Resonance self-shielding in the epi-thermal...

Institute for Reference Materials and Measurements (IRMM)Geel, Belgium

http://www.irmm.jrc.behttp://www.jrc.cec.eu.int

P. Schillebeeckx, K. Volev and M. Moxon

[email protected]

EC – JRC – IRMM, B - 2440 Geel, (B)

Resonance self-shielding in the epi-thermal region

Workshop on Nuclear Data for Activation Analysis

Trieste, Italy, 7 – 18 March 2005

Resonance self-shielding in the epithermal region , P. Schillebeeckx 15_03_05

Neutron resonances and NAA

• Basic principles of NAA

• Nuclear reaction theory

• Neutron resonances and NAA

• Resonance self-shielding for a parallel beam

• Resonance self-shielding for an isotropic beam


1. Basic principles of NAA


Neutron capture process

(I)NAA

PGAA

T1/2

Sn

Eγ

Resonances

Neu

tron

Ener

gy

Cross section

Eγ

nn* E

1AASE+

+=

XA *1A X+ γ++ X1An +

A+1X

NRCA


Capture cross section

10-2 100 102 104 10610-4

10-2

100

102

104

59Co(n,γ)

σ(n,

γ) /

barn

Neutron Energy / eV

• The probability that a neutron interacts with a nucleus is expressed as a cross section σ, which has the dimension of an area

• The unit of a cross section is taken as : 1 barn, 1 b = 10-24 cm2

• To calculate reaction probabilities we express the target thickness in atoms per barn :

mA : atomic massρ : density in g/cm3

t : thickness in cmn : target thickness in at/b

tm6022.0n

A

ρ=


Total reaction rate

The total reaction rate per atom:

depends on:

ϕ(En) the neutron flux and

σγ(En) the capture cross section

n0

nn dE)E()E(R ∫∞

γσϕ=

10-2 100 102 104 10610-8

10-4

100

104

Neutron flux per eV

Neutron Flux 59Co : σ(n,γ)

σ(n,

γ) /

barn

Neutron Energy / eV


Neutron flux in a thermal reactor

The neutron flux in a thermal reactor is asum of three components

• Maxwellian distribution with maximum at En = kT

• 1/E1+α distribution due to moderation process of the fast neutrons (epi-thermal spectrum)

• “Watt spectrum” of fission neutrons

At a neutron guide, the neutron flux can bedescribed by a Maxwellian distribution

(cfr. PGAA at Budapest, i.e no resonance shielding !)

10-3 10-1 101 103 105 10710-9

10-7

10-5

10-3

10-1 Total spectrum Maxw. (kT = 0.025 eV) 1 / E1+α

Fission

Neu

tron

flux

per e

V

Neutron Energy / eV


Reaction rate in a thermal reactor• The total reaction rate per atom is:

• To solve the integral one separates between the thermal and the epi-thermal region:

with ECd = 0.55 eV

• Conventions : Høgdahl & Westcott convention

fastdE)E()E(dE)E()E(R n

E

Enn

E

0nnn

3

Cd

Cd

+σϕ+σϕ= ∫∫

n0

nn dE)E()E(R ∫∞

σϕ=


Importance of resonance region

• Reaction rate

– σo : capture cross section at thermal – gw : Westcott g-factor (expresses the deviation of σγ from 1/v)– ϕt : thermal neutron flux– Gt : thermal self-shielding

– IR : resonance integral– ϕe : epi-thermal neutron flux– GR : resonance self-shielding

• Cd-ratio measurements

ReRwott IGgGR ϕ+σϕ≅

ReR

CdCd IG

RFϕ

=

⇒ σo, gw, IR, GR, FCd are influenced by the resonance structure


Transmission through cadmium

10-3 10-1 101 1030.0

0.5

1.0 e-nCd σt

Ideal

Tran

smis

sion

fact

or

Neutron Energy / eV

nCd = 4.63 10-3 at /b (1 mm)

ECd = 0.55 eV

10-3 10-1 101 103100

102

104

natCd + n

σ(n,

t) / b

arn

Neutron Energy / eV


2. Nuclear reaction theory


Energy differential cross sections

238U(n,tot) = 238U(n,n) + 238U(n,γ)

σtot = σn + σγ

• Thermal

• Resonance Region : D > ΓResolved Resonance Region : ∆R < DUnresolved Resonance Region : ∆R > D

