Disordered crystals

Let’s consider a disordered crystal

Long-range order butUnit cell content depends on Ruvw

Lattice of disordered A and B atoms𝐹𝐴 = 1 et 𝐹𝐵 = 2

Bragg’s spot|𝚺(𝒒)|2

1ère zone de Brillouin

Occurrence of Speckles

due to disorder‘‘Speckle pattern’’

𝐴 𝒒 =

𝑢𝑣𝑤

𝐹𝑢𝑣𝑤(𝒒)e−𝑖𝐪∙𝐑𝑢𝑣𝑤 =

𝑛

𝐹𝑛(𝒒)e−𝑖𝐪∙𝐑𝑛

𝐹𝑛 ≠ 𝐹𝑛′

𝐴 𝒒 2

Example of diffuse scattering

Bragg reflexion+

Diffuse scattering

Precession photograph ofreciprocal plane h+k+l=0

Crystal of C60 at 300 K

Cours2.ppt#22. Désordre 1-Effet de la température

General expressionof scattered intensity

Calculation of intensity

Speckle term only visible in coherent diffraction conditions

𝑨∗ 𝒒 𝑨 𝒒 =

𝑛𝑛′

𝐹𝑛∗𝐹𝑛′𝑒

−𝑖𝒒∙(𝒓𝑛′−𝒓𝑛) =

𝑛𝑚

𝐹𝑛∗𝐹𝑛+𝑚 𝑒

−𝑖𝒒∙𝒓𝑚

=

𝑚

𝑁(𝑚)1

𝑁(𝑚)

𝑛

𝐹𝑛∗𝐹𝑛+𝑚 𝑒

−𝑖𝒒∙𝒓𝑚

1

𝑁(𝑚)

𝑛

𝐹𝑛∗𝐹𝑛+𝑚 = 𝐹𝑛

∗𝐹𝑛+𝑚 + Δ𝑚

= Statistical average + fluctuations

𝐼 𝒒 =1

𝑣

𝑚

𝑉 𝒓𝑚 𝐹𝑛∗𝐹𝑛+𝑚 𝑒

−𝑖𝒒∙𝒓𝑚 +1

𝑣

𝑚

𝑉(𝑚)Δ𝑚 𝑒−𝑖𝒒∙𝒓𝑚

𝑉 𝒓𝑚

… 𝑆𝑡𝑎𝑡 = … 𝑡

−𝒓𝑚

Ergodic hypothesis:

How to measure speckle patterns?

1. X-ray beam is coherent enough: 2. Disorder must be static during the time of measurement3. Detector good enough to measure speckles.

- Distance on detector: 𝜆𝑑/𝑎

If these conditiosn are not fulfilled, the speckles are smoothed

Diffuse scattering

𝐷

T

S

Speckles in AuAgZn2F. Livet et al. 𝑎 < 𝜉𝑡 =

𝜆𝐷

𝜎et 𝛿 < 𝜉𝑙 =

1

2

𝜆2

∆𝜆

𝑎

𝑑

𝜆

𝑎

Diffuse scattering

𝐼𝐷(𝒒) :Diffraction

𝐼𝐷𝐷(𝒒) : Diffuse scattering

𝜙𝑛: deviation from average value Fn

𝐹𝑛∗𝐹𝑛+𝑚 = 𝐹𝑛

∗ + 𝛷𝑛∗ 𝐹𝑛+𝑚 +𝛷𝑛+𝑚

= 𝐹 2 + 𝛷𝑛∗𝛷𝑛+𝑚

𝛷𝑛 = 𝐹𝑛 − 𝐹𝑛

𝐼 𝒒 = 𝐹 2

ℎ𝑘𝑙

Σ(𝒒 − 𝑸ℎ𝑘𝑙2

𝑣2+1

𝑣

𝑚

𝑉(𝒓𝑚) 𝛷𝑛∗𝛷𝑛+𝑚 𝑒

−𝑖𝐪∙𝐫𝑚

Cours10_26_10_2011.pptx#1. 4

Disorder

Diffuse scattering:Deviation from perfect order

If very few correlations:

