NUCLEAR CHARGE DISTRIBUTION IN PROTON FISSION OF Th 2 32
by
Philip P. Benjamin, M. Sc.
A thesis subrnitted to the Faculty of
Graduate Studies and Research in partial fulfilrnent
of the requirernents for the degree of
Doctor of Philosophy
Chernistry Departrnent
McGill University,
Montreal, Canada
•
April, 1965
Chemistry
Ph.D.
Philip P. BenJamin, M.Sc.
NUC:t..EAR CHARGE DISTRIBUTION IN PROTON FISSION OF Th 23 2
AB8TRACT
The independant formation cross-sections of 30 min cs 130 1
6.SdCs132 , 2.9 hrCs
18 d Rb
86
1
134
m, 2.1 yrCs 1349, 13 dCs 136 , 32 minCs
138
.,
and cumulative formation cross-sections of 30 yr Cs 137 and
17 min Xe 138 produced in the fission Qf Th2 32 qy 2G by 85 Me v.
·ptetons · h.,a v e · been measured radiochemically. Charge distribution
curves were constructed at 20, 30 1 39., 50 1 57, 65, 75 and 85 Mev.
With increasing proton energy the se curves broaden and the most probable
charge shifts towards the Une of be ta stabUity. The se effects are interpreted in terms of fission-spallation competition. Detailed calculations
on the basis of two empirical postulates on charge distribution have been
attempted. Previously published experimental results on proton induced
fission of uranium have also been incorporated in these calculations to
make a comparative study. lt was also possible to determine the isomer
ratio Cs 134 m;cs 1349 in these experimenta and to compare the results with
theoretical calculations at the lowest energies studied. Sorne qualitative
conclusions could be drawn about the angular momentum distribution dur!ng
the fission process.
ACKNOWLEDGEMENTS
The author wishes to express his sincere gratitude to the
following persons and institutions:
Professer L. Yaffe for suggesting the problem and for
encouragement received from him throughout the course of this work.
Dr. N.T. Porile for helpful suggestions and criticisms on
the theoretical aspects of this problem.
Professer R.E. Bell for permission to use the Cyclotron and
Mr. R. Mills for his cooperation and assistance in carrying out the
irradiations.
Prof essor W. D. Thorpe for permission to use the 1. B. M.
computer, and to McGill Computing Centre staff for their cooperation.
Dr. J.H. Davies, Dr. Donald Marsden, Dr. G.B. Soho, Messrs.
J. Foster and Ramamoorthy and ali ether members of the radiochem istry group
for their assistance and cooperation in various ways.
The Chemistry Department, McGill University, for demonstratorships for the academie sessions 1960-64, and for support from the
United States Air Force.
The Ministry of Education in lndia for the award of a travel
grant to Canada.
To my frien.ds Dt.R.C. Shukla and Dr.A.Malviya for constant encouragement.
To my wife lvy for her patience and encouragement at ali stages
of this work.
TABLE OF CONTENTS
I.
Introduction ................................
(t
.............
,_,
•••
1
1
I.l General
I. 2 Fis sion mo dels ........................................... . 3
5
I. 3 Fission yield disllril:iutions ............................. o . . . . . . .
I.4 Charged particle induced fission ............................ .. 12
I. 5 Correlation of independent yields in
charged particle induced fission ............................... . 13
I. 6 Present work. •••.•••••••••••••••• ,. ....................... . 17
......................................... 0
II.
•
-0
••••••
Experimental • • • • • • .. • • • • • • • • • • • • ..... ..............................
21
II.l Target and irradiation • • • • • • • • • • • • .. • • • • • • • • • • • • • .. • .. .. • .. • • • •
II. 2 Chemical processing • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • .. .. • ..
II. 3 Radioactive measurements • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
II. 4 Decay curve analysis • • • • • • • • • • .. • • • • • • • • • • • • • • • • • • • • • • • •
II.5 Calculation of cross sections .......... ,. ..............
II. 6 Accuracy of results • • • • • • • .. • • • • • • • • • • • • • • • • • .. • • • • .. • .. .. • •
21
23
27
33
34
41
o. o.....
III. Observations and Results • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
o...........
45
III.l Cross sections and excitation functions ·,. ........
III. 2 Charge distribution curves • • .. • • .. • • • • • • • • • • • • • • • • • .. .. • • • • •
45
45
IV. Discussion ••••••••••••••••••••••••••••••••••••••••••••••••
83
IV .1 A study of the excitation functions ......................... .
IV. 2 Charge distribution parameters ............................ ..
IV. 3. 1 Calculation of most probable charge, ZP .••••••••••••••••
IV. 3 .-2 Comparison of calculated and· experimental values· ......... .
IV. 4 ~ Isomer ratio ••••••••••.•••••••••••..•••••••••••••••••••
IV. 4.1 Statement of the problem ••• • .......................... .
IV .. 4. 2 The ory of the calculation .............................. ..
IV. 4. 3 Interpretation of results ••••• ,. ............................ .
V.
83
86
90
120
124
124
128
148
Summ&ry ........................... .- • • • • • .. • • • • • • • • • • • • • .. • • • • 1 58
Bibliography ···~··••••••••••••••••••••••••••••••••••••••••••160
LIST OF FIGURES
1.
Target mounting arrangement • • .. • • • • • • • • • • • • • • • • • .. • • • .. • • .. • .. • • • • • 22
2.
Section of the chart of the nuclides in the Cs region .............. ,.
46
3.
Excitation ;f.u··n ç t i o r.t
for the independant formation
13
cross section oi ës ~. •.• .. • • • • .. • .. • • • • • .. • • .. • • • • • • • • • • • • .. • • • • • •
48
Excitation function for the independant formation cross
section of C s13 4m.. • • • • • • • • • • • • • • • • • • • • • • • • • .. • • • • . • .. • • • • • .. • • • •
49
4.
s.
Excitation function for the independant formation cross
section of cs134 rn+ g • • • • • • .. • • • • • • .. • • • • • • • • • • • • • • • • • • • • • • • • • 50
6.
Excitation function for the independant formation cross
section of cs136. •• .. • • ....... •• •• • •••••••• .... • • • •••• •• • .... ••••
51
Excitation function for the cumulative formation cross
section of csl37 • • • • • • • • • • • • • ••.••• •• • • • • • .. • • • • • • • • • • • • • • .. • •
52
7.
s.
Excitation func~hons for the independant formation cross
section of Cs 1 and cumulative formation cross section
ofXel38 ••••••••••••••••••••••••••••••••••••••••••••••••••
53
Excitation furs"bti on for the independant forma ti on cross
section of Rb
....................................................
54
10. Charge distribution curve at 20 Mev. • •••••••••••••••••••••••••
73
11. Charge distribution curve at 30 Mev. ••••••••••••••••••••••••••
74
12. Charge distribution curve at 39 Mev • ••••••••••••••••••••••••••
75
13 .. Charge distribution curve at 50 Mev. • •••••••••••••••••••••••••
76
14. Charge distribution. curve at 57 Mev. ••••••••••••••••••••••••••
77
9.
15~
16.
..........................
Charge distribution curve at 75 Mev • ..... ......................
Charge distribution curve at 65 Mev ..
17 .. Charge distribution curve at 85 Mev. • ••••••••••••••••••••••••
78
79
80
LIST OF FIGURES (Cont' d .. )
Page
18. Tota 1 lsobar i c cross section for A= 136 • • • • • • • • • • • • • • • • • • • • • • • • • • •
,J:l2
19. Energies at which the excitation functions
reach maxima (a) as the function of N/Z
(b) as a function of ( Z- ZA) ••••••••••• , •••• , , •• , ••••••••••• , •
8.§
20a Full width at half maximum of charge distribution
curve plotted against incident proton energy • • • • • • • • • • • • • • • • • • • • • •
8"7
20b Displacement of most probable charge Zp
towards beta stab il ity ZA · .................................... , •
87
21. Fission probabil ity of heavy nu cl ides as
function of moss number A ••• , •••••••••••• , • • • • • • • • • • • • • • • • • • • • •
98'
22. Decay scheme
~f
Cs
13
4m ••••• , •••••••••••••••••••••••••••••••••
125'·
23. Ratio of formation cross sections of the meta
stable and ground states of Cs 134 as a function
of proton bombarding energy • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
l26
24. Calculated cross section for compound nucleus formation
as a function of the resultant spin in the co"f:und
nucleus formed in the bombardment of Hî 23 with
19 and 24 Mev • protons •..•••••.•••••.• o . . . . . . . . . . .
Co •
•
•
•
25. Change in the probabil ity distribution of spins due
to prefission neutron emission a.t .19Mev. witho'= 3 ••••••• u
•
•
•
•
•
•
•
•
•
..........
1à7
1381
26. Change in the probability distribution of spins due
to prefission neutron emission at 19 Mev. with 0= 4 •••••••••••••••••• 139 ..
27. Change in the probabil ity distribution of spins due
to prefission neutron emission at 19 Mev. withG'= 5 •••••••••••••• •••• 140
28. Change in the probabll ity distribution of spins due
to prefission neutron emission at 24 Mev. with d = 3 •• ., •••••••••••••• 141
29. Change Ln the probabil ity distribution of spins due
to prefission neutron emission at 24 Mev. with d = 4 ••• , ••••••••••••• 142
LIST OF FIGURES
Page
(Cont' d.)
30. Change in the probabil ity distribution of spins due
to prefission neutron emission at 24 Mev, with U
=5................
143
31. Change in the probabil ity distribution of spins due
to prompt neutron emission:
(a) with spin eut off factorG = 3 •••••••••••••••••••••••••••••••• 151
(b) n
n
u
u
u
r;; =4 • • • • • • • • • • • • o • • • • • • • • • • • • • • • • • • • • 152
.)
Il
Il
Il
Il
Il
(1
-5
153
(c
-
................................ .
32. Coupling of anguler momentum of primary
fission fragments .••
o. . . . . . . . . . . . . . . . . . . . . . . .
(1
••••••••
33. (a) Axial symmetry, (b) axial assymetry, of primary
fission fragments
0
0
••••••••••••• 0
••
C! • • • • • Q • • • 0
o.............
• • • ('. 0
••
cr •
.g • • • • 0
155
155
I. INTRODUCTION
1.1
General
Techniques of radiochemistry led to the discovery of nuclear
fission (1, 2} and have contributed immensely towards elucidating the
mechanism of this process.
The characteristic features of the fission process are the
following:
1.
The distribution of nuclear mass in the fission of a
h~avy
nucleus is predominantly asymmetric. The probabillty for
symmetric fission, however, increases with excitation
energies and exceeds that of asymmetric fission at high
enough energies (3-8).
2.
The distribution of nuclear charge usually results in the
formation of fragments which are proton deficient. The heavier
of the heavy fi'ssion fragment group are in general more proton
deficient, (i.,eq more beta decays to stability) than the
lighter. The same trend is displayed by the light fragment
group (9}.
3.
The average number (v) of neutrons promptly emitted in fission
varies with the fission mode. vis larger for symmetr!c
fission than for asymmetrlc fission. In symmetric fission
one fragment seems to emit more neutrons than the other (10-14).
2
However there is no information regarding the variation of v
values with excitation energies above a few million electron
volts other than that given by Harding and Farley (15). They
interpreted their experimental data for uranium lrradiated
with 147 Mev. protons in terrns of the sarne v value
(~
2. .. 5}
as for thermal neutron induced fission of uranium.
4.
The total kinetic energy of fragments varies with the fragment
rnass ratio. lt shows a decrease in the symrnetric fission
region. This decrease appears to be cornpensated for by an
increase in fragment excitation energies as evidenced by
larger neutron emission in this region (16-2j ..
s.
The fission fragments are ernitted preferentially in certain
directions relative to the incident bearn. The precise farrn
of this angular distribution depends upon the target nucleus,
the rnass ratio of the fragments and the nature and energy of
the projectiles ( 22, 23).
6 ..
On the average about four prompt gamma rays per fragment
accornpany the fission act. The experimental determination
of the prompt photon spectra show that the total energy of
the ernitted photons per fission is about 8 Mev. and is
nearly the sarne for ail targets (22, 23).
No comprehensive theory is as yet available which can
explain all the singularities of the fission process. Our knowledge of
3
the fission process is based primarily on various nuclear models amd empirical
prescriptions which give a reasonable correlation of the large amount of experimental
data coll ected (9 u 10 u 24-42).
A general survey of the early work on fission is given by Turner (43) 1
Whitehouse (44) and Coryell and Sugarman (45). More recently high energy fission is
reviewed by Spence and Ford(46) and lo<w energy fission by Glendenin and
Steinberg(47) and Halpern(22) and both high and low energy fission by Hyde(23).
Wheeler(48) has given a chronological development of ali fission theories.
ln this thesis we shall be concerned only with the fission of thorium
induced by protons of energy range 20 to 85 Mev.
1.2
Fission Models.
The customary analysis of the 1iquid drop model (22,49-55) sets
the stability limits and \explains .fissionability of heavy nuclei, but it fails to
account for the details of the fission process, such as the asymmetry and fine structure
in moss distribut-ion. These detdls were then explained in terms of shell structure
effects(31
~32,.33,35,56)
statistical treatment
on the binding energies of the product nuclei. ln the
deve~oped
by Fong (37) the probability of fission is taken
to be proportional to the level densities of fragments just at the moment of their separation.
The level density(Cexp. 2 ~)
increases rapidly with excitation energy E ( C and a
are empirical constants) (57). Since excitation energies of fragments adjacent to
4
1
magic
1
nuclei were found to be particularly high, the probability of a fissLon
process in which a fragment adjoins
Fong could . reproduce
1
magic' nuclei is also particularly high.
the experimental mess yield curve of
235
U
by this
approach(37).
Paul (40) modified the statstical treatment by making expl icit
allowance for the nuclear forces which act between the fission fragments even ..
after
their separation. The two fission fragments forma ' two"7 particle~syst.lem'
with a nuclear interaction potentiel consisting of both attractive ~.-.nuclear forces
and repulsive coulomb forces. The relative fission probability is identified
with the probability of penetrating this potentiel barrier. The detailed
calculations could account for, among other things, the fission asymmetry.
Curie(34) and Hill ( 35) considered the possibility of
the existence of a nonfissionable inner core 1ike 2QCa40 to a·ccount for asymmetric
fission of heavy nuclei.
5
1.3
Fission Yield Distributions
When a nucleus undergoes fission, both its mass and charge can; ·'
divide in a number of ways, subjeçt to the following two requirements:
1.
The sum of mass numbers of the complementary fragments
before neutron emission is equal to the moss number of
the fissioning nuclide.
2.
The sum of the charges of the compLementary primary
fragments is equal to the charge of the fissioning nu cl ide.
For a given mass division there are several possibilities of charge
division and vice versa. Ali the primary fission fra.gments with a given mass number
will( after the emission of prompt neutrons) forma fission product beta decay chain
ending in the stable end product of that mass number ( except for the very s;mall
effects caused by delayed neutron emission(56)). The variation of yields of primary
fission fragments (independant yields) along a fission product decay chain of a
given mass number gives the charge distribution in fission.
Mass Distribution
Several critical summaries of fission yield studies have been
prepared (3-6). The low energy fission yields of heavy nuclei ( Z ~ 90)
give the familiar double humped moss yield curve. The peak of the heavy
fragment region is nearly fixed
i n rn a s s
for
the various fissioning nuclei,
while th·e.light .peak shifts to higher moss number with increasing ,moss of the fissioning
6
nuclei. The symmetric fission probability increases rapidly with excitation
energy. The posit:bns of the asymmetrlc fission peaks do not appear to change
appreciably with excitation energy, except for a slight shift to lower mass
because of the emission of more neutrons• The overall width of the mass
distribution curve, in fact, increases with excitation energy. As the relative
yield of symmetric fission increases. the relative yield of very asymmetric
fission also increases. The twin-peaked distribution characteristic of low
energy fission, eventually, givts way to a broad single-humped curve with a
rather flat top (23).
The fission mode for lighter elements like lead and bismuth is
found to be highly symmetric at moderate excitation energies (58,59). The
mass yield curves for fissiQp of intermediate elements such as radium and
'
actinium are intermediate in character.. In the fission of radium bombarded
with 11 Mev. protons, Jensen and Fairhall found a three humped mass yield
curve (60). This indicated separate symn .tric and asymmetric fission modes
of roughly equal probability (59· ·60). A similar type of behaviours has also
'
been reported in the
~ission
of radium with neutrons ranging in energy from.
3 to 21 Mev. (61) and the helium ion !nduced fission of u233 (62 1 63). More
recently a triple peaked mass distribution curve has been reported for the
fission of Th 232 indu ced by reoctor neutrons (6,4).
Charge Distribution
In contrast to the voluminous literature avaUable on the distribution
of nuclear mass in fission, present knowledge on the division of nuclear charge
7
between the primary fission fragments is sparse. Moss distributions are obtained from
measurements of cumulative yields of the lote members of fission product chains after
beta decay of early short lived members is substantially complete. On the other
hand charge distribution is obtained from primary yields of individuel fission products.
These, however, are usually far removed from beta stability so thot most of the
radioactive decay half lives are very short. Thus primary yields for most fission
products are difficult to measure and consequently relatively few have been
determined.
Direct measurements of independent yields of sorne fission fragments have
been made possible by the existence of shielded and semi-shielded nu cl ides.
A shielded nuclide is one which cannot be formed by beta decay because its
immediate precursor in the isobaric decay chain is stable (e.g. esl36). A semishielded nuclide is one which con be chemically isolated in a time shorter thon the
half 1ife of its beta decaying precursor (e.g. La 140).
However not ali fission product decay chains exhibit such characteristics
and not ali the shielded nuclides are formed in measurable yield. The very low
yields of the shielded isotopes moreover, can be influenced by any number of
small perturbations such as shell effects, the details of the distribution of
number of neutrons per fission, etc.
Shielded isotopes must necessarily
occur near the 1ine of beta stabil ity curves and since a stable configuration
8
may be a determinative factor in charge distribution the assumption that
the se are representative samples of the primary nuclides is not quite
indisputable.
Determination of independent yields of accessible nuclides in
chains of different mass number must be followed by correlation of these
data, in order to obtain the charge distribution. Correlation is customarily
effected by means of theoretical or semi-theoretical relationships that
give the most probable charge Zp, for a given mass number. A.
Zp, is
that charge value, net necessarily integral, at which the peak of the charge
distribution occurs. The distribution of independent yields around the most
probable charge is usually assumed to be a symmetrical function with the
same form for all mass chains and all fissile nuclides.
The various postulates that have been put forward to obtain a
value of the most probable charge, Zp, are summarized below:
1. Unchanged Charge Distribution (UCD) (65, 66)
One might expect by analogy to the simplest form of the ·
liqutçi· drop mode!, that the neutron to proton ratio of the light and heavy
fragments is identical to that of the fissioning nucleus. Implicit in this
postulate is the assumption that fission is a rapid process. The division
of the nucleus should follow in a time comparable to that for neutron 'em!sJSion
and thus be too fast to permit any rearrangement of nucleons between the
fragments.
9
Sorne high energy fission data ( 65 ) seem to support this postulate but
there is little evidence for this in low energy fission.
2. Minimum Potential Energy ( MPE)
Wigner ( 67 ) postulated thot the charge of the fissioning nucleus would
be distributed to give c minimum nuclear potential energy. Present ( 68)
postulated a minimum for the sum of nuclear potential energy plus the coulomb
energy of two spherical fragments in contact. These two spheres were assumed
to be polarizable with non-uniform proton density due to the tendency of the
chprge to collect at the surface of a body. The necessary parameters for the
cal cu lotion of Zp were taken from the Bohr..Wheel er moss equation ( 24 ) • Coryell
et al ( 69 ) appl ied Cameron' s ( 70 ) and levy' s ( 71 ) moss formulas to two
spherical fragments separated by a distance larger thon the sum of their radii
by about 15% ( to account for the experimental kinetic energy ) •
Thermal fission data gave sorne support to Present' s modified version
of MPE.
3.
Equal Charge Displacement (ECO)
Glendenin, Coryell 1 and Edward ( 72) postulated thot the most probable
charge (Z p) l and ( Zp) H of the two complementary fission fragments land H
qre equally displaced from their most stable charges (ZA)L and (ZA ) H.
10
1. 1
The ZA values were calculated from the Bohr-Wheeler (24) moss equation which
gives a continuous ZA 'function, and the Zp values were computed after emission of
(v) fission neutrons from equation 1-1 which reduces to the form:
L2
where
Af and lf = (lpL+
lpH) are the moss and charge of the fissioning nucleus;
L and H stand for 1ight and heavy fragments;
i
is the average number of fission
neutrons, and Z(Af ... ; _ A) refers to the most stable charge of the complementory
fragment.
