ix. x-ray diffraction
DESCRIPTION
IX. X-ray diffraction. 9-1. Production of X-ray. Vacuum, thermionic emission, high voltage, and a target. http://www.arpansa.gov.au/radiationprotection/basics/xrays.cfm. Energetic electron hitting a target. Braking radiation. Characteristic X-ray. Auger electrons. - PowerPoint PPT PresentationTRANSCRIPT
IX. X-ray diffraction9-1. Production of X-ray
http://www.arpansa.gov.au/radiationprotection/basics/xrays.cfm
Vacuum, thermionic emission, high voltage,and a target
Braking radiation CharacteristicX-ray
Auger electrons
Energetic electron hitting a target
Target
v0
v1
1121
20 /2/2/ hchmvmvE
v2
2222
21 /2/2/ hchmvmvE
(i) Braking radiation:
I
Vemv 2/20
swhchmvE /2/ max20
V1
V2
V2 > V1
Short wavelength limits
Excitationsource K
L1
L2
L3
K
L1
L2
L3
CharacteristicsX-Ray photon
k
L1
L2
L3
Augerelectron
Radiativetransition
NonradiativetransitionM} {
K K2 K1
1S
2S2P
3S
The figure in Note is wrong!
(ii) characteristic radiation
a typical X-ray spectrum
(iii) Cu K radiationCu K1 =1.54050 ÅCu K2 =1.54434 Å
Cu K =1.5418 Å
High angle line
Low angle line
Why we can resolve K1 and K2 doublelines at high angle?
sin2 hkld dddhkl cos2
tancos
sin22cos2 hkld
dd
tandd dtan
9-2. X-ray diffraction(i) Laue method: variable, white radiation, fixed(ii) Diffractometer method: fixed, characteristic radiation, variable(iii) Powder method: fixed, characteristic radiation, variable(iv) Rotating crystal method: fixed, characteristic radiation, variable
9-2-1. Laue method
(1) Laue conditionVon Laue derived the “ Laue conditions “ in1912 to express the necessary conditions fordiffraction.Assume that are three crystal lattice vectors in a crystal
cba , ,
)()( ****' clbkahaGaSSa hkl
hSSa )( '
Similarly, kSSb )( '
lSSc )( '
Very often , the Laue conditions are expressed as
hSSa )ˆˆ( ' kSSb )ˆˆ( '
lSSc )ˆˆ( '
/ˆ '' SS
/SS
Unit vector
The three Laue conditions must be satisfiedsimultaneously for diffraction to occur. The physical meaning of the 3 Laue conditionsare illustrated below.
1st Laue conditions
The path differencebetween the wavesequals to
)ˆˆ( ' SSaBDAC
ABa
The criterion for diffraction to occur ishSSa )ˆˆ( ' hSSa )( '
orinteger
For 1-dimensional crystal
Cones of diffracted beams for different h
Stereographic projection representation for 1-D crystalsThe projection of diffracted beams for h = 0 is a great circle if the incident beam direction is perpendicular, i.e.
aS
The projection of diffracted beams for h = 0 is a small circle if the incident beam direction is not perpendicular
2nd Laue condition
kSSb )( '
integer
For a two-dimensional crystal
The 1st and 2nd conditions are simultaneously satisfied only along lines of intersection of cones.
Stereographic projection for 2-D crystals
The lines of intersection of cones are labeled as (h , k)
3rd Laue condition
lSSc )( ' l: integer
For a single crystal and a monochromatic wavelength, it is usually no diffraction to occur
(2) Laue photographLaue photograph is performed at “ single crystal and variable wavelength ”.
(2-1) Ewald sphere construction
*'hklGSS
*'
2 hklGkk
or
white radiation : wavelength is continuoue;lwl and swl: the longest and shortest wavelength in the white radiation.
Two Ewald spheres of radius OA and OB form
swl
1OB
lwl
1OA
in between two spherical surface meets the diffraction condition since the wavelength is continuous in white radiation.
