HT2005: Rector PhysicsT09: Thermalisation1 SLOWING DOWN OF NEUTRONS Elastic scattering of neutrons....

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.


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


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


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


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


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


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


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


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


Differential Solid Angle

x

y

z

ez

ey

ex

r

Ω

d3r

φ

θ sind d d Ω

d

sin d


Hard Sphere Model

r

Total scatteringcross section σ = 2πr2

n

r

θ


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


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


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


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


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


( ) ( )

( )

( )

(

(

)

)

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


dA

A

E

dE

A

AA

E

E

sin1

2

1

cos21

20

2

2

0

E

E-dEEE0 E0

p(E;E0)

??

dEEpdp )()(


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


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


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


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


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


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


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


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


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


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


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


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


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

)


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


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.


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


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


E

t

dE

dt

assumed slowing-down

real slowing-down

Continues Slowing-Down


• 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


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


Lethargy Scale

coll un u

1 Number of collisions per 1 neutron to traverse

Energy

0E0Eu

1ln

u

u

Lethargy

1 collision

Lethargy


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


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


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


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


(u)

Eu

tscc

0(u)

q0(u)

q


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


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


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


The END


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

HT2005: Rector PhysicsT09: Thermalisation1 SLOWING DOWN OF NEUTRONS Elastic scattering of neutrons....

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Transcript of HT2005: Rector PhysicsT09: Thermalisation1 SLOWING DOWN OF NEUTRONS Elastic scattering of neutrons....