1 CH.VIII: RESONANCE REACTION RATES RESONANCE CROSS SECTIONS EFFECTIVE CROSS SECTIONS DOPPLER EFFECT COMPARISON WITH THE NATURAL PROFILE RESONANCE INTEGRAL.

1

CH.VIII: RESONANCE REACTION RATES

RESONANCE CROSS SECTIONS• EFFECTIVE CROSS SECTIONS• DOPPLER EFFECT • COMPARISON WITH THE NATURAL PROFILE

RESONANCE INTEGRAL

RESONANCE INTEGRAL – HOMOGENEOUS THERMAL REACTORS

• INFINITE DILUTION• NR AND NRIA APPROXIMATIONS

RESONANCE INTEGRAL – HETEROGENEOUS THERMAL REACTORS

• GEOMETRIC SELF-PROTECTION• NR AND NRIA APPROXIMATIONS• DOPPLER EFFECT

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VIII.1 RESONANCE CROSS SECTIONS

EFFECTIVE CROSS SECTIONS

Cross sections (see Chap.I) given as a function of the relative velocity of the n w.r.t. the target nucleus

Impact of the thermal motion of the nuclei!

Reaction rate:

where : absolute velocities of the n and nucleus, resp.

But P and : f (scalar v)

Effective cross section:

VdVPVvVvNvdvnRVv

)(|||).(|)(

VdVPVvVvvvV

eff )(|||).(|)(

dvvvNR effo)()(

dvvvnv 2

4

)()(

Vv ,

with

3

Particular cases

Let

1.

Profile in the relative v unchanged in the absolute v

2. slowly variable and velocity above the thermal domain

Conservation of the relative profiles outside the resonances

3. Energy of the n low compared to the thermal zone

Effect of the thermal motion on measurements of at low E

v

cVdVPVv

Vv

c

vv

V

eff

)(||.||

1)(

Vvvr

vvr

)()()(1

)( vVdVvPvv

vV

eff

rvc /

Vvr

v

cVdVVPV

vv

ste

V

eff )()(1

)( indep. of !

4

DOPPLER EFFECT

Rem: = convolution of and widening of the resonance peak

Doppler profile for a resonance centered in Eo >> kT ?

Maxwellian spectrum for the thermal motion:

Effective cross section:

|)(|.|| vv )(VP)(vv eff

VdekT

MVdVP kT

MV

2

2/3 2

.2

)(

rkT

vvM

rr

v

eff vdevvvkT

Mv

r

r

2

||2/3 2

)(1

.2

)(

rkT

vvM

rroeff dvevvvkT

Mv

r

2

)(2

2

2/1 2

)(1

.2

)(

rkT

vvM

kT

vvM

rroeff dveevvvkT

Mv

rr

2

)(

2

)(2

2

2/1 22

)(1

.2

)(

(Eo: energy of therelative motion !)

5

Approximation:

Let:

r

rr v

vvvv

2

22

,2

~ 2vE

2

2r

r

vE

M

EkT o4

mM

mM

r

EE

rro

oeff dEeEE

E

EE oE

rEr

2

2)~

(

)(~.1

)~

(

r

EE

rodEeE

r2

2)~

(

)(1

: reduced mass of the n-nucleus system

: Doppler width of the peak

6

COMPARISON WITH THE NATURAL PROFILE

Let:

21

1

yoa

pasin

os y

21

1

pasiot y

21

1

2

2/1

1

2

y

yg n

Jpaosi

24 Rpa

2/

or EEy

dyeyxyx

eff4

)( 22

)(2

)(

,2/

~

oEEx

dyey

xyx

4

)(

2

22

1

1

2),(

dyey

yx

yx

4

)(

2

22

1

2

2),(

Natural profile Bethe-Placzek fcts Doppler profile

and ( : peak width)

),( xoeffa

paeffsin

oeffs x

),(

paeffsioefft x ),(

),(2/1

xg nJpaoeffsi

24 Reffpa

7

Properties of the Bethe – Placzek functions

1. (low to)

Natural profiles

2. 0 (high to)

3.

Widening of the peak, but conservation of the total surface below the resonance peak (in this approximation)

dxx),(

21

1),(

xx

21

2),(

x

xx

4

22

.2

),(x

ex

and

8

VIII.2 RESONANCE INTEGRAL

Absorption rate in a resonance peak:

By definition, resonance integral:

Flux depression in the resonance but slowing-down density +/- cst on the u of 1 unique resonance

If absorption weak or = :

Before the resonance: because

duuuR a

Rés

a )()(

duu

uIas

a

Rés )(

)(

asa NIR

quuuq t )()()(

aspq

duu

uIt

pa

Rés )()(

p

a

NIqR

stept c

and

(as: asymptotic flux, i.e. without resonance)

I : equivalent cross section

(p : scattering of potential)

9

Resonance escape proba

For a set of isolated resonances:

Homogeneous mix

Ex: moderator m and absorbing heavy nuclei a

Heterogeneous mix

Ex: fuel cell

Hyp: asymptotic flux spatially constant too

At first no ( scattering are different), but as = result of a large nb of collisions ( in the fuel as well as in m)

Homogenization of the cell:

Resonance escape proba

mmpaap

p

a NI

q

Rqp

exp

p

iiIN

p

exp

V

VV popoop

111

V

VNIp o

pexp

Vo

V1

V

10

VIII.3 RESONANCE INTEGRAL – HOMOGENEOUS THERMAL REACTORS

INFINITE DILUTION

Very few absorbing atoms (u) = as(u)

(resonance integral at dilution)

NR AND NRIA APPROXIMATIONS

Mix of a moderator m (non-absorbing, scattering of potential m) and of N (/vol.) absorbing heavy nuclei a.

