improved formula for prompt fission neutron spectrum

Transcription

Journal of Nuclear and Radiation Physics, Vol. 6, No. 1&2, pp. 31-41
IMPROVED FORMULA FOR PROMPT FISSION
NEUTRON SPECTRUM
M. Aziza, M. O. Shakerb, A. Aboanberb, E. Massouda, and M. Slamaa
a
National Center for Nuclear Safety and Radiation Control, Atomic Energy
authority , Ahmed EL-Zomor st. P.O. Box: 7551 Nasr City , Cairo , Egypt
b
Mathematics Department, Faculty of Science, Tanta University, Tanta , Egypt
Rec. 21/7/2009
Accept. 15/3/2011
Madland-Nix model for the calculation of prompt fission neutron spectrum was
solved numerically for the determination of prompt fission spectrum of several fissile
nuclei. The model is based on the standard nuclear evaporation theory and considers
the effect of fission fragment motion and the distribution of residual nuclear
temperature on the prompt fission neutrons spectrum. The results of the model are
fitted into simple analytical formula which is easy to be applied in both shielding and
reactor core calculations. These results are compared with the empirical formula and
other theoretical ones. Good agreements were well found.
Keywords: Prompt Fission Neutron Spectrum, Madland–Nix Spectrum, Watt and
Maxwellian Spectrum, Experimental Spectrum, Exponential Fitting Formula.
INTRODUCTION
Prompt fission neutron spectra from neutrons induced fissions are important in
nuclear reactor applications and constitute the most important component of the source
term for nuclear reactor shielding calculations. Many experimental and theoretical
researches have been carried out for the determination of the spectrum mainly Watt
and Maxwell prompt fission neutron spectra. Madland-Nix Model spectrum which is
based on the standard nuclear evaporation theory accounts for two important physical
factors [1]:
1- The distribution of fission fragment residual nuclear temperature that results from
the initial distribution of fission fragment excitation energy and the subsequent cooling
of the fragments as neutrons are emitted.
M. Aziz et al.
32
2- The energy dependence of the cross section for the inverse process of compound
nucleus formation
In the following sections, different formula for the prompt fission neutron
spectrum such as Madland-Nix, Watt and Maxwillian are presented and solved
numerically, Madland–Nix model was used to determine the prompt fission spectrum
for different fissionable isotopes such as 235U, 238U, 239Pu, 240Pu, 241Pu, and spontaneous
fission of 252Cf. The results are fitted to a simple analytical formula, which can be used
in nuclear and reactor shielding applications [2, 3, 4].
PROMPT FISSION NEUTRON SPECTRUM (PFNS) AND
MODELS
1- Madland-Nix Spectrum
The standard nuclear evaporation theory is used to calculate the neutron energy
spectrum in the center of mass system of a given fission fragment and then transform it
to the laboratory system , taking into account that the average velocity of the light
fragment is higher than that of the heavy fragment. When the energy dependence of the
cross section for the inverse process of compound nucleus formation
is taken into
account, the neutron energy spectrum
in the laboratory system for a fission
fragment moving with average kinetic energy per nucleon
is obtained by [5].
(1)
where,
the temperature –dependent normalization constant is given by:
(2)
neutron energy in center of mass system,
laboratory neutron energy,
average kinetic energy per nucleon for the fission fragments,
compound nucleus cross section in center of mass system,
maximum temperature in units of (MeV),
temperature in units of energy (we include the Boltzmann constant into the definition
of temperature so that it has units of energy) (MeV).
The laboratory prompt fission neutron energy spectrum
is obtained by
evaluating the average spectra calculated for both neutron emission from the light (L)
and heavy (H) average fission fragments, namely [5, 6, 7]
(3)
IMPROVED FORMULA FOR PROMPT FISSION NEUTRON……
33
Since
is the sum of all the various cross-sections, its variation with energy
reflects the behavior of the individual component cross-sections. In particular, at low
energy, it behaves as
(4)
where ε is the neutron energy, and are constants and the two terms on the righthand side represent the cross-section for elastic scattering contribution and the crosssection for radioactive capture , absorption or whatever other exothermic reaction is
possible at this energy, respectively [8, 9, 10, 11].
