FEBRUARY, 1985

THE GRAPE CODE SYSTEM FOR THE CALCULATION OFPRECOMPOUND

AND COMPOUND NUCLEAR REACTIONS -GRYPHON CODE DESCRIPTION AND MANUAL-

BY

H. GRUPPELAAR AND J.M. AKKERMANS

Netherlands Energy Research F oundation

ECN does not assume any liability with respect to the use of, or for damages resulting from the use of any information, apparatus, method or process disclosed in this document.

Netherlands Energy Research Foundation ECN RO. Box 1

1755 ZG Petten (NH)

The Netherlands

Telephone (0)2246-4949

Telex 57211

ECN-164

s * ui

FEBRUARY, 1985 '

THE GRAPE CODE SYSTEM l=OR THE CALCULATION OFPRECOMPOUND

AND COMPOUND NUCLEAR REACTIONS -GRYPHON CODE DESCRIPTION AND MANUAL-

BY

H. GRUPPELAAR AND J.M. AKKERMANS*

* Nuclear Consultant, Urlusetraat 4, NL-3533 SN Utrecht,

The Netherlands

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Griffioen

- 5 -

"Will you, von't you, will you, won't you,

will you join the dance?

Will you, won't you, will you, won't you,

won't you join the dance?"

Lewis Carroll, Alice's adventures in Wonderland

(meeting the Gryphon)

ABSTRACT

The statistical exciton model following the master-equation approach

has been improved and extended for application as an evaluation tool of

double-differential reaction cross sections at incident nucleon ener­

gies of 5 to 50 MeV. For this purpose the code system GRAPE has been

developed, which combines a number of interesting features such as:

unified treatment of pre-equilibrium and equilibrium processes, renor-

malized exciton state-densities summing up to the back-shifted Fermi-

gas formula, a new model for the internal transition rates based upon

the nucleon mean free path in nuclear matter, angle-energy distribu­

tions based on intra-nuclear scattering in nuclear matter, account of

discrete-level excitations, a new model for Y-ray competition, inclu­

sion of multi-particle emission, and various sorting options with code

output in the new ENDF-VI format»

An important characteristic of the proposed model is that consistency

with equilibrium models has been demanded for the summed exclton-state

densities as well as for the particle and "y-ray emission cross sec­

tions. Consistency with the adopted state densities has also been

imposed upon the internal transition rates. A survey of the theory is

given and the structure of the GRYPHON code is described. This report

also contains a users' manual for GRYPHON.

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KEYWORDS

INELASTIC SCATTERING MEV RANGE 5-100

COMPUTER CALCULATIONS EXCITONS

P CODES CROSS SECTIONS

ANGULAR DISTRIBUTION PARTICLE-HOLE MODEL

PRECOMPOUND-NUCLEUS EMISSION LEGENDRE POLYNOMIALS

NEUTRON REACTIONS PROGRAMMING

EMISSION SPECTRA

i

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CONTENTS Page

ABSTRACT 5

1. INTRODUCTION: PURPOSE OF THE GRAPE LIBRARY 9

2. THEORIES, MODELS AND ALGORITHMS EMPLOYED IN GRYPHON 15

2.1. The master equation and the calculation of mean lifetimes 15

2.1.1. Master equation 15

2.1.2. Mean lifetime 16

2.1.3. Initial condition 18

2.2. Exclton-state densities and discrete-level excitation 20

2.2.1. Normalized Williams level-density formula 20

2.2.2. Discrete-level excitation 21

2.3. Internal transition rates 23

2.3.1. Phenomenologlcal parametrization of <M2> 23

2.3.2. Mean free path parametrization of <M2> 24

2.3.3. Evaluation of integration 24

2.3.4. Equilibrium exciton number 26

2.4. Particle emission rates 27

2.4.1. Q-factor 28

2.4.2. Complex-particle emission 28

2.5. Gamma-ray emission rates 29

2.6. Multi-particle emission 31

2.6.1. Organization in GRYPHON 32

2.7. Angular distributions (generalized exciton model) 36

2.7.1. Refraction kernel 37

2.7.2. Free-scattering kernel 38

2.7.3. Energy-averaged Klkuchl-Kawai kernel 39

2.7.4. Kikuchi-Kawai angle-energy correlated kernel 39

2.7.5. Summary of the generalized exciton model 40

2.7.6. Finite-size effects and angular-momentum cut-off 42

2.7.7. Additional symmetric component 44

2.7.8. Systematics of Kalbach and Mann 48

2.8. Energy grid and integration method 49

2.8.1. Non-equidistant energy grid 49

2.8.2. Triangular energy bins and integration procedure 50

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Раке

2.8.3. Integration problems in multi-particle emission 51 2.8.4. Consistency in multi-particle and Y~ray emission 53

2.9. Coupling to Hauser-Feshbach codes 54

3. STRUCTURE OF THE GRYPHON CODE 56

3.1. Main program 56

3.2. PRANG master-equation routine 59

4. USERS' MANUAL OF GRYPHON 62

4.1. Input description 62

4.1.1. First emission cycle 62 4.1.2. Higher emission cycles 70

4.2. The inverse-reaction cross section file INV 70 4.3. Output and scratch files 71

4.3.1. The SPECTR file 72 4.4. Sample problems 73

ACKNOWLEDGEMENTS 96

REFERENCES 97

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1. INTRODUCTION; PURPOSE OF THE GRAPE LIBRARY

In this report we discuss the interactive code library GRAPE, written

in FORTRAN-77 for the N0S/BE1 CDC operating system.

The purpose of GRAPE is to describe the non-elastic nuclear reaction

processes originating from light-ion (A < 4) induced reactions at inci­

dent energies from a few MeV upto at least 50 MeV, i.e. reactions in

which precompound-decay characteristics play a prominent role. A survey

of codes contained in the GRAPE library is listed in table 1.

The heart of the GRAPE library is formed by the pre-equilibrium program

GRYPHON and its subroutine PRANG. GRYPHON calculates:

1. Reaction cross sections for emission of particles (n,p,a,T,d,t) and

Y-rays.

2. Emission spectra of emitted particles and (primary) y-rays.

3. Angular distributions and double-differential cross sections for

emitted particles.

4. Multiple-emission cross sections and spectra for any combination of

the afore-mentioned ejectiles, by repeated calls of PRANG.

Both the pre-equilibrium and the equilibrium contributions to the

nuclear decay can be calculated, separately as well as in a unified

manner.

GRYPHON asks and reads most input from an interactive display (CRT),

except for the (variable) energy grid of the emission energies and the

inverse-reaction cross sections that are read from the file INV.

This is illustrated in figure 1. The main output file of GRYPHON is the

file SPECTR that contains the double-differential emission cross sec­

tions of all emitted particles or Y-rays for each excited state of the

composite or intermediate nucleus after each call of PRANG.

In multi-particle emission many intermediate nuclei with various excl-

tation energies occur and this information is sorted and combined by

means of a sorting routine, see figure 1.

For the sorting of the multi-particle emission results some auxiliary

codes may be used. The simplest of these codes, PSORT, only calculates

the total particle-production cross sections and their energy spectra

and Legendre polynomial coefficients. PSORT is a quick routine that

reads the (binary) output file SPECTR from GRYPHON and prints the final

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results. A similar function, with formatted output rather than print

output is performed by MCFRLN. The output of this code is easily trans­

formed with MF6EXT into the format of ENDF/B-VI.

Another sorting code, SPSRT1, selects more information from the SPECTR

file; it prints after each emission all corresponding cross sections,

particle or y-ray spectra and Legendre polynomial coefficients refer­

ring to the last emitted particle. Finally, the total particle-produc­

tion spectra are printed and a survey is given of all activation and

transmutation cross sections. A limitation of this code is that only

the spectra of the last-emitted particle are accumulated.

H The most general sorting code is SPSORT. Here the user specifies the

reaction type, that may be a single reaction sequence such as (n, npn)

or a (lumped) reaction type such as (n, nnp) + (n, npn) + (n, pnn). For

each emitted particle the spectra are accumulated, e.g., in the case of

(n, npn) the neutron spectrum finally contains two contributions and

the proton spectrum finally contains only one contribution. Also for

the (n, pnn) reaction two neutron spectra are accumulated.

More complicated lumped quantities such as the particle-production

cross sections could also be calculated with this scheme. If the user

is interested in e.g. the first neutron emission in the (n, 2n)-reac-

tion a special option is available to perform the sorting. The output

is stored in the new ENDF-VI format (files MTF * 3 and 6).

There are three other codes to process the output of GRYPHON: PRINTSP

to print the binary output file SPECTR, to be used for diagnostics; LEG

to convert the Legendre coefficients given in SPECTR into angular

distributions as a function of angle; and ANG that performs the same

conversion after input of Legendre coefficients by the user.

The remaining codes listed in table 1 are for preparing input or

nuclear constants (stored in data statements) of GRYPHON. Inverse-reac­

tion cross sections are calculated at the required energy grid by the

optical-model code OPTMDL, that prepares the file INV. For the calcula­

tion of level-density parameters from experimental information the

WPILG code is very useful (cf. sect. 2.2). The codes ANSIA, REFR,

CINESI and KIKAW have been used to study the eigenvalues of the refrac­

tion and intra-nuclear scattering kernels which control the angular

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distribution of emitted nucleons. Part of the refraction code REFR has

been included in GRYPHON. The eigenvalues calculated with C1NESI and

KIKAW have been included in data statements of GRYPHON.

The code system is still being further developed. In particular,

effects of angular momentum and parity conservation are being intro­

duced by means of a coupling of Hauser-Feshbach codes (PER1NNI and

GNASH) with GRYPHON, cf. Sect. 2.9. Recently an auxilliary code

(GROUPXS, to be published) has been developed to transform the center-

of-mass energy-angular distribution to the center-of-mass system.

The present report only describes the central code GRYPHON. The other

codes will be reported later. For operation of the codes a series of

CCL procedures are available (control-card language procedures).

The export version of the code system consists at present of the codes

GRYPHON, LEG, PSORT, SPSRT1 and WDILG.

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Table 1 Codes in the GRAPE library

Name Function

ANG Angular-distribution calculator; input: reduced Legendre coefficients

ANSIA Angular distribution of neutrons, refracted at a spherical surface [36]

CINESI Eigenvalues of averaged Kikuchi-Kawai scattering kernel, used in Chinese work [40]

GRYPHON* Double-differential reaction cross sections with account of рге-equilibrium and equilibrium effects, using the unified exciton model

KIKAW Eigenvalues of Kikuchi-Kawai scattering kernel, as applied in GRYPHON; cf. [36]

LEG* Angular distribution from Legendre coefficients given on the GRYPHON output file SPECTR

MCPRLN Collects double-differential cross sections for total ejectile emission from the GRYPHON output file SPECTR and stores the data in the "MacFarlane format" (preliminary version of ENDF-VI format)

MF6EXT Converts data in Macfarlane format (output of MCFRLN) into ENDF-VI format (files 3 and 6)

OPTMDL Optical-model code for the calculation of inverse-reaction cross sections. The code prepares the energy grid for outgoing particles and the inverse reaction cross sections on the

GRYPHON input file 1NV

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Table 1 (continued) Codes in the GRAPE library

Name Function

PRINTSP Prints the (binary) SPECTR file that is output of GRYPHON

PSORT* Simple sorting routine to print double-differential cross sections for total particle emission from the GRYPHON output file SPECTR (same function as MCFRLN, without format conversion to MacFarlane format)

REFR Legendre coefficients of neutrons, refracted at a spherical surface for given potential depths (ANSIA calculates the angular distribution)

SPSRT1* Sorts and prints energy-angle integrated cross sections, last-ejectile emission spectra and angular distributions for all possible reactions from the GRYPHON output file SPECTR. In addition the total emission spectra are printed.

SPSORT General spectrum sorting routine for energy-angle integrated cross sections, particle emission spectra and angular distributions for any given reaction type from the GRYPHON output file SPECTR. The output is in the ENDF-VI format.

WDILG* Fits level-density parameters g and Л to the experimental level scheme data and observed level spacing, using the Williams formula [12] renormallzed to the back-shifted Fermi-gas formula of Dilg et al. [17].

* Included in present export version of code system.

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GRYPHON

Sorting code

print

print

Figure 1. GRAPE-code system; overall scheme.

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2. THEORIES, MODELS AND ALGORITHMS EMPLOYED IN GRYPHON

2.1. The master equation and the calculation of mean lifetimes

Precompound-decay or pre-equilibrium reactions [l] can be considered as those reactions that occur on time scales intermediate with respect to direct reactions (~ 10~22 s, on the order of the nuclear transit time) and to compound-nucleus reactions (~ 10~18 s, when the nucleus has become equilibrated after a large number of intranuclear interactions and Bohr's amnesia assumption becomes valid). Consequently, precom­pound-decay reactions have characteristics that are reminiscent of both direct and compound reactions, such as pronounced high-energy tails in the emission spectra, smooth forward-peaked angular distributions and high nuclear level densities, thus allowing a statistical description.

2^1 l^Magter equation

In the theoretical description of pre-equilibrium reaction two models have become widely popular: the hybrid model [2,3] and the exciton model, the latter being Incorporated in the present GRYPHON code. The exciton model dates back to the basic work of Griffin (1966) [4]. The various formulations of the exciton model presently in use can all be derived from the master-equation approach to the exciton model, first proposed by Cline and Blann [5]. Our starting point is the master equation in the form proposed by Cline [6] and Ribansky et al. [7]:

dq(n,t)/dt « X+(n-2)q(n-2,t) + X"(n+2)q(n+2,t)

-[X+(n) + X"(n)-Hr(n)] q(n,t) . (1)

Here: - q(n,t) is the probability that the excited nuclear system is in exci­

ton state n at time t; - n is the exciton number, being equal to the number of particles p above the Fermi level plus the number of holes h below it;

- X+(n) and X~(n) a r e t h e Internal transition rates from n + n+2 (An - ±2), respectively (the terms containing X°(n), the transition rate with Дп - 0, cancel in Eq. 1);

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- w(n) is ths total emission rate from the state n, summed over all outgoing particles and energies.

The master equation (1) is a gain-loss equation and describes the tem­poral evolution of the exciton probability distribution q(n,t). The dynamics of the process according to the master equation is illustrated in Fig. 2. An interesting feature of the master-equation approach is the fact that it describes the dynamics for all times t, thus including the evaporative (equilibrium) compound-nucleus stage. Therefore, it enables to compute the emission cross sections in a unified way, with­out introducing arbitrary adjustments between equilibrium and pre-equi-librium contributions.

2.1.2. Mean lifetime

An important quantity for the calculation of emission cross-sections is the mean lifetime т(п) for the exciton state n, defined by:

CD

T(n) = ƒ q(n,t) dt . (2) о

Integration of the master equation over time [54,55] gives the equation for the mean lifetimes:

-q(n,t»0) - Х+(п-2)т(п-2) + А-(п-2)т(п-2)

- [X+(n)+A-(n)-»v(n)] т(п) , (3)

where q(n,t*0) is the initial condition for the process.

The average cross section for the emission of particle b at channel energy e is then obtained from:

do(a,b)/de - a I w.(n,e)t(n) , (4) а о n

where a denotes the incident particle, о is the composite-formation cross-section, w (n,e) is the average rate for emission of particle b b at energy e from the state n (see sections 2.4 and 2*5), and т(п) is obtained from solving Eq. (3).

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The GRYPHON code compute* the emission cross sections and $ >ectra according to Eq. (4) by calling the subroutine PRANG, the s-.-ucture of which is discussed in section 3.2. PRANG solves Eq. (3) for the mean lifetimes by calling its subroutine ТАК, which employs the following algorithm [8]. As a first step we take:

T' (n ) - q(n ,t-0) , (5.a) о о

т'(п) - q(n,t«0) + X+(n-2)T(n-2)h(n-2)T'(n-2),

(n - n +2, n +4, , N) , (5.b)

о о

where:

T(n) = [x+(n) + A"(n) + w(n)]"1 , (6.a) h(n) = [l-A+(n-2)T(n-2)A~(n)T(n)h(n-2)]-1 , (6.b)

h(no) - 1 , (6.c)

and n and N are the minimum and maximum exciton numbers. In the second о step the solution for the mean lifetimes is obtained according to:

T(N) - T(N)h(N)T'(N) , (7.a)

т(п) - T(n)h(n) [т'(ч)+А-(п+2)т(п+2)],

(n - N-2, N-4 n ) . (7.b) о

This algorithm produces, for any initial condition, results identical to the exact solution of the time-integrated master equation as discus­sed in Ref. [9]. We note that it gives the summed pre-equilibrium and equilibrium contributions to the nuclear decay. However, the program also allows to calculate the pure pre-equllibrium part by computing the mean lifetimes under the never-come-back assumption (i.e. A~(n) » A°(n) - 0 for all n). Then the lifetimes are given by [lO]:

n n-2 ., . ,/ ч r. q(k,t-0) „ A+0») т'(п) - I — 2 — — П . (8) k-n [A+(n)+w(n)] m-k [A+(m)4w(m)j Дк-2 Дт-2

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In this case, PRANG calls the subroutine PREQ instead of ТАК.

2.1.3. Initial condition

It remains to specify the initial condition. The program uses for the first emission cycle:

q(n, t«0) - 6n , (9) n.n

where n can be specified by the user by giving the initial hole number h (n »h +p; p » mass number of projectile). The default value is h «0, о о о о which means n «1 in the case of nucleon-induced reactions with inclu-o sion of the competition by y-ray emission. We notice that for particle scattering this initial condition is effectively the same as the usual condition n * 3 (2plh). For the higher emission cycles, the condition (9) is replaced by an Initial condition determined by the program itself in the previous cycle (see section 2.6). In addition, the program GRYPHON contains an option to compute the pure equilibrium contribution. In this case, Eq. (9) is replaced by the stationary solution of the master equation (1) (cf. the discussion in Refs. [5] and [lO]) according to:

q(n, eq.) - ш (E)/Z ш (E) . (10) n n n

Here, ш (Е) is the nuclear state density of exciton state n at the excitation energy E, to be discussed in the next section. We note that in Eq. (10) the summation over n is restricted to the physically allow­ed values of п(Дп»2).

The different options in the GRYPHON program mentioned above are controlled by the parameter NEQL which may be given as input by the user, see section 4. NEQL controls the options as follows:

- NEQL - 0: unified pre-equilibrium and equilibrium (Eqs. (5) to (7) and (9);

- NEQL - 1: pure equilibrium (Eqs. (5) to (7) and (10)); - NEQL > 2: pure pre-equilibrium (Eqs. (8) and (9)). - NEQL — 1 : Weisekopf-Ewing formula (future option, to be included).

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Figure 2. The GRYPHON Quadrille

"Will you, won't you, will you, won't you, will you join the dance? Will you, won't you, will you, won't you, won't you join the dance?"