Poppos ( 56 ) considered the effects of shell structure on the values of ZA
1
and calculated the Zp values before the emission of prompt neutrons. Kennett and
Thode ( 73 ) considered the possibil ity of shell effects on Zp as weil. They proposed
thot Zp remains close to 50 over several moss numbers in the vicinity of this closed
shell region.
This empirical , ECO
postulate gives predictions which at lo?t' energies
are reosonably concordant with the data and in fair accord with the theoreticol
calculations of Present ( 68 ). ln fact the ECO postulate has been generally accepted
as valid for low energy fission process.
The most complete and direct data on the most probable charge Zp were
recently published by Wahl et al ( 74) for low energy fission (thermal and
spontaneous) with a direct determination of Zp made for six chains each with
two or more independent yields. These data show thot the charge distribution
at low energies could be represented with a Gaussian curve:
11
P(Z)
=
1
2
[
.,
(C'T')-"2 Exp. -(Z- Zp) /C)
1.3
where C is an empirical constant which characterizes the half width of the
distribution. For mœ t of the available data on low energy fission the value
of c varies from
o.a
to 1.0,
P(Z) is the fractional inde pendent yield of
the fission product with atomic number
z.
More redently, Armbruster and Meister (75) have physically
separated by a magnetic field the various masses of fission products with
different mass number and have directly counted the number of beta decays ·
to the stable nuclides of these masses. These resulta, however, show a
wide discrepancy with those derived from radiochemical methods (74).
Carter, Wagman and Wyman (76) observed the energy spectrum
of x-rays in coincidence with fission fragments by using a thin Na! crystal
for K x-rays and a proportional counter for Lx-rays. The resolution of this
method is
on~y
fair but with improved technique this approach may give a
good picture of the ent!re nuclear charge distribution.
12
I. 4
Charned Particle Induced Fission
Reactions of nuclei with bombarding particles of energy not
exceeding about 50 Mev. (moderate energy) proceed through the formation
of a compound nucleus (49-55). As the bombarding energy increases the
mean free path of the projectile in nuclear matter increases and becomes
comparable to the diameter of the struck nucleus. The target nucleus
appears as a partially transparent collection of qua si-free nucleons to the
incoming projectile. The general scheme proposed for high-energy nuclear
reactions is the 11 cascade-evaporation" model first suggested by Serber (77).
The cascade process consista of collisions of the incident particle w!th
individual nucleons of the target nucleus and subsequent collisions of the
struck nucleons. This may result in the emission of "knock-on" particles.
The energy transfer to the residual nucleus depends upon the number of
collisions which the projectile makes within the nucleus and may vary from
the full energy of the projectile to a small fraction thereof. Within the
short period of time (lo- 22 seconds) when the cascade takes place, the
interactirtg target nuclei will be converted to a distribution of product
nuclei with a varlety of excitation energies. The excitation energy is then
dissipated by the subsequent evaporation of particles, much in the same
manner as in low-energy bombardments.
Particle evaporation and fission may be expected to compete in
a few or all stages of de-excitation process of these nuclei in the sense of
13
the following scheme
.) ( E)
t
(78~79):
Part!cle )
Emission
Fission
y/a' ( E),
j,
Fission
Particle ~ X'" '(E) 1.1
Emission
~
---41)
1
Fission
in which a nucleus of mass, A, excited to energy E evaporates particles
to form successively nuclei of masses A'# A11 # etc.., and energies E' # E" 11
•••
etc.~
with fission as an alternative to each particle evaporatiOn step.
The partiales ejected are most often neutrons. The relative probability for
neutron emission and fission is usually expressed in terms of their "widths"
denoted by fn and
ft
(79). For the heaviest elements (Z )90) the ratio
rn! [f decreases as the atomic number increases but is also subject to a
strong mass number dependance (the ratio increases with increasing A for a
given Z).
I.S
Correlation of Independant Yields in Charged Particle
Induced Fission
The interpretation of independant yield data from high energy
fission (E.> 50 Mev.) is rendered difficult by the fact that the identity
and excitation energy of the actual fissioning nucleus cannot be uniquely
defined, i.e.- there is a range of fissioning nuclei. The necess!ty of a
14
detailed knowledge of the charge distribution is all the more important
for high energy than for low energy fission. At low energies the most
probable fragment at each mass is neutron rich and is about 3 charge
units removed from the line of bata stability (72). The distribution of
primary yields about this most probable charge is so narrow that the
total fission yield for each mass can be obta!ned by measurement of
the yield of either the stable or long llved end products of the beta
decay chains. The situation is altogether different as the bombarding
energy is increased. Neutron deficient nuclides also begin to appear
among the fission products indicating a broadening of the charge
dispersion for a given mass number and a shift towards stability of
ZP (80). In order to construct a mass yield curve it will be necessary
to know the contributions from both sidas of beta stability to the total
chain yield of a given mass number.. This necessitates a detailed
knowledge of the isobaric yield distributions in the various mass regions.
Severa! attempts (81-87} have been made to correlats the
independant yield data of high and moderate energy fission by means
of the same postulates which have been proposed for low energy fission.
15
Alexander and Coryell (84) have decided in favour of ECD to correlate
the data from 13.6 Mev. deuteron induced fission of
u 238 and Th 2 32,.
The results of an analysis of data from proton induced fission of these
nuclei by Pate (81) indicated that ECD !s applicable to both high and
low energy fission equally weil. On the other hand a thesis study by
Gibson (8) on fission induced in the following cases:
1) ·pu
239
+ 20 Mev. deJ,ltèrons; • 2) u 233 + 23 Mev. deuterons
3) Np 237 + 31 Mev .. d~uterons; :4) Np237 + 46 Mev. He ions
indicated that the actual charge distribution may be intermediate to
the ECD and UCD mechanisms with a greater propensity toward the
latter. The same trend is reported by Chu and Michel (85) for 45.7 Mev.
and 24 Mev. helium ion induced fission of u235 and u238 though they
observe that the trend is more towards the ECD postulate in its original
form using a smooth ZA function that ignores pronounced shell affects (56).
Porile and Sugarman (86) have observed that ECD-type fission occurs
when bismuth is bombarded with 450 Mev. protons but that UCD-type
fission occurs in tantalum bombarded with protons of the same energy. Rudstam
and Pappas (87) have reported that a mass dependent mixture of the two hypotheses (UCD) and (ECD) would probably give a reasonable approximation of
the charge distribution in 170 Mev. proton induced fission of uranium. This
they suggested could be ascribed to symmetric fission taking place acccrding
16
to UCD postulate and asymmetric fission taking place via the ECD process (88).
Eismont also (89} has speculated about the occurrence of two different
modes of charge distribution for the asymmetric and symmetrlc fission
modes. Each of these is thought to be represented by the Gaussian form
given in Equation 1.3. However, Foreman (90) has concluded that for 45 Mev.
helium ion induced fission of Th 232 neither of these postulates !s adequately
valid.
Pate, Poster and Yaffe (80} · reported values for the inde pendent
yields of various isotopes of iodine and tellurium in the mass range 130 to
135 1 and energy range 8 to 87 Mev.,. for proton induced ·fission of thorium.
Their data pennitted conc rusions to be drawn regarding the variation of the
charge distribution with energy.., This work has been extended by Kjelberg
et al (91} to give infonnation about the independant yields of a few nuclides
in the mass range 72 (.A( 1401 but the scarcity of data did not pennit them
to draw any charge distribution curves. Frledlander, Friedman# Gordon and
Yaffe (92) have presented data on the independant yields of a number of
nuclides mainly in the mass range 125( A (140 1 formed in the proton induced
fission of uranium for an energy range 0.1 to 6.2 Gev. They have obtained
excitation functions as weil as charge distributions for each bombarding
energy. Davies and Yaffe (93) have extended this study to 20 to 80 Mev ..
proton bombarding energies, and for the mass range csl30 to csl38. They
have also examined the excitation functlons of the neutron rich nuclides
17
which peak in this energy region. The resulta of both studies gave an
empirical curve which. perhaps, relates universally the peak energies
of an independant yield excitation function to its neutron to proton ratio.
These peak energies are found to increase with decreasing neutron to
proton ratio of the varlous cesium and rubidium isotopes they have stud:l.ed.
This shows that as the bombarding energy is increased 1 products with
lower neutron to proton ratios become more prominent. Such a trend
has been earller postulated by Folger et al {94) and Lindner and Osborne (95)
and substant!ated by the data of Hicks and Gilbert (96) and Pate, Poster;
and Yaffe (80).
!.6 The Present Work
The purpose of the present work was to continue the study of
the dependance of nuclear charge distribution on excitation energy in :fission of
thorium wi th
proto~s
of energy range of 20 to 85 Mev. ComparisQn of
the data for Th232 with that for u238 should enable one to study the dependance of nuclear charge distribution on the identity of the target nucleus ..
The mass range of the present investigation was restricted to cesium
isotopes of ·
130 ( A
<:t3a.... : .
The element cesium is particularly
suitable for the study of charge distribution because of the existence of
a large number of shielded and semi-shielded isotopes. The lower limit
of this mass range is imposed by the
erterQ'Y range
of the investigation.
Cesium isotopes llghter than csl30 were not observed below 100 Mev.
18
The upper limit was imposed by the shortness of the half lives involved.
The cross-sections were measured radiochemically and excitation functions
are presented for the various nuclides. It was possible to separate rubidium
at the same time as cesium and an excitation function for the independent
formation cross-section of Rb86 is also presented. The cumulative
formation cross-sections of Cs 137 and Xe 138 {parent of Cs 138) could also
be determined. They were found to be extremely useful in defining the
shapes of the charge distribution curves.
Isomer Ratio
Nuclear isomers are different energy states of the same nucleus,
each having a different measurable lifetime (except that the ground state may
be stable). In a number of cases fission 1eads directly to the formation
of a product in an isomerie state. An important quantity in the description
of nuclear isomers in fission is the isomer ratioûH/dL whereO'H is the
formation cross-section of the high spin state ando' L that of the low spin
state of an independently formed fragment. The measurement of isomer
ratios of shielded nuclides (which are independently formed in fission)
provides an interesting possibility to investigate the distribution of
angular momentum in fission.
The problem of theoretical calculation of isomer ratios of
primary fragments in fission falls into three parts:
19
1. The calculation of the angular momentum of the nucleus
just prior to fission.
2. The division of this angular momentum between the two
pr'...mary fragments.
3. The calculation of the isomer ratio as the fragments de-excite.
Various approximations for Part 1 are cons:l.dered by severe! authors {97-100).
Part 2 is hard to determine. The angular momentum of the fission!ng nucleus
can appear in two places in fission:
a) the orbital angular momentum between the fragments,
b) the intrlnsic spin of the fragments.
Angular distributions of the fragments (22) indicate that most of the angular
momentum goes into (b). As to how it is divided between the fragments is
difficult to say. Once this is determined one may proceed to the third stage,
and use the method developed by Vandenbosch and Huizenga to calculate
isomer ratios in nuclear reactions where the initial spin distribution is
known (97 6 98) .. Alternatively• one may use the experlmentally observed
isomer ratio and reverse the normal procedure to deduce the pr!mary fragment
spin distribution. Comparison of this angular momentum distribution with
that of the fissioning nucleus should be instructive.
Very little work has been reported on this problem partly because
it is difficult to find isomerie pairs wh:!.ch are completely shielded. The
data available in the litera ture are mostly for low and medium· energy fission
(101-105).
20
cs134 provides one of the rare instances where it is possible
to measure the independant formation cross sections of a pair of nuclear
isomers produced in fission. Bath Cs 134 m ('tÎ
:=
2 .. 9 hr.) and Cs 134g
('tÎ = 2.1 yr .. ) are formed in the proton indùced fission of uranium (93,.105)
and thorium ... The 2 .. 9 hr. meta stable state is a primary product shielded
by stable Xe
134
am Ba 134 .. The 2.1 yr. ground state is produced bath as
a primary fragment and from decay of the meta stable state by isomerie
transition. In the course of the present work. it was possible to determine
the ratio of the independant formation cross-sections of the Cs 134 isomerie
pairs at various incident proton energies.
21
II. EXPERIMENTAL
II .1
Target and Irradiation
Thorium metal foil of about 80 mg/cm2 superficial density was used
as the target material.
Ail irradiations were performed in the internai circu-
lating bearn of the McGill Synchrocyclotron. Thê energy of b'ombard;mènt wa s
varied by inserting the target at various distances from the centre of the
cyclotron. Kirkaldy' s (106) standard curve of proton energy vs radius was used
to determine the energy of bombardment. The energy spread of the proton bearn
as reported by the Poster Radiation Laboratory was .!_ 2 Mev.
In all experimenta but the one at the lowest energy, foils were used
in stacks of three. The middle foil was chemically processed for the cesium
and rubidium yield determinations. Loss of fission fragments by recoil from
the target foil was compensated for by gain of recoils from the neighbouringfoils. Upstream from the thorium was a monitor foil, again with a guard foil
on both aides. To keep the energy degradation in the target stack within the
range of the proton bearn energy spread it was found advisable to use only
single foils for the low energy bombardment. A small correction for recoil
losa of fission fragments was made using the data given by Noshkin and
Sugihara. (107)
Recoil losa of Rb 86 was assumed to be the same as that of
Br 83 (8%) and a value of 5% was estimated for the Cs isotopes.
22
--- .... _..,
------~
_________ ,
....
1
1
1
:
~
1
"'0
::0
0
CD
TARGET
HOLDER
~ ··-·-
.... _,
. c.:!::::.:.:.-::_-:_-:fi~
HOLDER
MOUNTING
SCREW
1"11
TARGET
1
r--'
1
,
EDGE
1
1
______'-:,
1
1
PROTON
TARGET
CLAMPING
SCREW
Fig. l - Target Mounting Arrangement.
BEAM
23
The bearn intensity was monitored by using either of the following
reactions:
27
(p 1 3pn) Na
24
1.
Al
(for energies above 60 Mev. only)
2.
cu 65 (p,pn) cu 64 (throughout the bombarding energy range, i.e.
20 to 85 Mev.)
The excitation· functions of both the se reactions have been standar-
dized in this laboratory by S. Meghir (108) relative to the cl2(p, pn) cll data
reported by Crandall et all (109). The superficial density of the aluminum
foils was about 6. 5 mg/cm2 and that of the copper foUs about 42 mg/cm2.
The leading edge and the top and bottom edges of the foils were aligned carefully prior to bombardment by shearing with scissors after the foils had been
clamped in the target holder.
(Fig~
1).
The length of irradiations usually
varied from 5 to 30 minutes and the energy range covered was from 20 to 85
Mev.
II. 2
Chemical Processing
After removal of the target from the cyclotron probe and transfer to
the radiochemistry laboratory, the leading edge of the stack of foUs was carefully sheared from the target holder. This together with the preliminary alignment described above, ensured that the target and monitor foils received the
same proton flux.
Cesium
The separation and purification of cesium was accomplished by a
procedure similar to the one described by Evans (110). The target foil was
24
dissolved in concentrated hydrochloric acid. A few drops of ammonium silicofluoride
were added to dissolve the last traces of residue. Carriers for Cs
~20
mg.}, Rb (..-20 mg},
Ba {HS mg.), Sr, La, Ce, Y, and Zr ~2 mg each) were added and the insoluble
hydroxides and carbonates were precipitated with 12-N sodium hydroxide and 2-N
sodium carbonate solutions. After a further lanthanum hydroxide scavenge, the solution
was made slightly acidic with hydrochloric acid, cooled in ice, and a few drops of
Bil3 -Hl reagent were added. On vigorous scratching cesium was precipitated as
Cse Bi2 19. This was centri,fuged and the supernatant solution was set aside for
subsequent processing of rubidium. The CsJBi2l9 precipitate was washed with .... 0.2 N
hYdrochloric acid, cooled in ice, dissolved in hot dilute HCI and reprecipitated after
addition of more rubidium hold back carrier. The precipitate was then dissolved in
6 N nitric acid. lodine was removed by boiling. The solution was evaporated almost to
dryness, cooled and then bismuth was precipitated as oxynitrate by the addition of water.
About one ml. of concentrated nitric acid and excess alcohol were added to the supernate.
The cesium was precipitated as Cs2 PtCI6by the addition of excess chloroplatinic acid.
The precipitate was filtered, washed with absolute alcohol, then with ether, dried at
0
110 C, weighed for chemical yield determinations and then mounted for activity
measurements on cardboard planchettes. The whole procedure took about one hour.
ln some of the earl ier experiments the supernate after removal of bismuth was
made to a known volume. A known al iquot of this was taken on a VYNS film mounted
on aluminum ring, dried, and used for 4- n counting.
25
Rubidium
The solution containing the rubidium fraction was scavenged three
times with the addition of Cs carrier and subsequent precipitation of Cs3Bi 2I 9 •
Excess BU 3-HI reagent was removed by boiling with nitric acid to drive off
iodine and by precipitation of bismuth as oxynitrate. Rubidium was now
precipitated as Rb 2PtC1 6 which was washed with a 75% ethanol solution.
For further purification the complex was reduced to platinum with formic acid
and the excess formic acid was destroyed by boiling with hydrochloric acid.
A further CsgBi 2I 9 scavenge was performed and then complete removaLof all
cesium activity from the rubidium solution was tested for by a gamma-ray
spectrum analysis. · Rubidium was then reprecipitated as Rb 2 PtC16. The
precipitate was filtered, weighed and mounted for activity measurements.
Aluminum
No chemical separation of sodium from aluminum was effected ..
The aluminum foU was simply dissolved in the minimum amount of hydrochloric acid and the solution. made up to a known volume. A known aliquot
of this was taken on a VYNS film mounted on aluminum ring. When dried it
was covered with another VYNS film to keep the source intact (A1Cl 3 :being
deliquescent). This source was used for 4-TI'. counting.
Copper
An ion exchange technique was used to separate copper isotopes
from the other activities, which interfere with the disintegration rate measure-
26
ment of Cu
64
• These interfering activities were due to the isotopes of
Zn, Co 1 and Ni. The target copper foil itself could act as a carrier for the
copper activity.
The procedure adopted was that described by Kraus and Moore (111) ..
The copper foil was dissolved in a small volume of concentrated hydrochloric
acid to which a few drops of hydrogen peroxide were added. Excess hydrogen
peroxide was destroyed by boiling. The solution was heated to dryness and
the residue redissolved in a small amount of concentrated HCL This solution was adsorbed on a Dowex anion exchange column (200-400
mesh) pre-
conditioned with about 20 ml. of 4. 5 N hydrochloric acid. The elution step
then followed.
Cobalt and nickel contaminants were first eluted with
4. 5 N hydrochloric a cid, while the copper layer moved slowly towards the
bottom of the column. The yellow-green col or of the copper layer made
inspection of the elution easy and rendered unnecessary the use of exact
column dimensions, provided the resin column was sufficiently high
to adsorb ali the copper ions. The column used was about lScm. high and
0. 6c.m.. in diameter. When the copper reached the bottom of the column the
eluant was changed to 1. 5 N .hydrochloric acid. The copper was brought out
from the column within the next 5-10 ml. of eluate. Zinc was retained on
the column. To ensure purity of the copper fraction the first and the last
few drops of He copper eluate were rejected. The eluate was collected
directly into a volumetrie flask, made up to exact volume and suitable
aliquots
taken for the activi ty mea surements.
The chemical yield of copper
was determined by direct titration with ethylene diamine tetraacetic acid using
murexide as indicator (112)"
27
II. 3
Radioactiyity Measurements
The pertinent properties of the radio nuclides mea sured in tht.s
work and their rhodes of detection are given in Table I (113) • Four kinds
of instruments were used:
f
1. 4:](Counter
The 4...ll'counting equipment used i:s similar to that desctibed by
Pate and Yaffe (114a). The positive an6de voltage is supplied by a Nichols
high voltage supply (AEP-1000 7 B)while the cathodé is kept at gtound potential.
The two anodes, each acting as a separate unit of 2 ngeometry when the source
is kept in the centre of the sphere, are connnected in parallel to a preamplifier
(Atomic Instrument 205 B). The output from this is fed into an Atomic Energy
of Canada Limited amplifier discrimlnator (AEP 908). A Lambda regulated
power supply (model 28) is used to supply power to the preamplifier. To the
other units. power is supplied from a voltage regulated power supply through
a Sola oonstant voltage transformer.
