–

–

–

dS

dt> 0

–

p(r)

r

–

p(r)

r

–

p(r)

r

p(r)

r

p(r)

r

–

–

–

–

Structure factor for a gas

–

A- Determine the static structure factor for a monoatomic gas

of punctual non-interacting particles. B- Determine the static structure factor for a gas where the

particles have finite size and consequently an effective repulsion distance. Determine the maximum possible

density of such a gas. C- Study the low-Q limit of the structure factor. Should the

structure factor be null at the origin?

Exercise

–

θ ω ω

θ

σ Ω ω

–

ω

ρ(�r, t) =

N∑i=1

δ (�r − �ri(t))

–

ρ ρ

ρ ρ r

t ∼ τ

t » τ

t « τρ

ρ

ρ

0

α βα ≠ β

d2σ

dΩdω(2θ, ω) = N

k′

k

{|〈b〉|2 Sd( �Q, ω) + 〈|b|2〉Ss( �Q, ω)

}

Coherent and incoherent xs

–

Starting with the total scattering cross section and considering

the self and distinct contributions, derive the expression showing

explicitly the coherent and incoherent contributions.

Exercise

d2σ

dΩdω(2θ, ω) = N

k′

k{〈b〉|2 S( �Q, ω) +

(〈|b|2〉 − |〈b〉|2

)Ss( �Q, ω)}

–

σ

4π σ

4π

σ = σ σ

σcoh = 4π |〈b〉|2

σinc = 4π(〈|b|2〉 − |〈b〉|2

)

Static structure factor

–

Exercises

Find the relationship between the van Hove time-dependent

correlation function G(r,t) and the static structure factor S(Q).

Propose an expression for the static correlation function

using the macroscopic density of the material and the real

space pair distribution function g(r).

Using this expression, find the relationship between the static

structure factor S(Q) and and g(r).

Static structure factor

–

S( �Q) =

∫ +∞

−∞dω S( �Q, ω) =

∫d�r ei

�Q·�rG(�r, 0)

Static approximation

S(QQ)-1 & g(r)-1 become a FT pair

F ( �Q) = S( �Q)− 1

G(�r) = 4πρr [g(�r)− 1]

Definition

S( �Q)− 1 = ρ

∫V

d�r [g(�r)− 1] ei�Q·�r

ρ [g(�r)− 1] =1

(2π)3

∫d �Q

[S( �Q)− 1

]e−i�Q·�r

F ( �Q) =

∫V

d�rG(�r)

4πrei

�Q·�r

G(�r)

4πr=

1

(2π)3

∫d �Q F ( �Q) e−i�Q·�r

–

0 2 4 6 8 10 120

1

2

3

S(Q)

Q/Å-1

0 5 10 15 20 25

0

1

2

3

g(r)

r/Å

Isotropic structure factor

–

Exercises

Consider now the isotropic case and find the relationship

between the isotropic structure factor S(Q) and

the isotropic pair distribution function g(r).

A- Consider a well-ordered system with a single (sharp) peak

in the real space correlation function and calculate the corresponding structure factor. B- Discuss the changes you could expect in the structure factor if

you had a distribution of distances around a single mean-value. C- Discuss now what you should observe if you had multiple

distances, each one with a distribution around the mean value.

–

0 5 10 15 20 25

0

1

2

3

g(r)

r/Å

2D analogy 0 2 4 6 8 10 12

0

1

2

3

S(Q)

Q/Å-1

–

0 2 4 6 8 10 120

1

2

3

S(Q)

Q/Å-1

0 5 10 15 20 25

0

1

2

3

g(r)

r/Å

Δ

→

S(∞) = 1

S(0) = ρ χT kBT

–

0Q

4π/λ

f(Q) = b

f(Q)

f(Q) = b

f(Q)

Q =4π

λsin

(2θ

2

)Bragg law

(d2σ

dΩdE′

)=

k′

k

1

2π�

∑jj′

fj( �Q)fj′( �Q)

∫ ∞

−∞〈e−i �Q·�rj′ (0)ei �Q·�rj(t)〉e−iωtdt

Form factor

–

The Fermi’s pseudo-potential works well for the neutron-nucleus

interaction yielding to a constant scattering length.

Explain qualitatively the effect that a non-punctual potential

has on the structure factor.

Exercise

–

π ρ

0

1

g(r)

r

-2

0

D(r)

r

πρ

0

RDF(r

)

r

π ρ

∝ ρ

π ρ π ρ

Correlation functions

–

Exercises

Consider a system with a FSDP at 2.1Å−1 with a FWHM of 0.62 Å−1.

What could you say about the characteristics of the

corresponding pair distribution function?

