introduction to solid state physics chapter...

11Semiconductor Materials Lab. Hanyang University

Introduction to Solid State Physics

Chapter 2


• X선 회절• X-ray의 파장은 0.1~10Å 정도인 전자기파로 전자가 가속(감속)될 때 생기는 제동방사를 통해 생

기는 백색 X-ray

• 각 면에서 일어나는 반사는 정반사이지만 특별한 방향 q 에 대해서는 모든 면에서 반사된 파의위상이 같아져 강한 반사파가 된다.

• 한 격자면이 입사파를 전부 반사한다면 첫번째 면 만이 모든 파장의 파를 전부 반사

• 실제로는 각 면은 입사파의 10-3~10-5 정도를 반사 103~105개의 결정 면이 Bragg 반사파를 형성하는데 참여

• 중성자 회절• 중성자는 질량이 거의 양성자와 동일하고 전하를 갖지 않은 중성의 소립자

• 수 MeV 정도의 에너지를 갖음0.3~3Å

• 전자선 회절• 전자를 가속하여 얻어진 전자선도 de Broglie파(matter wave)로서 파동의 성격을 가짐으로써

그 회절을 이용하여 결정을 해석


X-Rays

•Evacuated tube

•Target

•Electrons

•Cathode

•X-ray

• 그림2.15 X-ray tube. 가속전압 V가 클수록전자가 빨라지고 X-선 파장은 짧아진다.

Cathode는 filament로 heating 되고 electron은thermionic emission에 의해 방출된다.

cathode(음극)와 anode(양극)사이에 큰 전압V를 인가하면 전자가 target쪽으로 가속하면 전자가 충돌.

X-Ray 발생.

(전자가 가속되는데 방해되지 않도록 tube는 진공)

In classical EM theory, Bremsstrahlung is predicted

when electrons are accelerated X-ray produced.

그러나 이론과 실제가 몇몇 일치하지 않는 것이 있음.

(그림 2.16, 2.17 참조) • 현대식 X-Ray tube


X-Rays

•Wavelength, nm

•Relative intensity

•0 •0.02

•0.04

•0.06

•0.08

•0.10

•2

•4

•6

•8

•10 •W

target

•30 kV

•50 kV

•20 kV

•40 kV

•Wavelength, nm

•Relative intensity

•0 •0.02

•0.04

•0.06

•0.08

•0.10

•2

•4

•6

•8

•10

•12

•Tungsten, 35kV

•molybdenum,35kV

• 그림2.16 몇 가속전압에서 텅스텐의 x-선 스펙트럼• 그림2.17 35kV의 가속전압에서 텅스텐과

몰리브덴의 x-선 스페트럼

• lmin (그림 2.16)이 있다는 것과

•특성 X-ray가 발생(그림 2.17)한다는 것은 기존의 고전론 이론으로 설명불가


•Real Space and Reciprocal Space

•A real space image is a map of the crystal structure

•A diffraction pattern is a map of reciprocal space

• reciprocal space ~ momentum space

•Real Space •Reciprocal Space


2. Reciprocal Lattice – Diffracton of waves by crystals

2.1 Diffraction of Waves by Crystals

1) Bragg Law : 2dsinθ=nλ - 결정 면으로부터 회절현상의 설명, 보강간섭의 조건- maximum λ=2d- 따라서 원자 크기 정도의 파장을 갖는 파동만이

Bragg scattering 을 일으킬 수 있다.결정구조 연구 : photons(x-ray), neutrons, electrons (입자의 파동성)

→simple explanation of the diffracted beams from a crystal

θθ

θ

dsinθ

d

2dsinθ =nλ

Constructive interference →

when the path difference is an integer

number of wavelengths λ

l must be less than 2d to see 1st order of diffraction.

)(24.1

nmkeVhc

Ell


Bragg law is a consequence of the periodicity of the lattice

Not refer to the composition of the basis of atoms associated with every lattice point.

Composition of the basis → determine the relative intensity of the various orders of diffraction (n)

2. Reciprocal Lattice – Diffracton of waves by crystals

Fig. 3 Sketch of a monochromator which by Bragg

reflection selects a narrow spectrum of x-ray or

neutron wavelengths from a broad spectrum incident

beam.


Si powder로부터얻은 x-선회절패턴- 각각의결정면간의간격에따라보강간섭이일어나는각도가달라진다.

Example of X-ray diffraction patterns of Si

Fig. 4. X-ray diffractometer recording of powdered silicon,

showing a counter recording of the diffracted beams. (courtesy of W. Parrish.)

2. Reciprocal Lattice – Diffraction of waves by crystals


2.2 Scattered wave amplitudeBragg 가 도입한 회절조건은 lattice points 에서산란되는파의

constructive interference를 정확히 설명

But we need a deeper analysis to determine the scattering intensity

from the basis of atoms

332211 auauauT

Any local physical property of the crystal is invariant under T

(전자수밀도(n(r)), 질량밀도, 자기모멘트밀도는모든 T에대해불변)

e number density n(r) is a periodic function of r with periods a1, a2, a3.

)()( rnTrn

Such periodicity creates an ideal situation for Fourier analysis

결정에서중요한것은전자밀도의 Fourier components와관련있다는것임.

(이같은주기성때문에 Fourier analysis를 역격자공간에적용가능)

2. Reciprocal Lattice – Scattered wave amplitude

1) Fourier Analysis Crystal = Lattice + Basis

d-spacings intensity


Fourier Analysis

Periodicity of the crystal results in a periodicity of electron density

n (r+T) = n (r)

Simplest case is for simple atoms in a line

n(x) n0 sin( 2a

x)

n(r)

electrondensity

a

8 Reciprocal

Such periodicity creates an ideal situation for Fourier analysis

결정에서중요한것은전자밀도의 Fourier components와관련있다는것임.

(이같은주기성때문에 Fourier analysis를 역격자공간에적용가능)

)()( rnTrn


일 차원(1-D) 에서 x 방향으로주기 “a”를 가지는함수 n(x)를 생각하자

n(x) in a Fourier series of “sin” & “cos”

0

)/2sin()/2cos()(p

PPo apxSapxCnxn

p : +ve integer

CP, SP : real constants, Fourier coefficients of the expansion

n(x) has a period of “a”

Fourier series

P

P apxinxn )/2exp()( → 1-Dimension (p~-ve, 0, +ve)


-(3)

n(x) n0 p

np exp[i2p

ax] → 1-Dimension (p>0)

전자수밀도 (n(x))


Factor 2π/a in the arguments ensures that n(x) has the period “a”

)2/2sin()2/2cos()( 0 papxSpapxCnaxn PP

)()/2sin()/2cos(0 xnapxSapxCn PP -(4)

2πp/a 를그결정의 reciprocal lattice 공간에있는한점 or Fourier 공간에있는한점이라한다.

