Transcript Thermalization

SLOWING DOWN OF NEUTRONS
•
•
•
•
Elastic scattering of neutrons.
Lethargy. Average Energy Loss per Collision.
Resonance Escape Probability
Neutron Spectrum in a Core.
HT2005: Rector Physics
T09: Thermalisation
1
Chain Reaction
n
β
235
92
U
n 0.1 eV
ν
Moderator
ν
235
92
U
γ
n 2 MeV
γ
235
92
U
β
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2
Why to Slow Down (Moderate)?
10 4
235
(barns)
10 3
U
10 2
fission
10
1
10 0
capture
10
-1
10 -2 -3
10
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
Energy (eV)
235
92
U n
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1
0
236
92
139
94
1

 56 Ba  36 Kr  3 0 n (84%)
U   236
7
U


(16%)
T

2.
4

1
0
yr

92
1
2

*
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3
Principles of a Nuclear Reactor
E
Leakage
N2
2 MeV
N2
k
N1
Fast fission
Resonance abs.
ν ≈ 2.5
Non-fissile abs.
1 eV
Slowing down
n n/fission
Energy
N1
Non-fuel abs.
Fission
200 MeV/fission
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Leakage
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4
Breeding
4
1
0
2
3
8
U
(barns)
3
1
0
2
1
0
t
o
t
a
l
1
1
0
0
1
0
c
a
p
t
u
r
e
1
1
0
2
1
0
3210 1 2 3 4 5 6 7
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
E
n
e
r
g
y
(
e
V
)
238
92
23.5min
2.3day
239
239
U  01n  239
U

Np


92
93
94 Pu
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5
10 4
239
fission
(barns)
10
3
10 2
Pu
capture
10 1
10 0
10 -1
10 -2 -3
10
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
Energy (eV)
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Energy Dependence
10 4
233
(barns)
10 3
U
10 2
10 1
fission
10 0
capture
10 -1
10 -2 -3
10
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
Energy (eV)
1
log    log E  const  
2
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1
E1 2
1
v
7
Breeding
10 4
232
(barns)
10 3
Th
10 2
10 1
capture
10 0
fission
10 -1
10 -2 -3
10
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
Energy (eV)
23.3min
27.4day
233
233
Th  01n  23390Th 
Pa


91
92 U
232
90
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Space and Energy Aspects
 cm2 
 s  Ω, E  Ω, E   

 sterad  eV 
z  Ω, E
dΩ
 Ω, E
r
dns   s  Ω, E  Ω, E  n  r, Ω  dΩdE
y
dns
 2ns
 s  Ω, E  Ω, E  

ndΩdE nΩE
x
Double differential cross section
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Differential Solid Angle
d
ez
θ
d3r
z
sin  d
ey
r
y
dΩ  sin  d d
Ω
ex
φ
x
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Hard Sphere Model
θ
r
Total scattering
cross section σ = 2πr2
n
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r
11
Hard Sphere Scattering
dθ
impact parameter
cross section σ(θ)
θ
b(θ)
n(r)
r
σ(θ) n is the number of neutrons deflected by an angle greater than θ
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Unit sphere r = 1
n
d  2 sin  d
d  2 bdb
Number of neutrons scattered within d d n
Angular density ns 

Area on the unit sphere
d
Number of neutrons scattered within d , dns 
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d
 n  d
d
13
Differential Cross Section
Number of neutrons
scattered within d
d
dns 
 n  d
d
dns 
dns
d
  s  Ω  Ω  
nd
d
Detector
n
 s    s  Ω  Ω  dΩ
 s  Ω  Ω    s  Ω  Ω 
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Elastic Scattering
μc  cos( )
μ0  cos()
vc
u
u0
v
U0
μ0 
 A2  1  2 A cos  
E  E0 

( A  1)2


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v cos   u cos   vc


U
v sin   u sin 
1  Aμ c
1  A2  2 Aμ c
E0 
1
E
0   A  1
  A  1

2
E0
E 
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Energy Loss
 A  1  2 A cos  
 A 1 
E  E0 
  E0 
  E  E0
2
( A  1)
 A 1 


2
2
θ = 180
 A 1 
 

 A 1 
E
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2
E
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θ=0
 E0  E  E0
 E0
E0
E
16
n(v)
 neutron 
 cm3  eV 
mv
E
2
2
 neutron 
 cm3  cm s 


