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Cross sections of level excitation for Mg, 52Cr, 58Ni, 60Ni and 93Nb in neutron inelastic scattering
Journalof Nuclear Energy Parts A/B. 1964.Vol. 18, pp. 645 to 655. PergamonPress Ltd. Printedin NorthernIreland
CROSS SECTIONS OF LEVEL EXCITATION FOR Mg, 52Cr, 58Ni, 6oNi AND g3Nb IN NEUTRON INELASTIC SCATTERING* D. L. BRODER, V. E. KOLESOV, and A. G. (Received
A.
I.
LASHUK,
I.
P.
SADOKHIN
DOVBENKO
15 April 1963)
Abstract-The neutron excitation cross sections of some levels of 84Mg, a6Mg, eaMg, 58Ni, 60Ni, 5aCr and QSNb were measured in the range of energies from 1 to 4 MeV. The experimental results were compared with the results of optical model calculations with a Woods-Saxon potential. INTRODUCTION
on the cross sections of neutron inelastic scattering can be obtained by various experimental methods. The method based on the measurement of the spectra of the y-radiation accompanying the process is widely used. This method enables us not only to establish the energy level systems of the nuclei, but also to determine the values of the excitation cross sections for individual levels. If we use separated isotopes as scatterers, we can interpret the experimental results uniquely. A theory enabling us to calculate the different cross sections of interaction of neutrons with atomic nuclei has not as yet been satisfactorily worked out. The greatest success is achieved when the cross sections are calculated for an optical model of the nucleus. Such a model, that is, a complex potential with a diffuse edge, makes it possible to describe satisfactorily not only the total cross section of neutronnucleus interaction, but also its individual components. The model is also used to calculate the cross sections of neutron inelastic scattering, and gives good agreement of the theoretical cross sections with the experimental in the cases where the nuclear level schemes and their quantum characteristics are known. In the present paper we examine the cross sections of level excitation in neutron inelastic scattering for nuclei of Mg, 52Cr, 58Ni, 60Ni and g3Nb. We compare the experimental results with the theoretical results calculated on the basis of an optical nuclear model with a Woods-Saxon potential. INFORMATION
THE
EXPERIMENT
For the source of monochromatic neutrons with energies up to 3 MeV we used the reaction 3H(p, n)3He; for energies above 3 MeV we used the reaction 2H(d, n)3He. These reactions were brought about with the aid of electrostatic and cascade generators. The scatter of neutron energies did not exceed *30 keV in the first case and f80 keV in the second case. To measure the y-ray emission accompanying the neutron inelastic scattering we used a spectrometer with a 40 x 40 mm NaI(T1) crystal, an FEU-13 photomultiplier and a 128-channel amplitude analyser. The measurements were made in an annular geometry(l). The resolution of the spectrometer for the O-661 MeV line of 13’Cs was about 11 * Translated by J. STUART from
Atomnayu
Energiya 645
16, 103 (1964).
646
D. L. BRODER, V. E. KOLEWV, A. I. LASHUK, I. P. SAD~KHIN and A. G. DOVBENKO
per cent. The energy calibration of the spectrometer and determination of the recording efficiency were effected as given by BRODERand LASHUK.~) We used a fission chamber with 23sU and an all-wave counter to check the uniformity of the neutron flux. The reference value for the y-emission cross section was the cross section of 0.53b for y-rays with E, = 084 MeV following inelastic scattering of l-2 MeV neutrons by iron nuclei. (2-4) We used a scatterer of iron enriched with the isotope S6Fe to 99 per cent to determine the absolute values of the y-emission cross sections of the different isotopes. The values obtained for y-emission cross sections were corrected for the dead time of the analyser, for the efficiency of recording of y-rays by the detector, and for their absorption by the scattering samples. The errors in connexion with repeated scattering of neutrons and with the difference in the angular distributions of y-rays arising from the iron and the samples were estimated. The error in the determination of the transverse cross sections was not greater than 15 per cent for strong y lines (1.37 and 183 MeV for Mg, 1.45 MeV for S2Cr and ssNi, and 1.33 MeV for soNi), or 20-25 per cent for the other lines. The cross sections for emission of y-rays following neutron inelastic scattering by chromium and nickel nuclei were studied using samples enriched with the isotopes s2Cr (to 99.5 %), ssNi (to 96.8%) and 60Ni (to 92.8 %) and samples with a natural mixture of isotopes. The emission cross sections obtained by measurements taken with the different isotopes coincided (to within the margin of experimental error) with those obtained for samples of the natural mixtures. CALCULATION OF THE CROSS SECTIONS The theory of the calculation of neutron inelastic scattering was worked out by HAUSERand FESHBACH.(~)It is based on formation of the compound nucleus and on Bohr’s hypothesis that the decay of a compound nucleus is independent of the way it was formed. It is assumed that the statistical approach is valid for a compound nucleus. The necessary transparencies Tl were calculated for an optical nuclear model with a diffuse edge of the Woods-Saxon form:
The numerical calculation method for the transparencies is given by DAY(‘O). The calculations were made with a computer. The cross sections of inelastic scattering were calculated for nuclei of 24Mg, 2sMg, 2aMg, s2Cr,58Ni,soNi and 93Nb. The parameters of (1) were the same for all the nuclei : 5 = 0.1. a = O-5 x lo-l3 cm , V, = 45 MeV, (2) The nuclear radius R was chosen in the form R = rOH3 x lo-l3 cm.