Diffusion scattering ~ N

Diffraction ~ N2 is much stronger

𝐼𝐷𝐷 𝒒 =1

𝑣

𝑚

𝑉(𝒓𝑚) Φ𝑛∗Φ𝑛+𝑚 𝑒

−𝑖𝒒∙𝒓𝑚

lim𝑚→∞

Φ𝑛∗Φ𝑛+𝑚 = 0, and then 𝑉(𝒓𝑚) ≅ 𝑉

𝐼𝐷𝐷 𝒒 = 𝑁

𝑚

Φ𝑛∗Φ𝑛+𝑚 𝑒

−𝑖𝒒∙𝒓𝑚

Two types of disorder

Displacementdisorder

Substitution disorder

Displacement disorder:phonons

N: number of unit cellsM: atomic mass𝒌: wave vector of the phonon modeeak: polarisation of the modeqak: normal coordinates

𝑟𝑛(𝑡) = 𝑟𝑛 + 𝑢𝑛(𝑡)

Harmonic theory• Interaction potential 𝑈, elastic constant 𝐶

Atomic displacements

𝑈 = 𝑈0 +𝐶

2

𝑛

(𝒖𝑛+1 − 𝒖𝑛)2

𝒖𝑛 𝑡 =1

𝑁𝑀

𝛼,𝒌

𝜺𝛼𝒌𝑞𝛼𝒌(𝑡)𝑒𝑖𝒌∙𝒓𝑛

𝑞𝛼𝒌𝑞𝛼−𝒌 =ℏ

2𝜔𝛼(𝒌)coth

ℏ𝜔𝛼(𝒌)

2𝑘𝐵𝑇 𝑘𝐵𝑇≫ℏ𝜔𝛼

𝑘𝐵𝑇

𝜔𝛼2(𝒌)

Phonons

k

𝜔(𝒌)

p/a-p/a

optical

acoustic

Longitudinal

LO

Transverse

TO X 2

LA

TA X 2

10 Thz

Harmonic theoryDebye approx:𝑉𝑘𝐷

3 = 6𝜋2

ℏ𝑘𝐷𝑣𝑠 = 𝑘𝐵𝑇𝐷T at which highest mode excited

Debye-Waller factor

One atom:

Harmonic crystal

Intensity decreased by factor 𝑒−2𝑊

is the Debye-Waller factorRe

Im

Re

N

𝑇 and 𝑞 large ⇒ 𝜃 large

𝐹𝑛(𝐪) = 𝑓 𝑒−𝑖𝐪⋅𝐮𝒏

𝑒−𝑖𝐪⋅𝐮𝒏 = 𝑒−1/2 𝐪⋅𝐮𝒏2= 𝑒−𝑊

𝐼 𝐪 = 𝑁2 𝐹𝑛(𝐪)2 = 𝑁2𝑓2𝑒−2𝑊

𝑒−𝑊

𝜃

Debye-Waller factor 2

Unit cell with n atoms in rj

Isotropic vibrations

Diffraction allows one to measure:

ID e -2W 𝐹𝑛(𝐪) =

𝑗

𝑓𝑗𝑒−𝑊𝑗𝑒−𝑖𝐪⋅𝐫𝑗

𝑊𝑗 =1

2𝐪 ⋅ 𝒖𝑗

2=1

2𝑞2 𝑢𝑗𝑞

2

𝑢𝑗2 = 𝑢𝑥𝑗

2 + 𝑢𝑦𝑗2 + 𝑢𝑧𝑗

2 = 3 𝑢𝑗𝑞2

𝑊𝑗 =1

6

4𝜋 sin 𝜃

𝜆

2

𝑢𝑗2 ≡ 𝐵𝑗,𝑇,𝑖𝑠𝑜

sin 𝜃

𝜆

2

𝐵𝑗,𝑇,𝑖𝑠𝑜 =8𝜋2

3𝑢𝑗2

Calculation of 𝑊

Equipartition theorem

Slow vibrations have large amplitude

𝑊 =𝑞2

𝑁

𝐤

𝑘𝐵𝑇

2𝑀𝜔2(𝐤)

