1836.063. 99.2. Y89(n,3n). Y87. 79,8h. 20,8. 388.53. 82.00. 484.805. 89.7. Y89(n,4n). Y86. 14,74h. 32,7. 1076.64. 82.00. Y89(n,5n). Y85. 2.68h. 42.633. 231.67.
A METHOD OF SPALLATION NEUTRON ENERGY SPECTRUM RECONSTRUCTION WITH YTTRIUM SAMPLE ACTIVATION S. Kilim, M. Bielewicz, B. Słowiński, E. StrugalskaGola, M. Szuta, A. Wojciechowski Institute of Atomic Energy, 05-400 Otwock-Swierk, Poland
Purpose of the work Looking for possibility to determine spallation neutron energy spectrum with
Y89 activation detector Why yttrium? Y89 the only one naturally occurring isotope – no overlapping reactions – easy to trace
- several residual nuclei Resulting isotopes relatively easy to identify
Stanislaw Kilim
Baldin seminar 4-9.10.2010
2
Y-89(n,?) reaction cross sections available in ENDFs
Y89(n,2n)Y88 a dominant reaction
Stanislaw Kilim
Baldin seminar 4-9.10.2010
3
Yttrium-89 activation reactions taken into account Reaction
Y89(n,g)
Y89(n,2n)
Produced Isotope Y90
Y88
T1/2
3.19h
106,65d
Reaction Threshold [MeV] -6.8570*
11,5
-line Energy [keV]
-line Intensity [%]
202.51
97.3
479.17
90.74
898.042
93.7
1836.063
99.2
388.53
82.00
484.805
89.7
Y89(n,3n)
Y87
79,8h
20,8
Y89(n,4n)
Y86
14,74h
32,7
1076.64
82.00
2.68h
42.633
231.67
84.00
4.86h
42.633
231.67
22.8
Y89(n,5n) Stanislaw Kilim
Y85
Baldin seminar 4-9.10.2010
4
Y89(n,xn) reaction residual nuclei example production. Here is, where the method idea comes from.
Production[nuclei/g/d]
P2-R3 - production experimental value
1,0x10
-4
8,0x10
-5
6,0x10
-5
4,0x10
-5
2,0x10
-5
88Y
87Y
86Y 85Y 90Y
0,0 -10
0
10
20
30
40
50
Reaction threshold energy [MeV]
I k Nt ir
E E, E dE k
thr, k
k = Y90, Y88, Y87, Y86, Y85
Ethr , k
Stanislaw Kilim
Baldin seminar 4-9.10.2010
5
Y89(n,xn) reaction residual nuclei example production – remarks P2-R3 - experimantal value fitting -4
1,2x10
-4
Production [nuclei/g/d]
1,0x10
88Y
-5
8,0x10
I(Ethr)
-5
6,0x10
-5
4,0x10
87Y
-5
2,0x10
86Y 85Y
90Y
0,0 -10
0
10
20
30
40
50
Reaction threshold energy [MeV]
1. The experimental values make a curve I(Ethr) 2. For the I(Ethr) curve the Ethr is a variable Stanislaw Kilim
Baldin seminar 4-9.10.2010
6
The method idea Each isotope production is a result of convoluted influence of spallation neutron spectrum and reaction cross section 0
20
40
60
80
100
Y89(n,g)Y90 CS Y89(n,n')Y89 CS Y89(n,2n)Y88 CS Y89(n,3n)Y87 CS Y89(n,4n)Y86 CS Y89(n,5n)Y85 CS P2-R3 spectrum
(E) 4
-4
2,0x10
-4
1,5x10
-4
1,0x10
2
k(E) 0 0
Ethr,k 40
20
I k Nt ir
60
NSpectrum (User)
Maxw_BoltzCS (User) (barn)
6
-5
5,0x10
80
0,0 100
Energy (MeV)
E E, E dE k
thr, k
Here the Ethr,k is a value
Ethr , k
Stanislaw Kilim
Baldin seminar 4-9.10.2010
7
The method idea
The idea is to transform one isotope production formula I k - nuclei/g/d N - Y89 nuclei/g t ir s E n/cm 2 /s/M eV/d σ k barn 10 24 cm 2
I k Nt ir
E E, E dE k
thr, k
Ethr , k
or E tir E
I k N
E k E, Ethr dE
E neutron fluence/cm 2 /MeV/d
Ethr , k
into a general one valid for any isotope
I Ethr N E E , Ethr dE
Here the Ethr is a variable
Ethr
and solve it. Stanislaw Kilim
Baldin seminar 4-9.10.2010
8
The generalized equation
I Ethr N E E , Ethr dE Ethr
(E) looked for, neutron fluence/cm2/MeV/g/d = n/cm2/MeV/g/d (E) is in fact r , E but dependence on r is not a subject Ethr – a variable parameter I(Ethr) and (E,Ethr) to be known and continuous functions. (E) 0 while E
The equation creates two issues • How to find an analytical form of I(Ethr) • Need to assume something about the (E,Ethr) Stanislaw Kilim
Baldin seminar 4-9.10.2010
9
I(Ethr) case on example sample (Ed = 4.0 GeV, position P2-R3) P2-R3 - experimantal value fitting
-4
1,2x10
-4
Production [nuclei/g/d]
1,0x10
88Y
-5
8,0x10
I(Ethr)
-5
6,0x10
-5
4,0x10
87Y
-5
2,0x10
86Y 85Y
90Y
0,0 -10
0
10
20
30
40
50
Reaction threshold energy [MeV]
I Ethr Ethr e
Ethr
with , and as fitting parameters Stanislaw Kilim
Baldin seminar 4-9.10.2010
10
Y89(n,xn) reaction cross section case available data – experimental and evaluated
Y89(n,2n)Y88
Y89(n,3n)Y87
Y89(n,4n)Y86
Stanislaw Kilim
Baldin seminar 4-9.10.2010
11
Y89(n,xn) reaction cross section case - basic assumptions and consequences
E Ethr E Ethr exp 3 kT E E , Ethr kT 2 0
E E thr E E thr
because it fits the best EXFOR database for Y89(n,2n)Y88 reaction cross section experimental data.
The same formula applies to all Y89(n,xn) reactions. The main difference between Y89(n,2n), (n,3n) and (n,4n) reactions is their threshold energy Ethr.
Stanislaw Kilim
Baldin seminar 4-9.10.2010
12
Notes on Y89(n,xn) reaction cross section analytical formula •
It resembles Maxwell-Boltzmann energy distribution function for ideal gas particles in equilibrium f(E) ~ sqrt(E)*exp(-E/kT).
•
Statistical model of Y89 nucleus applies, i.e. nucleons behave like an ideal gas particles. The only interactions between them are elastic collisions. In an equilibrium stage their energy is dispersed around mean energy kT.
•
Reaction CS proportional to f(E), (E)~f(E).