(3) The parameter r, was varied for the calculations, and that value was chosen for which the best agreement was obtained with the experimental data on the total cross sections of neutron interaction. It was found that the values of r, selected in this way
Cross sections of level excitation for Mg, ‘Wr, 6BNi, O”Ni and 03Nb
647
also gave good agreement between the calculated and measured cross sections of inelastic scattering. The results of calculation of the total cross section of inelastic scattering a,, and of individual excitation of levels for all the above-mentioned nuclei are shown in Figs. l-6.* These show for each nucleus both the cross sections of formation of a compound nucleus (T,calculated from the optical model (these cross sections may be regarded, in the case of high energies, as cross sections of neutron-nucleus inelastic interaction) and our experimental results for the excitation cross sections of individual levels. All the errors given are mean square errors of measurement. TABLE I.-GAMMA-RAY ENERGIES FOR WHICH EMISSION CROSS SECTIONS WERE MEASURED Element _____ =Mg
26Mg aeMg Wr
68Ni
60Ni
=Nb
Gamma-ray
energy (MeV) 1.37 1.60 1.83 0.94 1.45 0.6 1.00 1.33 1.45 1.80 0.86 1.33 2.2 0.74 0.81 0.96 1.08
The necessary nuclear energy levels with their spins and parities are taken, in general, from DZHELEPOV and PEKER ('); for certain nuclei they are taken from various papers.‘*-11) In cases where the spin and parity of a level were doubtful, calculations were carried out for various values of these quantities. The systems of levels on which the theoretical calculations were based are indicated in the figures. Table 1 shows the y-ray energies for which the emission cross sections were measured. DISCUSSION
OF THE
RESULTS
Magnesium
It is evident from analysis of the known level-schemes of the stable isotopes of magnesium(7*s) that y-rays with energies of 1.37 and I.83 MeV are caused by excitation of the first levels of the nuclei of 24Mg and 26Mg respectively, and y-rays with E, = I.60 MeV correspond to a direct transition from the third excited level of 25Mg to the ground state. Since the energy of the second level of the 24Mg nucleus is 4.122 MeV, our emission cross section for y-rays with an energy of 1.37 MeV, measured as far as 4 MeV, is equivalent to the excitation cross section of the l-368 MeV level and is the total cross section of neutron inelastic scattering for this isotope. As is evident from Fig. 1, the * In the figures the energies of the levels are indicated for the appropriate curves.
648
D. L. BRODER,V. E. KOLESOV,A. I. LASHUK, I. P. SADOKHINand
A. G.
DOVBENKO
calculated excitation function of the 2+ level (1.368 MeV) is in good agreement with the experimental results. The cross section of the emission of y-rays with an energy of 1.83 from the threshold up to 3 MeV can also be regarded as the excitation cross section of the 1.83 MeV level of zeMg. With higher energies, cascade transitions from higher levels to this
l!il 0 I.368 4.122 024
24 I2M8,2
26 12M814
Level 1.366 MeV
E,
MeV
FIG. l.-Cross section of neutron inelastic scattering on a4Mg and SsMg: ?? - - exand - - - - calculation (spin and parity of the 3.584 MeV perimental points; level Of and 0- respectively). level
become possible. The measured y-ray emission cross section may therefore be connected with other stimuli besides direct excitation of the 1.83 MeV level. However, good agreement of calculations and experiment is observed right up to 3.5 MeV (see Fig. 1). This is an indication that the probability of such cascade transitions in this energy range is negligible. Analysis is more complicated in the case of the emission cross section for y-rays with an energy of l-60 MeV. This cross section corresponds to the excitation cross section for the 1.61 MeV level of a5Mg only up to an energy of about 2 MeV. Excitation of the l-96 MeV level may lead to cascade transition by way of the 1.60 MeV
Cross sections of level excitation for Mg, Tr,
h*Ni, BoNi and OaNb
649
emission of y-rays with energies of 0.350 and 1.61 MeV.* A y-line with an energy close to this has been recorded in measurements of neutron inelastic scattering for magnesium u3). Thus the emission of y-rays with an energy of 1.60 MeV determines the excitation cross section of the two levels 1.610 and 1.960 MeV up to
level,
with
r iI
I.0
6 0.5
0
0.6
0.4 1 0.2
0
I.0
I.5
2.0
E “I
2.5
3.0
3.5
MeV
FIG. 2.-Cross section of neutron inelastic scattering on 26Mg: ?? - - experiment; and - - - - calculation (spin and parity of the 1.610 MeV level 7/2+ and 5/2+ respectively).
an energy of about 2.6 MeV. This is confirmed by the good agreement of the experimental data with the excitation cross section of these two levels (Fig. 2). Owing to the large number of levels at the higher energies the analysis of the experimental results ceases to be well defined. DZHELEPOV and PEKER”) give two possible spin values, 5/2+ and 7/2+, for the l-610 MeV level in 25Mg. Comparing calculation and experiment, we conclude that 7/2+ is the more likely. This is confirmed by the results of SHELINE and HARLAN. * With SHELINE and HARLANu*) however, the 1.610 and 1.960 MeV levels are referred to different rotational bands, on theoretical grounds. 4
650
D. L. BRODER, V. E. KOLESOV, A. I. LASHIJK, I. P. SADOKHIN and A. G. DOVBENKO
In calculating the cross sections, r,, was taken as 1.28 for all the magnesium isotopes. 52Cr According to the level system given by DZHELEPOVand PEKBRfor s2Cr, y-rays with an energy of 1.45 MeV arise in connexion with the transition of the nucleus from the first excited state to the ground state. Measurements show the emission of such y-rays to be already appreciable at an energy of 1.6 MeV. Therefore up to 2.5 MeV
2.430 I.460
o+ IO
I
Level
1460
MeV
”
o-5
0
05
IO
15 E “1
FIG.