The counter was operated in the proportional region with methane as
the counting gas. A gas flow rate of approximately 0,5 ml per second at
atmospheric pressure was maintained during meaaurements. The counter was
flushed for severa! minutes after inserting a shmple., before actual counting
began. The operating plateau regions for different nuclides were about 400 volts
•
.1
long and:had a slope of 0~.1% per IQO vQJts. A common operdf.rng.vottage.Jwas selected
28
TAJ.1LE I
Relevant Properties of RadioNuclides Measured and Modes of Detection Used
Nu elide
Half-life
cs130
30 minutes
05 132
6.5 days
05 1349
csl3flm
05 136
Radiation
Measured
p+
Branch
Abundance
Instrument U sed
0.46
Sll-511 Coincidence
670-Kev.~
0.98
1-.ra y Spectrometer
2.1 years
800-Kev.~
o.83
'6-ray Spectrometer
2.9 hours
,e"'tJ-
1
4 rr, and end win dow proportional counters
1~9
/.1-
1
41J1 and end window
days
)
proportional counters
csl37
30 yea.rs
661-Kev.!
o.84
'(-ray Spectrometer
csl38
32.2 min.
/] ..
1
End window propor-t1ona1 counter
Rb86
18.6 days
/1-
1
Ejnd. window proportiona1 counter
29
for ali nuclides at about 2700 volts. A setting of 15 volts was used on the
discriminator. A check of. the statistical behaviour of the counter was made
using Ra (D + E) standard source under the operating voltage and discriminator setting s indic a ted above be fore the actual opera ti on.
The counting rates were always below 105 cpm. The error due to
· resolution loss was negligible at this rate.
The superficial density
oP 4-:lt
2
sources varied from 70 to 2sorg.cm •
Corrections due to source mount were assumed negligible compared to self
absorption. The values of self absorption factors were taken from self_
absorption curves reproduced in Santry' s thesis (Il4b). The correction for
self absorption of the Cs 134 m conversion electrons was artived at using the
decay scheme calculations of Baerg et al (115';).
2. End Win dow Be ta Proportional Counter
The operating plateau voltage was kept at about 2600
volt~.
and
bias voltage was kept at about 20 volts. The counting rates were always
5
much less than 10 cpm. so that error due to dead Ume loss was negligible.
The background count varied- from 10 to 12 cpm. The counting efficiencies of
the various nuclides under study were obtained from the empirical efficiency
curves constructed by Dav:l.es and Yaffe (116) by calibration against a 4-rr
counter.
30
3. Scintillation Spectrometer
This instrument was used for the assay of gamma emitting nuclides.
The detecter is a hermetically sealed unit supplied by the Harshaw
Chemical Company consisting of a 3" x 3" sodium iodide crystal activated
with about 0.1% thallium iodide. This was coupled to a DuMont 6363
photomultiplier tube. The latter was operated at a voltage of 1100 volts
obtained from a Baird Atomic {Mode! 318) high voltage supply. The output
pulses from the photomultiplier tube are fed through a Hamner Electronics
Company preamplifier {Model N-351) to a variable gain linear amplifier
(Baird Atomic, Mode! 215). The amplifier is coupled to a 100 channel pulse
height analyzer (Cbmputing Deviees of Canada Limited, type AEP-2230)
through a cathode follower. The data collected by the pulse height analyzer
can be viewed on a cathode ray tube or can be recorded on a We stronics
recorder (mode! 2705) whichoaccepts analog signais. A decimal scaler
(C.D.C. Type 450) converts the analog signais to decimal form and the .
print-out is obtained on a Victor digitmatic printer programmed by a print
control unit (C.D.C. Type 460). The analyzer is equipped with a microammeter on which resolution !osses during measurement are indicated.
These !osses are due to the dead time of the analyzer which lies between 35
and 135 micro seconds depending on the pulse height. Resolution !osses do
not cause any distortion in the gamma ray spectrum but cause reduction in
its overall height.
31
In order to calculate from the observed gamma-ray spectra the
absolute gamma emission rate, it is essential to know the efficiency for
detecting a gamma ray of given energy. The detection efficiencies of gamma
rays of various energies were determined experimentally by Grant et al {ll'l).
Sources were used which were both gamma arrl beta emitters and whose
branching ratios and conversion coefficients are weil known, so that they
could be calibrated by means of their beta activity in a 4- n counter.
Since
these calibration curves were constructed for liquid sources a small correction
factor has to be applied to these efficiency values when solid cesium sources
were used.
This factor was determined by the following simple experiment.
A filter paper dise of the same type as that used for collecting the cesium
sources was fastened on a piece of scotch tape eut to proper size and mounted
on a cardboard planchette.. A small aliquot {20 ~) of a standard source of
Cs 13 7 was placed on the center of the filter dise and by careful addition of
one or two droplets of water 1t was allowed to spread to an area which was
roughly the same as that usually covered by the cesium source. An equal
aliquotl of the same source was taken in a vial described in reference (HZ)
and 2 ml. water added. The two sources were counted on the same shelf and
from the counting rate of the 660 Kev gamma ray a value of 1. 2 was arrived
at as the ratio of the solid source efficiency to that of the liquid source.
4. Positron Counter
The absorption and subsequent annihilation of positrons results in
the production of two quanta (511 Kev each) emitted at 180° to each other.
32
The number of 511 Kev gamma rays arriving in coïncidence at two detectors
placed in 1ine, one on each side of the source, gives a measure of the positron
intensity. By imposing pulse height and coïncidence conditions on the gamma rays
arriving at the two detectors, we con discriminate against ali gamma rays except
those arising from the genuine annihilation of a positron. To correct for accidentai
coïncidences one of the detectors is paired with a third detector placed at right angles
to the source, but with the same geometry with respect to it as the other two. The
detectors used were three Harshaw Nai(Tl) crystal! (1~ 11 x 1~"), Type 656. These
were mounted on 6342A type phototubes.
Sol id sources were prepared and mounted on filter dises ( source diameter
nearly 1.7 ems.). These are in turn mounted on aluminum cards using scotch tape
and sandwiched between copper absorbers of sufficient thickness to completely stop
ali the positrons. Thewhole set upwas then clamped in position at an angle of 45°
to each detector in arder to g ive the seme geometry for ali detectors. The detectors
were placed at about 8 ems. from the source. The operating voltages of the
phototubes were of the arder of 1000 v. The preamplifiers were maintained at
300 volts D.C. from a regulated Lambda Electronics Model-25 power supply.
The calibration of the counter was performed with a standard positron
emitter (Na22 ). The counting efficiency of the system was monitored each
ti me it was used •
•
33
II. 4
Decay Curve Analy sis
The cesium decay curves obtained from the proportional counters
were resolved manually giving three components with half lives of 32 min ..
(Cs 138), 2.9 hrs. (Cs 134), and 13 days (Cs
136
), and a long lived tail which
by gamma ray spectrum analysis was found to be composed of csl3 7 (tt= 30 yr.)
and Cs134) (tl= 2.1 yr.). The radiations of Csl3 2 , Csl34g, and Csl37 were
2
measured on the 100 channel gamma ray spectrometer.
670 Kev photopeak of csl32
(tf = 6. 5
The area under the
d.) was estimated after subtracting
the Cs136 contribution from the total gamma ray spectrum. The energies of
the se gammas are listed below (113).
TABLE II
Gamma Ray'Abundances of Cs 132 and Csl36
Nuclide
csl32
Radiation Measured (Energy in Kev .)
Abundance%
673
647
99
830
100
9
The gamma ray spectrum of cesium sources measured about one year
after irradiation showed two weak peaks at 650 Kev. and 800 Kev. The latter
oF
was a Cs1349 peak and the former a composite of the 660 Kev. peakACsl3? and
the 560 Kev. and 605 Kev. peaks of Cs1349. The Cs13 4 contribution to the
lower energy peak was determined from the 800 Kev. peak using the abundance
34
values listed in nuclear data sheets (Ill). These values are listed below.
From these data it was possible to determine Cs 134g and Cs 137 disintegration rate.
TABLE III
Gamma Ray Aburidances of Cs 134g and cs 137
Nuclide
Radiation Measured (Energy in Kev .)
Csl34g
(tf = 2.1 yr .}
. cs 137
(tf = 30 yr.)
Abundance%
797
801
605
570
72.5
10.5
98.0
25.5
660
84.0
The Na 24 activity induced in the aluminum monitor foUs was measuœd
in the 4-
n counter.
cu 64 fw-o·m copper monitor was estimated on' the gamma ray
spectrometer after chemical separation from other spallation products of copper.
II. 5
Calculation of Cross Sections
ln the thin target approximation, the differentiai equation expressing
the rate of change of the number of radioactive a toms of a nuclide, P 1 formed
during an irradiation is expressed as:
dN
dt
~·. .: 1 n.,-crp - Â:p. Nt.
If
...................... , • • • • • • • • . II. 5 .1
35
where
=
I
n,... =
Np
the proton flux, 1. e., the number of protons per cm2 per
second.
the number of target nuclei presented to this proton bearn.
=
formation cross-section of the nuclide P, in cm 2 •
=
the decay constant of nuclide P.
=
the number of a toms of nuclide P.
If the proton flux I is considered constant during the irradiation
period t
= t0 ,
imposing the condition. that at t
= 0, r.;p = 0,
equation 11. 5.1
ha s the solution:
0
0' p
A
-J.ptO\
-, ................................. II. s. 2
p
where Nop À:p is the disintegration rate D 0 p of the particular nuclide at the end
Np
=
Inr (1 - e
of irradiation. Therefore equation ll.5.2 may be written as;
Do p -_($_.p
In,~(1
- e - Àpto) . • . • • • • • • . • • • • • • • • • • • • • • • • • • Il • 5 .. 3 a
ln those irradiations which did not require considerations of parentdaughter relationships, {e.g., shielded nuclides,or cumulative yields of long
lived daughters with short lived parents) the formation cross sectionG"p ·may
be determined from equation II.S.3a if the disintegration rate of the nuclide P
at the end of bombardment is known. The proton flux I is determined by the
monitor reaction. Since the monitor and target are exposed to the same bearn
simultaneously, equation II. 5. 3a wh en applied to the monitor reaction give s:
o
_
D:M-
û't(InM(l-e
-ÀMto
· ) •••••••••••••••••••••• I1.5.3b
where the subscript M stands for the monitor. Dividing equation II, 5. 3a
by II.S.3b and rearranging terms gives:
36
oop.r
op =
where
F
and
0 oo
II.S.4
:6'M.nM
u.s. 5
oooM .. nr
=
0 o/(l _;
~o)
II. S. 6
since the same area of target and monitor foils is exposed to the proton
bearn the ratio nM/n'l' can be expressed as:
nM = (SD)M x (NA)M
n'(
(SD)"r x (NA)"?
x (A)'T
II. 5. 7
x (A) M
where:
SD
(NA)
A
Subscript~M
=
superficial density
= natural abundance
c;:
of the element
atomic weight
and T stand for the monitor and target respectively.
The elements thorium and aluminum are monoisotopic, copper
has two stable isotopes, cu63 (69%) and cu65 (31%) and the latter value
was taken into account in ali the calculations involving the copper monitor
reaction.
The monitor cross-section values used at each energy are
shown in Table 4. They have been interpolated from excitation functions
reported by Meghir (108).
In the case of csl38
formation from the parent Xe 138
<tl = 32.2
(tt = 17
min.) a correction for its
min.) during irradiation and prior
to separation has to be applied. Xenon being formed in the gaseous phase
37
TABLE 4
Pertinent Data for Each Bombardment
No.
En erg y
(Mev.)
Monitbr. 6"(mb>
td
Al
Min. Cu
D0 M( dis/min)
8
F
2.82 x
w- 8
1
40
30
260
1.53 x 10
2
50
30
212
6.25x10 7
5.65 x 10- 8
3
73
30
13.7
1.49x10 6
1.54 x 10- 8
4
65
30
13.8
8.o x 10
5
86
25
12.4
2.4 x 10 6
6
21
32
320
2.2 x 10
7
19
20
160
1.16 x 10 8
1.53 x Io- 8
8
30
20
400
2.25 x 10 8
1.98 x 1o-8
9
70
25
1.04x10 7
1.85 x 10- 8
10
57
30
1.45 x 10 8
2.15 x 10-8
11
85
30
3.52 x 10 7
6.6 x 10- 8
12
57
5
188
2.99 x 10 7
1.76x1o-8
13
39
5
272
3.90 x 108
1.94 x 10- 8
14
30
5
400
1.15 x 10 8
9.65 x 1o-9
15
20
5
240
3.00 x 107
2.22 x 1o-8
16
57
60
188
2.42 x 10
8
2.42 x 10- 8
17
85
5
140
2.07 x 10 7
1.87 x 10- 8
13.8
188
12.4
6
w- 8
w- 8
2.91 x
7.2 x
8
2.68 x 10- 8
(Cont'd.)
38
TABLE 4 {Cont' d,)
No.
Energy
(Mev.)
to
Min.
Monitor CSlmp)
·Al
Cu
D0 M(di s,Aniri)
F
18
20
5
240
3.62 x 10
7
1.84x1o-8
19
29
5
400
6.95 x 10 7
1.60 x 10-a
20
85
5
140
2.56x10 7
1.51 x to-8
21
57
5
188
3.09 x 10
7
1. 62 x 10-8
22
39
5
272
5,62 x 10 7
1.35 x 10- 8
23
65
5
168
3.19 x 10
7
1.46 x 10- 8
24
65
5
168
2.52 x 10 7
1.86 x lo- 8
_25
75
5
152
1.92 x 10 7
2.20 x 10
26
65
30
168
1.09 x 10~
2.16 x 10- 8
27
75
5
152
2.88 x 10 7
1.46 x 10- 8
28
39
5
272
4.84 x 10 7
1,56 x 10- 8
29
65
5
168
2.53 x 10 7
1.85 x lo-8
30
80
30
144
8.ss x 101
2.80 x 10-8
31
85
30
140
1.23 x 10 8
1.88 x 1o-8
32
75
5
152
2.37 x 10
33
45
30
238
1.92 x 10 8
to = duration of bombardment
F
is defined in Equation II. s.s.
D0 :M
disintegration rate of monitor per minute.
=
7
1.78xl0
-8
-8
2.oo x 1o-8
39
could be easily removed from the system by simply dissolving and boiling
the irradiated tho'l'ium foil with concentrated HCl. If the time of removal
of Xe after the end of irradiation is known, the disintegration rate of
independently formed Cs 138 at the end .of irradiation can be calculated as
follows:
Let:
~p
=
decay constant of parent xel38
)\ d
=
decay constant of the daughter csl38
Ûp
=
cumul-ative formation cross-section of xel38
(5
d
=
cumulative formation cross-section of csl38
N
=
number of target (thorium) nuclei
I
=
proton flux
Xe 0
=
number of Xe 138 nuclei present at the end of
irradiation
t
C s0 =
number of C s 138 nuclei present at the end of
irradiation.
If Xe is the number of Xel38 nuclei at any time, t, the net rate of production
of xe 138 is:
dXe
dt
=
NIUP -
and the amount present after a time of irradii:ltion, t 0
Xed
=
ÀGP (l-;)\pt")
p
II. s. 8
XeÀ p
,
is given by:
u.s.9
If Cs is the number of Cs 138 nuclei at any time, t, the net rate of production
of cs138 is given by:
40
dCs
· 1'1'
,,-!.
"" \ 't
\
= Nh.J d + Nl'llp (1 - e ~'P) -Cs/\d
dt
11.5.10
and the omount present ot the end of irradiation is:
11.5.11
During the inter.val 1 t'
of Xe
138
1
1
between the end of irradiation and fi nol removol
the amount of Cs
Cs
The omount af Cs
after o time 1 t•
1
138
138
formed from on initial arnount of Xe, Xe0
·- ' Àp • Xeo {\e- >.pt•-e ·-Àdt')
.
(Xeo)Àd..;,Àp
remoining from on original omount Cs 0 of Cs
,
is:
11.5.12
138
,
is:::
11.5.13
Th eref ore t he amount of Cs138 present when
xe 138 •1s dnven
•
aff •1s:
S'ubstituting the values of Cs0 and Xe0 _from equations 11.5.9 and 11.5.11 and,
further multiplying throughout by Àd and rearranging terms 1 the resulting
equation is:
·wh ère
and
11.5.17
41
Now using equation (II.S.3b) for the monitor reaction, equation (II.S,15)
can be transformed into the form:
Dd'
= ·~_g (Kl;p + K2Gd)
II. 5.18
where Dd' is the disintegration rate (CsÀ.d) after a Ume t' from the end of
bombardment and F is as defined in equation
u.s. s.
ln order to obtain the values ofG"; and ()~ two short
irradiations (5 minutes èluration each) were performed at each energy, In
one the targetsthorium was dissolved as quickly as possible after the end
of bombardment and in the second an·intervai of about one hour was. allowed
to elapse before dissolution. The ex1act time of dissolution was noted in
both cases. Both solutions were processed identically for the recovery of
Cesium. The yields of csl38 at the respective times of separatbn were
measured and each of these gave an equation of the form of equation
II. 5.14 and their solution gave the values of G""p and
II.6
Ùd•
Accuracy of Results
The uncertainty in the cross- section values determined in
these experimenta is brought about by two types of errors:
1. Random errors that affect the relative magnitude
of the cross-section values, and
2. Systematic errors which affect the magnitudes of the
cross-section values in the same way.
42
The latter would be primarily due to el'!'ors in the accepted half-lives,
decay-schemes, etc..
This, however,. is not included in the estimates
of the errors.
An uncertainty of t 5% was estimated for chemical yields
and ± 2% for dilution factors. A standard deviation of± 3% was experimentally ascertained for the measurements of superficial densities of the
target and monitor foils. Small uncertainties introduced by the nonuniformity and non-alignment of foUs were considered negligible compared
to ether uncertainties.
The random errors in the disintegration rates of the nuclides
studied vary according to the methods of measurement employed.
The uncertainty associated with gamma ray spectrum analysis
is usually the most serious. The errors in determining the photopoak areas
and the uncertainty in the decay curve analysis were taken into consideration. The total errer was estimated to vary fromt20 to!'40% depending upon
the activity of the species in question and that of the ethers present in
the sample.
An error of± 20% is assigned to positron decay curve analysis.
ln beta countlng experiments, an uncertainty of t 10% was assigned to the
decay curve analysis. For 4-lf. beta counting the main source of error is
the correction factor due to self absorption and source-mount absorption.
43
These errors depend upon the values of these factors, and wcre estimat ed to range
from
t
2 to
:t 10%.
An additional error of± 5% was assigned to the empirically
establ ished efficiency curves of the end-window beta-counter wh ich was callbrated with respect to the
4-lT beta counter. This takes account of the fact thot the
error in the efficiency factor depends on the thickness of the end window source.
The heavier the source the smaller is the error.
The error due to the energy spread (
:t 2 Mev) of the proton beam
is qu ite
appreciable at low energies ( 20 - 30 Mev \ where the slopes of the excitation
function of Cu monitor reaction are steep. As this error was difficult to evaluate
it was not accounted for in these calculations.
The estimated total uncertainly in the calculated cross-sections of each
nucl ide studied is quoted in Table No. 5
Where the results of more thon one determination of the seme nuclide are
given the standard deviation of the mean value is reported. The deviations
of individuel results from the averages were generally quite comparable to the
estimated errors for the individuel determinations.
44
TABLE 5
Errors
Method of
Mea surement
Nucllde
End window beta counter
Il
'
lt
~
.Il
-
. tt·
.Il
.
- Il
-
.
Total Error %
csl38
.t
22
lt .•.
Il
Xe138
± 22
JI .
Il·
c 5 136
± 20
Il
li
csl34m
± 23
Rb86
± 23
05 136
± 18
csl34m
± 21
csl34g+m
:!: 40
.
'
Il .
4-Jr. counter
Il
Gamma -ra y spectrometer
E~SO
Mev.
± 25 E?;SO Mev.
Il
Il
Il
± 40
E-.:5 50 Mev.
:!: 28 E ~50 Mev.
Il
Il
Positron counter
Il
:!: 25 E~ 70 Mev.
:!: 22 E "» 70 Mev.
+ 27
45
Ill. OBSERVATIONS AND RESUlTS
111.1
Cross-sections and Excitation Functions
A section of the nuclide chart showing the nuclides in the Cesium region
is shown in Figure 2. The pertinent data of each bombardment and the formation crosssections of each nuclide studied are reported in Tables 4, 6- 15. Excitation functions
of the various nu cl ides are g iven in Figures 3-9.
130
For the detection of Cs
(t~=30 minutes) five irradiations were performed,
in the energy range 57 to 85 Mev. ln ali cases except the ones at 80 and 85 Mev the
90° coïncidence counting l'Ote was nearly equal to the 180° coïncidence rate.
The counting data from the latter two runs showed wide scatter.