𝑊 =1

2𝑞2 𝑢𝑛

2𝑢𝑛2 =

1

𝑁

𝑛

𝑢𝑛2 =

1

𝑁

𝑘

𝑢𝑘2

1

2𝑀𝜔2(𝒌) 𝑢𝒌

2 =1

2𝑘𝐵𝑇

Einstein model 𝜔 = 𝜔𝑒

∝ 𝑞2𝑘𝐵𝑇

2𝑀𝜔𝑒2

Debye

Debye approximation

Si T >> TD Classical

1

1

0.5

0.5

with

𝑊 =sin𝜃

𝜆

26ℎ2𝑇

𝑀𝑘𝐵𝑇𝐷2

𝐼 ∝ 𝑒−2𝑊 ⇒ ln 𝐼 ∝ −𝑇

𝑊 =sin𝜃

𝜆

26ℎ2

𝑀𝑘𝐵𝑇𝐷

1

4+𝑇

𝑇𝐷Φ(𝑇𝐷𝑇)

élément C Al Cu Mo Ag Pb

TD (K) 3000 390 320 380 226 90

Φ𝑇𝐷𝑇

=𝑇

𝑇𝐷න0

𝑇𝐷𝑇 𝑦

𝑒𝑦 − 1𝑑𝑦

Example 1:Determination of TD

R.M. Nicklow and R.A. Young, Phys. Rev. 152, 591–596 (1966)

Intensity of Al (h00) reflexions

TD ~ 400±5K

Deviation due tozero-point vibrations

http://cornell.mirror.aps.org/abstract/PR/v152/i2/p591_1

Example 2: Lindemann criterion

Solide melts when:

Aluminiumf.c.c.

a=4.04 Å

Melting

𝑢2 = 10% 𝑑1𝑠𝑡 neigh

Example 3:

Simple etals:

Organic compounds:

Anisotropic B:𝑢2 depends

on the directions

𝑃𝐹6

𝐵𝑗,𝑇,𝑖𝑠𝑜 =8𝜋2

3𝑢𝑗2

𝑊𝑗 = 𝐵𝑗,𝑇,𝑖𝑠𝑜sin 𝜃

𝜆

2

𝑢2 = 0,05 − 0,2 Å

𝑢2 = 0,5 Å

Thermal ellipsoids

Influence of dimensionality

Integral is governed by divergence of:

Debye, isotropic

Harmonic theory

𝑊 = 𝑎𝐷𝑞2න𝑘

Influence of dimensionality-2

D=1

𝐿

𝑊 = 𝐴1න𝐿−1

𝑘𝐷 𝑑𝑘

𝑘2= 𝐴1(𝐿 − 𝑘𝐷

−1)𝑊 = 𝐴1න

𝐿−1

𝑘𝐷 𝑑𝑘

𝑘2

𝑊 = 𝐴2න𝐿−1

𝑘𝐷 2𝜋𝑘𝑑𝑘

𝑘2= 2𝜋𝐴2ln(𝐿𝑘𝐷)

𝑊 = 𝐴3න𝐿−1

𝑘𝐷 4𝜋𝑘2𝑑𝑘

𝑘2= 4𝜋𝐴3(𝑘𝐷 − 𝐿

−1)

𝑊 =1

2𝑞2 𝑢2 Si 𝐷 ≤ 2 𝑢2

𝐿→∞∞

D=3

D=2

𝑢2

Influence of dimensionality-3

No long-range order if D 2

If 𝐷 ≤ 2 𝑊𝐿→∞

∞

D=1

D=3

D=2

𝑒−𝑊~𝑒−𝐴1𝐿

𝑒−𝑊~𝐿−2𝜋𝐴2

𝑒−𝑊~exp −2sin2 𝜃

𝜆26ℎ2𝑇

𝑀𝑘𝐵𝑇𝐷2

Thermal Diffuse Scattering

Si 300 K

XRay //

XRay //

Experiment Simulation

M. Holt, Phys. Rev. Lett 83, 3317 (1999)

False colors,Log scale.