• High Energy Region : D < Γ

10-2 100 102 104 10610-2

102

106

88 90 92 940

5

10

15

Eo

Γ

D

Cro

ss-s

ectio

n / b

arn

Energy / eV

σt

σγ

σn


Energy Differential Cross Sections

10-2 100 102 10410-3

10-1

101

103

105

σt

σn

σγ

Cro

ss s

ectio

n / b

arn

Neutron Energy / eV

197Au + n

10-2 100 102 104 10610-4

10-2

100

102

104 σt

σn

σγ

Cro

ss s

ectio

n / b

arn

Neutron Energy / eV

56Fe + n

- No capture without scattering- Relative contribution of σn and σγ to σt may be different- Boundaries of the RRR differ


Importance of resonance structure

3.7 3.8 3.9 4.0 4.1 4.20.0000

0.0002

Tran

smis

sion

Neutron Energy / MeV

Transmission (dFe = 40 cm)

0

2

4

6

8

σ tot

/ ba

rn

IRMM 1993 ENDF/ B-VI

natFe + n


Resonance structure

A cross section as a function of En shows a resonant structure, which can be described by a Breit-Wignershape :

with

Γ natural line width (FWHM)

ER resonance energy

( ) 22Rn

t )2(EE1~

Γ+−σ

133 134 1350

4000

8000σt

Γ

σR

ER

σ t / b

arn

Neutron Energy / eV


Bohr’s hypothesis : Compound nucleus reaction

• Two step process(1) Formation of compound nucleus σC*

(2) Decay of compound nucleus Pr

• Partial cross section

( )( ) 22

Rn

n2n

n*C )2(EEkgE

Γ+−ΓΓπ

=σ

,...)f,,nr(r

r γ=Γ=Γ ∑

,...)f,,nr(P rr γ=

ΓΓ

=

r*Cr Pσ=σ

AX

-Sn

En

A+1X

σc*

γ

E*

AX +n

nn* E

1AASE+

+=


Breit – Wigner formula Resonance part of the cross section

• Total Cross Section (n,tot)

• Elastic Cross Section (n,n)

• Capture Cross Section (n,γ)

( )( ) 22

Rn

n2n

nt )2(EEkgE

Γ+−ΓΓπ

=σ

( ) ( )ΓΓ

σ=σ nntnn EE

wavenumberk;)1I2(2

1J2g n =++

=

( ) ( )ΓΓ

σ=σ γγ ntn EE


e.g. 109Ag s-wave at Eo = 134 eV(Eo, Γn, Γγ , Jπ)

133 134 1350

4000

8000σt

Γ

σR

ERσ t

/ ba

rn

Neutron Energy / eV

133 134 1350

4000

8000

ER

σR Γ

Γγ

Γ

σγ

σ γ / b

arn

Neutron Energy / eV133 134 1350

4000

8000

ER

Γn

ΓσR

Γ

σn

σ n /

barn

Neutron Energy / eV

Eo = 134 eVΓn = 0.093 eVΓγ = 0.106 eVJπ = 1-

g = 3/4

ΓΓ

⎟⎟⎠

⎞⎜⎜⎝

⎛ +=σ n

2

A

A

R

6

Rg

m1m

)eV(E10x608.2)barn(

γΓ+Γ=Γ n


Energy dependence of neutron width

• The energy dependence of the neutron width is due to the centrifugal-barrier penetrability, which depends on the angular momentum of the incoming neutron l and En

• The neutron width Γn depends on En

s-wave (l = 0)

p-wave (l = 1)

( )( ) 22

Rn

n2n

n )2(EEkgE

Γ+−

ΓΓπ=σ γ

γ

eV1E)E( n0

nnn Γ=Γ

22n

22nn1

nnn ak1ak

eV1E)E(

+Γ=Γ

10-3 10-2 10-1 100 101100

102

104

106

s - wave p - wave

σ(n,

γ) /

barn

Neutron Energy / eV

ER = 1 eVΓ

γ = 100 meV

Γn = 10 meV


3. Neutron Resonances and NAA


Influence of resonance structure on:

