# SLOWING DOWN by jennyyingdi

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• pg 1
```									         Neutron Slowing-Down

Vasily Arzhanov
Reactor Physics, KTH

Overview
•   Lethargy
•   Continues Slowing-Down Model
•   Discontinues Slowing-Down Model
•   Slowing-Down with Absorptions
•   Resonance Escape Probability
•   Neutron Spectrum in Reactor

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1
Principles of a Nuclear Reactor
E                                                                                              Leakage
N2
?
N1

N                              Fast fission

Slowing down
k≡ 2
ν n/fission
Energy

N1                             Resonance abs.
ν=?
Non-fissile abs.         Non-fuel abs.

?
Fission

?
Leakage
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Lethargy Variable
E0
E0
E0           α
∫ ln         E
p( E)dE
α
ξ ≡ ln       =                                    = 1+         ln α
Eo
Does not depend on energy!!!
E                    E0
1−α
∫
α Eo
p( E)dE
Eref              dE
u( E) ≡ u ≡ ln          ; du ≡ −
E                 E
E0      ⎛ E Eref ⎞                                                                       Eref ∼ 10MeV
= ln ⎜ 0        = u( E) − u( E0 ) ≡ Δu( E)
⎜ Eref E ⎟
ln
E                ⎟
⎝        ⎠

E0
u ⎯⎯⎯ u + ξ ,
1 coll
→                   on average
∫ Δu(E) p(E)dE                                α
ξ=
α  Eo
= Δu = 1 +     ln α                   →
u ⎯⎯⎯ u + Δumax , at most
1 coll
E0
1−α
∫
α Eo
p( E)dE                                                       E
Δumax = ln 0 = ln α −1
α E0
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2
Lethargy Scale
1 average collision

α E0                                E0
Energy
ΔE                           1−α
ΔE =          E0
2
u                                                                           ξ
Lethargy
Δumax = ln α −1
u                                                                           ξ
Lethargy
Δu                       u0

It is tempting to assume: the number of                                            Δu                 Strictly speaking
n1 =
coll
this is not true!!!
collisions per 1 neutron to traverse Δu is                                         ξ

HT2008                                      Slowing down                                                                         5

Warning Message
Eref                                    dE
u ≡ ln                  ; du ≡ −
E                                E

Average number       du       dE   dE   dE                                                  Average number
of collisions to        =      =    =                                                     of collisions to
traverse du     ξ       ξ E ΔE 1 − α E                                                traverse dE
2
1

0.9

0.8

ξ
0.7

1−α                            0.6

ξ=     Erroneous result!!!        0.5

2                             0.4

0.3

0.2
(1 − α ) 2
A −1⎞
2

α =⎛
0.1

0
⎜     ⎟
⎝ A +1⎠
0       0.1       0.2   0.3   0.4    0.5   0.6   0.7    0.8   0.9   1

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3
Average Number of Collisions
A very important             The average number of collisions made by a neutron in
quantity:                    traversing a lethargy interval u1 to u2 where 0 ≤ u1 ≤ u2 < ∞.
Δu
n1 =
coll

ξ       This formula is true for H!!

S.M. Dancoff has shown: the average gain in lethargy,               Δu   , by a neutron after
n collisions with nuclei of mass A is given by

Δu = n ⋅ ξ
For the general case, A > 1, the corresponding calculations are very complicated,
no simple approximating formulas have been found yet. However, it was shown
that this formiula is reasonably accurate, particularly when
Δu   u −u
= 2 −11                   1
Δumax ln α
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Frequency Function in Lethargy
the frequency function in
1
angle: p(θ ) = sin θ
2
1
cosine: p(η ) =
2
( A + 1)2         1
energy: p( E) =           =
4 AE0     ( 1 − α ) E0

p(u)du = − p( E)dE
Eref              Eref              e −(u−u0 )
u0 = ln          ; u = ln          ⇒ p(u) =
E0                 E                 1−α

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4
Slowing-Down Features
of Some Moderators
N=
(
ln E f Eth     ) = ln ( 2 ⋅ 10   6
0.025 )
=
18.2
ξ                     ξ                      ξ
Moderator           ξ          N               ξΣs          ξΣs/Σa
H 2O          0.927          19.6           1.36                 62
D 2O          0.510          35.7           0.18           5860
9Be           0.207          88.0           0.15               138
12C          0.158         115.3           0.06               166
238U          0.008       2171.6            .0040          0.011

N - number of collision to thermal energy
ξΣs - slowing down power
ξΣs/Σa - moderation ratio (quality factor)
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Infinite Medium Model
• Purpose: to describe neutron motion in
energy space, not in physical space.