*hklG
http://www.xtal.iqfr.csic.es/Cristalografia/archivos_06/laue1.jpg
(2-2) zone axisThe planes belong to a zone
We define several planes belong to a zone [uvw], their plane normal [hi ki li] are perpendicular to the zone axis, In other words,
0][][ iii lkhuvw 0 iii wlvkuh
In the Laue experiments, all the planes meet the diffraction criterion. Each plane meets the 3 Laue conditions.
ihSSa )ˆˆ( '
ikSSb )ˆˆ( '
ilSSc )ˆˆ( '
0)ˆˆ()( ' wlvkuhSScwbvau iii
0 iii wlvkuh
0)ˆˆ()( ' SScwbvau
for all the plane belonged to the zone [uvw].
If we define cwbvauOA
The path difference between the waves equals to
0)ˆˆ( ' SSOA
(A) for the condition < 45o
For all the planes (hikili) in the same zone [uvw]0 wlvkuh iii
0)ˆˆ()( ' wlvkuhSScwbvau iii
Therefore, all the diffracted beam directions Ŝ1', Ŝ2', Ŝ3', …Ŝi' are on the same cone surface.(B) for the condition > 45o
The same as well
9-2-2 Diffractometer method
Instrumentation at MSE, NTHU
Shimatsu XRD 6000
Rigaku TTR
Bruker D2 phaser
9-2-2-1 -2 scan
If a new material is deposited on Si(100) and put on the substrate holder in the diffractometer set-up, is always parallel to the Si(100) surface normal.
kk'
Any sample
1k
2k
'1k'
2k
1
2
1'
1 kk
2'2 kk
Rotate the incident beam at until meets the diffraction condition at
*hklG
122
Ewald sphere construction
1
21
*111 lkhG
Si (100) substrate
S
'S
o
*111 lkhG
This is equivalent to at least one crystal with(h1k1l1) plane normal in parallel with theSi(100) surface normal.
When the incident beam is rotated (rotatingthe Ewald sphere or rotating the reciprocallattice), will meet the diffractioncondition at
*222 lkhG
222
1
22
*111 lkhG
Si (100) substrate
S
'S
o
*222 lkhG
*222 lkhG
2
This is equivalent to at least one crystal with(h2k2l2) plane normal in parallel with theSi(100) surface normal.
If only one plane of the film is observed in the diffractometer measurement, the film is epitaxial or textured on Si(100).
(b) XRD spectrum of AlN deposited on Si(100)
20 30 40 50 600
5000
10000
15000
20000
25000
30000
AlN(101)AlN(100)
AlN(002)
RFpower=180wI=1.2(A)AC pluse=350(v)
Inte
nsity
(arb
.uni
t)
2 £c(degree)
A large number of grains with AlN(002) surface normal in parallel to Si(100) surface normal, but small amount of grains of AlN(100) and AlN(101) are also in parallel to Si(100) surface normal.
9-2-2-2 X-ray rocking curveX-ray rocking is usually used to characterize the crystal quality of an epitaxial or textured film.At a certain diffraction condition
*'111
2 lkhGkk
the incident and outgoing directions of X-ray are fixed and the sample is rotated by scan.
Careful scan around a specific diffractionpeak!
When the crystal is rotated by d, this is equivalentto the detector being moved by the same d. (?) The FWHM of the diffracted peak truly reflects thewidth of each reciprocal lattice point, i.e. theshape effect of a crystal. (by high resolution!)
2
S
'S
o
*111 lkhG
2(+)
o
*hklG
9-2-2-3. ScanThe scan is used to characterize the quality of the epitaxial film around its plane normal by rotating (0-2)
Example: the crystal quality of highly oriented grains each of which is a columnar structure shown below
Each lattice point in the reciprocal lattice will be expanded into an orthorhombic volume according to the shape effect. The scan can be used to confirm the orientation of the columnar structure relative to the substrate.
Shape effect
grain
( )Reciprocal
lattice points
2
S
'S
*hklG
2
S
'S
*hklG
2
S
'S
*hklG
0 360
9-2-4 Powder method (Debye-Scherrer Camera)
http://www.adias-uae.com/plaster.html
http://www.stanford.edu/group/glam/xlab/MatSci162_172/LectureNotes/06_Geometry,%20Detectors.pdf
(a) Ewald sphere constructionReciprocal lattice of a polycrystallinesample
B.D. Cullity
Reciprocal lattice become a series of concentricspherical shells with as their radius*
hklG
Powder can be treated as a very larger number of polycrystals with grain orientation in a random distribution
Intersection of Ewaldsphere with sphericalreciprocal lattice!
forms a cone resulting from the intersection between Ewald sphere and the spherical reciprocal lattice.