The dynamics of the precompound-decay process is described by a master equation, which is, under appropriate conditions, equivalent to a ran­dom (drunkard's) walk [ll]. Steps (i.e., transitions) are possible in the backward (Дп»-2), sideward (Дп-О), forward (ДпМ-2), as well as in the downward (escape through emission) directions. A detailed discus­sion of the dynamics of precompound decay seen as a stochastic, Markov!an process may be found in Refs. [lO] and [ll]»

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2.2. Exclton-state densities and discrete-level excitation

The subroutine PRANG of the GRYPHON code computes the relative exciton-state densities according to the Williams formula [12] by calling its subroutine STATED. These state densities are given by:

w /-Ч gfgU-ACp.h)]11'1 ш (Е) • Sir yr' 'f . t\\\ n ' p!h! (n-1)! » K '

where g is the single-particle level-density parameter (approximately equal to g = A/13 M e V 1 ) , и - E-A, E being the excitation energy and A being the (pairing)-energy shift. The quantity A(p,h) is the correction for the Paull exclusion principle and is given by Williaas [12] as

AW(p,h) « (p2+h2+p-3h)/4 . (12)

However, in the GRYPHON code we have chosen for a Pauli correction that is symmetric with respect to particle and hole excitations, as discuss­ed by Kalbach [13]:

A(p,h) - [p(p-l)+h(h-l)]/4 . (13)

Apart froe its p-h symmetry, we have opted for Eq. (13) rather than for the more commonly used Eq. (12), since the former yields more plau­sible results for states with low p and higher h. Such states frequent­ly occur in the higher-emission cycles. Williams has shown that the sum of Eq. (11) over all excitons is asymp­totically equal to [12]

I oiW(E) * — IT1 exp (2/aÏÏ) , (14.a)

n n m

with

a - £•% . (14.b)

2.2.1. Normalized Williams level-density formula

Equation (14) is the expression for a one-fermion gas. In (equilibrium)

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statistical-aodel codes it is usual to apply two-fermion gas expres­sions, which have a dependence U~5/4 instead of IT"1 as in Eq. (14.a). For these and other reasons we have proposed [14-16,21] to renormalize the Williams expression such that it coincides with the back-shifted Fermi gas formula as given by Dilg et al. [17]. This is to a very good approximation accomplished b> taking:

ы(п,Е) - f(U) u." .(U) , (15.a) p,h

f(Ü) « JL - , (15.b)

with the nuclear temperature t being given by:

U » at2 - t . (15.c)

This is the full expression employed by GRYPHON*. (We remark that if

t is set to zero in Eq. (15.b) this expression would coincide with the

Gilbert-Cameron formula above the matching energy [l8]). The level-

density parameters g and Д are given as input. They should be fitted to experimental information, e.g. by means of the code WD1LG (Table 1). The parameters g and A are very close to those of Dilg et al. [l7]. Therefore, their systematics could be used. An approximation of the systematics of g and Д has been coded in GRYPHON (default mechanism). Hence, GRYPHON ensures an approach as close as possible to the one followed in equilibrium statistical-model codes. However, it is easy in the present program to set f(U) equal to unity, in which case the computed exciton-state densities coincide with the ones standardly taken in precompound-decay (hybrid or exclton) model codes.

In the GRYPHON code an option has been introduced to account for dis­crete-level excitation [16] in a simple manner. The discrete nuclear levels must be given as input by the user, see further Sec. 4.1, up to an energy E . For residual excitation energies above U-B-E _ (B cut cut denoting the binding energies), the usual procedure for the continuum is followed. For residual excitation energies below this value

* Note: In applications discussed in sect. 2.3 we have u(n«0,B) - 6(E)

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discrete-level excitation is accounted for by «normalizing the exci-ton-state densities such that the state density summed over the allowed exciton numbers agrees with the number of discrete levels within the considered energy interval. Thus, in the case of discrete levels the exciton-state density at the energy grid point E. corresponding to an energy interval A^ is given by:

ЛП kl w ( n'V »1в,<п,Е.) » /К о -i \ , (16.a)

i Aj i «(n.Ej)

where о is the spin cut-off parameter of the spin distribution and к is the number of discrete levels within the energy interval A , which in the GRYPHON code is taken equal to A - E .-E . Of course, at the cut-off energy E and at zero this interval is appropriately reduced by the code.

For very low values of the residual excitation energy, Eq. (16.a) may not work well, since u(n,E) vanishes for E < A+A(p,h)/g. In this case all discrete levels within A. are assigned to the lowest possible exci­ton number, because their excitation corresponds to emission at very high channel energies, which usually stems from the simplest exciton states available. Thus, Eq. (16.a) is then replaced by

u»(d)(n,E.) - /2To T - 6n . (16.b) 1 Ai n,nmin

Note that for the first emission nfflln= nQ - a . In conclusion, the GRYPHON code allows for discrete-level excitation by a rather natural transfor­mation to a quasi-contlnuum level description. Of course, the equi-probabllity assumptions (etcetera) that are made in the description of the continuum, are also Introduced into the description of discrete-level excitation as proposed here.

The excitation-state densities are the quantities crucial to the calcu­lation of both the internal transition rates and the emission rates, which will be discussed in the following sections.

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2.3. Internal transition rates

The internal transition rates of the exciton model are calculated by invoking Fermi's golden rule [4]. Only transitions with An«+2, An«0, An*-2 are allowed as a result of the assumption that only binary colli­sions occur within the nucleus. The transition rates are then given by [21]:

+ interchange term p •»-»• h} (17.a)

l0# , rE

л , V 2 , h ( E " € ) M 2 , 0 ( E ) 21, ^ 2 s , . 0 p,h

+ interchange term p •«-»• h + -^—* (-> ' .г- <М2>ш. ,(е)} (17.b)

*-<»> - ƒ * t V 2 ' h ; l ( ? E ) " 2 ' l ( 0 • r <«2> »i,o<=> 0 p,h

+ interchange term p * * h } . (17.c)

g^j^l^^P^enomenologlcal^garametrization^of^^^

In Eq. (17) <M2> is the average squared matrix element for the residual

interaction, which is supposed not to depend on the transition type. In

the calculation of the above energy integrals it is mostly assumed that

the matrix element is independent of energy [19]. At the same time one

frequently encounters the (practically very successful) incident-energy

dependent parametrizatlon [20]

<M2> - c/A3E , (18.a)

where с is a constant. In GRYPHON this has been coded as

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<M2> - с/2197 g3E , (18.b)

which is equivalent to Eq. (18.a) for g = A/13 MeV 1; see subroutine TRANS, Sect. 3.2. However, the assumptions made in the evaluation of Eq. (17) are in our opinion quite arbitrary and inconsistent [2l].

2.3.2. Mean free path parametrlzatlon of <M2>

It seems to us a much better approach to obtain the matrix element from the characteristics of the nucleon mean free path in nuclear matter. The approach adopted in GRYPHON is essentially similar to the method used in the hybrid model [2]. Then we impose the condition

X +d,e) - ^- <M2> « 2 ) 1(0 * £ e + 0(e2) , (19)

where о * 1.4 x 1021 [2] and к is an adjustment parameter for the mean free path. For simplicity we neglect the higher-order terms in e. Comparison with Eqs. (11) and (15) then shows that the matrix element is not constant, but is roughly inversely proportional to the energy in disagreement with the practice of assuming <M2> constant in evaluat­ing Eq. (17) through Eq. (18)).

2^3.3^ Eyaluation^of integration

To perform the energy integration [2l] in Eq. (17), we note that the integrals are of the Laplace convolution type and therefore can be solved by applying the Faltung theorem. A complication is, however, that the solution cannot be obtained in analytical form as a result of the complicated energy dependence of the «normalization function f(U) in Eq. (15). To circumvent this difficulty we replace f(U) by the func­tion F(U), where F(U) is the asymptotic result for f(U) if aU » 1. This is a reasonable assumption for a very large part of the integra­tion domain. Then we have

f(U) * F(U) - (3U)"6, 0 - 3g/2 , (20)

which is inserted in Eq. (15). To obtain exciton-state densities corre­sponding to the back-shifted Ferni-gas model of Dllg et al. [17]

- 25 -

or to the Gilbert-Cameron [l8] expression, we must set S - 1/4. This has been done in the present version of the code. However, it is easy to recover the Williams formula (Eq. (17)) by setting б equal to 0 so that f(U) is equal to 1 in Eq. (15)).

Inserting Eqs. (19), (20) and (15) into Eq. (17) and applying the Laplace convolution theorem, we find upon inversion for the transition rates [16, 21]

X+(n) - r- . 3 .n(n~l)B(n-l-«,2-ö)(E') , (21.а)

while

X+(l) - 2-Е' , (21.b) к

X°(n) » 2- . e~6.^p(p-l)+4ph+h(h-l)](n-l)(n-2)B(n-2-ó,2-6)(E')~6, K 8

(21.c)

while

A ° < 2 ) " I * i [p(p-D+4ph+h(h-l)] , (21.d)

X"(n) - 2..e"6.i2ph(n-l)(n-2)2(n-3)B(n-3-6,2-6)(E,r1"6 , (21.e) к g

while

X"(3) - £ . p ph(n-l)(n-2)/E' . (21.f)

These are the equations calculated by the subroutine PRANG of GRYPHON.

Here, £' - U-A(p,h)/g and В denotes the Beta function, which is related to the Gamma function by B(x,y) - Г(х)Г(у)/Г(х+у). In GRYPHON the Gamma function is computed [2l] by employing its well-known recursion proper­ty, together with a simple, but very accurate polynomial approximation for Gamma function arguments between unity and two.

It is instructive to write down the result for 6*0, i.e. for the exci-ton-state densities according to Williams, Eq. (11). In this case Eq. (21) yields

- 26 -

X+00 - £ E' , (22.a)

X°(n) - g. . I [p(p-l)+4ph+h(h-l)] , (22.b)

1-СЛ = a X Ph(n-l)(n~2) * ' E" * g 2 * Ë"7 * (22.c)

Comparing these expressions with the usual ones [e.g. 1, 5-8, 19, 20,

22, 23] we see that the energy dependence is not changed, but the exci-

ton-state dependence differs. Roughly speaking, Eq. (22) should be

multiplied by 2/n to obtain the earlier results. This difference is

entirely due to the taking into account of the energy dependence of the

matrix element according to Eq. (19), instead of using Eq. (18), while

assuming the matrix element to be constant in Eq. (17). When the two-

fermion character of the nuclear Fermi gas is taken into account (6 •

1/4) also the energy dependence of the transition rates changes. On the

other hand, in all cases the relative value of the internal transition

rates with respect to each other do not change very much. Hence, the

equilibration process is not very much affected by these changes. How­

ever, the adjustment parameter for the mean free path k, that controls

the ratio between the emission and internal transition rates, must be

readjusted. Using the two-component expressions, б - 1/4, its value decreases to 1.5 to 2.0 instead of the higher values (k * 3 to 6) found in the standard exciton models (6 • 0). A more detailed discussion is given in Ref. [2l]. We attribute the remaining discrepancy to the neglect of geometry effects in the exciton model [2l] as in the hybrid vs. the geometry-dependent hybrid models [2,3]. In addition, the presence of collective-direct effects may play a role [2l]. Finally, we note that by taking E' - U-A(p,h)/g instead of U in Eqs. (21) and (22), the Paull exclusion principle has been taken into account in leading order. Methods to obtain higher-order Pauli corrections may be found in Refs. [23,24].

2.3.4. Equilibrium exciton number

A quantity of physical interest is the equilibrium exciton number n, defined by X+(n) • X~(n)« After equilibration the exciton probability distribution q(n,t) has a sharp peak at n. To obtain the pure pre-equi-

- 27 -

librium contribution to the reaction, the mean lifetimes are calculated from Eq. (8) with a cut-off at n. From Eq. (21) we obtain for the equi­librium exciton number

n = / •=-5 gE , (23.a)

if the Pauli correction A(p,h), Eq. (12) or (13), is included, whereas the neglect of the Pauli correction yields

n = / 2gE . (23.b)

The latter result corresponds to the Ericson [25] state densities (A(p,h) * 0 in Eq. (11)). The code GRYPHON employs Eq. (23.a). Note that this equation is independent of the value of б in Eq. (21).

2.4. Particle emission rates

The GRYPHON code considers emission of particles with A < 4. Their emission rates are calculated following the approach of Kalbach-Cline [5,6,26], who has proposed a derivation from detailed balance or micro­scopic reversibility which is a generalization of the procedure applied in the evaporation model. GRYPHON computes the particle emission rates, by calling the subroutine PRANG, according to

(2sb+l) o)(p-b,h,Eres) W b ( n ' ° " ^ T " ybe°b,inv(e) ы(р,п,Е) - V n ) V ( 2 4* a )

In GRYPHON the symbol W is used for the integral of w over a small energy bin Де:

Wb(n,e) - wb(n,e)ue , (24.b)

cf. sect. 2.8.

In Eq. (24.a) s , u and b are the intrinsic spin, the relative mass b b and the nucleon number of the ejectlle b, respectively, and E «fi­res B(b)-e is the excitation energy of the residual nucleus. The important n, p and a inverse-reaction cross sections a. M (e) are calculated ' r b,inv from the optical model and are read (together with the energy grid)

- 28 -

from the file INV, which is prepared by the optical-model code OPTMDL. For the other particles the inverse-reaction cross sections are calcu­lated from systematics by the subroutine DOSTR, using the approximation of Dostrovsky et al. [27], see further Sec. 3.1.2.

2.4.1. (^factor

The factor Q (n) accounts for the memory by the system in the first b stages of the reaction of the projectile type when emitting the parti­cle b. Strictly speaking, the Q-factor should be time-dependent, but this would necessitate the consideration of complicated inhomogeneous Markov processes. Instead, an exciton-number dependence is used, which seems an acceptable procedure because of the correlation between the exciton number and the time parameter. The Q-factor is obtained by Kalbach [26] from a combinatorial calculation

p-a . _ 1 „ p-a- i /IT +i\/p-ir - i \

д I О Ф Ф ] ( ;„ ) (» ; )• <".., where ». , v, , b (IT , v , a) are the proton, neutron and nucleon numbers of the ejectile (projectile). PRANG calculates Eq. (25.a) for n < n. At equilibrium, the correlation between projectile and ejectile should vanish and the Q-factor should become unity. This is also a consistency condition for the pre-equilibrium emission rates in order to go over into the decay rates of the evaporation model. Therefore, the program sets:

Q. (n) - 1.0 for n > n . (25.b) D

The latter equation is used for all n in the case of a pure equilibrium calculation, in the case of y-ray emission (except when direct and semi-direct y-ray transitions are adjusted by means of a paramete GD, see next section) and in the higher emission cycles. Note that Eq. (25.b) is the asymptotical limit of Eq. (25.a) for high n.

2.A.2. Complex-particle emission

Finally, the (pre-) formation factor Ф. for complex particles occurs ь

- 29 -

in the expression (24) for the emission rates. Essentially, it is nothing more than a fit parameter to account for effects not included in the exciton model, e.g. the existence of clusters within the nucleus and direct processes. The formation factor may be given as input by the user (see Sec. 4.1); in the case of neutrons, protons and gammas it is fixed to unity. Default values for a, ?, d and t are: 10, 10, 5 and 10, respectively. Evidently, a more advanced treatment of complex particle emission is called for. The emission rates (24.b) are calculated in the subroutine PRANG of GRYPHON by calling the routines SPMSS, MAS, ICHARG, STATED and DOSTR or INV. Together with Eqs. (21) and (9) or (10), Eq. (24) completely determines the results for the mean lifetimes, Eq. (3), and the emis­sion cross sections, Eq. (4).

2.5. Gamma-ray emission rates

Analogous to the particle emission rates, the continuum y-ray emission rates may be derived from the principle of detailed balance or micro­scopic reversibility, assuming that only El-transitions contribute. This yields [28-30]:

Z b(k + П,Е) ы(к,Е ) 2 k res

w (n,e) » о aKe(e) , v . . (26.a) Y n2h3c2 Yf3bs ü)(n,E)

The coefficients b(k • n,e) are the branching ratios that subdivide the

total absorption cross-section over the various exciton states accessi­

ble after capture [30J.

The various authors on precompound y-ray decay [28-30] differ, however,

in the results obtained for the coefficients b(k •*• n,e). Plyuiko and

Prokopets [28] allow transitions with к * n-2, n, n+2 and propose

b(n-2 * n,e) - b(n+n,e) - b(n+2 • n,e) - 1/3 , (27)

whereas Betak and DobeS [29] correctly point out that a transition with к - n+2 cannot result from a single Intranuclear interaction (given the way in which the particle emission rates are calculated). Therefore, they only consider к - n-2,n and suggest

b(n-2 •*• n,e) - 1; b(n •*• n,e) - g(p+h)/w(2,e) . (28)

- 30 -

Our objection [30] to Eq. (28) is that at equilibrium it does not agree with the decay rates of equilibrium statistical models. (On the other hand, one might argue that the equilibrium models should be changed in accordance with Eq. (28)). To obtain consistency with the Brink-Axel hypothesis as applied in current equilibrium statistical models, we must impose the condition [30]

E b (к + n,e) - 1 (29.a) n

for all allowed k. With this requirement we obtain Instead of Eq. (28) for the photo-absorption branching ratios [16, 30]

Ь ( П" 2 * n' E ) " g(n-2()!:(2,e)' b< n * n'E> " gn +8"(2,e) ' <29'b>

For "direct" y-ray emission in nucleon-induced reactions only the second expression (n«k*l) applies and n =1 is the initial exciton

о number. The "semi-direct" y-ray emission (n»3) consists of two terms. The strength of the combined direct and semi-direct transitions can be adjusted with the parameter GD. It also depends on the value of A+(l), controlled by FITMFP. In the example given in Ref. [30] the values FITMFP—650, GD-1 were used. The program GRYPHON allows to compute у-ray emission rates according to all of the three procedures outlined here. This is achieved through the input parameter MT (cf. Sec» 4.1) which determines the method of calcu­lation. MT - 0: Akkermans and Gruppelaar [30], Eq. (29) (default). MT - 1: Betfik and Dobes [29], Eq. (28). MT - 2: Plyulko and Prokopets [28], Eq. (27). The y-ray emission rates are calculated in the subroutine RHOG, cf. Sec. 3.1.2. The photon-absorption cross-section is calculated in the subroutine DOSTR, which assumes a Lorentzlan glant-dipole resonance form for this cross section. The user is warned that with MT - 2 not very realistic results are obtained with the present program, since the authors of Ref. [28] did not adopt the Lorentzian form, but the Welsskopf single-particle estimate (in addition to their incorrect inclusion of Дп - +2 emission processes).

- 31 -

The spectra obtained by the methods of Refs. [29] and [30] have the same shape and are quite acceptable, albeit that the method of Bet6k and Dobes yields results higher than those obtained with the method proposed by the present authors, in particular in the equilibrium region [16,30]. The equation for the photo-absorption cross-section used in DOSTR reads:

Г2е2 1

Y' 8 b S (e2-E2)2 + Г2е2 1 1

where Nj, Oj, Г1 and Ej are input parameters (see Sec. 4.1). These parameters refer in first instance to the composite nucleus, where Nj-1. The same parameters are also used for all other residual nuclei excited in multi-particle emission, with the possibility to renormalize о . by reading in different values for Nj. In this way the input remains relatively simple. If necessary a second Lorentzian could be added to Eq. (30) by modifying the input and the DOSTR routine.

2.6. Multi-particle emission

In Ref. [lO] we have rigorously demonstrated that, in the framework of the master-equation exciton model, multi-particle emission can be accounted for by just changing the initial condition and entering a new computation cycle (cf. Fig. 3). Thus, In the case of multi-particle emission the initial condition is determined by the previous excitation cycle and Eq. (9) Is replaced by:

q*(n* » n-b, t-0) - w.(n,e.) т(п) . (31.a) b b

Here, all quantities except q* refer to the nucleus from which the previous emission took place; b is the nucleon number of the particle emitted in the previous cycle. The starred quantities refer to the excited composite nucleus under consideration. It is understood that the computational cycle with the initial condition (31.a) takes as the excitation energy E* - E-B(b)-e.. The computation of the mean lifetimes and emission cross-sections for

- 32 -

the new cycle then proceeds as already described in the earlier sec­

tions with respect to Eqs. (3) and (4).

Accordingly, the computation of multi-particle emission follows in a

natural way the same scheme as utilized for the first emission and does

not pose any severe difficulties. We note that with the above-mentioned

scheme the multi-particle emission is only calculated for fixed ener­

gies of all previous particles (e.). It is more practical to perform b

the calculation for small energy-intervals or "bins" as described

below.

2.6.1._Organlzationin_GRYPHOH

In GRYPHON this is organized in the following way, see also Fig. 3.

GRYPHON calls PRANG to calculate quantities related to the first cycle,

using the initial distribution given by the Kronecker 6-symbol (9),

stored in array D1S. The emission spectrum

dTw(a»b;E»eb)

b

is calculated and printed. In addition the quantity

£ (a,b;E,e ) Де -»eb b a

is stored on file SPECTR. The summation over all bins Ae gives the a

total cross section a (a,b;E) that is also stored on the SPECTR file (see also Sect. 4.3.1). Meanwhile, the initial conditions for the second cycle have been calcu­lated, following the equation Eq. (31.a) with multiplication by Деь:

q*(n* - n-b, t-0) bt^ - Я. (nte.) т(п) . (31.b) b b b

Please note that the summation over all bins Де. gives the fraction of D

the total cross section, emitted by particle b. The initial distribution of Eq. (31.b) is stored on a file DIST1 for each particle b and incident energy E.