We bel ieve that this unusually large accidentai coïncidence rate is due to
138
Cs
138
• This nu cl ide is formed in mu ch larger yield (6 Cs
/6
130
Cs
';:iJ
40) and has the
130
138
same half 1ife as Cs • Examination of the decay scheme of Cs
reveals the presence
of a cascade of 560 and 460 Kev gamma rays in coïncidence. The channel settings on
the Nal detectors were such thot these gamma rays would partly contribute to 511 - 511
Kev coïncidence rate.
130
ln view of these difficulties we only repart upper limits for the Cs
cross section.
111.2
Charge Distribution Curves
The cross-section data given in Table 15 were used to obtain information
concerning the isobaric yield patterns for the moss range 13Q-I38 at proton energies
20, 30, 39, 50, 57, 65, 75 and 85 Mev. Points interpolated from excitation functions
•
were included in the 20, 50, and 57 Mev ourves. The cross-sections were plotted as
a function of N/Z rather thon the conventional parameter (Z- ZA). The use of the
46
Bol29
25h
•.,&•1<43,.1.25,
98 pl8,21,
, ..'!1,.!27
., ...
f29
Csl38
Csl39
.... ,
9.5m
y
...
lt'llôU
•••
Xel38
tl" 2.
r 42, ~).:
Fig. 2- Section of the Chart of the Nuclides in the Cs region •
•
Xe139
~-.
e.zo
1137
24s
66s
41s
17m
JJ'35
Csl40
•,.,
}'4tl. 30, Il,
1138
63s
47
simple parameter N/Z avoids any difficulties with ZA at shell edges (56). ln those
regions where data on independent formation cross-sections were not available, the
cumulative yields of Cs
137
138
and Xe
were found extremely useful in defining
the shapes of the curves. These curves were drawn on the assumption thot they are
symmetric about Zp·
The right hand dashed portion has been so drawn thot the
interpolated primary yields of each isobaric chain (A= 137 and A= 138) have to sum
to the measured cumulative yield for thot chain ( 80, 92 6 93 ). The N/Z values of
the isobars contributing to these cumulative yields are indicated near the bottom
of the curves. These curves are presented in Figures 10-17.
The assumptions impl icit in such a correlation are:
1. The mass yield curve in the mass range 130~A~l38 is essentially flat.
2. The distribution of yields along a chain is the seme for chains of different moss numbers.
These assumptions are probably reasonable over the narrow moss region
(130~A ~ 138) under consideration.
The cross-sections of unmeasured nucl ides can be estimated from these curves.
By summing the individuel independent yields of ali the isobars of a given moss number,
it is possible to deduce the total fission yield per moss number in the region
(130~A~ 138), as a function of bombarding energy. ln Table 16 are given the data
for total formation cross-section of the mass chain A= 136 as a function of energy and
they are plotted in Figure 18.
48
4.0.---------------------,
2.0
z
0
t-
u
w
(/)
1
(/)
(/)
~ 1.0
(.)
50
60
70
PROTON
80
90
100
ENERGY (Mev)
Fig. 3 - Excitation Function of Cs 132 for the Indepe~dent Formation
Cross-section of Csl32.
·
49
10----------------------------------------------~
1
!
-z
r'''l
co
<!
t-r,
...J
...J
'
(J)
c:::
I
~
~
:::
- 1.o;1t
t
z
0
1ü
w
'? 0.5
(J)
(J)
0
a::
ü
QI------~------~------~------~----~------~
10
30
PROTON
50
70
ENERGY
90
(Mev)
Fig, 4 - Excit'ltion Function for the Inde pendent Formation Cross-section
of Cs 34 m.
50
-ezn
a:
<(
m
_J
_J
-
7
:E
z 5
0
....
(..)
IJJ
en1
en
en
0
a:
(..)
20
30
40
PROTON
50
60
70
80
ENERGY (Mev)
['ig. 5 - Excitation Function for the Independant Formation Cross-section
of csl34m + g
90
51
30~------------------------------------------~
-z
25
( /)
0::
~20
:J
....J
-z
:E
0
1-
15
î
(.)
LLI
(/)
1
I
(/)
~lO
0::
(.)
5
o~_.----~~----~----~----_.
20
30
40
PROTON
50
______ ____._____
60
~
70
80
ENERGY (Mev)
Fig, 6 • Excitation Funotion for the Independant Formation Cross-Seotion
·
of Csl36 ·
·
~
52
80--~------------------------------------.
70
-z
Cl)
a::
<(
~50
_J
-z
:E
0
5w
40
Cl)
1
Cl)
Cl)
0
a::
(.)
30
20
lO
20
30
40
PROTON
50
60
ENERGY
70
80
<Mev)
Fig, 7 -Excitation Function for the Cumulative Formation Cross-section
of csl 37
53
40~-------------------------------------------,
( /)
z
œ:
<t
co 25
....J
....J
~
ô2o
t=
(.)
w
(/)
1
(/)
(/) 15
0
œ:
(.)
5
0~-----L-----L----~----~----~--~~--~~
20
30
40
PROTON
50
60
70
80
ENERGY <Mev)
Fig. 8 - Excitation Functions for the Independant Formation Cross-section
of csl38 and Cumulative Formation Cross•section of txel38
90
54
.e
1.5
-z
( /)
a::
<!
CD
...J
...J
l.O
~
z
0
l-
u
LJJ
(/)
1
(/)
(/)
0.5
0
a::
(.)
0.1
20
30
40
PROTON
50
60
70
ENERGY (Mev)
Fig. 9 - Excitation Function for the Independent Formation
Cross-section of. Rb86.
· ·
e.
80
90
55
TABLE 6
Formation Cross Sections of Cs138 (Independent)and Xe1 3 8 (Cumulative)_
-
No ..
En erg y
tl
(Mev.) _{min) D0 Cs 138(~tr'll tt)
dmb~·
kl
k2
xel38
cs138
18
15
20
20
5
70
1.15 x 10 8
6,.43 x 10 7
1 .. 199
1 .. 669
4.255
1.044
14.5:!: 3 .. 3
14
19
30
30
65
3
8
1.37 x 10
1,.53 x 10 8
1.790
0.931
1.164
4.443
22.2
28
22
39
39
75.3 4.84xl0~
4
5,.62x10
1.545
1.069
0.930
4.348
24.8 ±
21
12
57
57
3
69
1.08 x 10 8
3.8 x 107
0.931
1.694
0.443 15.9
1.067
23
29
65
65
75
5.5
3.72 x 10 7
2.45 x 10 7
1.549
1.259
0.937
4.209
32
25
75
75
5
62
7
7.66 x 10
8.14 x 10 7
1.199
1.874
4.255 12.9
1.242
± 2.8
7.2 ± 1.6
20
17
85
85
5
65
8.16 x 107
2.91 x 107
1.199
1.790
4.255 11.6
1.164
± 2.5
6.6 ± 1.4
t. 1
= Period of time
±5
s.s
± 3.5
12.5±3.7
30.6
±7
19.4 t 4.3
14.8 i: 3.3
10
± 2. 2
9.2
±2
elapsed between end of bombardment and separation of
Xenon.
: k 1 and k are defined in equations II. 5.16 and II. 5.17.
2
56
TABLE 7
csl37 Cumulative Formation Cross Section
No.
Enemv (Mev.,
no csl37
dis/min.
0 oo c 8 137
9
clmb
+ 19.4
48.5-
7
19
2.2 x 10 3
2.so x 10
8
30
2.84 x 10 3
3.10 x 10 9
56.0 ± 22
33
45
2,.24 x 10 3
1.78 x 109
35.6
:t 14.5
10
57
2.04 x 10 3
1.54 x 10 9
33.3
± 9.3
16
57
9.5 x 10 2
7.5 x 10 8
18.1±5
26
65
1.24 x 10 3
9.85 x 10 8
21.3 + 6
9
70
1.58 x 10 3
1.20 x 10 9
22.5
-±
11
85
3.20 x 103
2.42 x 10 9
16.0
± 4.5
5
86
2.88 x 103
2.62 x 10 9
18.9
± s.3
6.3
57
TABLE 8-.{1,
Independant Formation Cross Sections of Cs
Er~.c:i w~Y\.dow Proporti.on.d.l Cou.nter
No.
En erg y
{Mev.)
136
Do Csl;jo
dis/min.
4
7
19
7.8 x 10
18
20
4 .. 25 x 10 4
15
20
3.8 x 10
14
30
1.5 x lOS
19
30
1.04 x 10
8
30
2.74 x 10 5
28
39
2.08 x 10
22
39
2.62 x 10
13
39
33'
4
Doo Csl36
1.05 x
roB
5
5
± 0.34
1.7
2.28 x 10 8
4.2 +
... 0.84
2.02 x 10 8
4.5
± 0.91
7.8
± 1.6
8.12 x 10
5
d mb
5.62 x 10
3. 72 x 10
1.09 x 10
8
8
8
9
9.0
± 1.8
7.36
± 1.5
11.0
± 3.5
1.41 x 10 8
19.0 t 3.8
1.31 x 10 5
7 .o x 10 8
13.7
± 2.7
45
1.32 x 10 6
1.18 x 10
23.6
± 4.7
21
57
1.32 x 10 5
8.65 x 10 8
14.0
± 2.8
12
57
1.61 x 10
10
57
16
57
1.02 x 10
23
65
1.8 x 10
29
65
1.21xl0
5
9
8
12.5 ± 2.5
8.25 x 10 5
7.35 x 10 8
15.8 t 3.1
6
4.55 x 10 8
11.0 ± 2.2
9.,73xl0 8
14.0
6.52 x 108
11.2 + 2.2
s.8 x 108
12.3
± 2.5
15.1
± 3.0
5
5
5
26
65
6.5 x 10
24
65
1.61 x 10 5
7.12 x 10
8.65 x 10
8
± 2 .. 8
-
Cont'd.
58
TABLE 8-ê (Cont! d.}
No.
En erg y
(Mev.)
no cslJb
dis/min.
Doo c 5 136
10 8
dmb
9
70
5.15 x 10 5
s.s x
27
75
1.91 x 10 5
1.03 x 10
25
75
1.07 x 10 5
5.75 x 10
32
75
1.43 x 10 5
7.70xl0
30
80
4.65 x 10 5
4.15 x 10 6
20
85
1.62xl0 5
5.75 x
17
85
1.07 x 106
a. 7 x
11
85
1.57 x 10 5
1.4 x 10
31
85
6.55 x 10 5
5.9 x 10 8
11 + 2.2
5
86
1 .. 46 x 10 5
1.54xlo 8
11
7
8
8
10~
10 8
8
10.3 +
... 2.0
15.2.:!:: 3.0
12.7±2.5
13.7
t
11.6
± 2:.3
2.7
10.7t2.1
13.2 ± 2.6
9.3
* 1.8
..
t
2.2
59
TABLE 8-b
Independènt Formation Cross..:.section of Cs 136 ·
(4 if Counter)
No.
Energy (Mev.)
D°Csl36 dis/min.
d'ocsl36
6mb
2
40
7.5Sxlos
6.75 x 10 8
19.0:!: 3.4
2
50
2.98 x 10 5
2.66 x 10 8
15.1
t
4
65
s.ss x 10
15.5
± 2.8
3
73
1.25 x 10 6
5
5.31 x 10
8
1.12 x 10 9
2.7
17.4±3.1
60
TABLE 9-a
Independent Formation Cross-Section of Csl34m -:__ .•
(End Window PtoRQÎtional·Gounter)
No ..
Energy
(Mev)
nocsl34m
dis/min.
noo csl34m
ô mb
7
19
1.41 x 10 6
1.85 x 10 7
o.27 ± o.o6
18
20
2.84 x 105
1.42 x 107
0.26 :t 0.06
15
20
2.32 x 10
5
1.17 x 10 7
0.22 :t o.o5
14
30
1.69x1o6
8 .. s x 107
0.81
19
30
9.47 x 105
4.75xlo7
o. 76 ± 0.1 .
8
30
3.14 x 10 7
4.35 x 10 7
0.86 ':t 0.26
28
39
3.75xlo6
1.,88x10 8
2.9
22
39
4 .. 32 x 105
2.18x10
33
45
2.11 x 10 7
1. 77 x 108
3.5±0.82
21
57
4. 7 x 10 6
3.00 x 10 8
4.8! 1.1
12
57
6.0 x 10
8
2.36 x 10 8
4.1 ± 0.9
10
57
2.41 x 10 7
2.02 x 10 8
4.3 ± 0.1
23
65
7.48 x 10 6
3.76 x 10 8
,5.50
29
65
4.86 x 10 6
2.45 x 10 8
4.5±1.0
26
65
2. 7 x 10 7
2.26 x 10 8
4.8 ± 1.1
9
70
2.67 x 10 7
2.68 x 10 8
4.9
27
75
8.22 x 10 6
4.14 x 10 8
6.2 ± 1.4
8
± 0.18
± 0.67
3.8;!:0.9
± 1.2
± 1.1
61
TABLE 9-a (Cont' d.)
En erg y
No. (Mev.)
nocsl34m
dis/min.
noocsl34m
6
mb
25
75
5.52 x 10 6
2.78 x 10 8
32
75
6
6.31 x 10
3.17 x 10
3.0
80
2.4xl0 7
2.02 x 10 8
5.6 ± 1.3
2b
85
8.4 x 10
6
3.48 x 10 8
6 .. 5 ± 1.5
17
85
6.9 x 10
6
8
4.21 x 10
6.3±1•4
li
85
1.25 x 107
1.05x10 7
6.9 ± 1.5
3i
85
4.23 x 10 7
3.56 x 10 8
6.7±1.5
5
86
7.62 x 10 6
8.16 x 10 7
6.0±1.3
8
6.3 '!: 1.4
5.9 ± 1.3
62
TABLE 9-b
Independent Formation Cross-Section of csl34m
{4-n Countert ·
No.
En erg y
(Mev.)
DoCslJ4m
dis/min.
2
40
1.12 x 106
1
50
4.3 x 10
4
65
2.18 x 10
3
73
5.64 x 10
6
noo csl34m
9.4 x 10 7
3.62 x 10
7
7
1.86 x 10
4. 72 x 10
6mb
2.6±0.56
8
8
8
2.0
± 0.43
5.4 t 1.1
7.3:!:1.5
63
TABLE 10
Independent Formation Cross-section of Cs 134m + g
No.
Energ y
(Mev.)
o0 c s134 dis/min
0 oo0 sl34
6mb
7
19
4.42 x 10 2
3.52 x 10 7
0.50
t
8
30
9.95 x 10 2
7.95xl0 7
1.54
± 0.60
33
45
5.67 x 103
3.02 x 10 8
6.04
± 2.7
10
57
5.35 x 10 3
2.85 x 108
5.91
t 2.6
16
57
6.05 x 10 3
3.02 x 108
7.3
t
26
65
5.48 x 10 3
2.87 x 10
6.2
t 2.7
9
70
7 .o x 10 3
3.72 x 10 8
6.8 + 2.6
11
85
2.67 x 10 3
1.42 x 10 8
9.4 t 3.9
5
86
1.73x103
1.13 x 10 8
8.2
8
0.21
3.0
-
± 3.3
64
TABLE 11
Independent Formation Cross-Section of Os 132
No.
En erg y
(Mev.
D0 0 s 132 ' dis/min~
nc:o csl32
dcmb;
23
65
3.,3 x 10 4
9.0x10 7
1.3:!: 0.32
24
65
3.16 x 10 4
8.5 x 10 7
1.6:!: 0.40
26
65
1.25 x 10
5
5.6 x 108
1.4
29
65
2.98 x 10 4
9
70
1. 2 x 10
25
75
4.0 x 10
27
5
7.9 x 10
7
± o.as
1.2 ±
o.ao
6.5 x 108
1.2±0.,30
4
1.09 x 108
2.,4
75
6.0 x 10 4
1.64 x 108
2.4 ± 0.6
11
85
6.85 x 10
4
3.09 x 10 7
2.o4
17
85
6.09 x 10 4
1.64 x 10 8
3.4o ± o.85
20
85
7.99 x 10
8
3.28 ± 0.,82
5
86
6. 76 x 10 5
3.66 x 108
2.94 ± 0.73
4
2.16 x 10
± 0.6
t o.s1
65
TABLE 12
Independe!lt Formation Cross-section of Cs
No.
Energ y
(Mev.)
D°Cs 130 (dis/min~
30
80
7.9 x 10 6
31
85
1.36 x 10
1
130
Doo Cs130
1.14 x 10
7
1.97 x 10 7
6rmb)
o.32
± o.o9
0.37
* 0.10
66
TABLE 13
Independent Formation Cross-sèction of Rb 86
No.
Energy
(Mev.)
D 0 Rb86(dis/rnin~
2
Doo Rb86
d
6
mb
0.027 ± .006
7
19
9 .o x 10
6
21
1.56 x 10
8
30
6.72 x 10 3
1.32 x 10 7
0.26
2
40
6.2 x 10 3
8.05 x 10 6
o.45 t o.o1
1
50
3.82 x 10 3
4.96 x 10 6
0.31 ± 0.07
10
57
8 x 10 3
1.04 x 10 7
o.22 ± o.o5
4
65
1.58 x 10 4
2.5 x 107
0.73 ± 0.16
9
70
2.7x10 4
4.2 x 107
0.78 ± 0.18
3
73
2.94 x 10 4
3.82 x 10 7
o. 59 ±0.13
11
85
1.36 x 104
1.76x10 7
1.16
5
86
9.1 x 10
1.42xl0 7
1.02 ± 0.23
3
1.76xl0
4
1.89x10
6
o.os t o.o1
± 0.06
~
0.27
67
TABLE 14
Isomer Ratio
Enerqv (Mev.)
<5 Csl34mId Csl34g
19
1.17±0.52
l
30
:..26 ± o.s6
,
45
1.44 ± 0.65
1
57
2.15 ± 0.63
2
65
3.4±1.2
1
70
2.58 ± 0.87
1
85
2.78 ± 0.95
1
86
2.72 ± 0.92
1
Number of Determbations
.l.
68
TABLE 15
Independe::1t Formation C:-oss-sect:'.ons
Nu.:ü.ide
csl38
(!:J.depe:"!de:-:::)1
EP {Me7,.)
6 mb
N:::nbe!" of Determinations
± 3.,3
1
22 .. 2 ±
s.o
1
39
24.8 ±
s .. s
57
15.9±3 .. 5
1
65
12.5
± 2 .. 9
1
75
12.-.9
± i~B
1
85
11.6
i:
2.5
1
20
30.6±7.0
,
30
19.4
± 4.3
1
39
14.8
± 3.3
1
57
10.0
± 2.,2
1
6.5
9 .. 2 ± /
20
14,.5
30
,,
J..
Xel38
{Cumulati?e)
.J..
1
-
!
'
1
75
Î..,2:!:;l.,6
1
85
6. 6 i: 1.4
1
69
-
TABLE 15 (Cont'dtl
Nuclide
csl37
(Cumulative)
csl36
(Independen t)
='P (Mev.)
6
mb
Number of Dete..'"Irlinations
19
48.5
"!:
30
56.0
± 22
l
45
35~6
± 14.5
1
57
25.7t7.6
65
21.3
±6
70
22.5
:1;
6.3
l
85
16.0
± 4.5
l
86
18.9
± s.3
1
19
1.7
± 0.34
l
20
4.4
± 0.32
2
30
8.o
± o.s7
3
39
16.6
± 1.5
3
40
19.0 ± 3.8
l
45
23.6 ± 4.7
1
50
15.1 ± 2.7
1
57
13•3
±1
4
65
13.6
± o.1
5
70
10.3 t 2
1
73
17.4±3.1
1
19.;4
1
2
1
(Cont'd.)
70
TABLE 15 (Cont' d,)
Nuclide
136
08
(Inde pendent
Cont'd.)
csl34m
(Independant)
EP (Mev.}
·:d (mb'
Number of Determ!natirms::
75
13.8
±1
3
80
11.6
:1;
2
1
85
11.1
±1
4
86
11.0
± 2.2
1
19
0.27 ± o.o6
1
20
o.24
± o.o2
2
30
o.81 ± o.11
3
39
3.35
± 0.45
2
40
2.6
± o.s6
1
45
3.5:!: 0.82
1
50
2.0 ± 0.45
1
57
4.4
± 0.65
3
65
5.04
± 1.0
4
70
4.9±1.1
1
73
7.3
± 1.5
1
75
6.1 ± o.1
3
80
5.6 ± 1.3
1
85
6.6
± 0.2
4
86
6.0 ± 1.3
1
(Cont'd.)
71
TABLE 15 (Cont'd1 )
Nu elide
EP (Mev.)
