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'iECURITY CLASSIFICATION OF THIS PAGG (*hm /'«I» HnfaraifJ ä> iiäJsii REPORT DOCUMENTATION PAGE

1 REPORT NUMBER 2. OOVT ACCESSION NO

4. TITLE (mnd Subllllm)

The Development of a Neutron Evaporation Theory Code for the Thermal Fission of Uranium-235

7. AUTHOR(a)

Rosener, Thomas J.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

Student,HDQA, MILPERCEN (DAPC-OPA-E) 200 Stovall Street Alexandria, Va. 22332

II. CONTROLLING OFFICE NAME AND ADDRESS

HQDA, MILPERCEN, ATTN« DAPC-OPA-E 200 Stovall Street Alexandria, Va. 22332

READ INSTRUCTIONS BEFORE COMPLETING KORM

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5. TYPE OF REPORT » PERlOO COVERED

•inal Report 3 May 1985

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IB. SUPPLEMENTARY NOTES

Document is a thesis prepared at Rensselaer Polytechnic Institute, located in Troy, New York.

VL v*' ■ r i-

:». KEY WORDS (Ctmltmw an imrmrmm airfi If nacaaaary and Id»'ill» hy Mac* nummmt)

Fission; Prompt Neutron Spectrum,' Uranimu-235;Liquid drop modeli Semi-empirical mass formula; (/Ar.s«»s) i— F

10. AHTHACT rCwMBw* mm mwmnrmm tim\ II inmaw, aval 14—My my MM* III.IIIIJ

The thermal fission of Uranium-235 is studied by the de- velopment of a computer code which uses evaporation theory based on the WeissKopf model for neutron emission from individual fission fragments. This computer code calculates a prompt neutron spectrum, total gamma ray energy, and average prompt neutron kinetic energy for the thermal fission of Uranium-235. All possible fission fragment combinations are considered, weighted by their fractional yields

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SECUMTV CLASSIFICATION OF THIS PA.E ( />afa Erttar.tf)

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a,., | ^ » ^ «T-. «I ■ fcai m *** * mi * m m m M -» M-.-M—L! S_.l."._.._ ........ _... — ■

,1 \ t THE DEVELOPMENT OF A NEUTRON EVAPORATION

THEORY CODE FOR THE THERMAL FISSION OF URANIUM-235

X

X

by

Thomas J. Rosener

A Thesis Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment o* the

Requirements for the Degree of

MASTER OF SCIENCE

Approved:

D4faal<yR. Harris rhesis Advisor

Rensselaer Polytechnic Institute Troy, New York

May 1985

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I TABLE OF CONTENTS

Page

LIST OF FIGURES 111

ACKNOWLEDGEMENT ..... iv

ABSTRACT v

1. INTRODUCTION AND HISTORICAL PERSPECTIVE 1

2. THEORY 6

2.1 Mass Distribution 6 2.2 Charge Distribution 9 2.3 Mass and Binding Energy 14 2.4 Nuclear Evaporation Theory 24

3. CODE DEVELOPMENT 30

3.1 Block Data Subprogram 30 3.2 The Average Heavy and Light Masses 30 3.3 Determination of Fragment Kinetic Energy Constant . . 32 3.4 The Main Program Loop 32 3.5 Binding Energy Subroutine 37 3.6 Energy Subroutine 39 3.7 Subroutine for Mass Correction 44

4. DISCUSSION AND CONCLUSION 62

LITERATURE CITED 67

APPENDIX I - BLOCK DATA INPUT 69

APPENDIX II - COMPUTER LISTING 72

APPENDIX III - FRAGMENT MULTIPLICITIES 88

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LIST OF FIGURES

Page

Fig. 2.1 Mass Yield Curve for Uranium-235 7

Fig. 2.2 Average Binding Energy Per Nucleon for Naturally Occuring Nuclides 16

Fig. 2.3 Summary of Semi-empirical Liquid-Drop Model Treatment of the Average Binding Energy Curve ... 19

Fig. 2.4 Naturally Occurring Nuclides for Z < 83 20

Fig. 3.1 Effective Distance Between Fission Fragments .... 33

Fig. 3.2 A-Matrix for Mass Correction Subroutine 49

Fig. 3.3 Mass Correction for Neutron Number 52

Fig. 3.4 Mass Correction for Proton Number 53

Fig. 3.5 Flowchart for the Neutron Evaporation Code 54

Fig. 4.1 Determination of Adjustable Parameter RR 63

Fig. 4.2 Prompt Neutron Spectrum for U-235 64

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ACKNOWLEDGEMENTS

The author wishes to express his gratitude to Dr. Donald R.

Harris for the guidance and encouragement gained while working for

him on this monumental undertaking. Discussions with Dr. Ivor Preiss

of the Chemistry Department are also gratefully acknowledged.

Special thanks are extended to Elaine Verrastro for her pains-

taking word processing support. Finally, the author is especially

grateful to his wife Ann and daughter Teri for their patience during

the exhaustive work on this thesis.

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ABSTRACT

11 •

~? The thermal fission of Uranium-235 has been studied by developing

a computer code that uses evaporation theory based on the Weisskopf

model for neutron emission from individual fission fragments. This

computer code calculates a prompt neutron spectrum for the thermal

fission of Uranium-235, calculates the total gamma ray energy for the

fission of Uranium-235, and calculates the average prompt neutron kin-

etic energy. Until now, evaporation theory codes used average values

of mass and energy in determining prompt neutron spectra, and this

code was developed to use all possible fission fragment combinations

in order to eliminate this averaging. This approach 1s necessary to

compute certain measured data such as the probability that 0,1,2,... '"Us cs,

neutrons are emitted per fission. This code^itlUzes subroutines to

calculate rest masses, corrects the calculated rest mass to agree with

experiment, and to determine the emitted neutron kinetic energies.

Evaporation theory codes have predicted low total gamma energy

release in fission, but agree quite favorably with experiments in pre-

dicting the averge prompt neutron kinetic energy. The prompt neutron

spectrum obtained in this code has a high tall region, that 1s, below

0.5 MeV. The average kinetic energy of the neutrons agree very well

with experiment, and the total gamma energy is lower than experiment,

as previous evaporation theory codes have been. '

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CHAPTER 1

INTRODUCTION AND HISTORICAL PERSPECTIVE

Fission is the process which occurs when a fissionable nucleus

captures a neutron and splits into two lighter nuclei, increasing the

Cv, stability of the system. When fission occurs, up to a half-dozen

neutrons are emitted along with gamma rays, and at some time later,

beta particles and neutrinos. The fission fragments forme' as a re-

sult of fission are neutron rich and must decay to a more stable con-

figuration. One of the most important products of the fission proc-

ess is the prompt neutrons which are emitted within the first 10"17

seconds of the split of the parent nucleus.

In the design of nuclear reactors, it is important to know the

energy spectrum for the neutrons emitted as a result of the fission-

ing of the compound nucleus and the energy of the gamma rays emitted

from the fission process. The neutrons are necessary for the sus-

tainment of the fission process, and the gamma rays are a biological

hazard and require shielding.

James Terrell, in 1958, published his study entitled "Fission

Neutron Spectra and Nuclear Temperatures". His paper considered "the

agreement with experiment of various theoretical predictions as to

the energy spectrum of fission neutrons".1 He also gathered Informa-

tion concerning nuclear temperatures from the measured spectra, and

looked at how the neutron multiplicity (that 1s, the average number

of neutrons emitted in the fissioning process) varies with the exci-

tation energy. He also calculated a center of mass neutron spectrum

1

$

I and calculated the effect of anisotropy of neutron emission on the

laboratory spectrum. Terrell concludes that all experimental energy

distributions for fission neutrons are indistinguishable from

Maxwelliam distributions and shows that distributions of essentially

this form are predicted by Weisskopf's evaporation theory.

"Calculation from evaporation theory gives the observed distribution

of neutron numbers; but calculated total gamma-ray energies are too

low, unless gamma rays are emitted much more rapidly than commonly

assumed."! Neutron multiplicity in Terrell's paper 1s determined to

be 2.46 neutrons/fission.

In Terrell's paper, calculations are performed using an average

light mass and an average heavy mass. His average mass of the heavy

fragment to the light fragment 1s 1.47 for Uranium 235. A compila-

tion of the average energy of fission neutrons 1s presented 1n his

paper based on experimental methods. They are for U-235 + neutron:

Experimenter* Energy Range (MeV) I (MeV) Method

Cranberg & Nereson

0.18 - 3 1.9806 Time of Flight

Frye 4 Rosen 0.35 - 12 1.9806 Photoplate

Mulkhln 1 - 12 ... Photoplate

Nereson 0.4 - 7 — Photoplate

Watt 3.3 - 17 2.00 Proportional Counter

i

The more recent research on the prompt neutron energy spectrum

has been performed by David G. Madland and J. Rayford N1x, at Los ••V--

m

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Alamos Scientific Laboratory. Ag^in, average values of the light and

heavy fission fragments are used. They use standard evaporation

theory to calculate the neutron energy spectrum in the center-of-mass

of a given fission fragment, and then transform into the laboratory

system, taking into account that the average velocity of the light

fragment is higher than the average velocity of the heavy fragment.

The energy released in fission is based on seven heavy fission frag-

ments. These fragments are 2 Iodine isotopes, 3 Xenon isotopes and 2

Cesium isotopes. For the total average fission-fragment kinetic

energies, Madland and Nix used the experimental results of Un'k et

al., which are appropriate to thermal neutron induced fission, how-

ever, these are average values only. Their results show an average N£«

neutron multiplicity of 2.403 for thermal neutron induced fission of

Uranium-235. In the laboratory system, the average energy of the

prompt neutrons for 0.53 MeV induced fission and 0.60 MeV induced

fission is 2.046 and 2.048 respectively. No kinetic energy results

for thermal fission are given in their paper.

As seen in the literature, there is a need to apply first prin-

ciples to the fission process for all possible fragment combinations

in order to remove the averaging of fragment weight and kinetic

energy. The code developed in this thesis uses first principles on

all fission fragment combinations for the thermally induced fission

of Uranium 235. Major Items of interest are (1) to develop a neutron

energy spectrum for thermal fission of Uranium-235, (2) determine the

neutron multiplicity of Uranium-235, (3) determine the total gamma

energy available on the fragments. Total gamma energy is determined

■ v. • > '. »»»_»»•'

4

by the initial excitation energy in the case when no neutrons are

emitted, and the residual energy in the case when neutrons are emit-

ted, but energy remains which is below that required to emit a neu-

tron. This total gamma energy is weighted by the yield of the

particular nuclide and displayed as a result in the code.

Another development which this code will allow is the determina-

tion of the number of neutrons emitted from each possible fragment.

This allows one to compute, by knowing the fractional yield of the

nuclide and the number of neutrons emitted by the fragment, the con-

tribution of the neutron multiplicity from each individual fragment.

The description of fission theory in Chapter 2 is presented

largely in the form of quotations. The application to the code de-

velopment is presented in parallel to the theory. In brief, the

yield of each primary fission fragment is determined as the measured

A-chain yield multiplied by a Gaussian probability distribution 1n

charge. For a given primary fission fragment pair the rest mass

energies are computed and subtracted from the mass energy of tne

fissioning nucleus. This energy decrement appears as the sum of the

fragment excitation energies plus the kinetic energies of the sepa-

rating fragments. The latter 1s determined 1n ten;* of the fragment

charges and the distance RR between the fragments at the Instant of

scission. For an assumed RR the fragment kinetic energies are deter-

mined and, from the sum of excitation energies, a plausible assump-

tion results in the initial excitation energy on each primary fission

fragment. Next, computing the mass of the nucleus which would result

if neutron emission occurred, it is determined whether neutron

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emission is energetically possible. If so, the neutron kinetic

energy in the fission fragment frame of reference is determined by

evaporation theory, and we are again left with an excited nucleus.