3.-Cross
section of neutron
2.0 MeV
inelastic scattering calculation.
on Tr:
?? --
experiment;
the emission cross section for y-rays with Ey = I.45MeV corresponds to the excitation cross section of the 1.460 MeV level. Gamma rays with an energy of O-94 MeV were recorded only with neutron energies above 2.5 MeV. This is confirmed by the fact that a y-line with an energy of O-94 MeV arises in the case of cascade transition from the 2.430 MeV level by way of the l-460 MeV level to the ground state. Since no y-rays with an energy of 2.43 MeV were observed, the emission cross section of this y-line is the excitation cross section of the second, 2.430 MeV, level up to an energy of 3 MeV. By subtracting this cross section from the emission cross section of y-rays with an energy of l-45 MeV, we obtain the value of the excitation function of the l-460 MeV level in this energy range. Calculations made with the potential parameters given above and with r, = l-25 show good agreement of the theoretical and experimental excitation functions for the l-460 and 2.430 MeV levels up to. energies of 3.0 MeV (Fig. 3). The theoretical and experimental values for the total cross section of neutron inelastic scattering agree well.
Cross sections of level excitation for Mg, Wr,
68Ni, 60Ni and sSNb
651
All the y-lines observed for 5*Ni can be successfully analysed, starting from the experimentally established level sequence cg). Gamma rays with an energy of 1.45 MeV correspond to a transition from the first excited level to the ground state, and the cross section of their formation up to 2.5 MeV is the cross section of neutron inelastic scattering with excitation of the first level. The second level can decay with emission of one y-ray of 2.46 MeV or two cascade y-rays of 1.0 and 1.45 MeV. No y-rays with an energy of 2.46 MeV were found, but y-rays with an energy of 1.0 MeV were recorded, an appreciable emission of these being observed when E, > 2.5 MeV. The measurements showed that when the third (2.772 MeV) level is excited, emission of a single y-ray with the maximum possible energy does not occur, but a cascade transition by way of the first level is observed, with emission of y-rays with an energy of 1.33 MeV. It appears that a cascade transition by way of the second (2.458 MeV) level is unlikely. We can therefore say that the emission cross section of y-rays with an energy of I.33 MeV gives the excitation cross section of the third level. Excitation of the next two levels, which have energies very close in value (2.899 and 2.939 MeV) can lead to the emission of y-rays with an energy of about 2.9 MeV or to the emission of two y-rays each with an energy of 1.45 MeV in a cascade transition by way of the first level. Gamma rays with an energy of about 2.9 MeV were not observed. Cascade transitions by way of the second and third levels were also, apparently, unlikely. This being so, we can assume that the emission cross section of y-rays with an energy of 1.33 MeV is the excitation cross section of the 2.772 MeV level, and the emission cross section of y-rays with E, = 1.0 MeV is the excitation cross section of the second, 2.458 MeV, level of the 58Ni nucleus up to an energy of about 3.1 MeV. In the case of excitation of the 3.035 MeV level, assuming that this state decays by cascade transition by way of the second level, y-rays with energies of 0.6, 1-O and 1.45 MeV may be expected. Then the emission cross section for y-rays with an energy of O-6 MeV is the excitation cross section of the 3.035 MeV level, and the difference of the emission cross sections for y-rays with energies of 1.0 and O-6 MeV will be the excitation cross section of the 2.458 MeV level for energies above 3.1 MeV. The appearance in the spectrum of a y-line with an energy of 1.8 MeV can be explained if we assume that a transition from the 3.260 MeV level takes place by way of the first level. A cascade transition by way of the second level (2.458 MeV) appears to be unlikely, since no y-rays with an energy of 0.8 MeV are observed, such as should be produced with this decay. We can therefore assume that the emission cross section for y-rays with an energy of 1.8 MeV defines the excitation cross section of the 3.260 MeV level. This analysis of the experimental data shows that all the excited states of 5sNi decay mainly by way of the first, l-452 MeV, level. This means that the emission cross section for y-rays with an energy of 1.45 MeV is close to the total cross section of neutron inelastic scattering. For %Ni, the spins and parities are known only for the ground and first excited states.“) The energies of the higher levels are given by PARISand BUECHNER’~).We therefore confined our calculations to the first two excited levels, taking the spin and
652
D. L. BRODER, V. E. KOLESW, A. I. LASHUK, I. P. SADOKI-IIN and
A. G.
DOVBENKO
parity of the second level to be either 4+ or 2f. However, calculations of the cross sections with the parameters given above and with r,, equal to l-25, 1.32, l-35 and 1.37 did not give satisfactory agreement with the experiinental data. Figure 4 shows the
L---l
I
‘cLtll.452
2+
I
o+
,o
I
I
I
iiNiu, 04
I
I
I
-Level2772 MeV o,05 - E,=
I. 0
MeV
0
.p/y/.I
f
=
I.0
Level
I.452 /y~y-.__m-.I@* o.5 MeV T . 0 0 I.0
I.5 2.0
2.5 E"(
0
0
30
35
MeV
FIG.4.-Cross
section of neutron inelastic scattering on 5wi: 0 -- experiment; and - - - - calculation (spin and parity of the 2.458 MeV level 4f and 2+ re spectively); - * - * - curves drawn through the experimental points.
calculated
and experimental cross sections for r, = l-37. The discrepancy was even greater for the other values of r,.
The y-lines found here can also be explained on the basis of the level system of the Up to a neutron energy of 2.2 MeV, y-rays with an energy of l-33 MeV result from excitation of the first level. When the next two levels, which are close in energy (2.158 MeV and 2.285 MeV) and unresolved, are excited, the possibility arises of cascade as well as direct transitions to the ground state. Gamma rays with an energy of O-86 MeV, the emission of which becomes appreciable only when E, > 2.2 MeV, are the result of cascade transitions of the nucleus from these levels to the
WNi nucleus(7).