•
N. Bohr reaction model applies to Y89(n,xn) one, where the reaction has two stages – compound nucleus formation (10-22 s) and compound nucleus evaporation (10-18 – 10-16 s).
•
Y89(n,xn) reaction is caused by neutrons with energy larger than threshold energy – Ethr.
•
Y89(n,xn) reaction cross section is directly proportional to a time of flight through yttrium nucleus, i.e. inversely proportional to neutron speed, i.e. inversely proportional to the square root of neutron energy
•
Neutrons and protons are the two separated, not interacting gases
Stanislaw Kilim
Baldin seminar 4-9.10.2010
13
Y89(n,2n)Y88 reaction cross section fitting
kT
3 2
E
e
The fitting gives • ,kT - coefficients • Ethr – reaction threshold energy
E Ethr EXFOR Y89(n,2n)Y88 CS fitting
1,5
kT
Y89(n,2n)Y88 CS (barns)
E , Ethr
E Ethr
1,0
0,5
0,0 0
10
Ethr
20
30
Energy (MeV)
Stanislaw Kilim
Baldin seminar 4-9.10.2010
14
Y89(n,xn) reaction cross section looks like
E , Ethr
E Ethr
kT
3 2
E
e
E Ethr kT
Y89(n,2n)Y88 CS fitting coefficients ,kT applied to any Y89(n,xn) reaction Ethr – from elsewhere
Y89(n,xn) reaction cross section [cm2]
1.6E-024 Y89(n,xn) reaction cross sections Y89(n,2n)Y88 Y89(n,3n)Y87 Y89(n,4n)Y86 1.2E-024
8E-025
4E-025
0 0
20
40
60
Neutron energy [MeV]
Stanislaw Kilim
Baldin seminar 4-9.10.2010
15
Y89(n,xn) reaction cross section comparison: experimental data, MENDL-2 data base, and this work
Y89(n,xn) cross section [barn]
1.6 Y89(n,xn) reaction cross sections Y89(n,2n)Y88 CS exp data Y89(n,3n)Y87 CS exp data Sigma Y89(n,2n)Y88 ev data - SK Sigma Y89(n,3n)Y87 ev data - SK Sigma Y89(n,4n)Y86 ev data - SK MENDL-2 Y89(n,2n)Y88 CS ev data MENDL-2 Y89(n,3n)Y87 CS ev data MENDL-2 Y89(n,4n)Y86 CS ev data
1.2
0.8
0.4
0 0
20
40
60
80
100
Neutron energy [MeV]
Stanislaw Kilim
Baldin seminar 4-9.10.2010
16
Spallation neutron energy spectrum (E) determination basing on experimental data I(Ethr) and (E,Ethr)
I Ethr N E E , Ethr dE Ethr
Stanislaw Kilim
Baldin seminar 4-9.10.2010
17
Solving an integral equation for yttrium isotope production Using the mentioned earlier functions for (E,Ethr) and I(Ethr) the equation
I Ethr N E E , Ethr dE Ethr
becomes N
kT
Ethr 3 2
(E
thr
E )e
E kT
E E
dE Ethr e
Ethr kT
Differentiating twice both sides on Ethr one gets:
Ethr Stanislaw Kilim
kT
2
N 2 kT
1 2
2kT Ethr e Baldin seminar 4-9.10.2010
Ethr
Ethr 18
Solving a Volterra’s integral equation for Yttrium isotope production - continuation To fulfill request (E 0) = 0 and (E ) 0 must be
2kT kT
and (E) becomes
E
kT
2
N kT 2
Stanislaw Kilim
1 2
3 2
E e
Baldin seminar 4-9.10.2010
1 2kT E kT
19
2,0x10
-4
1,5x10
-4
1,0x10
-4
5,0x10
-5
2
Neutron energy spectrum [n/cm /MeV/g/d]
(E) example view (Ed = 4 GeV, P2-R3)
NSpectrum (User)
E
kT 2 N 2 kT
1 2
3 2
E e
1 2kT E kT
max E at E 3 2
0,0 0
10
20
30
40
50
60
70
80
90
100
Energy [MeV]
Stanislaw Kilim
Baldin seminar 4-9.10.2010
20
The method application example
The method has been applied to „Energy plus Transmutation” experiment results elaboration
Stanislaw Kilim
Baldin seminar 4-9.10.2010
21
„Energy plus Transmutation” project • International research project realised in JINR Dubna. – 12 states take part in – Started 1999
• Purpose of the project is to study transmutation on U/Pbassembly driven by accelerator NUKLOTRON. • Transmutation samples – 129I, 237Np, 238Pu, 239Pu • Activation detectors – Al, Ti, V, Mn, Fe, Co, Ni, Cu, Y, Nb, In, Dy, Lu, W, Au, Bi •
3He
counter
• SSNTD Stanislaw Kilim
Baldin seminar 4-9.10.2010
22
Yttrium samples location during irradiation
U natural
RADIUS
Pb
DEUTERON BEAM
R13.5 R10.5 R8.5 R6 R3 R0
U-nat Pb
B E A M A XIS
U-nat
P LA NE 1 0 P1
Stanislaw Kilim
P LA NE 2 11.8 CM P2
P LA NE 3 24.2 CM P3
P LA NE 4 38.4 CM P4
P LA NE 5 48.4 CM P5
Baldin seminar 4-9.10.2010
23
Experiments with „Energy plus Transmutation” set-up
• Proton beam:
• Deuteron beam:
• • • •
• Ed = 1.6 GeV
Ep = 0.7 GeV Ep = 1.0 GeV Ep = 1.5 GeV Ep = 2.0 GeV
= 0.8 GeV/nucleon
• Ed = 2.52 GeV = 1.26 GeV/nucleon
• Ed = 4.0 GeV = 2.0 GeV/nucleon
30 Y89 samples irradiated in each experiment Stanislaw Kilim
Baldin seminar 4-9.10.2010
24
Y-90 and Y-88 production spatial distribution comparison
6,0E-07 5,0E-07
2,0E-07
0,0E+00
0,0E+00 0,0
5
]
1,0E-07
]
1,0E-07 2 3 4 Detector plane number
3,0E-07
m s[c
2,0E-07
4,0E-07
m s[c diu
3,0E-07
5,0E-07
1,2E-06
11,8
24,0
