iitphys.iit.edu/~segre/phys570/13f/lecture_18.pdf · 2013. 10. 31. · today’s outline - october...

Today’s Outline - October 28, 2013

• PHYS 570 day at 10-BM

• Brief introduction to EXAFS

• Crystal Truncation Rods

• Lattice Vibrations

• Thermal Diffuse Scattering

• Debye Waller Factor

• Lorentz Factor, Extinction & Absorption

• Powder Diffraction

October 30, 2013 class in 111 Life Sciences(Chemistry Colloquium)

Homework Assignment #06:Chapter 6: 1,6,7,8,9due Wednesday, November 11, 2013

C. Segre (IIT) PHYS 570 - Fall 2013 October 28, 2013 1 / 23

PHYS 570 day at 10-BM

1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00

2 Activities

• Absolute flux measurement• Reflectivity measurement• EXAFS measurement

3 Make sure your badge is ready

4 Leave plenty of time to get the badge

5 Let me know when you plan to come!

1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00

2 Activities• Absolute flux measurement• Reflectivity measurement• EXAFS measurement

1 3 sessions:• 09:00 – 12:00• 13:00 – 16:00• 17:00 – 20:00

What is XAFS?

X-ray Absorption Fine-Structure (XAFS) is the modulation of the x-rayabsorption coefficient at energies near and above an x-ray absorption edge.XAFS is also referred to as X-ray Absorption Spectroscopy (XAS) and isbroken into 2 regimes:

XANES X-ray Absorption Near-Edge SpectroscopyEXAFS Extended X-ray Absorption Fine-Structure

which contain related, but slightly different information about an element’slocal coordination and chemical state.

E (eV)

77007600750074007300720071007000

Fe K-edge XAFS for FeO

XAFS Characteristics:

• local atomic coordination

• chemical / oxidation state

• applies to any element

• works at low concentrations

• minimal sample requirements

The x-ray absorption process

An x-ray is absorbed by anatom when the energy of thex-ray is transferred to a core-level electron (K, L, or Mshell).

The atom is in an excitedstate with an empty elec-tronic level: a core hole.

Any excess energy fromthe x-ray is given to anejected photoelectron

, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.

x−ray

Energy

photo−electron

Continuum

x−ray

Energy

photo−electron

Continuum

x−ray

Energy

photo−electron

Continuum

Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave

,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.

Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave

,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.

Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds

, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.

Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole

, cre-ating interference patternscalled XANES and EXAFS.

Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole

, cre-ating interference patternscalled XANES and EXAFS.

Any excess energy fromthe x-ray is given to anejected photoelectron, whichexpands as a spherical wave,reaches the neighboring elec-tron clouds, and scattersback to the core hole, cre-ating interference patternscalled XANES and EXAFS.

11500 12000 12500

EXAFS data extraction

normalize by fitting pre-edgeand post-edge trends

remove “smooth” µ0 back-ground

convert to photoelectron mo-mentum space

k =2π

√E − E0

weight by appropriate powerof k to obtain “good” enve-lope which clearly shows thatEXAFS is a sum of oscilla-tions with varying frequen-cies and phases

Fourier transform to get realspace EXAFS

11500 12000 12500

k =2π

√E − E0

11500 12000 12500

k =2π

√E − E0

11500 12000 12500

k =2π

√E − E0

0 5 10 15

k(Å-1

k =2π

√E − E0

0 5 10 15

k(Å-1

k =2π

√E − E0

0 2 4 6

The EXAFS equation

The EXAFS oscillations can be modelled and interpreted using aconceptually simple equation (the details are more subtle!):

χ(k) =∑j

NjS20 fj(k)e−2Rj/λ(k) e−2k2σ2

sin [2kRj + δj(k)]

where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .. . . over scattering paths for the photo-electron.

Nj : path degeneracy

Rj : half path length

σ2j : path “disorder”

S20 : amplitude reduction factor

k is the photoelectron wave number

fj(k): scattering factor for the path

δj(k): phase shift for the path

λ(k): photoelectron mean free path

Because we can compute f (k) and δ(k), and λ(k) we can determine Z, R,N, and σ2 for scattering paths to neighboring atoms by fitting the data.

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

where the sum could be over shells of atoms (Fe-O, Fe-Fe) or . . .

. . . over scattering paths for the photo-electron.

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

The EXAFS equation

χ(k) =∑j

sin [2kRj + δj(k)]

XANES edge shifts and pre-edge peaks

5460 5470 5480 5490 5500

V metal

LiVOPO4

The shift of the edge positioncan be used to determine thevalence state.

The heights and positions ofpre-edge peaks can also be re-liably used to determine ionicratios for many atomic species.

XANES can be used as a fin-gerprint of phases and XANESanalysis can be as simple asmaking linear combinations of“known” spectra to get com-position.