Cross sections of level excitation for Mg, Yk,
653
6SNi, BoNi and DaNb
ground state by way of the l-332 MeV level. The y-rays with an energy of 2.20 MeV which arise at a neutron energy of 2.7 MeV are caused by direct transition of the nucleus from these states to the ground state. The sum of the emission cross sections for y-rays with energies of O-86 and 2.20 MeV gives the excitation cross section of the 2.158 and 2.285 MeV levels.
0.3,
I
I
I
I
2.62
0
05
IO
I.5 2.0 EIF
FIG. 5.-Cross
2-5
3.0
3.5
Mev
section of neutron inelastic scattering on s0Ni: 0 - experiment; calculation.
___
The spins and parities of many of the states in the level system of 60Ni are unknown(7). This makes it difficult to calculate the cross sections and compare them with the experimental results. A spin and parity of 2+ was ascribed to the 2.627 MeV state. The 2.285 MeV level, which is close to the 2+ (2.158 MeV) level, and a number of levels in the 3.130 to 3.520 MeV range were not taken into account in the calculations. The calculations of the cross sections were carried out with the same parameters for the potential and the same values of r, as for %Ni. The results of calculation for the case of r, = l-37 and a comparison with the experimental data are shown in Fig. 5. A comparatively good agreement between theory and experiment is observed for both q, and the excitation functions of the first two levels. s3Nb The level system of the s3Nb nucleus is not yet precisely established. The fullest information on the energies of the levels is to be found in the paper by NATH et aI.
654
D. L. BRODER,V. E.
KOLESOV, A. I. LASHUK, I. P. SADOKHIN
and A. G.
DOVBENKO
The y-lines recorded by us with energies of O-74, 0.96 and 1.08 MeV correspond, at least up to neutron energies of l-2 MeV, to direct transition between the appropriate excitation levels and the ground state or a state with an energy of 29 keV. It is indicated that there are a great many cascade transitions by way of the 0.74 and O-96 MeV levels from the higher states, 1.28, 1.34 and 1.48 MeV. Thus the measured y-ray emission cross sections can be regarded as the excitation cross sections of the corresponding levels only up to E, G 1.3 MeV.
0
I 0. - Level
I
0.5
0
I
lr*
0.74 MeV
0.5.
4 ‘-‘;- _g
I.0
*
I.5
E “’
2.0
2.5
MeV
FIG.
6.-Gross section of inelastic scattering by niobium. Results of calculation for three sets of spins and parities: ----II ; - . - . - III ; 0 experimental I; points.
The excitation functions of individual levels of the 93Nb nucleus were calculated with r,, = 1.35. The level system used as the basis of calculation was taken from NATH et aZ.(ll) Calculations were made for three sets of spins and parities, and are shown in Fig. 6. The figure also shows the experimental points. It is difficult to draw any conclusions from a comparison of theory and experiment, since the spin and parity values of the levels are unknown. From our calculations, we prefer set II of
Cross sections of level excitation for Mg, 5eCr, 5*Ni, 6ONi and 93Nb
655
and parities (see Fig. 6). The fact that the values of the experimental emission cross sections are higher than those of the excitation cross sections of individual levels when En > 1.5 MeV for almost all the y-lines can be explained by cascade transitions by way of the said levels from higher levels. This also has reference to the total y-ray emission. The agreement of theory and experiment in this energy range for the l-08 MeV level may indicate that cascade transitions by way of this level are unlikely. This conclusion is in keeping with the data of NATH et aZ.(ll) Our calculations show a satisfactory agreement between the theoretical and experimental cross sections in all cases where the nuclear level systems and their spins and parities are known precisely. An exception is 58Ni. The divergence of the experimental and theoretical values of the excitation cross sections of the first level have not been satisfactorily explained here from the statistical point of view. The values obtained for the cross sections can be used in the construction of multigroup constants for the calculation of nuclear energy installations and their biological shields.
spins
authors thank SH. S. NIKOLAISHVILIfor his interest in this work, and V. V. BULYCHEVICH,A. N. SORBINOV,V. A. ROMANOVand A. P. KLIMOVfor seeing to the stable operation of the computer and accelerators.
Acknowledgments--The
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
BRODER D. L., LASHUK A. I. and SADOKHINI. P. Izv. Akad. Nauk SSSR Ser. fiz. 25, 309 (1961). LOEF I. VAN and LIND D. Phys. Rev. 101, 103 (1956). KIEHN R. M. and GOODMANC. Phys. Rev. 95,989 (1954). HUGHES D. and SCHWARTZ R. Neutron Cross Sections, BNL-325 (1958). HAUSER W. and FESHBACHH. Phys. Rev. 87, 366 (1952). YERMAKOV S. M., KOLESOV V. E. and MARCHUK G. I. In Symposium on Neutron Physics, p. 314, Moscow, Gosatomizdat (1961). DZHELEPOVB. S. and PEKER L. K. Modes of Decay of Radioactive Nuclei. Moscow-Leningrad, Izdatel’stvo Akad. Nauk SSSR (1958). ENDT P. and LEXJN C. VAN DER Nucl. Phys. 34,l (1962). PARIS C. and BUECHNER W. Comptes Rendus du Congress International de Physique Nucltfaire, Paris, 1958 DUNOB (1959). DAY R. In Symposium on the Nuclear Reactions at Low and Medium Energies, p. 149, Moscow, Izdatel’stvo Akad. Nauk SSSR (1958). NATH N. et al. Nucl. Phys. 14, 78 (1959). SHELINER. and HARLAN R. Nucl. Phys. 29, 177 (1962). ANDROSENKOA. L., BRODER D. L. and LASH~JKA. I. Atomnaya Energiya 9,403 (1960).