• Thermal capture cross section

• 1/v behaviour of the capture cross section

• Westcott gw - factor


σo and contribution of s-wave resonances #1

with σo in barn

ER, Γno, Γγ in eV

∑=

γ=

ΓΓ⎟⎟⎠

⎞⎜⎜⎝

⎛ +≅σ

N

1j2

Rj

jonj

2

A

A60,o E

gm

1m10x099.4l

10-2 100 10210-2

100

102

104

ER = 5.2 eV

σ0

= 91 b

109Ag(n,γ)

σ γ / b

arn

Neutron Energy / eV


σo and contribution of s-wave resonances #2

• If

• Additional contribution frombound states (negative resonances)

∑=

γ=

ΓΓ⎟⎟⎠

⎞⎜⎜⎝

⎛ +≅σ>σ

N

1j2

Rj

jonj

2

A

A60,oo E

gm

1m10.099.4l

10-3 10-1 10110-2

100

102

104

106 σ

γ , total

σγ for Er = 1 eV

σγ for Er = - 1 eV

σ(n,

γ) /

barn

Neutron Energy / eV

σo

En

AX

-Sn A+1X

σc*

γ

E*AX +n


1/v behaviour of capture cross section

• Capture cross section

• Neutron width for a s-wave neutron

• For En << ER witk kn2 ∝ En

( )nn

n v1

E1E =∝σγ

eV1E)E( n0

nnn Γ=Γ

( )( ) 22

Rn

n2n

n )2(EEkgE

Γ+−

ΓΓπ=σ γ

γ

10-3 10-2 10-1 100 101100

102

104

106

s-wave 1/v

σ(n,

γ) /

barn

Neutron Energy / eV

ER = 1 eVΓ

γ = 100 meV

Γn = 10 meV


Westcott gw - factor: deviation of 1/v behaviour #1

10-3 10-1 101 10310-2

100

102

104

σ(n,

γ) E

1/2 / b

arn

eV1/

2

Neutron flux at: 300 K

59Co(n,γ)

Neutron Energy / eV

gw = 1.0004

10-3 10-1 101 10310-2

100

102

104gw = 1.7579

Neutron flux at: 300 K

176Lu(n,γ)

σ(n,

γ) E

1/2 / b

arn

eV1/

2

Neutron Energy / eV

ER = 0.14 eVER = 132 eV


Westcott gw - factor: deviation of 1/v behaviour #2

The Westcott gw – factor is temperature dependent

10-4 10-2 100 10210-2

100

102

104

Neutron flux at: 10 K 300 K 1000 K

176Lu(n,γ)

σ(n,

γ) E

1/2 / b

arn

eV1/

2

Neutron Energy / eV

200 300 400 500 600 7001

2

3

4

5 176Lu(n,γ)

Wes

tcot

t Fac

tor,

g w (T

)Temperature / K

⇒ 176Lu(n,γ) temperature monitor


Data base of resonance parameters

• Evaluated data libraries (see IAEA INDC website)

– JEF– ENDF-B– JENDL– CENDL

• Compilation by S.F.Mughabghab

“Neutron Resonance Parameters and Thermal Cross Sections”

Part A & B

NNDC, BNL, 1984


Data base of σo, gw-factor and IR

Compilation by S.F.Mughabghab, BNL, USA

“Thermal neutron capture cross sections, resonance integrals and gw-factors”

INDC(NDS) – 440

February 2003


4. Resonance self-shielding for a parallel beam


Self-shielding and multiple scattering

Parallel neutron beam on a foil or a disc

• Study the basic effects

(parallel beam is not directly applicable to NAA, but experimental verification of procedures is possible)

• Influence of resonance structure on the self-shielding factor

• Doppler effect

• Total correction factors (due to self-shielding & scattering)

• Interference effects


Parallel beam on thin sample

• For a relatively thin sample, no scattering only self-shielding

• Infinitely thin sample

dxedx)x( x0x

tρσ−=ϕ=ϕ

dx)x()E()E(Rt

0nn ∫ ϕσρ∝ γ

( ) 1nσfornσn

e1nR tt

σn

thin

tr

<<σ=−

σ∝ γ

−

γ

γ

x

ϕ (x)

)E(n)e1()E(n)E(R

nt

)E(n

n0xn

nt

σ−

σϕ∝σ−

γ=

self-shielding factor


γ

γ

γ

n nn’nn’

n’’

∑=j

jRR

Ro R1 R2

Self-shielding and multiple scattering

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛θ−⎟⎟

⎠

⎞⎜⎜⎝

⎛+θ⎟⎟

⎠

⎞⎜⎜⎝

⎛+

= 22

n

A

nA

nn

'n sin

mmcos

mmmEE

t

n

n)e1(nR

t

σ−

σ∝σ−

γ


Self-shielding and multiple scattering factor• Calculation

– Analytical expressionsREFIT (parallel beam+ disc)SAMMY (parallel beam+ disc)

– Monte Carlo simulationsSAMSMC (parallel beam + disc)MCNP (no geometry limitations, use probability tables !!)