;:{$ (mb)
Number of Detenninatlons 1i
i
cs134m + g
(Independent)
08132
(Independent)
130
08
(lndependen ~
upper limits
19
o.so
± 0.21
1
30
1.54
± o.6
1
45
6.04±2.7
1
57
6.60
± 0.7
2
65
6.0 ± 2.7
1
70
6.8
± 2.6
1
85
9.4
± 3.9
1
86
8.2
± 3.3
1
65
1.4
± o.o6
4
70
1.2 ± o.3o
1
75
2.4±0.60
2
85
2.9
± 0.4
3
86
2.9
± 0.73
1
80
0.32
± 0.09
1
85
0.37 t. 0.10
1
(Cont'd.)
72
TABLE 15 (Cont'd 1 )
Nuclide
Rb86
Independant)
6
EP (Mev.)
mb
Number of Determinations
19
o.o21 ± o.oo6
l
21
o.oss
± o.ol3
1
30
0.26 ± o.o6
1
40
o.4s ± o.o1
1
50
o.31 ± o.o7
1
57
o.22 ± o.os
l
65
0.73±0.16
1
70
0.78
0.18
1
73
0.,59 ± O.lS
1
85
1.16
± 0.27
1
86
1.10
± 0.23
1
;:1::
73
Fig. 10 -Charge distribution curve .at 20 Mev. (The
right-hand (dashed) portion of the curve
has been drawn so that the sums of the
isobaric yields read off the curve approximate the measured cumulative yields.)
73a
50----------------------------------------------~
20 Mev
l.54
!
"" ....
/ /
1
-- '
.....
\
\
\
\
\
\
\
\
\
\
\
\
\
\
(J"(mb
\
\
\
\
\
\
\
\
.
cJ 37
.
X~
38
FROM CURVE
EXPERIMENTAL
47.5mb
52±21 mb
30.6mb
29.4mb
...
A •137
Cs
1
Xe
1
A •138
1
Te
1.
1
1
1
Xe.
I
1
.Te
0.1'-----'-----'- -----'-------...L ...----L..----.... .a.-___.
1.50
1.40
1.45
1.55
1.60
1.65
NIZ
. :-
"
.
,.
.
_
~
. · r
\
.
.·, , ,,
·~·
~
~
74
Fig. 11.- Charge distribution curve at 30 Mev. (The
right-hand (dashed) portion of the curves
has been drawn so that the sums of the
isobaric yields read off the curve approximate the measured cumulative yields.)
1
74a
40l
30 Mev
1.5 3
-'
1
~
Cst3e
.....
'\
\
Z UNITS---\
\
\
\
(J"
\
\
\
\
(mb)
\
\
1
l
1.0
l
1
A= 137
Xe
Cs
~
~
~r
A= 138
FROM
137
Cs
138
Xe
CURVE
1
1
1
Te
I
1
1
Xe
I
EXPERIMENTAL
49mb
56± 22mb
21.5mb
19.4±4.4mb
0.2
1.40
1.45
1.55
1.50
N/Z
1.60
1.65
75
Fig. 12 -Charge distribution curve at 39 Mev. (The
right-hand (dashed) portion of the curve
has been drawn so that the sums of the
isobaric yields read off the curve approximate the ineasured cumulative yields.)
40-----------------------------------------,
1.515
39Mev
Cs13BfL,'\
\
2.3
Z
\
UNITS
\
\
\
\
\
(J(mb ·
\
\
1
1
1
A•l37
Cs
1
1
1
Xe
•
cd
37
X~
38
1
Te
I
Xe
A• 138
\
I
1
Te
1
FROM CURVE
EX PERIMENT AL.
49mb
44.8t 20mb
17mb
14.8±3.3mb
0.1 1...-------L-------'--------'-----::--""--------~~
1.45
1.50
1.55
1.40
1.60
1.65
N/Z
76
Fig. 13 -Charge distribution curve at 50 Mev. (The
right-:-hand (dashed) portion of the curve
has been drawn so that the sums of the
isobaric yields read off the curve approximate the measured cumulative yields .)
7
1.51
t
50 Mev
..........
"'\ \
2.5
Z
~
UNITS
134
Cs
/
\
\
\
\
1
\
O'"(mb
A= 137 .
1
1
1
Cs
Xe
1
A=l38
I
Xe
1
1
FROM
C~
37
CURVE
EXPERIMENTAL
33mb
33± 9.2 mb
10 mb
12± 7.2mb
138
Xe
-----L----J...----'-----....__--___,-__,
0.1 .......
1.40
1.45
1.50
1.55
N/Z
1.60
1.65
77
Fig, 14- Charge distribution curve at 57 Mev. (The
right-hand (dashed) portion of the curve
has been drawn so that the sums of the
isobaric yields read off the curve approximate the measured cumulative yields.)
77a
30
57 Mev
1.50
l
c~3s
'
' "\
2.5
Z UNITS
-\
c 134
\
1
s
\
1
\
1
<Y (m b)
\
\
1
\
\
.,
1
1
1
!A•I37
\
1
1
1
I
Xe
Cs
Xe
A=l38
1
FROM
137
Cs
CURVE
T!
1
1
Te
1
EXPERIMENTAL.
32.5 mb
25.7± 7.6 mb
10 mb
10.0:!: 2 mb
138
Xe
O,l...._..___ _~-----'----...L-------L-----'--..a
1.65
1.60 .
1.50
1.55
1.45
1.40
NIZ
78
..
.; ;. F.; ;.oigo:.•,;......;l;....;S;....- Charge distribution curve at 65 Mev. (The
right-hand (dashed) portion of the curV-e
has been drawn so that the sums of the
isobaric yields read off the curve approximate the measured cumulative yields.)
ryo,.,
,u~,...;,.
30~----------------------------------------~
65Mev
2. B
Z UNITS
MJ-_..;_--
''\
~
(){mb)
1
1
1
1
FROM
138
0.1
1.40
1.45
\
\
I \
A=l38 Xe
Xe
\
1 \
Xe
1
137
\
\
\
A= 137 Cs
Cs
\
CURVE
\1
\I
EXPERIMENTAL
29.5mb
2B:t 7.Bmb
9.1 mb
9.2:t2mb
1.55
1.50
NIZ
1.60
1.65
79
1
Fig. 16 - Charge distribution curve at 75 Mev. (The
right-hand {da shed) portion of the curve
has been drawn so that the sums of the
isobaric yields read off the curves approximate the measured cumulative yields.)
'7 (') {)
1
•
30
75Mev
e
Csl38
........
'
2.9
' '\
~
Z UNITS
\
\
\
(J(mb)
\
\
\
1
\
\
1
1
1
\
A• 137
1
1
1
Cs
Xe
I
1
A•l38 Xe
FROM
137
Cs
Xe
0.1
1.40
1.45
CURVE
7.2:!: 1.6mb
7.6mb
1.55
1.50
NIZ
EXPERIMENTAL
17.8 t 5mb
24.1mb
138
1
I
1.60
1.65
80
Fig. 17 - Charge distribution curve at 85 Mev. (The
right-hand (da shed)· portion of the curves ·
has been drawn so that the sums of the
isobaric yields read off the curves approximate the measured cumulative yields.)
30r--------------------------------------85 Mev
e
c~3a
'
3. l
''\
"
\
Z UNITS
Csl32
()(mb)
1
1
1
1
\
\
\
\
\
./
1
1
\
\
\
1
A= 137 Cs
1
FROM
Cs 137
138
Xe
1.40
1.45
1
I
Xe
A•I3S
O.l
\
Xe
1
CURVE
EXPERIMENTAL
6.6t 1.5mb
6.2 mb
1.55
NIZ
1
16 :t 4.5 mb
22.3mb
l.50
I
1.60
1.65
81
TABLE 16
Total Isobaric Cross-sections
for Mass No, A= 136
rnb
Enerov
20
56.0
:!:
30
59.0
± 23,6
39
57.3 ± 22,9
50
43,2±17.3
57
40.7
65
41.7±16.7
75
40,5
:!: 16.2
85
38.8
± 15,5
22,4
± 16,3
82
/
Fig. 18 - Total I sobaric Oro s s- section for A = 13 6
82a
90
..,.
T
70-
60
·!"
C)
..
.
CJ(m b)
50
()
40-
(
)
...
()
0
...
.
30
.
.
.
...
""'
20
20
30
40
50
ENERGY <Mev)
60
70
80
83
IV. DISCUSSION
IV .1
A Study of the Excitation Functions
The initial rise of the excitation function of the shielded
nuclide
Cs 136 to a maximum followed by a graduai decrease with energy
may be qualitatively explained as follows: The maximum of the isobaric
charge distribution (i.,e .. the most probable charge Zp) for the mass number
A= 136 shifts to higher Z with increasing energy.
This would imply that
the cross-section for the independant formation of a nuclide close to
stability (Z greater than the maximum of the charge distribution) should
show an initialincrease with excitation energy as the most probable
charge, Zp, approaches the atomic number Z of the nuclide in question.
It should leve! off when Zp becomes approximately equal to the given Z
and then decrease as the maximum moves to higher Z( 80,81, 92-96)
The shape of the excitation function of the independently
formed csl38 may also be explained .in the same manner. The shielded
nuclides csl34, csl32,. and Rb86 clearly show the initial rise of their
independant formation cross-sections with bombarding energy. The maximum
proton energy available was too low to observe the maximum and subsequent
decrease. The csl3 4 excitation function appears to leve! off at about
80 Mev.
84
The increase in the formation cross-sections of neutron deficient products
with increasing bombarding energy is due to increasing energy deposition in the
struck nucleus which enhances neutron emission occuring either before or after
fission.
ln Figure 19 a the energies at which the excitation functions for the
independent formation of various products reach their maxima are plotted against
the N/Z values of the nuclides in question. The independent formation cross sections
131 132 134
J31
140
of 1 , 1 , 1 ' and Te from Pate et al (80) and La
from Kjelberg et al (91)
1
1
are also included. ln Figure l9b the same peak energies are plotted against (Z-ZA)·
If the rubidium datacirtto fit weil with the cesium isotopes on the same plot, it would
require thot on the (Z- ZA) plot Rb
86
excitation function must have a maximum at
about 70 Mev which clearly is not true as is evident from Figure 9. However from
the N/Z plot the peak energy for Rb 86 is far above 100 Mev which seems to be in
agreement with the trend of the excitation function of Rb 86 (Fig. 9).
ln the high energy fission of Uranium (91) it has also been observed thot
although the ' peak energies 1 of the excitation functions of the various cesium isotopes
fa li on a smooth curve in either N/Z or (Z - ZA ) plot, the rubidium .data fit weil
with the cesium points only on the N/Z plot of the cesium data. lt would appear,
therefore, thot the
with (Z- ZA ).
1
peak energies
1
are more closely related with N/Z thon
85
Fig. 19 - Energies at which the excitation functions reach
maxima (a) as the function of N/Z (b) as a function
, of (Z- ZJJ.. ZA values are those of Coryell (Ref.Il8).
200r---------------------~------------~------~
-
( b)
(a)
>-
(!)
0:::
w
w
z
~
13a
rl31
Cs
<(
w
Q.,
T.
0
131
.e a
-3
oii32
-2
Z
1.45
-1
1.50
1.55
NIZ.
- ZA
.
/
(tc\)
86
IV.2
Charge Distribution Parameters
The width il z, for the charge distribution curve at a given
energy wa s obtained by converting the appropria te N/Z ratios at half
maximum to Z values using the (N + Z) value of the cesium isotope nearest
to the peak position. Similarly the N/Z and (N + Z) values at the peak
positions were used to determine the displacement of the most probable
charge
ZP
from stability 2A_ (116) at different energies. The parametersLI Z
and {ZÀ - z;P,) are tabulated in Table 17..
The data from proton induced
fission of uranium (93) are also included. Their variation with energy of
bombardment is shown in Figure 20.
As the energy is increased the most probable charge Zp,
moves towards stability
(Z.p.)
for both uranium and thorium. At a given
energy ZA - Zp is smaller for thorium (N/Z
= 1. 577} than for uranium
(N/Z = 1. 587). Thus, the smaller the N/Z of the target nucleus, the smaller
the N/Z values of the most probable fragments. The width of the charge
distribution remains nearly constant up to about 50 Mev. and then it begins
to increa se wi th increa sing energy.
These effects may be qualitatively explained in terms of a
two step model for nuclear reactions. The projectile first undergoes a
rapid interaction with the target nucleus, which predominantly leads to the
formation of the compound nucleus for energies up to about 50 Mev. The
87
..
Fig. 20 (a) - Full width at half maximum of charge
distribution curve plotted against
incident proton energy.
Fig. 20 (b) -: Displacement of most probable charge,
Zp' towards beta stability, ZA•
.
•
•
4--------------------------------------------------------------(a)
3
.....
-- ....
.
--------o
-0
UNITS
2
OF Z.
1
''.
4t-•
238
o----o
U
• •
Th
232
<DAVIES
a
YAFFE
<PRESENT
WORK)
(DAVIES 6
YAFFE)
<PRESENT
WORK}
(b)
o----o U
•
3
238
• Th
232
zA -zp
2
•
co
20
30
40
50
60
ENERGY
70
(Mev)
80
90
100
110
-...:}
}1)
88
TABLE 17
Charne Dist."'ibution Parameters
' Target
Thorium
Proton Energy
{Mev.)
.
2.1
1.54
2.3
0.105
2.2
1.5;
2.3
39
0.107
2.3
1.51
1.8
50
0.11
2.5
1.51
1.8
57
0.115
2.5
1.5o
1.,6
65
0.130
2.8
1.49
1.3
75
0.133
2.9
1.49
1•3
85
0.140
3.1
1.48
1.2
20
0.10
30
·-
Uranium
Peak Position
ZA- ZD
NIZ
Full Width at Half Maximum
Az
N/Z
------~
-'··-~
~
-~----------
,_ ...
----~------
20
0.098
2.2
1.550
2.so
36
0.104
2.2
1.525
2.00
50
0.115
2.5
1.512
1.76
65
0.136
3.0
1.5os
1.6o
80
0.150
3.25
1.500
1.ss
89
observation that the width of the charge distribution remain nearly
constant (,.....2.2 charge units) for both uranium and thorium up to about
50 Mev. may be an indication that the fissioning nucleus is rather
uniquely defined in this energy region. At higher energies interaction
with individual nucleons associated with the prompt ejection of sorne
nucleons {nuclear cascade) becomes increasingly important. After this
initial stage has terminated the excitation energy is dissipated in a second
slower stage by neutron evaporation. In heavy nuclei fission is a competing
process. The lncrease in width of the charge distribution curves may be
rationalized in terms of a proliferation of both the nuclear cascade and
nucleon evaporation leadlng to a wider distribution of fissioning nuclei
and hence a wider distribution of N/Z ratios of the fission products.
89b
TABLE 17b
ZA values
Mass No.
(A)
(118) used in calculating Zp
Most
Stable
Charge
(ZA)
•
87
38.8
88
39.0
89
39.4
90
39.8
91
40.2
92
40.6
93
41.0
94
41.4
95
41.8
96
42.0
97
42.5
98
43.1
99
43.7
100
44.0
101
44.4
136
55.9
137
56 .. 4
138
56.6
139
56.9
140
57.3
141
5'7.7
__l
90
Calculation of Most Probable Charge, Zp
IV.3.1
An attempt has been made ~o calculate the most probable charge
of a fission fragment in the mass region of interest on the basis. of the ECO
and UCO postulates mentioned in Section 1. These values are then compared
with the experimentally determined
Zp values. rhe mass number chosen is A=136.
Compound nucleus formation isassumed wherever appropriate and elsewhere
the results of published calculations on the prompt nuclear cascade (119,120) are used.
ln ali
cases, the data published by Vandenbosch and Huizenga ( 121) on fission-
spoliation competition have a Iso been used.
The experi:mentol results of
Devies and Yaffe (93) on proton· induced fission of uranium were also included
in the present analysis to make a comparative study.
According io the ECO postulate (72) the most probable charge (Zp)j
of a fission fragment of initial mass number Ai is given by:
IV .3.1
where AF and
ZF
denote the mass number and :the charge of the fissioning nucleus,
Z(AF-Aï:)and ZAi are the most stable charges of the two complementary fission
fragments with mass numbers Ai and .(\ A F - Ai) respectively. Values of
ZAi
used . are those given by Coryell (118) and they are tisted in Table 17b.
Shell effects had
been included in the evaluation of these values.
According to the UCO postulate(65)
the most pro1bable charge is given by:
IV~.3.2
91
Since neutrons may be emitted from an excited nucleus prior
to fission the calculations are perforrned for a range of fissioning
nucle~~
.
The Zp values calculated in this fashion refer to the primary fragments.
The latter may subsequently emit additonal neutrons to lead to the observed
fission products. If the initial fragment emits 'n' neutrons immediately
after fission to give the final mass number Af (experimentally studied) then
Zp of Af is calculated from equations 1V.3.1 and IV .. 3.2 with Ai= (Af + n).
ln evaluating the value of •n• the following assumptions were made:
1.
On the average 2 'fission neutrons• are emitted promptly
regardless of energy or fission mode (15) .and. they are equ'ally divided between
the two complementary fragments. (These neutrons probably originate lrom
th è
2
deformation energy of the fissioning nucleus.)
The excitation energy of the fissioning nucleus is divided
between the fragments in the ratio of their masses.
3 ..
The average number of neutrons emitted from the primary fragments
due to this excitation energy is obtained by using an average neutron binding
energy value (7 Mev.)( 11) appropriate to the nuclides in question plus an
estimated average neutron kinetic energy (2 Mev.) (22).
On these assumptions, the fissioning nucleus divides into two
fragments such that one of them has a mas s number given by
E·*
AH=
136 +t+ -:
IV.3.3
where Ap is the ma ss number of the f!ssioning nucleus and
excitation energy in Mev.
E~
is its
This equation can be rearranged to give
. 137
IV.3.4
Et
(1--)
9Ap
The excitation energy
Ec of the compound nucleus is given
by the sum of the proton kinetic and binding energies. The residual
excitation energy
Er
after emission of a neutron is given by:
IV.-3.5
where Bn is the neutron binding energy, Ek = 21' !s the average k!netic
energy of the neutron .. 1' being the nuclear temperature. Details of the
calculqtlon are described in the section IV on !samer ratios. The proton
and neutron b!nc:Ung energies are taken from reference (.7 1). The calculation
of
Er is
carried out for successive neutron evaporation until the residual
excitation energy reaches a value be1ow the neutron b!nding energy•.
Table 18 eJhows the excitation energies of the various nuclei formed from the
compound nuclei by successive evaporation of neutrons, and the estimated
initial· fr'gment masses.
Incorporation of fission competition with spallat!on has to be
considered in any attempt to obta!n the details and mechanism of the nuc1ear
charge distribution between primary fragments of medium and h!gh energy
fission {121-125).
93
TABLE 18
Excitation Enemy Distribution and Estimated Initial Fission Fragment Mass
Fissioning Nuclei
A ' ~(Mev.)
Th232 +
20 Mev.
Protons
Th232 +
50 Mev.
Protons
Th232 +
57 Mev.
Protons
Bn (Mev.)
Ek= 2T
Fission Fragment
{Final Mass No.=136)
Initial Mass
233
25.6
6.6
1.6
138
232
17.4
4.2
1.3
138
231
11.9
6.9
0.83
137
230
4.2
s.8
233
ss.6
6.6
2.76
140
232
46.2
4.2
2.53
140
231
39.5
6.9
2.23
139
230
30.4
s.8
2.05
139
229
22.5
7.1
1.77
138
228
13.6
6.0
1.38
137
227
16.3
7.3
-
137
233
62.6
6.6
2.77
141
232
53.6
4.2
2.60
140
231
46.4
6.9
2.33
140
230
37.2
s.8
2.09
139
229
29.3
7.1
1.91
138
228
20.3
6.0
1.41
138
227
12.9
7.3
0.87
137
226
4.7
6.2
-
137
-
137
(Cont'd.)
94
TABLE 18 (Cont'd,)
Fis s!oninglf!uclei
(Mev,}
A
Bn (Mev,)
Ek = 2T
Fission Frfigment
(Final Mass No. 136
Initial Mass
1
u238 +
20 Mev.
239
25.4
6,38
1,58
138
Protons
238
17.4
5.39
1.34
138
237
10.7
6.70
0,74
137
236
2.8
s.71
-
137
u238 +
50 Mev.
239
55.4
6.38
2,56
140
Protons
238
46,4
5.39
2,34
139
237
38.7
6.70
2.29
139
236
29.7
5.71
2,00
138
235
22.0
7.02
1.73
137
234
13.3
4.70
1,35
137
-
137
'
233
7.2
7,54
95
The relative probability that a nucleus in a given energy state
de-excites via a specifie reaction path is usually described in terms of a
partial "width" of the state for:that reaction. The probability 'G' for fission
of a given nucleus at a given energy is expressed as a ratio:
where
rr
G
Fr 1lfEvaporation +Fr)
=
IV.3.6
is the fission width and revaporation is a sum of particle
emission widths.