- 33 -

In the second cycle GRYPHON reads data froa file DIST1 to overwrite array DIS with the initial distribution calculated in the previous cycle. Siailar quantities as before are calculated:

g l ^ - (a,b,c; E,eb,cc)AebAec , b с

ar <a»b»cï E » V A e b • b

At the saae tiae the initial distributions for the third cycle are

calculated and stored on file DIST2.

At higher cycles the saae calculations are perforaed with alternate use

of DIST1 and DIST2, see Fig. 3.

In addition to the quantities aentioned before, soae control paraaeters

are stored on the SPECTR file to be used as keys in the sorting codes.

This sorting is necessary since one is usually aore interested in the

quantities:

j|- (a.b.c; E,eb) b

and

~ (a,b,c; E, ec) . с

In order to facilitate this sorting the following keys are defined:

Particle history key: IPHIST « к + 10 к + 100 к + Energy history key : 1EHIST - j + 102 j • 104 j + The nuabers к and j are one- and two-digit nuabers, identifying the particles and energies, respectively. The used codes are as follows:

- 34 -

n j « 01 first energy

p j * 02 second energy

a •

T

d

t

Y j * 99 maximum energy

'or the emitted particles the Indices refer to the energy grid; for the

ncident particle the index refers to the order in which Incident ener­

gies have been read by GRYPHON.

- 35 -

GRYPHON (main code)

PRANG (subr~"»:ne)

f SPECTR j

Figure 3. Scheee for eulti-particle emission calculation with GRYPHON.

- 36 -

2.7» Angular distributions (generalised exciton model)

The GRYPHON code incorporates the extensive work on angular distribu­

tions carried out in recent years at ECN-Petten [8, 14, 31-36] with the

generalized exciton or fast-particle model due to Mantzouranis et al.

[37, 38] as a starting point.

In our angle-energy correlated model of pre-equilibrium angular distri­

butions the double-differential cross section is given by

d2o(a,b) ^ T J " \ Z wb(„,e)T(„,a) , (32)

n

where t(n,Q) is the mean lifetime of state n at emission angle Q. The

evolution of the solid-angle parameter Q is determined [37,38] by an

intra-nuclear scattering kernel G(Q,Q'), that plays a role with respect

to & that is equivalent to the role of the internal transition rates + X (n) in relation to the evolution of the exciton state n.

To avoid complicated solid-angle integrations, we have shown [3l] that

it is possible to transform to a Legendre polynomial representa­

tion:

t(n,Q) - ƒ q(n,a,t)dt - Z t (n) p (cos в) , (ЗЗ.а) о I *

q(n,Q,t) - £ nA(n,t) ?t (cos 0) , (33.b)

ƒ dS'6(fiva,)F.(cos 0') « u p (cos 0), 0 - ft-fl' , (33.c)

such that the Legendre coefficients С (n) of the mean lifetimes can be obtained from a set of time-integrated master equations, one for each I:

-n£(n,t-0) - илХ+(п-2Н£(п-2)+игА-(п+2)сА(п+2)

-[X+dO+A-OO-***0)^1-,, >A0(n>] 5 (n) • (34)

It is seer, that Eq. (34) has precisely the same structure as the time-integrated master equation (3) for the angle-integrated quantities.

- 37 -

Hence, it is solved in the same way as in Eqs. (5) to (7) by the sub­routine ТАК of PRANG. The complete algorithm is given in Eqs. (19) to (21) of Ref. [8]. Accordingly, also the calculation of angular distri­butions naturally fits into the same computational scheme.

2.7.1. Refraction kernel

The initial condition is in the present case given by [35,36]

n (n,t-0) - 6 p (2i+l)/4ir , (35) l n,n x ' о

with nQ«l for nucleon-induced reactions and p^»p4(E) following from

/dn'R(ft,Q,)P.(cos 0) - p P (cos 0) , (36)

where R is a kernel accounting for refraction of a neutral incident particle at the nuclear surface. R has been obtained from a geometrical-optics calculation [36]

R(Q,Q') - 2? ("-*)<"*-4 H(x-l/n) , (37.a) * (n2-2nx+l)2

where H is the Heaviside step function and x - cos 0. The refractive index n is defined as

n « V - — , (37.b)

where V is the real potential, which by default is computed from the expression of Becchetti and Greenlees for neutrons [39]:

V - V-a^ - 24(A-Z)/A (37.c)

with V • 56.3 MeV, a - 0.32 MeV"1 and E given in the center-of-mass system. It паз been shown [36] that the above refractive kernel may also be used to describe the refraction of the outgoing neutral beam, if Eq. (37.b) is evaluated at the channel energy e instead of the incident energy E. In this case the solution ?,(n) of Eq. (34) is simply replaced by:

- 38 -

С; (n) - p£(e)ct(n) (38)

In the absence of refraction the kernel R becomes a 6-function so that

1 , (39)

for all £ (no refraction).

The mentioned options can be controlled by the user through the input

variable NRFR of GRYPHON (see Sec. 4.1):

NRFR - 1 : Incoming and outgoing refraction, Eqs. (35) to (38)

(default).

NRFR - 0 : no refraction, Eq. (39) inserted in Eq. (35).

NRFR * -1: incoming refraction only, Eqs. (35) to (37).

Please note that all particles are treated without taking into account

Coulomb deflection and with the above-mentioned potential (37.c).

2.7.2.mFreg _ scattering „kernel

Finally, we discuss the important scattering kernel G, for which also

several options are available.

A first possibility is to consider free nucleon-nucleon scattering

which is assumed to be isotropic in the c m . system of the two nude-

one [37,38]. Then [8]

„free (fi,fl') - cos 0 H(ir/2-0)/* (40.a)

and

free

1 2/3

.A-l (-l)<*+2)/2 %\

2* a-l)(*+2)[(A/2)!]2

(A-0), (i-D, (i odd,* 1),

(£ even).

(40.b)

- 39 -

This kernel has been used In Refs. [8,37,38]. Note that it is complete­ly independent of energy.

2.7.3. Energy-averaged Kikuchi-Kawai kernel

A better approach is to account for the effects of Fermi motion and the Paul! principle within nuclear matter as suggested in Ref. [40]:

KK KK ~

where the expressions discussed by Kikuchi and Kawai [4l] for two-body collisions in nuclear matter are employed and Q" « 8-fl'. Here, the kernel has been averaged ovar the final energies (i.e., the energy after the interaction), but the kernel does depend on the Incident energy. In GRYPHON the corresponding eigenvalues u (П) are tabulated as a function of E.

A further improvement is obtained following our proposal of Refs. KK [14,35,36], by taking into account also the dependence of G on the

final energy, since the angular distribution is clearly correlated with the final energy. Formally, we then would obtain a convolution with respect to the energy similar to the one over the solid-angle parameter ft. However, we have shown [36] that from a numerical point of view it is sufficient to take into account the angle-enerpy Onitir.l and final) correlation by using the full Kikuchi-Kawai expr for the first interaction only. This is accomplished by identii> . -b the final energy with the energy e' * e+B of the emitted particle and replacing for n • n +2 (-3 for nucleon-induced reactions) the solution C.(n) of Eq. (34) by

У* (E,e)

with

- 40 -

j(d2a /de'dn)P (cos 0)dn W* (E,e) == - (42.b) * J(d2o /de'd0)dn

Our proposal to use the full Kikuchi-Kawai expressions has also been followed in Refs. [42] and [43], whereby Ref. [42] in addition incorpo­rated the full final-energy convolution in the higher exciton states. We explicitly point out that in all of these cases Eq. (33.c) remains applicable. It appears that the results of Refs. [42] and [43] are similar to ours [36]. For the calculation of Eq. (42.b) a linear approximation is followed [36]:

ЭИ* V* (E,e) - u*(E,E-l) + - ~ _ (e-E+1) . (42.c)

The coefficients ;i* and —r— are tabulated as a function of E (MeV) in GRYPHON (they were calculated at the energy E-l MeV).

2.7^5^_Summary_of the generalized exciton model

We may summarize the model in a slightly different notation by insert­ing Eq. (33.a) into Eq. (32) and using the replacement of Eq. (38) to obtain:

d2ded«b) * °a l Vn'e) Z « t< n ) pl ( e )V C O i 0> ' (43> n I

For nQ«l (nucleon-induced reaction) we easily find:

2£+l c t ( 1 ) " " T T P * ( E ) T ( 1 ) • (44*a)

Therefore we introduce the parameter £(n,E) defined by+

;* ( П ) " ^Р А(Е)т(п)С(п,Е) , (44.b)

to write Eq. (43) as [46]

+ Note that there is an obvious error in the definition of £ in Ref. [46].

- 41 -

* * j j " - oa I wb(n,OT(n) Z * g i ft(n)P (ce. 0) , (45..) n l

where

f (n) - 1 , (45.b) o

f£(l) - Р4(Е)рг(е) , (45.c)

f£(3) = p£(E) ^*(Е,е)ря(е) , (45.d)

f£(n>3) - Pjl(EH£(n,E)Pjl(e) . (45.e)

We note that Eq. (45.с) has no physical meaning since w.(l,e)«0 (no D

elastic scattering allowed) and we point out that £* (E,e) follows from Eqe. (42) and (44). In very good approximation (never-come-back

assumption) we have £*(E,e) - w* (E,e) . (45.f)

If the free-scattering kernel is used, Eq. (45.e) is adopted also for

n=3. Eq. (45) gives a summary of the model showing that the reduced

Legendre coefficients f. consist of three factors, referring to inci­

dent refraction, intra-nuclear scattering and outgoing refraction.

However, Eqs. (32) to (42) are closer to the actual programming in

GRYPHON.

We note that the expressions for refraction have been derived for neu­

trons only, whereas the expressions for intra-nuclear scattering hold

only for nucleon-induced reactions. Strictly speaking the model is only

derived for inelastic neutron scattering, but is expected to give

reasonably good results also for (p,p'), (n,p) and (p,n) reactions. For

a-emission the results are no more than indicative; for other particles

or f-rays no angular-distribution calculations are performed. The code

also gives angular distributions for multi-particle emission (using the

free nucleon-nucleon scattering kernel in the higher cycles); again the

results are only indicative. In most cases it is satisfactory to assume

lsctropy in secondary and higher emissions by setting LMAX • 0.

- 42 -

The input variable KERNEL of GRYPHON controls the options for the scat­

tering kernel as follows (see sec. 4.1):

KERNEL « 0; angle-energy correlated kernel for n « n +2;

KERNEL » 1; free scattering kernel, Eq. (40);

KERNEL « 2; averaged Kikuchi-Kawai kernel, Eq. (41).

2.7.6. Finite-size effects and angular-momentum cut-off

The above-mentioned Kikuchi-Kawai scattering kernel applies for infi­

nite nuclear matter. This kernel has a rather complicated angular

dependence which is illustrated by the fact that a good representation

by Legendre polynomial coefficients is obtained with no less than 40

coefficients. This fine-structure, however, is not observed in scatter­

ing from actual, finite nuclei, due to effects of refraction, finite

size and diffraction. In our model the refraction effects have been

treated by classical theory, using a refractive index representative

for nuclear matter. Diffraction effects could be interpreted as a

result of localization constraints as discussed by Mantzouranis et al.

[38] and recently by Blann et al. [43]. Due to these localization

effects the maximum orbital angular momentum is restricted to £ = kR max

where к is the nucleon wave number and R the nuclear radius. This corresponds to an angular resolution of ДО • 2ir/& or about 19° for I "9, which is a reasonable value for 14 MeV neutrons on medium-max mass nuclei (Nb). It is difficult to introduce this quantal effect into the present model. A practical solution is to truncate the Legendre polynomial series to a small number, which leads to a low angular resolution due to the reduced number of oscillations in the angular distribution. In the GRYPHON code this truncation method has been followed, the maximum order of Legendre coefficients for which data are given in the code being 6 at present. This number is based upon the required number of Legendre coefficients to represent the experimental angular distributions upto about 50 MeV (see systematics of Kalbach and Mann [44]).

A justification of this relatively small number of coefficients may also be given from the following considerations. In the spin-dependent exciton model [15] there are two mechanisms that restrict the angular

- 43 -

momenta to relatively low values. Firstly, there is the well-known

limitation of the orbital angular momenta due to the cut-off of the

optical-model transmission coefficients which leads to £ - 6 for 93Nb+n at 14 MeV. Secondly, the intrinsic spin cut-off of the level

density of composite and final states should be considered. This may be

quite important for low values of the exciton number, from which the

major contribution comes. The expression for the spin cut-off parameter

[15]

o2(n) = 0.24 n A 2 / 3

shows that, Independent of energy, the value of a2(n) is relatively

small. This means that for the major precompound emission, i.e. from

n«3 to n«2, there is a limitation of angular momenta from the spin cut­

off in the emission rates [15]. In particular the spin distribution of

the final n=2 states leads to a limitation, which corresponds for A-100

(o2»10) with J - 6.0, independent of energy. These considerations

show that a truncation of the Legendre polynomial expansion to a value

of about 6 can be considered as reasonable.

Following the above-mentioned experimental and theoretical justifica­

tion for a truncated Legendre-polynomial expansion, our problems are

still not fully solved. The question arises how to truncate. Should we

use the first I coefficients of the Kikuchi-Kawai kernel, or should max we try to make a "best" fit to this kernel with % coefficients? In

max the code (KERNEL»0) we give in data statements the first % coeffi-

max

cients, but the second approach seems to be preferable. It is in this

context interesting to note that the truncation leads to unphyslcsl

negative values of the scattering kernel at certain (backward) angles,

but also to increased scattering at angles around 180°. Due to refrac­

tion the distribution is further smoothed and usually negative values

are not encountered In the angular distributions. Still, it is not

physical to start with a negative scattering kernel. This could be

avoided by fitting the I coefficients to the Kikuchi-Kawai distribu-° max

tion with the constraint that the fitted kernel remains always posi­

tive. We expect that this will lead to a decrease in the ratio f./i2»

in accordance with experimental evidence. Preliminary calculations

indicate that this procedure may "solve" the backward-angle problem,

- 44 -

although the chain of arguments given above should be considered as rather tentative. Still, it gives some justification for the semi-empirical adjustments of the free-scattering kernel in ref. [8] and of the Kikuchi-Kawai scattering kernel in ref. [55]. In the last-mentioned case semi-empirical correction factors for the second-order coefficient have been used, based upon available experimental information. It has been checked that these adjustment factors assure positive or nearly positive scattering kernels. This option may be entered by requesting: KERNEL - -1: i.e., as KERNEL - 0, with semi-empirical correction

* factors for y2(E,E-l), cf. Eq. (42.c).

2*7.7» Additional symmetric component

The model discussed in the previous sections (KERNEL ж 0) leads to isotropy in the equilibrium limit. This corresponds with the assumption in the Hauser-Feshbach model that the spin distribution of the levels in the continuum is proportional to 2J+1, where J denotes the spin of the levels [47]. However, for the more realistic spin distribution

R(J) - exp [- •» u J , (46) 2o2 2o2

with a finite value of the spin cut-off parameter o, the HF model for continuum emission [48] predicts a symmetric angular distribution. This is a consequence of the conservation of angular momentum: the incoming angular momentum is absorbed by the compound nucleus, leading to a rotation around an axis perpendicular to the incident direction. Emis­sion from the compound nucleus then leads to an anisotropic distribu­tion, symmetric around 90° [49].

In the model that we have used (KERNEL • 0) there is angular-momentum conservation (in a classical sense) only for the component of emission at the first collision. After the first collision only the "leading" fast particle Is followed that collides with target nucleons in the Fermi sea. In each collision a recoil nucleon absorbs part of the ener­gy and of the angular momentum. After a long lapse of time all incoming angular momentum has been absorbed by the nucleus and a symmetric angu­lar distribution should result. However, also at an earlier stage, when a significant part of the incoming angular momentum has been exchanged, but still the energy is not yet equilibrated, a symmetric component

- 45 -

is to be expected. It was shown by Akkermans and Gruppelaar [10] that

indeed the characteristic times for these processes are different.

Consequently one might expect that emission from already relatively low

exciton numbers should have a symmetric component, in addition to the

forward peaked angular distribution.

In order to estimate the symmetric component one could start from a

spin-dependent exciton model or "unified" model, using the random-phase

approximation for the (pre-)compound contributions. The resulting

expression for the double-differential cross section has been given by

Plyuïko [50] and Fu [5l]. In the weak-coupling limit this expression

can be reduced to [52]

.2 г <*2><lJ> ' n 12a4(n-b)

under the condition

« 2>« 2> —- ~ « 1 , (47.b) оц(п-Ь)

where <*2> is the average value of the angular momentum of the incoming or outgoing particle (denoted by a and b respectively), weighted with the corresponding transmission coefficients:

<42> - £ £3T (E)/E « ( E ) . (48.a) * * I *

This can be parametrized by a linear approximation, valid for not too

small values of E (cf. [49])

<£2> - c(E-E )+2 (E in MeV) , (48.b)

where E c is a Coulomb energy correction

E - 0.75 z Z/A1/3 MeV . (49)

с

The spin cut-off parameter a is calculated according to [34]

.2/3 л ,,, c -—-, .2/3i o2 (n-b) - max [0.24 (n-b)A ' , 0.114 f /g"T.A ] (50)

- 46 -

where the last value given in square brackets is the equilibrium value (f is equal to unity by default).

In the equilibrium limit Eq. (47) reduces to the expression of Ericson and Strutinsky [49]. Therefore, it seems clear that for high values of n, where the previously discussed model leads to isotropy, Eq. (47) should be inserted. With the help of Eqs. (47), (48) and the equilibri­um value of o2 given in Eq. (50) we find for neutron inelastic scatter­ing at low-emission energies с that the incident-energy dependence of f2 approximately cancels and f2 is only a function of e (there is also a weak dependence from the nuclear mass). This is an interesting state­ment that was also found in the systematics [44]. However, the system-atics is based upon experimental data at higher values of e. Here we have shown with arguments based upon a simple theory that the statement of Kalbach and Mann holds at low values of e.

For low values of n there is a problem of how to determine the fraction of т(п) that contributes to the symmetric component. Denoting this fraction by r we could use for the double-differential cross section expression (45.a) with:

<l2Xt£> f (n) - рг(ЕН£(п)р£(е) + r(n) ^ 2 в . (51)

60 0*4 n-n.) D

Qualitatively we may say that r(n) will be large when the number of collisions is large. Therefore r(n) could be equated with the "round­about" fraction

r(n) - 1-т'(п)/т(п) , (52)

where t'(n) is the mean lifetime corresponding to the "never-come-back" assumption in the formulation of Eq. (8). This gives for 93Nb(n,n') at E - 14 MeV values of 0.08, 0.39, 0.67 and 0.96 for n - 3,5,7 and 9, respectively; for n > n the value of r(n) is set equal to 1.0.

Application of Eq. (51) leads to problems at very high incident and very high emission energies together with low values of n when the weak-coupling limit (47.b) is no longer valid. This could yield a very large and unrealistic symmetric component. Therefore, we limit the

sym value of f2 to at most 0.05, which is already beyond the validity of

Eq. (47.b). In the derivation of Eq. (47.a) it was assumed that j2(x) •

- 47 -

x2/15, that holds upto about j2(x) - 0.2 or f^" « 0.2/4 - 0.05. With these precautions, Eqs. (48-52) may give a first estimate of the sym­metric component.

GRYPHON performs the calculation of Eq. (51) only when с in Eq. (48.b) is positive. Additional print output is given, i.e. the average values

sym of r and f2

E *(п,е)т*(п) R<0 * 1 " T—7 г-7-Г- (53.a)

E w(n,e)T(n)

n

and E «(n.OTOOf*7"

*?*" ж " ,. , Г Т - ï — , (53.b) 2 E w(n,c)T(n)

n

sym where f2 is the second term of Eq. (51).

The input requirements of GRYPHON for this option are с (Eq. 48.b) and f (Eq. 50). The last-mentioned parameter may be used to adjust the equilibrium value of the spin cut-off parameter.

Finally, we make the following notes: (1) Due to the Coulomb barrier, the symmetric component is more impor­

tant for neutrons than for charged particles that are involved in the reaction.