(1-D에서는이점은 line (선)상에있게된다. 역격자점은 Fourier series에서 allowed terms 이다)

A term is allowed if it is consistent with the periodicity of the crystal as in Fig. 5,

other points in the reciprocal space are Not allowed in the Fourier expansion


•Figure 5. A periodic function n(x) of period a,

and the terms 2πp/a that may appear in the

Fourier transform

•n(x) = ∑ np exp(i2πpx/a) the magnitudes of

the individual terms np are not plotted

Real Space

Reciprocal Space or Fourier space or Momentum space


Series(4) → P

P apxinxn )/2exp()( → (5) 1 dimension

n(x) 가실수가되기위해 nn PP

*

→ (6)

)(*

복소공액의는nn PP

φ = 2πpx/a 이면식5의 –p항과 p항의합은

)sin(cos)sin(cos ii nn PP

sin)(cos)( nnnn PpPPi

→ (7)

식(6)이성립하면

sinIm2cosRe2 nn PP 와같아진다.=>n(x) is a real function


P: +ve, -ve, and zero

np is complex numbers

여기서 n(x) 가실수가되기위해서는


Inversion of Fourier Series

식(5)의 Fourier coefficient nP 가

a

P apxixndxan0

1 )/2exp()( -(10)

Substitute (5) in (10) to obtain

a

P

PP axppidxnan0'

'

1 /)'(2exp -(11)

p' = p 항에대해 exp(i 0)=1 이므로적분값은 a가됨

PPP nanan 1성립


If p'≠p the value of integral is

01)'(2

)'(2

ppieppi

a

P

P apxinxn )/2exp()( 1-D

성립안됨가)( 0n0)a(nan PP

1

P

-(5)


G

G riGnrn )exp()( 3-D Fourier series -(9) ))(exp(),,( G

zyxG zGyGxGinzyxn

3-D

주기 함수에 대한 Fourier analysis 를 3차원 function n(r) 에 대해 확장하려면, 결정을 변하지 않게 하는 모든 결정 translation vector T 에 대해 식(9) 가 불변인 벡터 G의 집합을 찾아야 한다.

결정에서 탄성 산란된 X-선의 진폭이 전자밀도의 Fourier계수 nG로 부터 결정된다.

같은 방법으로 식(9)의 inversion은

cell

1

CG r)iGexp(n(r)dVn V

VC= vol. of cell of the crystal

Electron density in real space is n(r)

Scattering amplitude in reciprocal space is nG이것이 역격자공간에서 회절점 즉 F


3D 역격자 공간의 Scattering Amplitude F와 관련


Electron Density in 1-D

The Fourier series relates the electron density in real space to the

scattering amplitudes in reciprocal space.

Mathematically we can invert this equation to get the scattering

amplitudes as a function of the real space electron density.

Real Space Reciprocal Space

(k space, momentum space, Fourier space)

Real SpaceReciprocal

Space

a

P apxixndxan0

1 )/2exp()(

n(x) n0 p

np exp[i2p

ax]


Electron Density in 3-D

The Fourier series relates the electron density in real space to the

scattering amplitudes in reciprocal space.

Mathematically we can invert this equation to get the scattering

amplitudes as a function of the real space electron density.

n(r) n0 G

nGeiGr

Real Space

nG 1V

dV n(r)eiGr

cell

Real SpaceReciprocal

Space

Reciprocal Space

(k space, momentum space, Fourier space)


Scattering amplitude in reciprocal space is nG

3D 역격자 공간의 Scattering Amplitude F와 관련


Real Space <--> Reciprocal Space

If we know nG from a

diffraction experiment we can

calculate the electron density

in real space. Tells us what

kind of atoms and where they

are.

This is what happens when

we identify an unknown

crystal structure with XRD.

n(r) n0 G

nGeiGr

nG 1V

dV n(r)eiGr

cell

If we know n(r) for a crystal

structure we can calculate the

scattering amplitudes nG in

reciprocal space. This tells us how a

wave (photons, phonons, electrons)

will interact with the crystal.

This is what happens when we

calculate the electronic band

structure of a crystal.




2) Reciprocal lattice Vectors

전자밀도(연속함수)의 Fourier해석을위해식(9)로부터

→Must find the vectors of Fourier Sum G

)exp( riGnG

This procedure forms the THEORETICAL basis of Solid State Physics.

Axis vectors 321 ,, bbb of the reciprocal lattice

321

321 2

aaa

aab

321

132 2

aaa

aab

321

213 2

aaa

aab

-(13)

321 ,, aaa

Primitive vectors of the crystal lattice

321 ,, bbb

Primitive vectors of the reciprocal lattice (Fourier 공간의격자)


각 면의 법선의 길이를 면간 거리의 역수에 비례하도록 잡으면, 그 법선의 끝은 격자배열을 형성한다.이 배열이 결정의 역격자임.

Real space에서는 lattice point가 lattice translation vector(T)로연결되어있지만Reciprocal space에서는역격자 points는 reciprocal translation vector(G)로연결되어있다.

Each point (hkl) in the reciprocal lattice corresponds to a set of lattice planes (hkl) in the real space lattice. The direction of the reciprocal latticevector corresponds to the normal to the real space planes. The magnitude of the reciprocal lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes.

The reciprocal lattice of a lattice (usually a Bravais lattice) is the lattice in which the Fourier transform of the spatial wavefunction of the original lattice (or direct lattice) is represented. This space is also known as momentum space

The Brillouin zone is a weigner seitz cell of the reciprocal lattice.

Reciprocal lattice --- Real lattice(hkl) point – (hkl) plane

Direction — real plane의 수직Magnitude(1/nm)--- 면간 거리의 역수

http://en.wikipedia.org/wiki/Real_space

http://en.wikipedia.org/wiki/Normal_(mathematics)

http://en.wikipedia.org/wiki/Reciprocal_length

http://en.wikipedia.org/wiki/Bravais_lattice

http://en.wikipedia.org/wiki/Fourier_transform

http://en.wikipedia.org/wiki/Momentum_space

http://en.wikipedia.org/wiki/Brillouin_zone


각 면의 법선의 길이를 면간 거리의 역수에 비례하도록 잡으면, 그 법선의 끝은 격자배열을 형성한다.이 배열이 결정의 역격자임.

1/a

This extends directly to 3-D with each point (kx, ky, kz) representing a wave

Each k is a point that represents a wave that constructively interferes.

x,t Asin2n

ax t

k=0 2/a 4/a-2/a-4/a

a

1/aa

Real SpaceReciprocal Space, k-space

when

k n2

a

Reciprocal Space (k-space)

1/a

This extends directly to 3-D with each point (kx, ky, kz) representing a wave

a

Reciprocal lattice represents the distances and

normal directions between planes of atoms in

real space. Miller Indices can be used to

identify the points in reciprocal space.