36,2
1,0E-06
4.0 GeV deuteron beam
diu Ra
4,0E-07
2.52 GeV deuteron beam 0,00 3,00 6,00 8,50 10,5 13,5
Ra
m]
s[c diu
0,00 3,00 6,00 8,50 10,5 13,5
6,0E-07
8,0E-07 0,0 3,0 6,0 8,5 10,5 13,5
48,4 cm
6,0E-07 4,0E-07 2,0E-07
B[nuclei/g/deuteron]
7,0E-07
B[nuclei/g/deuteron]
8,0E-07
9,0E-07
B[nuclei/g/deuteron]
1,0E-06
7,0E-07
1.6 GeV deuteron beam
1
Y-90 S1 spatial distribution based on line 202.51
Y-90 S1 spatial distribution based on gamma line 202.51 keV
8,0E-07
Ra
Y90
Y-90 spatial distribution based on gamma line 202.51 keV
0,0E+00 0,0
Distance from the front of the Pb-target
11,8 24,0 36,2 48,4 cm Distance from the front of the Pb-target
Y-88 S2 spatial distribution based on lines 898.042 and 1836.063 keV
Y-88 S2 spatial distribution based on lines 898.042 and 1836.063 keV
Y-88 S2 spatial distribution based on lines 898.042 and 1836.063 keV
1,0E-04 8,0E-05 6,0E-05 4,0E-05
4,0E-05 2,0E-05
1,4E-04
]
]
11,8 24,0 36,2 48,4 cm Distance from the front of the Pb-target
6,0E-05
1,6E-04
0,0E+00 0,0
11,8
24,0
36,2
48,4 cm
Distance from the front of the Pb-target
Ed = 1.6 GeV Stanislaw Kilim
4.0 GeV deuteron beam
m s[c diu Ra
0,0E+00 0,0
8,0E-05
2.52 GeV deuteron beam 0,0 3,0 6,0 8,5 10,5 13,5
m s[c diu
2,0E-05
1,0E-04
Ra
]
0,0 3,0 6,0 8,5 10,5 13,5
1,2E-04
Ed = 2.52 GeV Baldin seminar 4-9.10.2010
1,2E-04 1,0E-04 0,0 3,0 6,0 8,5 10,5 13,5
8,0E-05 6,0E-05 4,0E-05
B[nuclei/g/deuteron]
1,2E-04
B[nuclei/g/deuteron]
1,4E-04
B[nuclei/g/deuteron]
1,6E-04
m s[c diu Ra
Y88
1,8E-04 1.6 GeV deuteron beam
2,0E-05 0,0E+00 0,0
11,8 24,0 36,2 48,4 cm Distance from the front of the Pb-target
Ed = 4.0 GeV 25
Y-87 and Y-86 production spatial distribution comparison
2,0E-05
13,5
0,0E+00 0,0
11,8 24,0 36,2 Distance from the front of the Pb-target
0,0E+00 11,8
24,0
36,2
2,0E-05
1,5E-05 1,0E-05 5,0E-06
2,0E-05
]
Ed = 1.6 GeV
0,0 3,0 6,0 8,5 10,5 13,5
2,5E-05
cm
11,8 24,0 36,2 48,4 cm Distance from the front of the Pb-target
2,0E-05
s[ diu
]
0,0E+00
3,0E-05 4.0 GeV deuteron beam
Ra
cm
5,0E-06
] cm s[ diu
1,0E-05
Ra
s[ diu
Ra
1,5E-05
48,4 cm
Y-86 S1 spatial distribution based on gamma line 1076.64 keV
2,5E-05 2.52 GeV deuteron beam
0,0E+00 0,0
B[nuclei/g/deuteron]
0,0E+00 11,8 24,0 36,2 Distance from the front of the Pb-target
3,0E-05 B[nuclei/g/deuteron]
2,5E-05
Stanislaw Kilim
1,0E-05 0,0
Y-86 spatial distribution based on gamma line 1076.64 keV
3,0E-05
0,0
2,0E-05 13,5
48,4 cm
3,5E-05
0,0 3,0 6,0 8,5 10,5 13,5
3,0E-05 8,5 10,5
4,0E-05
1.6 GeV deuteron beam
4,0E-05 6,0
Distance from the front of the Pb-target
Y-86 spatial distribution based on gamma line 1076.64 keV
Y86
1,0E-05 0,0
48,4 cm
5,0E-05
3,0
B[nuclei/g/deuteron]
10,5
2,0E-05
6,0E-05
0,0
m] s[c diu Ra
8,5
3,0E-05
]
m] s[c diu Ra
4,0E-05
6,0
4,0E-05
m s[c diu
3,0
5,0E-05 2.52 GeV deuteron beam 0,0 3,0 6,0 8,5 10,5 13,5
4.0 GeV deuteron beam
B[nuclei/g/deuteron]
6,0E-05
0,0
7,0E-05
6,0E-05 B[nuclei/g/deuteron]
8,0E-05
8,0E-05
7,0E-05
1,0E-04
11,8 24,0 36,2 48,4 cm Distance from the front of the Pb-target
Ed = 2.52 GeV Baldin seminar 4-9.10.2010
0,0 3,0 6,0 8,5 10,5 13,5
1,5E-05 1,0E-05 5,0E-06
B[nuclei/g/deuteron]
1,2E-04 1.6 GeV deuteron beam
Y-87 S2 spatial distribution based on gamma lines 388.53 and 484.805 keV
Y-87 S2 spatial distribution based on gamma lines 388.53 and 484.805 keV
Ra
Y87
Y-87 S2 spatial distribution based on gamma lines 388.53 and 484.805 keV
0,0E+00 1
2 3 4 Detector plane number
5
Ed = 4.0 GeV 26
Y-85 production spatial distribution comparison
8,0E-06
1,2E-05 1,0E-05
2 3 4 Detector plane number
5
Ed = 1.6 GeV
Stanislaw Kilim
1,0E-06 0,0E+00 0,0
11,8
24,0
36,2
48,4 cm
Distance from the front of the Pb-target
Ed = 2.52 GeV
Baldin seminar 4-9.10.2010
8,0E-06
m]
0,0E+00
2,0E-06
]
2,0E-06
3,0E-06
9,0E-06 7,0E-06
4.0 GeV deuteron beam
s[c diu
4,0E-06
4,0E-06
m s[c diu
6,0E-06
5,0E-06
1,0E-05
Ra
8,0E-06
6,0E-06
2.52 GeV deuteron beam 0,00 3,00 6,00 8,50 10,5 13,5
Ra
m]
s[c diu
0,00 3,00 6,00 8,50 10,5 13,5
7,0E-06
6,0E-06 0,00 3,00 6,00 8,50 10,50 13,50
5,0E-06 4,0E-06 3,0E-06 2,0E-06
B[nuclei/g/deuteron]
9,0E-06
1,6E-05
B[nuclei/g/deuteron]
1,8E-05
B[nuclei/g/deuteron]
1.6 GeV deuteron beam
1
Y-85 spatial distribution based on gamma line 231.67 keV
Y-85 S1 spatial distribution based on gamma line 231.67 keV
1,4E-05
Ra
Y85
Y-85 S1 spatial distribution based on gamma line 231.67 keV
1,0E-06 0,0E+00 1
