5/15/2015 1 solid state physics 2. x-ray diffraction
TRANSCRIPT
![Page 1: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/1.jpg)
04/18/231
Solid State Physics
2. X-ray Diffraction
![Page 2: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/2.jpg)
Diffraction
04/18/232
![Page 3: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/3.jpg)
Diffraction
04/18/233
1 2 3sin , , , ...m
mW
![Page 4: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/4.jpg)
Diffraction
04/18/234
![Page 5: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/5.jpg)
04/18/235
Diffraction using Light
http://physics.kenyon.edu/coolphys/FranklinMiller/protected/Diffdouble.html
One Slit
Two Slits
Diffraction Grating
d
m sin
![Page 6: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/6.jpg)
Diffraction
04/18/236
The diffraction pattern formed by an opaque disk consists of a small bright spot in the center of the dark shadow, circular bright fringes within the shadow, and concentric bright and dark fringes surrounding the shadow.
![Page 7: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/7.jpg)
04/18/237
Diffraction for CrystalsPhotonsElectronsNeutrons
Diffraction techniques exploit the scattering of radiation from large numbers of sites. We will concentrate on scattering from atoms, groups of atoms and molecules, mainly in crystals.
There are various diffraction techniques currently employed which result in diffraction patterns. These patterns are records of the diffracted beams produced.
![Page 8: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/8.jpg)
04/18/238
What is This Diffraction?
![Page 9: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/9.jpg)
04/18/239
Bragg Law
nd sin2
William Lawrence
Bragg1980 - 1971
![Page 10: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/10.jpg)
04/18/2310
Mo 0.07 nmCu 0.15 nmCo 0.18 nmCr 0.23 nm
![Page 11: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/11.jpg)
04/18/2311
Monochromatic Radiation
![Page 12: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/12.jpg)
04/18/2312
Diffractometer
![Page 13: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/13.jpg)
04/18/2313
![Page 14: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/14.jpg)
04/18/2314
Nuts and Bolts
The Bragg law gives us something easy to use,To determine the relationship between diffractionAngle and planar spacing (which we already knowIs related to the Miller indices).
But…We need a deeper analysis to determine theScattering intensity from a basis of atoms.
![Page 15: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/15.jpg)
04/18/2315
Reciprocal Lattices Simple Cubic Lattice
1 2 3ˆ ˆ ˆa x a y a za a a
1 2 32 2 2
ˆ ˆ ˆx y z
******************************************G G G
a a a
The reciprocal lattice is itself a simple cubic lattice with lattice constant 2/a.
![Page 16: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/16.jpg)
04/18/2316
BCC Lattice1 1
1 22 2
13 2
ˆ ˆ ˆ ˆ ˆ ˆa ( x y z) a (x y z)
ˆ ˆ ˆ a (x y z)
a a
a
1 2 32 2 2
ˆ ˆ ˆ ˆ ˆ ˆy z x z x y
******************************************G G G
a a a
The reciprocal lattice is represented by the primitive vectors of an FCC lattice.
Reciprocal Lattices
310 1 2 3 2a a a a
![Page 17: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/17.jpg)
04/18/2317
FCC Lattice
1 11 22 2
13 2
ˆ ˆ ˆ ˆ ˆ ˆ( x y z) (x y z)
ˆ ˆ ˆ (x y z)
****************************
**************G a G a
G a
1 2 3
2 2 2ˆ ˆ ˆ ˆ ˆ ˆa y z a x z a x y
a a a
The reciprocal lattice is represented by the primitive vectors of an BCC lattice.
Reciprocal Lattices
30 1 2 3a a a a
![Page 18: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/18.jpg)
04/18/2318
Drawing Brillouin ZonesWigner–Seitz cell
The BZ is the fundamental unit cell in the space defined by reciprocal lattice vectors.
![Page 19: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/19.jpg)
04/18/2319
Drawing Brillouin Zones
![Page 20: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/20.jpg)
04/18/2320
Back to Diffraction
Diffraction is related to the electron density.Therefore, we have a...