Modern codes, such as FEFF9,are able to accurately computeXANES features.

5460 5470 5480 5490 5500

V metal

LiVOPO4

5460 5470 5480 5490 5500

V metal

LiVOPO4

5460 5470 5480 5490 5500

V metal

LiVOPO4

5460 5470 5480 5490 5500

V metal

LiVOPO4

Coordination chemistry

E (eV)

60506040603060206010600059905980

The XANES of Cr3+ and Cr6+ shows a dramatic dependence on oxidationstate and coordination chemistry.

For ions with partially filled d shells, the p-d hybridization changesdramatically as regular octahedra distort, and is very large for tetrahedralcoordination.

This gives a dramatic pre-edge peak – absorption to a localized electronicstate.

Coordination chemistry

E (eV)

60506040603060206010600059905980

The XANES of Cr3+ and Cr6+ shows a dramatic dependence on oxidationstate and coordination chemistry.

For ions with partially filled d shells, the p-d hybridization changesdramatically as regular octahedra distort, and is very large for tetrahedralcoordination.

This gives a dramatic pre-edge peak – absorption to a localized electronicstate.

Diffraction from a Truncated Surface

For an infinite sample, the diffractionspots are infinitesimally sharp.

With finite sample size, these spotsgrow in extent and become more dif-fuse.

If the sample is cleaved and left withflat surface, the diffraction will spreadinto rods perpendicular to the surface.

The scattering intensity can be ob-tained by treating the charge distri-bution as a convolution of an infinitesample with a step function in the z-direction.

CTR Scattering Factor

The scattering amplitude FCTR along a crystal truncation rod is given bysumming an infinite stack of atomic layers, each with scattering amplitudeA(~Q).

FCTR = A(~Q)∞∑j=0

e iQza3j

=A(~Q)

1− e iQza3=

1− e i2πl

this sum has been discussed previ-ously and gives

or, in terms of the momentumtransfer along the z-axis,Qz = 2πl/a3

since the intensity is the square of the scattering factor

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)=

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

e iQza3j

=A(~Q)

1− e iQza3=

1− e i2πl

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)=

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

e iQza3j

=A(~Q)

1− e iQza3=

1− e i2πl

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)=

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

e iQza3j

=A(~Q)

1− e iQza3

=A(~Q)

1− e i2πl

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)=

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

e iQza3j

=A(~Q)

1− e iQza3=

1− e i2πl

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)=

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

e iQza3j

=A(~Q)

1− e iQza3=

1− e i2πl

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)=

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

e iQza3j

=A(~Q)

1− e iQza3=

1− e i2πl

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

e iQza3j

=A(~Q)

1− e iQza3=

1− e i2πl

ICTR =∣∣∣FCTR

∣∣∣2 =

∣∣∣A(~Q)∣∣∣2

(1− e i2πl) (1− e−i2πl)=

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl)

Dependence on Q

When l is an integer (meeting the Laue condition), the scattering factor isinfinite but just off this value, the scattering factor can be computed byletting Qz = qz + 2π/a3, with qz small.

ICTR =

∣∣∣A(~Q)∣∣∣2

4 sin2 (Qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (πl + qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (qza3/2)

∣∣∣A(~Q)∣∣∣2

4(qza3/2)2=

∣∣∣A(~Q)∣∣∣2

Dependence on Q

ICTR =

∣∣∣A(~Q)∣∣∣2

4 sin2 (Qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (qza3/2)

∣∣∣A(~Q)∣∣∣2

4(qza3/2)2=

∣∣∣A(~Q)∣∣∣2

Dependence on Q

ICTR =

∣∣∣A(~Q)∣∣∣2

4 sin2 (Qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (qza3/2)

∣∣∣A(~Q)∣∣∣2

4(qza3/2)2=

∣∣∣A(~Q)∣∣∣2

Dependence on Q

ICTR =

∣∣∣A(~Q)∣∣∣2

4 sin2 (Qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (qza3/2)

∣∣∣A(~Q)∣∣∣2

4(qza3/2)2=

∣∣∣A(~Q)∣∣∣2

Dependence on Q

ICTR =

∣∣∣A(~Q)∣∣∣2

4 sin2 (Qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (qza3/2)

∣∣∣A(~Q)∣∣∣2

4(qza3/2)2

∣∣∣A(~Q)∣∣∣2

Dependence on Q

ICTR =

∣∣∣A(~Q)∣∣∣2

4 sin2 (Qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (qza3/2)

∣∣∣A(~Q)∣∣∣2

4(qza3/2)2=

∣∣∣A(~Q)∣∣∣2

Dependence on Q

ICTR =

∣∣∣A(~Q)∣∣∣2

4 sin2 (Qza3/2)

∣∣∣A(~Q)∣∣∣2

4 sin2 (qza3/2)