The process is repeated until neutron emission is not energetically

possible, after which the total gamma deexcitation energy is record-

ed; the gamma emission spectrum is not computed. Finally, the neu-

tron kinetic energy is transformed to the laboratory coordinate

system. The calculations are repeated for all fission fragment pairs

and for various values of RR. The best value of RR is chosen to give

the best agreement with measured data.

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CHAPTER 2

THEORY

I

2.1 Mass Distribution

"One of the most characteristic features of fission is the

asymmetric division of the fissioning nucleus, and for many years our

most complete knowledge of the mass division came from radiochemical

research."2 "I'hat particular fragment nuclides are produced by the

given nucleus is a matter of chance and the range of gross fission

products is roughly from bromine to barium in the periodic table.

The concentration of fission nuclides depends on the atomic mass and

the distribution curve has a curious saddleback shape (see Fig. 2.1).

These are two well-defined maxima, at A » 95 and A a 140 (for the

fission of U-235), roughly. The total yield is 200% since there are

two fragments per fission. The total number of Identified fission

nuclides 1s about 300, Including nearly 200 different ß" emitters."3

"The first work on fission yield and the introduction of the

concept of fission yield were due to Fermi and his co-workers at

Columbia."2

"The determination of the fission yield of a specific species

consists of a number of steps,

1. A measured amount of non-rad1oactive carrier material of a

given fission product element 1s added to a solution of

uranium in which a known number of fission events has

occurred.

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i VV K ft

i 80 100 120 140 160

Mass number, A, of fission product 180

F1g. 2.1 Mass yield in the low-energy fission of U-235 by thermal neutrons (solid curve). The asymmetry 1s reduced when fission takes place from more highly excited levels of the compound nucleus. The dotted curve shows the mass yield for Th-232 ♦ 37 MeV alpha particles, which also Involves the compound nucleus U-236. (From R.D. Evans, The Atomic Nucleus, p. 393)

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2. If it is necessary, chemical treatment is given this

solution to insure complete isotopic exchange of the stabls

and radioactive isotopes of the element.

3. The solution is subjected to an analytical procedure to

separate the element from the solution in a state of

chemical and radioactive purity.

4. The fractional recovery of the inert carrier is determined t

by some quantitative analytical method. The chemical

recovery of the tracer element is assumed equal to that for

the inert carrier material.

5. The radiation of the purified radioelements are measured to

identify the isotopes and to determine the absolute amounts

of each species. Correction is made for radioactive decay

from the time of fission to the time of counting.

6. From the counting data, the chemical yield data, and the

known number of fission events the fission yield is

calculated. The fission yield is defined as the percentage

of fissions leading to the formation of a measured product.

It is to be noted that the radiochemical results do not in

general give the independent yield of the specific isotope measured.

Usually, the experimentally determined yield is the cumulative yield

of the specific isotope, including any precursors which have

undergone decay to the specific isotope before the chemical isolation

occurred."2

"The largest error in determination of a fission yield lies in

the evaluation of the absolute disintegration rate of the radioactive

rv"

•>.

species. In order to correct for the effects of absorbers [such as

the window of the detector, the material, the source covering, etc.],

it is usual to establish empirically the initial half-thickness of

the radiation emitted by the source by means of the determination of

an absorption curve. In many instances such a curve is complex. The

establishment of an initial half-thickness, then, becomes a matter of

interpretation and individual judgment".^

"In thermal neutron fission of most nuclei, an asymmetric mass

split, with a ratio of heavy-to-light fragment mass (M^/ML) of about

1.4, is much more likely than a symmetric mass split. With

increasing bombarding energy the valley between the two humps in the

fission-yield curve gradually fills in, so that, for example, with 14

MeV neutrons on U-235, the peak to valley ratio is only about 6

(compared to about 600 for thermal neutrons). When the bombarding

energy reaches about 50 meV, the fission-yield curves for the highly

fissile elements (Z > 90) exhibit single bread humps with the valley

completely filled in. It has recently been established that for the

very heaviest elements, that is in the region of fermium, the

probability of a symmetrical mass split again increases relative to

the asymmetric modes, until, for the thermal-neutron fission of

Fm-257, symmetric fission predominates."5

2.2 Charge Distribution

"An important part of the information that one would like to

have about the fission process is the division of nuclear charge

between the primary fission fragments. Unfortunately, to determine

m 10

this is a difficult experimental problem and the available data are

limited. The reason for this difficulty is that the primary

fragments are so far from beta stability that most of them have very

short half-lives. Hence, by the time the necessary chemical

separations have been carried out, the primary products have been

completely converted into different elements. This is not true in

the case of shielded nuclides, and their fission yields are of

necessity independent rather than cumulative chain yields. A

shielded nuclide is one which cannot be formed by beta decay because

the isobaric nuclide of the next lower atomic number is stable."2

"Several postulates have been made concerning the manner in

which the nuclear charge of the fissioning nucleus distributes

itself between the two fragments:

(1) The unchanged charge distribution (UCD) or constant charge

ratio (CCR) postulate predicts that the two fragments

will have he same charge density (Z0/A0) as Lhe

fissioning nucleus. This type of charge division leads to

chain lengths which are longer in the light fragment.

Approximately equal chain lengths are observed in

low-energy fission.

(2) The postulate of minimum potential energy (MPE) proposed by

R.D. Present (1947), postulated that nuclear charge dis-

tributes itself between fragments such that a minimum is

achieved between the nuclear potential energy and the

Coulomb energy. This corresponds to maximizing the

excitation energy. The postulate predicts chain lengths

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of the light fragments which are shorter than those of the

heavy fragments.

(3) The postulate of maximum energy release (MER) was proposed

by Kennett and Thode (1956) to explain better the primary

yields of nuclei in the region of Z=50. The maximum

energy release is calculated as the difference between the

mass of the fissioning nucleus and the masses of the

primary fission products and neutrons emitted. The most

probable nuclear charge which results from this method

depends upon the mass formula that is used to determine

the primary fission product masses.

(4) The postulate of equal charge displacement (ECD) is

essentially an empirical correlation of low-energy

experimental data. The two basic assumptions of this

postulate are (a) the most probable charges for

complementary fission products be an equal number of

charge units away from the valley of beta stability and

(b) the distribution function about the most probable

charge is symmetric and the same for all mass splits."6

"The postulate that the most probable division of charge in

fission is the division that gives rise to fragments of equal

effective chain length (equal charge displacement) may now be taken

as a working hypothesis of the variation in nuclear charge among the

primary fission products. It should be emphasized that this is

essentially an empirical hypothesis. Its validity may be based on

the fact that there is competition between the effects of splitting

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of the nucleus to give a charge distribution not to greatly different

from that of the fissioning nucleus, with perhaps slightly asymetric

charge distribution as suggested by Present, (Phys. Rev. 72:7 (1947))

and the effect of splitting to give minimal potential energy."7

"The shape of the distribution function which represents the

relative probability P(Z) of forming a product with atomic number Z

is assumed to be a Gaussian given by the relationship:

P(Z) = [l/(Clr)i/2] exp[-(Z-Zp)2/c] (2.1)

where c is a constant empirically determined."6

"This relationship holds for all mass numbers, with the param-

eter c = 0.79±0.14 (for 235U fission by thermal neutrons). This

corresponds to a full width at half maximum of 1.50±0.12 charge

units. (Wahl et al. "Nuclear-Charge Distribution in Low-Energy

Fission," Physic?! Review 126, 1112 (1962))."5

"From the assumption of equal charge displacement, one may write

'ZAL ■ V * (ZAH " W (2-2>

where Z^ refers to the most stable charge, Zp to the most probably

charge, and the subscripts L and H refer to the light and heavy

complementary primary products, respectively. The following

relationships are valid for binary fission:

(2.3)

and

ZPL + ZPH a Z0

AL + AH * A0 " v (2.4)

where v is the number of neutrons emitted in the mass split. The

most probable charge of a fission product of mass A 1s given by:

'.V

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13

ZP = ZA 'J^h + Z(Art - v - A) ■ Zo) (2.5)

Thus the Zp function is defined in terms of the most stable

masses."6

Wahl was able to determine the primary and cumulative yields of

nine short lived isotopes of krypton and xenon by measuring the

fractional cumulative yields of inert gases produced as fission

products and captured in a thin layer of barium stearate powder.

From this work, Wahl developed an empirical curve for Zp. "Several

of Wahl's students have contributed newer data on independent and

cumulative yields which are significant for an analysis of the equal

charge displacement hypothesis in its various formulations; the

comprehensive summary paper on this work is a key paper in the

literature on charge distribution."2

In order to code the equal charge displacement hypothesis a few

variables must be defined.

AN(IA) is the mass of the IAth nuclide.

ZN(IA.IZ) is the charge number of the IZtn nuclide with mass

number AN(IA).

ZA(IA) is the starting Z for the IA*n chain.

NZ is the number of charge steps off the line of stability.

The Gaussian curve may be coded as follows:

The probability distribution is given by:

1 SQRT(CC*3.14159)

CC = .79

Cl = 1.0/SQRT(CC*3.14159)

* EXP(-((ZN(IA,IZ)~ZP(IA))**2)/CC) (2.6)

>*.■-■.

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YA(IA)/200.0 is the chain yield divided by 200%, since the yield

is given by thi number out of 200 events that it will occur. The

independent yield is given as the product of the chain yield/200.0

and the Probability Distribution for the specified Z-chain.

YZ(IA.IZ) = (YA(IA)/200.0)*C1*EXP(-((ZN(IA,IZ)-ZP(IA))**2)/CC)

(2.7)

After all yields are determined, the yield weighted average mass

may be determined by summing up all of the products of yield and mass

as:

I [(YA(IA)/200.00)*AN(IA)] IA

and then divided by the sum of all yields:

YA(IA)/200.00

(2.8)

(2.9)

So, the yield weighted average of all masses can be written:

I (YA(IA)/200.00)*AN(IA)

(TOST) = . IA

1/ (VA(IA)/2ÖÖ.ÖÖ) (2.10)

This averaging is performed for two groups. First, all nuclides with

4 < A < 114 are considered in the light group. Second, all nuclides

with 115 < A < 235 are considered in the heavy group. From this

averaging process, the Average heavy fragment and average light

fragment mass may be determined.

2.3 Mass and Binding Energy

"The liquid-drop model provides reasonable explanations for many

phenomena which are inaccessible to the shell model. These phenomena

are mainly:

15

• >>.

(1) Substantially constant density of nuclei with radius

R « Ai/3.

(2) Systematic dependence of the neutron excess (N-Z) on A5/3

for stable nuclides.

(3) Approximate constancy of the binding energy per nucleon

B/A, as well as its small but definite systematic trends

with A (see Fig. 2.2).

(4) Mass differences in families of isobars and the energies of

cascade ß transitions.

(5) Systematic variation of a decay energies with N and Z.

(6) Fission by thermal neutrons of U-235 and other odd-N

nuclides.

(7) Finite upper bound Z and N of heavy nuclides produced in

reactions and the non-existence in nature of nuclides

heavier than [Pu-244].

which involve the masses and binding energy of nuclear ground levels;

the energetics of ß-decay, a decay, and nuclear reactions; and the

energetics of nuclear fission."8

• "The basic premise of the liquid drop model of complex nuclei is

that the number of nucleons in a typical nucleus is sufficiently

large that the individuality of nucleons may be disregarded. While

the treatment of such a system is most rigorously given by using the

methods of quantum statistical mechanics, it 1s possible to arrive at

three components of the expression for the energy of a complex

nucleus by simple classical considerations. These components are the

volume binding energy, the surface energy, and the coulomb energy."9

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E 3 C

8 10

«0 m uo»pnu/A*ft u| v/ff

fO CM

«/I

o •CO

</) II >-> OK CTi

"D en •p- +J OJ i— <0 o • 3UQ. C i—

at o </) C «/» 3

•r- 01 l-< T- 3 <J O OI 3 o x: z O 4->

o

"S ia c *J (. o < 3'i- *J +J aj

c o »- «<*- » •M C C

i~ « > OEui

o • a» a» oe

i— o» ü c E 3 <0 Q C JE (. o u.