2 3 4 Detector plane number
5
Ed = 4.0 GeV
27
Function I(Ethr) in various points of R = 3 cm axis for Ed = 1.6 GeV -5
6,0x10 Maxw_Boltz1 (U ser)
Model
-5
-4
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
Equation
4,0x10
Reduced Chi-Sqr
1,0x10
0,4835 0,99762
Adj. R-Square
Isotope production [nuclei/g/d]
Value
Standard Error
P1-R3
kappa
1,5091E-5
1,50323E-6
P1-R3
eps
6,87252
0,00163
P1-R3
eta
7,19196
0,11985
Model
Maxw_Boltz1 (U ser)
Equation
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
Reduced Chi-Sqr
0,66416
Adj. R-Square
0,99923
8,0x10
-5
3,0x10
P2-R3
kappa
P2-R3
eps
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
2,52526E-6
6,87723
0,00172
8,25657
0,12129
0,88154
Value
-5
4,0x10
Standard Error
P3-R3
kappa
1,67797E-5
1,77906E-6
P3-R3
eps
6,88345
0,00298
P3-R3
eta
8,47577
0,16772
-5
6,0x10 88Y
88Y
88Y
-5
2,0x10
-5
4,0x10
-5
2,0x10
87Y
87Y -5
1,0x10
87Y
-5
2,0x10
86Y
86Y 86Y
0,0
85Y
90Y
0 -5
3,0x10
20
0,0
40
Model
Maxw_Boltz1 (U ser)
Equation
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
2,0x10
20
0,0
40
Model
Maxw_Boltz1 (U ser)
Equation
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
-6
Reduced Chi-Sqr
0,99582 kappa
8,17116E-6
1,23741E-6
P4-R3
eps
6,88646
0,00478
P4-R3
eta
8,7489
0,25199
6,0x10
40
Reaction threshold energy (MeV)
1,2204
Value
-6
20
0,99558
Adj. R-Square Standard Error
P4-R3
85Y
90Y
0
Reaction threshold energy (MeV)
8,0x10
Value
-5
90Y
1,495
Adj. R-Square
85Y
0
Reaction threshold energy (MeV)
Reduced Chi-Sqr
Isotope production [nuclei/g/d]
0,99829
Adj. R-Square
Standard Error
3,10778E-5
eta
P2-R3
Maxw_Boltz1 (U ser)
Equation Reduced Chi-Sqr Value
-5
Model
Standard Error
P5-R3
kappa
2,19847E-6
3,43645E-7
P5-R3
eps
6,88801
0,00628
P5-R3
eta
9,62742
0,31827
88Y 88Y -6
87Y
4,0x10 -5
1,0x10
P1-R3 P2-R3 P3-R3 P4-R3 P5-R3 fitting
87Y -6
2,0x10
86Y
86Y
85Y
85Y
0,0
90Y
0
0,0 20
40
Reaction threshold energy (MeV)
90Y
0
20
40
Reaction threshold energy (MeV)
Ed = 1.6 GeV, axis R3 - yttrium isotope production dependence on reaction threshold energy at various axial positions
Stanislaw Kilim
Baldin seminar 4-9.10.2010
28
Function I(Ethr) in various points of R = 3 cm axis for Ed = 2.52 GeV -5
-5
Isotope production [nuclei/g/d]
3,0x10
-5
7,0x10 Model
Maxw_Boltz1 (U s er)
Equation
y = kappa*(x+ep s )*exp(-(x+eps )/ eta)
Reduced Chi-Sqr
1,94838
Adj. R-Square
0,98978
-5
6,0x10 Value
-5
2,0x10
P1-R3
kappa
P1-R3 P1-R3
6,0x10
Standard Error
1,06576E-5
1,81457E-6
eps
6,87542
0,00446
eta
7,25584
0,27117
Model
Maxw_Boltz1 (U ser)
Model
Maxw_Boltz1 (U ser)
Equation
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
Equation
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
Reduced Chi-Sqr
0,54633
Reduced Chi-Sqr
Adj. R-Square
0,99838
-5
5,0x10
1,27385 0,9969
Adj. R-Square Value
Value
Standard Error
Standard Error
P2-R3
kappa
2,44615E-5
1,43203E-6
P3-R3
kappa
1,74205E-5
1,53456E-6
P2-R3
eps
6,87791
0,00171
P3-R3
eps
6,87955
0,00281
P2-R3
eta
7,6122
0,09755
P3-R3
eta
8,31379
0,17071
-5
4,0x10
-5
4,0x10
88Y
88Y
88Y
-5
3,0x10 -5
-5
1,0x10
2,0x10 -5
2,0x10
87Y
87Y
87Y -5
1,0x10 0,0
85Y
90Y
0
20
Model
Maxw_Boltz1 (U ser)
Equation
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
-5
3,0x10
Reduced Chi-Sqr
Isotope production [nuclei/g/d]
90Y
20
1,0x10
kappa
3,0327E-7
P4-R3 P4-R3
eps
6,87709
0,00104
eta
8,21335
0,06058
Equation
y = kappa*(x+ep s)*exp(-(x+eps)/e ta)
0
8,0x10
20
40
Reaction threshold energy (MeV)
0,98603 Value
-6
85Y
90Y
2,65835
Adj. R-Square
Standard Error
1,00007E-5
Model
Maxw_Boltz1 (U ser)
Reduced Chi-Sqr
0,99967
P4-R3
0,0
40
Reaction threshold energy (MeV)
-5
1,2x10
-5
Value
-5
85Y
0
0,09205
Adj. R-Square
2,5x10
0,0
40
Reaction threshold energy (MeV)
-5
3,5x10
86Y
86Y
86Y
Standard Error
P5-R3
kappa
3,67302E-6
8,03557E-7
P5-R3
eps
6,88242
0,0149
P5-R3
eta
8,86895
0,538
-5
2,0x10
88Y
88Y
-6
6,0x10 -5
1,5x10
P1-R3 P2-R3 P3-R3 P4-R3 P5-R3 fitting
87Y -6
4,0x10 -5
1,0x10
87Y -6
2,0x10
-6
5,0x10
86Y
86Y
0,0
85Y
90Y 0
20
Reaction threshold energy (MeV)
40
85Y
0,0
90Y
0
20
40
Reaction threshold energy (MeV)
Ed = 2.52 GeV, axis R3 - yttrium isotope production dependence on reaction threshold energy at various axial positions
Stanislaw Kilim
Baldin seminar 4-9.10.2010
29
Function I(Ethr) in various points of R = 3 cm axis for Ed = 4.0 GeV Maxw_Boltz1 (U ser)
Model
Maxw_Boltz1 (U ser)
Model
