HT2005: Rector Physics T09: Thermalisation 1
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 2
Chain Reaction
ν
β
γ
γ
β
ν
n
23592 U
2 MeVn
0.1 eVn
23592 U
23592 U
Mo
der
ato
rM
od
erat
or
HT2005: Rector Physics T09: Thermalisation 3
Why to Slow Down (Moderate)?
139 94 156 3
23
235 1 236 *92 0 9 6 7
92 1
6 0
2
2
Ba Kr
U (16%) 2.U U
4 1
)
0
3 (84%nn
T yr
HT2005: Rector Physics T09: Thermalisation 4
Principles of a Nuclear Reactor
1
2
N
Nk
n/
fissi
onN1
N2Leakage
Fast fission
Resonance abs.
Non-fuel abs.
Leakage
Non-fissile abs.
Fission
Slo
win
g d
ow
n
Ene
rgy
E
2 MeV
1 eV
200 MeV/fission
ν ≈ 2.5
HT2005: Rector Physics T09: Thermalisation 5
Breeding
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(b
arns
)238 U
capture
total
23.5min 2.3day238 1 239 239 23992 0 92 93 94U U Np Pun
HT2005: Rector Physics T09: Thermalisation 6
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(bar
ns)
239 Pu
capture
fission
HT2005: Rector Physics T09: Thermalisation 7
Energy Dependence
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(b
arns
)
233 U
capture
fission
1 2
1 1 1log log
2E const
E v
HT2005: Rector Physics T09: Thermalisation 8
Breeding
23.3min 27.4day232 1 233 233 23390 0 90 91 92Th Th Pa Un
10-3 10-2 10-1 100 101 102 103 104 105 106 107
Energy (eV)
10-2
10-1
100
101
102
103
104
(b
arns
)
232 Th
capture
fission
HT2005: Rector Physics T09: Thermalisation 9
Space and Energy Aspects
,E Ω
r
x
y
z
,EΩ
2cm
, ,sterad eVs E E
Ω Ω
2
, , ,
, ,
s s
s ss
dn E E n d dE
dn nE E
nd dE n E
Ω Ω r Ω Ω
Ω ΩΩ Ω
Double differential cross section
dΩ
HT2005: Rector Physics T09: Thermalisation 10
Differential Solid Angle
x
y
z
ez
ey
ex
r
Ω
d3r
φ
θ sind d d Ω
d
sin d
HT2005: Rector Physics T09: Thermalisation 11
Hard Sphere Model
r
Total scatteringcross section σ = 2πr2
n
r
θ
HT2005: Rector Physics T09: Thermalisation 12
rb(θ)
θ
impact parameter
cross section σ(θ)
dθ
n(r)
σ(θ) n is the number of neutrons deflected by an angle greater than θ
Hard Sphere Scattering
HT2005: Rector Physics T09: Thermalisation 13
2d bdb 2 sind d
Unit sphere r = 1
Number of neutrons scattered withinAngular density
Area on the unit spheres
d d nn
d
n
Number of neutrons scattered within , s
dd dn n d
d
HT2005: Rector Physics T09: Thermalisation 14
Detector
n
Differential Cross Section
ss s
dn ddn
nd d
Ω Ω
s s
s s
d
Ω Ω Ω
Ω Ω Ω Ω
Number of neutrons
scattered within
s
d
ddn n d
d
HT2005: Rector Physics T09: Thermalisation 15
Elastic Scattering
2
0 2
1 2 cos
( 1)
A AE E
A
c 0μ cos( ) μ cos( )
c0 2
c
1 Aμμ
1 A 2Aμ
u0
U0U
u
vc
v
sin sin
cos cos c
v u
v u v
E
EA
E
EA 0
00 11
2
1
HT2005: Rector Physics T09: Thermalisation 16
Energy Loss
0 0E E E
22
0 0 02
1 2 cos 1
( 1) 1
A A AE E E E E
A A
21
1
AA
θ = 0
θ = 180
E0E0EE E
HT2005: Rector Physics T09: Thermalisation 17
( ) ( )
( )
( )
(
(
)
)
dvn E n v
dEdE
n v n
n E dE n v
Edv
dv
3 3
2
( ) (
2
)
2
neutron neutroncm eV cm c
n E n v
m s
mvE v mE
E v
E+dE v+dv
( ) ( ) ( ) 2 ( )
1( ) ( ) ( )
dE mvdv
dEn v n E mvn E mE n E
dvdv
n E n v n vdE mv
Change of Variables
( )n E dE ( )n v dv
Ene
rgy
Vel
ocit
y
HT2005: Rector Physics T09: Thermalisation 18
dA
A
E
dE
A
AA
E
E
sin1
2
1
cos21
20
2
2
0
E
E-dEEE0 E0
p(E;E0)
??