O.JE

«141 C Xw «

O *J CZ «A •*- c o < • o

>}0t JE OlJS O»

at 3 5 ecu ai

«/» «/> o> «/> —

£|« •a ^ C <t- 00

£ • c vo

«OH

«- o

> 3 O

C\i

CM

\V."

R ift

M-

&

r*c

17

In addition to the three terms which may be given by classical

considerations, there are also additional terms in the semi-empirical

mass formula. These terms are called the symmetry terms, which

Meyers and Swiatecki include in the volume term, the correction term

to the Coulomb energy term and also the pairing energy term. These

terms are discussed individually:

(1) Volume Energy Term: "The first and dominant term,

proportional to A and thus to the nuclear volume, expresses

the fact that :he binding energy is in first approximation

proportional to the number of nucleons. This is a direct

consequence of the short range and saturation character of

the nuclear forces. The saturation is almost, though not

entirely, complete when four nucleons, two protons and two

neutrons, interact, as is indicated by the large observed

binding energies of '♦He, 12c, 160 and so on."-

(2) Surface Energy Term: " '!*• nucleons at the surface 'feel

less interaction' than nucleons deep inside; compared to

the latter, the external medium, vacuum, deprives them of

more than half their partners. The volume energy term

over-estimates the total binding, since the nucleus is

finite. The lost binding is proportional to the number of

nucleons at the surface, that is to the surface area. By

assuming the nuclear drop to be a sphere, its surface area

is given by 4*r02A2/3 and the correction to be applied can

be written as (AB)S 3 -asA2/3 where as is a constant."10

With increasing nuclear size, the surface-to-volume ratio

1 lvi>"•. *.•'•.< ■V. r\:«r•

" . * ■. *% " *• »^- " » - " - ... - . „ »"* k ■

J:JT "4».TrTrvwa.'ins:wt^T.™ r W""S^TT^."f'-'lit,;WT"'" TfTTf"^.T'"-!I:J*

K

18

decreases, and therefore this term becomes relatively

unimportant."5

(3) Coulomb Energy Term; "The above constitute the main

feature of an electrically neutral drop. We must now

consider the Coulomb interaction which acts between

protons. This is a repulsive action, and therefore its

contribution decreases the binding energy. It is

responsible for the slow decrease of B/A beyond A 2 60, as

Z increases (Fig. 2.3). For all light nuclei, this

contribution is small since the strong interaction is

stronger than the electromagnetic interaction. If this

contribution becomes appreciable for high Z, it is because

the Coulomb interaction is of 'infinite' range, and

therefore acts over the whole volume, whereas the strong

interaction acts over a region of radius close to 1.5

Fermis."10 "The electrostatic energy of a uniformly

charged sphere of charge q and radius R is 3/5 q2/R and

since q a Ze and R » rQ&/* for a nucleus of radius R and

atomic number Z, we can write its electrostatic energy as

(3e2/5r0)Z2A-i/3. The coefficient (3e*/5r0) * 0.717 MeV

corresponds to an r0 value of 1.205 Fermis. Because of its

Z2 dependence, the coulomb energy becomes increasingly

important as Z increases and accounts for the fact that all

stable nuclei with Z>20 con ain more neutrons than protons

(see Fig. 3.4) despite the symmetry energy that maximizes

nuclear binding for N»Z."5

• <•'»-'-.-"- ■*-•"•

" \X:' '.' :' :" ' i«..-.. :.<. t. -,.-1

Kiasmm JT-JT.-W wn T-H i": wm vu xm. "-"iia n v «a f ■ i — gPSW■" "J"1 ST I 'If. Wi 'I1 W If W1W B< 1l« 1HTW !W' 1^ 1 !"■ \ WP^^iWTWT

19

L'r L L-.V

^

16

14 •-Volume energy

1 777777777777%

Asymmetry energy

&%**PlxP I

j-L

127 ► « Bk 24$

30 60 90 120 150 180 Mass number A

210 240 270

Fig. 2.3 Summary of the semi-empirical 11qu1d-drop-model treatment of the average binding energy curve. Note how the decrease 1n surface energy and the Increase In coulomb energy conslplre to produce the maximum observed in B/A at A~60. (From R.O. Evans, The Atomic Nucleus, p. 382)

5w

-'••°-■%%>*]

20

S 10 IS 20 25 30 33 40 *5 SO S5 SO SS 70 75 30 S3 90

•»' 9 « C C- U » «Hi* It *r 34 r«» * I» % üQU«v»MSrrcC«S Ci !« b Tk M I

* C * J k«rk*i»N»h«tO)UOt!l

F1g. 2.4 The naturally occurring nuclldes for Z < 83. The solid line shows the course of Z0, which Is the bottom of the mass energy valley, or the "line of 8 stability". The chemical elements are Identi- fied by their symbols along the Z axis. (From R.D. Evans, The Atomic Nucleus» p. 287)

•■\v-

■•" ■■»■

ft^-'iS' \*

.-•;• V V V

i.V.» . > **h• . •

21

(4) Coulomb Energy Correction Term: "Nuclei are not uniformly

charged, but have charge distributions with diffuse

boundaries. The diffuse boundary gives rise to a

correction to coulomb energy (lowering it), and this is

expressed in the fourth term of the semi empirical mass

formula."5 This correction term, introduced by Meyers and

Swiatecki is written as: (ir2e2/(2r0)) • (d/r0)2Z2/A, where

d is given by 0.5461 Fermis, based on the Stanford electron

scattering experiments. So this correction term can be

written: 1.211 Z2/A.

(5) Symmetry Energy Term: The correction term proportional to

(N-Z)2/A is included in both the volume energy term and the

surface energy term in order that "the surface and volume

energies vary in a parallel manner and, in particular,

vanish simultaneously for a large neutron excess.

(Conventinal formulae without a composition-dependent

surface energy correspond to a surface tension that retains

its full standard value even when the neutron excess has

made the volume binding vanish, and all cohesion has been

lost").1* "It reflects the observation that for a given A

the binding energy due to nuclear forces (I.e.,

disregarding the Coulomb effect) is greatest for the

nucleus with equal number of neutrons and protons and

decreases symmetrically on both sides of N-Z. The simplest

functional form expressing these empirical facts is a term

in (N-Z)2. The A~i dependence of the symmetry energy comes

• •.- -.•'■•.•.-••".• - • ■••-.• •>. ■- V>J>\1«% • " • >" -*-% •* -% • • • •> ' i ••••-•• .- • • . - .-.-..• ■ .». -. • . . .• •.> .• • ■ *.• •-• .«".- ••.-• \«\« v • »S>>J> " »\> v\« v '\-% - v v "-"-•.■••'••••■••••-••■• ■ • • • " - ,"• --..-/• •"* -' -•>."- ■ .• ."^.'» > - V'-J> - >V- .*• ."w /•> Vv> k> .*»y•¥* .'• ".y\%>\v.-.y.\%\ ..v.- .\v, •.*.-% V , -. *%* •

■ * ■ ■» •- -• »~ *>-

22

about because the binding energy contribution per

neutron-proton pair is proportional to the probability of

having such a pair within a certain volume (determined by

the range of nuclear forces), and this probability in turn

is inversely proportional to the nuclear volume. The extra

stability of N=Z nuclei comes about at least in part

through the Pauli exclusion principle since two identical

nucleons cannot be in the same energy state, the lowest

state for a given number of nucleons is attained (in the

absence of Coulomb forces) for equal numbers of neutrons

and protons."5

(6) Pairing Energy Term: This correction term is due to the

coupling of pairs of identical nucleons. "This term is a

quantitative expression of the fact that binding energies

for a given A depend somewhat on whether N and Z are even

or odd. So called even-even nuclei (even Z-even N) are the

stablest and for them the pairing energy term may be taken

as +11/A1/2; for even-odd (even Z-odd N) and odd-even

(odd Z-even N) nuclei, the pairing energy term is zero; for

odd-odd nuclei (odd Z-odd N) the pairing energy term is

-ll/Ai/2. The greater stability of nuclei with filled

energy states is apparent not only 1n the larger number of

even-even nuclei but also in their greater abundance

relative to the other types of nuclei. On the average,

elements of even Z are much more abundant than those of odd

Z (by a factor of about 10). For elements of even Z the

,-f. .'■V

"•V

/.'

.■-!

23

isotopes of even mass (even N) account in general for about

70 to 100 percent of the element. The general shape of the

binding-energy curve (Fig. 2.3) with the maximum at A * 60

comes about through opposing trends with mass number of the

relative contributions of surface energy (decreasing with

A) and coulomb and symmetry energies (increasing with

A)."5

The semi-empirical mass formula can be written as:

EB - h A [1 - k(Y) ] - '2 A2/3 t1 " km 1

C3 Z2 A-l/3 + c4 Z2 A-l + 5 (2.11)

"where Eß is the binding energy, that is, the energy

required to dissociate the nucleus into its constituent

nucleons, and A, Z and N are the mass number, proton number

and neutron number respectively. With Ej$ expressed in MeV

the coefficients take on the following values:

ci = 15.677 MeV

c2 = 18.56 MeV

C3 = 0.717 MeV

c4 = 1.211 MeV

k = 1.79

6 = pairing term.

The equation contains only six empirically adjusted

parameters, yet this simple equation yields binding

energies that agree with experimental values for each of

äftftfoft&^^^^^

■•!>

vf«.

24

the approximately 1200 nuclides of known mass to within

l3ss than 10 MeV, and very much better than that for most.

This represents a remarkable success for the liquid-drop

model of the nucleus."5

The mass M of a neutral atom whose nucleus contains Z

protons and N neutrons is:

M = ZMH + N Mn - EB (2.12)

where Eg is the binding energy, MH is the mass of the

hydrogen atom and Mn is the mass of the neutron. By

determining the binding energy for a nuclide, and knowning

its constituents, the mass may easily be determined.

2.4 Nuclear Evaporation Theory

"The application of quantum mechanics to the calculation of

nuclear reactions heavy nuclei gives rise to extremely complicated

expressions which cannot be treated by ordinary methods, since there

is no approximate solution for this complex many-bodied problem on

account of the intense interaction between the constituents of atomic

nuclei. On the other hand Bohr has pointed out that just the extreme

facility of energy exchange gives rise to a characteristic

simplification of the course of every nuclear process initiated by a

collision between a particle or a light quantum and a nucleus. It

consists in the possibility of dividing the process into two

well-separated stages. The first is the formation of a compound

nucleus in a well-defined state in which the incident energy is

shared among al1 the constituents, the second is the disintegration

■ - . • . - . • . - ►. ■ .> - ► . ■ _»% .% . • .• -v . * . • « - . • j.^ - i-J»VJ> . ■ . -_>> •-■.-••.•»-.'•.•.• .»>•.•• --•«>.- . ,-.• . - . • .> . •> r. ■•»>

25

of that compound system, which can tc treated as quite independent of

the first stage of the process. This conception has been extremely

fruitful in the treatment of all kinds of nuclear reactions.

Qualitative statistical conclusions about the energy exchange between

the nuclear constituents in the compound stüLe have led to simple

explanations of many characteristic features of nuclear reactions.

In particular, the use of thermodynamical analogies has proved very

convenient for describing the general trend of nuclear processes.