𝐼𝐷𝐷 𝒒 = 𝑁

𝑚

Φ𝑛∗Φ𝑛+𝑚 𝑒

−𝑖𝒒∙𝒓𝑚

TDS calculation-1

One atom per cell

Harmonic theory:

At first order,

displacement-displacement correlations

Φ𝑛∗Φ𝑛+𝑚 = 𝐹𝑛

∗𝐹𝑛+𝑚 − 𝐹2

Φ𝑛∗Φ𝑛+𝑚 = 𝑓

2 𝑒−𝑖𝒒∙(𝒖𝑛+𝑚−𝒖𝑛) − 𝑒−2𝑊

𝑒−𝑖𝒒∙(𝒖𝑛+𝑚−𝒖𝑛) = 𝑒−1/2 (𝒒∙(𝒖𝑛+𝑚−𝒖𝑛))2= 𝑒−2𝑊𝑒 (𝒒∙𝒖𝑛+𝑚)(𝒒∙𝒖𝑛)

Φ𝑛∗Φ𝑛+𝑚 = 𝑓

2𝑒−2𝑊 (𝒒 ∙ 𝒖𝑛+𝑚)(𝒒 ∙ 𝒖𝑛)

TDS calculation-3

+k-k

Qhkl Qhkl q

~1/𝑘2

• ~𝑁: diffuse scattering• 𝑘𝐵𝑇: thermal scattering• (𝒒 ∙ 𝒖)2: geometrical factor, (mode selection)• All a modes contribute to the same k

𝐼𝐷𝐷 𝒒 = 𝑸ℎ𝑘𝑙 + 𝒌 = 𝑁𝑓2𝑒−2𝑊𝑘𝐵𝑇

𝛼

(𝒒 ∙ 𝒖𝛼𝒌)2

𝐼𝐷𝐷~(𝒒 ∙ 𝒖𝛼𝒌)2

~1

𝜔𝛼2(𝒌)

Example of TDS

ComparisonX (-)-neutrons(o)

M. Holt, Phys. Rev. Lett 83, 3317 (1999)

Harmonic theory:Born-von Karman model using

constant forcesup to 6th neighbours

Si 300 K

𝐼𝐷𝐷(𝒒) = 𝑁𝑓2𝑒−2𝑊𝑘𝐵𝑇

𝛼

(𝒒 ∙ 𝜺𝛼𝒌)2

𝑀𝜔𝛼2(𝒌)

Substitution disorder

Alloy or solid solution AxB1-x

No information on correlations

• Case of total disorder

Laue scattering:

𝑓 = 𝑥𝑓𝐴 + (1 − 𝑥)𝑓𝐵

𝐼𝐷(𝒒) = 𝑁2 𝑥𝑓𝐴 + 1 − 𝑥 𝑓𝐵

2

𝐼𝐷𝐷 𝒒 = 𝑁 Φ02 = 𝑁 𝑓2 − 𝑓 2

𝐼𝐷𝐷 𝒒 = 𝑁𝑥(1 − 𝑥) 𝑓𝐴 − 𝑓𝐵2

𝐼 𝒒 = 𝑁𝑥(1 − 𝑥)(𝑓𝐴−𝑓𝐵)2

𝑚

(1 −𝑝𝐴 𝑚

𝑥)𝑒−𝑖𝒒∙𝒓𝑚

Correlations

𝑝𝐴(𝑚) A

B

Short-range order:

Conditional probabilities𝑝𝐴(𝑚): probability of having A in 𝒓𝑚 given B at 0𝑝𝐵(𝑚): probability of having B in 𝒓𝑚 given A at 0

AB pairs = BA pairs

Warren-Cowley parameters

⟹ 𝑥𝑝𝐵 𝑚 = (1 − 𝑥)𝑝𝐴(𝑚)