The set of reciprocal lattice vectors determines the possible x-ray reflections.
![Page 21: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/21.jpg)
04/18/2321
The difference in path length of the of the incident wave at the points O and r is sinrThe difference in phase angle is rk
sin2
r
For the diffracted wave, the phase difference is k r ****************************
So, the total difference in phase angle is r)kk(
![Page 22: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/22.jpg)
04/18/2322
Diffraction Conditions Since the amplitude of the wave scattered
from a volume element is proportional to the local electron density, the total amplitude in the direction k is
dVen
dVenf
i
i
r )k(
r kk
)r(
)r(
kkk
![Page 23: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/23.jpg)
04/18/2323
Diffraction Conditions When we introduce the Fourier
components for the electron density as before, we get
( k) r i ss
s
f n e dV
ks Constructive
Interference
![Page 24: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/24.jpg)
04/18/2324
Diffraction Conditions
kkk
k ks
ks
nd sin2
2 2
2
(k )
or 2 k
s k
s s
![Page 25: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/25.jpg)
04/18/2325
r
cell ( r ) i s
sF N n e dV NS
Diffraction Conditions For a crystal of N cells, we can write down
)rr()r(1
j
s
jjnn
![Page 26: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/26.jpg)
04/18/2326
r
cell j
r
( r r )
( )j
i s
s jj
i s i s
jj
S n e dV
e n e dV
Diffraction Conditions The structure factor can now be written as
integrals over s atoms of a cell.
( )
i s
j jf n e dV
Atomic formfactor
![Page 27: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/27.jpg)
04/18/2327
Diffraction Conditions Let
Then, for an given h k l reflection
1 2 3a a aj j j jr x y z
1 2 3 1 2 3r ( a a a ) ( a a a )j j j j
j j j
s h k l x y z
hx ky lz
2 j j ji hx ky lz
s jj
S f e
![Page 28: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/28.jpg)
04/18/2328
Diffraction Conditions For a BCC lattice, the basis has identical
atoms at and
The structure factor for this basis is
S is zero when the exponential is i × (odd integer) and S = 2f when h + k + l is even.
So, the diffraction pattern will not contain lines for (100), (300), (111), or (221).
)0,0,0(),,( 111 zyx ),,(),,( 21
21
21
222 zyx
)1( 2 lkhiG efS
![Page 29: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/29.jpg)
04/18/2329
![Page 30: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/30.jpg)
04/18/2330
Diffraction Conditions For an FCC lattice, the basis has identical
atoms at
The structure factor for this basis is
S = 4f when hkl are all even or all odd. S = 0 when one of hkl is either even or
odd.
0 and ,0 ,0 ,000 21
21
21
21
21
21
)1( khilhilkiG eeefS
![Page 31: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/31.jpg)
04/18/2331
![Page 32: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/32.jpg)
04/18/2332
Structure Determination
222
lkh
ad
Simple
Cubic
2222
2
4sin lkh
a
When combined with the Bragg law:
![Page 33: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/33.jpg)
04/18/2333
(degrees) sin2 ratios hkl
11.44 0.0394 1 100
16.28 0.0786 2 110
20.13 0.1184 3 111
23.38 0.1575 4 200
26.33 0.1967 5 210
29.07 0.2361 6 211
34.14 0.3151 8 220
36.53 0.3543 9 300, 221
38.88 0.3940 10 310
X-ray powder pattern determined using Cu K radiation, = 1.542 Å
![Page 34: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/34.jpg)
04/18/2334
Structure Determination (310)
angstroms 88.3
104
)5420.1(3940.0
4sin
2
2
2222
2
aa
lkha
![Page 35: 5/15/2015 1 Solid State Physics 2. X-ray Diffraction](https://reader035.vdocuments.net/reader035/viewer/2022062221/56649cef5503460f949bd405/html5/thumbnails/35.jpg)
04/18/2335