The energy stored in the compound nucleus can in fact be compared

with the heat energy of a solid body or a liquid, and, as first

emphasized by Frenkel, the subsequent expulsion of particles is

analogous to an evaporation process."12

In the fission process, a neutron with a given amount of kinetic

energy combines with a heavy nucleus to form a compound system.

After the formation of the compound nucleus the disintegration stage

may be treated separately in most cases from the formation stage "in

a sense that the mode of disintegration of the compound system

depends only on its energy, angular momentum, and parity, but not on

the specific way in which has been produced. If the incident

particle comes within the range of forces, its energy is quickly

shared among all the constituents well before any re-emission can

occur. The state of the compound system is then no longer dependent

on the way it was formed."2

Consider the reaction a + X*c*Y + b, where c is the compound

nucleus. For the evaporation theory to be used to express the

process after formation of the compound nucleus, the assumption is ^MhtUI

1 • « •\> • W> J» .^ . • . »L ■ .^ ,T» V* • • - • - • . • • » • . • . ' . ■ . - . ' öü'QL«f»."r«."i\."r^.". * •'- "\-V" - ■ - ■ • • • "V"- • • • - •. •» • • ■

SVfWnvl • * K- o v *■•«,%-.'-•."•"-"• ■ •■•■'.• • - • . *_ •-

- -. •■ •. ». •■ •• . •. •. -. \ N ST •. % % ». •.

.* ■-* ■.'•'•.*•"• ■ ' ■-* •-'%.• O •!.*• * ' v' • * • ' •

• ••"•*_» ••*-*. •»•»"> "»Xi^i " » • » *\ * j. • Oj!

&

KV

26

made that "the probability that the compound nucleus c decays into

channel 0 is given by:

K2 a if) G(ß)-—£—£ where (2.13)

I ** ac(y) Y

Gc(s) = Branching ratio (relative decay probability) for

the compound nucleus c to decay through channel g.

Kß = ÜSßl = Xß"1; channel wave number in channel ß,

öC(0) - Cross section for the formation of the compound

nucleus through channel ß.

I K 2ac(y) = Sum of K2oc extended over all channels y»

Y Y

into which c can decay.

and therefore can be determined from the cross sections, ao"2^

The energy distribution of the emitted particles is first

studied. "The kinetic energy in channel ß is given by:

H • % * Q«B (2-14)

a = entrance channel

ß ■ exit channel

Qaß = Q value for the reaction « Ea' + Ea" - Eß' - Eg" (single '

Indicates states of nuclei and double ' indicates the states

of the incident or emerging particle.)

Since most of this energy is the kinetic energy of the emitted

particle b, and only a small part is the recoil energy of the

residual nucleus Y, we consider eß the kinetic energy of b.

fiffi

)■£•■

27

We write here: eg = egy - Eg*. where egy = e + Qab and e = ea0

is the kinetic energy in the entrance channel (both a and X in their

ground states), and represents the maximum value of eg, which occurs

if the residual nucleus Y remains in its ground state. Eg* is the

excitation energy of Y after the reaction and the difference between

the energy of Y after the reaction and the energy of the ground state

of Y. To every level Eg* of the residual nucleus there corresponds

an energy eg of the outgoing particle, as long as Eg* < egy."13

The formation cross section ac generally increases with in-

creasing channel energy (owing to better penetration of the barrier),

except for neutrons with i = 0 where oc decreases, but very slowly

(proportionally to k"1 or e"1/2). In all cases the product k2oc

which appears in Eq. (2.13) is a monotonically increasing function of

the channel energy e. If the energy is high enough, a great many

levels of the residual nucleus can be excited. The energ' distribu-

tion of the particles becomes continuous when the energies Eg* are

closer together than the definition of the energy in the incident

beam, or closer than the energy resolution of the experimental ar-

rangement for detection of the reaction products. The shape of the

distribution function Gb(e)de for the number of particles b emitted

between e and e + de is then given by:

Gb(e)de = I Gc(a) (2.15)

e<e<e + de ß

where the sum is extended over channels ß whose energy lies within

the energy interval de. The number of terms in this sum is given by

Ota

• - ■"- ■ «>• -% )> »'■ V-V\V". V«Vv ■', i,N"l% \<\:\'.\\ o.j.% \\\: ,\ \-\V- . -V-V-.% .-.\%V- . 1. .••"„-• vr *-"-•' -- *-" ••" V ••' •

V.*-:-. -' 1 £i..~\ r.f

lv\

UN

■Y\

28

the number of levels of the residual nucleus Y with an excitation

energy Eg* between E and E - de, where E = eby - e. We call this

number Wy(E)de, and Wy(E) the level density.1-^

The level density can be used to determine the relative

intensity distribution of the outgoing particles Ib(e)de by the

relationship:

Ib(e)de = constant e oc(s) wyUby " £) de (2.16)

wnere acU) = ac(e) is a function ot the channel energy e ■ eg.

"We obtain a rough estimate of the shape of Ib(e), especially

for neutrons, by introducing the following Taylor expansion of the

logarithm of the level density:

n(E) = log W(E) (2.17)

around the maximum energy eby by which the residual nucleus Y can be

excited.

v (ebY - e) - v (eb.) - e (||) (E«eby) + . . . (2.18)

If the expansion is used to approximate W(egy - e) 1n the relation-

ship for Ib(e)de, we can abosrb the factor coming from n (ebY - c)

into the constant and get:

Ib (e)de a constant e oc(e) exp - r-r—r dE (2.19)

where the function o is determined by:

1 da 9 * dE (2.20)

0 has the dimensions of an energy and can be interpreted as a nuclear

temperature. This is the well-known thermodynamical relation between

entropy (n) and temperature (G)."^

y^iv."!..-^-.'.^.-; ■ . J '.>AJ -'"tilriVV; jr».

29

IM

W

The application of the temperature concept to nuclear reactions

can be understood in the following way: The incident particle 'a'

enters the target nucleus and forms a highly excited compound

nucleus. The excitation energy can be considered heat energy

delivered by the impact of 'a' on the target. The heating of the

compound system causes an evaporation of neutrons or other particles

and the energy distribution of theemitted neutrons is similar to the

Maxwell Distribution. The potential barrier distorts the Maxwell

distribution for charged particles by depressing the emission of low

energy particles.

The temperature which determines the Maxwell distribution 1s

given by 0 (ebY):

1 9 UKV) bY' Td77dTT ; E = e bY (2.21)

It is the temperature of the residual nucleus Y at the excita-

tion energy ebY (i.e.. the temperature of the residual nucleus after

emission). In the case of the fissioning of U-236 Into two frag-

ments, the excitation energy of the fragments will be the important

factor in determining the energy allowed for neutron emission.

-.1-

, ■ _"* .v t • i « „ ». • M»» -

,v," " • ■ -

r-v.

£?: >>','

:v%

CHAPTER 3

CODE DEVELOPMENT

The code to determine the prompt neutron spectra and neutron

multiplicity for the thermal fission of U-235 is comprised of a main

program, a block data subprogram, a mass correction subroutine, a

binding energy subroutine, and a neutron kinetic energy subroutine.

3.1 Block Data Subprogram

The Block Data Subprogram is used ti initialize the large

arrays, AN(IA) ZP(IA), YA(IA), and ZA(IA), which are the 95 values of

the atomic number of the IA-th nuclide chain, the most probable

charge number of the IA-th nuclide chain, the chain yield of IA-th

nuclide chain, and the starting proton number for the IA-th chain.

The values for AN(IA) range from 71.0 to 165.0, which is the atomic

number range for the fission fragments of U-235. The values of

ZP(IA) were taken from Table 52.10, Reference [7], the values for

YA(IA) were found in Reference [14], The values of ZA(IA) were taken

from the chart of the nuclides by examining each nuclide chain and

extracting the stable species of highest proton number (lowest

neutron number). These values are passed to the main program in a

common statement called COMMON/BLK/.

3.2 The Average Heavy and Light Masses

The charge number of the IZ-th nuclide with mass number AN(IA)

can be placed into an array ZN(IA.IZ) by using two 00-L00PS over IA

(number of nuclide chains) and IZ (number of nuclides in each chain)

30

fo£-Aii^^k^c;..'■.-•■■■y-^^^ ^v:i:ri-:^vV^vv;v^-,-'.--.-:

31

and setting ZN(IA,IZ) equal to the starting nuclide of the chain

(ZA(IA) - IZ + 1) for IZ = 1, (NZ = the number of nuclides in the

chain), so that the proton number of the IA-th chain starts with

ZA(IA) and decreases by 1 until all nuclides of the chain have a

proton number associated with it.

Knowing the chain yield for the IA-th chain (YA(IA)), the yield

of the IA, IZ-th nuclide can be calculated by using the Gaussian

distribution formula:

YZ(IA.IZ) = (YA(IA)/200.00)*Cl*exp(-((ZN(IA,IZ)

- ZP(IA))**2)/CC) (3.1)

where CC is an empirical constant approximately equal to 0.79 and Cl

is equal to 1.0/SQRT(CC*3.14159). All other terms are discussed

above. The yield is divided by 200.0 because the fragment shows up

twice in the loop. One time a fragment is the primary fragment, and

another time it is a complementary fragment.

By using another DO LOOP, the product (PR(IA)) of the yield and

the mass number can be determined for all IA-chains. Atomic number

115 is chosen to be the cutoff point between light fragments and

heavy fragments. Within this do loop, all products (PR(IA)) are

summed for the light fragment group, and for the heavy fragment

group, and the chain yields for each group are also summed. The

yield weighted average mass then can be calculated by dividing the

sum of the products, PR(IA), by the sum of the yields. In mathe-

matical notation:

I M. * YA./200.0 i i 1

M. ave ■~T

YA./200.0 (3.2)

->» * * - * -r •-■ * ft - ft * - m *■ » * ft m ft ft ft ft_* _ft

il • • '', *•. O v* iS O "L* V" -.* •." -." »." ••" «." -." ". -." • f * - •?L»T» • ~ • *■%.>■ . • • • •> »»••.-.-»■••' » • •

^.-•fv-- % .-• --.•-, .* V * -' •" -f m

32

W.

The weighted averages are then printed as output, to give the average

heavy fragment mass and the average light fragment mass.

3.3 Determination of Fragment Kinetic Energy Constant

The formula for the electrostatic repulsion of two charged

spheres is used, with minor modification, to approximate its fragment

kinetic energy (EF).

EF, Z(l) * Z(2) * e2 (3.3)

tota1 RR[(A(l)l/3 + (A(2))i/3]

where Z(l), A(l) are the proton number and atomic mass number of

fragment 1 and Z(2), A(2) are the proton number and atomic mass num-

ber for fragment 2, e is the elementary charge, and RR is the effec-

tive distance (see Fig. 3.1) between the centers of the fragments at

the moment the neck of the fissioning system splits and coulomb

repulsion forces the fragments to repel away from each other. The

value for RR is determined by plotting results for different values

and obtaining an empirical "best fit". The value for e 1s

4.80298 x 1010 electrostatic units. The conversion constant 1s used

to transform ergs to MeV 1.60219 x 10-6 ergs/MeV. Therefore we can

state:

EF4 Z(D * Z(2) * Constant (3.4)

total RR[(A(l))i/3 + (A(2))i/3]

where the constant 1s equal to 1.439899 and RR is in Fermis (10-13

centimeters).