The resulting temperature distribution is approximately triangular in shape, with a
moderately broad high-temperature cutoff. Terrell observed that if this diffuse cutoff is
replaced by a sharp cutoff, so that
approximated by the triangular distribution
(5)
then the maximum temperature
is related to the initial total average fission
fragment excitation energy
approximately by [3, 5]
(6)
The approximate validity of this expression is based on specific relationship
between the fission-fragment neutron separation energy and the width of the initial
distribution of fission-fragment excitation energy. For the level density parameter ,
we use the relationship
where is the mass number of the fissioning
nucleus [4, 12].
The initial distribution of total fission-fragment excitation energy is
approximately Gaussian in shape, with a total average value that is given by [4, 13]
(7)
Here,
is the average energy release,
and
are the separation (situation)
and kinetic energies of the neutron inducing fission, and
is the total average
fission-fragment kinetic energy. For spontaneous fission, both
and
in Eq. (7) are
zeros [4,14].
34
M. Aziz et al.
2- Watt and Maxwellian Spectrum
Two early representations of the prompt fission neutron spectrum, which are still
used today, are the Maxwellian and Watt spectrum representations with parameters
that are adjusted to optimally reproduce the experimental spectrum for a given
fissioning system at a given excitation energy. The Maxwellian spectrum is given by
[2].
(8)
is the neutron energy,
effective Maxwellian temperature,
is the maximum temperature
The relation between Maxwellian temperature and maximum temperature [5, 15, 16] in
laboratory system is given as
(9)
is laboratory Maxwellian temperature,
is average kinetic energy per nucleon of the light fragment,
is average kinetic energy per nucleon of the heavy fragment.
Watt Spectrum is given by:
(10)
is the laboratory neutron energy,
is the average kinetic energy per nucleon,
is the effective Watt temperature [17, 18, 19].
The relation between the effective Watt temperature and maximum temperature for
laboratory system is given as
(11)
3- Experimental Spectrum
The empirical formula for prompt neutron fission spectrum for 235U is given by
[8, 20, 21]
(12)
the empirical formula for prompt neutron fission spectrum for
21]
35
239
Pu is given by [8, 20,
(13)
RESULTS AND DISCUSSION
1- Comparison with Experimental Spectrum
A Fortran 90 computer program was the written to solve the mathematical
models for Madland-Nix, Maxwell, and Watt spectrum. The results of these
calculations are compared with the experimental model. Figure 1 illustrates the neutron
energy spectrum
for 235U versus laboratory neutron energy (MeV) for fission
induced by incident neutron energy 2.5E-8 MeV for Madland-Nix , Maxwell , Watt
and experimental models The results indicates that MNM model is higher in the high
energy zone which implies safer reactor shielding calculations.
Figure 2 illustrates the neutron energy spectrum
for 239Pu versus neutron
energy (MeV) for fission induced by incident neutron energy 2.5E-8 MeV for
Madland-Nix, Maxwell, Watt and experimental models
Neutron Energy Spectrum, N(E) (1/MeV)
1
235
U + n (E n= 2 .5 E - 8 M e V )
0 .1
0 .0 1
1 E -3
1 E -4
P r e s e n t C a l c u l a ti o n
M a x w e l li a n S p e c t r u m
W a tt S p e c t r u m
E x p e r im e n t S p e c t r u m
1 E -5
1 E -6
1 E -7
0
5
10
15
20
L a b o r a to r y N e u tr o n E n e r g y , E (M e V )
Figure 1. Prompt fission neutron spectrum with different models for the fission of 235U induced
by 2.5E-8 MeV neutrons.
M. Aziz et al.
36
1
239
P u + n ( E n = 2 .5 E - 8 M e V )
0 .1
0 .0 1
1 E -3
P r e s e n t C a lc u l a ti o n
M a x w e l l i a n S p e c tr u m
W a tt S p e c tru m
E x p e rim e n t S p e c tru m
1 E -4
1 E -5
1 E -6
0
5
10
15
20
L a b o r a to r y N e u tr o n E n e r g y , E (M e V )
Figure 2. Prompt fission neutron spectrum in the laboratory system for the fission of
induced by 2.5E-8 MeV neutrons.