How many coordination shells could you observe in such a system?

Taking into account the repulsion region in the interaction

potential, show that the low-r region of density function is

proportional to the density.

–

αβ

α   

α α

b̄2

⎛⎝ FS1(Q)

FS2(Q)FS3(Q)

⎞⎠ =

⎛⎝ c2Xb

2X1 c2Yb

2Y1 2cXcYbX1bY1

c2Xb2X2 c2Yb

2Y2 2cXcYbX2bY2

c2Xb2X3 c2Yb

2Y3 2cXcYbX3bY3

⎞⎠

⎛⎝ FXX(Q)

FYY(Q)FXY(Q)

⎞⎠

–

Weighting factors

–

Exercise Consider a binary system for which you have performed three

different total scattering experiments with three different isotopic

compositions. A- Write down the expression of the weighting factors matrix B- Evaluate this matrix for the case of the Ag2Se, if you used the

following isotopic compositions:

107Ag-natSe, 109Ag-76Se and natAg-76Se.

Assume that for each substitution, you have replaced 100% of

the atoms by its corresponding isotopes.

C- Invert the matrix to obtain the partial structure factors

from the three experimental structure factors.

–

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

)()()(

3211.01654.01559.02272.01654.00780.02706.00706.02594.0

)()()(

AgSe

SeSe

AgAg

S3nat76

S210976

S1107nat

QFQFQF

QFQFQF

Samples: 107Ag2natSe, 109Ag2

76Se, natAg276Se

Isotope b (fm) σσa (b) σ

s (b)

nAg 5.922 24.6 4.99107Ag 7.64 14.6 7.44109Ag 4.19 35.4 2.55

nSe 7.97 4.55 8.3176Se 12.2 33.1 18.7

Silver chalcogenidesAg2X

Network formersAsX or As2X3

+

X= S, Te or Se

107F76 + 1.0

NF76 + 0.5

107FN

Fast-ion conductor or semiconductor glasses

–

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

−

−

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

)()()(

30.2900.2509.1063.2922.3211.850.2231.1717.12

)()()(

S3nat76

S210976

S1107nat

AgSe

SeSe

AgAg

QFQFQF

QFQFQF

gSeSe(r) + 3

gAgAg(r)

gAgSe(r) + 6

2.6 Å

4.6 Å

2.8 Å

SSeSe(Q) + 3

SAgAg(Q)

SAgSe(Q) - 3

nAgSe = 9.3nSeSe = 13.0nAgAg = 5.3

–

0 4 8 12 16 200

1

2

3

g ij(r)

r (Å)

AgSe

AgAg

SeSe

Ionic

b̄2 ΔFγ(Q)

c2γ(b2γ1 − b2γ2)

= Fγγ(Q) +

∑nα �=γ cαbα Fαγ(Q)

cγ (bγ1 + bγ2)

First difference method

–

Substitution ----- γγ: γ1, γ2

Important! We change scattering lengths but nnot composition

b̄2 F (Q) =

n∑α=1

n∑β=1

cαcβbαbβ Fαβ(Q)

b̄2 Fγ1(Q) = c2γb2γ1 Fγγ(Q) + cγbγ1

n∑α �=γ

cαbα Fαγ(Q) +

n∑α,β �=γ

cαcβbαbβ Fαβ(Q)

b̄2 Fγ2(Q) = c2γb2γ2 Fγγ(Q) + cγbγ2

n∑α �=γ

cαbα Fαγ(Q) +

n∑α,β �=γ

cαcβbαbβ Fαβ(Q)

b̄2 ΔFγ(Q) = c2γ(b2γ1 − b2γ2) Fγγ(Q) + cγ(bγ1 − bγ2)

n∑α �=γ

cαbα Fαγ(Q)

Correlation function of atom γγ with all other

components

small

–

Ω

⇒

…

–

Second difference method

–

Partial structure factor for pairs γ and δ

New substitution δδ: δ1, δ2

b̄2 Δ2Fγδ(Q) = cγcδ(bγ1 − bγ2) (bδ1 − bδ2) Fγδ(Q)

Fγδ(Q) =b̄2 Δ2Fγδ(Q)

cγcδ(bγ1 − bγ2) (bδ1 − bδ2)

b̄2 ΔFγδ1(Q) = c2γ(b2γ1 − b2γ2) Fγγ(Q) + cγcδ(bγ1 − bγ2) bδ1 Fγδ(Q) + cγ(bγ1 − bγ2)

n∑α �=γ,δ

cαbα Fαγ(Q)

b̄2 ΔFγδ2(Q) = c2γ(b2γ1 − b2γ2) Fγγ(Q) + cγcδ(bγ1 − bγ2) bδ2 Fγδ(Q) + cγ(bγ1 − bγ2)

n∑α �=γ,δ

cαbα Fαγ(Q)