(2) In Ref. [46] the "roundabout fraction" was mentioned, but not used. Instead, a "multi-step compound" or "bound-level" fraction [44] was used, with similar characteristics:

rbound<>'h> - И ^ - А ^ 1 • <5*>

where В is the binding energy. If с (Eq. 48.b) is entered with a negative sign Eq. (52) is replaced by Eq. (54) in the code. The present approach (Eq. 52) is more in the spirit of the generalized exclton model.

(3) In Ref. [46] the first term of Eq. (51) was reduced with the "never-come-back" fraction l-r(n). On second thought this seems

- 48 -

double counting, since the «aster equation already follows the

complete process. The point is to find a correction for the isotro­

pic term that is Implicitly used for "roundabout*' processes.

2.7.8. Systematica of Kalbach and Mann

Another possibility to estimate the fraction of symmetric emission

comes from the work of Feshbach et al. [51], who introduced the

distinction between multi-step-direct (MSD) and multi-step-compound

processes (HSC). It was postulated that these processes proceed through

unbound and bound states, respectively. For the MSC reaction mechanism

the random-phase approximation was assumed to be valid, leading to a

symmetric angular distribution; for the MSD reaction mechanism a

distinct forward-peaked angular distribution was predicted. It was

assumed that the MSD and MSC branches are independent of each

other.

Kalbach [4l] has introduced some of these ideas in the exciton model by

defining Internal transition rates from unbound to unbound, bound to

bound, unbound to bound and bound to unbound states and by limiting the

emission to unbound states only. In Kaloach's model the MSC definition

of emission is based upon the processes that have passed through at

least one bound state and eventually through one unbound state. This

definition was made since in the exciton model internal transitions are

treated independently from emission. Kalbach has confirmed that the MSC

and MSD mechanisms are almost uncorrelated.

In the systematica of Kalbach and Mann [44] the fraction of (symmetric)

MSC emission is used as follows:

f^e) - [1 - R ^ e ) ] fj y e t(0 (odd i) , (55.a)

ft(e) - fjy , t (c) (even I) , (55.b)

syst where I ' (e) has been obtained from experimental data of reactions

predominated by the MSD process. At high outgoing energies e the frac­

tion IL», is usually small, even at E - 14.6 MeV (about 3% in the case

- 49 -

of 93Nb+n); at lower outgoing energies this fraction Increases.

We note that Eq. (55.b) aeans that Kalbach and Mann assume that the t«2 component for MSD is equal to that for MSC. This is quite arbitrary and leads to (too) high values of f2 at low values of c. At high emission energies the generalised exclton model [35] is in good agreement with the systeaatics.

In GRYPHON there is an option (KALBACH*1) to simulate the KM-systeaat-ics by equating ^.«.(O vith Eq. (53). This gives results that are quite close to the original Rf-systeaatlcs, without the need to distinguish between MSC and MSD processes. In this option additional print output is given. For higher emission cycles the option KALBACH-1 should not be used.

2.8. Energy grid and integration method

Below we list some specific features of GRYPHON, the knowledge of which may be helpful to the potential user of the code. a. The energy grid used in GRYPHON is taken over from the optical-model

calculation that produces the inverse cross sections that are read from tape by the subroutine INV.

b. This energy grid does not need to be equidistant, but may have a varying bin site.

с Integration is performed according to the trapezoidal rule instead of using rectangular bins. At the endpoints Integration corrections are applied such that there is no loss of flux,

d. On input, quantities are to be given in the laboratory system, whereas the output quantities are given in the center-of-mass system.

2.8.1. Non-equidistant energy_grid

The energy grid is read by GRYPHON from the file INV that contains the (positive) values of e and ofl l n v<0» о l n y ( e ) , <*a i n v< e) needed In Eq. (24). The maximum number of energies 1 should not exceed 99. It

max is recommended to use a relatively fine energy grid at low values of e in order to represent the evaporation peak with sufficient accuracy. The energy grid ei, e2, , e. is the same for all emitted

Jmax particles.

- 50 -

2.8.2. Triangular energy bins and integration procedure

In most codes rectangular energy

perform the energy integration:

In most codes rectangular energy bins with width Ae are used to

e IX

ƒ f(e)de - Z f(e )Ae , (56.a)

0 j 3 3

where Ae. • e.-e, ..In GRYPHON a linear-linear interpolation scheme is J J J"1

used, i.e. integration is performed with the trapezoidal rule. This is

achieved by adopting Eq. (56.a) with

A C j - 0.5 ( e ^ - e ^ ) • <56.b)

For the end points, assuming f(0) » f(e ) * 0, the widths are defined

as:

A€j » 0.5 Ej , (56.c)

Ae - 0.5 (e - e .) . (56.d) max max max-1

Eq. (56-a-d) may be interpreted as a summation over triangular energy

bins, as It? shown in Fig. A. For functions f(e) * g(e)ci>(e), where ш is a strongly "arying function of e, a still better approximation might be

ƒ g(e)u(e)de - I g(e.) n (e)Ae , (57.a) 0 i

with

nt(e) - ƒ w(e)de/Ae1 . (57.b)

Aei

The latter approximation was followed in the integration over discrete

levels (Sect. 2.2.2) in the energy range from e to e , where n. cut max l

represents the average number of states in Ae . For the

- 51 -

calculation of n. In the intervals containing e . and e appropriate 1 cut max integration corrections are made in GRYPHON.

da de

Figure 4. The trapezoidal rule for integration in GRYPHON as a summa­tion over angles.

2.8^3. Integration problems in multi-particle emission

If we consider the reaction specified by (a;b,c) and (E; е., е ) the GRYPHON code first calculates

do dT ( a» b ) Дек к (58.a)

for all physically accessible energy bins Ле. . This information is used in the next cycle to compute

dTdTT ( a ' b ' c ) Д е Л • к Я,

(58.b)

provided that secondary emission is energetically possible, i.e. с should be less than a certain energy e ,. Problems arise for the

maxl

- 52 -

last bin with energy below e ., as can be seen from the illustration maxi

below.

da de

ei £2 ез ei» ее э maxi " max Figure 5. Problem of integration at the end points. Without corrections

there will be a loss of flux (dotted area).

In this case the triangular bin is also limited to e , (hatched maxi

area), neglecting the remaining contribution (dotted area). This means that part of the incoming cross section

do , . . . ТГ (a'b) Дек к

is not used for further particle decay. This part is attributed to Y ~ deray in GRYPHON by a renormalisation of the f-ray transition rates. We note that by this method the last bin of secondary emission seems to be underestimated. However, the probability for secondary decay from the region e. •*• e .is smaller than the corresponding transition c. + e for this bin, which may compensate for this effect; see figure 6. This figure shows that satisfactory results can only be obtained when the bin width is relatively small near e . (and e ). This is particu-

maxl max larly important for neutron emission, for which there are no Coulomb barriers. We note that for (secondary) у-emission there are of course no thresholds (e 'maxl

e ), at least in a continuum representation. max

- 53 -

к-1 т

'maxi

'k+1

' Л - ' l

(ground state)

-£+1

max (ground state)

Figure 6. No particle emission is possible from the region e to maxi

ек+Г

2.8.4. Consistencg^ln^multi-garticle^and Y'ray^emisslon

In the code GRYPHON the su* 11 emission cross sections is equal or very close to the total incoming cross section. The difference is printed (typically it is on the order of 10~6). Problems may occur in multi-particle or y-ray emission if no further emission is possible because of vanishing level densities in all residual nuclei. Further­more, the user may exclude further emission when the incoming cross section is below a specified value (EMISMIN), which is set equal to 1 mb for particles and a factor of 100 smaller for Y~*ays by default.

When the user selects the option that Y~ray competition is not consid­ered (I0T(7) • 0) the code automatically assumes that the neutron threshold Is the lowest particle-emission threshold; no charged-parti­d e emission is allowed when it is impossible to emit a neutron. This approximation is made because otherwise states below the neutron separation energy would -in the absence of y-гау competition- be de-excited completely by particle emission. In this option the code aseumee that no y-vay emission is possible from the energies above the neutron separation energy and that only yray emission (without calcu­lating the spectral distribution) occurs below that energy.

- 54 -

Thus, in this case the code calculates all particle emission spectra, but only an energy-integrated Y-ray emission cross section. Again, the sum of all emission cross sections agrees with the incident cross section.

2.9. Coupling to Hauser-Feshbach codes

The spin-dependent, unified exciton model can be denoted as [15]:

Z w (п,е)т (n) do(a,b) _ Jn n - i r _ - o : , (,t-o) , (59)

JÏÏ Z w(n) T (n)

jn

where т is the solution of a master equation integrated over time, like Eq. (3), with all quantities indexed by the spin J and parity П of the composite state and with initial condition:

qJII(n , t«0) = (cJ I I /o )6 . (60) M o' ' a a' nn о

At equilibrium Eq. (60) coincides with the Hauser-Feshbach (HF) expres-JIl sion. Assuming that the spin-parity population of т (n) is independent

of n it can be shown [l5] that the following substitution in the HF formula leads to consistency with the spin-independent exciton model:

1П p(I,n,E ) - Z <n"b? u)(n-b,Eres)pb(n) (61)

n R (n)

with

Q (n)t(n) -Шс<п'Е> Шс(п,Е)

b The (normalized) coefficients p (n) are calculated by GRYPHON and are given on file COF. They are read by the (modified) HF codes PERINNI and

- 55 -

GNASH-ECN. In applications reported so far [57] we have assumed that

the spin distribution functions R are independent of n, which is rather

too simple. Further study on this topic is in progress.

- 56 -

3. STRUCTURE OF THE GRYPHON CODE

3.1. Main program

The structure of the GRYPHON code is depicted in Fig. 8. After initial­ization of those variables that do not change during execution of GRYPHON, input is requested through an interactive procedure. Next, GRYPHON calculates the various nuclear quantities needed for the cross-sections computations by calling a number of subroutines and functions that are described below. Finally, the desired cross-sections etc. are computed and given to output by calling the routine PRANG. If calcula­tion of multiple-emission processes is required, the program returns to the input procedure (vhich is strongly simplified in this case). Execution of the program is terminated interactively by the user or, automatically, if the nuclei under consideration have been deexcited. The input descriptions of GRYPHON is given in Sec. 4.

GRYPHON calls the following subroutines and functions:

a. GAMMAF Calculates the Gamma function Г(х) for 1 < x < 2 through a very accurate polynomial approximation (absolute error < 5xl0~5). This function is used for the calculation of the internal transition rates, see Sec. 2.3.3, Eq. (21).

b. MASS Selects the mass of the incoming and outgoing particles.

с BNDG Calculates the binding energy of a particle in a given nucleus. To this end BNDG calls MASS, EXCESS (gives the mass excess in amu of a nucleus) and ICHARG (see below).

d. ICHARG Selects the electric charge of the incoming and outgoing particle types.

e. IUV Reads the energy grid and the inverse cross section for the most important particles (n,p,a), needed to calculate the emission rates (see Sec. 2.A), for a given composite nucleus from file INV. If a

- 57 -

nuclide is not found, a nearby nuclide is taken* Note: the inverse reaction cross sections for the other particles and for Y-ray emis­sion are calculated by DOSTR.

f. REFRACT Calculates the Legendre coefficients for the refraction at the nuclear surface of the incident and/or outgoing particles, which are employed in the calculation of the angular distributions (see Sec. 2.7.1. Eqs. (36) and (37)). REFRACT calls DCADRE, an IMSL integration routine. The integrand is provided by the function FIN, which calls the function P giving the Legendre polynomials.

Warning; A numerical integration routine, such as DCADRE, should be supplied by the user.

g. REPLACE Selects the level-density parameters g and A, the normalization factor Nj for the photo-absorption cross section and the set of discrete levels for the composite and residual nuclei occurring in a given calculation cycle of GRYPHON, from the data given at input for a set of nuclides (see Sec. 4.1).

h. GDILG Calculates the single-particle level-density parameter g according to the systematics of Dilg et al. [17] (back-shifted Fermi gas model). Calls FINT, a linear Interpolation routine.

i. PDILG Calculates the energy shift Д according to systematics based upon Dilg et al. [17] (back-shifted Fermi gas model). Calls FINT.

j. PRANG Computes emission cross sections, spectra and angular distributions according to the master-equation exclton model for (pre)compound decay. See Sec. 2 for the theory employed and Sec. 3.2. for its structure.

- 58 -

(initializations)

(interactive input)

(computation of

nuclear variables

needed for

calculations in

PRANG)

(cross-section

calculations)

GAHHAF

MASS

BNDG

ICHARG

INV

REFRACT

REPLACE

GDILG

PDILG

PRANG

EXCESS 1

ICHARG I

MASS I

_ A DCAD DCADRE I 1 '

FINT H H"" I

]

Figure 8. Structure of the GRYPHON code.

- 59 -

3.2. PRANG master-equation routine

PRANG is the central subroutine called by GRYPHON and computes cross sections, spectra and angular distributions according to the (general­ized) master-equation approach of the pre-equilibrium exciton model. A survey of the theory has been given in Sec. 2. The structure of PRANG is shown in Fig. 9.

The main part of the PRANG routine performs the following actions: - calculation of the emission rates, including the associated Q-factor

(see Sec. 2.4); - calculation of the internal transition rates (see Sec. 2.3); - calculation and printout of cross sections, spectra and angular

distributions (see Sees. 2.1 and 2.7).

PRANG calls the following subroutines and functions:

a. MASS Selects the mass of the incoming and outgoing particle types.

b. ICHARG Selects the charge of the incoming and outgoing particle types.

с MDIF Selects, for each computational cycle of GRYPHON, the difference (p-h) between the particle and hole exciton numbers.

d. DOSTR Estimates the inverse-reaction cross sections, needed for the emis­sion rates of 3He, d and t particles using the Dostrovsky approxima­tion [27]. In the case of Y-ray emission, a Lorentzian form is assured for the photo-absorption cross section. Cf. Sees. 2.4 and 2.5.

Warning: In the case of n, p, a DOSTR should not be used and the Inverse cross sections must be read from tape, produced by an optical-model calculation.

e. STATED Calculates the particle-hole exciton state densities, according to the Williams formula [12], see Sec. 2.2, Eq. (11). Note: this routine does not include the «normalization factor f(U), defined in Eq. (15.b).

- 60 -

Note also that the Pauli correction of Williams [12] (Eq. (12)) has been replaced by the expression of Kalbach [13], see Eq. (13).

f. SPMS Computes the multiplication factor (2s+l)u for the emission rates, cf. Sec. 2.4, Eq. (24).

g. RHOG Calculates the gamma-ray emission rates (see Sec. 2.5).

h. TRANS Calculates the internal transition rates with the phenoaenological parametrization of <M2> of Kalbach according to Eq. (18.b), see Sect. 2.3.1. In this subroutine the transition rates are calculated as in the previous PRANG code, according to the formulae employed by Betak [45], i.e. with corrections for the finite hole depth and with the Williams state density formula [l2] (without renormaliza-tion to the back-shifted Fermi-gas model).

i. NHEAD Prints the heading for each program run.

j. ТАК Solves the time-integrated (generalized) master equatior for an arbitrary initial condition, according to Eqs. (5) to (7) of Sec. 2.1.

k. PREQ Computes the mean exciton-state lifetimes in the case of the never-come-back approximation, see Sec. 2.1, Eq. (8). In this case, PREQ is called instead of ТАК.

1. KMSYST Gives the Legendre coefficients of the angular distributions of emitted particles according to the systematics of Kalbach and Mann [44].

- 61 -

GRYPHON

Г~Т (initiali-zat ions) 4.

(inverse cross-sections)

(state densities)

(emission rates)

(transition rates)

print

(solut ion of the master equations)

(spectra, ang.dist.)

print

4

H:

MASS

ICHARG

MDIF

DOSTR

STATED

SPMS

RHOG

TRANS

NHEAD

ТАК

PREQ

KMSYST

H

Figure 9. Structure of the routine PRANG.

- 62 -

4. USER'S MANUAL OF GRYPHON

4.1. Input description

The program GRYPHON requests interactive "list directed" (free format)

input, each input stateaent to be ended by a slash. The files INPUT and

DISPLY should be connected with the terminal. Print-output will appear

on file OUTPUT. Batch-mode operation is possible by disconnecting the

files INPUT and DISPLY.

The input for the higher-emission cycles is strongly simplified as

compared to that for the first emission. Therefore, they have been

separately treated below. Input parameters that are not given on input

by the user are set to the default values by the program.

Please note that additional input is required from file INV (Sect.

4.2).

4.1.1. First emission cycle

The simplest input consists of only very few parameters: the specifica­

tion of the reaction and of the incident energy.

These quantities are underlined in the following table.

- 63 -

a. Card set 1 (Specification of projectile and ejectiles)

Input variable

IPHIST

IOT(I),1-1,7

Explanation

Gives incident particle types • 0: target excited by previous

emission (i.e., hig'aer emission cycle)

• 1: neutron; « 2: proton; • Э: alpha (**He); - 4: tau (3He); - 5: deuteron (2H); » 6: triton (3H); « 7: gamma; >10: follow only one particular

reaction sequence, indicated by the "reaction history code" (see Sect. 2.6.1);

< 0: stop execution of GRYPHON. Gives outgoing particle types to be considered. IOT(I)>0: a particle of type I (see explanation of IPHIST) is emitted; output for this particle is given; I0T(I)-0: no emission of particle type I is allowed; I0T(I)<0: competition through emission of particle type I is included in the calculations; however, no output for this particle type is given. Note: competition of n,p,a is always included.

Default

0

К З : I0T(I)=-I I>3:I0T(T)«0

- 64 -

b. Card set 2 (Reaction mechanism)

Input variable

NEQL

Explanation

Controls (pre-)equilibrium option of calculation, see discussion in Sec. 2.1: «0 :unified pre-equilibrium plus

equilibrium calculation; «1 :pure equilibrium calculation

(initial condition of Eq. (10));

»-l:Weisskopf-Ewing formula (future option)

>2: pure pre-equilibrium calcula­tion (never-come-back assump­tion: mean lifetimes according to Eq. (8); similarly for the angular distributions).

Default

0

c. Card set 3 (Target specification)

Input variable

ZT AT

Explanation

Proton number of target nucleus Mass number of target nucleus

Default

0 0

- 65 -

d. Card set 4 (y-ray parameters)

Input variable

GSIGRl GGAMRl GERl GD

MT

Explanation

Giant-dipole resonance parameters for the calculation of the photo-absorption cross-section referring to the composite nucleus according to Eq. (30): o^mb) T1 (MeV) Ej (MeV) Direct and semi-direct reaction multiplier * 0: method of Ref. [30], Eq.

(29), for the y-ray emission rates;

* 1: method of Ref. [29], Eq. (28);

• 2: method of Ref. [28], Eq. (27).

See further Sec. 2.5.

Default

0a> 0a) 0a> 1.0

0

If on input a,, Г. or E, are less or equal than zero the following values are taken from the systematics by GRYPHON (A and Z relate to the composite nucleus): Г. - 5.5 MeV, E, - 163.*та=2Т/А4/3 MeV,

* 75.Z(A-Z)/Ari mb.

e. Card set 5 (Control parameter for residual nuclei information)

Input variable

NGP

Explanation

Possibility for NGP nuclei, to be specified in card set 6, to change the level-density parameters g and A, to define the photo-absorption constant Nj and to give the set of discrete levels (see f). Maximum of NGP is 50

Default

0

- 66 -

f. Card set 6; up to 2*NGP cards (Residual nuclei inforaation) For each N, where N - 1, NGP, one card (8 fields) should be reserved to specify the desired changes for an indicated nucleus, giving the parameters listed below. If discrete levels are to be con­sidered, they should be given through an additional card.

Input variable

IZZ(N)

INN(N) GG(N)

PP(N) ECT(N)

GNORM(N)

LDSC(N)

SDSC

Explanation

Proton number of nucleus for which changes are to be made. Mass number of this nucleus. Level-density parameter g for this nucleus. Energy shift д for this nucleus Determines for this nucleus the maximum emission energy to be considered according to e_„,. «

a max E-B-ECT(N) (in the absence of discrete levels). Otherwise, it is the energy up to which the discrete levels are given. Gives the photo-absorption normalization parameter Nj of Eq. (30) for this nucleus. Gives the total number of discrete levels to be considered for this nucleus (max. 30). Gives the spin cut-off parameter of discrete levels.