239
Pu
2- Analytical Formula for the Present Calculation (MNM Model)
The complex Madland-Nix model which is given by equation (1) is fitted into a
simpler analytical function to be easy for application in both shielding and core
calculations for different isotopes such as 235U, 239Pu, 238U, 252Cf, and 233U. The
analytical function will take the following form:
(14)
All the fitting coefficients
and , j = 1, 2, 3 and errors for fissioning nuclei in
the laboratory system can be determined by using Originlap program [13].
Figure 3 illustrates Neutron energy spectrum
versus energy (MeV) for
fissioning of 235U with thermal neutron energy 2.5E-8 MeV. The figure compares
between three types of spectrum, Present calculation (MNM model), fitting formula
and experimental results. Table 1 contains the fitting coefficients
and
where
j=1,2,3.
fissioning of 239Pu with thermal neutron energy 2.5E-8 MeV. The figure compares
between three types of spectrum, Present calculation (MNM model), fitting formula
and experimental results. Table 2 contains the fitting coefficients
and
where
j=1,2,3.
235
1
U + n (E n= 2.5E-8 MeV)
1
0.1
0.01
1E-3
Present Calculation
Exponential Fitting Formula
Experiment Spectrum
1E-4
1E-5
0
5
10
23 9
0 .0 1
1 E -3
1 E -4
Pres en t C a lc ulation
Ex p o n en tial Fittin g Form u la
Ex p erim en t Sp ectru m
1 E -5
1 E -6
15
0
5
Error
0.050
0.050
0.001
Error
0.000
0.001
0.001
235
20
U, the correlation coefficient R2 =
Parameter
B1
B2
B3
Table 2. The value of parameters in equation (14) for
0.99986.
Value
1.232
-1.079
-0.146
15
Figure 4. Prompt fission neutron spectrum in
the laboratory system for the fission of 239Pu
0.99987.
Parameter
A1
A2
A3
10
L ab or atory N e utr on E ne rg y, E (M e V )
Figure 3. Prompt fission neutron spectrum
in the laboratory system for the fission of
235
U induced by 2.5E-8 MeV neutrons.
Value
1.450
-1.270
-0.154
P u + n (E n = 2.5E -8 M e V )
0.1
Laboratory Neutron Energy, E (M eV)
Parameter
A1
A2
A3
37
239
Value
1.470
0.870
0.053
Error
0.010
0.010
0.001
Pu, the correlation coefficient R2 =
Parameter
B1
B2
B3
Value
1.610
0.902
0.057
Error
0.001
0.001
0.001
252
spontaneous fission of Cf, The figure indicates present calculation (MNM model )
and fitting formula. Table 3 contains the fitting coefficients and where j=1,2,3.
fissioning of 238U with fast neutron energy 2 MeV. The figure compares between
Present calculation (MNM model), fitting formula. Table 4 contains the fitting
coefficients and where j=1,2,3.
M. Aziz et al.
38
1
1
238
Cf
252
0.1
0.01
Present Calculation
1E-3
U + n (E n= 2 M eV )
0.1
0.01
1E -3
P resent C alculation
E xponential Fitting Formula
1E -4
1E -5
1E-4
0
5
10
0
15
5
Figure 5. Prompt fission neutron spectrum
in the laboratory system for the spontaneous
fission of 252Cf.
Value
1.040
-0.890
-0.138
Error
0.020
0.020
0.002
Value
1.380
-1.220
-0.155
Error
0.040
0.040
0.001
252
Cf, the correlation coefficient R2 =
Parameter
B1
B2
B3
0.99987.
Parameter
A1
A2
A3
15
Figure 6. Prompt fission neutron spectrum in
the laboratory system for the fission of 238U
induced by 2 MeV neutrons.
0.99985.