∣∣∣A(~Q)∣∣∣2

4(qza3/2)2=

∣∣∣A(~Q)∣∣∣2

Absorption Effect

Absorption effects can be in-cluded as well

e iQza3je−βj

=A(~Q)

1− e iQza3e−βj

This removes the infinity andincreases the scattering pro-file of the crystal truncationrod

Absorption Effect

e iQza3j

e−βj

=A(~Q)

1− e iQza3e−βj

Absorption Effect

e iQza3je−βj

=A(~Q)

1− e iQza3e−βj

Absorption Effect

e iQza3je−βj

=A(~Q)

1− e iQza3e−βj

Absorption Effect

e iQza3je−βj

=A(~Q)

1− e iQza3e−βj

Density Effect

The CTR profile is sensitive to the termination of the surface. This makesit an ideal probe of electron density of adsorbed species or single atomoverlayers.

F total = FCTR + F top layer

=A(~Q)

1− e i2πl

+ A(~Q)e−i2π(1+z0)l

where z0 is the relative dis-placement of the top layerfrom the bulk lattice spacinga3

This effect gets larger forlarger momentum transfers

Density Effect

=A(~Q)

1− e i2πl

+ A(~Q)e−i2π(1+z0)l

Density Effect

=A(~Q)

1− e i2πl

+ A(~Q)e−i2π(1+z0)l

Density Effect

=A(~Q)

1− e i2πl

+ A(~Q)e−i2π(1+z0)l

Density Effect

=A(~Q)

1− e i2πl

+ A(~Q)e−i2π(1+z0)l

Density Effect

=A(~Q)

1− e i2πl

+ A(~Q)e−i2π(1+z0)l

Lattice Vibrations

Atoms on a lattice are not rigid but vibrate. There is zero-point motion aswell as thermal motion. These vibrations influence the x-ray scattering.

For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn. Computing the intensity:

⟨∑m

f (~Q)e i~Q·(~Rm+~um)

f ∗(~Q)e−i~Q·(~Rn+~un)

⟩=∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

⟨e i~Q·(~um−~un)

⟩The last term is a time average which can be simplified using theBaker-Hausdorff theorem,

⟨e ix⟩

= e−〈x2〉/2⟨

e i~Q·(~um−~un)

⟩=⟨e iQ(uQm−uQn)

⟩= e−〈Q

2(uQm−uQn)2〉/2

Lattice Vibrations

For a 1D lattice, we replace the position of the atom with itsinstantaneous position, ~Rn + ~un where ~un is the displacement from theequilibrium position, ~Rn.

Computing the intensity:

⟨∑m

⟩=∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

⟨e ix⟩

= e−〈x2〉/2⟨

e i~Q·(~um−~un)

⟩= e−〈Q

2(uQm−uQn)2〉/2

Lattice Vibrations

⟨∑m

⟩=∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

⟨e ix⟩

= e−〈x2〉/2⟨

e i~Q·(~um−~un)

⟩= e−〈Q

2(uQm−uQn)2〉/2

Lattice Vibrations

⟨∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

⟨e ix⟩

= e−〈x2〉/2⟨

e i~Q·(~um−~un)

⟩= e−〈Q

2(uQm−uQn)2〉/2

Lattice Vibrations

⟨∑m

⟩=∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

The last term is a time average which can be simplified using theBaker-Hausdorff theorem,

⟨e ix⟩

= e−〈x2〉/2⟨

e i~Q·(~um−~un)

⟩= e−〈Q

2(uQm−uQn)2〉/2

Lattice Vibrations

⟨∑m

⟩=∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

⟨e ix⟩

= e−〈x2〉/2

⟩= e−〈Q

2(uQm−uQn)2〉/2

Lattice Vibrations

⟨∑m

⟩=∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

⟨e ix⟩

= e−〈x2〉/2⟨

e i~Q·(~um−~un)

= e−〈Q2(uQm−uQn)2〉/2

Lattice Vibrations

⟨∑m

⟩=∑m

f (~Q)f ∗(~Q)e i~Q·(~Rm−~Rn)

⟨e ix⟩

= e−〈x2〉/2⟨

e i~Q·(~um−~un)

⟩= e−〈Q

2(uQm−uQn)2〉/2

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

Q〉eQ2〈uQmuQn〉 = e−MeQ

2〈uQmuQn〉

= e−M[1 + eQ

2〈uQmuQn〉 − 1]

Substituting into the expression for intensity

I =∑m

f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i

~Q·~Rn

The first term is just the elastic scattering from the lattice with the

addition of the term e−M = e−Q2〈u2

Q〉/2, called the Debye-Waller factor.

The second term is the Thermal Diffuse Scattering and actually increaseswith mean squared displacement.