Default

0 (n.a) 0 (n.a)

-10b> -10b>

oc>

1.0

0

2.5

Only if LDSC(N) > 0, the following input should be given In addition (one card, LDSC(N) < 30 fields)

EDSC(N,LDSC(N)) Array of dimension LDSC(N) giving the set of discrete levels for this nucleus in increasing order, up to the energy ECT(N) (MeV).

30*0c)

b^ If GG(N) and PP(N) are not given on input, i.e., are equal to -10.0, the parameter g and Д are taken from the systematica of Dllg et al. [17].

c) If LDSC(N)-0 (no discrete levels), ECT(N) is set by the program to the value ECT(N) • PP(N)+1./GG(N), If It has not been set to a higher value by the user. This is done in order to avoid a region where the exciton-state densities are not well defined (cf. Eq. (11)). Warning: If LDSC(N) > 0 (presence of discrete levels), the user should ensure that the eet of discrete levels is given up to ECT(N). Only then the program is able to smoothly join the number of levels at the point where the continuum region goes over into the discrete region.

- 67 -

It may be helpful here to point out that the subroutine REPLACE auto­matically selects from card set 6 the quantities required with respect to the composite and residual nuclei occurring in the computational cycle that is to be executed.

g. Card set 7 (Precompound model-parameters)

Input variable

NOUT

EMISMIN

FITMFP

FKME1

FORM(I),1-1,7

NHO

Explanation

Print switch - 0: emission and transition

rates, lifetimes, cross-section, spectra and angular distributions are printed;

Ф 0: no printout Minimum cross-section (mb) below which further decay is not con­sidered in the next cycle. Mean free path multiplier к of Eq. (21). If negative the transition rates are calculated according to Ref. [45]; its absolute value being equal to the matrix element constant с of Eq. (18); suggested value:-650 Multiplication factor for the n > n0 internal transition rates only (In this way it is possible to separately consider the contribution of n-nQ). (Pre-)fornation factor Ф^ in the calculation of the emission rates Eq. (24). For neutrons, protons and gammas it is fixed to unity.

Initial-hole number. It is recommended not to change this parameter from its default value

Default

0 (1 for higher emission cycles)

1.0 (0.01 for у)

1.8

1.0

1-3:10(a) 1-4:10(т) 1-5:5 (d) I-6:10(t) Otherwise 1.0 0

- 68 -

h. Card set 8 (Angular-distribution parameters)

Input variable

LMAX

NRFR

KERNEL

KALMANN

V

AV

Explanation

Maximum order of Legendre poly­nomials to be considered for the angular-distribution calculation (max. 6). For multi-particle emission LMAX-0 is often quite reasonable.

Option for refraction (cf. Sec. 2.7) »1 : incoming and outgoing

refraction;

=0 : no refraction;

•-1: incoming-wave refraction only.

Option to determine the intra­nuclear scattering kernel G of the generalized exciton model for the angular distributions (cf. Sec. 2.7); » 0: Kikuchi-Kawai kernel with

angle-energy correlation for nenQ+2;

=-1: idem, with semi-empirical corrections for second-order coefficient

* 1: free-scattering kernel, Eq. (40);

* 2: averaged Kikuchi-Kawai kernel, Eq. (41).

Option to calculate the results of the Kalbach-Mann systematics [44] • 0: normal operation

(generalized exciton model); • 1: Kalbach-Mann systematics is

given (parameter NRFR and KERNEL are inactive then).

Don't use KALMANN-1 at higher emission cycles! Value of real potential depth for refraction in Eq. (37.c), expressed in MeV; default value from Ref. [39] for neutrons. Parameter for energy dependence of refraction potential in MeV'1;

default value from Ref. [39]

for neutrons.

Default

0

1

0 (1 for higher emission cycles)

0

56.3

0.32

1

- 69 -

h. (continued)

CA2

FSG

If different from 0, a symmetric component is added; the absolute value of CA2 is equal to the constant с of Eq. (48.b). The plus or minus sign of CA2 determines whether Eq. (52) or (54) is used, respectively. Factor f in Eq. (50) (only needed when a symmetric component is added).

0.0

1.0

j. Card set 9 (Incident-energies and energy-dependent quantities)

Input variable

EPART

SIGNE

FIT(L), L-l, LMAX

Explanation

Laboratory energy (MeV) of the Incident particle. Non-elastic cross section (mb); by default the composite forma­tion cross section is calculated from the inverse-reaction cross section, given on file INV. Fitting factors for the eigen­values V (only for test calcula-

X» tionb).

Default

0.0 0.0 (means о )

CI

1.0

1 The input of card set 9 may be repeated for an arbitrary number (< 100) of incident energies, ended by a slash. This completes the input description of GRYPHON for the first-emission cycle.

- 70 -

4.1«2. Higher emission cycles

In the case of the higher emission cycles (i.e., multi-particle emis­

sion, see Sec. 2.6), the input for GRYPHON is strongly simplified,

since many quantities are now determined by the program itself. In this

case, the following cards are requested (only the reaction specifica­

tion is actually needed):

a. IPHISX, IOT(I), 1-1,7 Cf. card set 1; note that IPHIST must be nega­

tive, zero or > 10 here.

b. NEQL Cf. card set 2; at low incident energies

NEQL-1 could be used, but this does not

decrease the calculation time- (For this

reason, an option NEQL»-1 giving the

Weisskopf-Ewing formula will be Included in

the near future).

c. LMAX Cf. card set 8; in most cases it is sufficient

to assume isotropy In all higher cycles

(LMAX=0).

d. NOUT, EMISMIN Cf. card set 7; the print output, however, is

suppressed by default for all higher cycles;

NOUT = 1.

4.2. The Inverse-reaction cross section file INV

Before running the code, a (binary) file* with inverse-reaction cross

sections should be prepared. This file also contains the energy grid

used for representing the emission spectra. The definition of these

quantities follows from Eq. (24.a). Please note that e in Eq. (24.a)

refers to the emission energy in the center-of-mass system. The file

should be prepared with an optical-model code; it should contain all

necessary composite nuclei entering the calculation. However, when

information for one or more nuclei is lacking, the code GRYPHON auto­

matically selects the data for the nearest nucleus. The file is

organized as follows:

* In the export version of the code the file IN1/ is a coded file with

"list-directed" output (free format).

- 71 -

The file INV

INPUT VARIABLE

N

EN(j),J=l, N

Al, Zl

SGl(l,j),J=l,N

SGl(2,j),J=l,N

SGl(3,j),J=l,N

EXPLANATION OF QUANTITIES

Number of energies (N < 100)

Energies in MeV ordered from low to high.

The minimum energy should be positive, the

last energy should be high enough to

perform the calculations (usually higher

than the incident energy!).

Mass number and charge number of composite

nucleus entering the calculations.

Inverse-reaction cross sections for

neutrons (mb).

Inverse-reaction cross sections for

protons (mb).

Inverse-reaction cross sections for

alphas (mb).

This information may be repeated for an arbitrary number of other

composite nuclei; it is not necessary to order the values of Al and

Zl.

The file INV is a binary file* that contains unformatted data in the

above-mentioned structure. It is read by the subroutine with the same

name.

4.3. Output and scratch files

The GRYPHON code utilizes the following files:

TAPE 1: INPUT, to be connected to a terminal;

TAPE 2: INV, inverse-reaction cross sections (binary*);

TAPE 4: SPECTR, calculated spectral information (binary);

* Coded file In export version of code.

- 72 -

TAPE 6: DISPLY, to be connected to a terminal; TAPE 7: COF, coefficients for coupling with HF code; see Sect. 2.9 and

Ref. [15]; TAPE 8: OUTPUT, print output; TAPE 11: DIST1 .exciton distribution for next cycle TAPE 12: DIST2 or from previous cycle, cf. Sect. 2.6 (binary).

The input files INPUT and INV have been discussed In the previous sections; the files DISPLY and OUTPUT give control output and print output, respectively. The file COF is required for the coupling to HF codes, see Sect. 2.9. The two scratch files DIST1 and DIST2 have already been introduced in Sect. 2.6. The main output file, to be used by sorting codes is the file SPECTR.

4.3.1. The SPECTRflie

This binary file contains one record with the energy grid, followed by a large number of records with spectral information: EN (1, ..., 99), IPHIST, IEHIST, IA1, IZ1, IA2, IZ2, L, J , jf, EF, EP, SIGR, SIG, SPEC (jj jf)

These quantities are explained below: EN - energies from adopted energy grid; IPHIST - reaction-history code upto last-emitted particle

(Sect. 2.6.1); IEHIST - energy-history code upto previously emitted particle

(Sect. 2.6.1); IA1, IZ1 - mass and charge number of previous nucleus or

target; IA2, IZ2 - mass and charge number of residual nucleus; L * Legendre order (-1 means angle-integrated); j., j. • Initial and final indices of emission energies; с * final energy of emitted particle; SIGR ' a , the non-elastic reaction cross section;

a SIG - sum over all elements of array SPEC; SPEC (J4, ..,Jf) " spectrum of last-emitted particle.

For the reaction history code 112 or (n,np) and energy history code 12 (first Incident energy E , second grid energy e ) :).* meaning of SIG

1 n,2

- 73 -

and SPEC is as follows:

SIG • j ~ (n,np) Ae2 . n

SPEC( j) - -=- £ (п,п,р)Де2Де . n,2 p,j J

4.4. Sample problems

Two simple sample problems are chosen to illustrate the execution of the code GRYPHON. The input shown below refers to neutron-induced reac­tions on 93Nb at incident energy of 14.5 MeV. For moet of the input the default options have been selected. In sample problem 1 the angle-integrated cross sections are calculated. In the first and second cycles the ejectiles are neutrons (1), protons (2), a-particles (3) and Y-rays (7). The interactive GRYPHON input and output is shown on the next page. The user input has been underlined; a slash means: default values. The print output of GRYPHON is only given for the results obtained after the first cycle by default, see pages 75 to 81 • The results are sorted by the code SPSRT1 of which the interac­tive input and output is reproduced on page 82 • The output on the CRT screen is given only for the total neutron production cross section and emission spectrum. The print output of SPSRT1, shown on pages 82 to 91 gives all computed cross sections and the emission spectra of the last particle. In addition the total production cross sections and spectra are given and (on page91 ) the total transmutation cross sections. A reduced print of the output of SPSRT1 (not given) contains only the energy-integrated cross sections. There are several other sorting codes with different possibilities, cf. Table 1.

- 74 -Sample problem 1 (Interactive GRYPHON I/O)

«t««*4t«*<tt«M*Mtt GRYPHON i t i H i i M H i H H f H i INPUT PREPARATION FOR FRANS FREE FORMAT INFUT, END WITH / GIVE INC0niN6 AND GUTGCING PARTICLE NUMBERS 0=TAR6ET IN EXCITED STATE. ^NEUTRON, 2=PRDTON, 3=ALPHA, 4=TAU ,5=DEUTER0N, t>=TRIT0N, 7=GAHNA (INSTEAD OF О THE PARTICLE HISTORY NAY BE ENTERED) A NEGATIVE I I O I I N 6 PARTICLE NUMBER TERMINATES RUN A NESATIVE OUTSOINS PARTICLE NUMBER - 4 , - 5 . - 4 , O R - 7 MEANS ONLY COMPETITION INCLUDED .'NO OUTPUT) COMPETITION OF N,P,A IS ALWAYS INCLUDED

'M Ч * | У ;

SIVE 1 , 1 FOP PURE EQUILIBRIUM OR PURE P R E E B Ü I L I B R I U H = i

SIVE I .A OF TARGET = 4 1 . 9 3 /

GIVE 3.D.R. PARAMETERS FOR 6ABNA ENISSICN OR SLASH

GSISR (MB) ,uS«1R (BEV) ,5ER !HEV! ,GD ,1ETHOD - j _

POSSIBILITY TG 3VERKRITE LEVEL DENSITY PARAMETERS

DEFAULT SETTINGS: SYSTEMATICA OF W.DILG ET AL.

POSSIBILITY TO RENORBALIZE PHOTO-ABS XSECT.(DEFAULTS.0)

DEFAULT SETTING: VALUES FOR COMPOUND NUCLEUS

POSSIBILITY TO GIVE DISCRETE LEVELS { M I . 30)

READ NUMBER OF CHANGES =£

'Ш «AY OVERWRITE N0UT,E!1I3>1IN,FITHFF,FKBE1,FQRHFACTQRS, Ш = / GIVE LBAI .NRFR,KERNEL,M.BANN.V ,HV ,CH2 ,FS6 ? - i

GtvÊ~INCWlN6 ENERGY <«EV», «ON-ELASTIC «SECT (MB),

FIT PARAMETERS ANGULAR DISTRIBUTION:

DEFAULT «SECT = C2NPÖÜND FORMATION «SECT FOR N,P,A

= CQSTRQVSKV APPRO)!. FOR OTHER PARTICLES.

E ; I > , s i S N E i i ; , F I T H S = U . 5 /

E C ; , S I 6 N E C ) , F I T ! L i =/

« t t t * m * f t m m t t t GRYPHON » t # t m » * m # M # # * « t

INPUT PREPARATION FOR PRAN6

FREE FORMAT INPUT. END WITH /

GIVE INC0MIN6 AND'OUTGOING PARTICLE NUMBERS

•MARSET IN EXCITED STATE, !=NEUTRON, 2=PR0TON,

3=ALPHA, 4=TAU ,5=DEUTERQN, 6=?RIT0N, 7=GAMHA

(INSTEAD OF 0 THE PARTICLE HISTORY MAY BE ENTERED)

A NESATIVE INCOMING PARTICLE NUMBER TERMINATES RUN

A NEGATIVE 0UT60INB PARTICLE NUMBER - 4 , - 5 , - 6 , O R - 7 MEANS

ONLY COMPETITION INCLUDED (NO OUTPUT)

COMPETITION OF K,P,A IS ALWAYS INCLUDED

=0. 1 . 2 . 3 . 7 /

GIVE 1,> 1 FOR PURE EQUILIBRIUM OR PURE PREESUILIBRIUB -l_

SIVE LMAX=_/

GIVE NOUT.EMISMIN, E1.3E / : £

ш * ш * * * » * * ш м * » GRYPHON * » # # » # • # * * * # # # • * » * • # INPUT PREPARATION FOR PRAN6 FREE FORMAT INPUT. END WITH ,' SIVE INCOMING AND OUTGOING PARTICLE NUMBERS (MARSET IN EXCITED STATE, l=NEUTRCN, 2=PRCT0Nt

3=ALPHA, 4*TAU ,5=DEUTER0N, fe=TftITCK, 7=6AMHA 'INSTEAD OF 0 THE PARTICLE HISTORY MAY BE ENTERED) A NEGATIVE INCCMIN6 RETICLE NUMBER TERMINATES RUN A NEGATIVE 0UT60IN6 PARTICLE NUMBER - 4 , - 5 , - 6 , O R - 7 MEANS ONLY COMPETITION INCLL'CED (NO OUTPUT! COMPETITION OF N,P,A 13 fLWAVS INCLUDED •-M

53*35/6' s:*3ss!' »3+3/89' »3*368i' *3*3Z*3' »3+Э/9Г »3+Э8И' £3+3£SB' £3+3839' £3+369*' £3+3*S£' £3+3693' £3+3S03' £3+3851' £3+3£3I' 33+38/6' 33+3108' 33+360/' 33+3/£*'

*3+3*/Z' •3*33/1' *3+33£t' *3+Э90Г £3+3858' £3+3669' £3+3/95' 23+395»' £3+3£9£' £3+3£83' £3+3/13' £3+3191' £3+3511' 33+38//' 33+348»' 33+3U3' 33+3/3Г IJ+30*fr-00+3000'

S3+3/»6' 53+382!' *3+3!SS' *3+36Z3' •3+3951' £3+3616' £3+31SS' 23+3322' £3+3861' £3+3911' 33+3359' 33+36*2' 33+3»/Г 13+36//' 13+3663' 03+3268' 03+33/Г 61+3/W 00+3000'

00+3000' 33+3833' 33+3»££' ZZ+303*' 33+336»' ZZ+32S5' 33+3209' 33+3»*9' 33+Э//9' 33+310/' 33+381/' 33+383/' 3Z+312/' 33+393/' 33+3*!/' 33+3569' 33+33/9' 33+3*99' 33+3/2*'

££+33££' 13+3131' 81+3033' /I+369I' 91+358»' 91+3£S3' 91+3181' 91+3£91' 91+3663' il+3313' 81+395!' 81+3258' 61+3»9£' 03+3121' 03+361*' 13+393!' 13+30ВГ 33+3531' 61+3/8/'

££+33££' 13+3131' 81+3033' /1+349Г 9I+3SB»' 91+3£S3' 91+3181' 91+345Г 91+339!' 91+398!' 9!+34£3' 9I+3*»£' 91+3095' /1+350Г /J+3££3' /1+38*9' 8I+3I»3' 41+3831' 61+3/8/'

00*3000' 00*3000' 00+3000' 00*3000' 00*3000' 00*3000' 00+3000' 00+3000' O0+3O00' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' O0+3OÖ0'

00+3000' 00*3000' 00*3000' 00*3000' 00*3000' 00*3000' 00*3000' 00*3000' 00*3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000'

00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000' 00+3000'

00+300C' ("''•3000' 00+3000' 00+3000' 90-31»!' 30*3281' 50*3906' 80+3i.»£' 0I+36»9' 31+3295' »1+3£S!' 91+358!' /1+3531' /1+3S35' 81*351!' 81+30/1' 00+3000' 00+3000' 00+3000'

00+3000' 00+3000' 00+3000' 00+3000' IT-3860-

00+333!' SO+3/t!' 80+300»' I!+30£!' £1+3531' »!+30Z*' S!+3»06' /1+311!' 81+3101' 81+222/' 61+309»' 03+3653' 13+343!' 00+3000'

00+3000' 00+3000' ÖÖ+300C' 00+3000' 10+3»1!' 80+35/*'

»l+306£' 91+3i£!' /l+3£6!' 81+3*5!' Bl+3/»8' 61+3192' 03+363!' 03+30!*' 13+313!' 13+3»5£' 33+331!' 00+3000'

L'.