CROSS SECTIONS OF LEVEL EXCITATION FOR Mg, 52Cr, 58Ni, 6oNi AND g3Nb IN NEUTRON INELASTIC SCATTERING* D. L. BRODER, V. E. KOLESOV, and A. G. (Received
A.
I.
LASHUK,
I.
P.
SADOKHIN
DOVBENKO
15 April 1963)
Abstract-The neutron excitation cross sections of some levels of 84Mg, a6Mg, eaMg, 58Ni, 60Ni, 5aCr and QSNb were measured in the range of energies from 1 to 4 MeV. The experimental results were compared with the results of optical model calculations with a Woods-Saxon potential. INTRODUCTION
on the cross sections of neutron inelastic scattering can be obtained by various experimental methods. The method based on the measurement of the spectra of the y-radiation accompanying the process is widely used. This method enables us not only to establish the energy level systems of the nuclei, but also to determine the values of the excitation cross sections for individual levels. If we use separated isotopes as scatterers, we can interpret the experimental results uniquely. A theory enabling us to calculate the different cross sections of interaction of neutrons with atomic nuclei has not as yet been satisfactorily worked out. The greatest success is achieved when the cross sections are calculated for an optical model of the nucleus. Such a model, that is, a complex potential with a diffuse edge, makes it possible to describe satisfactorily not only the total cross section of neutronnucleus interaction, but also its individual components. The model is also used to calculate the cross sections of neutron inelastic scattering, and gives good agreement of the theoretical cross sections with the experimental in the cases where the nuclear level schemes and their quantum characteristics are known. In the present paper we examine the cross sections of level excitation in neutron inelastic scattering for nuclei of Mg, 52Cr, 58Ni, 60Ni and g3Nb. We compare the experimental results with the theoretical results calculated on the basis of an optical nuclear model with a Woods-Saxon potential. INFORMATION
THE
EXPERIMENT
For the source of monochromatic neutrons with energies up to 3 MeV we used the reaction 3H(p, n)3He; for energies above 3 MeV we used the reaction 2H(d, n)3He. These reactions were brought about with the aid of electrostatic and cascade generators. The scatter of neutron energies did not exceed *30 keV in the first case and f80 keV in the second case. To measure the y-ray emission accompanying the neutron inelastic scattering we used a spectrometer with a 40 x 40 mm NaI(T1) crystal, an FEU-13 photomultiplier and a 128-channel amplitude analyser. The measurements were made in an annular geometry(l). The resolution of the spectrometer for the O-661 MeV line of 13’Cs was about 11 * Translated by J. STUART from
Atomnayu
Energiya 645
16, 103 (1964).
646
D. L. BRODER, V. E. KOLEWV, A. I. LASHUK, I. P. SAD~KHIN and A. G. DOVBENKO
per cent. The energy calibration of the spectrometer and determination of the recording efficiency were effected as given by BRODERand LASHUK.~) We used a fission chamber with 23sU and an all-wave counter to check the uniformity of the neutron flux. The reference value for the y-emission cross section was the cross section of 0.53b for y-rays with E, = 084 MeV following inelastic scattering of l-2 MeV neutrons by iron nuclei. (2-4) We used a scatterer of iron enriched with the isotope S6Fe to 99 per cent to determine the absolute values of the y-emission cross sections of the different isotopes. The values obtained for y-emission cross sections were corrected for the dead time of the analyser, for the efficiency of recording of y-rays by the detector, and for their absorption by the scattering samples. The errors in connexion with repeated scattering of neutrons and with the difference in the angular distributions of y-rays arising from the iron and the samples were estimated. The error in the determination of the transverse cross sections was not greater than 15 per cent for strong y lines (1.37 and 183 MeV for Mg, 1.45 MeV for S2Cr and ssNi, and 1.33 MeV for soNi), or 20-25 per cent for the other lines. The cross sections for emission of y-rays following neutron inelastic scattering by chromium and nickel nuclei were studied using samples enriched with the isotopes s2Cr (to 99.5 %), ssNi (to 96.8%) and 60Ni (to 92.8 %) and samples with a natural mixture of isotopes. The emission cross sections obtained by measurements taken with the different isotopes coincided (to within the margin of experimental error) with those obtained for samples of the natural mixtures. CALCULATION OF THE CROSS SECTIONS The theory of the calculation of neutron inelastic scattering was worked out by HAUSERand FESHBACH.(~)It is based on formation of the compound nucleus and on Bohr’s hypothesis that the decay of a compound nucleus is independent of the way it was formed. It is assumed that the statistical approach is valid for a compound nucleus. The necessary transparencies Tl were calculated for an optical nuclear model with a diffuse edge of the Woods-Saxon form:
The numerical calculation method for the transparencies is given by DAY(‘O). The calculations were made with a computer. The cross sections of inelastic scattering were calculated for nuclei of 24Mg, 2sMg, 2aMg, s2Cr,58Ni,soNi and 93Nb. The parameters of (1) were the same for all the nuclei : 5 = 0.1. a = O-5 x lo-l3 cm , V, = 45 MeV, (2) The nuclear radius R was chosen in the form R = rOH3 x lo-l3 cm.
