today’s outline - november 04, 2013csrri.iit.edu/~segre/phys570/13f/lecture_20.pdf · today’s...

Today’s Outline - November 04, 2013

• Bragg & Laue Geometries

• Reflection for a Single Layer

• Kinematical Approach for Many Layers

• Darwin Curve

• Dynamical Diffraction Theory

Homework Assignment #5:Chapter 5: 1, 3, 7, 9, 10due Monday, November 11, 2013

C. Segre (IIT) PHYS 570 - Fall 2013 November 04, 2013 1 / 17

• Darwin Curve

Lattice Vibrations Review

Recall that the inclusion of atomic vibration resulted in a scatteredintensity with two terms.

I =∑m

f (~Q)e−Me i~Q·~Rm f ∗(~Q)e−Me−i

~Q·~Rn

2〈uQmuQn〉 − 1]

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.

This second term is what we will focus on now.

I =∑m

~Q·~Rn

I =∑m

~Q·~Rn

I =∑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

Bragg & Laue Geometries

symmetric

asymmetric

symmetric

asymmetric

symmetric

asymmetric

symmetric

asymmetric

symmetric

asymmetric

symmetric

asymmetric

Scattering Geometry

Consider symmetric Bragg geometry

We expect the crystal to diffract in anenergy bandwidth defined by ∆k

This defines a spread of scattering vec-tors such that

ζ =∆Q

called the relative energy or wavelengthbandwidth

Q=mG(1+ζ)

Scattering Geometry

ζ =∆Q

Q=mG(1+ζ)

k k’

Scattering Geometry

ζ =∆Q

Q=mG(1+ζ)

k k’

Scattering Geometry

ζ =∆Q

Q=mG(1+ζ)

Dynamical Diffraction - Darwin Approach

The Darwin approach treats a perfectcrystal as an infinite stack of atomicplanes. This is fundamentally equivalentto the Ewald and von Laue approaches.

For a single thin slab of density ρ andthickness d � λ, the reflected and trans-mitted waves are

g =λroρd

sin θ

for a layer of unit cells ρ = |F |/vc and

S = −igT(1− igo)T ≈ e−igoT

g =[2d sin θ/m]ro(|F |/vc)d

sin θ=

2d2rovc|F |

g =λroρd

sin θ

sin θ=

2d2rovc|F |

g =λroρd

sin θ

sin θ=

2d2rovc|F |

g =λroρd

sin θ

S = −igT

(1− igo)T ≈ e−igoT

sin θ=

2d2rovc|F |

g =λroρd

sin θ

sin θ=

2d2rovc|F |

g =λroρd

sin θ

sin θ=

2d2rovc|F |

g =λroρd

sin θ

sin θ=

2d2rovc|F |

g =λroρd

sin θ

2d2rovc|F |

g =λroρd

sin θ

sin θ=

2d2rovc|F |

since vc ∼ d3 then g ∼ ro/d ≈ 10−5

the transmitted beam depends on

go =λρat f

0(0)ro∆

sin θ

which can be rewritten

go =|Fo ||F |

where Fo is the forward scattering factorat Q = θ = 0

2d2rovc|F |

go =λρat f

0(0)ro∆

sin θ

go =|Fo ||F |

2d2rovc|F |

go =λρat f

0(0)ro∆

sin θ

go =|Fo ||F |

2d2rovc|F |

go =λρat f

0(0)ro∆

sin θ

go =|Fo ||F |

2d2rovc|F |

go =λρat f

0(0)ro∆

sin θ

go =|Fo ||F |

Kinematical Reflection

If we now extend this model to N layers we can use this kinematicalapproximation if Ng � 1.

Proceed by adding reflectivity from each layer with the usual phase factor

rN(Q) = −igN−1∑j=0

e iQdje−igo je−igo j = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)where the x-rays pass through each layertwice

these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d we are interested in small de-viations from the Bragg condition:

ζ =∆Q

=∆λ

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

ζ =∆Q

=∆λ

rN(Q) = −igN−1∑j=0

e iQdje−igo je−igo j

= −igN−1∑j=0

e i(Qd−2go)j

ζ =∆Q

=∆λ

rN(Q) = −igN−1∑j=0

e iQdje−igo je−igo j

= −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

where the x-rays pass through each layertwice

ζ =∆Q

=∆λ

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

ζ =∆Q

=∆λ

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

these N unit cell layers will give a recip-rocal lattice with points at multiples ofG = 2π/d

we are interested in small de-viations from the Bragg condition:

ζ =∆Q

=∆λ

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

ζ =∆Q

=∆λ

Multiple Layer Reflection

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

The term in the phase factor now be-comes

Qd − 2go = mG (1 + ζ)2π

G− 2go

= 2π(m + mζ − goπ

rN(Q) = −igN−1∑j=0

e i2π(m+mζ−go/π

= −igN−1∑j=0

e i2πmje i2π(mζ−go/π)

= −igN−1∑j=0

1 · e i2π(mζ−go/π)

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

Qd − 2go

= mG (1 + ζ)2π

G− 2go

rN(Q) = −igN−1∑j=0

= −igN−1∑j=0

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

Qd − 2go = mG (1 + ζ)2π

G− 2go

rN(Q) = −igN−1∑j=0

= −igN−1∑j=0

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

Qd − 2go = mG (1 + ζ)2π

G− 2go

rN(Q) = −igN−1∑j=0

= −igN−1∑j=0

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

Qd − 2go = mG (1 + ζ)2π

G− 2go

rN(Q) = −igN−1∑j=0

= −igN−1∑j=0

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

Qd − 2go = mG (1 + ζ)2π

G− 2go

rN(Q) = −igN−1∑j=0

= −igN−1∑j=0

rN(Q) = −igN−1∑j=0

e i(Qd−2go)j

Q=mG(1+ζ)

Qd − 2go = mG (1 + ζ)2π

G− 2go

rN(Q) = −igN−1∑j=0

= −igN−1∑j=0

This geometric series can besummed as usual

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

[sin(πN[mζ − ζo ])

sin(π[mζ − ζo ])

]|rN(ζ)|2 → g2

2(π[mζ − ζo ])2

This describes a shift of the Bragg peak away from the reciprocal latticepoint, the maximum being at ζ = ζo/m

The kinematical approach now breaks down and we need to develop a newtheory for dynamical diffraction.

First, let’s explore how the intensity would vary using the kinematicalexpression

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

|rN(ζ)|2 → g2

2(π[mζ − ζo ])2

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

|rN(ζ)|2 → g2

2(π[mζ − ζo ])2

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

|rN(ζ)|2 → g2

2(π[mζ − ζo ])2

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

|rN(ζ)|2 → g2

2(π[mζ − ζo ])2

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

|rN(ζ)|2 → g2

2(π[mζ − ζo ])2

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

]|rN(ζ)|2 → g2

2 sin2(π[mζ − ζo ])

ζo =goπ

=2d2|Fo |πmvc

rN(Q) = −igN−1∑j=0

e i2π(mζ−go/π)

|rN(ζ)| =

]|rN(ζ)|2 → g2

2(π[mζ − ζo ])2

today’s outline - november 04, 2013csrri.iit.edu/~segre/phys570/13f/lecture_20.pdf · today’s...

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