3.4 The Main Program Loop

The greatest portion of the main program consists of two

D0-L00PS, one over IA (number of nucllde chains) and IZ (number of

is RSi»:ai

s3

33

(a) Uranium 235 + neutron * Uranium 236*

Vv I

(b) Upon separation during fission process a "neck is formed"

<r RR 4

(c) The two fragments are separated by an "effective distance' RR determined in this program to be 1.835 fermis (1.835 x 10-13 centimeters)

Fig. 3.1 Effective Distance Between Fission Fragments

>.•'. >.'•*.

it- ̂̂ .>.•».. j»> *■-»■-»■

-J~S...£AJ .'. - . i........»,. i.Ur'

34

W

nuclides in the chain), which in effect performs all operations in

the program for each possible fission fragment released in the binary

fission (thermal neutron induced) of Uranium-235. The index IA is

transformed to ID = NA - IA + 1 (where NA = total number of chains)

so that the heaviest chain is used first, and IZ is transformed to

IE * NZ - IZ + 1 (where NZ = total number of nuclides in the chain),

so the smallest proton number is used first, and hence the largest

neutron number nuclide is used to facilitate the neutron emission

scheme used in the code.

The atomic number of fragment 1 is stored in A(l) and is equal

to AN(ID). The atomic number of fragment 2 is stored in A(2) and is

equal to 236.0 - AN(ID) so that the two fragments add up to the com-

plex nucleus U-236's total number of nucleons which is 236. The pro-

ton number of fragment 1 is stored in Z(l) and is equal to ZN(ID.IE)

and the proton number of fragment 2 is equal to 92.0 - ZN(ID,IE) so

that the sum of the proton numbers is equal to 92.0 which is the pro-

ton number for the complex nucleus U-236. The binding energy sub-

routine is called and the rest mass for fragment 1 and fragment 2 is

returned so that the energy released (ER) in fission can be deter-

mined by subtracting the rest mass of the two fragments from the rest

mass of the complex nucleus. By using Z(l), Z(2), A(l), A(2), the

fragment kinetic energy can be determined for the two fragments under

consideration. In the main program, all ER and all EF are summed and

an average ER and average EF can be computed after completion of the

DO LOOP. The incident neutron, being of thermal energy, adds .025 eV

to the system. Although this is an insignificant amount, it is in-

•• •■.■■.•.•.%*.■ -.■•-.■ •■ ..-•/-•.

.vv-v- V-,v.v. •.■>•-• ■. * - •. • v • • ■• • c, • r -%J ». -. - .v . •. •. -. • ', ■*."?' "■*«0 «.'• '■>.• v"

I

35

eluded. The total excitation energy (EE(ID.IE)) of the fragments can

be determined by:

EE(ID.IE) = ER + EN - EF (3.5)

where EN is the incident neutron energy. The values for EE(ID.IE)

are also summed up in order to determine an average excitation energy.

The binding energy subroutine is called a second time and enters

the subroutine with the atomic mass numbers and proton numbers of

fragment 1 and fragment 2. For each of these fragments, the rest mass

is determined, and the rest mass RM(I), neutron separation energy SEN,

SEN2 and pairing energy T5(I) is determined for 9 successive neutron

emissions in order to form neutron decay chains which can be examined

to determine the number of neutrons emitted within the allowable

energy of the system.

Fragment 1 (the heavy fragment) is examined first. The excita-

tion energy on the heavy fragment (EEH) can be computed by multiplying

the excitation energy of the system by the fragment atomic mass number

and dividing by the atomic mass number of U-236.

EEH * EE(ID,IE)*A(1)/236.GQ (3.6)

Here, it is assumed tht all 236 nucleons are in thermodynamic equi-

librium and on the average, each nucleon shares an equal amount of

residual energy in the nucleus. The separation energy of a neutron

(SEN) can be computed by:

SEN » 939.553 - (RM(I) - RM(I + 1)) (3.7)

where 939.553 is the rest mass of the neutron, RM(I) is the rest mass

of the fragment and RM(I + 1) is the rest mass of the next fragment 1n

the neutron decay chain.

'.v-c

•Vü

.* s f .* ••

$

36

The maximum kinetic energy (EKMAX) of the emitted neutron from

the heavy fragment is the excitation energy of the heavy fragment

minus the separation energy of the neutron minus the pairing energy:

EKMAX = EEH - SEN - T5(I) (3.8)

The energy subroutine is called to determine a neutron energy,

based on evaporation theory, for the emitted neutron. If the neutron

is emitted, the multiplicity NU(ID.IE) for the fragment is increased

by one. The excitation energy for the fragment (EEH) is now equal to

the maximum kinetic energy of the emitted neutron minus the kinetic

energy (EK(ID,IE,I)) determined from the energy subroutine. This

process is continued until the excitation energy (EEH) is less than

the sum of the separation energy of the neutron and the pairing energy

of the fragment. When this occurs, the residual excitation energy

(EEH) can be set equal to the energy available for Gamma Decay (GEH)

from the heavy fragment.

The complementary (light) fragment also is considered in the same

manner as the heavy fragment. In this portion of the program, the

excitation energy of the light fragment is EEL, and the separation

energy is SEN2.

The index counter (I) ranges from 1 to 20. The numbers 1 through

10 are used for the heavy fragment decay chain and number 20 down to

11 are used for the light fragment decay chain.

The Gamma decay energy for the light fragment is called GEL. The

total energy available for gamma emission (GF) is then:

GE = GEH + GEL (3.9)

37

The neutron multiplicity can be multiplied by the fractional

yield of the nuclides and summed over all nuclides to get a yield

weighted neutron multiplicity. Also, a yield weighted gamma energy is

obtained.

This completes the main loop, and the values of the average

release energy (ER), fragment total kinetic energy (EF), excitation

energy (E), Gamma energy (GE) and the yield weighted neutron

multiplicity is sent to output.

In order to produce a neutron kinetic energy spectrum, 20 energy

bins are established in 0.5 MeV increments. The number of neutrons in

each of these bins are stored in NN(1) through NN(20). This data

forms the basis for a spectrum of the number of neutrons per energy

bin versus the energy of the bins.

3.5 Binding Energy Subroutine

The binding energy subroutine utilizes the Meyer-Swiatecki mass

formula to compute the rest mass and binding energies for the fission

fragments in order to determine the release energy of the fission

process. The rest mass of the fragments and the successive daughters

of the neutron evaporation decay chain is computed in order to

determine the separation energy of the emitted neutron.

The subroutine is entered with the atomic number A(I) of the

heavy (I - 1) fragment and light (I a 2) fragment, Z(I) of the heavy

fragment and light fragment, from which the neutron number U(I) is

determined (U(I) » A(I) Z(I)).

-»'-»^-^ v. n» v. 1. s.■*_• ^£»i^£»4\i^<C_ *. _

i-T-,

38

FT'

The first time this routine is called, counter J is set equal to

2, denoting that the loop for rest mass calculation will be run twice,

once for the heavy fragment, once for the light fragment.

The rest mass loop consists of computing the volume energy term

Tl, the surface energy term (T2), the coulomb energy term (T3), the

coulomb correction term (T4) and the pairing energy term T5(I). The

pairing energy term is placed in a vector to be used in the main

program for determining the maximum kinetic energy of the emitted

neutron (EKMAX = EEH - SEN - T5(I)). The pairing term can be one of

three different relationships depending on whether A(I) is odd or even

and whether Z(I) is odd or even, so a test is performed to determine

whether A and Z are odd or even and the appropriate pairing term rela-

tionship is used. The Binding energy (BE) is determined by summing

the five terms and the units are in MeV. In order to determine the

binding energy in amu (BEA) the binding energy is divided by 931.5016

MeV per amu. The rest mass 1n MeV is equal to the negative of the

binding energy in MeV plus the proton number times the rest mass of

the proton plus the neutron number times the rest mass of the neutron.

RM(I) * BE(I) + 2(1) * 938.7906 + (A(I) - Z(I)) * 939.553 (3.10)

The rest mass in amu is determined by dividing the rest mass in MeV by

931.5016 MeV per amu. In order to apply the rest mass corrections 1n

MeV for proton and neutron number as determined in the rest mass cor-

rection subroutine (MASCOR) the terms YB(IMC) (neutron mass correc-

tion) and YD(IMA) (proton mass correction) are added to the rest mass

in MeV (RM(I)). The subroutine returns to the main program (after the

m* L^ilj^L y ■/■•■:- •.^.■vy--^'- . -.- • • -.>.. v ->Vv v\- v •-• v -\-" • w • • - • • • • -N*■ -'■ ^^•.v^V^^•^>^>V^\%VAV^>^l>V;•.V.^^:^^^>^^^.>^^•^•^••^ ■.j.yt.j---«-j

39

# first call) with the rest masses and pairing energies of the heavy and

light fragments.

The second call of the binding energy subroutine enters with the

atomic number AN(ID) and proton number ZN(ID.IE) of the heavy frag-

ment. The neutron number U(I) is determined by subtracting the proton

number from the atomic number. The heavy fragment and its nine

successive evaporation decay daughters are stored in A{1) through

A(10) with corresponding Z(l) through Z(10) (proton numbers) and U(l)

through U(10) (neutron numbers). The complementary light fragment is

stored in A(20) with a corresponding Z(20) and U(20). The nine

successive decay products are stored in A(19) through A(ll) with

corresponding Z(19) through Z(ll) (proton numbers) and U(19) through

U(11) (neutron numbers). The rest mass is then determined for each of

these fragments, requiring a 20 count loop, and this information is

passed back into the main program so that the separation energy for

the heavy fragment (SEN = 939.553 - (RM(I) - RM(I + 1)) and the

separation energy for the light fragment (SEN2 « 939.553 - (RM(I) -

RM(I - 1)) can be determined for use in the primary loop of the main

program.

3.6 Energy Subroutine

The energy subroutine uses an evaporation theory model of deter-

mining the kinetic energy of the emitted nucleon. This subroutine 1s

based on Its work of I. Oostrovsky, Z. Fraenkel and G. Friedlander,

which 1s based on the model of Weisskopf, and uses a Monte Carlo

selection of the kinetic energy based on the evaporation spectrum.

'1

*•" v' :•:^•:■.■:•.•:■■^•v:■/^^.^•:•^•:■.^■^^^v:^^v.•:••^v•A^yv^>>>

ifl&^aiij^^^

40

The required input to the subroutine is the Atomic Number of the

fission fragment Al(II), the proton number of the fission fragment,

Zl(II), the neutron number of the fission fragment Ul(II), the

excitation energy of the fission fragment (EEH for the heavy fragment

and EEL for the light fragment), and the separation energy of the

fission fragment (SEN for the heavy fragment, and SEN2 for the light

fragment). The subroutine initially determines the level density

parameter AU by the following empirical relationship:

AU = A((l. - 1.3 * THETA/A1(I))**2) (3.11)

where A = Al(II)/20.0, the atomic number divided by 20.0. A is the

uncorrected level density parameter. THETA is the neutron number

minus the proton number divided by the atomic number (THETA = (Ul(II)

- Zl(II))/Al(II)). A test is performed to determine if Al(II) and/or

Zl(II) 1s odd or even so that the appropriate pairing term DELTA can

be determined.

If the subscript II < 10, this denotes the heavy fragment, and if

II > 11, this denotes the complementary light fragment. Initially,

the heavy fragment 1s used and the maximum kinetic energy RJ1 for the

neutron emitted 1s computed by:

RJ1 = EEH - SEN - DELTA (3.12)

The neutron kinetic energy selection scheme uses the substitution

XX » EK(ID2, IE2, II) + BETA where EK is the kinetic energy of the

emitted neutron and BETA 1s an empirical formula

BETA = (2.12 * A ** (-2/3) - 0.050)/(0.76 + 2.2 * A ** (-1/3))

(3.13)

VN',

ft

41

"With the inverse cross section as given by ojaq = ct(l + 3e)

where a = 0.76 + 2.2 * A ** (-1/3) and 3 given as BETA above, and ac

and ag are the capture and geometric sections respectively, and the

level density is given by:

W(E) = C*EXP (2. *(AU(EE - DELTA)) 8** 1/2)

where DELTA = 0 for odd-odd nuclei, and DELTA * 0 for other types.