E

dEEI effa

Rés

)( o

oeffa

Réso Edyy

E 2)(

2

aa N

mtt N mpap N

11

Microscopic cross sections per absorbing atom

NR approximation (narrow resonance)

Narrow resonance s.t. and

i.e., in terms of moderation, qi >> ures:

By definition:

mEE '

Nm

m

mtt

tot N

mpap

p N

aEE '

)()()( uuuF t

')'()'(1

)('

duuue

uF sii

uuu

qui i

aspii

uuu

qui

due

i

'1

'

asp

)(

)(

u

u

tot

p

as

duu

uI

tot

pa

Rés

NR

)(

)(

12

NRIA approximation (narrow resonance, infinite mass absorber)

Narrow resonance s.t. but

(resonance large enough to undergo several collisions with the absorbant wide resonance, WR)

for the absorbant

Thus

Natural profile

with and

mEE ' 0' aEE

asmsat uuuuuF )()()()()(

duu

uI

ma

ma

Rés

NRIA

)(

)(

)()( uuK

12 *

**

o

o

EI

o

mpaNR

o

mNRIA

13

Remarks

NR NRIA for ( dilution: I I)

I if ([absorbant] )Resonance self-protection: depression of the flux reduces

the value of I

Doppler profile

with

Remarks

J(,) if (i.e. T ) Fast stabilizing effect linked to the fuel T

),( *** J

EI

o

o

dxx

xJ

),(

),(

2

1),(

dx

x

xo ),(

),(

E

as

T I p keff T

14

15

Choice of the approximation?

Practical resonance width: p s.t. B-W>pa

To compare with the mean moderation due to the absorbant

p < (1 - a) Eo NR

p > (1 - a) Eo NRIA

Intermediate cases ?

We can write with =0 (NRIA),1 (NR)

Goldstein – Cohen method :

Intermediate value de from

the slowing-down equation

no

pam

pama

pam

as

u

)(

mas

soo

uuu

quas

t duu

ueu

uo

')'(

)'(1

)()(

'

16

VIII.4 RESONANCE INTEGRAL – HETEROGENEOUS THERMAL REACTORS

GEOMETRIC SELF-PROTECTION

Outside resonances (see above):

Asymptotic flux spatially uniform

with

(Rem: fuel partially moderating)

In the resonances:

Strong depression of the flux in Vo

Geometric self-protection of the resonance

Justification of the use of heterogeneous reactors (see notes)

p

as

q

Vo

V1

VV

VV popoop

111

I

mmpaapoo

17

NR AND NRIA APPROXIMATIONS

Hyp: k(u) spatially cst in zone k; resonance o 1

Let Pk : proba that 1 n appearing uniformly and isotropically at lethargy u in zone k will be absorbed or moderated in the other zone

Slowing-down in the fuel ?

NR approximation

qo, q1 >>

Rem: Pk = leakage proba without collision

)()( uuV otoo ')'()'()'( duuuuuKV osoo

u

quoo

')'()'()'( 11111

duuuuuKV s

u

qu

)1( oP

)()( uuV otoo

1P

aspoo

u

quoo duuuKVPo

')'()1(

asp

u

quduuuKVP 1111 ')'(

1

18

Reminder chap.II

Relation between Po and P1p1V1P1 = to(u)VoPo

19

Thus

Wigner approximation for Po :

with l : average chord length in the fuel (see appendix)

NRIA approximation

since the absorbant does not moderate

oto

poo

as

o Pu

Pu

)(

)1()(

tooP

1

1

to

po

as

o u

1

1)(

*

*

t

p

NNmm

m

/1**

**

*mt

tt N

**

*mpa

pp N

du

u

uI

mt

pa

Rés

NR*

*

)(

)(

aspasmm

u

quoomao VPduuuKVPuuVm

111')'()1()())((

)(

)(

)( u

uP

u to

potoo

to

po

with

20

Thus

Pk : leakage proba, with or without collision

If Pc = capture proba for 1 n emitted …, then

Wigner approx:

INR, INRIA formally similar to the homogeneous caseEquivalence theorems

ma

ao

ma

m

as u

uP

u

u

)(

)(

)(

)(

oct

sco PPPP

1

t

tcP

1 )(1

1

maoP

*

*

)(

)(

ma

m

as u

u

du

u

uI

mt

ma

Rés

NRIA*

*

)(

)(

and

21

DOPPLER EFFECT IN HETEROGENEOUS MEDIA

NR case: without Wigner, with Doppler

Doppler, while neglecting the interference term:

NRIA case: same formal result with and

duu

uuPdu

u

uI

t

ptao

Rést

pa

Rés

NR

)(

)()(

)(

)(

duu

uuuPdu

u

u

mt

ptato

Résmt

pa

Rés

)(

)()())((

)(

)(

dxx

xlxP

EJ

EI to

o

o

o

oNR

),(

)),(())((

2),(

2

)),(

1()),(( x

NxN ppot o

mpa

)),(

1( x

t

),,(),(

tL

EJ

E o

o

o

o

dx

x

xxtPtL o

),(

)),(())1),(((

2

1),,(

2

o

m mNt

with

22

Appendix: average chord length

Let : chord length in volume V from on S in the direction

with : internal normal ( )

Proportion of chords of length : linked to the corresponding normal cross section:

Average chord length:

),( sr

dSrnV si

S

),(.

dSni

rs

.),(

dSdn

dSdn

di

S

i

rs

.

.

)( ),(

S

V

dSdn

dSdnr

di

S

is

S

o

4

.

.),(

)(

0.0),( is nifrin

sr

oao P