• Medium is
– homogeneous
– isotropic
– infinite in extent
• Neutron sources
– are uniformly distributed in space
– emit neutrons isotropically

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5
Basic Quantities
Eref
mv 2 2 = E;        u ≡ log
E

φ ( E) ≡ v ⋅ n( E) ⎡# ( cm 2 ⋅ s ⋅ J ) ⎤
⎣                   ⎦

unit volume per unit time
φ ( E)dE = Total track length per energies between E and Eof dE.
neutrons with kinetic                            +

number
Σ s ( E) = Expectedof energyof elastic-scattering collisions between
neutrons          E and nuclei per unit distance of travel.

Σ a ( E) = Σγ ( E) + Σ f ( E);       φ ( E)dE = φ ( v)dv = −φ (u)du

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Continues Slowing Down
E

The actual (discontinues)
Slowing
slowing down neutron-energy
down track
track is very difficult to handle.

Distance
E

Much attention has been devoted
Slowing down track                                    to find a continues function which
in a medium of                                    will serve as a good average over
heavy nuclear mass                                    all the possible step functions.

Distance
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6
Continues Slowing Down
Model
• The energy loss is continues in time process.
• Neutron energy changes are due to collisions
occurring in arbitrary small (differential)
decrements.
• To describe this process the concept of slowing
down density, q(E), is defined.
• The model is fairly accurate for heavy nuclei.
• On the contrary, the model is not that useful for
light nuclei; a more accurate description is
needed.

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Slowing Down Density
Energy          It does
not
q0       E0          count

q(E) = Number of neutrons per unit volume per
unit time whose energies are changed from          E        q(E)
some value above E to some value below E.

• Neutrons are introduced at energy E0                  0       It counts
uniformly in space and continuously in time.
• Neutrons are removed at zero energy.
• Medium is a pure scatterer, Σa = 0.

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7
Energy
No leakage
No absorption
q( E) = 0;    E > E0
No accumulation

q( E0 ) = q0             Source:  q0             E0

dqin = dqout
E + dE

E
dqout

Removal                0

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Slowing Down Density
Solution
Energy
q( E + dE) = dqin + q
q( E) = q + dqout
dqin
q( E + dE) = q( E)
E + dE                                      dq( E)
=0
q                      dE
E

dqout                           q( E) = q0

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8
Flux Solution in Lethargy
Total number of scatterings per
Σ s (u)φ (u)du =        unit volume, per unit time due to
neutrons with lethargy u about du

Number of neutrons per                     Average number of
unit volume, per unit time                   scatterings made by a
that slow past lethargy u                   neutron in traversing du

du       Total number of scatterings per
q(u)        =   unit volume, per unit time due to
ξ        neutrons with lethargy u about du

du                             q0
Σ s (u)φ (u)du = q(u)                        φ (u) =              const
ξ                          Σ s (u)ξ
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Flux Solution in Energy
dE
du = −
E
φ ( E)dE = −φ (u)du

The continues slowing                          q0      q     1
down model predicts:            φ ( E) =             = 0 ∝
ξΣ s ( E)E ξΣ s E E
φ (u)                               φ ( E)

u                                 E

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9
Discontinues Slowing Down
Model
•   It accounts for discontinuity of neutron trajectory.
•   Infinite medium model.
•   Target nuclei are at rest (E > 1 eV).
•   Scattering is isotropic in CoM (E < 1 MeV).
•   System is at steady state.
•   Neutron source is uniform and monoenergetic:
q0 neutrons of E0 appear per unit volume and
time from uniformly distributed sources
throughout the medium.