(100)

b

(010)

(110)

1/b

Real Space

Reciprocal Space

(100)

(110)(010)

(220)

Reciprocal lattice --- Real lattice

(hkl) point – (hkl) plane

Direction — real plane의수직Magnitude(1/nm)--- 면간거리의역수


the real space lattice vectors ai and reciprocal lattice vectors bi are connected by:

G = v1b1 + v2b2 + v3b3

b1 2a2 a3

a1 a2 a3

b2 2a3 a1

a1 a2 a3

b3 2a1 a2

a1 a2 a3

T = n1a1 + n2a2 + n3a3

a1

a2

T

Real Space

b1

b2

G

Reciprocal Space


jiji ab 2jiif

ji 1

jiifji 0

-(14)

Reciprocal lattice vectors332211 bvbvbvG

-(15)

Every crystal structure has two lattices

①. crystal lattice

②. reciprocal lattice

결정의 diffraction pattern 은 reciprocal lattice 의 map이다

현미경상 image는 real space에서 crystal structure의 map이다

Two lattices 간에는 식 13의 관계가 성립

If rotate the crystal → rotate both the direct lattice & the reciprocal lattice

Direct lattice →dimension of [length]

Reciprocal lattice → dimension of [1/length]

Reciprocal lattice 는 Fourier 공간에 있는 격자

파수 vector (wave vector), k, 는 항상 Fourier 공간에서 그릴 수 있다

Fourier공간에 있는 각 position 은 description of a wave를 뜻하게 된다.

G로 정의되는 점은 특히 “중요한 의미”



Fourier series에서 는 reciprocal lattice vector 이므로, 전자밀도에대한 Fourier 급수는

(crystal translation vector)에대해불변

G

T

G

G TGirGinTrn )exp()exp()(

-(16)

1]exp[ TGi

을증명하기위해 (14)를이용

)]()(exp[]exp[ 332211332211 auauaubvbvbviTGi

)](2exp[ 332211 uvuvuvi

332211 uvuvuv 는정수

)()( rnTrn

-(17)


G

G riGnrn )exp()(


n(r) n0 G

nGeiGr

G

G riGnrn )exp()(

n(x) n0 p

np exp[i2p

ax]

P

P apxinxn )/2exp()(

nG 1V

dV n(r)eiGr

cell

Real Space Reciprocal SpaceReal SpaceReciprocal

Space

3-D Fourier series

1-D Fourier series

If we know nG from a diffraction experiment we can calculate the electron density in real space(n(r)).

If we know n(r) for a crystal structure we can calculate the scattering amplitudes nG in reciprocal

space.

cell

aP xa

pixndxn )exp()(

21




Face Centered Cubic (FCC)

[FCC의역격자가 BCC의결정격자와같음]



Body Centered Cubic (BCC)


FCC Reciprocal lattice

Real Space primative vectors

Reciprocal Space vectors

bi 2a j ak

ai a j ak


6 Reciprocal

BCC Reciprocal lattice

Real Space primative vectors

Reciprocal Space vectors

bi 2a j ak

ai a j ak


6 Reciprocal

FCC Reciprocal lattice

Brillouin Zone for the FCC crystal structure is bounded by the bisectors to

the 8 nearest neighbor points (hexagonal faces) and 6 next nearest

neighbors (square faces). The symmetry of the reciprocal lattice is BCC.

6 Reciprocal

BCC Reciprocal lattice

Brillouin Zone for the BCC crystal structure is bounded by the bisectors

to the 12 nearest neighbor points (rhombohedral faces). The symmetry of

the reciprocal lattice is FCC.

6 Reciprocal


3) Diffraction Conditions

Theorem : reciprocal lattice vector 가 X 선의반사를결정G

그림6을보면 r 만큼떨어져있는 volume

element로부터탄성산란된파동은위상차때문에아래위상인자 만큼 다르다:

한 volume element로 부터 산란되는파의 진폭은 그 vol. element가 있는위치의 전자밀도(n(r)(=local econcentration))에 비례한다

The total amplitude of scattered wave in

the direction of is proportional to

))'(exp( rkki

'k

))'(exp()( rkkidVrn

- Fig 6


Phase factor

n(r)dV: Crystal vol.

nG 1V

dV n(r)eiGr

cell


The amplitude of electric or magnetic field vectors in the scattered electromagnetic

wave is proportional to

그림6을보면 r 만큼떨어져있는 volume element로부터탄성산란된파동은위상차때문에아래위상인자 만큼 다르다:

한 volume element로부터 산란되는 파의 진폭은그 vol. element가있는위치의 전자밀도(n(r)(=local e concentration))에 비례한다

The total amplitude of scattered wave in the direction of is proportional to

))'(exp( rkki

'k

))'(exp()( rkkidVrn

)exp()())'(exp()( rkirdVnrkkirdVnF -(18)

When k+Δk=k’ -(19)Scattering amplitude

Δk : change in wavevector (=scattering vector)


Crystal vol. Phase factor

Phase factor

n(r)dV: Crystal vol.

nG 1V

dV n(r)eiGr

cell


Introduce the Fourier components (9) of n(r) to obtain the scattering amplitude

G

G rkGindVF ])(exp[ -(20)

If the scattering vector (Δk) = particular reciprocal lattice vector (G)

Δk=G -(21) (Δk = 파동벡터의 변화, scattering factor)

(20) 식의 지수함수가 1이 되고 20식은 F=VnG 가 된다.

(Δk 가 G와 같지 않다면 F is negligibly SMALL(F=0), 회절조건이 아님)

2. Reciprocal Lattice

k'

k

Δk

G

G riGnrn )exp()(-(9)

)exp()( rkirdVnF




탄성산란 (elastic scattering) 에서 모멘텀 (ħω)은 보존된다. (X-ray는 탄성산란, AES, XPS등은 비탄성산란)

따라서 산란파의 진동수 ω’=ck’ 는 입사파의 진동수 ω=ck 와 같아진다.

∴ lkl=lk’l and k2=k’2 to find or Gk

'kGk

02') 2222 GGkorkkGk( -(22). If is a reciprocal lattice vector, so is –

(ignore –ve sign)G

G

22 GGk -(23) condition for diffraction (= Bragg condition)

2. Reciprocal Lattice-(9)


G= h b1 + k b2 + l b3 방향에수직인 parallel lattice plane

→ d(hkl) = 2π/ |G|

2 k· = G2 maybe written → 2(2π/λ)sinθ = 2π/d(hkl)

or 2d(hkl)sinθ = λ (hkl maybe contain common factor n)

2dsinθ=nλ -(24)

Laue equation white X-ray (power 법: 단색 X-ray, powder)

Single Xtal (rotating Xtal 법)

Δk =G (회절이론)은 Laue 방정식으로도나타낼수있다

Take the scalar product of both Δk ≠ with a1, a2, a3

from 14&15

G

G

33 2 vπka 22 2 vπka 11 2 vπka -(25)


이식은회절조건을만족시키는

scattering factor (Δk)를어떻게

결정하는지알려준다.

Wave-Particle Duality

To understand diffraction at a more fundamental level (and therefore get more

information from diffraction experiments) we will see that diffraction can also

be analyzed in terms of the wave properties of the x-ray.