• Definitions

– Self-shielding without scattering : GR,0

– Self-shielding + 1 scattering : GR,1

– Self-shielding + 1,2 ,... scattering : GR,2

∫ ∫∫

∫ ∫∫

Ω=ΦΣΩ

ΩΦΣΩ

=

γ

γ

t

0

E

E

t

0

E

ER

)E,,0x()E(dxddE

)E,,x()E(dxddEG

2

1

2

1


Examples : 56Fe(n,γ), 59Co(n,γ) and 197Au(n,γ)

Reaction ER / eV Γn / eV Γγ / eV Γ / eV ∆D / eV

56Fe + n 1147.4 0.056 0.680 0.736 1.425 59Co + n 132.0 5.150 0.470 5.620 0.470

197Au + n 4.9 0.015 0.124 0.139 0.050

10-2 100 102 104

59Co(n,γ)

Neutron Energy / eV10-2 100 102 10410-4

10-2

100

102

104

σ(n,

γ) /

barn

197Au(n,γ)

Neutron Energy / eV10-2 100 102 104

56Fe(n,γ)

Neutron Energy / eV


Self-shielding + multiple scattering for 56Fe(n,γ) #1

ER = 1.15 keV in 56Fe θ En / keV En’ / keV

90o 1.192 keV 1.15 keV 180o 1.235 keV 1.15 keV

90o & 180o 1.280 keV 1.15 keV

1m1mEE90

A

An

'n

o

+−

=⇒=θ

2

A

An

'n

o

1m1mEE180 ⎟⎟

⎠

⎞⎜⎜⎝

⎛+−

=⇒=θ

θ = 90o

θ = 180o

1100 1150 1200 1250 130010-4

10-3

10-2

10-1

100

101

102

R R0

R1

R2

Yie

ld p

er a

tom

/ ba

rn

Neutron Energy / eV

56Fe(n,γ) nFe = 1x10-3 at/b (t = 0.12 mm)


Self-shielding + multiple scattering for 56Fe(n,γ) #2R

eact

ion

yiel

d [a

rb. u

nits

]

Neutron Energy / eV Neutron Energy / eV

nFe = 1.075 10-2 at/b, t = 1.3 mm

R0

R

R1R2

Rea

ctio

n yi

eld

x 10

3


Self-shielding + multiple scattering for 197Au(n,γ) #1

Rea

ctio

n yi

eld

[arb

. Uni

ts] o Exp.

- REFIT

2 4 6 810-2

100

102

104

R R0 R1 R2

Yiel

d pe

r ato

m /

barn

Neutron Energy / eV

197Au(n,γ) nAu = 6x10-4 at/b (t = 100µm)


Self-shielding + multiple scattering for 197Au(n,γ) #2R

eact

ion

yiel

d [a

rb. u

nits

]

Neutron Energy / eV


Influence of the resonance structure on GR #1

>σ<−σ− ≠>< tt nn ee

...)var2n1(ee t

2nn tt −σ+≈>< >σ<−σ−

100 120 140 1600.0

0.4

0.8

1.2

5x10-6 at/b 5x10-5 at/b 5x10-4 at/b (55.0 µm)

GR

,0(E

n)

Neutron Energy / eV

59Co(n,γ)59Co foil (1-e-n σt) / nσt GR,0

thickness <σt> σt,max

at /b µm 472 b 10539 b

5x10-6 0.55 1.00 0.97 0.99 5x10-5 5.50 0.99 0.78 0.88 5x10-5 55.0 0.89 0.19 0.46


Influence of the resonance structure on GR #2

0 40 80 12010

15

20 This evaluation Gregoriev et al. Uttly et al. Phoenitz et al. Iwasaki et al.

σ(n,

tot)