3t.

il

Ы /: 53 IL

13 61 M

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5! £1 II 6 L

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IS 10+30' 03"-69'S ttttt J£'Z

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- Si -

- 76 -

Print output of GRYPHON (cont«)

INVERSE REACTION CROSS SECTIONS

ENCR8Y NEUTRONS PROTONS ALPHAS HEUUft-3 OEUTERONS TRITONS 6MNAS

Л00Е-О5 Л00Е-04 Л00Е-03 Л0ОЕ-О2 .200E-02 .5O0E-O2 .700E-02 .10OE-O1 .200E-01 .500E-01 .700E-01 Л00Е*О0 .200E+00 .300E*O0 .400E*00 .500E*00 .600E+00 .700E+00 .800E+00 .900E*00 Л00Е*01 Л20Е*01 Л50Е*01 .200E*01 .230E*01 .300E*01 .400E*01 .SO0E*01 .600E*01 .700E*01 .800E*01 .900E+01 .100E*02 .110E+02 Л20Е*02 Л30Е*02 Л40Е+02 .150E*02 Л60Е*02 Л80Е*02 .200Е*О2 .220Е*02 .240Е*02 .260Е*02 .280Е*02 .ЗООЕ+02 .320Е+02 .340Е*02 .360Е*02 .380Е*02 .400Е*07

.406Е+06 Л28Е*06 .40SE*0S Л31Е*05 .955Е*04 .655Е*04 .582Е+04 .524Е+04 .455Е*04 .438Е*04 .445Е+04 .454Е+04 .452Е*04 .424Е+04 .391Е*04 .361Е*04 .335Е*04 .314Е+04 .297Е+04 .283Е*04 .273Е*04 .258Е*04 .245Е*04 .236Е*04 .229Е+04 .221Е*04 .208Е+04 .203Е*04 .200Е*04 Л97Е*04 Л91Е*04 Л86Е*04 Л82Е*04 Л79Е+04 Л77Е+04 Л76Е*04 Л76Е*04 Л75Е+04 Л75Е+04 Л71Е*04 Л67Е*04 .16ЭЕ+04 .163Е+04 Л60Е*04 .157Е+04 Л55Е+04 Л52Е*04 Л50Е*04 Л48Е*04 Л43Е*04 Л43Е*04

.000Е*00

.000Е*00

.0О0Е*ОО

.000Е*О0

.0О0Е*О0

.000Е*00

.0О0Е*0О

.О00Е*0О ,ОООЕ*0О .0О0Е*0О ,00ОЕ*О0 .0ООЕ*0О .00ОЕ*ОО .000Е*00 .ОООЕ*00 .000Е*00 .000Е*ОО .000Е*0О .0О0Е*ОО .000Е*00 .ОООЕ*00 .492Е-03 Л54Е-03 .683Е-02 .810Е-01 ,457Е*00 Л72Е*02 .816Е+02 .216Е*03 .380Е*03 .328Е*03 .650Е*03 •748£*03

.829Е*03

.900Е*03

.962Е*03 Л02Е*04 Л06Е*04 Л09Е*04 Л15Е*04 Л19Е*04 Л23Е*04 Л26Е*04 Л28Е*04 Л29Е+04 Л30Е*04 Л31Е*04 Л31Е*04 Л31Е*04 Л31Е*04 Л31Е*04

.ооое+оо

.0О0Е*ОО

.000Е+00

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+00

.000Е+00

.ОООЕ+00

.000Е+00

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+ОО Л23Е-03 Л40Е-03 .461Е-02 .669Е-01 .631Е*01 .337Е+02 Л14Е*03 .253Е+03 .4I3E+03 .567Е+03 .705Е+03 .В26Е+03 .103Е+04 .U8E+04 ЛЗОЕ+04 Л40Е+04 .148Е+04 .155Е+04 Л61Е+04 Л66Е+04 Л70Е+04 Л73Е+04 . 176Е+04 Л79Е+04

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00 •ОООЕ+ОО .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+ОО .ОООЕ+ОО .ОООЕ+ОО .ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО .ОООЕ+ОО .ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00 •ОООЕ+00 .ОООЕ+00 .ОООЕ+00 •ОООЕ+00 •ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+ОО ,ОООЕ+ОО .ОООЕ+00 .ОООЕ+00 •ОООЕ+00 ,ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО ,ОООЕ+00 .ОООЕ+ОО ,ОООЕ+00 ,ОООЕ+ОО .ОООЕ+ОО ,ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО

.ОООЕ+00

.ОООЕ+ОО •ОООЕ+00 .ОООЕ+ОО •ОООЕ+00 •ОООЕ+ОО .ОООЕ+00 •ОООЕ+ОО .ОООЕ+00 •ОООЕ+ОО .ОООЕ+00 •ОООЕ+ОО .ОООЕ+00 .ОООЕ+ОО •ОООЕ+00 .ОООЕ+00 .ОООЕ+00 .ОООЕ+ОО •ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+ОО .ОООЕ+00 .ОООЕ+ОО ,ОООЕ+00 .ОООЕ+00 ,ОООЕ+ОО .ОООЕ+ОО .ОООЕ+ОО .ОООЕ+00 ,ОООЕ+00 .ОООЕ+ОО ,ОООЕ+00 •ОООЕ+00 •ОООЕ+ОО .ОООЕ+ОО

•955Е-13 •955Е-11 •953Е-09 .955Е-07 .382Е-06 •239Е-05 .468E-0S •955Е-05 .ЗВ2Е-04 .239Е-03 .468Е-03 .953Е-03 .382Е-02 .860Е-02 Л53Е-01 .239Е-01 .344Е-01 .469Е-01 •613Е-01 .777E-0I .960Е-01 .139Е+0О .218Е+00 .391Е*00 .620Е+00 •908Е+00 Л69Е+01 .279Е+01 .432Е+01 .642Е+01 •932Е+01 Л34Е+02 Л92Е+02 •277Е+02 •404Е+02 •603Е+02 .919Е+02 Л42Е+03 .215Е+03 .313Е+03 Л99£*03 Л08Е+03 •646Е+02 .430Е+02 .308Е+02 .233Е+02 ЛВЗЕ+02 Л49Е+02 Л24Е*02 Л03Е*02 •899Е+01

- 77 -

Print output of GRYPHON (cont . )

SOLUTION OF THE TINE-INTEGRATED MASTER EQUATION

INITIAL DIST. (1EAN LIFETINE !S)

.IOOE+01

.ÖO0E+0O

.OOOE+00

.OOOE+00

.0ÖOE+0O

.OOOE+00

.Ö0OE+0O

.OOOE+00

.OOOE+00

.000E+Ö0

.OOOE+00

.OOOE+00

.OOOE+00

.OOOE+00

.OOOE+00

.OOOE+00

.OOOE+00

.Ö0ÖE+00

.Ö00E+00

.I29E-21

.1I7E-21 Л21Е-21 .200E-2I •209E-20 .183E-19 .760E-19 .I59E-18 .177E-I8 Л10Е-18 .J90E-19 .794E-20 .928E-21 .609E-22 .215E-23 .380E-25 .290E-27 .700E-30 .•79E-41

- 78 -

Pr in t output of GRYPHON ( c o n t . )

N E U T R O N C H A N N E L

ENER6IT CROSS-SECT. 0.001 0.01 ö.i 1 15 100 CH(HEV) ШВ/PtEV) X X X X I X

XXXUtXmXiXXXXXXXiXXXXXXXXXlUXXXXiXXXnXXXX mmmmmmmnxunmmmmmmuumm ххххххххххххххххххххххххххххххххххххххххххххиххшххшхххххххххх тттттнииитттхтихипитхтхтхититиити хппптпттттхттхттитттптпттттпттт хххшхшххххххххххшххххххххххххххшшхтххшхххххххххххшхххххххххххххххххх шшхххххшшхххххххххшххххххххххххшхххххшхххххххххххххххххххххххххххххххххх xnminmmmmmmnnnmnmnmmmmmmmmmimmimmmi шшхшххххххххххххххххххххххххххххххшххххххшхххххххххххххххшххххххххххххххххххххххх ХХХШХХХХХ1ХХХХХХХХХШХХХХХХХХХХХХХ1ХХХШШХХШХШХИХХ11ХХХХХХХХХХХХХХХХХХХХХХХ1ХХХХ1ХХХШХ1 ШШХХХХХХХХХХХХХХХХХХХХХХХХХХХШХХХХШХХХХХХШХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХ хшхххшхххххххххххххххххххххххххххшхххххтхтхтхххххххххххххххххххххххххххххххххххшхххххнххххх шшшххххххххххххххххшхххххшххххххххшххххххххххххххххшххххххххххххххххххххххххххххххххххххххшх mimmmmmnnmmmxnnmnummmmmxmmmmuuummmimmmmnmm ХХХШХХХХХХХХХХХХХХ1ШХХХХХШХШХХХХХХХХХХХХХШХШХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХШХХ xxxnxxmxxxxxxxxxxixxxxxxxxxxxxxxxxxxxxxxxxmxixxxmxnxxixxxxxxxxxxxxxxxxxxxxxxxxxixxxmxmxmxnxxx шшхшххххххххххшхххшххххшхххххххххххххшхххххшхххххххххххххххххххххххххххххххххххххххххххшхх ХХХХХХХШХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХШХШХШХШХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХШХХХХХХ ХХШХЩХХШШХХХХХХХХХХХХШХШХХХХХХХХШХШХХХШШХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХтХШХХ ХХХШШХХШХХХХХШХХХХХХХХХХХХХХХХХХ:ШХХШХШХШШХХ1ХХХХХХХХХХХХШХХХХХХХХХХ1ХХХ1ХХХ1ХХХХШШХХ шшхшххшххххххххххххххххххххххххшххшхшхххххххххххххххшххххххххххххххххххххххххххххххшхххххх хххтхшххкххххххшххххххххххххххххххшхшххшхтхшхххххшхххххххххххххххххххххххххххххххшхххххх imxmxnmmxnnmimunmmmmummnmmnuuummimmmmmiimmmmm mumxmxnmmmmmmmnmmmmmnmmnmmnnnumxmmmmmmmmm Ш1ШШХХШХХШХШХХХХХХХХХШХХХХШХХХХХХШХ1ХХХХХХХХШХХХХХХХХХХ1ХХХХХХХХХХХХХХХХХХХХХХХХХХХШП ххшххшхххххххххтхххшхшхшххххххххтххшхшхшххшхшххххххххххххххххххххххххххшхшхш тттхтптттпттиптитхтхттхттхтжттттхититттп хххтхтхтххххххтххтхиххтххххтитхтхтхииинпттхтххтхттххххххх хххххххпшшххххххххххххххххххшххххшшхххшхшшххххххшххххххххххххххххххххххххх ХХХШХШХШХХХХШХХХХХХХХХХХШХтХШХШХХНХХШХШХтХХШХХХХХХХХХХХХХХХХХХХХХХ хххххххххххххххххххххххххххххххххшххххххххххххнхххххххшхххххххшхххххххххшхххххххх ХХХШХШХХХХХХХХХХХХХХХХХХИХХХШХШ11ХХХШХШХШХШХ1ХХХШХХ1ХХХНХХХХХХХХХХХХ хххххххшхххххххххххххххххххшхшххххххххшххшхххххшххххххххххххххшхххххххххх хххшхшхшхшхтххххххшххшхтххххшхшххшхшхшххшхххххнххххххххх mmmmnmmmïummxmxxxummumnummmumiixuuxxm хххшхшххпшххшхшхххххххшхшххххтххшхшхшхтххшхшхххх

CALCULATED REACTION CROSS SECTION - 1701. «В FINAL ENERGY= 13.7' flEV

.10Ö0E-Ö5

.1Ö00E-04

.1000E-03

.1Ö00E-O2

.2000E-02

.50ÖOE-02

.7C0OE-O2

.1000E-01

.2ÖÖ0E-01

.5ÖOOE-01

.7000E-01

.1000

.2000

.3000

.4000

.5000

. öOOO

.7000

.8000

.9000 1.000 1.200 1. JOO 2.000 2.500 3.000 4.000 5.0Ö0 6.000 7.0C0 9.000 9.000 10.00 11.CO 12.00 13.00

.2107

.6657 2.101 6.809 9.866 16.90 21.00 26.91 46.36 108.3 151.2 214.2 337.9 496.1 554.6 581.3 53B.8 585.0 574.7 560.7 544.8 510.2 455.9 363.9 276.6

109.2 65.97 46.79 37.47 31.92 27.59 23.46 19.05 13.97 7.682

- 79 -

Print output of GRYPHON (cont . )

P R O T O N C H A N N E L

ENERGY CROSS-SECT. O.rtOl 0.01 0 .1 1 10 100

CKÏNEVI iflB/HEV) X X X X X X

.1Ö00E-05Ï).

.1000E-040.

.1ÖÖOE-030.

.100OE-02O.

.2000E-020. .5000E-Ü20.

.7000E-020.

.1000E-010. •2000E-010. .5000E-010. .70ÖOE-010. .1000 0 . .2000 0 . .3000 0 . .4000 0 . .5000 0 . .6000 0.

.7000 ö. .8000 0. .9000 0 .

1.000 Ö.

1.200 .1736E-05 1.500 .5068E-04 2.000 .1336E-02 XXXXX

2.500 .1668E-Ö1 ХХХШХХХХХХХХХХХХХХХХХХ 3.000 .6922E-01 mmumummummmmm 4.000 1.317 ШХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХШХШХХХХХХХХХХХХХХШХХ 5.000 3 .153 m m x x x x x x x x x x x x x x x x x x x x x x x x x x i x x x x x x x x x x x x i x x x i x x x x x x x x x x x x a x x x x x i.ooo 4.580 mimnummuummmxmxuummmmmmmmunmm 7.000 5.160 mmmmmmumnuuumuummtmmmmmummxum S.000 5 .359 ХХХШХХШХХХХХХХХХХХХХХХХХХХХХХ1ХХХХХХШХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХ 9.000 5.416 uuummummmnmnmummuuummmmuummnm lo.oo 5.276 mimmummwmmmwmmmxmmnmmummmmui 11.00 4 .880 ХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХ1ХХХШХХХХХХХХХШХ1ХХХХХХХХХХХ 12.00 4 .180 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXIXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX 13.00 3 .081 ХХХШХХХХХ1ХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХХШХШХШХ1ШХШХХХХ 14.00 1.403 xxxixxxxxxxxxxxxxxxxxxxxxxxxxxxtmxxxxmxxxxxxxxxxmxxxxxxx

CALCULATED REACTION CROSS SECTION -- 43.50 fIB FINAL ENERGY* 14.48 HEV

- 80 -

Print output of GRYPHON (cont . )

A L P H A C H A N N E L

ENERGY CROSS-SECT, л,>01 0.01 0.1 1 10 CH(HEV) (ИВ/HEVi I I X X I

. W 0 0 H

.IO00E-C m. 140.

.1000E-030.

.IOOOE-020.

. :CCOE-( )20.

.5000E-020. •7000E-020. ЛО00Е-01О. •2000E-010. .5Ö00E-010. .7000E-010. . 1Ö0Ö . 2000 . 3000 .4000 .5000 .6000 .7000 .8000 .9000 1.000 1.200 1.500 2.000 2.500 3.000 4.000 5.000 6.000 7.000 8.000 9.000 10.00 11.00 12.00 13.00 14.00 15.00 16.00 18.00

0. 0. 0. Ö. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. Ö. , ,

.

1305E-05 1090E-03 .1771E-Ö2 .1175E-01 .4709 .9950 1.240 .9402 .4872 .1940 .6319E-Ö1 .1720E-01 26UE-03

XXXX ummnmumm mumummmunmmmnmmmmmuu mmmmmmnmmmnmmummmmmmn xxxxxxxmxxxxxixxxxxxxixxxxxxxxxxxxxxxxxmxxxxxixxxxxxxixxx mmmmnmumnmmmmmmmmmmmn mmmnmmumummmmumimmmn umummmmmummmmmxmn ummmmmmmmmnnm mmmmmmmm

CALCULATED REACTION CROSS SECTION •- 4,431 ffl FINAL ENERGY* 18.44 «EV

- 81 -

Print output of GRYPHON (cont.)

S А П П A C H A N N E L

ENERGY CRQSS-SECT. 0.001 0.01 0.1 1 CMHEV) (HB/HEV) X I X X

.IOÖOE-05 .49UE-25

.1000E-04 .4911E-21

.1000E-03 .491 IE-17

.1Ö00E-O2 .4908E-13

.2000E-02 .734BE-12

.5000E-02 .3059E-10

.7000E-02 .1174E-09

.1000E-01 .4879E-09

.2Ö00E-01 .7754E-08

.50Ö0E-01 .2968E-06

.7000E-01 .1125E-05

.1000 .4591E-05

.2000 .6356E-04

.1000 .3235E-Ö3

.4000 .9517E-03

.5000 .2161E-02 mm

.óooo .4162E-02 n m m u n

.7000 .7158E-02 ХХХ1ХХХШХ1ХХХХХ

.8000 .1133E-01 ШХХХШХХХХХХХХХХХХ

.9000 . 1 Ó B 1 E - Ö 1 XXXXXXXXXXXXXXXXXXXXXXXX i.oo^ .2374E-01 mmummmmnmn 1.200 .4216E-01 ххххштхххштнштшхх i.soo .3133E-01 ттитттпттжитти 2.000 .1723 XXXXXXXXXXXXXX*'' 'XXXXXXXXXtXXXXXXXXXXXXXXXX 2.500 .2304 ттжтхи-иитххштпттпии.п з.ооо .3659 штшшшххшшххххтшшпштшхш 4.000 .5327 xxxiixxmmnnnmmnmmmmxmmxmxt.il 5.000 .5647 nmmxmxxmnxxxxiinmxximximxmxxxxxxxxxx i . ооо .5073 xxmmnmmmmiminiimmnmmmmm 7.000 .4079 mummimmmmxmuummmxxmmm \000 .3045 Х Х Ш Х Х Х Х Х Х Х Ш Х Х Х Х Х Х Х Х Х Х Х ; . лХХХХХХХХХХХХХХХХХХХ 9.000 .2184 aiumumxwmmumuummmimi ïo.oo .1571 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx»x ïi.oo .1213 xxxixxixxxxxxxxxxxxxxxxxxxxxxrxxxxxxxxxxx 12.00 .1098 xxxxxxmxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 13.00 .1234 Ш П П К Ш ( П Н П Ш П ! И Н П Ш 1 Ш Ш 14.00 .1679 mxxmxxmxxxxxxxxxxxxxxxxxnxxxxxnxnii 15.00 .2565 итттиттиттттхтхиитш ib. со .4024 ххшххтххтхтххшхххххххххххххххххххххххххххх i8.oo .6683 тттптттититтиихиитиишии :о.оо .5094 mimmxxmxxxxxmxxxxxxxxxxxxxxxxxmxxxxnxxxxx

CALCULATED REACTION CROSS SECTION = 6.889 NB FINAL ENERGY' 21.58 HEV

TOTAL iNLOHINB CP0SS SECTION = 1756. И8 TOTAL GUT60IN6 CROSS SECTION = 1756. N8 DIFFERENCE = .4160E-02 ЯВ

- 82 -

Interactive SPSRT1 I/O

INPUT IPROPT. ISUP: IPROPT: 1 (= COMPLETE OUTPUT) OR

2 1= PARTIAL OUTPUT FOS PO AND PI ONLY) ; ISUP : 1 TO SUPPRESS OUTPUT OF "PECTRA

0 OR / (SLAShi TO PRINT SPf.TRA 61VE IPROPT,ISUP:U> SPECIFY FOR WHICH IOUT TOTAL EHISSIDN MILL BE ВISPLAYED;

0=NO DISPLAY REQUIRED l'NEUTRON 2>PROTON 3*ALPHA 4-TAU 5=DEUTER0N 6*TRIT0N 7«6AW№ U SPECIFY ENER6Y INDEX 1

TOTAL NEUTRON EMISSION : SI6HA - 3114.96 MB.

E (HEV) SI6HA (ИВ/NEV) E IMEV) SIBNA IMB/HEV) E (HEV) SIGMA (NB/HEV) .IOE-05 .782120 .10E-04 2.47159 .10E-03 7.79832 .10E-02 25.2647 .20E-02 36.6651 .50E-02 62.5778 .70E-02 77.709) .10E-01 99.4106 .20E-01 170.410 .50E-01 392.272 .70E-01 542.104 .10 756.789 .20 1304.27 .30 1587.27 .40 1674.51! .50 1632.44 .60 1586.14 .70 1512.03 .80 1425.62 .90 1335.41 1.0 1245.99 1.2 1074.33 1.5 841.084 2.0 57i.046 2.5 372.169 3.0 242.120 4.0 113.741 5.0 66.1244 6.0 46.7958 7.0 37.4751 8.0 31.9162 9.0 27.5883 10. 23.4581 11. 19.0460 12. 13.9741 13. 7.68224

MAXIMUM EMISSION ENER8Y» 13.7946 HEV SPECIFY ENER6Y INDEX 0

STOP NORMAL END SPSPTi

(CP time - 2.1 s ee s . )

- 83 -

Print output of SPSRT1

(and spectra of last-emitted ejectile)

mt*mtmt4t<*mt«tfttttt«t*» t REACTION CROSS SECTIONS t I H I H H H I H H H H I I H H H t H I H

NEüTRQN EMISSION INC0MJN6 ENER6V = 14.500 MEV

NB 93 (N,N) NB 93 SI6HA = 1701 .17 KB. (Emission after first cycle)

E (MEV) SIGMA (M8/MEV1 E (REV) SIGMA (MB/NEV) E (HEV) S16MA (MB/HEVi

. 10F-05

.1ПЕ-02

.70E-Ö2

.50E-01

.20

.50

.30 1.2 i. J 5.0 8.0 11.

.210670 6.80860 21.0046 108.325 387.929 581.322 574.674 510.132 I7Ó.639 65.9716 31.9160 19.0460

MAXIMUM EMISSION ENERBY=

.10E-04

.20E-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0 9.0 12.

13.7946

.665743 9.8B584 26.9107 151.196 496.140 588.789 560.724 455.905 203.198 46.7901 27.5883 13.9741

MEV

.10E-03

.50E-02

.20E-01

.10

.40

.70 1.0 2.0 4.0 7.0 10. 13.