y = kappa*(x+eps)*exp(-(x+eps)/eta)
Model
y = kappa*(x+eps)*exp(-(x+eps)/eta)
Equation
-5
5,0x10
Reduced Chi-Sqr
5,24772
Adj. R-Square
0,98214
-4
Isotope production [nuclei/g/d]
Equation
1,2x10 Value
kappa P1-R3
Reduced Chi-Sqr
4,80382E-6
6,87492
0,00624
eta
7,52632
0,33997
0,878
Value kappa
-4
P2-R3
1,0x10
-5
Reduced Chi-Sqr
-5
0,99884
Adj. R-Square
Standard Error
1,83395E-5
eps
Standard Error
3,62404E-5
2,94316E-6
eps
6,88067
0,00288
eta
8,48746
0,12934
6,0x10
1,06853 0,99786
Adj. R-Square
Value kappa P3-R3
4,0x10
Standard Error
2,04508E-5
1,97008E-6
eps
6,88915
0,00443
eta
8,48386
0,15257
88Y
88Y -5
88Y
8,0x10
-5
4,0x10
-5
3,0x10
-5
6,0x10 -5
2,0x10
-5
4,0x10
-5
2,0x10
87Y 87Y
-5
1,0x10
87Y
-5
2,0x10
86Y
86Y 86Y
0,0
85Y
90Y
0
20
0,0
0,0
90Y
0
20
40
Energy (MeV)
Equation
Reduced Chi-Sqr
1,44892
Adj. R-Square
0,99583
Reduced Chi-Sqr
kappa
2,42811 0,99509
Adj. R-Square Value
Standard Error
1,11636E-5
1,35822E-6
eps
6,89488
0,00629
eta
8,53758
0,19627
Value
-5
1,5x10
kappa P5-R3
Standard Error
5,07318E-6
8,36614E-7
eps
6,87985
0,00542
eta
8,95644
0,29287
88Y
88Y
-5
P1-R3 P2-R3 P3-R3 P4-R3 P5-R3 fitting
-5
2,0x10
1,0x10
87Y
87Y
-6
5,0x10
86Y
86Y
85Y
85Y
90Y
0
40
y = kappa*(x+eps)*exp(-(x+eps)/eta)
Equation
0,0
20
Maxw_Boltz1 (U ser)
Model
y = kappa*(x+eps)*exp(-(x+eps)/eta)
P4-R3
90Y
0
Energy (MeV)
-5
2,0x10
Maxw_Boltz1 (U ser)
Model
85Y
85Y
40
Energy (MeV)
-5
4,0x10
Isotope production [nuclei/g/d]
Maxw_Boltz1 (U ser) y = kappa*(x+eps)*exp(-(x+eps)/eta)
Equation
0,0 20 Energy (MeV)
40
90Y
0
20
40
Energy (MeV)
Ed = 4 GeV, axis R3 - yttrium isotope production dependence on reaction threshold energy at various axial positions.
Stanislaw Kilim
Baldin seminar 4-9.10.2010
30
Ed = 1.6; 2.52 and 4.0 GeV - Neutron flux energy spectra comparison at various R3-axis positions - common Y-scale. 2,0x10
-4
1,5x10 Neutron spectrum
P2-R3 Ed = 1.6 GeV P2-R3 Ed = 2.52 GeV -4 P2-R3 Ed = 4.0 GeV 2,0x10
P1-R3 Ed = 1.6 GeV P1-R3 Ed = 2.52 GeV -4 P1-R3 Ed = 4.0 GeV 2,0x10
-4
-4
-4
1,5x10
-4
1,0x10
-5
5,0x10
1,5x10
-4
1,0x10
-5
5,0x10
1,0x10
5,0x10
0,0
-4
-5
0,0 0
10
20
30
40
50
60
70
80
90 100
0,0 0
10 20 30 40 50 60 70 80 90 100 110
Energy (MeV)
10
20
30
-4
1,5x10
-4
-4
1,0x10
1,0x10
BEAM
-5
60
70
80
90 100
R 13.5 R 10.5 R 8.5 R6 R3 R0
U-nat Pb
BEAM AXIS
U-nat
-5
5,0x10
50
RADIUS
-4
40
Energy (MeV)
P5-R3 Ed = 1.6 GeV P5-R3 Ed = 2.52 GeV P5-R3 Ed = 4.0 GeV
-4
2,0x10
1,5x10 Neutron spectrum
0
Energy (MeV)
P4-R3 Ed = 1.6 GeV P4-R3 Ed = 2.52 GeV P4-R3 Ed = 4.0 GeV
-4
2,0x10
P3-R3 Ed = 1.6 GeV P3-R3 Ed = 2.52 GeV P3-R3 Ed = 4.0 GeV
5,0x10
0,0
0,0 0
10
20
30
40
50
60
Energy (MeV)
70
80
90 100
0
10
20
30
40
50
60
70
80
90 100
PLANE 1 0 P1
PLANE 2 11.8 CM P2
PLANE 3 24.2 CM P3
PLANE 4 38.4 CM P4
PLANE 5 48.4 CM P5
Energy (MeV)
Ed = 1.6; 2.52; and 4.0 GeV - Neutron flux energy spectra comparison at various R3-axis positions - common Y-scale.
Stanislaw Kilim
Baldin seminar 4-9.10.2010
31
Ed = 1.6; 2.52 and 4.0 GeV - Neutron flux spectra comparison at various P2 radial positions - common Y-scale P2-R0 Ed = 1.6 GeV P2-R0 Ed = 2.52 GeV P2-R0 Ed = 4.0 GeV
-4
4,0x10
Neutron spectrum
3,0x10
-4
2,0x10
-4
1,0x10
3,0x10
-4
2,0x10
-4
1,0x10
2,0x10
1,0x10
0,0
-4
-4
-4
0,0 0
10 20 30 40 50 60 70 80 90 100 110
0,0 0
10
20
30
Energy (MeV)
50
60
70
80
90 100
3,0x10
-4
2,0x10
-4
1,0x10
2,0x10
1,0x10
0,0 10
20
30
40
50
60
Energy (MeV)
10
20
30
70
80
90 100
40
50
60
70
80
90 100
P2-R13.5 Ed = 1.6 GeV P2-R13.5 Ed = 2.52 GeV P2-R13.5 Ed = 4.0 GeV
-4
4,0x10
-4
3,0x10
-4
-4
2,0x10
-4
1,0x10
-4
-4
0,0 0
0
Energy (MeV)
P2-R10.5 Ed = 1.6 GeV P2-R10.5 Ed = 2.52 GeV P2-R10.5 Ed = 4.0 GeV
-4
4,0x10
-4
3,0x10 Neutron spectrum
40
Energy (MeV)
P2-R8.5 Ed = 1.6 GeV P2-R8.5 Ed = 2.52 GeV P2-R8.5 Ed = 4.0 GeV
-4
4,0x10
P2-R6 Ed = 1.6 GeV P2-R6 Ed = 2.52 GeV P2-R6 Ed = 4.0 GeV
-4
4,0x10
-4
-4
3,0x10
P2-R3 Ed = 1.6 GeV P2-R3 Ed = 2.52 GeV P2-R3 Ed = 4.0 GeV
-4
4,0x10
0,0 0
10
20
30
40
50
60
70
80
90 100
Energy (MeV)
0
10
20
30
40
50
60
70
80
90 100
Energy (MeV)
Ed = 1.6; 2.52 and 4.0 GeV - Neutron flux spectra comparison at various P2 radial positions - common Y-scale
Stanislaw Kilim
Baldin seminar 4-9.10.2010
32
Ed = 1.6, 2.52 and 4.0 GeV parameter kappa comparison Ed = 2.52 GeV - Kappa
Ed = 1.6 GeV - Kappa
1,2E-04
6,0E-05
1,0E-04
5,0E-05
8,0E-05
4,0E-05
6,0E-05
3,0E-05
4,0E-05
2,0E-05
2,0E-05
Ra d
iu.