Parameter
A1
A2
A3
10
L aborato ry N eutron E n ergy, E (M eV)
Laboratory Neutron Energy, E (MeV)
238
Value
1.730
0.890
0.056
Error
0.010
0.010
0.001
Parameter
B1
B2
B3
Value
1.490
0.860
0.053
Error
0.010
0.010
0.001
fissioning of 233U with thermal neutron energy 2.5E-8 MeV. The figure compares
between Present calculation (MNM model), fitting formula. Table 5 contains the fitting
coefficients and where j=1,2,3.
0.99986.
Parameter
A1
A2
A3
Value
1.310
-1.150
-0.149
Error
0.040
0.040
0.001
233
Parameter
B1
B2
B3
Value
1.540
0.880
0.055
Error
0.010
0.010
0.001
1
233
39
U +n (E n= 2.5E-8 MeV)
0.1
0.01
1E-3
Present Calculation
1E-4
0
5
10
15
Laboratory Neutron Energy, E (MeV)
Figure 7. Prompt fission neutron spectrum in the laboratory system for the fission of
233
U
CONCLUSION

Madland-Nix model (MNM) for the calculation of prompt fission neutron spectrum
was solved numerically for different fissile and fissionable nuclei such as 235U, 239Pu,
238
U, 233U, and the spontaneous fission spectrum of 252Cf.
 The results of the MNM were compared with the standard Watt formula, Maxwellian
distribution and the empirical formula.
 It was found that MNM model illustrates good agreement with the empirical formula
and shows higher values in the range of energy [2-20MeV] which indicate safer
shielding calculations.
 The results of MNM model for the previous nuclei are fitted in the form of three
exponential summations and the constants of these functions are determined.
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40