Particle Wave

Needed to determine the

intensity of diffraction

Visualize the direction

of diffraction

dVeikr

eik’r

k

k’

k

8 Reciprocal

What does this mean physically?

k = G

Incoming

Wave

Momentum

Out going

Wave

Momentum

Recoil

Momentum

of the Crystal

What really happens is the wave comes in and its momentum is transfered

to the lattice (vibrating electrons) that then turns the momentum into the out

going wave. The crystal can only allow certain momenta given by the

points in the reciprocal lattice. Note the mass of the crystal is very large so

the recoil is very small (this is called radiation pressure).

out

in

recoil

7 Reciprocal

Why We Call It Reciprocal Space

Atoms

a

Wave

l

x,t Asin kxt

k 2

l

a nl

But a wave can only go through the crystal if it constructively interferes

Therefore transmitted wave depends on the reciprocal of the lattice spacing

x,t Asin2n

ax t

k 2n

a

8 Reciprocal


Overview of Scattering from Crystals

8 Reciprocal

dV

eikr

eik’r

F dV n(r)eikrei k r

electrondensity

incomingwave

outgoingwave

Last lecture we saw how the particle view can be used to find the scattering directions k’. But we need to use the wave view to find the scattering intensities F for each direction. We know two things:

1) the scattering intensity must be proportional to the electron density n(r)

2) momentum of incoming and outgoing waves must be conserved.

We calculate the amount of scattering dF from a small part of the crystal dV and then integrate over the entire crystal to get the total scattering F.





8 Reciprocal

dV

eikr

eik’r

F dV n(r)eikrei k r

electrondensity

incomingwave

outgoingwave

FG N dV n(r)eiGr

cell

NSG

The problem becomes easier when we realize that the crystal is build up from identical unit cells. Therefore, we can calculate the scattering from a single unit cell then multiple by the number of unit cells in the crystal.



8 Reciprocal

Now you can see why the properties of waves in the crystal depend primarily on the structure (symmetry) and the composition is only (an important) detail.

FG N dV n(r)eiGr

cell

NSG

SG f jeiGrj

j

Now each atom and bond type scatter differently, so we need factor in the strength of scattering fj from each individual atom in the unit cell.


There are several different forms of the diffraction condition. These all

apply for the elastic (energy conserved) scattering of any type of wave.

These forms are derived in Kittel.

2. k = G

3. 2k G = G2

a1 k = 2v1 a3 k = 2v3a2 k = 2v2

4. Laue Equations

5. Brillouin Equation

k (1/2G) = (1/2G)2

visualized by the Brillouin Zones

1. Bragg’s Law 2dsin = nl

The Brillouin equation shows us exactly why the Brillouin Zone is so very important. Any wave-vector k that terminates on the Brillouin zone constructively interferes in the crystal.

Brillouin Zones

1/2 G

k (1/2G) = (1/2G)21/2 G

k

k (1/2G) = (1/2G)2

1/2 G

k

k (1/2G) = (1/2G)2

Wave-Particle Duality

PHY 407 Solid State Physics

Now remember that x-ray has a wavelength (wave property) and a momentum

(particle property). This wave-particle duality means that we can view the

scattering from either the wave view or the particle view. Today we will examine

the particle view and then next lecture study the wave view. Each gives us a

different perspective on the same problem.

dVeikr

eik’r

7 Reciprocal

Wave View Particle View

k’

k

k-k’

Diffraction Condition


k

k’

k

Momentum conservation

What is the vector k?

In the particle view we see the incoming x-ray photon as having a momentum

given by hk. Of course during diffraction momentum must be conserved.

k - k’ = k

We will prove next class that k = G, the

reciprocal lattice vectors

But we could expect this result. The reciprocal

lattice is the set of all wave vectors that can diffract

in a crystal. Therefore to diffract, both k and k’

wave vectors must end on a reciprocal lattice point.

Since G are simply the vectors connecting

reciprocal lattice points we expect k = G

k’

k G

7 Reciprocal

What does this mean physically?

k = G


Incoming

Wave

Momentum

Out going

Wave

Momentum

Recoil

Momentum

of the Crystal

What really happens is the wave comes in and its momentum is transfered

to the lattice (vibrating electrons) that then turns the momentum into the out

going wave. The crystal can only allow certain momenta given by the

points in the reciprocal lattice. Note the mass of the crystal is very large so

the recoil is very small (this is called radiation pressure).

out

in

recoil

7 Reciprocal

Diffraction Conditions

The fundamental diffraction condition is: k = G


Bragg’s Law 2dsin = nl

We can then use this to derive several other forms of the diffraction condition.

They are all equivalent so we can choose the form that simplifies the math.

k’

k G

k1/2 G

sin G

2

k

Remember G, k defined as

G 2n

d

k 2

l

2dsin = nl

Doing the algebra gives

7 Reciprocal

Diffraction Conditions


a1 k = 2h a3 k = 2la2 k = 2k

Laue Equations

Brillouin Equation

k (1/2G) = (1/2)G2


where (h,k,l) are the Miller Indicies for the diffracting plane

and the ai are the real space lattice vectors

k’

useful in single crystal diffraction (especially TEM)

k

k’y

ky

k = G

k (1/2G) = (1/2G)2

dot both sides with 1/2 G

2k (1/2G) = (1/2)G2

* |k| = |k’| because energy is

conserved in x-ray diffraction.

(elastic scattering)

note* |k| = |ky| + |k’y| = 2k

7 Reciprocal

The Brillouin equation shows us exactly why the Brillouin Zone is so very

important. Any wave-vector k that terminates on the Brillouin zone

constructively interferes in the crystal.

Brillouin Zones

1/2 G


k (1/2G) = (1/2G)2

1/2 G

k

k (1/2G) = (1/2G)2

1/2 G

k

k (1/2G) = (1/2G)2

7 Reciprocal

Diffraction ConditionsAll these forms are equivalent and apply for the elastic (energy conserved)

scattering of any type of wave.


1. k = G

3. 2k G = G2

a1 k = 2h a3 k = 2la2 k = 2k

4. Laue Equations

5. Brillouin Equation

k (1/2G) = (1/2G)2


2. Bragg’s Law 2dsin = nl

7 Reciprocal

Connections


2

Int

BZ

(110)

(100)

(hk0)

Energy

(110) (hk0)(100)

Allowed Waves

Bands

Diffracted Waves

Band Gap

We’ll see in chapters 4-8 that knowing the BZ tells us which waves are allowed in the

crystal and which are in the energy gap (Bragg diffracted)

Knowing the crystal structure gives the BZ. Then for a given wave vector k magnitude

we can determine angles for Bragg diffraction. However, we need to use the wave

picture to find the intensities of the diffracted peaks (next lecture)

7 Reciprocal


11 2 vπka → tell us that Δk lies on a cone about the direction

→ tell us that Δk lies on a cone about the direction

→ tell us that Δk lies on a cone about the direction

1a

22 2 vπka 2a

33 2 vπka 3a

따라서반사가일어나기위해서는 Δk가 3개의방정식을모두만족해야함

즉 k가 3개의원추가동시에만나는교선위에있어야한다

매우어려운조건(결정방향과 X-선을조직적으로변화시켜야함)

이현상을이해하는데도움을주는작도법 Ewald 작도법

이작도법은 3차원에서회절조건을만족시키는조건을잘나타냄.



회전결정법 (단색 X-ray을사용하여단결정분석,

결정의격자형태격자상수결정에편리)

(단색 X-ray을사용하여다결정분말결정분석, 단결정이필요없은분석)

(연속X-ray을사용하여단결정분석, 결정의방위를결정하는데편리)


• Vector : 입사 x-ray방향, 역격자점에서끝남• k의 출발선을기준으로반경 k=2π/λ 인

구를그림• 이 구가역격자점과만날때회절선이생긴다.

