Chapter 9 – COHERENT AND INCOHERENT NEUTRON SCATTERING

9:1. COHERENT AND INCOHERENT CROSS SECTIONS

9:2. SPIN INCOHERENCE

9:3. COHERENT SCATTERING CROSS SECTION

9:7. CONTRAST FACTORS

9:11. TABULATED SCATTERING LENGTHS AND CROSS SECTIONS

9:4. ISOTOPIC INCOHERENT SCATTERING CROSS SECTIONS

9:12. NEUTRON TRANSMISSION

20:5. ESTIMATION OF INCOHERENT SCATTERING

inccoh d

d

d

)Q(d

d

)Q(d

2N

j,iij

2 br.Qiexp}b{d

)Q(d

d

)Q(d

V

N

d

)Q(d

9:1. COHERENT AND INCOHERENT CROSS SECTIONS

222 bbb

Microscopic cross section (in barns):

Macroscopic cross section (in cm-1):

coherentpart

incoherent part

Q-dependent Q-independent

Standard deviation:

9:2. SPIN INCOHERENCE

spin operators

2(I-1/2)+1=2I

2(I+1/2)+1=2I+2I + 1/2

eigenvalues

number ofdegenerate states

I - 1/2

2

1I

incidentneutron of spin 1/2

scatteringnucleus of

spin I

1I2

1I

2I4

2I2W

1I2

I

2I4

I2W

12I

Ib1)b(IbWbW}b{

12I

Ib1)b(IbWbW}b{

22222

bWbW}b{bc

)bb(WW}b{}b{b 22i

DERIVATIONS

Coherent scattering length:

Incoherent scattering length:

Average over spins:

9:3. COHERENT SCATTERING CROSS SECTION

9:7. CONTRAST FACTORS

2

B

B

A

A2BA

2

v

b

v

b

Contrast factor:

2BBAA

2nmc )bfbf(4}b{4)BA(

1ff BA

Coherent cross section:

With the fractions:

cc V

N

Macroscopic (cm-1) and microscopic (barns) cross sections:

9:4. ISOTOPIC INCOHERENT SCATTERING CROSS SECTIONS

)B(f)A(f)BA( iBiAspinnmi

)}b{}b({4)BA( 22isotopenmi

Spin incoherence cross section:

Isotopic incoherence cross section:

9:11. TABULATED SCATTERING LENGTHS AND CROSS SECTIONS

Element bFermi

c

Barn

i

Barn

s

Barn

a

Barn

H -3.739 1.757 80.30 82.057 0.333

D 6.671 5.592 2.05 7.642 0.000

C 6.646 5.550 0.001 5.551 0.003

N 9.36 11.01 0.50 11.51 1.90

0 5.803 4.232 0.000 4.232 0.000

F 5.654 4.232 0.001 4.233 0.000

Na 3.63 1.66 1.62 3.28 0.530

Si 4.149 2.163 0.004 2.167 0.171

P 5.13 3.307 0.005 3.312 0.172

S 2.847 1.017 0.007 1.024 0.53

Cl 9.577 11.526 5.3 16.826 33.5

9:12. NEUTRON TRANSMISSION

aicast

)dexp(T t

Total cross section:

scattering absorption

coherent incoherent

t: total cross section

d: sample thickness

Neutron transmission:

20:5. ESTIMATION OF INCOHERENT SCATTERING

4)Tdλ(I

I

dΩ

Σd i

0

ii

1n

ni

i d!n

1

d4

1

dΩ

Σd

)dexp(

)dexp(1

d4

1

dΩ

Σd

i

ii

T

T1

d4

1

dΩ

Σd i

it

Incoherent scattering cross section:

Including multiple scattering:

Resuming the series:

When incoherent scattering dominates:

Way to estimate the incoherent cross section:

V.F. Sears, “Neutron Scattering Lengths and Cross Sections”, Neutron News 3, 26-37 (1992).

A scattering length density calculator can be found online at: http://www.ncnr.nist.gov/resources/sldcalc.html

REFERENCES

COMMENTS

-- Coherent scattering probes the sample structure (Q-dependent). Incoherent scattering does not contain information about the structure (Q-independent).

-- Scattering length densities and contrast factors can be estimated using the online calculator.