Disregarding the probability of emission of charged particle as
unlikely {119) in the energy region of interest this reduces to:
G =
where
rFt{(n
+ rr)
ht is the neutron evaporation width.
The dependence of G on the
fissioning nucleus is quite pronounced and has been the topic of severa!
investigations (121-125). This dependence is shown !n Figure 21 and is
based on the data by Vandenbosch and Huizenga (121) (Table 19}.
For example we will now cons! der 20 Mev. proton induced
fission of thorium:
Pa233
t
Fission
'Y\t
Pa232 ·'Y\,
>t
'(t.~ J'a230
7 J..
Pa231
}J..
Fission
Fission
Fission
From Table 19 the G V41ues for these nuclei (assumed to be energy
are, respectively, 0.2a. 0.34 1 0.40,
o.so.
indepénd~nt}
It is assumed that the formation
probab!lity of Pa233 (compound nucleus) is 1. The f!ssfon probability · .
96
TABLE 19
Fission Spallation Competition
Nucleus
Pa
Th
2
z /A
rn; TF
233
35.4
2.5
0.28
232
35.69
1.9
0.34
231
35.85
1.5
0.40
230
36.00
1.0
0.50
229
36.16
0,.8
0.,.56
228
36.32
0.6
0.62
227
36.48
0.45
0.69
226
36.64
0.30
0.77
225
36.80
0.25
o.8o
224
36.97
0.18
o.85
223
37.13
0.12
0.89
232
34.91
18.0
o.os
231
35.06
13.0
o.o7
230
35.22
9.0
0.10
229
35.37
s .. o
0.16
228
35.53
3.5
0.22
227
35.68
2.2
0.31
226
35.84
1.8
0.36
225
36.00
1.2
0.45
224
36.16
o.8
0.55
Mass No. A._
G =rp;trn + rF)
(Cont'd.)
97
TABLE 19 (Cont' d.}
z 2/A
fn/rF
G
= fp/t~ +
Nucleus
Mass No. A
Np
239
36.20
1.0
o.so
238
36.35
o.s
0.56
237
36.50
0.5
0.66
236
36.70
o.4
o.11
235
36.81
0.3
0.77
234
37 .o
0.2
o.83
233
37.2
o.o5
0.95
232
37.38
0.03
0.98
238
35.6
4.0
0.20
237
35.79
3.0
0.25
236
35.9
2.0
0.33
235
36.0
1.5
o.4o
234
36.2
1.0
o.so
233
36.3
0.8
o.s6
232
36.5
o.s
0.66
231
36.68
0.4
0.72
u
rr>
98
,.
Fig. 21 - .Fission probability of heavy nuclldes as
function of mas s number A.
'
~--------------------------------------------~----~~
(.\J
.e
99
is 0,28 1 and the neutron evaporation probability iso. 72. The latter value
is equal to the formation probability of Pa
232
.. Since the G value . of Pa 2 32
is 0.34, its fission probability is o. 72 x 0.34 = 0.245 and the neutron
evaporation probability is o. 72 (1-0.34) = 0.,475 .. The calculation is continued
until the last stage of neutron evaporation is
reached~
The ZP values calculated on the basis of ECD and UCD are now
weighted by the fission probability of each fissioning nucleus and the weighted
average Zp is determined. Similar calculations were performed for 50 and 57 Mev.
proton induced fission of thorium and 20 and 50 Mev. proton fission of uranium,
The results are given in Table 20.
Th 232 + 85 Mev, Protons and u238 + 80 Mev, Protons
The experimental data for these systems will be compared with an
analysis starting with the results of the prompt cascade for 82. Mev. protons
incident upon u 238 {119), The difference in energy of bombardment is
neglected and the difference in target characteristics is taken 1nto account
by introduction of appropriate shifts in charges and masses of the reaction
products,
The total number of cascades followed in the prompt interaction
calculation was 1258 in addition to 78 events in which nuclear transparenoy
is !ndicated. The excitation energy spectrum is also available from the
calculations in reference 119.. This is generally a continuous distribution
rao
TABLE 20
Most Probable Charge Zp (20-57 Mev.)
Mass No. r 'f'n+fF)Fonna• Fission Neutron
A of
ti on
Proba- EvaporaFissioning
Proba- bility
t!on ProNucleus
billtv
babil!tv
Th232 +
20 Mev.
Protons
233
0.28
l
0.28
0.72
.52.5
53.8
232
0.34
0.72
0.24
0.47
52.9
53.9
231
0.40
o.4t
0.19
o.21
52.8
53.9
230
0.50
0.27
0.14
o.l4
53.0
53.9
Weighted Average Zp
52.8
53.9 53.s
-
Th232 +
50 Mev.
Protons
Zp
Expert•
{A= 136) mental
ECD UCD
233
0.28
1
0.28
o.72
53.6
54.,6
232
0.34
0.72
0.24
0.47
53.5
54.8
231
0.40
0.47
0.19
0.27
53.,4
54.7
230
o.so
0.27
0.14
0.14
53.2
54.5
229
0.56
0.14
0.08
o.os
53.5
54.7
228
0.62
0.06
o.o4
o.o2
53.4
54.6
227
o .. 69
o.o2
0.02
o.oo7
53.6
54.7
53.4
54.;7
-
Weighted Average Zp
{Cont'd.)
54.2
101
TABLE 20 (Gont1 d,)
Mass No. fp{fn +fFJForma- Fission Neutron
Proba- EvaporaA of
ti on
Fissioning
Proba- bility
tion Prob!lity
Nucleus
b!lity
ExperiZp
(A= 136)
mental
ECD
UCD
-
Th232 +
57 Mèv.
233
o.28
1
0.28
0.72
53.5
ss.o
Protons
232
o.34
0.72
0.24
0.47
53.5
54.8
231
0.40
0.47
0.19
0.27
53.8
55.1
230
o.5o
0.27
0.14
0.14
53.7
54.9
229
· o.56
0.14
o.o8
o.o6
53.5
54.7
228
0.62
o.o6
0.04
o.o2
53.7
55.0
227
0.69
o•o2
0.,02
o.oo7
53.7
54.7
226
0.77
o.oo7
o.oo5
o.oo2
53.9
5S.l
53.5
54.8
-
Weighted Ave:rage;:zp
u238 +
20 Mev.
239
0.50
1
o.so
o.so
52.4
53.7
Protons
238
o.56
o.so
0.28
0.22
52.6
53.7
237
0.66
0.22
0.14
o.oa
52.5
53.7
236
0.71
o.. o8
0.056
0.024
52.8
54.0
We!ghted Average Zp 52.4
53.7
-
u238 +
50 Mev.
239
0.50
1
0.50
0.50
53.2
54.5
Protons
238
0.56
0.50
0.28
0.22
53.1
541114
237
0.66
0.,22
0.14
o.oa
53.3
54.5
236
0.71
o .. os
o.s6
0.024
53.1
54.4
235
0.77
o.o24
0.19
o.oos
53.3
54o6
234
0.83
o.oos
0.004
0.001
53.2
54.5
233
0.95
0.001
0.009
o.ooo1
53.4
54.6
53.2
54.5
-
Weighted Average Z,..,
.
54.4
53.3
54.2
102
•
except in the case of the compound nuclei ( Pa
233
and
u239 ) which
are excited to
unique energies. Table 21 gives the cascade products., their relative frequencies
and their average excitation
<er~ergies.
The evaporation calculation for heavy nuclei (120) excited to 75 Mev
indicate thot 99.7% of the evaporated nuclei are neutrons. Consequently charged parti cie
evaporation will be neglected. For each cascade product the spectrum of residuel
excitation energies was divided into a number of bins and an average excitation energy
was taken for each binu as indicated in Table 21. This procedure simplified the
calculation without introducing any significant distortion of the results.
The evaporation reaction proceeds until
the residuel excitation
ener_gy reaches a value below the neutron binding energy, as described earlier
.(.
Tables 22-24) •
Fission is assumed to compete with neutron evaporation in these
cascade products in the same manner as in the low energy reactions.
average
Zp value of the fission fragment (
The
final mass number A=136 )
calculated fo..r each cascade product is weighted with the relative frequencies
of each cascade reaction and the final weighted average of Zp is compared
with experimental values for both thorium and uranium ( Tables 25 and 26 ).
Contributions from ( P, 2P) and
•
are neglected in these calculations •
( P, P2N) cascade reactions
103
TABLE 21
Prompt Interaction Data on u238 + 82 Mev, Protons
Excitation
En erg y
Range
{Mev.)
RELATIVE FREOUENCIES
Average
Compound (P,N) (P 1 2N} (P 1 P') (P~PN) KP,P2N) (P 1 2P)
Np238 Np237 u238 u 37 u236 Pa237
Excitation Nucleus
Np(239)
Energy of
Bins (Mev..)
0-9.38
6.4
0-18.8
9
0-28.1
14
18.8-37,5
28
9,38-56.3
35
28.1-46.9
37
37.5-56.3
46
56.3-65,7
61
56.3-75.0
65
-
1
22
21
55
63
16
92
152
40
224
520
Total No.
520
453
Average Excitation (Mev.)
88,9
52.5
Transparenciés::
3
27
88,9
Incident Protons:
22
1336
78
27
217
37
1
3
34,7
40,7
22.5
6.4
13.7
104
TABLE 22
Excitation Energ;t: Distribution and Estimated Initial Fissign Fragment Mass
Fissioning Nuclei Bn(Mev.) E = 2'f
A
. (fAev ~)
- t ....
Th232 +
85 Mev.
protons.
Compound
nucleus
formation
u238 +
80 Mev.
protons.
Compound
nucleus
formation
233
90.6
Fission Fragment
(!'inal. Jy1a~J:i ~9~ A~~-4.§
Initial Mass
6.61
3.52
142
232
80.5
4.19
3.34
142
231
73.0
6.92
3.18
141
230
62.9
5.8o
2.95
141
229
54.2
7.11
2.76
140
228
44.34
6.00
2.49
139
227
35.85
7.31
2.24
139
226
26.31
6.19
1.90
138
225
18.21
7.50
1.60
138
224
9.11
6.38
1.10
137
223
1.63
7.69
239
85.40
6.38
3.38
142
238
75.64
5.39
3.18
141
237
66.07
6.70
2.99
141
236
56.38
5.71
2.76
140
235
47.91
7.02
2.55
140
234
38.34
4.70
2.29
139
233
31.35
7.54
2.07
139
232
21.74
6.42
1.73
138
231
13.59
7.73
1.37
137
230
4.49
6.62
105
TABLE 23
Excitation Energy Distributions and Initial Fission Fragment Mass
th23.i:.+ BS Mev. Protons
,
Cascade . Fissioning Nucleus. Bn(Mev.) Ek
Frequency Mass Excitation
No .. A Energy E
(Mev.}
(PiN)
22
55
152
224
232
9 .. 0
4 .. 2
231
3~7
6.9
= 2T Fission Fragment (Final
Mass No .. 136)
Initial Mass No.
1.11
137
137
232
28
4.2
1.96
138
231
2L.84.
6 .. 9
1.74
138
230
13 .. 2
5 .. 8
1.38
137
229
6.0
7.1
137
232
46
4.2
2.52
139
231
39.3
6 .. 9
2.32
139
230
30.1
s. a
2.04
139
229
22.2
7.1
1.78
138
228
13.4
6.0
1.37
137
227
6.0
7.3
137
232
65
4.2
2.98
141
231
57.8
6.9
2.82
140
230
48 .. 1
s.a
2.59
140
229
39.7
7.1
2.36
139
(Coned.)
106
TABLE 2:.<...(0ont' d,)
Cascade F!ssion!ng Nucleus Bn(Mev.) Ek
Frequency Mass Excitation
No. A Energy E
(Me~.7
(P,N)
224
= 2'!' Fis sion Fragment (Final
Mass No. 136)
Initial Mass No.
.)
228
30 .. 2
6.0
2.06
139
227
22.2
7.3
1. 76
138
226
13.1
6.2
1.36
137
7.5
0.88
137
225
5.6
(P 1 2N)
27
{P ,PN)
21
16
231
34.7
6.9
2.23
139
230
25.6
5.8
1.88
138
229
17.9
7.1
1.58
138
228
9.2
6.0
1.13
137
231
14.0
3.68
1.39
137
1.13
137
230
8.92
6.82
229
0.97
5.28
231
37
3.68
2.26
139
230
31.06
6.82
2.08
139
229
22.16
5.28
1.76
138
228
15.12
7.02
1.45
138
227
6 .. 6.5
5.47
0.97
137
226
0.21
7.21
232
9
6.29
231
1.6
3.68
137
1
(P, P )
22
1.1
137
(Cont1 d.)
107
TABLE
aa
(Cor:.t'!it)
bascade F!ssiord:1g N1:.cleus Bn(Mev .. ) Ek = 21'
Freq'l.!ency Mass Excitation
E:1e:rgy E
No.A
(Mev.)
Fission Fragment
(F!~al Mass No. 136)
Initial Mass No.
{P ,.P')
63
92
40
232
28
6.29
1.96
138
231
19 .. 7.5
3.68
1.65
138
230
14.42
6.82
1.41
137
229
6.19
5.28
1.36
137
232
46
6.,29
2.52
140
231
37.2
3.68
2.26
139
230
31.25
6.82
2.08
139
229
22.35
5.28
1.76
138
228
15.31
7.02
1.46
138
227
6.83
5.47
1.29
137
226
o.a1
7.21
137
232
61
6.29
2.9
141
231
51.8
3.68
2.68
140
230
45.45
6.82
2.51
140
229
36.12
5.28
2.24
139
228
28.6
7.02
2.00
138
227
19.58
5.47
1.66
138
226
8.45
7.21
5.67
137
225
0.13
5.67
137
:108
TABLE 24
Excitation
E::1erg~
Distributions and Initial Fiss!on Fragment Mass
u23 8 + 80 Mev, Protons
Cascade Fissioning Nucleus Bn(Mev .) Ek.
Frequency Mass Excitation
No, A Energ y
(Mev.)
(P, N)
22
55
152
224
238
9
5.39
237
2.5
6,70
= 2'1."
Fission Fragment
(Final Mass No, 136)
Initial Mas s No,
137
L.l
137
238
28
5 .. 39
1,94
138
237
20,67
6,70
1,67
138
236
12,3
5,71
1,29
137
235
5.3
7,02
137
238
46
5,39
2.48
139
237
38.1
6,70
2,27
139
236
29.16
5.71
1,98
138
235
21.46
7,02
1,71
138
234
12,73
4.70
1,32
137
233
6,7
7.54
137
238
61
5,39
2,86
140
237
52.75
6-.70
2,61
140
236
43,42
5.71
2,42
139
235
35,29
7,02
2.2
139
(Cont'd,}
109
TABLE 24 {CQnt'd1 }
Cascade Fissioning Nucleus
Frequency Mass Excitation
No.A En erg y
(Mev.)
(P rN)
26,.07:
234
224
(P 1 2N)
27
Bn(Mev.) Ek
= 2T
Fission Fragment
{Final Mass No. 136)
Initial Mass No.
4 .. 70
1.91
138
233
19 .. 46
7.54
1.63
138
232
9.29
6.42
1.11
137
231
0.66
7.73
137
237
34.7
6.70
2.16
139
236
25.84
5.7
1.87
138
235
18.27
7.0
1.57
138
234
9.70
4.7
1.15
137
233
3.85
7.5
137
(P ,PN)
21
237
236
16
14
7.33
5.29
1.37
137
6.38
0.99
137
237
37
5.29
2.24
139
236
29.47
6.38
2.00
138
235
21.09
5.6
1.70
138
234
13.79
6.71
1.37
137
233
5.71
4.30
o.88
137
232
0.53
7.44
137
(Contd.)
llO
TABLE 24 !Cont'd•)
Cascade Fissioning Nucleus Bn(Mev .) Ek = 2'!'
Frequency Ma.ss Excitation
No.A Energ y
(Mev .. )
(P~r>
63
92
40
238
9
6.06
237
4,.04
5.29
1.1
Fission Fragment
(Finql Mass No. 136)
Initial Mass No.
137
137
238
28
6.06
1.94
138
237
20
5.,29
1 •. 64
138
236
13.07
6.38
1.33
137
235
5.36
5.61
":"
137
238
46
6.06
2 .. 48
139
237
37.46
5.29
2.25
139
236
29.92
6.38
2.00
138
235
21.54
5.61
1.71
138
234
14.22
6.71
1.39
137
233
6.12
4.30
0.91
137
238
61
6.06
2.86
140
237
52.08
5.29
2.65
140
236
44.14
6.38
2.46
139
235
35.30
s.s1
2.19
139
234
27.50
6.71
1.94
138
233
18.85
4.30
1.6l
138
232
12.94
7.44
1.33
137
231
3.17
5_.90
137
e
e
TABLE 25
Most Probable Charge
Th232 + 85 Mev t Protons
Cascade
Nucleus Ï p;ff;, + f:) Formation Fission
Neutron
Frequency
F Probabillty Probability Evaporation
Probabll!tv
Mass No. A-136
ECD
zP
UCD
. (P,.N)..
22
55
152
232
0.34
1
0.34
0.66
52.5
53.6
231
0.40
0.66
0.26
0.40
52.8
53.9
232
0.34
1
0.34
0.66
52.9
54.0
231
0.40
o.66
0.26
o.4o
53.1
54.3
230
o.so
0.40
0.20
0.20
52.9
54.1
229
0.56
0.20
0.11
o.o9
53.1
54.4
232
0.34
1
0.34
0.66
53.2
54.5
231
0.40
0.66
0.26
0.40
53.4
54.7
230
o.so
0.40
o.2o
0.20
53.7
54.9
229
o.s6
0.20
o.n
0.09
53.5
54.7
228
o.62
0.09
o.oss
o.o3s
53.3
54.6
227
o.4s
o.o3s
0.024
0.014
s3.s
54.7
(Cont'd.)
.....
.....
.....
e
e
TABLE 25
r
Cascade
Nucleus rr{f: + fFormàtion
Frequency
n
F Probability
(Cont'd~}
Fission
Neutron
Probab!lity Evaporation
Probability
Mass No. A-136
z
ECD
p UCD
(PTN)
224
232
0.34
1
0.34
0.66
54.1
55.2
231
0.40
0.66
0.26
0.40
53.9
55.1
230
0.50
0.40
0.20
0.20
54.1
55.4
229
o.s6
0.20
o.11
0.09
53.9
55.1
228
0.62
0.09
o.055
0.035
54.1
55.4
227
0.69
0.035
o.o24
0.014
53.9
55.2
226
0.77
0.014
o.o11
0.003
53.7
55.2
225
o.8o
o.oo3
0.0024
0.0006
53.9
ss.s
231
0.40
1
0.40
0.60
53.4
54.7
230
0.50
0.6
0.30
0.30
53.3
54.5
229
o.s6
0.3
0.17
0.13
53.5
54.7
228
0.62
0.13
o.o8
o.os
53.3
54.6
(P,2N)
27
{Cont'd.)
!-r'
.,_,.
N
e
e
TABLE 25 {Cont'd!t}
Cascade
Nucleus
Frequency
--~
rF{(;+
~
fF )Formation
Fission
Neutron
Mass No. A-136
Probability Probability Evaporation
Zp
Probability ECD
UCD
(P ,PN}
21
16
231
o.o7
1
o.o1
0.93
52.3
53.3
230
0.10
o.93
o.o93
0.837
52.5
53.6
229
0.16
0.-83
0.134
0.703
52.6
53.8
231
o.o7
1
0.07
0.93
53.1
54.1
230
0.10
0.93
0.093
0.837
53.3
54.4
229
0.16
o.83
0.134
0.703
53.1
54.1
228
-.20
0.70
0.140
0.560
53.3
54.5
227
0.26
o.56
0.145
0.42
53.1
54•2
232
o.o5
1
o.o5
0.95
S2.0
53.1
231
o.o7
0.95
o.os6
0.88
52.3
53.3
232
o.o5
1
o.o5
0.95
52.4
53.5
231
o.o7
0.95
o.o66
0.88
52.6
53.7
230
0.10
o.88
0.088
0.80
52.5
53.6
229
0.16
0.80
0.128
0.67
52.7
53.8
(P ,P')
22
63
(Cont'd.)
.