Θ는 Bragg 각이고이것은 Ewald가 고안해냈음

Fig 8. visualize the nature of accident to satisfy the diffraction condition in 3-D

Reflection from a single plane of atoms takes place in the direction of the lines of intersection of two cones. (2-D의 경우)

- 3차원같이우연히일치될필요는없음

→ 2D의경우 low energy electron diffraction이 중요

역격자점

- Fig. 8 The points on the left-hand side are

reciprocal lattice points of the crystal


의미하는것공간에서역격자가kG


Ewald construction Equivalent to Bragg condition

2kGcos(90-)=G2 2ksin=G 2dsin=l


Ewald Construction

The Ewald construction helps us visualize the diffraction condition.

1. Chose a starting point in the lattice.

2. Draw a vector (AO) in the incident

x-ray direction of length 2/a

3. Make a circler = 2/a centered at A.

Every point intersected by the circle

satisfies the diffraction condition.

4. Draw a vector (AB) to the intersection,

this is the scattered wave k’.

5. Draw a vector (OB) to the intersection,

this is the translation vector G.

6. Draw a line (AE) perpendicular to (OB),

this gives the angle of the diffraction

(the angle in Bragg’s Law).

k = G


회절조건에관한표현법을 Brillouin이처음발표 (고체물리에서가장널리쓰이는회절조건을제시)

오늘날 energy band 이론과 elementary excitation 등을설명하는데사용

Brillouin Zone : defined as a Wigner-Seitz primitive cell in the reciprocal lattice

→ gives a vivid geometrical interpretation of the diffraction condition

22 GGk Divide both sides by 42)

2

1()

2

1( GGk

2.3 Brillouin Zones

1,2 평면은 GC와 GD를수직 2등분한선.

원점 O 에서 평면1에도달하는임의의 vector k1

은 회절조건을만족시킴 2

1 )2

1(

2

1CC GGk

수직이등분선 (1,2면)

→ forms a part of the zone boundary

→ X-ray beam will be diffracted

& diffracted beam will be in the direction Gk

1 2

Brillouin construction은 결정내에서 Bragg 반사가가능한 wavevector k를 나타낸다

Fig. 9a

2. Reciprocal Lattice – Brillouin Zones

Fig. 9a

Reciprocal

lattice points

near the point

O at the origin

of the

reciprocal

lattice


The set of planes (bisectors of the reciprocal lattice vectors) – very important

원점에서출발하여이들면에서끝나는 wave vector를 가지는 wave는모두회절조건을만족시킨다.

또한이들평면은 Fig 9b 와 같이 Fourier 공간을갈라놓는다 (divide)

→ central square → primitive cell of the reciprocal lattice

= Wigner Seitz cell of the reciprocal lattice

1st Brillouin Zone

Fig. 9b Square reciprocal lattice with reciprocal lattice vectors shown as fine black lines



1st Brillouin Zone → the smallest volume entirely enclosed by planes

that are the perpendicular bisectors of reciprocal lattice

Fig 10 → 1st Brillouin Zone of an oblique lattice in 2-D

Fig 11 → linear lattice in 1-D

Zone boundaries →

(historically B.Z. 은 X-ray에서쓰이지않았던 language이지만

그러나지금은 B.Z.은결정의전자 energy band 구조연구에 “아주중요”특히 1st B.Z.

ak

Linear crystal lattice

a

Reciprocal lattice

b

ak

ak

0 k→

Fig. 11 Crystal and reciprocal lattices in one dimension.Fig. 10 Construction of the first Brillouin

zone for an oblique lattice in two dimensions.



1) Reciprocal lattice to SC (Simple Cubic) lattice

Primitive translation vectors of SC

yaa ˆ2

zaa ˆ3

xaa ˆ1

zyx ˆ,ˆ,ˆ : orthogonal vectors (직교단위 vector)

Vol. of cell →3

321 aaaa

Primitive translation vectors of the reciprocal lattice

xab ˆ)/2(1

yab ˆ)/2(2

zab ˆ)/2(3

(27b)

(27a)

SC의 역격자는격자상수가 2/a인 SC이다.

Lattice constant of reciprocal lattice = 2/a

1st B.Z. 의 boundaries → planes normal to the 6 reciprocal lattice vectors

321 ,, bbb → At this midpoints

xab ˆ)/(2

11

yab ˆ)/(2

12

zab ˆ)/(2

13

6개평면의한변의길이 2π/a 체적3)/2( a

→ this cube is the 1st B.Z. of the SC crystal lattice



2) Reciprocal lattice to bcc lattice

Primitive translation vectors of bcc

Primitive translation of the reciprocal lattice

)ˆˆ)(2

(1 zya

b

)ˆˆ)(2

(2 zxa

b

)ˆˆ)(2

(3 yxa

b

)ˆˆˆ(2

12 zyxaa

)ˆˆˆ(2

13 zyxaa

)ˆˆˆ(2

11 zyxaa

-(28)

-(30)

FCC lattice is the reciprocal lattice of the BCC lattice

Fig.12 Primitive basis vectors of the

body-centered cubic lattice

The volume of primitive cell

3

3212

1aaaaV

-(29)



)ˆˆ)(/2( zxa )ˆˆ)(/2( zya

]ˆ)(ˆ)(ˆ))[(2

( 213132332211 zyxa

bbbG vvvvvvvvv

-(31)

가장짧은 G vector들은 12개의 vectors

)ˆˆ)(/2( yxa

Primitive cell of the reciprocal lattice → described by321 ,, bbb

Vol. →3

321 )/2(2 abbb

Fig 13 → regular rhombic dodecahedron (사방 12면체)

Solid state physics 에서는역격자의중앙에있는

Wigner Seitz cell을 1st B.Z. 으로택하는관례가있다

General reciprocal lattice (v1,v2,v3)

Fig. 13. First Brillouin zone of the

body-centered cubic lattice



3) Reciprocal lattice to fcc lattice

Primitive translation vector of the fcc (see Fig 14)

3

3214

1aaaaV

)ˆˆ(2

12 zxaa )ˆˆ(

2

13 yxaa )ˆˆ(

2

11 zyaa -(34)

Vol. of primitive cell

Fcc의 reciprocal lattice

)ˆˆˆ)(2

(1 zyxa

b

)ˆˆˆ)(2

(2 zyxa

b

)ˆˆˆ)(2

(3 zyxa

b

BCC lattice is reciprocal to the FCC lattice

Vol. of primitive cell of the reciprocal lattice is3)/2(4 a

Shortest G’s are 8 vectors )ˆˆˆ)(2

( zyxa

Fig. 14. Primitive basis

vectors of the face-centered

cubic lattice.