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Collision Intervals
Source
E0

E              1st collision interval
Neutron energy

αE0
2nd collision interval
α2E   0
3d       collision interval
α3E0

A −1⎞
2

α =⎛
0
⎜       ⎟
Removal                                                      ⎝ A +1⎠

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10
Solution in 1st Interval
Σ ( E′ ) φ ( E′ )
E0
q
Equation         Σs ( E) φ ( E) =         ∫ (1 − α ) E′ dE′ + (1 − α ) E
s                             0

E                                           0

Collision density          F1 ( E) ≡ Σ s ( E ) φ ( E )

F1 ( E′)
E0
q0
Equation           F1 ( E) =    ∫ (1 − α ) E′ dE′ + (1 − α ) E
E                                        0

1 ( 1−α )
q0         ⎛ E0 ⎞
φ1 ( E) =                                                         α E0 ≤ E ≤ E0
(1 − α ) E0Σs ( E) ⎜ E ⎟
Solution                                                                     ;
⎝ ⎠

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Solution in Other Intervals
α n−1 E0                 Eα
Fn ( E′)dE′        F ( E′)dE′
Recurrence relation            Fn ( E) =        ∫E
+ ∫ n −1
(1 − α ) E′ α n−1E (1 − α ) E′
0

α ( 1−α )
q0 ( E0 )               ⎧ α 1 (1−α ) ⎛ E ⎞
⎪                             1 ( 1−α ) ⎪
⎫
F2 ( E) ≡ Σ s ( E)φ ( E) =                            ⎨           ln ⎜   ⎟ + ⎡1 − α
⎣
⎤⎬
⎦
( 1 − α ) E1 (1−α )      ⎪ ( 1 − α ) ⎝ α E0 ⎠
⎩                                        ⎪
⎭

Eα
F∞ ( E′)dE′                               q
Fn ( E) ⎯⎯⎯ F∞ ( E) =
n→∞
→                  ∫E   ( 1 − α ) E′
⇒ F∞ ( E) ≡ Σ s ( E)φ∞ ( E) = 0
ξE
q0
φ∞ ( E) =
ξ EΣ s ( E)

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11
Collision Density in Energy
80

70
q0
Asymptotic form F∞ ( E) =
60
ξE
50

40

30                               F3 ( E)
F2 ( E)
20                                                               F1 ( E)

10

0
0     0.1       0.2         0.3     0.4     0.5       0.6    0.7      0.8   0.9       1
E E0
4th           3rd         2nd interval                  1st interval

α4 α3                  α2                   α                                   1
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Solution in Lethargy
Collision density in lethargy may be obtained by
(1) Repeating the derivation in lethargy scale

(2) Using the relationship:                       F( E)dE = − F(u)du;                    du = − dE E

q0 = 1 ⇒ Normalized density or Placzek functions

1 α (1−α )u
F1 (u) =          e
1−α
u
α (1−α )( u − n ln1 α )         1                                      α (1−α )( u −u′)
Fn+1 (u) = Cn+1e                                 −
1 −α        ∫ α F (u′ − ln 1 α )e
n ln1
n                                         du′

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12
Collision Density in Lethargy
2.6

2.4                                                            A= 4                             1
F∞ (u) =       = 2.35
ξ
2.2

2.0
F3 (u)
1.8                                    F2 (u)

1.6              F1 (u)
A= 2                             1
1.4                                                                                  F∞ (u) =       = 1.38
ξ
1.2

A= 1                             1
1.0                                                                                  F∞ (u) =       =1
0
ln
1
2 ln
1
3 ln
1                  ξ
α                           α                 α
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Slowing Down in Mixtures
Mixture of nuclides:                 Ai , Ni ,σ s ,i
2
⎛ A −1⎞                                                   αi
Σ s ,i = Niσ s ,i                 αi = ⎜ i      ⎟                                   ξi = 1 +          ln α i
⎝ Ai + 1 ⎠                                              1 − αi

qi (u) = ξi Σ s ,i (u)φ (u)
q(u) = ∑ qi (u) = ∑ ξ i Σ s ,i (u)φ (u) = ξ ⋅ Σ s (u)φ (u)
i                        i

∑ξ Σ       i   s ,i   (u)     ∑ ξ Σ (u)       i     s ,i
ξ =    i
=       i

Σ s (u)                 ∑ Σ (u)   i
s ,i

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13
Slowing-Down with Absorption
Σa
Absorption                   Pr {Absorption} =
Σt
Each collision
Σs
Scattering                   Pr {Scattering} =
Σt

Neutrons may be removed from the slowing-down stream.
Further complication: neutrons may appear due to fission.
Consequence: q(E) ≠ const
First step: Slowing-down in absorbing medium of pure hydrogen.