..
03
68,5 P4 P5 10,5 13,5 P1 P2 P3
Ra d
0,0E+00
l Ax ia
iu.
..
1,0E-05 03 0,0E+00 6 8,5 P5 P4 10,5 13,5 P1 P2 P3 ition l pos Ax ia
7,0E-05 6,0E-05 5,0E-05 4,0E-05 3,0E-05 2,0E-05 1,0E-05 0,0E+00
l Ax ia
Kappa
RADIUS
03 6 8,5 P4 P5 10,5 13,5 P1 P2 P3
Stanislaw Kilim
..
tion posi
Ed = 4 GeV - Kappa
Ra d
iu.
BEAM
R 13.5 R 10.5 R 8.5 R6 R3 R0
U-nat Pb
BEAM AXIS
U-nat
tion posi
PLANE 1 0 P1
Baldin seminar 4-9.10.2010
PLANE 2 11.8 CM P2
PLANE 3 24.2 CM P3
PLANE 4 38.4 CM P4
PLANE 5 48.4 CM P5
33
Ed = 1.6, 2.52 and 4.0 GeV parameter eta comparison Ed = 2.52 GeV - eta
Ed = 1.6 GeV - eta
Ra diu .
10
10
8
8
6
6
4
4
2
2
03 0 68,5 P4 P5 10,5 P3 13,5 P1 P2
..
Ra d
iu.
l Ax ia
posi
tion
4
Eta
6
BEAM
R 13.5 R 10.5 R 8.5 R6 R3 R0
03
68,5 P4 P5 10,5 13,5 P1 P2 P3
tion
U-nat Pb
BEAM AXIS
0
l Ax ia
Stanislaw Kilim
posi
U-nat
2
..
0
RADIUS 10 8
iu.
68,5 P4 P5 10,5 13,5 P1 P2 P3
l Ax ia
Ed = 4 GeV - eta
Ra d
03
..
posi
tion
PLANE 1 0 P1
Baldin seminar 4-9.10.2010
PLANE 2 11.8 CM P2
PLANE 3 24.2 CM P3
PLANE 4 38.4 CM P4
PLANE 5 48.4 CM P5
34
Ed = 1.6, 2.52 and 4.0 GeV parameter eps comparison Ed = 1.6 GeV - eps
Ed = 2.52 GeV - eps
6,90
6,89 6,88 6,88 6,87 6,87 6,86 6,86 6,85 6,85
6,89 6,88 6,87
iu.
..
6,85 03 6,84 6 8,5 P4 P5 10,5 P3 13,5 P1 P2
l Ax ia
Ra d
iu.
posi
iu.
03
..
l Ax ia
tion posi
Eps
RADIUS
68,5 P4 P5 10,5 13,5 P1 P2 P3
6,92 6,91 6,90 6,89 6,88 6,87 6,86 6,85 6,84
l Ax ia
Stanislaw Kilim
6 8,5 10,5 13,5
tion
Ed = 4 GeV - eps
Ra d
03
..
P2 P3 P4 P5
Ra d
P1
6,86
posi
BEAM
R 13.5 R 10.5 R 8.5 R6 R3 R0
U-nat Pb
BEAM AXIS
U-nat
tion
PLANE 1 0 P1
Baldin seminar 4-9.10.2010
PLANE 2 11.8 CM P2
PLANE 3 24.2 CM P3
PLANE 4 38.4 CM P4
PLANE 5 48.4 CM P5
35
Ed = 1.6, 2.52 and 4.0 GeV E(max) comparison Ed = 2.52 GeV - E(fimax)
Ed = 1.6 GeV - E(fimax)
Ra d
iu.
03
..
68,5 P4 P5 10,5 13,5 P1 P2 P3
16 14 12 10 8 6 4 2 0
Ra d
iu.
l Ax ia
posi
tion
iu.
03
..
posi
tion
Energy of maximum [MeV]
RADIUS
68,5 P4 P5 10,5 13,5 P1 P2 P3
14 12 10 8 6 4 2 0
l Ax ia
Stanislaw Kilim
36 8,5 P4 P5 10,5 13,5 P1 P2 P3
l Ax ia
Ed = 4 GeV - E(fimax)
Ra d
0
..
14 12 10 8 6 4 2 0
posi
tion
BEAM
R 13.5 R 10.5 R 8.5 R6 R3 R0
U-nat Pb
BEAM AXIS
U-nat
PLANE 1 0 P1
Baldin seminar 4-9.10.2010
PLANE 2 11.8 CM P2
PLANE 3 24.2 CM P3
PLANE 4 38.4 CM P4
PLANE 5 48.4 CM P5
36
Ed = 1.6, 2.52 and 4.0 GeV E(max) comparison Plane P2 - Energy of spectrum maximum comparison
14
1.6 GeV 2.52 GeV
12
4.0 GeV
10 P1-R3
P2-R3
P3-R3
P4-R3
P5-R3
Energy of maximum [MeV]
Energy of maximum [MeV]
Axis R3 - Energy of spectrum maximum comparison
14 1.6 GeV
10 P2-R0
Axial position
P2-R3
P2-R6
P2-R8.5 P2-R10.5 P2-R13.5
RADIUS
Radial position
Ed = 4 GeV, Energy of spectrum maximum
Energy of maximum [MeV]
2.52 GeV 4.0 GeV
12
14,00
P1 P2
12,00
BEAM
R 13.5 R 10.5 R 8.5 R6 R3 R0
P3
U-nat Pb
BEAM AXIS
U-nat
P4
10,00 0
3
6
8,5
Radial position
Stanislaw Kilim
10,5
13,5
P5
PLANE 1 0 P1
Baldin seminar 4-9.10.2010
PLANE 2 11.8 CM P2
PLANE 3 24.2 CM P3
PLANE 4 38.4 CM P4
PLANE 5 48.4 CM P5
37
Energy of spectrum maximum P1
R0
R3
R6
P5
Efmax
Efmax Error
Efmax
Efmax Error
Efmax
Efmax Error
Efmax
Efmax Error
1.6
10,13
2,33
14,17
0,16
14,75
0,36
14,58
0,26
14,27
0,36
12,78
0,27
2.52 4.0
10,74
0,17
12,78
0,07
12,59
0,06
12,67
0,17
13,55
0,11
1.6
10,79
0,18
12,38
0,18
12,71
0,25
13,12
0,38
14,44
0,48
2.52
10,88
0,41
11,42
0,15
12,47
0,26
12,32
0,09
13,30
0,81
4.0
11,29
0,51
12,73
0,19
12,73
0,23
12,81
0,29
13,43
0,44
1.6
12,10
0,19
2.52
11,08
0,15
12,10
0,07
12,48
0,11
12,90
0,47
13,49
0,65
1.6
11,60
0,35
2.52
11,01
0,25
11,93
0,11
11,97
0,16
12,42
0,39
13,09
0,74
1.6
11,48
0,21
2.52
10,18
0,34
11,33
0,23
12,51
0,20
12,36
0,26
13,30
1,17
1.6
11,30
0,21
2.52
10,99
0,08
11,45
0,16
12,34
0,51
12,49
0,51
13,07
0,68
4.0
R13.5
P4
Efmax Error
4.0
R10.5
P3
Efmax
4.0
R8.5
P2
Ed [GeV]
4.0
Stanislaw Kilim
10,41
10,81
10,68
11,00
0,11
0,13
0,19
0,15
Baldin seminar 4-9.10.2010
38
The method error discussion • Typical errors like count statistics, deuteron fluence error, sample mass error and so on are of less importance; •Y89(n,2n)Y88 reaction cross section approximation (fitting) error of much more importance, but difficult to assess; •Y89(n,xn) reaction cross section approximation by Y89(n,2n)Y88 parameters makes additional error; •Approximation of infinite number of ideal gas particles with 50 neutrons (statistical model of nucleus); •Relativistic effects?
