a for the o t h e r n u c l e i a r e f r o m B o w m a n e t al. *).
33 and for constant p r o t o n numbers of 26 and 28 suggest the effect of a closed shell at a nucleon number of 28, but other groups of nuclei do not substantiate this. I f shell effects are present they are certainty less marked than for the spectrographic data. 3.4. C O M P A R I S O N
OF THE SPECTROGRAPHIC
AND NEUTRON
RESONANCE
DATA
In columns 2 and 3 of table 4 we compare the values of g obtained for a particular nucleus through the two experiments; the neutron resonance data generally give lower values of g. It is possible that the magnitude of g does in fact decrease with increasing excitation energy, resulting from the excitation of nucleons from well
A. ASPINALL et al.
638
+
7
+
+
2> +
A
o
/
//
o
/M
• D
19,,
2O •
g;" 28 32
Neutron numbec
i
I
34
8
•
~>7f 6
"t" 5~,
3
•
',
•
.,,,,,,
o
•
210
O,
•
'
•
Q
212
2r4
N 21 25 27 29 31 33
hw
"O •
' 26 2 Proton number
• o • o •
3
Fig. 4. T h e v a r i a t i o n w i t h p r o t o n a n d n e u t r o n n u m b e r o f the single-particle level de ns i t y 9, o b t a i n e d f r o m the n e u t r o n r e s o n a n c e data.
TABLE 4 C o m p a r i s o n o f the values o f g f r o m spectrogra phi c a n d n e u t r o n re s ona nc e d a t a N e u t r o n re s ona nc e d a t a Nucleus
K 4° K 42 C a 4~ Ca 45 V~ Mn b8 Fe 55 Fe 5~ Ni 69 Ni ~ Cu ~
Spectrographic data
5.4 6.2 5.9 6.9 5.8 7.8 7.1 8.4 8.2 8.8 8.1
D i s p e r s i o n c't 5.0 5.2 4.7 5.3 6.8 6.8 5.1 6.2 5.2 6.2 7.6
D i s p e r s i o n ct 5.5 5.9 6.0 6.8 7.5 7.5 6.2 8.0 6.5 8.0 8.8
LEVEL DENSITIES
639
below the Fermi level. The concept of an average single-particle spacing does not readily lend itself to speculation on such a variation and it is worthwhile therefore to look for other ways of accounting for this discrepancy. The first possibility is, as suggested by Lang and Le Couteur 3), that the pairing energy decreases with increasing excitation energy; such a reduction follows naturally from the use of a superconductivity model 22). Whilst retaining this as a possibility, we feel that the type of calculation used in our analysis would become unduly complicated. As a second alternative it is possible to remove the discrepancy by allowing the moment of inertia to change more rapidly than given by the form of eq. (7). This is justifiable because knowledge of the form of the dispersion of total angular momenta is extremely sparse; little consistency is revealed among the many attempts to derive this parameter from experimental data lO-15). If, as shown in column 4 of table 4, the moment of inertia at neutron binding energies is assumed to be given by eq. (8) the values of g from these data are similar to those from the spectrographic data. This moment of inertia corresponds approximately to the classical rigid body value for a nuclear radius of 1.4 A ~r fm. These calculations, which are in agreement with similar ones made by MacDonald 9), indicate that the moment of inertia used to interpret neutron resonance data should be approximately 4 times that used for spectrographic data; a ratio of only 1.7 is obtained using the methods of Lang and Le Couteur. 4. Discussion
The following anomalies are revealed by the present analysis: (1) Neutron shell effects but no proton shell effects are observed with the spectrographic data. (2) The shell effects observed with spectrographic data are stronger than any that may be present with the neutron resonance data. (3) Using the model of Lang and Le Couteur, the values of g for the spectrographic and neutron resonance data are not consistent. Further investigations are required to clarify these points. At present, analyses of spectrographic data must be confined to low excitation energies so that the effects of lack of resolution are not important. It is desirable that the resolution be improved so that reliable data can be obtained at higher excitation energies. It is equally important to discover whether at reasonable excitation energies a (d, p) reaction and (p, p') reaction leading to the same nucleus populate the same levels; if the same levels are not populated then the scope of spectrographic work is restricted more than it appears at present. Experiments should be performed with magnetic spectrographs to provide data on the magnitude of the proton shell effects, the nuclei chosen being such that the neutron number is kept constant but the proton number varied such that it is at or near the