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Slowing-Down with Absorption
in Hydrogen
Scattered out                        Scattered in

Σ s ( E′) φ ( E′)
E0
q0
Previous result:    Σs ( E)φ ( E) =      ∫                      dE′ +
E      ( 1 − α ) E′          ( 1 − α ) E0

E0
Σ s ( E′ ) φ ( E′ )           q0
Now:    Σt ( E ) φ ( E ) =   ∫                          dE′ +
E
E′                    E0
F ( E)

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14
Solution for Hydrogen
d           Σ ( E)                     q      ⎛ E0 Σ ( E′ ) dE′ ⎞
F( E) = − s         F( E) ⇒ FH ( E) = 0 exp ⎜ − ∫ a           ⎟
dE          Σt ( E ) E                   E     ⎝ E Σt ( E′ ) E′ ⎠
⎜                 ⎟

q0          ⎛ E0 Σ a ( E′ ) dE′ ⎞
φH ( E) =           exp ⎜ − ∫               ⎟
EΣt ( E )     ⎜ E Σt ( E′ ) E′ ⎟
⎝                   ⎠

⎛ E0 Σ a ( E′) dE′ ⎞
q( E) = EF( E)                qH ( E) = q0 exp ⎜ − ∫              ⎟
⎝ E Σt ( E′) E′ ⎠

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Non-Absorption Probability
qH ( E) qH ( E)
q0        E0      qH (E0 ) = q0      pH ( E; E0 ) =             =
qH ( E0 )   q0

⎛ E0 Σ a ( E′ ) dE′ ⎞
pH ( E; E0 ) = exp ⎜ − ∫               ⎟
⎜ E Σt ( E′ ) E′ ⎟
⎝                   ⎠
E         qH ( E)
⎛ u Σ a ( u′ ) du′ ⎞
pH ( u; 0 ) = exp ⎜ − ∫              ⎟
⎜ 0 Σ ( u′ ) ⎟
⎝        t         ⎠
0

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15
Slowing-Down in General Case
Neutrons                Neutrons

⎧ E0 Σ s ( E′ ) φ ( E′ )           q0
⎪∫                       dE′ +             ;       α E0 < E < E0
⎪ E ( 1 − α ) E′               (1 − α ) E0
Σs ( E) φ ( E ) = ⎨E α
Previous
⎪ Σ s ( E′ ) φ ( E′ )
result
⎪ ∫ ( 1 − α ) E′ dE′;                                      E < α E0
⎩E

⎧ E0 Σ s ( E′ ) φ ( E′ )           q0
⎪∫                       dE′ +             ;      α E0 < E < E0
⎪ E ( 1 − α ) E′               (1 − α ) E0
Present
Σt ( E ) φ ( E ) = ⎨ E α
⎪ Σ s ( E′ ) φ ( E′ )
situation
⎪ ∫ ( 1 − α ) E′ dE′;                                     E < α E0
⎩E
There is no analytic solution!
HT2008                                    Slowing down                                       31

⎡ 1u         Σ a (u′)         ⎤
q(u) = q(0) exp ⎢ − ∫                      du′⎥
Approximations:                                   ⎣ ξ 0 Σ s (u′) + Σ a (u′) ⎦
1) Asymptotic range
2) Small absorption                                   ⎡ 1 E0       Σ a ( E′)     dE′ ⎤
q( E) = q(0) exp ⎢ − ∫                          ⎥
⎣ ξ E Σ s ( E′) + Σ a ( E′) E′ ⎦

q0               ⎡ 1u        Σ a (u′)        ⎤
φ (u) =                           exp ⎢ − ∫                    du′⎥
ξ [ Σ s (u) + Σ a (u)]     ⎣ ξ 0 Σ s (u′) + Σ a (u′) ⎦

q0                 ⎡ 1 E0       Σ a ( E′)     dE′ ⎤
φ ( E) =                              exp ⎢ − ∫                          ⎥
ξ [ Σ s ( E) + Σ a ( E)] E     ⎣ ξ E Σ s ( E′) + Σ a ( E′) E′ ⎦

HT2008                                    Slowing down                                       32

16
Non-Absorption Probability
⎡ 1u         Σ a (u′)         ⎤
p(u) ≡ exp ⎢ − ∫                      du′⎥
⎣ ξ 0 Σ s (u′) + Σ a (u′) ⎦
⎡ 1 E0       Σ a ( E′)     dE′ ⎤
p( E) ≡ exp ⎢ − ∫                          ⎥
⎣ ξ E Σ s ( E′) + Σ a ( E′) E′ ⎦
Also called resonance
More generally                                                                           escape probability
⎡ 1 u2 Σ (u′) ⎤
p(u1 → u2 ) ≡ exp ⎢ − ∫ a        du′⎥                               Σa
⎢ ξ u1 Σt (u′)
⎣                 ⎥
⎦
⎡ 1 E2 Σ ( E′) dE′ ⎤
p( E1 ← E2 ) ≡ exp ⎢ − ∫ a            ⎥
⎢ ξ E1 Σt ( E′) E′ ⎥
⎣                  ⎦                                                     E

HT2008                                                Slowing down                                          33

Resonance Escape
⎡ 1 E0      Σ a ( E)     dE ⎤
p( E ← E0 ) = exp ⎢ − ∫                       ⎥
⎣ ξ E Σ a ( E) + Σ s ( E) E ⎦
10 4
235
U
10 3

10 2
σ (barns)

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)
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17
Simplified Derivation
No absorption: q(u) = const                                    Energy         Lethargy
E/α           u–lnα-1

E           u
Relative change          Probability of                                         u+du
E ‒ dE
in q due         absorption
absorptions          upon a
collision

dq du   Σa                                   ⎡ 1u        Σ a (u′)        ⎤
− =    ⋅                        q(u) = q0 exp ⎢ − ∫                    du′⎥