(3) The parameter r, was varied for the calculations, and that value was chosen for which the best agreement was obtained with the experimental data on the total cross sections of neutron interaction. It was found that the values of r, selected in this way
Cross sections of level excitation for Mg, ‘Wr, 6BNi, O”Ni and 03Nb
647
also gave good agreement between the calculated and measured cross sections of inelastic scattering. The results of calculation of the total cross section of inelastic scattering a,, and of individual excitation of levels for all the above-mentioned nuclei are shown in Figs. l-6.* These show for each nucleus both the cross sections of formation of a compound nucleus (T,calculated from the optical model (these cross sections may be regarded, in the case of high energies, as cross sections of neutron-nucleus inelastic interaction) and our experimental results for the excitation cross sections of individual levels. All the errors given are mean square errors of measurement. TABLE I.-GAMMA-RAY ENERGIES FOR WHICH EMISSION CROSS SECTIONS WERE MEASURED Element _____ =Mg
26Mg aeMg Wr
68Ni
60Ni
=Nb
Gamma-ray
energy (MeV) 1.37 1.60 1.83 0.94 1.45 0.6 1.00 1.33 1.45 1.80 0.86 1.33 2.2 0.74 0.81 0.96 1.08
The necessary nuclear energy levels with their spins and parities are taken, in general, from DZHELEPOV and PEKER ('); for certain nuclei they are taken from various papers.‘*-11) In cases where the spin and parity of a level were doubtful, calculations were carried out for various values of these quantities. The systems of levels on which the theoretical calculations were based are indicated in the figures. Table 1 shows the y-ray energies for which the emission cross sections were measured. DISCUSSION
OF THE
RESULTS
Magnesium
It is evident from analysis of the known level-schemes of the stable isotopes of magnesium(7*s) that y-rays with energies of 1.37 and I.83 MeV are caused by excitation of the first levels of the nuclei of 24Mg and 26Mg respectively, and y-rays with E, = I.60 MeV correspond to a direct transition from the third excited level of 25Mg to the ground state. Since the energy of the second level of the 24Mg nucleus is 4.122 MeV, our emission cross section for y-rays with an energy of 1.37 MeV, measured as far as 4 MeV, is equivalent to the excitation cross section of the l-368 MeV level and is the total cross section of neutron inelastic scattering for this isotope. As is evident from Fig. 1, the * In the figures the energies of the levels are indicated for the appropriate curves.
648
D. L. BRODER,V. E. KOLESOV,A. I. LASHUK, I. P. SADOKHINand
A. G.
DOVBENKO
calculated excitation function of the 2+ level (1.368 MeV) is in good agreement with the experimental results. The cross section of the emission of y-rays with an energy of 1.83 from the threshold up to 3 MeV can also be regarded as the excitation cross section of the 1.83 MeV level of zeMg. With higher energies, cascade transitions from higher levels to this
l!il 0 I.368 4.122 024
24 I2M8,2
26 12M814
Level 1.366 MeV
E,
MeV
FIG. l.-Cross section of neutron inelastic scattering on a4Mg and SsMg: ?? - - exand - - - - calculation (spin and parity of the 3.584 MeV perimental points; level Of and 0- respectively). level
become possible. The measured y-ray emission cross section may therefore be connected with other stimuli besides direct excitation of the 1.83 MeV level. However, good agreement of calculations and experiment is observed right up to 3.5 MeV (see Fig. 1). This is an indication that the probability of such cascade transitions in this energy range is negligible. Analysis is more complicated in the case of the emission cross section for y-rays with an energy of l-60 MeV. This cross section corresponds to the excitation cross section for the 1.61 MeV level of a5Mg only up to an energy of about 2 MeV. Excitation of the l-96 MeV level may lead to cascade transition by way of the 1.60 MeV
Cross sections of level excitation for Mg, Tr,
h*Ni, BoNi and OaNb
649
emission of y-rays with energies of 0.350 and 1.61 MeV.* A y-line with an energy close to this has been recorded in measurements of neutron inelastic scattering for magnesium u3). Thus the emission of y-rays with an energy of 1.60 MeV determines the excitation cross section of the two levels 1.610 and 1.960 MeV up to
level,
with
r iI
I.0
6 0.5
0
0.6
0.4 1 0.2
0
I.0
I.5
2.0
E “I
2.5
3.0
3.5
MeV
FIG. 2.-Cross section of neutron inelastic scattering on 26Mg: ?? - - experiment; and - - - - calculation (spin and parity of the 1.610 MeV level 7/2+ and 5/2+ respectively).
an energy of about 2.6 MeV. This is confirmed by the good agreement of the experimental data with the excitation cross section of these two levels (Fig. 2). Owing to the large number of levels at the higher energies the analysis of the experimental results ceases to be well defined. DZHELEPOV and PEKER”) give two possible spin values, 5/2+ and 7/2+, for the l-610 MeV level in 25Mg. Comparing calculation and experiment, we conclude that 7/2+ is the more likely. This is confirmed by the results of SHELINE and HARLAN. * With SHELINE and HARLANu*) however, the 1.610 and 1.960 MeV levels are referred to different rotational bands, on theoretical grounds. 4
650
D. L. BRODER, V. E. KOLESOV, A. I. LASHIJK, I. P. SADOKHIN and A. G. DOVBENKO
In calculating the cross sections, r,, was taken as 1.28 for all the magnesium isotopes. 52Cr According to the level system given by DZHELEPOVand PEKBRfor s2Cr, y-rays with an energy of 1.45 MeV arise in connexion with the transition of the nucleus from the first excited state to the ground state. Measurements show the emission of such y-rays to be already appreciable at an energy of 1.6 MeV. Therefore up to 2.5 MeV
2.430 I.460
o+ IO
I
Level
1460
MeV
”
o-5
0
05
IO
15 E “1
FIG.