/ b

Neutron Energy / keV

0 40 80 1201.00

1.02

1.04

GR

,2

Neutron Energy / keV

HARFOR (analytical model) SAMSMC (Monte Carlo) MCNP (probability tables) MCNP (average cross sections)

232Th(n,γ) n = 1.588x10-3 at/b t = 0.5 mm


Influence of the Doppler effect #1

4 5 60

10

20

30

40 σ

γ

300 K 1000 K

Yie

ld p

er a

tom

/ kb

arn

Neutron Energy / eV

197Au(n,γ)

n = 6x10-6 at/bt = 1 µm

4 5 60

10

20

30

40 σ

γ

10 K 300 K 1000 K

Yie

ld p

er a

tom

/ kb

arn

Neutron Energy / eV

197Au(n,γ)

n = 6x10-5 at/bt = 10 µm


Influence of the Doppler effect #2

Temperature Correction factor GR,2 t = 1 µm t = 10 µm

10 K 0.71 300 K 0.97 0.73

1000 K 0.98 0.76

4 5 60.0

0.5

1.0

1.5

Temperature 10 K 300 K 1000 K

GR(E

n)

Neutron Energy / eV

197Au(n,γ)


Reaction yields and correction factors for 56Fe(n,γ)

1100 1150 1200 1250 130010-2

100

102

σγ

1x10-3 at/b (0.12 mm) 1x10-2 at/b (1.2 mm) 1x10-1 at/b (12 mm)

Yiel

d pe

r ato

m /

barn

Neutron Energy / eV

56Fe(n,γ)

ER = 1147.4 eVΓn = 0.056 eVΓγ = 0.680 eVΓ = 0.736 eV∆D = 1.425 eV

Foil thickness Correction factor at / b mm GR,0 GR,1 GR,2

1x10-3 0.12 0.97 0.99 0.99 1x10-2 1.2 0.78 0.86 0.87 1x10-1 12.0 0.23 0.35 0.47


Reaction yields and correction factors for 59Co(n,γ)

50 100 150 200100

101

102

103

σγ

5x10-6 at/b 5x10-5 at/b 5x10-4 at/b

Yie

ld p

er a

tom

/ ba

rn

Neutron Energy / eV

59Co(n,γ)


Foil thickness Correction factor at / b µm GR,0 GR,1 GR,2

5x10-6 0.55 0.99 1.02 1.03 5x10-5 5.5 0.88 1.05 1.08 5x10-4 55.0 0.46 0.67 0.86


Reaction yields and correction factors for 197Au(n,γ)

2 4 6 8101

103

105

σγ

6x10-6 at/b 6x10-5 at/b 6x10-4 at/b

Yiel

d pe

r ato

m /

barn

Neutron Energy / eV

197Au(n,γ)


Foil thickness Correction factor at / b µm GR,0 GR,1 GR,2

6x10-6 1.0 0.96 0.97 0.97 6x10-5 10.0 0.69 0.73 0.73 6x10-4 100.0 0.25 0.26 0.27


Interference effects, Uranium #1

0 5 10 1510-5

10-3

10-1

101

103

238U(n,γ) 235U(n,γ) + 235U(n,f) 234U(n,γ)

Y

ield

per

U a

tom

/ ba

rn

Neutron Energy / eV

LEU PowderLEU Powder with 0.048 at/b U

238U 96.97 wt%

235U 3.00 wt%

234U 0.03 wt%


Interference effects, Cd-ratio measurements #2

0 100 200 30010-2

10-1

100

101

102

103

σγ 59Co

σt natCd

e-nσt for 1mm

Cro

ss s

ectio

n / b

arn

Neutron Energy / eV

ReR

CdCd IG

RFϕ

=


Interference effects, Cd-ratio measurements #3

0 100 200 3000.8

1.0

1.2 nCo = 0.0001 at/b

RC

o in

Co

+ C

d / R

Co

Neutron Energy / eV

59Co(n,γ) in a mixture of 59Co and natCd

ReR

CdCd IG

RFϕ

=

FCd is influenced by the resonance structure of the cross sections


5. Resonance self-shielding for an isotropic beam


Isotropic beam (NAA applications)

• Experimental data

• Only self-shielding, no scattering & Doppler effect

• Self-shielding + Doppler

• Self-shielding + scattering

• Universal method


Experimental GR factors, F. De Corte et al.