2.10063 16.8978 46.3609 214.239 554.642 584.975 544.768 363.928 109.198 37.4738 23.4581 7.68224

NB 93 (N,NN1 NB 92 SIGMA = 1*04.99 MP.

E (MEV) SI6HA (MB/DEVI E (HEV) SISHA (MB/MEV)

.10E-05

. 10E-02

.70E-O2

.50E-01

.20

.50

.80 1 T i . 1

2.5 5.0

.564361 1C.2773 56.0054 2S0.533 906.883 1043.54 845.965 561.915 95.0951 .179891

MAXIMUM EMISSION ENER6Y=

.10E-04

.20E-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0

5.45217

1.78344 26.4476 71.6086 386.363 1081.72 990.615 770.495 383.836 38.6687 0.

MEV

E (MEV)

.10E-03

.50E-02

.20E-01

.10

.40

.70 1.0 2.0 4.0 7.0

SIBMA IHB/HEV)

5.62700 45.1158 122.538 536.41? 1111.66 921.232 697.729 206.350 4.465B1 0,

£ iHEV)

. 10E-05

.lOE-02

.70E-02 -50E-Ö1 ,20 ,50 ,80 1.2 2.5

SIGMA (MB/MEV)

.460542E-02

.148588

.453724 2.17255 5.98740 7 07*??

2.45853 .783802 .228676E-02

E (MEV)

.10E-04

.20E-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0

SIGMA (MB/MEV)

.145535E-ÖI

.215377

.578324 2.92836 5.51880 3.47476 2.00612 .263602 .625306E-03

E (MEV)

.10E-03

.50E-02

.20E-01

.10

.40

.70 1.0 2.0 4.0

SI6MA (MB/HEV)

.459139E-01

.366262

.979361 3,93659 4,35847 2.95488 1.60711 .661285E-01 0.

MAtlHUM EMISSION ENERGY 3.30337 MEV

- 84 -

Print output of SPSRTl ( cont . )

NB 93 (N,AN) V Б9 SI6RA = 1.21707 RB.

E (REV)

.löE-05

.I0E-02

.70E-02

.50E-01

.20

.50

.30 1.2

SI5HA (HB/HEU)

. L49772E-02

.483078E-01

.147243

.695790 1.S1980 1.46615 .686265 .923476E-01

E (REV)

.10E-04 • 4 ik..

.70E-01

.30

.60

.90 1.5

SIGNA (HI/REV)

.473292E-02

.700005E-01

.187508

.931970 1.38*39 1.194" .47999G .345417E-03

E (НЕУ)

.10E-Ö3

.50E-02

.20E-01

.10

.40

.70 1.0 2.0

SI8RA IRB/REV)

•149312E-Ö1 .118932 .316569 1.24078 1.71457 .927311 .312083 0.

RAXIRUn ENISSION ENER6Y= 1.91168 REV

NB 93 !N,6N) NB 93 SISRA =

E (REV) SI6RA (RB/REV)

.10E-05

.10E-02

.70E-02

.50E-01

.20

.50

.80 1.2 2.5 5.Ö 8.0 11.

.987226E-03

.318902E-01

.980904E-01

.495245 1.64805 2.14011 1.84063 1.36196 .433157 .221672E-01 .258437E-03 .654117E-06

RAXIRUH EMISSION ENERGY^

3.52097

E (REV)

. 10E-04

.2ÖE-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.Ö 6.0 9.0 12.

13.2999

NB.

SIGNA (RB/REV)

.ЗИ975Е-02

.462806E-01

.125485

.684474 2.00837 2.06890 1.70239 1.07968 .252446 •571817E-02 .439430E-04 .379084E-07

E (REV)

. 1ÖE-..U

.5^-02

.20E-Ö1

.10

.40

.70 1.0 2.0 4.0 7.0 10. 13.

SI6RA (RB/REV)

.984337E-Ö2

.789899E-01

.215116

.955684 2.14050 1.96241 1.57234 .701781 .780013E-01 .I30233E-02 .621872E-05 .493812E-09

HEV

TOTAL NEUTRON ENISSION : SIGNA - 3114.96 RB.

E (REV)

.10E-05

.10E-02

.70E-02

.50E-01

.20

.50

.80 1.2 2.5 5.0 8.0 11.

SIGMA (HB/REV)

.782120 25.2647 77.7091 392.272 1304.27 1632.44 1425.62 1074.33 372.169 66.1244 31.9162 19.0460

MIRUR EMISSION ENER6Y=

E (REV)

.10E-04

.2ÖE-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0 9.0 12.

13.7946

SIGflA (RB/REV)

2.47159 36.6651 99.4106 542.104 1587.27 1586.14 1335.41 841.084 242.120 46,7958 27,5883 13.9741

REV

E (REV)

.1ÖE-03

.50E-02 •20E-01 .10 .40 .70 1.0 2.0 4.0 7.0 10. 13.

SI6RA IHB/HEV)

7.79832 62.5778 170.410 756.789 1674.52 1512.05 1245.99 571.046 113.741 37.4751 23.4581 7.68224

- 85 -

Print output of SPSRTl (cont»)

PROTON EHISSION INCONtNB £N£K6Y = 14.500 HEV

N8 93 iN,P) ZR 93 SIGHA = 43.500B

E (HEV) SIGMA (HB/HEV) E (REV)

NB. (Emission a f ter f i r s t cycle)

SIGHA (HB/flEV) E (HEV) SI6HA (HB/HEV)

.IOE-05

.10E-02

.70E-02

.50E-01

.20

.50

.80 1.2 1 Г L.J

5.0 8.0 11. 14.

0. 0. 0. 0. 0. 0. 0. , , 7

5

173564E-05 166766E-01 .15313 .35878

4.87993 1 .4u27b

.10E-04

.20E-02

. IOE-01

.70E-01

.30

.60

.90 1.5 3.0 6.0 9.0 12. 15.

0. 0. 0. 0. 0. 0. 0.

.506B39E-Q4

.692169E-01 4.5B046 5.41642 4.18009

0.

.10E-03

.50E-02 •20E-01 .10 .40 .70 1.0 2.0 4.0 7.0 10. 13. 16.

0. 0. 0. 0. 0. 0. 0. Л83638Е-02 1.31734 5.16037 5.27564 3.08123

0. ПАП HUH EMISSION ENERSY= 14.4774 REV

93 (N,NP) ZR 92 SIGHA = .205185 MB.

E (HEV)

.IOE-05

.10E-02

.70E-02

.50E-01

.20

.50

.30 1.2 2.5 5.0

SIGMA (ИВ/HEV)

0. Ö. Ö, 0. 0. 0. l ) .

.228777E-05

.791577E-02

.557140E-01

E (HEV)

.10E-04

.20E-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0

SIGHA (HB/HEV)

0. 0. 0. 0.

i/f

0. .530837E-04 .2Ö5789E-01 .399893E-02

E (HEV)

.10E-03 •50E-02 .20E-01 .10 .40 .70 1.0 2.0 4.0 7.0

SI6HA (HB/HEV)

0. 0. 0. 0, 0. 0. 0. Л31838Е-02 .126797

0. HAXIHUH EMISSION ENER6Y= 6.301O2 HEV

NB 93 (N,PP) Y 92 "GNA = .262221E-08 HB,

£ (HEV)

.IOE-05

.10E-02

.70E-02

.50E-01

.20

.50

.80 1.2

SISHA (HB/HEV)

0. 0. 0. 0. 0, Ö.

0. ,

'

201758E-09

E (HEV)

.1ÖE-04

.2OE-02

.1OE-01

.70E-01

.30

.60

.90 1.5

SIGHA

0. 0.

0. 0, 0. 0. 0.

(HB/HEV)

.299583E-09

E (HEV)

.10E-03

.50E-02

.20E-Ö1

.10

.40

.70 1.0 2.0

SISHA (HB/HEV)

0. 0. 0. 0. 0. 0. 0.

,548605E-0e HAXIHUH EMISSION ENERGY* 2.39388 HEV

- 86 -

Print output of SPSRTl (cont . )

E (HEVi

.10E-05

.10E-02

.70E-02

.50E-01

.20

.50

.80 1.2 2.5 5.0 9.0 11.

SI6HA (KB/HEV)

0. 0. 0-0. 0. Ö. 0.

.683606E-08

.385847E-04

.242285E-02

.234684E-03 Л640В1Е-05

E -HEV)

.IOE-04

.20E-Ö2

.1ÖE-01 •70E-01 .30 .60 .90 1.5 3.0 6.0 9.0 12.

SI6HA Ш8/ЙЕ¥>

0. 0. 0.

с. J.

0. 0.

Л72408Е-06 .132537E-03 .176149E-02 .55B246E-04 .159598E-06

E iHEV)

ЛОЕ-ОЗ .50E-02 .20E-01 .10 .40 .70 1.0 2.0 4.0 7.0 10. 13.

5I5I№ iHB/HEVi

0. 0. 0. 0. 0. 0. 0. .512965E-Ö5 .168061E-02 .766288E-03 .1060WE-04 .584796E-08

HAXIHUH EHISSION ENERSY= 13.9827 HEV

TOTAL PROTON ENISSION : SIGMA = 43.7130 N8.

E (HEV)

.10E-05

.10E-02

.70E-02

.5ÖE-01

.20

.50

.80 1.2 2.5 5.0 8.0 11. 14.

SISnB IHB/HEVi

0. 0. 0. 0. 0. o. 0.

.403044E-05

.246310E-01 3.21127 5.35901 4,87993 1.40276

E (HEV)

.10E-04

.20E-02 Л0Е-01 .70E-01 .30 .60 .90 1.5 3.0 6.0 9.0 12. 15.

SIGHA (HB/HEV)

0. 0. 0. 0. 0. 0. 0.

Л03940Е-03 .B99283E-01 4.58482 5.41647 4.18009

0.

E (HEV)

. 10E-03

.50E-02

.20E-Ö1

.10

.40

.70 1.0 2.0 4.0 7.0 10. 13. 16.

SI6HA (HB/HEV)

0. 0. Ö. 0. 0. 0. 0.

•315990E-02 1.44582 5.16114 5.27565 3.08123

0. ПАХ IHUH EHISSION ENERGY- 14.4774 HEV

- 87 -

P r i n t output of SPSRTl ( c o n t . )

ALPHA ERISSION INC0HIN6 EMER6Y = 14.500 NEV

*B ' 3 (N.A) Y «0 SISHA : 4.43068 (IB. ( E m i s s i o n a f t e r f i r s t c y c l e )

E («У; SISBA СИВ/BEV) E (BEVi SIGBA (BB/BEV) E (BEV) SIBBA (BB/BEV)

.10E-05 0. .10E-04 0. .10E-03 0. Л0Е-02 .W-92 .50E-01 .20 .50 .80 1.2 2.5 5.0 8.0 11. 14. 18.

0. 0. 0. 0. 0. 0. 0. 0.

.180474E-05

.117455E-01 i.24043 .194029 .261076E-03

.20E-02

.10E-01

.70E-01

. 30

.60

.90 t.5 3.0 6.0 9.0 12. 15. 20.

0. 0. 0, 0. 0. 0. 0. 0.

.108959E-03

.470889

.940208

.6319UE-01 0.

.50E-02

.20E-01

.10

.40

.70 1.0 2.0 4.0 7.0 i i \

13. 16.

0. 0. 0. 0. 0. 0. 0. 0.

AT, -P£-02 .994965 .487228 .171961E-01

0. BAüIBÜH EBÏSSIW ENER6Y= 18.4442 BEV

KB <>Z IN.NA) Y 89 SIGBA = .236854E-02 BB.

E (BEVi SISBA fBB/BEV! E (BEV) SIGBA (BB/BEV) E (BEV) SIGBA (BB/BEV)

.10E-05 Ö. . 10E-04 0. .10E-03 0.

.IOE-02 0. .20E-02 0. .50E-02 0.

.7OE-02

.50E-01

.20

.50

.80 * » i

2.5 5.0 8.0

0, 0. 0. V ,

0. •). 0. • , 142054E-05 307226E-03

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0 9.0

0. 0. 0. 0, 0, 0. 0. , t

315279E-04 181805E-02

.20E-01

.10

.40

.70 1.0 2.0 4.0 7.0 10.

0. 0. 0. 0. 0. 0. 0.

.174108E-03

.392722E-04 BAX IBUB EMISSION ENER6Y= 10.8438 BEV

NB 93 fN.PA) SR 89 SIGBA - .155051E-09 BB.

E IBEV) SISBA (BB/BEV) E (BEV) SIGBA (BB/BEV) E (BEV) SIBBA (BB/BEV)

.10E-05

.IOE-02

.7OE-02

.50E-01

.20

.50

.80 1.2 2.5 5.0

0. Ö.

0. Ö. 0. 0. 0. 0. 0. , 940877E-10

.10E-04

.20E-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0

0. 0. 0. 0. 0. 0. 0. 0. 0. , 609629E-1Ö

•10E-03 -50E-02 •20E-01 .10 .40 .70 1.0 2.0 4.0 7.0

0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

BAIIBUB EBISSI0N ENERGY* 7.26455 BEV

- 88 -

Print output of SPSRTl (cont . )

NB 93 !N,6Ai 1 90 SISNA = .350805E-Ö3 HB.

E iHEV)

.IOE-05

.IOE-02

.70E-02

.50E-01

.20

.50

.30 1.2 1 Г i., J

5.0 8.0 11. 14.

SI6HA

0. Ö. 0. Ö. 0. 0. 0. 0. 0.

.239

.331

.904

WI8/HEV)

307E-08 500E-706E-

.231053E-

•05 -04 -05

E !HEV'<

.10E-Ö4

.20E-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0 9.0 12. 15.

SI6HA (HB/HEV)

0. 0. 0. 0. 0. 0. 0. 0. Ö.

.840726E-07

.850570E-04

.412523E-04

.29Ы58Е-06

E iflEV!

.IOE-03

.50E-02

. ' " - - • )1

./« 1.0 2.0 4.0 7.0 10. 13. 16.

SISHA iHB/HEV!

0. 0. 0. 0. 0. 0. 0. 0. 0.

.800447E-Ö6

.115309E-03

.I18869E-04

.132972E-07 ПАПНШ1 EHISSION ENERSY= 17.9655 HEV

TOTAL ALPHA EHISSION : 5IGKA = 4.43340 ПВ.

E iHEV)

.IOE-05

.IOE-02

.70E-02 •50E-01 .20 .50 .80 1.2 i., J

5.0 8.0 11. 14. 18.

SIGHA iHB/HEV)

0. 0. Ö. 0. 0. 0. 0. 0. 0.

1 1

322826E-05 120560E-01

1.24052 194031

• 26M76E-03

E Ш )

.10E-04

.20E-02

.10E-01

.7ÖE-01

.30

.60

.90 1.5 3.0 6.0 9.0 12. 15. 20.

SI6HA (NB/HEV)

0. 0. 0. 0. 0. 0. 0. 0. 0.

, . , ,

0.

140571E-03 472792 940249 631919E-01

E (HEV)

.1ÖE-03 •50E-Ö2 .20E-Ö1 .10 .40 .70 1.0 2.0 4.0 7.0 10. 13. 16. 22.

SIGMA (HB/HEV)

0. 0. 0. 0. 0. 0. 0. 0. 0.

. 194548E-02

.995117

.487240

.171961E-01 0.

PtAXINÜH EMISSION ENERGY- 18.4442 HEV

TAU EMISSION INCQHIN6 ENER6Y = 14.500 HEV

• NO DATA ON TAU EHISSION t

DEUTERON EMISSION INCOniNG ENERGY = 14.500 HEV

* NO DATA ON OEUTEPON EHISSION *

TRITON EHISSION INCOMING ENER6Y - 14.500 HEV

* NO DATA ON TRITON EHÜSION •

- 89 -

Print output of SPSRTl (cont . )

5hMA EMISSION INC0HIN6 ENER6Y = 14.500 MEV

93 iN,6) NB 94 SI6KA = 4 .88909 KB. (Emission after first cycle)

E (HEV> SlbKA IHB/rtEV) E (HEV) SIGKA (HB/BEV) E iKEV) SI6RA (HB/HEV)

.IOE-05

. 10E-02 •70E-02 .50E-OI .20 .50 .35 1.2 n г

5.0 3.0 t t

14. 18.

ИАХ1И1Ш EdISS

.49Ц40Е-25

.490814E-13

.117374E-Ö9

.296B46E-06

.&B5587E-0*

.216070E-02 Л13255Е-01 •421643E-01 .2S0401 .504722 .304530 .121320 .167901 .668311

№ ENERGY

Л0Е-04 .20E-02 .10E-01 .70E-01 .30 .60

1.5 3.0 6.Ü 9.Ö 12. 15. 20.

.49U37E-21

.784779E-12

.487877E-09 Л12505Е-05 .323488E-03 .41623AE-02 Л68145Е-01 .813273E-01 .385855 .507320

.109839

.256454

.509449 21.5753 «EV

»0E-;E-

•03 •02

.20E-01

.50

.40

.70 1.0 2.0 4.0 7.0

13. 16.

.491108E-17

.305942E-10

.775402E-08

.459108E-05

.951736E-03

.715774E-02

.2373B6E-01

.172337

.532710

.407881 • Ы/У/V Л23402 .402360

0.

N8 93 (N,N6) N8 93 SïGtfA = 294.994 UB.

!HEV)

IOE-05 10E-02 70E-02 50E-01 20 50 80 .2

:.5 i.O I.Ö 1.

SI6KA fMB/nEV)

0. 0. 0. 0. 0. 214.0«9 ri С J Tff

36.7549 61.2319 21.8439 1.74003 .214530E-01

с (KEV)

ЛОЕ-04 .20E-02 .10E-ÖI .70E-01 .30 .60 .90 1.5 3.0 6.0 9.0 12.

SIGMA (NB/NEV)

0. 0. 0. 0. 94.5064 79.6870 21.6664 45.0732 59.8677 9.82844 .304188 .372937E-02

E (KEV)

ЛОЕ-03 .50E-02 .20E-01 .10 .40 .70 1.0 2.0 4.0 7.0 10. 13.

3I6HA (HB/KEV)

0. 0. 0. 0. 345.993 40.7624 26.8499 52.5743 41.5250 4.26582 .891370E-01 .290529E-03

М П HUH EniSSION ENER6Y= 13.9409 KEV

NB 93 (M,PS) 2R 93 SIGMA = 39.0918 KB.

E 'HEV)

ЛОЕ-05 Л0Е-02 .70Е-02 .50Е-01 .20 .50 ,80 1 •> 4 » 4. 1 С

5,0 8,0 11.

SI6KA ! И 6 Ш )

0, 0. 0. 0. 0,

14.0307 i.72371 12.7469 10.5104 1.22979 .«3449Е-01 .628366Е-06

E iHEV)

ЛОЕ-04 .20Е-02 Л0Е-01 .70Е-01 .30 .Ы) .90' 1.5 3,0 6.0 9,0 12,

SIGHA (NB/KEV)

0. 0. 0. 0.

3.90846 9,23990 9.48947 11.2705 7.76496 .786625 .996753Е-03

0.

E (HEV)

ЛОЕ-03 .50Е-02 .20Е-01 ЛО ,40 .70 1.0 2.0 4.0 7.0 10. 13.

S16HA (ПВ/HEV)

0. 0. 0. 0. 21.4965 6.96488 10.1560 12.8803 3.47889 .117342 .515319Е-04

0. WK ШИК EMISSION ENERBYÏ U.60'''d HEV

- 90 -

Print Output of SPSRTI ( c o n t . )

SB ' 3 iN.AB» 1 =0 5I5HA = 11.4893 UB.

Л0Е-О5 .IOE-02 . 7 OE-O: ЛОЕ-0 .20 .50 .80 • f

2.5 5.0 8.0

m\m

ВИД iHB-'ïtCVï

.1U541E-U:052E-

.Ln'Hl-.S^ïe't-4.397Г b.i7 :>(.,! 1.29324 1.93909 2.37084

С / С П ! 4

• w U J i D . '

.551061E-

• i * •07 •04 • • ' • ' ;

•02

£ <*flE

Л0Е-.20E-* r.