The following equation is obtained for neutron emission,

g m r2A 2/3

PX(e)de n n 0 n

h2/3 exp {-2*LAUQ(EE - DELTAQ))** |}*

e*a (1 + Sjp^) * exp {2 * [AUn * (EE - SEN - DELTAn - e)]** |}de

(3.14)

where the subscript 0 and n refer to the original and residual nucleus

respectively and e is the kinetic energy of the neutron emitted. To

simplify this notation, the substitution XX s e + BETA is made. By

differentiation, the value XMAX for which PX(XX) is a maximum may be

found to be:

XMAX = [(AU * RJ1 + 0.25) ** 0.5 - 0.5J/AU (3.15)

and now PX(XX) may be normalized to PX(XMAX) » 1; it then becomes:

XX PX(XX) XMAX exp {AU * XMAX - L(AU * (RJ1 - XX)) ** 1/2J}"16

(3.16)

"The actual choice of a particular value of XX involves the

consecutive drawing of two numbers between 0 and 1. The first random

number RND(l) is used to choose a value of XX between 0 and RJ1 (the

maximum possible kinetic energy for the given particle:

XX = RND(l) * RJ1 (3.17)

■ .-

?s

•\v

1 \' •

42

The second random number RND(2) determines whether the value of XX is

to be accepted. If PX(XX) > RND(2), accept the value of XX. If

PX(XX) < RND(2), reject the value of XX and select another set of

random numbers."15

If XX - BETA is negative, another set of random numbers is

selected to prevent negative energies which are not physically

meaningful.

"The range of possible kinetic energies of the outgoing particle

was divided into three regions, and a different procedure was used for

each, (a) For the region XX < 2 * XMAX, the normalized PX(XX) was

computed from Eq. (3.16) and the value compared with RND(2). If the

computed value was smaller than RND(2), it was rejected and a new

random number set (RND(l), RND(2)) is chosen. If the computed value

of PX(XX) was greater than RND(2), the kinetic energy chosen by the

first random number RND(l) was accepted and its value computed:

EK =■ XX - BETA (3.18)

(b) In the second region, 2 * XMAX < XX < 15 * XX, use was .nade of

the fact that, for any positive value of XX, the probability given by

a Maxwell distribution:

XX (3.19) PX(XX) "xMlxexP tl " (XX/XMAX)]

is greater than (or for XX ■ XMAX equal to) that given by

Eq. (3.16)."15 For this region the same procedure used in (a) above

was used but PX(XX) was calculated using Eq. (3.19).

£77. rJ

43

1

(c) For the region XX > 15 * XMAX, values were all rejected

because of the very small probability of neutrons being found in this

region.

Once the value of XX is accepted, then the kinetic energy EK(ID2,

IE2, II) is calculated by EK(ID2, IE2, II) = XX - BETA.

The value for EK(ID2, IE2, II) is in the center of mass

coordinate system and must be transformed into the laboratory system.

In the main program, the correction factors (for the heavy

fragment, CL1 and for the light fragment CL2) are computed and passed

into the subroutine by a common statement.

RM(2) LL1 " RM(1) (RM(1) + RM(2))

CL2 = 939;553 * RM(1)

* EF (3.20)

(3.21) * * v ' * re RM(2) (RM(1) + RM(2)) tr

where RM(I) is the rest mass of the heavy (1) or the light (2)

fragment and EF is the total kinetic energy of the fragments and

939.553 is the rest mass of the neutron.

The laboratory kinetic energy of the neutron being emitted is

then:

EK(I02, IE2, II) - CL1 + EK(ID, IE, II) + 2.0 * SQRT(CL1 * EK(ID2,

IE2, II)) * COS(ANGLE) (3.22)

where ANGLE is a randomly distributed angle between 0 and 2w:

ANGLE » RND(3) * 6.283185 . (3.23)

The light fragment also has a laboratory system kinetic energy:

EK(I02, IE2, II) - CL2 + EK(ID, IE, II) + 2.0 * SQRT(CL2 *

EK(I02, IE2, II)) * COS(ANGLE)

te

,v.yr

*-Vi

,v V JV

44

Once these neutron kinetic energies are determined, a common

statement passes them back into the main program and the new residual

energy (EEH, EEL) is found.

3.7 Subroutine for Mass Corrections

The MASCOR subroutine performs a cubic spline fit of selected

data points from the mass correction graphs for proton and neutron

number as plotted by Myers and Swiatecki. The graph used for this

subroutine is the experimental shell effect graph, Mass

(experimental)-Mass (liquid drop), and the discrepancy between the two

values is in MeV). The routine is performed for both neutron number

and proton number and these correction terms are then transferred to

to main program and back into the binding energy subroutine where the

values are added to the rest mass (in MeV) terms.

If the set of known points in increasing order are (UX(1),

YM(1)), (UX(2), YM(2)) (UX(NN), YM(NN)), then a spline fit is

accomplished by connecting each pair of adjacent points with a section

of a third-degree polynomial, matching up the sections so that the

first and second derivatives are continuous at each point. Let ZK(1),

ZK(2) ZK(NN) be the values of the second derivative at the

points. Then between points (UX(K), YM(K)) and (UX.K+1), YM(K+1)),

the second derivative has the value

YM" - ZK(K) UX^*a([ ' UX ♦ ZK(K+1) UX p/^*0 (3.24)

where DK » UX(K+1) - UX(K)"16 (3.25)

' - ' > -V • « \« » -V •.-.•.^.-.-.•••.'.•.,».».^.^.v.

... i_-. .

LW

45

Integrating Eq. (3.24) will result in getting a first derivative:

YM' = - ZK(K) [(UX(K+1) - UX)2/2DK] + ZK(K+1) [(UX - UX(K))2/2DK]

+ CN(1) (3.26)

where CN(1) is the constant of integration.

Integrating once more will result in the equation of the curve:

YM = ZK(K) [(UX(K-l) - UX)3)/6DK] + ZK(K+1) [(UX - UX(K))3/6DK]

+ CN(1) * UX + CN(2) (3.27)

and again CN(2) is the constant obtained by integration.

Knowing that the curve passes through (UX(K), YM(K)) and

(UX(K+1), YM(K+1)), the values of the integration constants may be

obtained.

YM(K) = (ZK(K) * (DK ** 2)/6) + CN(1) * UX(i) + CN(2) (3.28)

YM(K+1) » (ZK(K+1) * (DK**20/6) ♦ CN(1) * UX(1) + CN(2) (3.29)

from which the values of CN(1) and CN(2) are:

CN(1) = [(YM(K+1) - YM(K))/DK] - [ZK(K+1) - ZK(K)) * DK/6] (3.30)

CN(2) ' [(YM(K) * UX(K+1) - YM(K+1) * UX))/DK] -

[(ZK(K) * UX(K+1) - ZK(K+1) * UX(K)) * DK/6] (3.31)

If the values for CN(1) and CN(2) are substituted into Eq. (3.27)

the following expression is obtained

YM » [ZK(K)(UX(K+1) - UX)3/6DK] ♦ [ZK(K+1)(UX - UX(K))3/60K]

♦ L(UX(K+1) - UX)(YM(K)/DK - ZK(K) * DK/6)J

♦ L(UX - UX(K)) (YM(K+1)/DK - ZK(K*1) * DK/6)] (3.32)

All the values in Eq. (3.32) are known with the exception of ZK(K) and

ZK(K+1), which are the second derivatives at the endpoints of the

interval.

' . * - «. . > 1 « .. ■ JlAJ 111! -* - '- *-• -r-r .• . ■ . •V""- . . '• " - * .' \ - • • % • • • ' -"-*■.•.■".

IS"

46

"One condition which will help determine these values is that the

slope at (UX(K), YM(K)) as determined from Eq. (3.26) must be the

same as that determined by the corresponding formula for the interval

(UX(K-l), YM(K-l)) to (UX(K), YM(K)). When the value of CN(1) from

Eq. (3.30) is used in Eq. (3.26) the equation becomes:"16

YM' = - [ZK(K)(UX(K+1) - UX)2/2DK] + [ZK(K+1)(UX - UX(K))2/2DK]

+ [(YM(K+1) - YM(K))/DK] - [(ZK(K+1) - ZK(K))DK/6] (3.33)

For the interval preceding this one:

YM' = - [ZK(K-1)(UX(K) - UX)2/2DK(K-1)] + [ZK(UX

- UX(K-1))2/2DK(K-1)] + [(YM(K) - YM(K-1)/DK(K-1)]

- [(ZK(K) - ZK(K-l)) * DK(K-l)/6] (3.34)

At the point (UX(K), YM(K)) the following relationship is derived by

setting Eq. (3.33) equal to Eq. (3.34).

ZK(K-l)(DK(K-l)/6) + ZK[(DK(K-1) + DK)/3] + ZK(K+1) • (DK(K)/6)

s [(YM(K+1) -YM(K))/DK] -[(YM(K) -YM(K-1))/DK(K-1)]

(3.35)

For each of the interval points, K ■ 2,3 NN-1 there is an

equation like Eq. (3.35). "There ire NN-2 equations in the NN

unknowns ZK(1), ZK(2), ... ZK(NN). Two or more conditions may be

specified in order to determine these equations completely. It is

customary to place some additional condition on ZK(1) and ZK(NN), the

values of the second derivative at the endpoints. There are several

reasonable choices for these values, and the particular choice will

influence the shape of the fit, especially near the endpoints. We

will require the third derivative to be continuous at (UX(2), YM(2))

and (UX(NN-l), YM(NN-l))."16

.-. ■. - * *. •.

k"> O •

i£

47

YM'" = - ZK(K)/DK(K) + ZK(K+1)/DK(K) (3.36)

If the values for K = 1 and K = 2 are equated, as well as for K = NN-2

and K = NN-1, we obtain:

- ZK(1)/DL(1) + ZK(2)(1/DK(1) + 1/DK(2)) - ZK(3)/DK(2) = 0

(3.37)

and

- ZK(NN-2)/DK(NN-2) + ZK(NN-l)(l/DK(NN-2) + 1/DK(NN-1))

- ZK(NN)/DK(NN-1) = 0 (3.38)

The above two equations along with Eq. (3.35) constitute NN

equations in NN unknowns for ZK(1), ZK(2), ... ZK(NN). Upon solving

these equations and the ZK(K) determined we can use Eq. (3.32) to

find any value of YM for a given vzlue of UX in the interval (UX(1),

UX(NN)).

"Spline fit interpolation in the table of values (UX(1), YM(l))t

(UX(2), YM(2)), .... (UX(NN), YM(NN)) consists of determining which

two points the given value of UX lies between and then finding YM by

the formula:"16

YM ' CN(1,K)*(UX(K+1) - UX)3 + CN(2,K)*(UX - UX(K))3 +

CN(3,K)*(UX(K+1) - UX) + CN(4,K) (UX - UX(K)) (3.39)

where the constants CN(1,K), , CN(4,K) can be determined by solving

Eqs. (3.35), (3.37) and (3.38) for ZK(1), ZK(2) ZK(NN):

CN(1,K) - ZK(K)/6DK(K) (3.40)

CN(2,K) = ZK(K+1)/6DK(K) (3.41)

CN(3,K) - YM(K)/DK(K) - ZK(K)*DK(K)/6 (3.42)

CN(4,K) » YM(K+1)/DK(K) - ZK(K+1) * DK(K)/6 (3.43) V.