역격자중앙에있는 cell의 경계는이들 8개 vector을 수직 2등분함으로결정됨→이때생기는 8면체의꼭지는 6개의다른역격자 vector의 수직 2등분면에의해잘려진다

)ˆ2)(2

( ya

)ˆ2)(2

( za

)ˆ2)(2

( xa

)ˆ2)(2

( xa

1st B.Z. → Fig 15 (14면 존재, 꼭지가잘린팔면체)

14면중 4각형으로되어있는 6개면을연장하면

길이가 (4π/a)가되고 Vol. 이 된다.

는 역격자 vector중 하나임 )( 32 값이기때문bb

3)4(a

Fig. 15. Brillouin zones of the face-centered cubic lattice.

The cells are in reciprocal space, and the reciprocal lattice

is body centered.



2.4 Fourier Analysis of the basis

회절조건 (Δk=G)이만족되면 scattering amplitude는 식 18에 의해결정됨

Crystal이 N개의 cell 로 되어있을경우

cell

GG NSriGrdVnNF )exp()( -(39)

GS : structure factor

Defined as an integral over a single cell, with r=0 at one corner

한 cell내에 있는 원자로 부터 나온 총 산란된 파의 intensity

Electron concentration n(r) as the superposition of electron concentration

functions nj associated with each atom j of the cell

rj= vector to the center of atom j

nj(r-rj) → contribution of that atom to the electron concentration at r

2. Reciprocal Lattice – Fourier Analysis of the basis

)18()exp()())'(exp()( rkirdVnrkkirdVnF

Crystal = Lattice + Basis

d-spacings intensity


The total e concentration at r due to all atoms in the cell is the sum

s

j

jj rrnrn1

)()( -(40)

SG in equation 39 maybe written as integrals over the s atoms of a cell

j

jjG riGrrdVnS )exp()(

j

jj iGdVnriG )exp()()exp(

Atomic form factor (fj)원자에의존 (basis에같은원자혹은서로

다른원자의경우를다룸 Diamond, ZnS or GaAs등)

Where j

rr

Scattering power of the jth atom in

the unit cell


-(41)

구조인자(SG)는 cell 내 S개원자에대한적분

(Basis에관한것으로서Geometric SF는 basis에있는원자의위치에관한것Atomic form factor 은각원자의 scattering power, 방향에따른효율 )

Lattice


)exp()( iGdVnf jj

j

jjG riGfS )exp(

Define the atomic form factor(fj) as

-(42)

Combine (41) & (42) to obtain the structure factor of the basis

-(43)

)()( 321332211 azayaxbbbrG jjjj vvv

Usual form of this for atom j

321 azayaxr jjjj -(44)

)(2 321 jjj zyx vvv -(45)


)41()exp()()exp( j

jj iGdVnriG

Lattice Basis

SG eiGrj dV n j ()eiG

j

eiGrj

j

f j ()

LatticeBasis


(43) becomes

j

jjjjG zyxifS ](2exp[)( 321321 vvvvvv

S need not be real → S·S* is real

“SG 가 zero이면 G가역격자 vector라고해도산란강도는Zero가된다.”

만약우리가 cell을택할때 primitive cell을택하지않고 conventional cell을택한다면?

→Basis (단위구조)는변하지만 physical scattering 은변화가없다.

식 39에의해

N1(cell) X S1(basis) = N2(cell) X S2(basis)

-(46)


Primitive Cell (bcc) = Conventional Cell (bcc)

N X 1 = 1/2N X 2

basis Lattice


1) Structure Factor of the bcc lattice

bcc basis → identical atoms at (000) & )2

1

2

1

2

1(

Thus S(v1v2v3)=f(1+exp(-iπ(v1+v2+v3))

f: atomic form factor

S=0 whenever the exp has -1

S=0 when v1+v2+v3=odd integer

S=2f when v1+v2+v3=even integer

Metallic Sodium(Na) → bcc structure

→Diffraction patterns do not contain (100) (300) (111) or (221)….

But (200) (110) (222) will be present

Bragg 법칙이만족되는경우라도단위정의원자의특별한배열때문에회절이일어나지않는경우가존재


•Bragg law is a consequence of the periodicity of the lattice•Not refer to the composition of the basis of atoms associated with every lattice point.


What is the physical interpretation of the result that the (100) reflection vanishes?

π

π1st plane

2nd plane

3rd plane

Phase difference 2π

bcc에서 (100) 반사가일어나지않는이유는

인접면간의위상차가 π이므로두면에서

반사하는파의진폭은 0111 ie

(100) Reflection normally occurs (cubic cell 에서위상이 2π 다를때)

But in bcc, the intervening plane is equal in scattering power to the other planes

→canceling the contribution

→cancellation of the (100) reflection occurs in the bcc lattice

because the planes are identical in composition



2) Structure factor of the fcc lattice

Basis of the fcc structure →identical atoms at

식 46은

)2

1

2

10(),

2

10

2

1(),0

2

1

2

1(),000(

)])(exp[)](exp[)](exp[1(),,( 213132321 vvivvivvifvvvS

If all indices are even integers

If all indices are odd integers→ S=4f

If one of integer is even & two odd

If one odd & two even→ S=0


Scattering

Scattering intensity F is proportional to the electron density and the phase

difference between the incoming and outgoing wave.

F dV n(r)eikrei k r


k

k’

k

dVeikr

eik’r k = k - k’

F dV n(r)eikr

phase difference

momentum difference

electron

densityincoming

wave

outgoing

wave

8 Reciprocal

Intensity of Scattering

Remember we already linked n(r) to the scattering amplitude nG.

F dV n(r)eikr

F dV nGei(Gk)rG


dVeikr

eik’r

k = k - k’

n(r) n0 G

nGeiGr

What we just did there was moved from

real space n(r) to reciprocal space nG. We

want to see what the wave sees.

8 Reciprocal

Calculating this integral as a function of G, we find that the integral is

non-zero only when


Therefore, we only see diffraction when:

k = G

F dV nGei(Gk)rG

kG

VnG

F = VnG for k = G

F = 0 otherwise

k = G


F


8 Reciprocal

Fourier Analysis of the Basis

The diffraction condition tells us the direction (angle) of the

diffracted wave, but we still need to calculate the intensity of

the diffracted beam from the crystal FG.

FG N dV n(r)eiGr

cell

NSG

SG is called the structure factor.

It’s the total scattered intensity from

all the atoms in one cell.

PHY 407 Solid State Physics8 Reciprocal

Structure Factor

The structure factor SG is due to scattering from each

individual atom in a cell.

SG dV n(r)eiGr

cell

n(r) n j

j

(r rj )

SG dV n j (r rj )eiGr

cell

j

SG eiGrj dV n j ()eiG

j

eiGrj

j

f j ()


Lattice Basis

8 Reciprocal

Structure Factor

SG f jeiGrj

j

SG(v1,v2,v3) f jei2 (v1x j v2y j v3z j )

j

rj = xja1 + yja2 + zja3G = v1b1 + v2b2 + v3b3

Inserting the reciprocal vector G and the atomic position rj

Remember that we defined bi to be orthogonal to aj therefore


Simple Cubic Diffraction

For a simple cubic lattice we have 1

atom per cell located at the origin

rj = (0,0,0).

SG f jei2 (v1 0v2 0v3 0)

j

f

Naturally the scattering from a cell with one atom is equal to

the scattering from that atom SG = f.