.1-:"
1-:"
w
e
e
TABLE 25 {Cont' g.}
ascade Nucleusfp{fn + j) Formation
Fission
Neutron
requency
F Probability Probability Evaporation
Probability
(P ,P')
92
40
Mass No. A:.l36
z
ECD
P UCD
232
0.05
1
o.o5
0,95
53.2
54.3
231
0.07
0.95
0.066
o.88
53.1
54.1
230
0.10
o.88
o.088
o.8o
53.3
54.4
229
0.16
o.8o
0.128
o.67
53.1
54.1
228
0.22
0.67
0.145
o.52
53.3
54.5
227
0.31
0.52
0.161
0.36
53.1
54.2
226
0.36
0.36
0.13
0.23
53.3
54.5
232
o.o5
1
o.o5
0.95
53.7
54,7
231
o.o7
0.95
0.066
o.88
53.4
54,5
230
o.1o
o.88
o.o88
o.8o
53.7
54.7
229
0.16
o.8o
0.128
0.67
53.5
54.5
228
0.22
0.67
0.145
0.52
53.3
54.5
227
o.26
0.52
0.161
0.36
53.5
54.6
226
0.36
0.36
0.13
0.23
53.3
54•5
225
0.45
........
0.23
0.10
0.13
53.5
54*7
(Cont'd.)
1
......
,b.
e
e
TABLE 25 (Cont'd1 )
Cascade Nucleus
Frequency
Compound
Nucleus
520
rp{rn + rp) Formation
Maas No. A-136
Fission
Neutron
Probability Probability Evaporation
zP
UCD
Probability ECD
233
0.28
1
0.28
o.12
54.2
55•4
232
0.34
0.72
0.24
0.47
54.2
55.5
231
0.40
0.47
0.19
0.27
54.3
55.5
230
o.so
0.27
0 .. 14
0.14
54.0
55.4
229
o.s6
0.14
o.. o8
0.06
54.2
55.5
228
0.62
o.o6
o.o4
o.o2
54.0
55.4
227
0.69
o.o2
o.o2
o.oo7
54.3
55.6
226
0.77
o.oo7
o.oo5
0.002
54.1
55.5
225
o.ao
0.002
0.0016
o.ooo4
54.3
55.8
224
0.85
0.0004
0.0003
0.0001
54.3
55.8
Gross Wèightéd A'vètage "Zp · -
53.8
54.8
-
Experimental ZP
-
54.6
t-!..
~
(.Il
e
e
TABLE 26
u23S + 80 Mev. Protons
Cascade Nucleus
Frequency
rp;frn. + /F)Formation
Fission
Neutron
Probability Probability Evaporation
Probability
Mass No .. A-136
ECD
zP
UCD
(P,N)
22
55
152
238
0.56
1
o.56
0.44
52.2
53.6
237
0,.66
0.44
0.29
0.15
52.4
53.7
238
0.56
1
0.56
0,.44
52.6
54.0
237
0.66
0.44
0.29
0.15
52.9
54.1
236
0.71
0.15
0.106
0.44
52.7
54.0
235
0.77
0.044
0.034
0.010
52.9
54.2
238
0.56
1
0.56
0.44
53.1
54.4
237
0.66
0.44
0.29
o.15
53.3
54.5
236
0 .. 71
0 .. 15
0.106
0,.044
53.1
54.4
235
0.77
0.044
0.034
0 .. 010
53.3
54.6
234
0.83
o.cno
0.0083
0.0017
53.1
54.5
233
0.95
0.0017
0.0016
0.0001
53.3
54.6
(Cont'd,.)
:~
......
O'l
e
e
TABLE 26 (Cont' d.}
Cascade Nucleus
Frequency
(P ,N)
224
rpl{ rn+ rF )Formation
Fission
Probability Probability
Mass No. A-136
Neutron
Evaporation
Probability
ECD
zP
UCD
238
0.56
1
o.56
0.44
53.4
54.8
237
0.66
0.44
0.29
0.15
53.6
54.9
236
0.71
0.15
0.106
0.044
53.4
54.8
235
0.77
0.044
0.034
0.010
53.6
55.0
234
0.83
0.010
0.0083
0.0017
53.5
54.9
233
0.95
0.0017
0.0016
0.0001
53.6
55.1
232
0.98
0.0016
0 .. 0015
0.0001
53.5
55.0
237
0.66
1
0.66
0.34
53.3
54.5
236
0.71
0.34
o.24
0.10
53.1
54.4
235
0.77
0.10
0.077
0.023
53.3
54.6
234
0.83
0.023
0.019
0.004
53.1
54.6
233
0.95
o.oo4
o.oo3
0.001
53.3
54,6
(P, 2N)
27
(Cont'd.)
.,.:..
.....
-....!
e
e
TABLE 26 (Cont'd.)
~ascade Nucleus
requency
(P 1 PN)
21
16
f pl(fn + f p)Forrnation
Fission
Neutron
Mass No. A-136
Probabillty Probability Evaporation
ZP
Probability ECD
UCD
237
0.25
1
0.25
0.75
52.0
53.2
236
0.33
0.75
0.25
0.50
52.4
53.5
237
0.25
1
0.25
0.75
53.0
54.0
236
0.33
0.75
0.25
0.50
52.8
54.2
235
0.40
0.50
0.20
0.30
53.0
54.1
234
0.50
0.30
0.15
0.15
52.8
54.0
233
0.56
0.15
0.08
0.07
53.0
54.1
232
0.66
0.07
0.04
0.03
53.2
54.2
238
o.ao
1
0.20
0.80
51 .. 7
53..0
237
o.2s
0 .. 80
0 .. 20
0.60
52.0
53.2
238
0.20
1
0.20
0.80
52.4
53.4
237
0.25
o.8o
0.20
0.60
52.6
53.6
236
0.33
0.60
0.20
0.40
52.8
53.5
(P 1 P')
22
63
235
0.40
0.40
0.16
0.24
52.6
53.7
234
o.so
0.24
o.12
0.12
52!8
54.0
(Cont'd.)
;
1
......
. ......
Q)
e
e
TABLE 26 {Cont'd.)
Cascade Nucleus
Frequency
92
40
ff"ifn + fF) Formation
Mass No. A-136
Fission
Neutron
Probab!lity Probability Evaporation
zP
Probability ECD
UCD
238
0.20
1
0.20
o.8o
52.8
53.8
237
0.25
o.8o
0.20
0.60
53.0
54.0
236
0.33
0.60
0.20
0.40
52.8
54.1
235
0.40
0.40
0.16
0.24
53.0
54.4
234
0.50
0.24
0.12
0.12
52.8
5461
233
o.56
0.12
0.07
0 .. 05
53.8
54.2
232
0.66
o.
o.
o.
238
0.20
1
0.20
o.8o
53.2
54.2
237
0.25
o.8o
0.20
0.60
53.4
54•5
236
0.33
0.60
0.20
0.40
53.2
54.2
235
0.40
0.40
0.16
0.24
53.4
54•5
234
o.• 50
0.24
0.• 12
0.12
53.2
54.4
233
o.56
0.12
o.o7
o.o5
53.4
54.5
232
0.66
o.q~.
o.o3
0.02
53.2
54,2
231
0.72
0.02
o.o1
o.o1
53.4
54.5
(Cont'd.)
1
.....
.....
w
•
e
TABLE 26 (Cont' d.l
Cascade Nucleus
Frequency
Compound
Nucleus
r;_!;{r::: + r; ) Formation
n
Neutron
Mass No. A-136
Fission
F Probability Probability Evaporation
zP
UCD
Probabili tv ECD
239
o.so
1
0.5
0.50
54.0
55.1
238
0.56
0.5
0.28
0.22
53.8
55.1
237
0.66
0.22
0.14
o.oa
53.8
55.3
236
0.71
o.oa
0.056
0.024
53.8
55.2
235
0.77
0.024
0.019
0.005
54.1
55 .. 4
234
0.83
o.oos
0.004
0.001
54.0
55.3
233
0.90
0.001
0.0009
0.0001
53.8
55.2
232
0.97
0.0001
0.0009
0.00007
53.6
55.1
231
1.03
0.00007
o.ooo7
o.
54.0
ss.4
Gross We!ghted Average ZP
53.4
54.6
-
-
Experimental ZP
54.,4
........
N
0
12Œ>
Compuison af Qüculated and Elxi>t:rimental Values of Zp
ln arder to focilate comparison wah ex:pedment the calculated and
experimental Z values are Hsted in Table 26b.
p
The maximum unce::-to!nity in the estimation of the prlmary fragment
moss numbeu- is estimated to be about :::. ! moss unit. This would alter the
calculated Zp values by ·àbout .-: OA charge unit for bath UCD and ECD calculations.
in addition an erron- of;: 0.5 charge unit is assigned to ZA values in the case of
ECD calculations. This makes an uncertainity of about .i 0.63 charge unit in ECD
values. For the experimental values of Zp an errer of± 0.25 charge unit is estimated
for E E; 50 Mev and.±. 0.5 charge unit for E ~50 Mev.
TABLE 26b
Comparison of Calculated and Experimental ZP values.
Target
Th232
u238
Energy
(Mev)
'
zP
Correlated
ECD
UCD
20
52.8
53.9
52.3
53.5
50
53.4
54.7
51.7
54.2
57
53.5
5A.8
51.5
54.4
85
53.8
54.8
-
54.6
20
52.4
53.7
52.5
53,3
50
53.2
54,5
52.0
54.2
85
53.4
54.6
·-
Note: Column No. 5 (Zp Correlated) is explained later.
~
Experimental
54.4
121
În view of the large experimental errors and the uncertainty in
the cal culated values of Zp , it is not plausible to draw any defini te
conclusions. However sorne general trends could be noted from Table 26b.
The experimental values of Zp appear to be between .ECO
and UCD values for both uranium and thorium up to about 57 Mev. At
80 and 85 Mev these values are closer to those predicted by UCO thon ECO.
However, at these energies the charge distribution curves become mu ch
broader and ther.efore the uncertainty in the peak positions·(defining Zp }
becomes much larger. This makes it difficult to discriminate correctly
between the two postulates.
As the excitation energy increases, both the ECO and UCO
values of Zp increase. This increase is more rapid in the energy region
20 to 50 Mev thon 50 to 85 Mev. The calculated Zp values for thorium
fission are si ightly larger thon those for uranium fission. These results
are in conformity with the trend of the experimental data.
The charge distribution has been experimentally studied by
Aagard et al (87b} for moss number about 130 in 170 Mev proton fission of
uranium. They find thot the most probable neutron to proton ratio is 1.49.
This value falls between the UCO and ECO values calculated by Rudstam
et al (87a)
Çolby and Cobble (82) have observed thot their data on 2Q-40
~233
235
238
, and U
correlated better with
Mev He ion induced fission of U 1 U
UCO rule than with ECO.
l
'
122
Pate{81) used detailed calculations which have been made both of the prompt
initial nuclea r interaction(l19) and of the evaporation process as it would occur in the
absence of fission competition(l20). At this stage two different situations were considered
to incorporate fission-spoliation competition:
i. Fission takes place after complete de-excitation by neutron emission
) •e • ,
lF/ (r: + Ïr=)
is more dependent on en erg y th an on nu cl ear ·
;
lype'.(Z~A) •
ii·
rf:/(ln +
JF)
increases sharply from zero to unity as
Z~A
ratio is raised through sorne critical value (Zf:A = 36), i.e.,
rFA rn + rF )
is essentially independent of energy.
For both situations the ECD and UCD postulates were applied after the emission
of 2.5 fission neutrons for each fission act. Thesze
cal cul at ions indicated thot for energies
up to 480 Mev fission proceeds predominantly via a process described by the ECD postulate
either during or at the end of de-excitation in various nuclides with z2/A greater thon~ 36.0.
Since in low energy data Z fA =36 corresponds to a
[";;1(
fn
+f;:):valuè çf
O.p,: ·i·t<wœr::.:cr:
argued thot in nuclei with Z 2/A greater than 36, fission competes effective! y with
parti cl e emission in a fash .ion inde pendent of energy.
However, it must be pointed out thot both in the treatment and the trend of the
results the present analysis differs from thot of Pate {81) on proton fission of thorium. ln the
present treatment the energy independence of ( (f /(fr,+ rF) was assumed at the very
outset
•
and
the
p~u b. 1. j
sh e d
v a 1u e s
of
rF / ( ~
+
fF )
123a
were incorporated at every stage of neutron evaporation after the prompt
interaction. For each fissioning nucleus the calculated Zp values were
weighted with the fission probabil ity and the gross weighted average Zp,
was determined using the relative frequencies of each cascade process.
The fission neutrons (V.:::: 2 ) were assumed to be em itted at the moment
of fission, one from each fragment.
For low and medium energy fission Coryell et al (69) have
proposed a method which involves th-a use of zp• s from thermal neutron
fission of U
235
as reference values for correlation with other types of
fission. This procedure is based on the difference of Zp' s for the two types
of fission calculated by the ECD postulate. This method avoids the need for
explicity calculating ZA, which relies on assumptions about mass surface,
and the need for details about neutron emission. rhey have derived the
following equations:
=
=
IV. 3. 6
0.5
(2f -
Zf
+ 0 • 19 ( v T Superscript
11
ref.
vr
ref.
) -0.21 (AF- Af
)
ref.)
IV. 3. 7
ref' 1 apply to values from the thermal neutron fission of U 235. AF
is the mass number of the initial compound nucleus, and vT is the average
number of neutrons per fission (including both pre and post fission neutrons)
Table 26 (d), column 5, gives the Zp values calculated
according to this method for U and Th up to 57 Mev. The vr values were
estimated on the assumption that for a given excitation energy the total number
of evaporated neutrons ( prior to and after fission) is approximately independent
123b
of when fission actually takes place. This is due to the fact that neutron binding
energies concerned are comparable. The values of vT were then taken from
Table (18). As before it was assumed that one prompt neutron was also emitted
by each fragment. The
v~ef.
was taken as 2.5 (15).
From Table (26d) it is evident that the Coryell correlation method
is not successful for either
238
u
232
for the energy region 20-57 Mev 1 as
or Th
the data of Colby and Cobble (82) on 40 Mev helium ion induced fission of
uranium isotopes would also indicate
*
*.
This has been pointed out by Wolfsberg in his recent paper (Phys. Rev. 137,
B929, 1965) which appeared after completion of this thesis.
124
•
IV.4
lsomer Ratio
IV .4. 1
Stotement of the Probl em.
Figure 22 shows the decoy scheme of Cs 134m (113). From on
exominotion of the relative
·shielded nuclide Cs
134m
population of high and low spin isomers of the
as given in Figure 23, it oppeors thot the high spin
·meta stable stote (8-) is fovoured at ali energies relative to the low spin groun~ stote(4-+}.
These results ore in agreement with the results
on the isomerie poirs.
. 95g
Nb - -
No95m and
of a s t u d y by H a g e b o (1 03)
117g
ln.
117m
- ln
.
obtoined from the fiss1on
of Pb, Bi, Th, U, induced by 20-157 Mev protons. The dota on the isomer ratio
Br SOm/ Br SOg reported by Haller and Anderson (1 04) in proton induced fission of
uranium ore olso in agreement with these results. Between 60 and 80 Mev of
incident proton energy, the results of Devies and Yoffe (105) on the esl34
isomer ratio from
u238
are in IOirly good agreement with the present results, but
the much larger ratios of the high spin to the low spin isomer reported by them
at lower energies (20-60 Mev) could not be observed in the proton induced
fission of thorium. lt must be. pointed out 1 however, thot sizeable errors are
introduced in the determination of cross-section values of Cs
134
isomers at low
energies due to extremely low cross-sections for formation of this nu cl ide.
A formol ism bosed on the statistical madel for cal culoting
isomer ratios in nuclear reactions hos been developed by Vandenbosch
and Huizenga (97,98).
Using this model an attempt is made to cal cu lote
125
Fig, 22 - Decay scheme of Csl34m
125a
..
.
..
,..,
0
0
•
0
-
w
0
0
•
0
•
•"
_.,
~
-
rt)
-
(/)
<..)
..
.c
m•
(\J
-J
COl
••
•
(\J
- --
+
LO
+
"itl
126
1
Fig. 23 -Ratio of formation cross sections of the
meta stable and gromd states of csl34
as a function of proton bombarding energy.
126a
5
T
o~--~----~--~----~----~--~----~--~
10
20
30
40
ENERGY
50
(Mev)
60
70
80
127
the isomerie yield ratio csl3 4m;cs 134g resulting from proton induced
fission of thorium. The computations were performed with the IBM 7040
computer at McGill University using a program provided by the above
authors (126).
The energy range considered is restrlcted to 19 to 24 Mev.
due to lack of availability of proton transmission coefficients beyond this
range. Assuming compound nucleus formation. the initial spin distribution
is first calculated. Then the spin distribution following neutron emission
is determined. After this stage, the calculation of the spin distribution of
the fission fragments is made on the ba sis of the assumption that the spin
distribution of the fissioning nucleus is preserved intact in the fission fragments. The latter then de-excite by neutron and gamma emission to give the
final spin distribution. Two different calculations are performed;
1.
assuming fission before neutron evaporation and,
2..
fission.
occurring after complete de-excitation by
neutron emission.
In both cases the emission of 2 'fission neutrons' is assumed which are
equally divided between the two complementary fragments at the moment of
fission. S!nce there was no agreement of the experimental data with either
case it was decided to calculate what initial spin distribution of the primary
fragment is required to account for the observed isomer ratio and compare
with that of cases (1) and (2) above.
12.8
IV,4.IJ: Theorv of the Calculation
(a) Initial BRin Distribution of the Compound Nucleus Pa233
Since a h!gh energy projectile can bring into a nucleus various
amounts of angular momentum the compound nucleus produced in such
reactions contains a wide distribution of spins. The formation crosssection for a compound nucleus of spin J0 produced by a projectile of energy E
1s given by the equation (57, 12 7} :
2
U(Jc'
E} = 1f
~
~ L_
S
\~· h
=1 I - s/
+$
~L----=
l =[Je - S 1
2~ + 1
T,t(E)
(2 s + 1)(2! + 1)
IV.4.1
~ •
de Broglie wave1ength of the incoming projectile
1
=
s
= intrins!c
S
= Channel spin = s + I
Tl(E)
=
intrinsic spin of the target nucleus
spin of the projectile
( Vector sum )
batTier transmission coefficient for a projectile of angular
momentum
1. and energy E.
The maximum angular momentum of the. system is the vect or sum:
J0 (max) = ~max+ s + 1 and !s either integral or half
integral depending upon the intrinsic spins,
J (max) 1s half integral fer the system (Th2 32 + p) since 1 = 0
c
and s
=t.
129
The probabil!ty P(Jdfor the compound nucleus to have spin
J0 is given by:
00
PUe) =ô(Jc' E ) / L ()(Je' E)
:t
=0
The proportionality constant 11~ 2 !s set equal to one since
we are interested mainly in a normal!zed spin distribution P (Je)• However,
when ·the ab solute cross-section as a function of Je is des:ired (Fig. 24) the
-n-'1;.2
actual value of
is put in.
rrr~ 2 is expressed in units of lo-27 cm 2
so that the partial cross-sections appear in uni ts of millibams.
(b} Spin Dist.'"ibution Following: Neutron Emission
This distribution depends upon two factors:
(1) the density of available levels
of the product nucleus
(2) the amount of angular momentum car.ried away by the neutron (l') •
The relative probability for a prlmary fragment with angular momentum ]0
to emit a neutron with orbital angular momentum l' and spin s't=:i) leading
to a final state with angular momentum }fis given by:
p(1F
)
1
=
p(JF)
Ir+ s'
)
s
~
J0 +
T J...• (E)
~s'\i9F=sl
where T1, (E} is the transmission coefficient of the emitted particle of
energy E and angular momentum l'.
fUt) is the dens:lty of levels of spin ]ff
and is predicted theoretically to have the form (128 1 129) 1
130
IV .4.4
where dis called the spin eut off factor, and is given by the relationship (130)
IV.4 .. 5 .
...Jis the nuclear moment of inertia
T is the nuclear temperature.
The normalized spin It produced from an initial spin J0 is the
product of the initial normalized spin distribution of J0 and the probability
for an initial state J0 to emit a neutron leading to a final state of spin Jf•
The total normalized yield of Jf is computed by summing over all values of
Parameters Necessary; for the Calculation
1. Transmission Coefficients
(a) Protons
The angular momentum brought into the system by the incoming
projectile and the associated transmission coefficients T(l) are obtained
from the square weil coefficients tabulated by Feshbach et al (131) using, .a
radius parameter of 1. 5
fermit~
(Table 27).
(b) Neutrons
The neutron transmission coefficients used in this work are the
square well coefficients of Feld et al (132). The coefficients are given
•
in graphical formas a function of the parameter X which is defined in terms
of the nuclear radius Rand the neutron kinetic energy En:
131
TABLE 27
Proton Transmission Coefficients
.
E
1
19, Mev.