In some sense the method tests itself saying that where both , kT and are fitting parameters.
2kT kT
Discrepancy between the direct fitting value and calculated using , kT should be an error measure of the method. While direct fitting value ranges from 6.87 to 6.91 MeV the , kT calculated from ranges from 7.44 to 8.5. This suggests the method error to be of order of 10-20%.
This still doesn’t explain the high energy of the spectrum maximum!
Are there any other errors of the method? Note: These are the experimental values. Yurevich’s measurement of spallation neutron spectrum (but induced by protons) gave high average neutron energy too.
Stanislaw Kilim
Baldin seminar 4-9.10.2010
39
What else could be done Done
To be done
Reaction
Threshold energy [MeV]
Reaction
Threshold energy [MeV]
Y89(n,g)Y90
-6,857
Y89(n,He4)Rb86
-0,6916
Y89(n,2n)Y88
11,6048
Y89(n,p)Sr89
0,7183
Y89(n,3n)Y87
21,0645
Y89(n,d)Sr88
4,8995
Y89(n,4n)Y86
33,008
Y89(n,2a)Br82
7,0633
Y89(n,5n)Y85
42,633
Y89(n,np)Sr88
7,1494
Y89(n,T)Sr87
9,8104
Y89(n,He3)Rb87
10,0774
Y89(n,2p)Rb88
11,7317
Y89(n,2d)Rb86
23,4207
Y89(n,dT)Rb85
25,8424
But how?
Stanislaw Kilim
Baldin seminar 4-9.10.2010
40
Conclusions •
The method is simple, the results are coherent.
•
Surprising is that using threshold detector with few threshold energies one can say so much about the spectrum in the entire energy range.
•
According to the method there’s no big difference between Ed = 1.6, 2.52 and 4.0 GeV spectra. The spectrum seems to have a kind of saturation at these energies of deuterons.
Stanislaw Kilim
Baldin seminar 4-9.10.2010
41
References: 1. 2. 3. 4. 5. 6.
7. 8.
M.I. Krivopustov et al., JINR Preprint R1-2000-168, Dubna, 2000// Kerntechnik 2003, 68, p.p. 48-55// JINR-Preprint E1-2004-79, Dubna, 2004. Martsynkevich B. A. et al. „Unfolding of Fast Neutron Spectra in the Wide Energy Range (up to 200 MeV) in Heterogeneous Subcritical Assembly of an Electronuclear System „Energy plus Transmutation” – in Russ. – JINR report P1-2002-65. ANL/NDM-94 “Evaluated Neutronic Data File for Yttrium”, A.B. Smith, D.L. Smith, P. Rousset, R.D. Lawson, and R.J. Howerton, January 1986 Yu.N. Shubin, V.P. Lunev, A.Yu. Konobeyev, A.I. Dityuk „Cross-Section Data Library MENDL2 to Study Activation and Transmutation of Materials Irradiated by Nucleons of Intermediate Energies” – INDC(CCP)-385 Distrib.: G M.I. Krivopustov, M. Bielewicz, S. Kilim et al., JINR Preprint R1-2007-7, Dubna, 2007 S. Kilim et al., Spallation Neutron Energy Spectrum Determination with Yttrium as a Threshold Detector on U/Pb-assembly „Energy plus Transmutation”, p 343-352; Progress in High Energy Physics and Nuclear Safety – Edited by V. Begun, L.L. Jenkovszky, A. Polański, Springer, 2009 “Study of Deep Subcritical Electronuclear Systems and Feasibility of their Application for Energy Production and Radioactive Waste Transmutation” – E&T RAW Collaboration – JINR Dubna preprint E1-2010-61. Yurevich et al. //PEPAN Lett. 2006. V. 3. P. 49.
Stanislaw Kilim
Baldin seminar 4-9.10.2010
42
Thank You for attention
Stanislaw Kilim
Baldin seminar 4-9.10.2010
43
Background slides
Stanislaw Kilim
Baldin seminar 4-9.10.2010
44
Neutron spectrum determination stages
EXFOR (barns)
1,5
a)
Y89(n,2n)Y88 CS fitting
1,0
Y89(n,2n)Y88 reaction CS EXFOR data fitting to get parameters and kT
0,5
0,0 10
15
20
25
30
Energy (MeV)
Reaction CS [barn]
6
Y89(n,g)Y90 CS sim Y89(n,n')Y89 CS sim Y89(n,2n)Y88 CS sim Y89(n,3n)Y87 CS sim Y89(n,4n)Y86 CS sim Y89(n,5n)Y85 CS sim
b)
4
Any Y89(n,xn) reaction CS determination using and kT parameters and Ethr
2
0 0
10
20
30
40
50
60
70
80
90
100
Energy (MeV) C18 experiment fitting
c)
-4
1,2x10
-4
Production
1,0x10
-5
8,0x10
88Y
-5
6,0x10
Yttrium isotope production fitting to get parameters , and
-5
4,0x10
87Y
-5
2,0x10
86Y
0,0
85Y
90Y
-10
0
10
20
30
40
50
Reaction threshold energy [MeV] P2-R3 n. spectrum
d) -4
NSpectrum (User)
2,0x10
-4
1,5x10
-4
1,0x10
-5
Spallation neutron spectrum determination as A function of , kT, and .