BCC Structure Factor

For a body centered cubic lattice we have 2 identical atoms

per cell located at:

r1 = (0,0,0)

r2 = (1/2, 1/2, 1/2)

SG f jei2 (v1x j v2y j v3z j )

j

SG = f [1+exp(-i(v1+v2+v3))]

SG = 0 for (v1+v2+v3) = odd

SG = 2f for (v1+v2+v3) = even

In other words, some of the possible diffraction peaks are

missing from the XRD spectrum.


FCC Structure Factor

For a face centered cubic lattice we have 4 identical atoms

per cell located at:

r1 = (0,0,0)

r2 = (0, 1/2, 1/2)

r3 = (1/2, 0, 1/2)

r4 = (1/2, 1/2, 0)

SG f jei2 (v1x j v2y j v3z j )

j

SG = f [1+exp(-i(v2+v3))

+ exp(-i(v1+v3))

+ exp(-i(v1+v2))]

SG = 0 for (v1,v2,v3) all odd

or all even

SG = 4f for (v1,v2,v3) mixed

even and odd.


Scattering from Atoms

Now we need to calculate fj, the atomic form factor. This is where the

composition and bond types come into the problem.

fj is the scattering power of the jth atom

in the unit cell

f j dV n j ()eiG

The atomic form factor involves the number of electrons and their

distribution around the atom n(r) and also the wavelength and angle of

scattering of the out going radiation.

properties of the

electron density

in the atom

properties of the

x-ray waves



(1) 단순격자

단위정에원자 1개만이원점(000)에위치함

구조인자는 F = ƒe 2πi(h•0+k•0+l•0) = ƒe 2πi(0) = ƒ , 따라서 F2= ƒ2

이와같이 F2는 h, k, l에무관계이며모든반사에대하여모두같은값을갖는다

(2) 저심격자(C공간격자)

단위정에 2개의같은원자가 000, ½ ½ 0 에위치함

F = ƒe 2πi(0) + ƒe 2πi(h/2+k/2) = ƒ[1+ e πi(h+k)] , e nπi = (-1)n n:정수항상 (h+k)= 정수이므로 F 는실수h,k가비혼합지수일때 e πi(h+k)=1 이므로 F=2ƒ 따라서 F2=4ƒ2

h,k가혼합지수일때 e πi(h+k)=-1 이므로 F=0

어느경우라도 l 지수의값은구조인자에영향을주지않음즉, 반사가일어나지않는데이러한현상을 Extinction Rule(소멸칙)이라한다

●구조인자의계산


(3) 체심격자(I공간격자)단위정에 2개의같은원자가 000, ½ ½ ½ 에있음

F = ƒe 2πi(0) +ƒe 2πi(h/2+k/2+l/2) = ƒ[1+e πi(h+k+l)]

<h+k+l = 2n (n:정수)일때> F = 2ƒ , F2=4ƒ2

<h+k+l ≠ 2n 일때> F = 0

(4) 면심격자(F공간격자)

단위정에 4개의같은원자가 000, ½ ½ 0, ½ 0½ , ½ 0½ 에위치함

F = ƒe 2πi(0) +ƒe 2πi(h/2+k/2) +ƒe 2πi(h/2+l/2) +ƒe 2πi(l/2+h/2)

= ƒ[1+e πi(h+k)+ e πi(k+l) + e πi(l+h)]

<비혼합지수일때> F=4ƒ 따라서 F2=16ƒ2

<혼합지수일때> F=0

반사는 (111),(200),(220) 등에서일어나지만, (100),(110),(112)등에선일어나지않음즉, 구조인자는단위격자의모양이나크기에관계없으며다만인자의위치에만관계된다는것을알아야한다


(5)조밀육방정단위정에 2개의같은종류의원자가 000, 1/3 2/3 1/2에위치함

F = ƒe 2πi(0) +ƒe 2πi(h/3+2k/3+l/2)

= ƒ{1+e 2πi[(h+2k)/3+l/2]}

[※(h+2k)/3+l/2 값이분수값을가질수있으므로 F는 복소수로본다]

따라서 |F|2 = 4ƒ2 cos2π[(h+2k)/3+l/2]

h, k, l로잡을수있는모든값에대한정리는다음과같다

h+2k l F2

3n 홀수 0

3n 짝수 4ƒ2

3n±1 홀수 3ƒ2

3n±1 짝수 ƒ2 <n : 정수>


(6)NaCl 구조• 서로다른원자로되어있는화합물구조• 단위정에 4개의 Na 원자와 4개의 Cl원자가다음자리에위치함

Na : 000 , ½ ½ 0, 0½ ½ , ½ 0½

Cl : ½ ½ ½ , ½ 00, 0½ 0, 00½

• 각원자에대한원자산란인자가구조인자의식에들어가야한다

F = ƒNa[e 2πi(0) + e 2πi(h/2+k/2) + e 2πi(k/2+l/2) + e 2πi(l/2+h/2)]

+ ƒCl[e 2πi(h/2+k/2+l/2) + e 2πi(h/2) + e 2πi(k/2) + e 2πi(l/2)]

= ƒNa[1+ eπi(h+k) + e πi(k+l) + e πi(l+h)]

+ ƒCl eπi(h+k+l) [1+ e-πi(k+l) + e-πi(l+h) + e-πi(h+k)]

= [1+ eπi(h+k) + e πi(k+l) + e πi(l+h)][ƒNa + ƒCleπi(h+k+l)]=[lattice term][basis term]

• Na 원자와 Cl 원자의위치는각각면심병진(Face centering translation)에의하여관계되어지며즉, 첫째괄호가이면심병진에해당하고 NaCl이면심격자를가지고있다는것을나타낸다혼합지수일때 F = 0

비혼합지수일때 F = 4[ƒNa + ƒCleπi(h+k+l)]

h+k+l=2n일때 F2=16(ƒNa+ƒCl)2

h+k+l≠2n일때 F2=16(ƒNa -ƒCl)2


(7)ZnS (Zinc blend) 구조

• Zinc blend 형 ZnS는입방정이며, 격자상수는 5.41Å

• 단위정에 4개의 Zn원자와 4개의 S원자가각각,

Zn : ¼ ¼ ¼ + 면심병진S : 0 0 0 + 면심병진

• 혼합지수의면에대해 F = 0

비혼합지수일때 F = 4[ ƒS + ƒZnl(πi/2)(h+k+l) ] 이며, 공액복소수를곱하여

|F|2 = 16 [ ƒS + ƒZn(πi/2)(h+k+l) ] [ ƒS + ƒZne-(πi/2)(h+k+l) ]

|F|2 = 16 [ ƒS2 + ƒZn

2 + 2ƒSƒZncos(π/2)(h+k+l)]

(h+k+l)이홀수일때 |F|2 = 16 ( ƒS2 + ƒZn

2 )