Energy
n( E)
n(v)dv
n(E)dE
E+dE
v  2mE
E
dE  mvdv
dE
n(v)  n( E)
 mvn( E)  2mE  n( E)
dv
dv
1
n( E)  n( v)

n( v)
dE mv
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Velocity
Change of Variables
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v+dv
v
n( E)dE  n(v)dv
dv
n( E)  n(v)
dE
dE
n( v)  n( E)
dv
17
??
p(E;E0)
E0
E0
E
E-dE
E
p( )d   p( E )dE
E 1  A2  2 A cos

E0
1  A2
dE
2A

sin d
2
E0
1  A
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Quantum mechanics + detailed nuclear physics analysis conclude
Elastic scattering is isotropic in CM system for:
• neutrons with energies E < 10 MeV
• light nuclei with A < 13
p( )d 
The area of the ring rd
Total surface area of the sphere

2  r sin   rd 1
 sin  d
2
4 r
2
dE
2A
2A

sin

d



2p( )d
2
2
E0
1  A
1  A
1  A 
p( E)dE   p( )d 
4A
p( E) 
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2
dE
E0
1
 p( E  E0 )
E0 (1   )
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19
Post Collision
Energy Distribution
1
P   E
1
E0 (1   )
p(E)  p(E  E0 )
E
E0
E0
E0
E
1 
E0
2
1
E 
E0
2
E0 
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Average Logarithmic
Energy Loss
 E0
E
  ln 0 
E

Eo
ln
E0
p( E)dE
E
 E0

p( E)dE
 A  1 ln A  1

 1
ln   1 
1
2A
A 1
2
Eo
 1

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for A  1
2
2
A
3
for A  10
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Average Logarithmic
Energy Loss
0
10
Average lethargy gain  and 



-1
10

-2
10
0
10
1
10
2
10
Mass number A
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1.2

2
A
2
3
 A 1
  1
1.0
2
ln
A 1
A 1
0.8

2A
Exact
Approx.
0.6
0.4
0.2
0.0
0
2
4
6
8 10 12 14 16 18
A
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Number of collision
required for thermalisation:
2 106
E0
ln
ln
E  0.025  18.2
N


For non-homogeneous medium:


N 

N
i
s ,i i
i
i
s ,i
i
Average cosine value of the
scattering angle in CM-system
c  cos 
p( )d   p( c )dc 
1
1
sin  d   dc
2
2
1
1
p (  c )   c 
2

c
pc (  c )d c
0
1
1
 p ( )d
c
c
c
1
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Average Cosine in Lab-System
1
E0
 0    0 p(  c )d c    0 p( E0  E )dE
1
μ0 
E 0
A 1 
1  
0 
31    
1
2




3
 A 1 
 

A

1


1  Aμ c
1  A2  2 Aμ c
2
2
0  cos  
3A
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Material
A
α
0
1H
1
2
4
0
0.111
0.360
0.667
0.333
0.167
6
9
10
0.510
0.640
0.669
0.095
0.074
0.061
H2O
12
238
*
0.716
0.938
*
0.056
0.003
0.037
D2O
*
*
0.033
2D
4He
6Li
9Be
10B
12C
238U
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Slowing-Down Features of Some
Moderators
Moderator
ξ
N
ξΣs
ξΣs/Σa
H 2O
0.927
19.7
1.36
62
D 2O
0.510
36
0.180
5860
Be
0.209
87
0.153
138
C
0.158
115
0.060
166
U
.0084
2170
.0040
0.011
N - number of collision to thermal energy
Ss - slowing down power
Ss/Sa - moderation ratio (quality factor)
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Neutron Velocity Distribution
kB = 1.381×10-23 J/K = 8.617×10-5 eV/K
v+dv
v
Velocity space:
4πv2dv
Probability that energy level
E=mv2/2 is occupied:
p( E)  e
n(v)  n0
HT2005: Rector Physics
E