0 200 400 6000.80

0.85

0.90

0.95

1.00 94Zr(n,γ) 96Zr(n,γ) 99Mo(n,γ)

GR

Thickness / µm

Foil

0 200 400 600

99Mo(n,γ) Wire (diameter) Foil (thickness)

Thickness / µm


Theoretical calculation, discussion #1

ttanconsBoltzmanthekwithm

EkT4

A

RD =∆

Reaction ER / eV Γn / eV Γγ / eV Γ / eV ∆D / eV

56Fe + n 1147.4 0.056 0.680 0.736 1.425 59Co + n 132.0 5.150 0.470 5.620 0.470 94Zr + n 2243.0 1.230 0.097 1.327 1.545 96Zr + n 301.0 0.215 0.258 0.473 0.560

197Au + n 4.9 0.015 0.124 0.139 0.050


Theoretical calculation, discussion 96Zr(n,γ) #2

Foil GR for 96Zr(n,γ) GR,0 (theory, no scattering) Experiment

Thickness (µm) No Doppler Trubey*

With Doppler Roe*

De Corte

100 0.969 0.979 0.978 125 0.963 0.975 0.973 250 0.928 0.956 0.951 500 0.888 0.925 0.921

*Taken from P. De Neve, Thesis, Mol, 1992

De Corte 87, Aggregraatsproefschrift Hoger Onderwijs, Universiteit Gent, 1987

⇒ Importance of Doppler effect increases with thicknessespecially for ∆D ≥ Γ

Γn = 0.21 eVΓγ = 0.26 eVΓ = 0.47 eV∆D = 0.56 eV


Theoretical calculation, discussion 94Zr(n,γ) #3

⇒ Importance of multiple scattering for Γn > Γγ

Γn = 1.23 eVΓγ = 0.10 eVΓ = 1.33 eV∆D = 1.55 eV

Foil GR for 94Zr(n,γ) thickness GR,0 with Doppler Experiment

(µm) Roe* De Corte 100 0.996 0.985 125 0.996 0.983 250 0.992 0.969 500 0.986 0.951

*Taken from P. De Neve, Thesis, Mol, 1992

De Corte 87, Aggregraatsproefschrift Hoger Onderwijs, Universiteit Gent, 1987


Theoretical calculation, discussion 59Co(n,γ) #4Foil GR for 59Co(n,γ)

Thickness (µm)

Theory (Ro + R1) No Doppler

Lopes and Avila

Experiment Eastwood and Werner

10.2 0.810 0.840 22.9 0.690 0.700 25.4 0.675 0.630 50.8 0.554 0.590 91.4 0.455 0.460

101.6 0.438 0.450 M.C. Lopes and J.M. Avila, Nucl. Sci. & Eng. , 104 (1990) 40

T.E. Eastwood and R.D. Werner, Nucl. Sci. & Eng., 13 (1962) 385

Γn = 5.15 eVΓγ = 0.47 eVΓ = 5.62 eV∆D = 0.47 eV

⇒ Importance of multiple scattering for Γn > Γγ


Universal curve #1

Express GR as a function of the variable

E. Martinho et al., “Universal curve of epithermal neutron self-shielding factors in foils, wires and spheres”, J. Appl. Rad. And Isot., 58 (2003) 371

foilsfortyspheresforRywiresforRy

y)E(z rt

===

ΓΓ

Σ= γ


Universal curve #2

2p0

21res A

)z/z(1AA

G ++

−=

foilsfort5.1yspheresforRy

wiresforR2y

y)E(z rt

===

ΓΓ

Σ= γ


Summary• The importance of the resonance structure for NAA

– Thermal capture cross section– 1/v behaviour of cross sections– Westcott gw - factor, gw(T)– Self-shielding and multiple scattering corrections, GR

• A first estimate of the correction factor for self-shielding and scattering is given by E. Martinho et al.

• More accurate correction factors can be calculated by analyticalexpressions and Monte Carlo simulations if all relevant effects are accounted for

– Resonance structure (e.g. MCNP use probability tables)– Neutron scattering– Doppler effect

• Solution:F. De Corte et al., J. Radioanal. and Nucl. Chem., 179 (1994) 93

“In general, the best way to solve the problem of self-shielding is to avoid it ”

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Resonance self-shielding in the epi-thermal...

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