q ξ Σa + Σs                                 ⎣ ξ 0 Σ s (u′) + Σ a (u′) ⎦

Number of
collisions per
one neutron in
traversing du
HT2008                              Slowing down                                      35

Energy Dependent Flux
φ ( E)                           q0                           Infinite medium,
φ ( E) =                                  no absorption,
ξΣ s E                        constant Σs

q0 p( E)             Infinite medium
φ ( E) =
ξ [ Σ s ( E) + Σ a ( E)] E     with absorption

Eres                      E0

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18
Lethargy Dependent Flux

Σ a (u)

φ (u)

q0 p(u)
φ (u) =
ξ [ Σ s (u) + Σ a (u)]

0
u

HT2008                          Slowing down                                                        37

Neutron Velocity Distribution
kB = 1.381×10-23 J/K = 8.617×10-5 eV/K

v+dv                     Probability of being at
v                                      energy level E=mv2/2 :
Velocity                                                                                 E
−
space:                       4πv2dv                   p( E)dE = e                    k BT
dE

mv 2
Neutrons are at                         4π v 2              −
thermal equilibrium   n( v ) = n0                   3
e       2 k BT

with medium
( 2π kBT   m)   2

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19
Maxwell Distribution for
Neutron Density
2
⎛ v ⎞
4 v 2 − ⎜ v0 ⎟
Thermal spectrum                       n( v) = n0        e⎝ ⎠                         Hard spectrum
π v03

1

0.9
The most                                           0.8

probable
2 k BT                       0.7

velocity: v MP = v0 =                              0.6
m       0.5

0.4

and                        2                        0.3
mv
energy:         E0 =      = k BT
0                        0.2
2                           0.1
v0
0
0   1000   2000       3000    4000   5000       6000   7000   8000

HT2008                                           Slowing down                                                           39

Maxwell Distribution
for Neutron Flux
⎛ v ⎞
2
2kBT        m
4 v 3 −⎜ v0 ⎟
φ ( v) = n0 3     e⎝ ⎠ =
vMP = v0 =                   = 2200   ( 20 °C )
m          s
v0 π                                                            ∞

∫ vn(v)dv
2
⎛ v ⎞
4 v3     −⎜ ⎟
2
= n0 v0 4    e    ⎝ v0 ⎠
=                            v=     0
=          v0 = 1.128v0
v0 π                                                         ∞
π
⎛ v ⎞
2                                            ∫ n(v)dv
4 v 3 −⎜ v0 ⎟                                                 0
= φ0 4     e⎝ ⎠                                                                           3 2
v0 π                                                                        v2 =        v0
2
mv 2 3
= kBT
mv 2                                                       2   2
Don’t forget :            E=                             dE = mvdv
2
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20
Under the Neutron Life-Time
E

10 MeV
E
E      −
φ ( E) = φ0           e k BT ;                      kBT ≈ 2.2 MeV;    T ≈ 2.5 ⋅ 1010 K
( kBT )
2

0.1 MeV
q0 p( E)
φ ( E) =
ξ [ Σ s ( E) + Σ a ( E)] E
1 eV
E
−
E
φ ( E) = φ0                 e       k BT
;    kBT ≈ 0.025 eV;     T ≈ 300K
(k T )
2
B
0

HT2008                                            Slowing down                                                41

Life Time
How long time does the neutron exist under
Slowing-down time = ts                              slowing-down phase respectively as thermal?

Number of                                   Number of
collisions in du                                 collisions in dt

du       dE 2dv vdt       2λ dv
=     =   =    ⇒ dt = s 2
ξ        ξE ξv   λs        ξ v

2λs ( v) dv 2λ ⎛ 1 1 ⎞
v0
2 1                                            v(1 eV) = 1.39 · 106 cm/s
ts =   ∫ ξ v2 ≈ ξ s ⎜ v1 − v0 ⎟ ≈ ξΣs v1
v1                ⎝     ⎠
v(0.1 MeV ) = 4.4 · 108 cm/s

λa        1
tth =            =               Thermal life-length
v        Σav
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21
Neutrons Slowing-Down Time
and Thermal Life-Time
Material   ts [μs]          tth [μs]
H2O         1              2×102
D2O         8              1.5×105
Be         10             4.3×103
C        25              1.2×104

HT2008                Slowing down              43

The END

HT2008                Slowing down              44

22

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