(日本东北大学固体物理学课件)Structure of Solids

2. Structure ttice and reciprocal lattice Localization of X-ray analysis
Structural factor and atomic form factor
SPring 8
Solid state physics (SAITO)
Plot relation between intensity and ２θ of a material which consists of two elements (NaCl, GaAs). We can assume that atomic form factor is proportional to atomic number.
Body-centered lattice doesn’t have(1,0,0) , (1,1,1) spots
Problem : Derive rule of structure factor of face-centered lattice. In order to find phase transition point (temperature) which has structural phase transition from facecentered lattice to body-centered one, which spot should we focus? Plot the intensity vs 2θ.
square lattice
body-centered lattice
Solid state physics (SAITO)
2
Reciprocal lattice x0 x1 x 2 xn1 x n
Wave for lattice constant a ~eika (:integer)
Invariant for parallel k k 2
Problem : Derive rule of structure factor of diamond lattice. Which spot should we
observe to distinguish from face-centered lattice. Plot graph of intensity vs 2θ.
1
Crystal lattice (translation symmetry：space group)
Crystal：Solid (atoms place periodically) Unit cell：Minimum unit of periodicity
Solution : independent of how to set unit cell →high symmetry is better
X-ray
sample
k out
(various crystal orientations) – Change wave length of X-ray
(emitted light, SOR)
Measurement of Δk , presumption of n
→ determine G(h,k,l)
10
Structure factor
k in kout
θ
θ k
Body-centered lattice doesn’t have (1,0,0) spot.
Scattering amplitude of X-ray (Born approximation)
： n(r) Potential of the crystal
i 1 1
j 1
Ri : lattice vector
fi
:
Atomic
m
form
factor
NSG
SG fj expiG {rj}
j1
S Solid state physics (SAITO)
G:
structure
factor
11
Structure factor calculation m SG fj expiG {rj} j1
7
Calculation of wave length of particle beam
de Broglie wave length
h h
m, v mass
velocity
mv p
Energy
hc
Emv2 h2 h 2 2m
1
2
E
2mE
h 6 . 6 10 34 [J ・s] m e 9 . 1 10 31 [kg] m n 1 . 68 10 2 7 [ kg] e 1 . 602 10 19 [C] c 2 . 9979 10 8 [ m/s]
k in
k out
k2ksin
θ θ k kGn
G(h,k,l)
Intensity of X-ray
film
→ symmetry of crystal, lattice constant
Laue’s condition kGnis satisfied
– Use powder samples
k in
Diamond GaAs NaCl
Solid state physics (SAITO)
CsCl hcp wurtzite
12
Atomic form factor fjnj(r)ex iG pr()d r
Integrate by polar coordinate
f j nj (r)exp(iGr)dr
This condition is common when n=1
(h,k,l) Miller's indices
G h b 1 k b 2 lb 3 (h ,k ,l) h , k , l Negative number
Solid state physics (SAITO)
9
X-ray structure analysis
Body-centered lattice：Two atoms(A,B) in a unit cell
rj (0,0,0),(1 2,1 2,1 2)
fAfBf
a
G SG hfb (1 1 e k b i2 (h k llb ))3 20f
hkl odd hkleven
Body-centered lattice
– Primitive lattice vectors
a1 a (
3 , 1 ,0)
2
2
a2 a (
3 , 1 ,0) 22
b1
2
a1
a2 a3 (a2 a3)
Amplitude of bi →not 2π/a
ac ( 3 , 1 , 0) (0, 0,1)
2
22
a2c (
3 2
,
1 2
Energy λ
λ
λ
1eV 100eV 10keV
(electromagnetic (electron) (neutron)
wave)
1.24μm
12Å
0.28Å
124Å
1.2Å 0.028Å
1.24Å
0.12Å 0.0028Å
Solid state physics (SAITO)
electron microscope transmission (TEM)
Light (X ray) → X-ray diffraction
AC
d
Electron → Electron diffraction
B
Neutron → Neutron diffraction
h h
m v mass
velocity
mv p
de Broglie wave length
Solid state physics (SAITO)
2
Assumption of the structure and lattice constant → reproduce the intensity
intensity
(1,0,0)
Reproduction＝R-value
Debye-Scherrer method
(2,1,0) (1,0,1)
Solid state physics (SAITO) 2θ or Δ k
Unit cell in reciprocal lattice Brillouin zone
Solid state physics (SAITO)
3
ai
,
bi
of
non-orthogonal
latticeb1
2 a2a3
a1(a2a3)
ai bj 2ij
hexagonal lattice
Reciprocal lattice vectors
8
Laue’s condition kGn
Shift of wave number Δk and reciprocal lattice
k 2 k sin
k in kout
sin n
2d
θ
θ
k
2 k sin 2 n
d
G n
kGn |kin||kout|k2 Diffraction is happened when
a3
X-ray (particle beam) diffraction
Analysis of crystal structure
→ symmetry of crystals (space group)
Bragg condition 2dsinn sin n
2d