(h+k+l)이 2의홀수배일때 |F|2 = 16 ( ƒS - ƒZn )2

(h+k+l)이 2의짝수배일때 |F|2 = 16 ( ƒS + ƒZn )2


(8)Si (Diamond)구조• 단위정에 8개의 Si원자가각각,

Si : ¼ ¼ ¼ + 면심병진Si : 0 0 0 + 면심병진

• 혼합지수의면에대해 F = 0

비혼합지수일때 F = 4[ ƒSi + ƒSie(πi/2)(h+k+l) ] = 4 ƒSi[ 1 + e(πi/2)(h+k+l) ] 이며, 공액복소수를곱하여

|F|2 = 16 ƒSi2[ 1 + e(πi/2)(h+k+l) ] [ 1+ e-(πi/2)(h+k+l) ]

<혼합지수일때> F=0

<비혼합지수일때> F=4ƒ 따라서 F2=16ƒ2 [ 1 + e(πi/2)(h+k+l) ] [ 1+ e-(πi/2)(h+k+l) ]

h+k+l=4m 일경우 |F|2 = 64 ƒSi2

h+k+l=4m+2 즉 2의홀수배일때 |F|2 = 16 ( ƒSi - ƒSi ) 2=0

h+k+l=4m+/-1 일경우 |F|2 = 16 ( ƒSi2+ ƒSi

2)=32 ƒSi2


Structure factor of the FCC lattice


All FCC structures

NaCl, ZnS, and Diamond structures

KCl and KBr structures

Fig. 17: FCC 구조를 하는 KCl과 KBr은 모두 홀수거나 짝수일때 회절peak가 나오지만 KCl의 경우 K 원자와 Cl 원자의 전자수가 같아 Simple Cubic lattice와 같은 회절을 한다.(scattering amplitude 가 전자수가 같아서 거의 같다 f(K+) = f(Cl-) Structure factor의 fK+ = fCl-)

FCC

FCC 이여야 하지만 Simple Cubic과 같이 보임.

S(v1v2v3)=S(fcc lattice) x S(basis)

S(fcc lattice)

= ƒe 2πi(0) +ƒe 2πi(h/2+k/2) +ƒe 2πi(h/2+l/2) +ƒe 2πi(l/2+h/2)

= ƒ[1+e πi(h+k)+ e πi(k+l) + e πi(l+h)]

S(basis)= [ƒNa + ƒCleπi(h+k+l)] , [ ƒS + ƒZnl(πi/2)(h+k+l) ],

[ ƒSi + ƒSie(πi/2)(h+k+l) ]

h+k+l=2n일 때 F2=16(ƒK+ƒCl)2

h+k+l≠2n일 때 F2=16(ƒK -ƒCl)2= 0 즉 h+k+l이 2n일때만회절이된다는것은 Simple Cubic과같은회절


3) Atomic Form Factor (f) This is where the composition and bond types come into the problem.

fj; is a measure of the scattering power of the jth atom in the unit cell

(Unit cell 내 j번째 원자의 scattering power의 척도, 같은 원자와 다른 원자 차이 즉DC의 경우 같은 원자가 격자점에 있고 ZnS의 경우 다른 원자가 격자점에 있음)

→ involve the number and distribution of atomic electrons and the wavelength and angle of

scattering of the radiation

한 원자로부터 scattered radiation은 원자내의 interference effects를 고려해야 함

(원자에 의해 어떤 방향으로 산란될 경우의 효율을 나타냄)

)exp()( riGrdVnf jj

With the integral extended over the electron concentration associated with a single atom

r make an angle α with G G·r=Grcosα


-(49)

properties of the x-ray waves

properties of the electron densityin the atom


)cosexp()()(cos2 2 iGrrndrdrf jj

After integration over d(cosα) between -1 & 1

iGr

eernrdr

iGriGr

j

)(2 2

Zrrndrf jj

2)(4

If the same total e density were concentrated at r=0

Only Gr=0 lim sinGr/Gr = 1

Gr

Grrrndrf jj

sin)(4 2

fi 는원자가가진전자의총수(Z)와같아진다

# of atomic electrons


-(50)

-(51)


Scattering by an atom

X-ray beam 이원자와만나면…

→ 각전자들은 Thomson equation 에따라 X-ray beam 을 coherent 하게 scatter 한다.

Thomson 에의하면, 원자핵(양자)도 + 의전하를띄고있으니까 oscillation 을해야하나 e 의질량에비해원자핵의

질량이크기때문에진동을거의하지않는다.

→ 따라서원자의전자만이 coherent scattering 에관여한다.

2

1

mI

※ 그러면원자번호가 Z 인 atom의경우전자 1개에의한 amplitude의 Z 배가되는가?

만약 2θ = 0이면그렇지만, scattering 방향이다르면그렇지않다.

Space에서 atom속의 electron 들은 space에서서로다른위치에놓여 crystal 에서와같음(규칙성은없다.)

1

2AD

C B

2θ = 0인경우두 ray 는 in-phase에있어전자가 Z 개인경우amplitude는 1개전자에 의한것에 Z 배가된다.

만약 2θ = 0이아닌각도로 scatter 되면,

CB-AD 만큼의 path difference 가생긴다. 이차이는 one wavelength 보다작다.

따라서, ray ①과 ray ②사이에서는 partial interference가생긴다.

→ ※ 그래서똑바로나간 beam (2θ = 0)의 amplitude의보다다른방향으로 scattered 된 beam 의 amplitude 는항상작다.


pointaatlocalizedeonebyscatteredaptituderadiation

atomaninondistributieactualbyscatteredaptituderadiationofratiof

G=0인 경우 fj=Z 가 다시 된다

X선 회절에서 관측되는 고체속의 전자분포는 자유원자의 전자분포와 거의 같다.

Fig.18 : fcc결정의 반사가 되는 면을 표시 (부분적으로 홀,짝 No reflection)


Fig. 18 Absolute experimental atomic scattering

factors for metallic aluminum

f가 sin/l에비례하므로어떤원자에서도전방으로산란될때는( =0) f=Z가되며 가커질수록개개의전자에의하여산란된파는위상이맞아지지않는경우가커진다.

또 가일정하더라도 l가짧아지면행로차가파장에비해커지므로산란 X선사이의간섭이커진다.

λ가짧아지고, 2θ값이커질수록, atomic scattering factor는작아진다.


2.5 Quasicrystals

First observed in 1984, cannot be indexed

to any Bravais lattice

→have symmetries intermediate between

a crystal and a liquid

Al-Mn(14%) Icosahedron(정20면체)

Small Mn atoms are each surrounded by 12 Al atoms

Arranged at the corners of an icosahedron

Quasicrystals are intermetallic alloys & very poor

electrical conductors & nearly insulators with band

well known band gap

Fig. 19. A quasicrystal tiling in two

dimensions, after the work of Penrose.

2. Reciprocal Lattice - Quasicrystals


Quasicrystal : intermetallic alloy, poor electrical

conductors nearly insulator

Great interest intellectually in expanding the

definition of crystal lattice

Fig. 20 → computer generated diffraction pattern

with 5 fold symmetry

Fig. 20. Photograph of the calculated Fourier transform (diffraction pattern) of an

icosahedral quasicrystal along one of the fivefold axes, illustrating the fivefold symmetry

2. Reciprocal Lattice - Quasicrystals

introduction to solid state physics chapter...

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