3.-Cross
section of neutron
2.0 MeV
inelastic scattering calculation.
on Tr:
?? --
experiment;
the emission cross section for y-rays with Ey = I.45MeV corresponds to the excitation cross section of the 1.460 MeV level. Gamma rays with an energy of O-94 MeV were recorded only with neutron energies above 2.5 MeV. This is confirmed by the fact that a y-line with an energy of O-94 MeV arises in the case of cascade transition from the 2.430 MeV level by way of the l-460 MeV level to the ground state. Since no y-rays with an energy of 2.43 MeV were observed, the emission cross section of this y-line is the excitation cross section of the second, 2.430 MeV, level up to an energy of 3 MeV. By subtracting this cross section from the emission cross section of y-rays with an energy of l-45 MeV, we obtain the value of the excitation function of the l-460 MeV level in this energy range. Calculations made with the potential parameters given above and with r, = l-25 show good agreement of the theoretical and experimental excitation functions for the l-460 and 2.430 MeV levels up to. energies of 3.0 MeV (Fig. 3). The theoretical and experimental values for the total cross section of neutron inelastic scattering agree well.
Cross sections of level excitation for Mg, Wr,
68Ni, 60Ni and sSNb
651
All the y-lines observed for 5*Ni can be successfully analysed, starting from the experimentally established level sequence cg). Gamma rays with an energy of 1.45 MeV correspond to a transition from the first excited level to the ground state, and the cross section of their formation up to 2.5 MeV is the cross section of neutron inelastic scattering with excitation of the first level. The second level can decay with emission of one y-ray of 2.46 MeV or two cascade y-rays of 1.0 and 1.45 MeV. No y-rays with an energy of 2.46 MeV were found, but y-rays with an energy of 1.0 MeV were recorded, an appreciable emission of these being observed when E, > 2.5 MeV. The measurements showed that when the third (2.772 MeV) level is excited, emission of a single y-ray with the maximum possible energy does not occur, but a cascade transition by way of the first level is observed, with emission of y-rays with an energy of 1.33 MeV. It appears that a cascade transition by way of the second (2.458 MeV) level is unlikely. We can therefore say that the emission cross section of y-rays with an energy of I.33 MeV gives the excitation cross section of the third level. Excitation of the next two levels, which have energies very close in value (2.899 and 2.939 MeV) can lead to the emission of y-rays with an energy of about 2.9 MeV or to the emission of two y-rays each with an energy of 1.45 MeV in a cascade transition by way of the first level. Gamma rays with an energy of about 2.9 MeV were not observed. Cascade transitions by way of the second and third levels were also, apparently, unlikely. This being so, we can assume that the emission cross section of y-rays with an energy of 1.33 MeV is the excitation cross section of the 2.772 MeV level, and the emission cross section of y-rays with E, = 1.0 MeV is the excitation cross section of the second, 2.458 MeV, level of the 58Ni nucleus up to an energy of about 3.1 MeV. In the case of excitation of the 3.035 MeV level, assuming that this state decays by cascade transition by way of the second level, y-rays with energies of 0.6, 1-O and 1.45 MeV may be expected. Then the emission cross section for y-rays with an energy of O-6 MeV is the excitation cross section of the 3.035 MeV level, and the difference of the emission cross sections for y-rays with energies of 1.0 and O-6 MeV will be the excitation cross section of the 2.458 MeV level for energies above 3.1 MeV. The appearance in the spectrum of a y-line with an energy of 1.8 MeV can be explained if we assume that a transition from the 3.260 MeV level takes place by way of the first level. A cascade transition by way of the second level (2.458 MeV) appears to be unlikely, since no y-rays with an energy of 0.8 MeV are observed, such as should be produced with this decay. We can therefore assume that the emission cross section for y-rays with an energy of 1.8 MeV defines the excitation cross section of the 3.260 MeV level. This analysis of the experimental data shows that all the excited states of 5sNi decay mainly by way of the first, l-452 MeV, level. This means that the emission cross section for y-rays with an energy of 1.45 MeV is close to the total cross section of neutron inelastic scattering. For %Ni, the spins and parities are known only for the ground and first excited states.“) The energies of the higher levels are given by PARISand BUECHNER’~).We therefore confined our calculations to the first two excited levels, taking the spin and
652
D. L. BRODER, V. E. KOLESW, A. I. LASHUK, I. P. SADOKI-IIN and
A. G.
DOVBENKO
parity of the second level to be either 4+ or 2f. However, calculations of the cross sections with the parameters given above and with r,, equal to l-25, 1.32, l-35 and 1.37 did not give satisfactory agreement with the experiinental data. Figure 4 shows the
L---l
I
‘cLtll.452
2+
I
o+
,o
I
I
I
iiNiu, 04
I
I
I
-Level2772 MeV o,05 - E,=
I. 0
MeV
0
.p/y/.I
f
=
I.0
Level
I.452 /y~y-.__m-.I@* o.5 MeV T . 0 0 I.0
I.5 2.0
2.5 E"(
0
0
30
35
MeV
FIG.4.-Cross
section of neutron inelastic scattering on 5wi: 0 -- experiment; and - - - - calculation (spin and parity of the 2.458 MeV level 4f and 2+ re spectively); - * - * - curves drawn through the experimental points.
calculated
and experimental cross sections for r, = l-37. The discrepancy was even greater for the other values of r,.
The y-lines found here can also be explained on the basis of the level system of the Up to a neutron energy of 2.2 MeV, y-rays with an energy of l-33 MeV result from excitation of the first level. When the next two levels, which are close in energy (2.158 MeV and 2.285 MeV) and unresolved, are excited, the possibility arises of cascade as well as direct transitions to the ground state. Gamma rays with an energy of O-86 MeV, the emission of which becomes appreciable only when E, > 2.2 MeV, are the result of cascade transitions of the nucleus from these levels to the
WNi nucleus(7).