640
A. ASPlNALL et al.
closed shells for the numbers 20 to 28. These experiments would also give additional information on the values of # near the maxima and the minimum already observed. Additional direct comparisons of the spectrographic and neutron resonance data are dearly desirable. The method of comparison used here is inadequate because the energy interval between the two sets of data is too large. The experiments with spectrographs must be extended under conditions of constant resolution, to excitations approaching neutron binding energy for the nuclei to which the neutron resonance technique is applicable. If the spectrographic data are thus extended, a simple extrapolation for all isotopes to the neutron binding energy will enable reliable comparisons o f differences to be made. 5. Conclusions It has been demonstrated that the data now available from experiments with magnetic spectrographs can yield useful information on nuclear level densities and we suggest that much more can be readily obtained. It is desirable that additional experiments should be performed at least with a constant resolution but preferably with improved resolution so that the following information may be obtained: (i) the relative level density for isotopes and isotones, (ii) the distribution of level spacings so that it may be possible to correct for lack of resolution, (iii) the level density at excitation energies approaching neutron binding energy, so that changes in the single-partide level density and pairing energy may be investigated and comparisons made with the neutron resonance data.
We are grateful to Drs. N. MacDonald and R. Middleton of A.W.R.E., Aldermaston, for some interesting and stimulating discussions and to Dr. Smare of the Physics Department, Bradford, for his encouragement. References 1) T. D. Newton, Can. J. Phys. 34 (1956) 804 2) A. C. W. Cameron, Can. J. Phys. 36 (1958) 1040 3) D. W. Lang and K. J. Le Couteur, Nuclear Physics 14 (1959) 21 4) T. Ericson, Nuclear Physics 6 (1958) 62 5) T. Ericson, Nuclear Physics 11 (1958) 481 6) C. D. Bowman, E. G. Bilpuch and H. W. Newson, Ann. of Phys. 17 (1961) 319 7) A. C. Douglas and N. MacDonald, Nuclear Physics 13 (1959) 382 8) D. W. Lang, Nuclear Physics 26 (1965) 434 9) N. MacDonald, private communication 10) N. MacDonald, A. W. R. E. report N R / P -- 2/61 11) C. T. Hibdon, Phys. Rev. 114 (1959) 179, 124 (1961) 500 12) T. Ericson and V. Strutinski, Nuclear Physics 8 (1958) 284 13) J. H. Carver and G. A. Jones, Nuclear Physics 19 (1960) 184 14) J. R. Huizinga and R. Vandenboseh, Phys. Rev. 120 (1960) 1305, 1313 15) J. H. Carver, G. E. Coote and T. R. Sherwood, Nuclear Physics 37 (1962) 449 16) A. Aspinall, G. Brown and S. E. Warren, Nuclear Physics 46 (1963) 33 17) W. W. Bueehner, M.I.T.L.N.S. Progress Reports and appropriate publications 18) J. A. Harvey and D. J. Hughes, Phys. Rev. 109 (1958) 471 19) D. L. Allan, Nuclear Physics 24 (1961) 274 20) I. Dostrovsky, Z. Fraenkel and G. Friedlander, Phys. Rev. 116 (1959) 683 21) A. Stolovy and J. A. Harvey, Phys. Rev. 108 (1957) 353 22) D. W. Lang, Nuclear Physics 42 (1963) 353 23) D. J. Hughes, B. A. Magurne and M. K. Brussel, B.N.L. 325 second ed.