A A ."•

48

By making the following definitions

PK(K) = DK(K)/6; K = 1, 2, ... NN-1 (3.44)

EL(K) = (YM(K) - YM(K-1))/DK(K-1), K = 2, .... NN (3.45)

BK(1) = 0 (3.46)

BK(K) = EL(K+1) - EL(K); K = 2 NN-1 (3.47)

BK(NN) = 0 (3.48)

Then we can write the system of Eqs. (3.35, (3.37) and (3.38) in

matrix form as:

AZ=b (3.49)

where A is defined in Fig. 3.2.

Making use of the special nature of the above matrix the ZK(I)

can be found and then the constant CN(I,J) determined by the routine

on page below.

START

INPUTUK(K),YM(K),K»1,...,NN

DK(K)=UX(K+1)-UX(K)

PK(K)=DK(K)/6

El(KHYM(K+l)-YM(K))/DK(K)

BK(K)«EL(K)-EL(K-1) K*2 NN-1

AC(1,2)*-1-DK(1)/DK(2)

AC(1,3)»DK(1)/DK(2)

AC(2,3)»PK(2)-PK(1)*AC(1,2)

AC(2,2)»2(PK(1)+PK(2))-PK(1)*AC(1,2)

AC(2,3)=AC(2,3)/AC(2,2)

BK(2)»BK(2)/AC(2,2)

AC(K,K)»2(PK(K-1)+PK(K))-PK(K-1)*AC(<-1,K)

-.'.'

it '-•-

£ 49

BK(K)=BK(K)-PK(K-1)*BK(K-1)

AC(K,K+1)=PK(K)/AC(K,K)

AC(NN,NN-l)=l+DK(NN-2)/DK(NN-l)+AC(NN,NN-l)*AC(NN-l,NN)

AC(NN,NN)=-DK(NN-2)/DK(NN-l)-AX(NN,NN-l)*AC(NN-l,NN)

BK(NN)=BK(NN-2)-AC^NN,NN-l)'(BK(NN-l)

ZK(NN)=BK(NN)/AC(NN,NN)

ZK(K)=3K(K)-AC(K,K+1)*ZK(K+1); K-M-l,..,2

ZK(1)=-AC(1,2)*ZK(2)-AC(1,3)*ZK(3)

C(1,K),ZK(K)/6*DK(K)

C(2,K)=ZK(K+1)/6*DK(K)

C(3,K)=YM(K)/DK(K)-ZK(K)*PK(K)

C(4,K)=YM(K+1)/DK(K)-ZK(K+1)*P(K)

STOP"16

In the MASCOR subroutine the proton mass correction graph is

interpolated first for twelve known points used as input and all

interpolated values for the mass correction from proton number 22 to

proton number 80 as output. The neutron mass correction graph is

■f;\ interpolated with twelve known points as input and the values for the

mass correction for all neutron numbers from 31 to 120 as output. The

E output is transferred to the main program in a common statement in the

$ arrays YC(IP) for neutron number IP, and YE(IP) for proton number IP.

V- The main program transfers these corrections to the binding

•_ energy subroutine and there the values are added to the rest mass

£«S RM(I) of the nuclide with the proton and neutron numbers matching the

index of the correction factors. If for instance a nuclide (I) has a

V. 50

proton number 40 and a neutron number 60, the correction factors

YC(60) and YE(40) will be added to RM(I) of the nuclide I.

:\ .Vi5

..'.Y.^.lM1

ft*

51

a.

!.\--

CO

O —' a.

CM

a CM

a.

a. + C\J

a. CVJ

C\J

o

o

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a. +

a. CM

z i*: ^>. . a. I-H + + CM CM

z 1 Z 0) z z c *_^ «■«* ■r-

^ * 4J Q. a 3 *_* ^ O CM rH

3 ^-* oo ^->* CM CM 1 c

1 Z o z z •*»* z - - *J *_* M o ^ a 0) Q. ■•>» i_

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</» «/>

• • £ • •

o • • <«- • •

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a* a.

it

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0)

■* »* - *- • k * - ■ *

y,.v •>.'■>

-.-..-•.•. >• .^ .-.-..-•. •.-.-• L-- .•-.- k"- -I- »> ;-•»% .*• ^-".vj

-A- J^^-J

52

t

► <i

I— 1

I [

s-

i : r.4«&-_

8-

c o (- I 4-> 01 3 C <V-r- C 4->

3 «- O O f_ H- .O

3 • </) "öi c ■a p

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«.■."•.-. \ • -. ■." •." -." -." -.' v' •-• •.' •.' -., %" -.",*.• CiC V v' %' v" -." •." •.* -.' -." V \" -. -.* -.' -." O •. ■%. \ •- ■-* ■-■ -. N \ % "v\V -• %* %Vi -V- •" - V- "• •"• "V- ■*« ■"- •"•> ^iiVi^ "Wi-".^,.,\ t£.-, .•„.- ,-..•. ■..-. .*„.-..-. s .-. -■„ --.-••.■. -\-'--% >",-".-•. ■?.iJ-\«ft. A «yjtoH

54

BLOCK DATA INPUT AN(IA), ZP(IA), YA(IA), ZA(IA)

INITIALIZE THE ZN(IA.IZ) ARRAY; ZN(IA.IZ) = ZA(IA) - IZ + 1

COMPUTE THE INDEPENDENT YIELD OF ZN(IA.IZ)

CALCULATE THE WEIGHTED AVERAGE MASS OF THE

LIGHT AND HEAVY FRAGMENT

LVJ CALL MASS CORRECTION SUBROUTINE

Fig. 3.5 Flowchart for the neutron evaporation code,

55

ID = NA - IA + 1, IE = NZ - IZ + II

SELECT FRAGMENT 1 AND 2 A(l) = AN(ID), Z(l) = ZN(ID.IE)

A(2) = 236.0 - AN(ID), Z(2) » 92.0 - ZN(ID.IE)

I CALL BIND

|ER = RMU236 - RM(1) - R"M[2T

Tr Z(l) * Z(2) * CONSTANT

I DETERMINE THE CENTER OF MASS

TO LABORATORY CORRECTION FACTORS

SUM = SUM + EF SUM2 - SUM2 + ER

|£E(!Ö,te) - ER ♦ EN - EF| WHERE EN » .00000025 MeVl

NÜ(tÖ.lE) - Ö— NU (96 - ID,IE) » 0 (MULTIPLICITY » 0)

I

IGAMMA RAY ENFRGY »0.0

••„-'- V N't". %. '. V ■ •.-...-.■.••.. .

, A <.% i-. o. A L>..S i< ^y.-. .y.v .'.:.-.v..v. ,••.-.-:.• ,.-..!■..•» ■-- .- - ■ - » - A<> A v. •.■.:\L'AKI

>Y

56 Ä

SUM3 = SUM3 + EE(ID.IE)

I NU(ID.IE) = Q|

I.

ICALL BIND

COMPUTE THE I EXCITATION ENERGY ION THE HEAVY FRAGMENT!

COMPUTE SEPARATION ENERGY OF THE NEUTRON FROM FRAGMENT!

NU(ID.IE) * NU(ID.IE) + 1

I JEEH * EKMAX - | INEUTRON ENERGY!

•j^ .••

■ L% .

.> .!• -■>•

V". »-• >^ «> .*• ,

•>:. - % *. v ^ *. i *. • ■-*r*--*.iii-'-k? * •jL.> ;<y»--.'

57 W t

CV,

-W

#

V*.

(COMPUTE EXCITATION ENERGY j ON LIGHT FRAGMENT

I

COMPUTE SEPARATION ENERGY j OF NEUTRON FROM LIGHT FRAGMENTf

DETERMINE GAMMA ENERGY

JDETERMINE YIELD (WEIGHTED MULTI- PLICITY ON EACH FRAGMENT |NU(96 -(ID,IE) =

NU(96 - ID,IE) + 1

DETERMINE YIELD| WEIGHTED TOTAL | GAMMA ENERGY I

EEL - EKMAX - NEUTRON ENERGY

1

m

V,

ÜÜ • -

.'.•.-.'.'. T- " •■ *. O - " >■" s. -O •■■«."• 1 - '_•."_ J ^_ «a. »_ J1- Aj rf_- ■£- i_ £- «L_ if. -w. C. C. £» !

* ,^> .*■ L « ."» _"■ -*"* » H * . -. ^. - . - - '« *«' * » * j *>*■•> '>* * ,

■." • , *-" % \* *•* *•

58

BINDING ENERGY SUBROUTINE

DETERMINE FRAGMENT NEUTRON NUMBER

J=2

RETURN TO MAIN PROGRAM

DETERMINE NEUTRON NUMBER ON FRAGMENT AND 9 SUCCESSIVE DECAY DAUGHTERS

1=1

DETERMINE BINDING ENERGIES OF FRAGMENTS

DETERMINE REST MASS OF FRAGMENTS

YES

\ v."»-."-"-*-'' %.*

&^:vi£lv::^ A. •-•• -•A'.V.

.J..-I

Energy Subroutine

FROM MAIN PROGRAM

RJ1 = EEH - SEN - DELTA

59

COMPUTE BETA

w ■

]NO NEUTRONi I EMISSION r

RETURN TO MAIN PROGRAM

YES

O- SELECT THREE RANDOM NUMBERS

XX - RND(1)*RJ1

I

©

- -■ ■■*-** -■»«-■ ■ *.*■ s...,*. i;.: x. ..•>.**:- *.\ '■V

;.... "- - v— <Lim ,\,m i

v-.-v."-.-*v3 i ^ • " » ■ » " a • <*

V .- V "-" V V] - V- \>V- % \N ,I • ft> - » » * '' . • ft] ' ->A> * .> /* -1 ! S — 1_ _ t- %*« <*__ Ö

60

CONVERT TO LABORATORY COORDINATES

XX PX = j^ * EXP LAU * XMAX

SQRT(AU*(RJ1 - XX))]

REPEAT FOR LIGHT FRAGMENT RETURN TO

MAIN PROGRAM

A*

U

n , * -

j» _• j» Vi-. % s'.v •» '•

61

MASS CORRECTION SUBROUTINE

V

&

FROM MAIN PROGRAM

DETERMINE CONSTANTS CN(1-4,KI)

KI ■ Kl + If NO YES

COMPUTE MASS CORRECTIONS USING CUBIC

EQUATION

NO

| RETURN TO | MAIN PROGRAM I

"I

RETURN TO MAIN PROGRAM

. "* _'- . "• ,/*.

CHAPTER 4

DISCUSSION AND CONCLUSION

The code developed in this thesis calculates the total gamma

energy released from the thermal fission of U-235, produces a prompt

neutron spectrum, and determines the neutron multiplicity for the

thermal fission of U-235. The code was run varying the adjustable

parameter RR (see Fig. 4.1) in order to determine the optimum results.

When the optimum values were obtained, the spectrum of prompt neutron

energies was formed (see Fig. 4.2).

Using the value of 1.835 for RR, the total gamma energy is deter-

mined to be 3.2285 MeV/fission, and the average energy per neutron is

calculated to be 1.986 MeV/neutron. During the processing of the

code, it is interesting to edit the number of neutrons released by

each of the individual fragments. The results of this edit are found

in Appendix I. Another advantage of this code is that actual values

of the energy released in fission are computed for each fragment,

versus the use of average values for a heavy and light fragment,

thereby increasing the accuracy of the computed excitation energies.

The prompt neutron spectrum produced in the program results in a

spectrum that shows an abundance of very low energy neutrons in the

first energy bin (that is less than 0.5 MeV). This can be explained

by the fact that evaporation theory allows for the emission of any

neutron from a fragment that has enough residual energy greater than

the separation energy plus the pairing energy. No compensation for

allowed or not allowed transitions based on angular momentum

considerations is accounted for.

62

tf& v.V /„AA.-.I-.,:.. -' '-• v' 1* '-"' •-'■•-" ■ «-'•-'» *«_•,«■•-.'.»■■»v. -.. •-. :., -. . •.,. a- i..