dEEpdp )()(
HT2005: Rector Physics T09: Thermalisation 19
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
2
( )
2 sin 1sin
4 2
Theareaof theringrdp d
Totalsurfaceareaof thesphere
r rdd
r
2
0
1( ) ( )
4
A dEp E dE p d
A E
00
1( ) ( )
(1 )p E p E E
E
2 20
2 2sin 2 ( )
1 1
dE A Ad p d
E A A
HT2005: Rector Physics T09: Thermalisation 20
EE0 E0
0
1
(1 )E 0( ) ( )p E p E E
Post Collision Energy Distribution
1 P E
0E
E
0 0
0
1
21
2
E E
E E
HT2005: Rector Physics T09: Thermalisation 21
0
0
02
0
ln ( )1 1
ln 1 ln 1 ln1 2 1
( )
o
o
E
E
E
E
Ep E dE
E AE AE A A
p E dE
1 for 1
2for 10
23
A
AA
Average Logarithmic Energy Loss
HT2005: Rector Physics T09: Thermalisation 22
Average Logarithmic Energy Loss
100
101
102
10-2
10-1
100
Mass number A
Ave
rag
e le
tha
rgy
ga
in
an
d
HT2005: Rector Physics T09: Thermalisation 23
0 2 4 6 8 10 12 14 16 180.0
0.2
0.4
0.6
0.8
1.0
1.2
Exact Approx.
A
21 1
1 ln2 1
A AA A
223
A
HT2005: Rector Physics T09: Thermalisation 24
Number of collision required for thermalisation:
For non-homogeneous medium:
Average cosine value of the scattering angle in CM-system
2.18025.0
102lnln
60
EE
N
,
,
i s i ii
i s ii
N
N
0
)(
)(
2
1)( 1
1
1
1
ccc
cccc
cc
dp
dp
p
cos
( ) ( )
1 1sin
2 2
c
c c
c
p d p d
d d
HT2005: Rector Physics T09: Thermalisation 25
3
2
1
0 113
1
A
0
2cos
3
A
1
1
0000
0
0
)()(E
E
cc dEEEpdp
Average Cosine in Lab-System
21
1
AA
c0 2
c
1 Aμμ
1 A 2Aμ
HT2005: Rector Physics T09: Thermalisation 26
Material A α 0
1H 1 0 0.6672D 2 0.111 0.333
4He 4 0.360 0.1676Li 6 0.510 0.095
9Be 9 0.640 0.07410B 10 0.669 0.06112C 12 0.716 0.056
238U 238 0.938 0.003
H2O * * 0.037
D2O * * 0.033
HT2005: Rector Physics T09: Thermalisation 27
Moderator ξ N ξΣs ξΣs/Σa
H2O 0.927 19.7 1.36 62
D2O 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 energys - slowing down powers/a - moderation ratio (quality factor)
Slowing-Down Features of Some Moderators
HT2005: Rector Physics T09: Thermalisation 28
kB = 1.381×10-23 J/K = 8.617×10-5 eV/K
Velocity space:
v+dvv
4πv2dv
Probability that energy level E=mv2/2 is occupied:
2
2( ) B B
E mvk T k Tp E e e
22
20 3
2
4( )
2
B
mvk T
B
vn v n e
k T m
Neutron Velocity Distribution
HT2005: Rector Physics T09: Thermalisation 29
The most probable velocity:
mTk
vv BMP
20
Tkmv
E B2
20
0and corresponding energy:
dvev
vndvvn v
v2
0
30
2
0
4)(
0 2000 4000 6000 8000
0,0
0,2
0,4
0,6
0,8
1,0