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‫‪41‬‬
‫……‪IMPROVED FORMULA FOR PROMPT FISSION NEUTRON‬‬
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‫‪MeV by Watt Spectrum Fit" Nuclear Science and Engineering 106, 345‬‬‫‪352(1990).‬‬
‫‪[20] Lamarsh J. R. "Introduction to Nuclear Reactor Theory" Second Printing,‬‬
‫‪Addison-Wesley Publishing Company, Inc., (1972).‬‬
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‫‪Van Nostrand Reinhold Company, New York, (1981).‬‬
‫اﻟﺼﯿﻐﺔ اﻟﻤﺤﺴﻨﺔ ﻟﺤﺴﺎب اﻟﻄﯿﻒ اﻟﻨﯿﻮﺗﺮوﻧﻲ اﻟﻔﻮري اﻟﻨﺎﺗﺞ ﻣﻦ اﻹﻧﺸﻄﺎر اﻟﻨﻮوي‬
‫‪1‬ﻣﺼﻄﻔﻰ ﻋﺰﯾﺰ‪2 ،‬ﻣﺤﻤﺪ ﻋﻤﺮ ﺷﺎﻛﺮ‪2 ،‬أﺣﻤﺪ أﺑﻮ ﻋﻨﺒﺮ‪1 ،‬إﯾﻤﺎن ﻣﺴﻌﻮد و ‪1‬ﻣﺴﻌﺪ ﺳﻼﻣﮫ‬
‫‪3‬‬
‫‪ 1‬اﻟﻤﺮﻛﺰ اﻟﻘﻮﻣﻲ ﻟﻸﻣﺎن اﻹﺷﻌﺎﻋﻲ واﻟﺮﻗﺎﺑﺔ اﻹﺷﻌﺎﻋﯿﺔ‬
‫ﺷﺎرع أﺣﻤﺪ اﻟﺰﻣﺮ ص‪ .‬ب ‪ 7551‬ﻣﺪﯾﻨﺔ ﻧﺼﺮ‪ ،‬اﻟﻘﺎھﺮة‪ ،‬ﻣﺼﺮ‪.‬‬
‫‪ 2‬ﻗﺴﻢ اﻟﺮﯾﺎﺿﯿﺎت‪ ،‬ﻛﻠﯿﺔ اﻟﻌﻠﻮم‪ ،‬ﺟﺎﻣﻌﺔ ﻃﻨﻄﺎ‪ ،‬ﻣﺼﺮ‬
‫‪.‬‬
‫ﺗﺸﻜﻞ ﻧﯿﻮﺗﺮوﻧﺎت اﻟﻔﻮرﯾﺔ ﻟﻺﻧﺸﻄﺎر اﻟﻨﻮوي أھﻢ ﻣﺼﺎدر اﻹﺷﻌﺎع ﻟﺤﺴﺎﺑﺎت دروع اﻟﻤﻔﺎﻋﻼت اﻟﻨﻮوﯾﺔ ‪ .‬وإن اﻟﺘﺤﺪﯾﺪ‬
‫اﻟﺪﻗﯿﻖ ﻟﻄﯿﻒ ﻧﯿﻮﺗﺮوﻧﺎت اﻟﻔﻮرﯾﺔ ﻟﮭﻮ اﻟﮭﺪف ﻓﻲ ﺣﺴﺎﺑﺎت درع اﻟﻤﻔﺎﻋﻞ ‪ .‬وﻟﻘﺪ اﺻﺒﺢ ﻃﯿﻒ وات )‪ (Watt‬ﻟﻠﻨﯿﻮﺗﺮوﻧﺎت اﻟﻔﻮرﯾﺔ‬
‫ﻹﻧﺸﻄﺎر اﻟﯿﻮراﻧﯿﻮم ‪ ، 235U‬ﻣﻊ ﺗﺤﺪﯾﺪ اﻟﻤﻌﺎﻣﻼت ﺑﻮاﺳﻄﺔ ﻛﺮاﻧﺒﺮرج و آﺧﺮﯾﻦ )‪ (Cranberg et. al.‬ﻣﻌﯿﺎرا ﺻﻨﺎﻋﯿﺎ‬
‫اﻓﺘﺮاﺿﯿﺎ ﻓﻲ أوﺳﺎط ﺗﺪرﯾﻊ اﻟﻤﻔﺎﻋﻼت ‪.‬‬
‫ﺣﺴﺎﺑﺎت ھﺬا اﻟﺒﺤﺚ ﺗﺘﻀﻤﻦ اﻟﺤﻞ اﻟﻌﺪدي ﻟﻄﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ )‪ (Madland-Nix‬ﻟﻠﻨﯿﻮﺗﺮوﻧﺎت اﻟﻔﻮرﯾﺔ اﻟﻨﺎﺗﺠﺔ ﻣﻦ‬
‫اﻹﻧﺸﻄﺎر اﻟﻨﻮوي ﻣﺴﺘﺨﺪم ﻃﺮﯾﻘﺔ روﻣﺒﺮج )‪ (Romberg‬ﻟﻠﺘﻜﺎﻣﻞ‪ .‬ﺗﻢ ﺗﺼﻤﯿﻢ ﺑﺮﻧﺎﻣﺞ ﺣﺎﺳﻮﺑﻲ ﺑﻮاﺳﻄﺔ ﻟﻐﺔ اﻟﻔﻮرﺗﺮن‬