AA ∶ 𝑥 1 − 𝑝𝐵 𝑚 → 𝑓𝐴2

AB ∶ 𝑥𝑝𝐵 𝑚 → 𝑓𝐴𝑓𝐵AB ∶ (1 − 𝑥)𝑝𝐴 𝑚 → 𝑓𝐵𝑓𝐴BB ∶ (1 − 𝑥) 1 − 𝑝𝐴 𝑚 → 𝑓𝐵

2

𝐹𝑛∗𝐹𝑛+𝑚 =

𝑥 1 − 𝑝𝐵 𝑚 𝑓𝐴2 + 𝑥𝑝𝐵 𝑚 𝑓𝐴𝑓𝐵 + 1 − 𝑥 𝑝𝐴 𝑚 𝑓𝐵𝑓𝐴 + (1 − 𝑥) 1 − 𝑝𝐴 𝑚 𝑓𝐵

2

Example

Local order such thatAB pairs are favoredpA(m) A

B

ℎ0 1 2 3

1

1/2

𝑆(𝒒)

Tendency to doubleperiodicity

𝐼 𝒒 = 𝑁𝑥(1 − 𝑥)(𝑓𝐴−𝑓𝐵)2 1 + 2(1 −

𝑝𝐴 1

𝑥) cos(2𝜋ℎ)

𝑝𝐴 1 > 𝑥

Conclusion

• Substitution disorder:

𝐼𝐷𝐷 ~ (𝑓𝐴 − 𝑓𝐵)2

• Visible at small angles• Only electron density contrast at small angles

• Displacement disorder:

𝐼𝐷𝐷 ~ (𝒒. 𝒖)2

• Invisible at small angles𝒒. 𝒖 is too small for interferences to occur

Structural phase transitions Order parameter definition:

𝜂𝒌𝑐𝑒𝑖𝜑: order parameter

𝒌𝑐: critical wave vectorbelongs to the 1st BZ

Displacive Order-desorder

Order parmater 𝑈𝒌𝑐:

Displacement amplitude:

Order parameter :

Site probabilityIsing (pseudo-)spins

𝜂𝑛 = 𝜂𝒌𝑐 cos(𝒌𝑐 ∙ 𝒓𝑛 + 𝜑)

𝑼𝑛 = 𝜺𝒌𝑐𝑈𝒌𝑐 cos(𝒌𝑐 ∙ 𝒓𝑛 + 𝜑)

𝑇𝐶

ExamplesDisplacive transitions:

Order-disorder : • Alloy A0.5B0.5

TC

Critical wave vectors (1/4,0)

• FerroelectricZone center

Not a special point

• Displacive modulation (Peierls)

ab

Zone boundary

𝑇𝐶

𝒌𝐶 = 0 𝒌𝐶 =𝒂∗

2+𝒃∗

2

𝑼𝑛 = 𝜺𝒌𝑐𝑈𝒌𝑐 cos(𝒌𝑐 ∙ 𝒓𝑛 + 𝜑)

𝜺𝒌𝑐 = 𝒂

𝑆𝑛 = 𝑆𝒌𝑐 cos(𝒌𝑐 ∙ 𝒓𝑛 + 𝜑)

𝒌𝐶 =𝒂∗

4

Cours7.ppt#17. BaTiO3Cours7.ppt#17. BaTiO3

Displacive transition

Ordre parameter fluctuations

Susceptibility associated to order parameter

𝑢𝛼𝑘𝑐: composante principale

𝜒(𝒌𝒄) diverges at the transition temperature

𝐼𝐷𝐷 𝒒 = 𝑸ℎ𝑘𝑙 + 𝒌 = 𝑁𝑓2𝑒−2𝑊(𝒒 ∙ 𝜺𝒌)

2 𝑢𝒌𝑢−𝒌

𝑼𝑛 = 𝜺𝒌𝑐𝑈𝒌𝑐 cos(𝒌𝑐 ∙ 𝒓𝑛 + 𝜑)

𝒖𝑛 =1

𝑁

𝛼𝒌

𝜺𝛼𝒌𝑢𝛼𝒌𝒆𝒊𝒌⋅𝒓𝑛

𝒖𝒌 = 𝜒(𝒌)𝒉−𝒌

Fluctuation-dissipation

Example of phonons:

Par le théorème d’équipartition de l’énergie

𝒖𝒌𝒖−𝒌 − 𝒖𝒌 𝒖−𝒌 = 𝑘𝐵𝑇𝜒(𝒌)

1

2𝑀𝜔2(𝒌) 𝑢𝒌

2 =1

2𝑘𝐵𝑇

𝜒 𝒌 =1

𝑀𝜔2(𝒌)