. 70Б-

.30 sQ

.90 1.5 3.0 ó.Ö 9.0

"VI

•04 •02 .*,\

•01

5IG.4A i.NB/NEV)

.Ш534Е-:3

.: Ь396Е-0й

.10ö56iE-03 181902

8.49030 3.64909 1.26221 2.49515 2.гз<: . 1761,3 .57O571E-03

E iHEV)

.lOE-03

.50E-02

.20E-01

.10

.40

.70 1.0 2.' ».ч 7.0 10.

SI6NA (HB,'№V

.И1492Е-1!

.631710E-05

. 162299E-02

.617015 8.95470 1.92448 1.4541' 2.74'.<?6 1.23)05 .372415E-01

0. EMISSION ENERGY^ 5.36394 flEV

3 (N,86)

E iflEV)

Л0Е-05 -IOE-02 .70E-02 .50E-01 .20 .50 .80 1.2 2.5 5.0 3.0 11. 14. 18.

NB 94 SI8HA =

SI6t!ft (ИВ/NEVl

•376776E-19 .372472E-07 .S34457E-04 .130424 4.53614 1.16:84 .731092 1.26321 .362970 ,61697óE-01 .1S2B64E-02

Л246В9Е-03 .931977E-05 .7683Ö5E-07

iKIflUN EMISSION ENER6Y-

3.36720

E (KV)

.10E-04

.20E-02

.10E-01

.70E-01

.30

.60

.90 1.5 3.0 6.0 9.0 12. 15. 20.

21.0753

ЧВ.

3ISHA (HB/BEV)

.376737E-15

.5B9134E-06

.335657E-03

.391899 5.03965 .588332 .891854 .758005 .328162 .138031E-01 .608620E-03 .552377E-04 .357962E-05 .Ш236Е-08

NEV

E ifIEV)

.10E-03

.50E-02

.20E-01

.10

.40

.70 i.O 2.0 4.0 7.0 10. 13. 16. -л

SIGHA (ЯВ/ЯЕУ

.376348E-11

.222301E-04

.477820E-02 1.11738 2.BBB23 .586675 1.04137

7 7 Т С Ц

.181841

.416003E-02

.280516E-O3

.231606E-0*

.114618E-05 0.

TOTAL GAWIA EMISSION SI6RA 355,832 «В, (after 2 cycles)

E (PIEV)

.IÖE-05

.IOE-02

.70E-02

.5OE-01 ,20 .50 .80 1.2 2,5 5,0 8.0 П. 14. 18.

SI6HA IHB/NEV)

.4B8317E-19

.483525E-07

.109394E-03

.184081 8.98348 235.766 ^j.i769 52.7462 74.7564 24.2654 2,09704 .142900 .167910 .668311

WIHUH EMISSION ENER6YS

E (BEI/)

.10E-04

.20E-02 •10E-Ö1 .70E-01 .30 .60 .90 1.5 3.0 6.0 9.0 1 ? U i

15. 20.

21.5753

SIGHA (UB/KV)

.488274E-15

.76603IE-06

.442218E-03 ,573802 111,945 93.1685 32.3267 59,6782 70.3801 10.9143 .524678 .113624 .256458 .509449

WEV

E (MEV)

.10E-03

.50E-02

.20E-01

.10

.40

.70 1.0 2.0 4.0 7.0 10. 13. 16. i l

SISHA («B/ffl

.487841E-11

.290472E-04

.640120E-02 1.73439 379.333 50.1456 39.5252 68.7025 46.9485 4,85244 .246548 .123707 .402361

0.

- 91 -

Print output of SPSRTl (cont.)

S0RT12 EMISSION IMCCfllMG EKERSY = 14.500 HE'.' 444444444444444*44444444*4444444»

» :QTftL REACTION CROSS SECTION t 444*44444444444*4*44*444444*44444

SIGNE - 1756.00 № . SUHSISR = 1755.99 ,, 4444444444444444444444444444*4444 4 NET PRODUCTION CROSS SECTIONS * ft.rtt . . . . . (tranamutation cross sections) 4444444444444444*4444444444444444 '

SYffiffl. 5IGHA ( № . )

NB 94 MB 93 NB 92 ZR 93 ZR 92 Y 92 Y 90 V 89

SR 89

3.36071 299.501 1404.99 39.4483 4.26472 .262221E-08 3.21395 1.21944 . 155051E-09

STOP NORBflL END SPSRTl

- 92 -

Sample problem 2 illustrates the calculation of angular distributions. In order to limit the output the number of ejectiles is restricted to 1 (neutrons) and the order of Legendre polynomials is restricted to I

* л. in the secondary emission isotropy has been assumed (& = 0 ) . max

The input by the user is underlined; a slash means: default values. The output of GRYPHON is not reproduced; it is the same as in sample problem 1, however, restricted to neutrons and with a listing of the Legendre-polynomial coefficients after the first emission. The sorting of the results is performed with SPSRT1 (page 94) and MCFRLN + MF6EXT (page 95). Note that the total neutron emission is lower by the amount of (n,pn), (n,an) and (a,Yn) which sum up to 8.8 mb. The Legendre coefficients with I > 0 on page 94are defined by the series:

A 1 Л m a X

d d a - b d f - + Д a*Vc o s 9> . <59> £=1

whereas on page 95 the coefficients are defined according to the ENDF-

VI conventions. The data are stored under MT = 3, i.e. the non-elastic

cross section. This cross section is equal to 1756.0 mb at 14.5 MeV and

is given on file MF-3. On file MF-6 first the particle >xelds are

given. In our sample problem 2 we only consider neutrons, the yield is

у =• 1.7689. This means that the total neutron production cross section amounts 1.7689 x 1756.0 » 31062.2 mb. Next, the normalized energy- and angular-distribution coefficients follow. These are defined by:

d2o ov_ ™ x 21+1 _ p , ft, didfi-2T e

2 :f t — W C 0 8 e ) '

(60)

with

e ' max ƒ f de - 1. .61) о

Output in this format is also obtained if the code SPS0RT is used. In that case all quantities defined for ENDF-VI can be sorted and stored.

Sample problem 2 (Interact ive GRYPHON I/O) mtHMtmtt t tmt GRYPHON <«tmm«*t«tmtit INPUT PREPARATION FOR PRAN6 FREE FORMAT INPUT, END KITH / 6IVE INC0RIN6 AND OUTGOING PARTICLE NUMBERS 0=TAR6ET IN EXCITED STATE. 1*NEUTR0N, 2-PROTDN, 3*ALPHA, 4=TAU ,S-DEUTERON, 6=TRIT0N, 7=6AHMA (INSTEAD OF О THE PARTICLE HISTORY NAY BE ENTERED» A NE6ATIVE INC0MIN6 PARTICLE NUMBER TERMINATES RUN A NEGATIVE 0UT60IN6 PARTICLE NUMBER -4,-5,-6,0R -7 MEANS ONLY COMPETITION INCLUDED (NO OUTPUT) COMPETITION OF N,P,A IS ALNAYS INCLUDED '1,1,-7/

SIVE 1,)1 FOR PURE EQUILIBRIUM OR PURE PREEQUILIBRIUM '± 6IVE Z.A OF TAR6ET Ml.93/ 6IVE 6.D.R. PARAMETERS FOR 6AHHA EMISSION OR SLASH 6SI6R (MB) ,66AMR INEV) .SER (MEV) ,SD .METHOD * ± POSSIBILITY TO OVERWRITE LEVEL DENSITY PARAMETERS DEFAULT SETTIN6S: SYSTEMATICS OF N.DIL6 ET AL. POSSIBILITY TO RENORHALIZE PHOTQ-ABS XSECT.(DEFAULTS.0) DEFAULT SETTIN6: VALUES FOR COMPOUND NUCLEUS POSSIBILITY TO 6IVE DISCRETE LEVELS (MAI. 30) READ NUMBER OF CHAN6ES *£ YOU HAY OVERNRITE NOUT.EMSHIN.FITHFP.FKHEl.FORHFACTORS, MHO=_/ 6IVE LHAX,NRFR,KERNEL,KALMANN,V,AV,CA2,FS6

? =4/ 6IVE INCOMING ENER6Y (REV), NON-ELASTIC XSECT (MB), FIT PARAMETERS AN6ULAR DISTRIBUTION: DEFAULT XSECT - COMPOUND FORMATION XSECT FOR N,P,A

' DOSTROVSKY APPROX. FOR OTHER PARTICLES. E d ) , SIGNE(l), FIT(L) =14.5/ E(2), SI6NE(2), FIT(L) =£

mtmttmttHtm 6RYPH0N ««mmtmttmm INPUT PREPARATION FOR PRAN6 FREE FORMAT INPUT, END NITH / 6IVE INCOMING AND OUTGOIN6 PARTICLE NUMBERS O'TARSET IN EXCITED STATE, 1«NEUTR0N, 2*PR0T0N, 3*ALPHA, 4*TAU ,5»DEUTER0N, 6«TRIT0N, 7«GAMMA (INSTEAD OF 0 THE PARTICLE HISTORY MAY dE ENTERED) A NEGATIVE INC0MIN6 PARTICLE NUHBER TERMINATES RUN A NEGATIVE 0UT60IN6 PARTICLE NUMBER -4,-5,-6.OR -7 MEANS ONLY COMPETITION INCLUDED (NO OUTPUT) COMPETITION OF N,P,A IS ALWAYS INCLUDED '0,1,-7/

6IVE 1,>1 FOR PURE EQUILIBRIUM OR PURE PREEQUILIBRIUH ^ 6IVE LHAX=0/ 6IVE NOUT.EMISHIN, ELSE l\l_

шшнмшшш GRYPHON **»»«»*»«*»»**tttHi INPUT PREPARATION FOR PRAN6 FREE FORMAT INPUT, END WITH / 6IVE INCOMING AND OUTGOING PARTICLE NUMBERS 0-TAR6ET IN EXCITED STATE, 1«NEUTRON. 2«PR0T0N, J«ALPHA, 4*TAU ,5»DEUTER», 6-TRITON, 7*6AMMA (INSTEAD OF 0 THE PARTICLE HISTORY MAY BE ENTERED) A NEGATIVE INCOMING PARTICLE NUMBER TERMINATES RUN A NEGATIVE OUTGOING PARTICLE NUMBER -4,-3,-*,0R '7 MEANS ONLY COMPETITION INCLUDED INO OUTPUT)

- 94 -

Interactive SPSRT1 I/O INPUT IPROPT. ISUP: IPMPT: 1 1= COMPLETE OUTPUT) OR

2 1= PARTIAL OUTPUT FOR PO AND PI ONLY) ; ISUP : 1 TO SUPPRESS OUTPUT OF SPECTRA

O OR / (SLASH) TO PRINT SPECTRA GIVE IPROPT,ISUP;1,0/ SPECIFY FOR NNICH IOUT TOTAL EHISSION HILL BE DISPLAYED:

0*N0 DISPLAY REWIRED l=NEUTRDN 2=PR0TDN 3*ALPHA 4-TAU 5=DEUTER0N 6=TRIT0N 7=6AHHA l SPECIFY ENERGY INDEX I

TOTAL NEUTRON EHISSION : SI6HA * 3106.16 ПВ.

E (HEV) SI6PM (HB/HEV) .IOE-05 Л0Е-02 .70E-02 .506-01 .20 .30 .80 1.2 2.5 5.0 8.0 11.

.775030 25.0359 77.0101 388.908 1294.81 1624.86 1420.64 1072.10 371.734 66.1022 31.9160 19.0460

HAXIHUH EHISSION ENER6V

E (HEV) SI6HA (HB/HEV) .106-04 .20E-02 .10E-01 .70E-01 .30 .60 .90 1.5 3.0 6.0 9.0 12.

2.44918 36.J334 98.5193 537.559 1577.86 1579.40 1331.22 839.740 241.867 46.7901 27.5883 13 ^ 4 1

« 13.7946 HEV

E (HEV) SI6NA (HI/HEV) .10E-03 .50E-02 .20E-01 .10 .40 .70 1.0 2.0 4.0 7.0 10. 13.

7.72763 62.0137 168.899 750.656 1666.31 1506.21 1242.50 570.278 113.663 37.4738 23.45B1 7.68224

TOTAL NEUTRON EHISSION: SI6HA« 24.8566 «B/SR

LE6ENDRE COEFFICIENTS FOR L * 1

E (HEV) Sim (HB/HEVtSR) E (HEV) SI6HA (HB/HEVtSR) E (HEV) SIBRA MB/HEVfSR) .IOE-05 .106-02 .70E-02 .50E-01 .20 .50 .80 1.2 2.5 5.0 8.0 11.

HAXIHUH

.139781E-03

.453532E-02

.141368E-01

.766301E-01

.317827

.627136

.810881 1.02514 1.70418 2.37143 2.50777 1.79884

ЫЩЩ lllfhlT

.106-04 ,206-02 .106-01 .706-01 .30 .60 .90 1.5 3.0 6.0 9.0 12.

.441828E-03

.639970E-02

.181871E-01

.109195

.446124

.694967

.864631 1.19136 1.88824 2.32662 2.35539 1.36568

: \um te_

.106-03

.50E-02

.206-01

.10

.40

.70 1.0 2.0 4.0 7.0 10. 13.

.139519E-02

.113390E-01

.3172616-01

.159448

.546532

.755075

.917744 1.46643 2.15293 2.57107 2.12113 .770775

- 95 -

Output in ENDF-VI format

4.109SE+04 9.2900E*00 0 99 0 04193 3 3 1 .ООООе+00 .0ОО0Е+0О О 0 1 14193 3 3 2

1 2 4193 3 3 3 1.4500Е+07 1.7560Е*00 4193 3 3 4

4193 3 О S 4193 0 0 6

4.W93E+04 9.2W0E+00 О О 1 24193 А 3 7 1.0000Е+00 1.0000Е»00 0 5 1 14193 6 3 В

1 2 4193 6 3 9 1.4500Е+07 1.7689Е»00 4193 6 3 10 .ООООЕ+00 .ООООЕНЮ 1 2 1 14193 6 3 11

1 2 4193 6 3 12 .ООООЕ+ОО 1.4500Е+07 О S 228 384193 6 3 13 .ООООЕ+ОО .ООООЕМ» .ООООЕ+00 .ООООЕ+ОО .00О0Е*ОО .ООООЕ+004193 6 3 14

1.0000Е+00 2.4951Е-10 1.8850Е-13 9.6885Е-15 1.2191Е-18-4.5В50Е-164193 6 3 13 1.0000Е*01 7.8849Е-10 5.9582Е-13 3.0646Е-14 1.2190Е-17-1.4489Е-154193 6 3 16 1.0000Е»02 2.4878Е-09 1.8814Е-12 9.6994E-I4 1.21В5Е-16-4.5721Е-154193 6 3 17 1.0000E+03B.0601E-09 6.1I63E-12 3.1762E-13 1.2567E-15-I.4827E-144193 6 3 1В 2.00Ö0E+03 1.1697Е-08 В.899вЕ-12 4.6421Е-13 2.59ИЕ-15-2.1339Е-144193 6 3 19 5.0000Е+03 1.9965Е-08 1.5291Е-И 8.0461Е-13 7.0674Е-15-3.6877Е-144193 6 3 20 7.0000Е*03 2.4793Е-08 1.90ME-1I 1.0076Е-12 1.0447Е-14-4.5889Е-М4193 6 3 21 1.0000Е*04 3.1717Е-08 2.4526Е-И 1.3034Е-12 1.6Ю5Е-14-5.8883Е-144193 6 3 22 2.0000Е»04 5.4376Е-08 4.2784Е-И 2.3061Е-12 3.999ВЕ-14-1.0199Е-134193 6 3 23 5.0000Е»04 1.2521Е-07 1.0334Е-10 5.7301Е-12 1.5488Е-13-2.4207Е-134193 6 3 24 7.0000Е*04 1.7306Е-07 1.4725Е-Ю 8.282JE-12 2.6302Е-13-3.4133Е-134193 6 3 25 1.0000Е+05 2.4167Е-07 2.1503Е-10 1.2315Е-11 4.6338Е-13-4.9105Е-134193 6 3 26 2.0000Е*05 4.1685Е-07 4.2861Е-Ю 2.572iE-ll 1.3393E-12-9.3333E-134193 6 3 27 З.ООООЕ+05 5.0798Е-07 6.О16ОЕ-10 3.7451Е-11 2.3501E-12-1.2494E-124I93 6 3 28 4.Q000E+05 5.3645Е-07 7.3703Е-Ю 4.7344Е-П 3.385?Е-12-1.4374Е-124193 6 3 29 5.0000Е+05 5.23ИЕ-07 Г.4571Е-10 3.5873Е-И 4.4156Е-12-1.5886Е-124193 6 3 30 6.0000Е+05 5.0848Е-07 9.3718Е-Ю 6.3529Е-И 5.4438Е-12-1.6676Е-124193 6 3 31 7.0000Е*05 4.8491Е-07 1.0183Е-09 7.0695Е-11 6.4828Е-12-1.7102Е-124193 6 3 32 8.0000Е+05 4.3736Е-07 1.0935Е-09 7.7651Е-11 7.3469Е-12-1.7266Е-124193 6 3 33 9.0000Е+05 4.2857Е-07 1.1660Е-09 8.4578Е-11 8.6498Е-12-1.7226Е-124193 6 3 34 1.0000Е+06 4.0001Е-07 1.2376Е-09 9.1618E-U 9.8О34Е-12-1.7014Е-124193 6 3 33 1.2000Е+О6 3.4515Е-07 1.3824Е-09 1.0633Е-10 1.22W-1H.6119E-124193 6 3 36 1.5000Е»06 2.7035Е-07 1.6066Е-09 1.3024Е-10 1.6531E-U-1.3553E-124193 6 3 37 2.0000Е+06 1.8360Е-07 1.9776Е-09 1.7341Е-10 2.4755E-U-5.5440E-134193 6 3 38 2.5000Е+О6 1.1968Е-07 2.2982Е-09 2.1618Е-10 3.3728Е-11 7.2527Е-134193 6 3 39 3.0000E+06 7,7867Е-0? 2.5464Е-09 2.5535Е-10 4.2765E-S! 2.4020Е-124193 6 3 40 4.0000Е+06 3.6393Е-08 2.9033Е-09 3.2626Е-10 6.0887E-U 6.6449Е-124193 6 3 41 3.0000Е+06 2.1281Е-08 3.1979Е-09 3.9685Е-10 В.0278Е-11 1.1968Е-114193 6 3 42 6.0000Е+06 1.5064Е-08 3.4073Е-09 4.6138Е-Ю 9.9474E-U 1.8018Е-И4193 6 3 43 7.0000Е+06 1.2064F-08 3.4671Е-09 3.0773Е-10 1.1521Е-10 2.3946Е-114193 6 3 44 8.0000Е+06 1.0273Е-08 3.381ВЕ-09 3.3091Е-10 1.2578Е-Ю 2.9014E-U4193 6 3 43 9.0000Е+06 8.В818Е-09 3.17АЗЕ-09 3.3054Е-10 1.3038Е-10 3.2675Е-И4193 6 3 46 1.0000Е+07 7.3321Е-09 2.8604Е-09 3.0488Е-10 1.2808Е'Ю 3.4334Е-114193 6 3 47 1.1000Е+07 6.1317Е-09 2.4238Е-09 4.4966F-'0 Ы729Е-10 3.3303M14I93 6 3 48 1.2000Et07 4.4989Е-09 1.В417Е-09 3.564бь .* 9.S291E-U 2.84I1E-U4193 6 3 49 1,30ООЕ*О7 2.4732Е-09 1.0394С-09 2.0893Е-1О 5.7089Е-11 1.7737Е-114193 6 3 30 l.3795E*07 ,OO00E*0Ö .ÖOOOE'OO .MOOE'OO .OOÖOE+00 .000OE+0O4193 6 3 31

4193 6 O 52

- 96 -

ACKNOWLEDGEMENTS

The present code is the result of a development starting from the code of E. Betak [45], with a large number of improvements and extensions by F. Luider [54,55], С. Costa [l4,35,3b], D. Nierop [l4], H.L. Oudshoorn, and the authors. An important part of the work was directed towards obtaining tools for nuclear-data evaluation for fusion-reactor technology. This activity is supported by the European Commission in the framework of the B2-project on Fusion Technology. Som«? of the work was carried out as part of a research contract between the Netherlands Energy Research Foundation and the International Atomic Energy Agency at Vienna.

- 97 -

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- 99 -

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