thermal spectrum
"hard" spectrum
Neutron Velocity (m/s)
n(v)
Maxwell Distribution for Neutron Density
HT2005: Rector Physics T09: Thermalisation 30
dvev
v
dvev
vvn
dvev
vndvv
vv
vv
vv
2
0
2
0
2
0
40
3
0
40
3
00
30
3
0
4
4
4)(
Tkvm
vv
vv
dvvn
dvvvn
v
mTk
vv
B
BMP
2
3
2
2
3
128.12
)(
)(
2
2
20
2
00
0
0
0
Don’t forget :2
2
mvE dE mvdv
Maxwell Distributionfor Neutron Flux
HT2005: Rector Physics T09: Thermalisation 31
TkE
TkE
eTk
EEME
B
B
TkE
B
B
2
)()(
0
2
0 2 4 60,0
0,1
0,2
0,3
0,4
0,5
n(E)
(E)
Neutron Energy (E/kBT)
n(E)
, (E
)
HT2005: Rector Physics T09: Thermalisation 32
Tkmvmvmv
mvmvmvTkvm
E
mTk
vdvvnn
v
Bzyx
zyxB
B
2
1
2
1
2
1
2
12
1
2
1
2
1
2
3
2
32
3)(
1
222
2222
0
20
0
2
Neutron flux distribution:
( )E dEE
k Te dE
B
E
k TB
0 2
For thermal neutrons
00 3
0
3
0
)()(vv
th dvvvndvv
Average Energy of Neutrons
HT2005: Rector Physics T09: Thermalisation 33
Average cosine of scattering angle:
CM :
0
2
3 cos
ALAB-system:
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.
tr is defined as effective mean free path for non-isotropic scattering.
HT2005: Rector Physics T09: Thermalisation 34
scos scos
tr s s s s s
n cos cos cos . . . . . cos
2 3
s
tr
1 coss
tr
1 costr s
Transport Mean Free Path
1 1s tr
s tr
Information regarding the original direction is lost
HT2005: Rector Physics T09: Thermalisation 35
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
HT2005: Rector Physics T09: Thermalisation 36
E
t
dE
dt
assumed slowing-down
real slowing-down
Continues Slowing-Down
HT2005: Rector Physics T09: Thermalisation 37
• 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 (ncm-3 s-1)
• Assuming no or weak absorption (without resonances)
• Neutrons of zero energy are removed from the system
Energy
E q(E)
Slowing-Down Density
E0Q
0
HT2005: Rector Physics T09: Thermalisation 38
Lethargy Variableref
ref
Eu E u E
E ( ) ln ; 10MeV
o
o
ref
ref
E
EE
E
EE Eu E u E u E
E E E
u E p E dEE
uE
p E dE
u u
u u u
Eu
E
0
0
0 00
0
1 coll
1 collmax
10max
0
ln ln ( ) ( ) ( )
( ) ( )
ln 1 ln1
( )
, on average
, at most
ln ln
dEdu
E
HT2005: Rector Physics T09: Thermalisation 39
Lethargy Scale
coll un u
1 Number of collisions per 1 neutron to traverse
Energy
0E0Eu
1ln
u
u
Lethargy
1 collision
Lethargy
HT2005: Rector Physics T09: Thermalisation 40
Energy Lethargy
Eref 0
Energy Dependence
E u
q(u)
u+du
E+dE
1( )E
E s a
Qp EE
E E E
( )
( )( ) ( )
E/α u 1ln