Cross sections of level excitation for Mg, Yk,
653
6SNi, BoNi and DaNb
ground state by way of the l-332 MeV level. The y-rays with an energy of 2.20 MeV which arise at a neutron energy of 2.7 MeV are caused by direct transition of the nucleus from these states to the ground state. The sum of the emission cross sections for y-rays with energies of O-86 and 2.20 MeV gives the excitation cross section of the 2.158 and 2.285 MeV levels.
0.3,
I
I
I
I
2.62
0
05
IO
I.5 2.0 EIF
FIG. 5.-Cross
2-5
3.0
3.5
Mev
section of neutron inelastic scattering on s0Ni: 0 - experiment; calculation.
___
The spins and parities of many of the states in the level system of 60Ni are unknown(7). This makes it difficult to calculate the cross sections and compare them with the experimental results. A spin and parity of 2+ was ascribed to the 2.627 MeV state. The 2.285 MeV level, which is close to the 2+ (2.158 MeV) level, and a number of levels in the 3.130 to 3.520 MeV range were not taken into account in the calculations. The calculations of the cross sections were carried out with the same parameters for the potential and the same values of r, as for %Ni. The results of calculation for the case of r, = l-37 and a comparison with the experimental data are shown in Fig. 5. A comparatively good agreement between theory and experiment is observed for both q, and the excitation functions of the first two levels. s3Nb The level system of the s3Nb nucleus is not yet precisely established. The fullest information on the energies of the levels is to be found in the paper by NATH et aI.
654
D. L. BRODER,V. E.
KOLESOV, A. I. LASHUK, I. P. SADOKHIN
and A. G.
DOVBENKO
The y-lines recorded by us with energies of O-74, 0.96 and 1.08 MeV correspond, at least up to neutron energies of l-2 MeV, to direct transition between the appropriate excitation levels and the ground state or a state with an energy of 29 keV. It is indicated that there are a great many cascade transitions by way of the 0.74 and O-96 MeV levels from the higher states, 1.28, 1.34 and 1.48 MeV. Thus the measured y-ray emission cross sections can be regarded as the excitation cross sections of the corresponding levels only up to E, G 1.3 MeV.
0
I 0. - Level
I
0.5
0
I
lr*
0.74 MeV
0.5.
4 ‘-‘;- _g
I.0
*
I.5
E “’
2.0
2.5
MeV
FIG.
6.-Gross section of inelastic scattering by niobium. Results of calculation for three sets of spins and parities: ----II ; - . - . - III ; 0 experimental I; points.
The excitation functions of individual levels of the 93Nb nucleus were calculated with r,, = 1.35. The level system used as the basis of calculation was taken from NATH et aZ.(ll) Calculations were made for three sets of spins and parities, and are shown in Fig. 6. The figure also shows the experimental points. It is difficult to draw any conclusions from a comparison of theory and experiment, since the spin and parity values of the levels are unknown. From our calculations, we prefer set II of
Cross sections of level excitation for Mg, 5eCr, 5*Ni, 6ONi and 93Nb
655
and parities (see Fig. 6). The fact that the values of the experimental emission cross sections are higher than those of the excitation cross sections of individual levels when En > 1.5 MeV for almost all the y-lines can be explained by cascade transitions by way of the said levels from higher levels. This also has reference to the total y-ray emission. The agreement of theory and experiment in this energy range for the l-08 MeV level may indicate that cascade transitions by way of this level are unlikely. This conclusion is in keeping with the data of NATH et aZ.(ll) Our calculations show a satisfactory agreement between the theoretical and experimental cross sections in all cases where the nuclear level systems and their spins and parities are known precisely. An exception is 58Ni. The divergence of the experimental and theoretical values of the excitation cross sections of the first level have not been satisfactorily explained here from the statistical point of view. The values obtained for the cross sections can be used in the construction of multigroup constants for the calculation of nuclear energy installations and their biological shields.
spins
authors thank SH. S. NIKOLAISHVILIfor his interest in this work, and V. V. BULYCHEVICH,A. N. SORBINOV,V. A. ROMANOVand A. P. KLIMOVfor seeing to the stable operation of the computer and accelerators.
Acknowledgments--The
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
BRODER D. L., LASHUK A. I. and SADOKHINI. P. Izv. Akad. Nauk SSSR Ser. fiz. 25, 309 (1961). LOEF I. VAN and LIND D. Phys. Rev. 101, 103 (1956). KIEHN R. M. and GOODMANC. Phys. Rev. 95,989 (1954). HUGHES D. and SCHWARTZ R. Neutron Cross Sections, BNL-325 (1958). HAUSER W. and FESHBACHH. Phys. Rev. 87, 366 (1952). YERMAKOV S. M., KOLESOV V. E. and MARCHUK G. I. In Symposium on Neutron Physics, p. 314, Moscow, Gosatomizdat (1961). DZHELEPOVB. S. and PEKER L. K. Modes of Decay of Radioactive Nuclei. Moscow-Leningrad, Izdatel’stvo Akad. Nauk SSSR (1958). ENDT P. and LEXJN C. VAN DER Nucl. Phys. 34,l (1962). PARIS C. and BUECHNER W. Comptes Rendus du Congress International de Physique Nucltfaire, Paris, 1958 DUNOB (1959). DAY R. In Symposium on the Nuclear Reactions at Low and Medium Energies, p. 149, Moscow, Izdatel’stvo Akad. Nauk SSSR (1958). NATH N. et al. Nucl. Phys. 14, 78 (1959). SHELINER. and HARLAN R. Nucl. Phys. 29, 177 (1962). ANDROSENKOA. L., BRODER D. L. and LASH~JKA. I. Atomnaya Energiya 9,403 (1960).