•53

NEUTRON NULTIPLICITY-X-X-X

AVE NEUTRON ENERGY"Y~Y-Y

TOTAL.GAMMA ENERGY-Z-Z-Z

'„•

3 in

s..

3 er»

(Si M X er

i >- 8

8

8 i.78 1.80

—♦—

1.82 RR

1.84 1.86

.v. V,

Fig. 4.1 This graph was used to determine a value for the adjustable parameter RR. The line labeled with X's represents the neutron multiplicity and units of the Y-ax1s are neutrons/ fission. The line labeled with Y's represents the average neutron energy and the units for the Y-axis are MeV/neutron. The line labeled with Z's is the total gamma energy and the units for the Y-ax1s are MeV/fission. The optimum value is determined to be RR » 1.835.

■>.■,. - »v r'/>.'vv \™ ■» % *■* *» * "- *•

fr^^aa ■•-*> ja ^ " a -■-»>» .>»>^ly.v.->-.v.^-.>..v- ••■■-'

64

ys-;

l> • I

PROriPT NEUTRON SPECTRUM FOR U-235

EACH ENERGY BIN = 0.5 MEV

1.00 2.00 3.00 4.00 5.00 0.00 NEUTRON ENERGY

t.oo 10.00

Fig. 4.2 This prompt neutron energy spectrum was generated from the code developed in this thesis. Each energy bin 1s 0.5 MeV. Each square represents the number of neutrons found in each particular energy bin.

££ fr.Nw.V-.--- •vv.

65

The gamma energy is very low. As Terrell noted in his paper, the

value of the gamma energy he obtained is approximately 4.9 MeV. This

code obtains a value of 3.2285 MeV and the actual values based on a

simplified model is approximately 6.7 MeV. This leads one to believe

that gamma ray emission may compete more strongly with neutron

emission, and that evaporation theory does not adequately treat the

gamma emission problem adequately. Because there is great distortion

involved in the fission process, a rapidly varying electric field may

actually induce gamma emission instead of neutron emission.

Since the excitation energy is computed by using the energy re-

lease and the total fission fragment kinetic energies, any error in-

duced into these values can cause an error in the excitation energy.

For example, the parameter RR is held constant for all fission frag-

ments. In reality, because of the varying sizes of the fission frag-

ments, this parameter will vary, so the total kinetic energy will

differ from the values compued in the code.

Another consideration which can cause a lower kinetic energy of

the fission fragments is that the nucleus will "cool" with time so

that the kinetic energy of the fragments decrease by the time the

second neutron is emitted, leaving less residual energy for that neu-

tron, and this can result in more residual energy for gamma emission.

The average kinetic energy of prompt neutrons emitted in the

thermal fission of U-235 is very well predicted in the code. Terrell

determined a value of 1.935 MeV which favorably agrees with experi-

mental values found by Cranberg and Leachman. This code predicts a

value of 1.986 MeV/neutron.

*.v. ••. •■

sS£l<^ ,\<* ». .">/-. ..*.-..f.y.y»-..L-r..^

•.v,v.v.\v .u(. l". < !A .'■

-«■"- *■"•'■ '■•■-■■ -

66

In summary, this code was developed in order to use all possible

fission fragment combinations for the thermal fission of U-235 in

order to predict a prompt neutron spectra, average prompt neutron

energy, and total gamma energy. An additional feature of the code is

its ability to determine the individual neutron multiplicity of each

fragment produced in fission.

. ■•. •...».„ 1. v. *•'--'-•

LITERATURE CITED

1. Terrell, James, "Fission Neutron Spectra and Nuclear Temperatures", Physical Review, Vol. 113, Number 2, 1959.

2. Hyde, Earl K., The Nuclear Properties of the Heavy Elements, Vol. Ill, Prentice Hall Inc., Englewood Cliffs, 1964.

«

3. Littlefield, T.A., Atomic and Nuclear Physics, Van Nostrand and Company LTD., Princeton, 1968.

4. Steinburg, E.P. and Freeman, M.S., "Summary of the Results of Fission Yield Experiments," Radiochemical Studies: The Fission Products, Book 3, Coryell, Charles D. and Nathan Sugarman, Editors; McGraw-Hill Book Co., New York, 1951.

5. Friedlander, Gerhart, et al., Nuclear and Radiochemistry, John Wiley and Sons, New York, 1981.

6. Gindler, J.E. and Huizenga, J.R., "Nuclear Fission", Nuclear Chemistry, Yaffe, L. - editor, Academic Press, New York, 1968.

7. Glendenin, L.E., Coryell, CD. and Edwards, R.R., "Distribution of Nuclear Charge in Fission", Radiochemical Studies: The Fission Products, Coryell, Charles D., and Nathan Sugarman, Editors; McGraw-Hill Book co., New York, 1951.

8. Evans, Robley, D., The Atomic Nucleus, McGraw-Hill Book Company, New York, 1955.

9. Green, Alex, E.S., Nuclear Physics, McGraw-Hill Book Company, New York, 1955.

10. Valentin, Luc, Subatomic Physics: Nuclei and Particles, Vol. I, North-Holland Publishing Company, Amsterdam, 1981.

11. Meyers, W.D., and W.J. Swiatecki, "Nuclear Masses and Deformations", Nuclear Physics, Vol. 81, North Holland Publishing Co., Amsterdam," 1966.

12. Weisskopf, V., "Statistics and Nuclear Reactions", Physical Review, Vol. 52, August 1937.

13. Blatt, John M. and Weisskopf, V., Theoretical Nuclear Physics, Springer-Verlay, 1952.

14. Crouch, E.A.C., "Fission Product Yields from Neutron Induced Fission", Atomic Data and Nuclear Data Tables, Vol. 19, 1977.

67

- * - " -> . - ,~- .""» J"» »T> »^

v v * • \- v > v^* O

»1

68

15. Dostrovskey, I., Fraenkel, Z., Friedlander, G., "Monte Carlo Calculations of Nuclear Evaporation Processes", Physical Review, Vol.. 116, Number 3, 1959.

16. Pennington, Ralph H., Introductory Computer Methods and Numerical Analysis, MacMillan Company, Canada, 1970.

17. Madland, David 6. and Nix, R.J., New Calculation of Prompt Fission Neutron Spactra and Average Prompt Neutron Multiplicities, LA-ÜR-81-2968, Los Alamos, 1981.

L>V*.- vVV> .-V A'VVW/W>W'V/V.<W'A\»>V .■\«v'«■ v*v >.> sis;*?'.'. <• *« • ■M ,: :• .. U » - «r%

V V V

Ü

>-.'

&

■J.

i APPENDIX I

i BLOCK DATA INPUT

The following data is contained in the code as input and found in block data format. It is used in order to determine the independent yields of the individual nuclides in order to yield averge the neutron multiplicity and total gamma energy computed in the code.

I

k- 69

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70

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71

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v.

APPENDIX II

COMPUTER LISTING

The following computer listing is a code to determine the neutron multiplicity, total gamma energy, prompt neutron spectrum, and neutron kinetic energies for the thermal fission of Uranium-235.

AW

>£■

72

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o Ld z o to O Ld < et > ^ z — < Ld O O > w X Z — CC

u z k- O — 1- UI z — z h- UZ w cc Ld Ld 1- 3 ZZUI

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APPENDIX III

FRAGMENT MULTIPLICITIES

V;

The following data is a compilation of the individual neutron multiplicities for each frag- ment produced in the thermal fission of Uranium- 235. A1.Z1 are the atomic number and proton num- ber of fragment 1, and A2.Z2 are the atomic number and proton number for fragment 2. NU represents the neutron multiplicity and the first one is for fragment 1 and the second one is for fragment 2. Only those fragment splits where one (or more) neutrons is emitted by either or both fragments is printed.

"V.

88

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89

o o o o o o

o o o

o o o

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O O O

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O o o

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o o o

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o o o

O O O O o o

CO o\ CO CM

o ro

o CM ro

o ro

CM ro ro

o ro

0\ CM

CM CO ro

o ro

ro ro

CM ro ro

o ro

ro ro

CM ro ro

o ro

IT ro ro ro

II II CO CM N N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM N

II CM

N

II CM N

II CM N

II CM N

II CM N

II CM rsl

II CM N

II CM Si

II CM N

II II CM CM N N

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ii •S II sO n in n so II iTl II J' n iTl II .* II r» II so II iTl II 3 n in it a II ro II SO II lf\ II 9 II ro II lf\ II 3 II ro II CM ii r» II CM CM CM CM CM CM CM CM CM CM CM 2 CM CM CM CM CM CM 5! CM CM CM CM CM % < < < < < < < < < < < < < < < < < < < < < <

— 3 iTs SO SO so r- r» oo 09 CO CO ox 0> O o o O O 1" *— — — CM o o O O o O o O o o O o o o o o — o — © — o- o -o- o -o -o -O o o o o o o o o o o O o o o o o o o o o o o © o o o ~o —o~ o ~o —o — o —o ~»o — o ~o wO wO — © ~o —o wO — o ^o~ o wO — o —o —o wO . 3 • 3 -3 • 3 ■ 3 -3 3 • 3 -3 • 3 • 3 • 3 -3 3 • 3 • 3 -3 ■ 3 -3 • 3 -3 ■ 3 • 3 • 3 • SZnriMZ — ZCMZroZ — ZCMZOZ ••- ZlMZnzoz — ZIMZOVZOZ t"* ZMZOtZOZ — ZCMZCOZOs SO sO SO SO SO SO so so SO so SO SO SO SO SO lT> SO SO SO IA SO so SO IT« if\

II — II — II — II — II — II — II — II — II — H — II — II — II — II — II — II — II — II — II — II — II — II — II — II — II

fM N it II

N N 1

N 1 II

N N II

N II

N II 1

N N I II 1

N N N II II

M 1

N II 1

N 1

N N II II

N 1

N N 1 II

N 1

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-S^v V % 'S*. -» •- - .«■

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90

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II II II II II II II II II II II II II II II II II II II II II II II II II NMM(\II\INMMI\INN!\IMNNNMKN(\II\II\IMNN NNNNNNNNNNIMNNNNNNNNNNNNNN

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II CO II CO II «0 II CO II 00 II «0 II CO II «0 II «0 II CO II CO II CO II «0 II (0 II 00 II CO II CO II «0 II CO II <0 II CO II 90 II «0 II «0 II oo

\0 ii m ii .» n \0 ii ift ii J» ii to ii M ii \o ii IA ii Jt it to ii w ii i» it \o ii tr< ii 9 ii to ii r>- ii vo it if» u * I to n CM II r- II

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«- II w II f- II .- II w II f- II - II O II »- II •- II •- II r- II •» II *- II CM II CM II CM II «- II «- II t- II CM II «- II •" II O II *- II

NNNNNNNNNNNNNNNNNNNNNNNNN II II II II II II II II II II II II II II II II II II II II II II II II II

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w H w || —. II w || >- II — || w || —• II — II w || w || w || — || w || w || — || w || w || w || w || » || w || w || ~• || «- || 3»-3.-3«-3.-3»-3 — 3«-3-3.-3»-3«-3»-3*-3-3«-3 — 3»-3»-3.-3»-3 — 3»-3«-3f-3»- x<x<x<x<x<x<x<z<x<x<x<x<z<x<x<x<x<x<x<x<x<x<x<x<x<

91

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ro ro fO fO fO to to to fO rt to fO W fo W to fO f^ fO fO fO fO fO fO to

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NO II in II & II m || 03 II r» II NO II lA II * II C- II NO || IA n a II to II N » K II NO H IA II * II to II CNJ II ON II 00 II f- II NO II (UI\IIMNNMN(\IMIMril(VNIVN(\IÄ{Nt«IMA|ftl(UIVIN

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