kBT
4 v
 2 kBT
e
mv 2

2 kBT
2
m
3
2
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e
mv 2

2 kBT
28
Maxwell Distribution for
Neutron Density
thermal spectrum
"hard" spectrum
and corresponding
energy:
vMP
1,0
2kBT
 v0 
m
mv02
E0 
 k BT
2
0,8
0,6
n(v)
The most
probable velocity:
0,4
0,2
 v 
2
4v 2  v0 
n(v)dv  n0
e
dv
3
 v0
0,0
0
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2000 4000 6000 8000
Neutron Velocity (m/s)
29
Maxwell Distribution
for Neutron Flux
4v 3
 (v)dv  n0 3
e
v0 
3
4v
 n0 v0 4
e
v0 
3
 v

 v0
 v

 v0
4v
 0 4
e
v0 
 v

 v0
Don’t forget :
HT2005: Rector Physics



2
v MP
dv 
2 k BT
 v0 
m




2



dv 
v
 vn(v)dv
0


 n(v)dv
2
2

v0  1.128v0
0
dv
v2 
mv2
E
2
dE  mvdv
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3 2
v0
2
mv 2 3
 k BT
2
2
30
0,5
 ( E)  M ( E) 
E
e
2
k BT 

E
k BT
0,4
n(E)
E0  k BT
0,3
n(E), (E)
E  2 k BT
(E)
0,2
0,1
0,0
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0
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2
4
Neutron Energy (E/k T)
B
6
31
Average Energy of Neutrons

1
3
kT
v 2   n(v)dv  v02  3 B
n0 0
2
m
mv 2 3
1
1
1
E
 k BT  mvx2  mvy2  mvz2
2
2
2
2
2
1
1
1
1
2
2
2
mvx  mvy  mvz  k BT
2
2
2
2
Neutron flux distribution:
For thermal neutrons
HT2005: Rector Physics
( E )dE   0
 th 

E
 kB T 
2
e
3 v0
 3 v0
0
0
E
kB T
dE
 (v)dv   vn(v)dv
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Average cosine of scattering angle:
 0  cos  
LAB-system:
CM :
2
3A
 c  cos  0
The consequence of µ0  0 in the laboratory-system is that the neutron
scatters preferably forward, specially for A = 1 i.e. hydrogen and
practically isotropic scattering for A = 238 i.e. Uranium, because µ0  0
i.e.   90o in average. This corresponds to isotropic scattering.
ltr is defined as effective mean free path for non-isotropic scattering.
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Transport Mean Free Path

Information regarding the
original direction is lost


ls
lscos
lscos2
lt
r
l tr  l s  l s cos   l s cos   l s cos  . . .. . l s cos 
2
ltr 
ls
1  cos 
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Ss 
1
ls
 Str 
3
1
ltr
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
n
Str  S s 1  cos 

34
Slowing-Down of Fast Neutrons
• Infinite medium
• Homogeneous mixture of absorbing and
scattering matter
• Continues slowing down
• Uniformly distributed neutron source Q(E)
Φ(E) = [n/(cm2×s×eV)]
Φ(E)dE = number of neutrons
with energies in dE about E
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Continues Slowing-Down
assumed slowing-down
E
real slowing-down
dE
dt
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t
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36
Slowing-Down Density
Energy
• q(E) - number of neutrons, which per
cubic-centimeter and second pass energy
E. If no absorption exists in medium, so:
q(E) = Q; Q - source yield (ncm-3 s-1)
• Assuming no or weak absorption
(without resonances)
• Neutrons of zero energy are removed from
the system
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Q
E0
E
q(E)
0
37
Lethargy Variable
u( E)  u  ln
Eref
E
; Eref
du  
10MeV
dE
E
 E0 Eref 
E0
ln
 ln 
  u( E)  u( E0 )  u( E)

E
 Eref E 
 E0
  ln
E0
 u 
E
 u( E)p( E)dE
Eo
 E0

p( E)dE
 1

1
ln 
Eo
1 coll
u 
u   ,
on average
1 coll
u 
 u  umax , at most
E0
umax  ln
 ln  1
 E0
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Lethargy Scale
1 collision
Energy
 E0
u

u
ln  1

E0
Lethargy
Lethargy
u
n1coll 
u

Number of collisions per 1 neutron to traverse u
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Energy Dependence
Energy
Lethargy
Total number of collisions in dE
N coll   S s ( E)dE  S s (u)du
Eref
0
E/α
u  ln  1
Number of neutrons crossing u
q(u)
Total number of collisions in du
q(u)
E
E+dE
N coll  q(u)n
coll
1
u
u+du
S s ( E)dE  Q
Infinite medium,
no losses,
constant Σs
HT2005: Rector Physics
( E)
1
E
du
dE
 q(u)
 Q