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

Q〉eQ2〈uQmuQn〉

= e−MeQ2〈uQmuQn〉

= e−M[1 + eQ

I =∑m

~Q·~Rn

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

2〈uQmuQn〉

= e−M[1 + eQ

I =∑m

~Q·~Rn

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

2〈uQmuQn〉

= e−M[1 + eQ

I =∑m

~Q·~Rn

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

2〈uQmuQn〉

= e−M[1 + eQ

I =∑m

~Q·~Rn

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

2〈uQmuQn〉

= e−M[1 + eQ

I =∑m

~Q·~Rn

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

2〈uQmuQn〉

= e−M[1 + eQ

I =∑m

~Q·~Rn

Lattice Vibrations

⟨e iQ(uQm−uQn)

⟩= e−Q

2〈u2Qm〉/2e−Q

2〈u2Qn〉/2eQ

2〈uQmuQn〉

= e−Q2〈u2

2〈uQmuQn〉

= e−M[1 + eQ

I =∑m

~Q·~Rn

Thermal Diffuse Scattering

ITDS =∑m

~Q·~Rn

The TDS has a width deter-mined by the correlated dis-placement of atoms which ismuch broader than a Braggpeak.

These correlated motions arejust phonons.

A 0.5mm Si wafer illumi-nated by 28keV x-rays froman APS undulator were usedto measure the phonon dis-persion curves of silicon

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

ITDS =∑m

~Q·~Rn

ITDS =∑m

~Q·~Rn

ITDS =∑m

~Q·~Rn

ITDS =∑m

~Q·~Rn

ITDS =∑m

~Q·~Rn

ITDS =∑m

~Q·~Rn

Properties of the Debye-Waller Factor

For crystals with several differenttypes of atoms, we generalize theunit cell scattering factor.

B jT = 8π2〈u2

for isotropic atomic vibrations

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

fj(~Q)e−Mj e i~Q·~rj

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

In general, Debye-Waller factors can be anisotropic

B jT = 8π2〈u2

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

B jT = 8π2〈u2

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

B jT = 8π2〈u2

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

B jT = 8π2〈u2

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

B jT = 8π2〈u2

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

B jT = 8π2〈u2

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

B jT = 8π2〈u2

〈u2〉 = 〈u2x + u2

y + u2z 〉

= 3〈u2x 〉 = 3〈u2

F u.c. =∑j

2Q2〈u2

sin2 θ〈u2Qj〉

Mj = B jT

(sin θ

B isoT =

3〈u2〉

The Debye Model

The Debye model can be used tocompute BT by integrating a lin-ear phonon dispersion relation upto a cutoff frequency, ωD , calledthe Debye frequency.

BT is given as a function of theDebye temperature Θ.

BT =6h2

mAkBΘ

[φ(Θ/T )

]φ(x) =

∫ Θ/T

eξ − 1dξ

BT [A2] =

11492T[K]

AΘ2[K2]φ(Θ/T) +

AΘ[K]

The Debye Model

BT =6h2

mAkBΘ

[φ(Θ/T )

]φ(x) =

∫ Θ/T

eξ − 1dξ

BT [A2] =

11492T[K]

AΘ2[K2]φ(Θ/T) +

AΘ[K]

The Debye Model

BT =6h2

mAkBΘ

[φ(Θ/T )

φ(x) =1

∫ Θ/T

eξ − 1dξ

BT [A2] =

11492T[K]

AΘ2[K2]φ(Θ/T) +

AΘ[K]

The Debye Model

BT =6h2

mAkBΘ

[φ(Θ/T )

]φ(x) =

∫ Θ/T

eξ − 1dξ

BT [A2] =

11492T[K]

AΘ2[K2]φ(Θ/T) +

AΘ[K]

The Debye Model

BT =6h2

mAkBΘ

[φ(Θ/T )

]φ(x) =

∫ Θ/T

eξ − 1dξ

BT [A2] =

11492T[K]

AΘ2[K2]φ(Θ/T) +

AΘ[K]

Debye Temperatures

BT =11492T

AΘ2φ(Θ/T )

diamond is very stiff and Θdoes not vary much withtemperature

copper has a much lowerDebye temperature and awider variation of thermalfactor with temperature

A Θ B4.2 B77 B293

(K) (A2)

C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond

Debye Temperatures

BT =11492T

AΘ2φ(Θ/T )

A Θ B4.2 B77 B293

(K) (A2)

C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond

Debye Temperatures

BT =11492T

AΘ2φ(Θ/T )

A Θ B4.2 B77 B293

(K) (A2)

C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond

Debye Temperatures

BT =11492T

AΘ2φ(Θ/T )

A Θ B4.2 B77 B293

(K) (A2)

C∗ 12 2230 0.11 0.11 0.12Al 27 428 0.25 0.30 0.72Cu 63.5 343 0.13 0.17 0.47∗diamond

iitphys.iit.edu/~segre/phys570/13f/lecture_18.pdf · 2013. 10. 31. · today’s outline - october...

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