0
0.799
0.89
1
0.799
o.88
2
0.758
0.865
3
0.698
0.837
4
0.604
0.790
5
0.452
0.725
6
0.276
0.599
7
0.127
0.430
8
0.381 x 10-l
0.236
9
0.770 x 10- 2
0.977 x 10-l
10
0.134 x 10- 2
o.256 x 10-1
11
0.190 x 10-3
0.53 x 10- 2
24 Mev.
-3
12
0.500 x 10
13
0.560 x 10- 4
132
x= o.22R
{En
A nuclear radius parameter of 1.5 fermis was used in the present
calculations ..
The energy distribution•d neutrons emerging from an excited
nucleus is predicted theoretically to be of the fonn (57)
N (En) 0( En
exp~ { -
En/'1')
IV.A.7
where N(En) is the relative number of neutrons of kinetic energy En and'(
is the nuclear temperature of the residual nucleus defined by:
1/1'
r
=
dln{f (A, E)) / dE
IV .4.8
·(Al E) i"s .the leve! density and is related to the excitation energy and
the number of nucleons within the nucleus. It is approximated by the
expression (57)
f (A
1
E) = c exp (2../i.E)
c and a are constants dependent upon the mass number A. Substituting
Equation IV.4. 9 into :~quation IV .4. 8 and carry.ing out the d!fferentiation
yields
T=/Ë/i
IV .4.10
T !s evalu"ated at the maximum excitation energy of the residual nucleus
(~)and
E#E#compound nucleus - Bn
r -
IV .4.11
where Bn is the binding energy of the neutron. From the energy di.str!bution
of
~quation IV .4. 7
Ek
=
we obtain (5.7)
2'1'- the average neutron energy
IV .4.12
i33
The constant a is not unambiguously defined. Recent work
indicates its value lies in the range A/12 to A/ 8 Mev. -1 (133-136). Bishop (136)
has shown that the calculation is quite insensitive to this parameter. The
calculations done in this work use a value of A/8 Mev.- 1• Neutron binding
energies were taken from those ofrRiddell (711). :The T(l) values are given
in Table 28.
For the primary fragment it has been estimated that not more
than 2 neutrons are likely to be emitted and their average kinetic enetgy
is estimated to be abru t 2 Mev. (22). The transmission coefficients of
the neutrons em!tted from the primary fragments are given i!l Table 29 •
The use of the average value of kinetic energy (En) instead of
the kinetic energy spectrum has been found to be a good approximation in
calculations of isomer ratios (97 ,98,136).
The Spin Cut-off Parameter
Ericson (130) has shown that the sp!n eut-off parameter is
related to the moment of inertia of the nucleus
through the relationship
given in Equation Iv.4.5 wherecf is the nuclear temperature defined by
Equation IV. 4.10.
,j
rigid
As suming j to be tha t of a rigid sphere:
=~
mAR 2
IV .4.13
where rn is the nucleon mass, A is the mass number of the nuclide in
question, and Ris the nuclear radius.
134
TABLE 28(a)
Neutron Transmission Coefficients
(Th~32 + 24 Mev, P,)
t/ F,:k (Mev.)
N1
1.78
N2
1,51
N3
1.16
0
0 .. 74
0,72
0,70
1
o. 72
0,66
0,62
2
0,60
0.52
0.48
3
0,36
0,28
0,22
4
0.11
0,07
0,04
5
0.,013
o.o8 x 10-1
-
TABLE 28(b)
Neutron Transmission Coefficients
{Th 232 + 19 Mev 1 P}
R, j&}.Név.)
Nl
1.6·
N2
1. 3
N3
0.83
0
0,74
0,70
0.68
1
0.70
0,62
0,60
2
0,60
o.so
0,42
3
0,35
0,25
0,15
4
0,11
o.s4 x Io- 1
0,022
5
0,013
0,04 x 10-1
-
135
TABLE 29(a)
Neutron Transmission Coefficients ·
(Pr-lmary Fragment )
Cs
Ek (Average)
135·
= ...
')
Mev.
1..
T
0
0.70
1
0.62
2
0.48
3
0.24
4
0.04
TABLE 29(b)
Neutron Transmission Coefficients
(P!"imary Fragment' )
Gsl36
Ek (Average) = 1.5 Mev.
i
T
0
0.64
1
0.55
2
0.36
3
0.12
4
0.,01
1.36
Hence
IV .4.14
A number of investigations (97, 98, 101, 133, 136) have shown a value of
about 0.3 to
o.6
of the rigid body value is necessary in order to obtain
agreement between experimental results and calculations. In the present
calculations
values were set at 4 t 1 which corresponds to about 0.3
times the rigid body value (97 • 98, 136).
The results of the calculation of the spin distribution of the
compound nucleus before and after neutron emission are plotted in Fig s .• (25-30)
for
19 and 24 Mev. protons. The root mean square angular momentum of
the compound nucleus be fore and after neutron emission is given in Table 30 ..
In fission the angular momentum of the compound nucleus can
appear in two places:
(a) as orbital angular momentum between the fragments
(b) as intrinsic spin of the fragments.
Angular distributions (22, 137 1 138) !ndicate that the orbital
angular momentum between the fragments is small so that 1t is likely that
(b) is of primary importance. This fact should be reflected in the isomer
ratio.
Schematically the primary fragments have a particular spin
distribution and decay as follows:
Primary fragments
I Distribution
137
Fig. 24- Calculated cross section for compound nucleus :formation
as a function of the resultant spin ·in the compound
nucleus formed in the bombardment of Th232 with
19 and 24 Mev. protons.
137a
232
Th
+ p
240
200
160
CJ(mb)
120
80
40
0
2
8
10
138
Fig. 25 - Change ,i.n the probability distribution of
spins due to prefission neutron emission.
Energy of bombardment :::. 19 Mev.
Value of spin eut off f~or a-= 3
.
0
•
Jr~
0
Nn
= 0
4.47
---X
Nn
=1
4.05
---•Nn
= 2
3.75
.
/·
.20
1
1
1
.15
1
1
1
P{J)
1
1
.10
J
t
.
.05
1
1
e
i
0
2
4
J
6
8
10.
139
Fig. 2 6 - Change in the probability distribution of
spins ·due to prefission neutron emission.
Energy of bombardment 19 Mev.
Value of spin eut off factor 0" = 4.
Jj2
<)
•
---o
Nn = 0
---X
Nn
=
1
/
4.47
4.5 0
4.52
1
139a
•
)
!
i
1
t!
0.20·-1
1
'1:
''~
1
l
i
'
h
0.15;-
1.
'
!
$
1~
~
1i
1
P{J)
i
1
1
1
O.IOr
1
1
o.osl
1
:' 0 .........._-;0~-~2:--...J4:----~6---8L--...JIO~-J
J
140
Fig. 2 7 - Chang.e on the probability distribution of .
spins due to prefission neutron emission.
Energy of bombardment = 19 Mev.
Value of spin eut off factor 6 = 5.
jJ2
0
0
Nn
=0
4.47
x
x Nn
4.70
•
•
=1
=2
Nn
4.90
.
.20
.15
P(J)
.10
.05
2
4
J
6
141
Fig. 28 - Change in the probabiltty distribution
of spins due to prefis sion neutron emis sion.
Energy of bombardment = 24 Mev.
Value of spin eut off f p (}' = 3.
0
0
Nn
=
0
x
x.
Nn
=
1
4.69
•
•
Nn
=
2
4.24
.. 20
'
.16
P(J) .12
.08
.04
0
2
4
6 .
J'
8
10
12
142
Fig. 29 - Change on the probability distribution of
spins due to prefission neutron emission.
Energy of bombardment = 24 Mev.
Value of spin eut off fac~ a= 4.
.
=
" J
0
0
N
0
5.4 7
x
x
Nn = 1
5.20
•
•
Nn
n
=
2
5.10
.20
.16
.12
P(J)
.08
.04
J.
o~~~~--~--~--~--~~~
0
2
4
6
J
.
8
10
12
143
Fig. 30 - Change in the probability distribution
of spins due to prefission neutron emission.
Energy of bombardment = 24 Mev.
Spin eut off factor (J" = 5.
jJz
0
o Nn
=
0
s. 4 7
x
i. Nn
=
1
5.50
•
• Nn
=
2
5.52
.20~--------------------------------~
.16
.12
P(J)
.OB
.04
Q.
2
4
6
J
8
10
12
144
TABLE 30
Root Mean Square Angular Momentum of Fission!ng Nuclei (Th 2 32 (P 1 11f)}
En erg y
~
19 Mev.
24 Mev.
'~
3
~rr
4
5
3
4
5
0
4.47
4.47
4.47
5.47
5.47
5.47
1
4.05
4.50
4.70
4.69
5.20
5.50
2
3.75
4.52
4.90
4,.24
5.10
5.52
stands for spin eut-off parameter
stands for number of neutrons emitted 1 before fission.
At this stage, it was assumed that the angular momentum
distribution of the fissioning nucleus is preserved in fission. Two
extreme cases of fission were then considered:
1. Fission takes place before any de-excitation by neutron emission
2. Fission occurs after complete de-excitation by neutron emission.
About 2' 1fi'.S'Sion neutrons 1 are assumed to be emitted (15)
which are equally divided between the two fragments. Now the deexcitation of the primary fragments by the emission of neutrons and gamma
rays is taken into consideration.
The distribution after the emission of the first neutron is
obtained as follows. For every half integrat value of Ic in the original
distribution the relative probabilit!es P(Jf• Ef) are computed as before
using Equation IV.4.3 for all possible Jf values. For subsequent neutron
evaporation the calculations are repeated for every value of Jf•
The spin distribution follow!ng gamma ray emission is
calculated on the assumption that no gammas are emitted unt!l part!cle
evaporation is complete and that the probab!lity of decaying from a state
Ji to }fis proportional to the density of states of spin Jf• Considering
only d!pole radiation, the relative probabilities for a nucleus of spin Ji
to decay to states w!th
Jt-
1,
1t and 1t + 11
is determined by the leve!
density factor given in equation IV.4.4 provided J1 1
o.
According to
146
this assumption the total normalized yield of state Jf is given by:
r1 + l
D(Jf)
Jfjli-l\
Maximum value of Jf is Jf
max
=Itmax+
l, [
is the multipolarity of gamma
emission (equal to 1). FJt is the normalized initial spin distribution following
the evaporation of last particle from the primary fragment ..
The number (Ny) of prompt gamma rays per fission fragment observed
experimentally is almost four (22). However 1 as Warhanek and Vandenbosch
(101) point out, the observed average of four gamma rays may consist of contribution of those fragments with very high angular momentum which must emit
a large number of gamma rays to reach a final state. The present calculations
were performed for Ny = 3 and 4 ..
It is assumed that, upon emis.sion of the next-to-last gamma ray,
only two states are available for population, these states being the high and
low spin states of Cs134 , and that the isomer populated is that involving the
least spin change. The results of the calculation are summarized in Table 31 ..
The calculated values do not exceed 0.2 while the experimental value is about
1.2.
(Table 14).
We shall new proceed to estimate what value of the average angu-
lar momentum of the primary fragments is required to reproduce the observed
isomer ratio.
i41
TABLE 31
cs134.~/.cs134g - lsomer Ratio
{a)
.F.J.yion B,èfcq,J.:.e:excttatjgn_
...
~-~-"
.
--·
Energy =+19 Mev.
H
. J
= 4.47
cl
.
Nn
2
(b)
=4
~'(
'
....L
3
0.19
0.211
experimental
isomer ratio 1.2
Fission After De-excitation bv Emission of 2 Neut:'ons.
Jf2
r::l
Nn
= 4.5
= 4
NI-(
.
1
N
n
=
_L_
0.20
4
-
$cperimental
fsomer rat!o
·-
1.2
number of prompt neutr'ons emitted from p..""'!ma."'Y
fragment.
number of gammas emltted.
spin eut-off factor.
148
It is assumed that the primary fragments are formed with an
angular momentum distribution which has the same functional form (I 01)
as that of the nuclear leve! density:
P{J) = (2J + 1) Exp-J (J + l)
2b2
where bis a free parameter which we want to determine. This distribution
has a root mean square angular momentum (
ff
= b J2}.
Four different
guesses were made for the values of b. In trial 1 1t was aasumed that
j12 = 2. 64 which is about half that of the compound nucleus (Table 30)
1
as a check on the procedure. In the subsequent trials the values of
/J2 ranged from two to three. times that of the compound nucleus.
The range
of values of (f was 4 t 1 and calculations were performed with Nn
=1 and
Nn = 2 for the primary fragment. In each case emission of three as weil
as four gammas were considered. The results are tabulated in Table 32.
Figure 31 shows the spin distribution P{J) for
2 neutrons fot different values of
IV.4.1II
ff =
10 with emission of
cr ( = 4 ± 1).
Interpretation of Results
Any one of the different combinatibns of parameters for the primary
fragments as given in Table 33 gives fairly good agreement with the experimental resulta.
These calculations, although approximate, indicate that distributions with much larger root mean square angular momentum than that
of the initial compound nucleus have to be assumed in order to explain
Ï49
TABLE 32
I som er Ratio
o'cs134m;d cs13.4g
ff = 2.64
Triall
...
'
Nn
.... N~
tT-:-> 3 ..
'
4
.....
s.
~p~i~ental
Value
1
3
0.012
0.016
.
o.o53
1
4
0.013
0.017
0.066
2
3
-
0.033
0.094
2
4
-
0.041
0.108
'
""'1.2
"
Trial 2
"ff= 7
1
3
0.278
0.513
0.677
1
4
0.239
0.488
0.669
2
3
0.204
0.470
0.672
2
4
-
0.447
0.666
Trial 3
..
.,....., 1.2
ff=9
1
3
0.579
0.968
1.24
1
4
0.479
0.908
1.21
2
3
0.430
0.848
1.18
2
4
0.354
0.798
1.15
.....,1.2
{Cont'd.)
....
150
TABLE 32 (Cont'd,)
Nn
Ny
Trial4
3
if-,
ff=
10
4
5
Experimental
Value
1
3
0.773
1.,25
1.59
1
4
0.666
1.17
1 .. 54
2
3
0.550
1,08
1.46
2
4
0 .. 472
1.02
1.42
r:5
Spin eut-off factor.
Nn
Number of prompt neutrons emitted from primary fragment.
N't
Number of prompt gamma s.
..-1.2
Fig. 31 - Change in the probability distribution
of spins due to prompt neutron emission ..
· 31 (a) spin eut off factor 0" = 3
31 (b) spin eut ofLfactor cs = 4
31 (c) spin eut ofLfactor 0' = 5
0---0
Distribution calculated using
Eq. (IV. 4 • 16) with f5
.
= 10
·
x---x
·---·
JI-
Calculated distribution of
spins after the emission of one neutron.
Calculated distribution of spins
after the emission of the neutrons.
151
0
IJ
rn
.
_,
.
<")
co
tJ)
-ri
J:.'-1
0
co
0
--a.
'J
<.0
q
(\,1
q
152
-.
t\1
0
-.
ro
q
-1-:>
a.
<D
0
..
v
0•
t\1
0
.
153
0
tl
.--1
t-:>';
til
co
'"(,..
C\J
0
0
0
--:
co
0
0
.
tO
q
--a....,..
0
v
0
0
C\J
q
0
0
•t-l
~
154
TABLE 33
ComEarison of Calculated and Experimental Values of cs134 Isomer futio
~~"""'
Nn
Ny
r·'
'
~2
..
-- ----.
....
__
,
______ ... _...,______
-
.,.._... _,..,
Gmfûg
Ûm jû g
Calculated
Experimental
1
'
3
4
10
1 ~25
1
4
4
10
L17
1
3
5
9
1.,24
1
4
5
9
1.,21
2
3
5
9
1 .18
2
4
5
9
1.15
2
3
4
10
LOS
l'ln
=
Number of neutrons emitted from the primary fragment,
Nv
=
Number of prompt gammasJ
=
Hoot mean square angular œ.omentum"
=
Spin eut-off parame ter~
=
Cs134 Isomer ratioo
i
R
r-'
·.J
Gr::
,.._,1,2
lSS.
the observed isomer :ratio at 19 to 24 Mev. It would appear therefo:re that
angular momentum is somehow generated du.'ll'!ng the fission p:rocess itself.
That is 1 the total angular momentum is of course conserved, but the fragments can each be spinning in opposite directions so that even though the
total spin is small, the individual fragment spins are large. This is
illus·-~
trated in tre vector triangle given in Figure 32,. The vector addition. accou::tts
for the high spin primary fragments p:roduced from a low sp!n compound
nucleus.
FIGURE 32 - CoupHng.of An9ula:r Momentum
Jf' Jf 1 = spins of fragments
Je
= spin
of fissioning
nucleus
This opposite allgnment of spins of the primary fragments may be explained
in terms of the following physical
p!ctur~, (102 1
139). It may be assumed
1
that the primary fragments lmmediately after scission present sorne axial
asymmetry along the separation Une. Then the transverse component of
the coulomb repulsion may cause :relatively large
in~..nsic
spins opposite
in direction and perpend!cula:r to the direction of motion of the fragments (140
141 )
FIGURE~3
1
(a)
'·
(b)
Axial Symmetry
Axial Assymetry
156
The rotational energy
Er
=
J(J2 +}
112
IV.4.17
(J
associated with the root mean square angular momentum
of the primary fragment is about 1 Mev.
.../fZ
= 10
This is not unreasonable in
view of the fact that the magnitude of the total excitation energy taken
away from each fragment by gamma emission is about 3 to 4 Mev.
As a number of measurements of the Cs
134
!somer ratio in
fission have been reported, it seemed of interest to correlats the results
with the angular momentum of the compound nucleus. A simple calculation
can be performed on the assumption that the angular momentum of the
compound nucleus is equal to the orbital angular momentum brought in by the
incident particle. The latter can be estimated by tl'e sharp eut-off
approximation, giving
j
c
2
~·(fc.m 1'12
B)R2
whereftis the reduced mass, Ec.m is the energy in the centre of mass
system, Ris radius of the compound nucleus, and B the coulomb bamer.
The value of the radius parameter was taken as l.Sf. These values of
Jj~ 2
are given in Table 34 together with the values of the isomer ratio
cs134 m/Cs 134g in react!Ol'l where compound nucleus formation is indicated.
The table includes the experimental results of Da.vies and Yaffe{93), and
Vandenbosch and Warhane~k {lOl)à. Except for the surprisingly large values
157
reported by Davis and Yaffe {l'OS) at low energie.'S, a reasonably good correlation is found between §and the experimental isomer ratio.
TABLE 34
'\
Energ::l {Mev 1 )
Th232 + p
u238 + P
lf""2
" c
•
1
•
dcs134m~sl34g
Isomer Ratio
20
4.2
1 .. 2
45
8.7
1.4
65
11
20
4
45
8.6
Rëfe:-e:1.ce
Number
(Present
work)
3.4
20
(105)
6
65
11
Np237 + d
21
6
1.6
(101)
Np237 + He4
42
11
4.7
(101)"
u235 + He4
42
11
3.3
(101)
4
158
•
v.
SUMMARY
Th 232 was irradiated with protons of energy range 20-.85 Mev.
The independant formation cross-sections of the fission-fragments 30 min
Cs130 1 6.5 d Cs132 1 2.9 hr Cs
134
m1 2.1 yr Cs134g # 13 d Cs 136 1 32 min Cs 138 ,
18 d Rb 86 and the cumulative formation cross-sections of 30 yr csl37 and
17 min Xe 138 were measured radiochemically. These data were used to
construct charge distribution curves at 20, 30 1 39 1 50, 57 1 65 1 75 and 85
Mev. tt was observed that with increasing proton energy these curves broaden
and the most probable charge shifts toward the llne of beta stability. These
effects could be interpreted in terms of fission spallation competition.
Detailed calculations on the basis of ttEqual Charge Displacement" and
"Uniform Charge Distribution" postulates have been attempted for the entire
energy range studied. The previously published experimental data on proton
induced fission of uranium have also been incorporated in these calculations
to make a comparative study. 8oth experimental and calculated values
indicate that the most probable charge of a fission fragment of given mass
number lies a little closer to the Une of beta stability in the case of thorium
fission than uranium. 1t is suggested that this is due to -the lC7Ner neutronproton ratio of the target thorium compared to uranium. The general trends
of the results seem to indicate that the most probable charge of a fission
fragment in the mass region A= 130 to 138 lies between the values predicted
•
by the two empirical postulates •
159
lt was also possible to determine the ratio of the independant
formation cross-sections of the shielded isomerie pair cs 13 4m/C s134g
in the se experimenta. Theoretical calculations showed that at about
20 Mev. the calculated angular momentum of the compound nucleus is not
large enough to account for the observed isomer ratio. The primary fragments
appear to possess large intrinsic spins in opposite directions.
160
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