5,0x10
0,0 0
10
20
30
40
50
60
70
80
90
100
110
Energy (MeV)
Fig. ??. Spallation neutron energy spectrum reconstruction stages a) Y89(n,2n)Y88 reaction cross section (CS) data fitting to get "alpha" and" kT" parameters b) generalized Y89(n,xn)reaction CS construction c) isotope production data fitting to get generalized isotope production function I(E ) Stanislaw Kilim Baldin seminar 4-9.10.2010 d) Neutron energy spectrum reconstyruction. thr
45
Y89(n,2n)Y88 reaction CS EXFOR data various fittings
Y89(n,2n)Y88 CS (barns)
1,5
EXFOR fitting with y=alfa/sqrt(kT)/sqrt(x)* ((x-Ethr)/kT)*exp(-(x-Ethr)/kT)
a)
1,0
0,5
Model
Maxw_BoltzC S (User)
Equation
y=alfa/sqrt(kT )/sqrt(x)*(x-Et hr)/kT*exp(-(x -Ethr)/kT)
Reduced Chi-Sqr
0,00929
Adj. R-Square
0,88299
E E88
kT 2 3
Value
Standard Err
Y89(n,2n)Y88 alfa Y89(n,2n)Y88 kT
41,7125
1,16615
8,02267
0,26505
Y89(n,2n)Y88 Ethr
11,8439
0,05482
E
E Ethr,k exp kT
0,0 0
5
10
Y89(n,2n)Y88 CS (barns)
1,0
Model
Maxw_BoltzS QCS (User)
Equation
y=alfa/sqrt(kT )/sqrt(x)*((x-Et hr)/kT)^2*exp( -(x-Ethr)/kT)
25
EXFOR fitting with y=alfa/sqrt(kT)/sqrt(x)* ((x-Ethr)/kT)^2*exp(-(x-Ethr)/kT)
Value
kT
Standard Erro
Y89(n,2n)Y88 alfa Y89(n,2n)Y88 kT
21,6892
0,52543
4,33228
0,13764
Y89(n,2n)Y88 Ethr
10,5980
0,11516
0,0 0
5
10
15
20
25
Energy (MeV) c)
Equation
0,5
Reduced Chi-Sqr
0,01045
Adj. R-Square
0,86833 Value 52,9453
2,71832
18,6042
1,20092
Y89(n,2n)Y88 Ethr
12,5678
0,03074
0
5
10
15
Model
Maxw_BoltzCS N (User)
Equation
y=alfa/sqrt(kT)/ sqrt(x)*((x-Ethr)/ kT)^n*exp(-(x-Et hr)/kT)
kT 2
20
25
30
Value Y89(n,2n)Y88
alfa
Y89(n,2n)Y88
kT
Y89(n,2n)Y88
Ethr
Y89(n,2n)Y88
n
Standard Error
43,45287
1,90045
9,11047
0,96073
12,16393
0,08633
0,84421
0,06974
kT 2 3
0,0 0
5
10
15
E
E E88 n
0,89293
Adj. R-Square
E Ethr,k exp kT
E Ethr,k exp kT
EXFOR fitting with y=alfa/sqrt(kT)/sqrt(x)* ((x-Ethr)/kT)^n*exp(-(x-Ethr)/kT)
0,0085
Reduced Chi-Sqr
0,5
E
E E88 3
Energy (MeV) d)
1,0
3 2
0,0
1,5
2
30
Standard Erro
Y89(n,2n)Y88 alfa Y89(n,2n)Y88 kT
Reduced Chi-Sqr
Adj. RSquare
0,00929
0,88299
Statistics
Statistics
Reduced Chi-Sqr
Adj. RSquare
0,0112
0,85882
Statistics
Statistics
Reduced Chi-Sqr
Adj. RSquare
0,01045
0,86833
EXFOR fitting with y=alfa/sqrt(kT)/sqrt(x)* sqrt((x-Ethr)/kT)*exp(-(x-Ethr)/kT)
Maxw_BoltzC SSQRT (User ) y=alfa/sqrt(kT )/sqrt(x)*sqrt(( x-Ethr)/kT)*ex p(-(x-Ethr)/kT)
Model
1,0
E E88
0,85882
Adj. R-Square
Statistics
30
0,0112
Reduced Chi-Sqr
0,5
1,5
Y89(n,2n)Y88 CS (barns)
20
Energy (MeV) b)
1,5
Y89(n,2n)Y88 CS (barns)
15
Statistics
20
25
30
E
E Ethr,k exp kT
n
Statistics
Statistics
Value
Reduced Chi-Sqr
Adj. RSquare
0,84421
0,0085
0,89293
Energy (MeV)
Y89(n,2n)Y88 reaction CS EXFOR data various fittiings
Stanislaw Kilim
Baldin seminar 4-9.10.2010
46
Y89(n,2n)Y88 CS EXFOR data various fittings – cont. alfa Value Y89(n,2n)Y88 41,71254 CS alfa Value Y89(n,2n)Y88 21,68922 CS alfa Value Y89(n,2n)Y88 52,94533 CS alfa Value Y89(n,2n)Y88 43,45287 CS
Stanislaw Kilim
alfa Standard Error 1,16615
kT Value
alfa Standard Error 0,52543
kT Value
alfa Standard Error 2,71832
kT Value
alfa Standard Error 1,90045
8,02267
4,33228
18,60426
kT Value 9,11047
kT Standard Error 0,96073
kT Standard Error 0,26505
Ethr Value
kT Standard Error 0,13764
Ethr Value
kT Standard Error 1,20092
Ethr Value
Ethr Value 12,16393
11,84393
10,59808
12,5678
Ethr Standard Error 0,08633
Baldin seminar 4-9.10.2010
Ethr Standard Error 0,05482
Statistics Reduced ChiSqr 0,00929
Statistics Adj. RSquare 0,88299
Ethr Standard Error 0,11516
Statistics Reduced ChiSqr 0,0112
Statistics Adj. RSquare 0,85882
Ethr Standard Error 0,03074
Statistics Reduced ChiSqr 0,01045
Statistics Adj. RSquare 0,86833
n Value 0,84421
n Standard Error 0,06974
Statistics Reduced Chi-Sqr 0,0085
Statistics Adj. RSquare 0,89293
47
Evaporation mechanism of Y89(n,xn) reaction
ENERG Y
Incident neutron energ y
E MITTE D NE UTRONS E NE RGY S P E CTRUM E MITTE D NE UTRONS
0
-V0 "REAL" NUCLEUS
Stanislaw Kilim
F ERMI G AS MO DEL NUCLEUS
F ERMI G AS MO DEL O F EXCIT ED CO MPO UND NUCLEUS
Baldin seminar 4-9.10.2010
F ERMI G AS MO DEL O F RESIDUAL NUCLEUS AND EVAPO RAT ED NEUT RO NS
48