coll s s
collcoll
ss
dE
u
du
dE QE dE
N EdE u du
qu
du dEN qu
Q
n
EE E
qu QE
1
( ) ( )
(
Total number of collisions in
Number of neutrons crossing
Total number of collisions in
( ) ( )
)
( ) ( )
Infinite medium, no losses,
constant Σs
HT2005: Rector Physics T09: Thermalisation 41
Neutron spectrum
E
(E)
refE dEu du
E Eu du E dE
u
s
ln ;
( ) =- ( )
Q( )=E (E)=
u
(u)
E0.025 eV
2015
1050
10 MeV
HT2005: Rector Physics T09: Thermalisation 42
a
a s
du dEE
a a
a s a s
du dEE
ua
a s
a a
a s a s
du
dq du dEq E
q qe
0
0
Probability for absorption per collision:
Number of collisions per a neutron in du or dE:
Probability for absorption in du or dE:
Absorption in du causes a relative change in q:
Resonance AbsorptionEnergy
uE
Lethargy
u+du
Eua a
a s a sE
u E dEdu
u u E E EqEp E E e e
qE
0
0
( ) ( )1 1( ) ( ) ( ) ( )
00
( )( )
( )
E/α u–lnα-1
E+dE
HT2005: Rector Physics T09: Thermalisation 43
Eua a
a s a sE
u E dEdu
u u E E EqEp E E e e
qE
0
0
( ) ( )1 1( ) ( ) ( ) ( )
00
( )( )
( )
Resonance Escape
HT2005: Rector Physics T09: Thermalisation 44
(u)
Eu
tscc
0(u)
q0(u)
q
HT2005: Rector Physics T09: Thermalisation 45
How long time does the neutron exist under slowing-down phase respectively as thermal?
Slowing-down in time - ts:Number of collisions in du:
2du dE dvE v
Number of collisions in dt:s
s
vs s
ssv
vdt dvdt
v
v dvt
v v v v
0
1
2
21 0 1
2
2 ( ) 2 1 1 2 1
v(1 eV) = 1.39 · 106 cm/s v(0.1 MeV ) = 4.4 · 108 cm/s
Thermal life-length - tt :1a
ta
tv v
Life Time
HT2005: Rector Physics T09: Thermalisation 46
Material tfast
(s)
tthermal
(s)
H2O 1 200
D2O 8 1.5105
Be 10 4300
C 25 1.2104
Neutrons Slowing-Down Time and Thermal Life-Time
HT2005: Rector Physics T09: Thermalisation 47
0 2
10
( )
2.2
2.5 10
B
Ek T
B
B
EEdE e dE
kT
kT MeV
T K
( )( )
E dEQp E
EdE
s
( )
.
E dEE
k Te dE
k T eV
T K
B
E
k T
B
B
0 2
0 025
300
(1) Fission neutrons - fast neutrons (10 MeV-0.1 MeV)
(2) Slowing-down neutrons – resonance neutrons (0.1MeV - 1 eV)
(3) Thermal neutrons (1eV - 0.)
Under the Neutron Life-Time
10 MeV0.1 MeV1 eV0
(1)(2)(3)E
HT2005: Rector Physics T09: Thermalisation 48
The END
HT2005: Rector Physics T09: Thermalisation 49
0 0E E E
2
0 2
2
0 0
1 2 cos
( 1)
1
1
A AE E
A
AE E E
A
21
1
AA
θ = 0
θ = 180
( ) ( )n E dE n v dv
2
; 22
( ), ( )
mvE v mE
n E n v
E v
E+dE v+dv
( ) ( ) ( ) 2 ( )
1( ) ( ) ( )
dE mvdv
dEn v n E mvn E mE n E
dvdv
n E n v n vdE mv