E
dE
Q
 ( E) 
E
S s E
Qp( E)
( E) 
 Ss ( E)  Sa ( E) E
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40
Neutron spectrum
(E)
Eref
E
dE
u  ln
; du  
E
E
(u)du=-(E)dE
Q
(u)=E(E)=
 Ss
(u)
0
u
5
10
15
20
E
10 MeV
HT2005: Rector Physics
0.025 eV
T09: Thermalisation
41
Resonance Absorption
Probability for
absorption per collision:
Sa
Sa  Ss
Lethargy
u–lnα-1
E
E+dE
u
u+du
Number of collisions per
a neutron in du or dE:
du
Probability for absorption
in du or dE:
S a du
S a dE

Sa  Ss 
Sa  Ss  E
Absorption in du causes
a relative change in q:


dE
E
Energy
E/α
dq
S a du
S a dE


q Sa  Ss 
Sa  Ss  E
u
S
du
 Sa Sa s 
q  q0 e 0
u
Sa ( u )
du
 Sa ( u ) Ss ( u )
1

q( E)
p( E  E0 ) 
e 0
q( E0 )
HT2005: Rector Physics
T09: Thermalisation
e

1

E0

E
Sa ( E )
dE
Sa ( E ) Ss ( E ) E
42
Resonance Escape
u
Sa ( u )
du
 Sa ( u ) Ss ( u )
1

q( E)
p( E  E0 ) 
e 0
q( E0 )
e

1

E0

E
Sa ( E )
dE
Sa ( E ) Ss ( E ) E
10 4
235
(barns)
10 3
U
10 2
fission
10
1
10 0
capture
10
-1
10 -2 -3
10
10 -2
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
Energy (eV)
HT2005: Rector Physics
T09: Thermalisation
43
tsc~c

(u)
0(u)
(u)
q0
q
E
u
HT2005: Rector Physics
T09: Thermalisation
44
Life Time
How long time does the neutron exist under
slowing-down phase respectively as thermal?
Slowing-down in time - ts:
Number of collisions in du:
du


vdt
Number of collisions in dt:
ls
dE 2dv

E v
 dt 
2ls dv
 v2
2ls ( v) dv 2ls  1 1 
2 1
ts  





2

v

v
v
S s v1
0 
 1
v1
v0
v(1 eV) = 1.39 · 106 cm/s
v(0.1 MeV ) = 4.4 · 108 cm/s
Thermal life-length - tt :
HT2005: Rector Physics
tt 
la
v

1
Sav
T09: Thermalisation
45
Neutrons Slowing-Down Time
and Thermal Life-Time
Material
HT2005: Rector Physics
H2O
tfast
(s)
1
tthermal
(s)
200
D2O
8
1.5105
Be
10
4300
C
25
1.2104
T09: Thermalisation
46
Under the Neutron Life-Time
(3)
(2)
(1)
E
0
1 eV
0.1 MeV
10 MeV
E
(1) Fission neutrons - fast neutrons
(10 MeV-0.1 MeV)

E
k BT
(E)dE   0
e
dE
2
 kBT 
k BT  2.2 MeV
T  2.5  1010 K
(2) Slowing-down neutrons –
resonance neutrons (0.1MeV - 1 eV)
(3) Thermal neutrons
(1eV - 0.)
( E )dE 
Qp( E )
dE
 SsE
( E )dE   0

E
kB T 
2
e
E
kB T
dE
k B T  0.025 eV
T  300K
HT2005: Rector Physics
T09: Thermalisation
47
The END
HT2005: Rector Physics
T09: Thermalisation
48
 A2  1  2 A cos  
E  E0 

2
( A  1)


 A 1 
E0 
  E  E0
 A 1 
2
 A 1 
 

A

1


θ = 180
2
θ=0
 E0  E  E0
HT2005: Rector Physics
mv 2
E
; v  2mE
2
n( E), n(v)
E+dE
E
v+dv
n(E)dE  n(v)dv
v
dE  mvdv
dE
 mvn( E)  2mE  n( E)
dv
dv
1
n( E)  n( v)

n( v)
dE mv
n(v)  n( E)
T09: Thermalisation
49