–

Δ×

Δ

→

Δ

–

Δ

τ = (θ ϕ)/60 × 100%

–

–

Ispin inc(Q) =3

2ISF(Q)

Icoh(Q) + Iisotope inc(Q) = INSF(Q)− 1

2ISF(Q)

–

–

→

→

→

→→

–

Q =4π

λsin

(2θ

2

)Face d-spacing (Å)

λ (Å) Flux (107 n cm–2 s–1)

Filter

Si111 1.807 0.7 5.0 Ir Cu220 1.278 0.5 4.5 Rh Cu331 0.829 0.35 0.3 Non

–

–

θ Φ π σ σ ε

    

    

∞

–

    

    

Beam stop

Void sample position

 

  Sample-like Cd specimen

– = θ ≈ θ θ θ

σ

–

Θ

ω ω

�2Q2

2m= 2E + �ω − 2

√E2 + �ωE cos 2θ

I(2θ) = C Φ0 Nσ

4π

∫ Emax

−∞

k′

kS( �Q, ω) ε(k′) dω

θ

ε∞

–

  ω ω ⎯→ ω  ε ω ω 

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

λ (Å)

ε0

c1

c2

c3

3/8

C1 = 1− α/2

eα − 1

C2 =3

8− α (α+ 3)

8 (eα − 1)

C3 =α(α+ 1)

4 (eα − 1)

S(Q) =1

ε0

1

b2coh

(dσ

dΩ

)corr

+

(1 +

b2incb2coh

)(C1δ − C2δ

2 + C3δγ − m

2M(δ + γ)

)−− b2incb2coh

  ε  ε ∝  ε α

–

ω

IcorrS (2θ) =1

αS,SC(2θ)

(IS(2θ)− IBS (2θ)−

αC,SC(2θ)

αC,C(2θ)

(IC(2θ)− IBC (2θ)

))−Δ

–

θ

0 5 10 15 20 25

0

1

2

3

g(r)

r/Å

–

General D.I. Page, The structure of liquids by neutron diffraction, p. 173, Chap. 8 in Chemical applications of thermal neutron scattering, B.T.M. Willis (Ed.), Oxford University Press (1973). J.G. Powles, The structure of molecular liquids by neutron scattering, Advances in Physics 22, 1 (1973). A.C. Wright, The structure of amorphous solids by x-ray and neutron diffraction, Advances in Structure Research by Diffraction Methods 5, 1 (1974). P.A. Egelstaff, Classical fluids, p. 405, Chap.14 in Methods of experimental physics, vol. 23, part B, Academic Press (1987). P. Chieux, Introduction to accurate structure factor measurements of disordered materials by neutron scattering, J. Molec. Struct. 296, 177 (1993). G. J. Cuello, Structure factor determination of amorphous materials by neutron diffraction, J. Phys.: Condensed Matter 20, 244109 (2008) H. E. Fischer, A. C. Barnes, P. S. Salmon, Neutron and x-ray diffraction studies of liquids and glasses, Reports on Progress in Physics 69, 233-299 (2006) Examples

See the D4 web site (www.ill.fr/YellowBook/D4/pub/) for a list of publications arisen from D4 expts. Visit the Disordered Materials Group at ISIS (www.isis.stfc.ac.uk/groups/disordered-materials/)

–

Examples with useful information about data treatment D.M. North, J.E. Enderby and P.A. Egelstaff, The structure factor for liquid metals: I. The application of neutron diffraction techniques, J. Phys. C 1, 784 (1968). H. Bertagnolli, P. Chieux and M.D. Zeidler, A neutron-diffraction study of liquid acetonitrile: I. CD3C14N, Molec. Phys. 32, 759 (1976). J.L. Yarnell, M.J. Katz R.G. Wenzel and S.H. Koenig, Structure factor and radial distribution function for liquid Ar at 85 K, Phys. Rev. A 7, 2130 (1973). Corrections G. Placzek, Phys. Rev 86, 377 (1952).

A.K. Soper and P.A. Egelstaff, Multiple scattering and attenuation of neutrons in concentric cylinders: I. Isotropic first scattering, Nucl. Instr. Meth. 178, 415 (1980). I.A. Blech and B.L. Averbach, Multiple scattering of neutrons in vanadium and copper, Phys. Rev. 137, A1113 (1965). H.H. Paalman and C.J. Pings, Numerical evaluation of x-ray absorption factors for cylindrical samples and annular sample cells, J. Appl. Phys. 33, 2635 (1962).