Calculation of the intensity

Fluctuation-dissipation

𝑇 > 𝑇𝑐

𝑇 < 𝑇𝑐

𝐼𝐷 𝒒 = 𝑸ℎ𝑘𝑙 + 𝒌 = 𝑁𝑓2𝑒−2𝑊(𝒒 ∙ 𝜺𝒌)

2(𝑘𝐵𝑇𝜒 𝒌 − 𝑢𝒌 𝑢−𝒌 )

𝐼𝐷𝐷 𝒒 = 𝑁𝑓2𝑒−2𝑊(𝒒 ∙ 𝜺𝒌)

2𝑘𝐵𝑇𝜒 𝒌

𝑢𝒌 = 0

𝐼𝐷𝐷 𝒒 = 𝑁𝑓2𝑒−2𝑊(𝒒 ∙ 𝜺𝒌)

2𝑘𝐵𝑇𝜒 𝒌

+𝑁2𝑓2𝑒−2𝑊(𝒒 ∙ 𝜺𝒌)2 𝑈𝒌𝑐

2𝛿𝐾(𝒌 = ±𝒌𝑐)

𝑢𝒌𝐶 = 𝑁𝑈𝒌𝑐

Qhkl

Ornstein-Zernike

Lorentzian shape

x : correlation length

T>Tc

Qhkl

+𝒌𝑐−𝒌𝑐

T

Critical exponents

Temperature behavior of:

TTc• Susceptibility: 𝜒(𝑘𝑐) ~ 𝑇 −𝑇𝑐

−𝜸

• Correlation lengths: 𝜉 ~ 𝑇 −𝑇𝑐−𝝂 SRO

QLRO

LRO

Example : Order-disorder in AuAgZn2

T< 351.1°C T> 351.1°C

Au/Ag

Zn

Cubicf.c.c.

F. Livet et al. Phys. Rev. B 66, 134108 (2002)

2nd order pahse transition

𝒌𝐶 =𝒂∗

2+𝒃∗

2+𝒄∗

2

FluctuationsDiffuse scattering (1/2,1/2,1/2)

Ising 3Dg =1,24

n = 0,63

h = 0,04

h = 0,03

c(q)~ q-2+h

c~(T-Tc)-g

c-1/g ~(T-Tc)

g =1,242

x ~(T-Tc)-n

x-1/ n~(T-Tc)

n = 0,709

TC+4°C TC+0,13°C

TC+4°C

TC+0,08°C

Example:‘‘Blue bronze’’ K0.3MoO3

b

Octahedra MoO6

Potassium

(Rubidium)

E. Bervas, thesis (1984)

Tp=183 K

c

a

Blue bronze

𝑋𝑌 3𝐷𝛾 = 1,316𝜈 = 0,669𝛽 = 0,346

At T=183 K: appareance of sattelite reflexionsat the critical wave vector:

𝜒~(𝑇 − 𝑇𝑐)−𝛾

𝛾 = 1,33(4) 𝜈 = 0,68(5)

𝛽 = 0,31(5)

𝒌𝐶 = 0,748 𝒃∗ + 0,5 𝒄∗

𝜉~(𝑇 − 𝑇𝑐)−𝜈

𝐼~(𝑇𝑐 − 𝑇)𝛽

Determination of interatomic potential

Ex: Ising model

Within mean field appr.the susceptibility is:

Fit of 𝜒(𝒌) gives the 𝐽′𝑠

𝐻 = −𝑖𝑗𝐽𝑖𝑗 𝜎𝑖𝜎𝑗

𝜒(𝒌) =𝛽

1 + 𝛽𝐽(𝒌)

With,𝐽 𝒌 = 2𝐽𝑎 cos 𝒌 ∙ 𝒂 + 2𝐽𝑏 cos 𝒌 ∙ 𝒃 + 2𝐽𝑐 cos 𝒌 ∙ 𝒄

ExampleBragg

Diffuse scattering

IsotropeJi=Jj

Anisotrope (1D)100xJi=Jj

Local order

Difficultto see

in real space