‫)‪ (Fortran‬ﻟﻠﺤﻞ اﻟﺮﯾﺎﺿﻲ ﻟﻨﻤﺎذج ﻣﺎﻛﺴﻮﯾﻞ )‪ ،(Maxwell‬وات و ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ‪ .‬ﻣﻦ ﻣﻘﺎرﻧﺔ اﻟﻨﺘﺎﺋﺞ اﻟﺘﺠﺮﯾﺒﯿﺔ ﻣﻊ أﻃﯿﺎف‬
‫ﻣﺎﻛﺴﻮﯾﻞ و وات و ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ أﺗﻀﺢ اﻷﺗﻲ‪:‬‬
‫‪ ‬أن ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ ﻣﻄﺎﺑﻖ ﻟﺪرﺟﺔ ﻛﺒﯿﺮة ﻣﻊ اﻟﻄﯿﻒ اﻟﺘﺠﺮﯾﺒﻲ‪.‬‬
‫‪ ‬أن ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ أﻛﺜﺮ أﻣﺎﻧﺎ ﻣﻦ أﻃﯿﺎف وات وﻣﺎﻛﺴﻮﯾﻞ وذﻟﻚ ﺣﯿﻦ أﺳﺘﺨﺪاﻣﺔ ﻓﻲ ﺣﺴﺎب درع اﻟﻤﻔﺎﻋﻞ‪.‬‬
‫أﻣﺎ ﻋﻦ اﻟﺴﺒﺐ اﻟﺬي ﺟﻌﻞ ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ ﯾﺤﺼﻞ ﻋﻠﻲ ھﺬه اﻟﻨﺘﯿﺠﺔ ھﻮ أن ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ ﯾﺄﺧﺬ ﻓﻲ اﻻﻋﺘﺒﺎر اﻷﻣﻮر‬
‫اﻵﺗﯿﺔ‪:‬‬
‫‪ ‬ﺣﺮﻛﺔ ﻧﻮاﺗﺞ اﻻﻧﺸﻄﺎر‬
‫‪ ‬ﺗﻮزﯾﻊ درﺟﺔ ﺣﺮارة ﻧﻮاﺗﺞ اﻻﻧﺸﻄﺎر‬
‫‪ ‬اﻟﻤﻘﻄﻊ اﻟﻌﺮﺿﻲ ﻟﻠﺘﻔﺎﻋﻞ اﻟﻌﻜﺴﻲ‬
‫ﺗﻢ اﺳﺘﻨﺘﺎج ﺻﯿﻐﺔ ﺗﺤﻠﯿﻠﯿﺔ ﺟﺪﯾﺪة ﻟﻠﻄﯿﻒ اﻟﻨﯿﻮﺗﺮوﻧﻲ اﻟﻔﻮري اﻟﻨﺎﺗﺞ ﻣﻦ اﻹﻧﺸﻄﺎر اﻟﻨﻮوي وذﻟﻚ ﻣﻦ ﺧﻼل ﻣﻼﺋﻤﺔ‬
‫ﺣﺴﺒﺎﺗﻨﺎ )ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ( ﻷﻗﺮب داﻟﺔ أﺳﯿﺔ وﺳﻤﻰ ھﺬا اﻟﻄﯿﻒ اﻟﺠﺪﯾﺪ ﺑﺄﺳﻢ اﻟﻄﯿﻒ اﻷﺳﻲ اﻟﺠﺪﯾﺪ )‪ .N(E‬وﻣﻦ ﻣﻘﺎرﻧﺔ‬
‫اﻟﻄﯿﻒ اﻷﺳﻲ اﻟﺠﺪﯾﺪ ﻣﻊ ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ ﯾﺘﺒﯿﻦ اﻷﺗﻲ‪:‬‬
‫‪ ‬اﻟﻄﯿﻒ اﻷﺳﻰ اﻟﺠﺪﯾﺪ ﻗﺮﯾﺐ ﺟﺪا ﻣﻦ ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ‬
‫‪ ‬اﻟﻄﯿﻒ اﻷﺳﻰ أﻛﺜﺮ أﻣﺎﻧﺎ ﻋﻨﺪا أﺳﺘﺨﺪاﻣﺔ ﻓﻲ ﺣﺴﺎب درع اﻟﻤﻔﺎﻋﻞ ﻣﻦ ﻃﯿﻒ ﻣﺎدﻻﻧﺪ ‪ -‬ﻧﯿﻜﺲ و ﻣﻦ اﻟﻄﯿﻒ اﻟﺘﺠﺮﯾﺒﻲ‪.‬‬
‫وﺗﺨﻠﺺ اﻟﻨﺘﺎﺋﺞ اﻟﺒﺤﺜﯿﮫ ﻟﮭﺬة اﻟﺪراﺳﮫ إﻟﻲ أن " اﻟﻄﯿﻒ اﻷﺳﻰ اﻟﺠﺪﯾﺪ أﯾﺴﺮ وأﻛﺜﺮ أﻣﺎﻧﺎ ﻓﻲ ﺣﺴﺎب درع اﻟﻤﻔﺎﻋﻞ "‪.‬‬

improved formula for prompt fission neutron spectrum

Transcription

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دﻟﯿﻞ اﻟﻨﺘﺎج اﻟﻌﻠﻤﻲ ﻋﻀﺎء ھﯿﺌﺔ اﻟﺘﺪرﯾﺲ ﻗﺴﻢ اﻟﻜﯿﻤﯿﺎء ﻷ