'S
The intersection between the concentric spherical shells of in radius and the Ewald sphere of radius are a series of circles (recorded as lines in films)
*hklG
/12/ k
ExampleDebye-Scherrer powder pattern of Cu made with Cu K radiation
1 2 3 4 5 6 7 8line hkl h2+k2+l2 sin2 sin (o) sin/(Å-1) fCu
12345678
111200220311222400331420
3481112161920
0.13650.18200.3640.5000.5460.7280.8650.910
0.3690.4270.6030.7070.7390.8530.9300.954
21.725.337.145.047.658.568.472.6
0.240.270.390.460.480.550.600.62
22.120.916.814.814.212.511.511.1
1 9 10 11 12 13 14
line |F|2 P Relative integrated intensityCalc.(x105) Calc. Obs.
12345678
78106990452035003230250021201970
86
122486
2424
12.038.503.702.832.743.184.816.15
7.52 3.562.012.380.710.482.452.91
10.04.72.73.20.90.63.33.9
VsSssmwss
111200220311222400331420
cossin2cos1
2
2
(a) Structure factor (Cu: Fm-3m, a = 3.615 Å)
GNSF s
j
rGijG
jefS
2
Four atoms at [000], [½ ½ 0], [½ 0 ½],[0 ½ ½]
s
j
lwkvhuijG
jjjefS )(2
)021
21(2)000(2 lkhilkhi fefe
)21
210(2)
210
21(2 lkhilkhi
fefe
)1( )()()( lkilhikhi eeef
fSG 40GS
If h, k, l are unmixedIf h, k, l are mixed
222 16|| fNF
(b) sin2 hkld
alkh
dhkl
222
22sin
)(4
sin 2222
22 lkh
a
Small is associated with small h2+k2+l2.
2
222
2
1a
lkhdhkl
2
222
2
1a
lkhdhkl
3111ad
542.1sin3
615.32542.1sin2 111111111 d
24.0542.13694.0sin 111
o111111 68.213694.0sin
136.0sin 1112
Example: 111
0 0.1 0.2 0.3 0.429 27.19 23.63 19.90 16.48
sin
1.2263.2390.19
2.03.090.19
24.03.0 111111
CuCu
ff
7814)4( 21112111 CufF
}111{ p = 8 (23 = 8)
05.12cossin2cos1
1111112
1112
753370I
Cuf
(c) Multiplicity P is the one counted in the point group stereogram.
In Cubic (h k l){hkl}{hhl}{0kl}{0kk}{hhh}{00l}
P = 48P = 24P = 24P = 12P = 8P = 6
(3x2x23)(3x23)(3x2x22)(3x22)(23)(3x2)
How about tetragonal?
(d) Lorentz-polarization factor
cossin2cos1
2
2
(i) polarization factorThe scattered beam depends on the angle ofscattering. Thomson equation
2
202
22
420
0 sinsin4
rKI
rmeII
I0: intensity of the incident beam
27
0 CmKg104 K: constant
: angle between the scattering direction and the direction of acceleration of the electron
r
P
e
incident beam travel in directionxRandom polarized beam yEy ˆ zEz ˆ
zEyEE zy ˆˆ
220
00
222 IIIEEE zyzy
The intensity at point Py component = yOP = /2
20 rKII yPy
z component
2cos220 r
KII zPz
= zOP = /2 -2
2cos22020 r
KIrKIIII zyPzPyP
Polarization factor
22cos1 2
20
rKIIP
(ii) Lorentz factor
2sin
1cos2sin
1
(ii-1) factor due to grain orientation or crystal rotation2sin
1
The factor is counted in powder method and in rotating crystal method.
First, the integratedintensity )2( Id
BIId max)2(
(a) sin
1max I
Crystals with reflection planes make an angleB with the incident beam Bragg conditionmet intensity diffracted in the direction 2B.But, some other crystals are still diffracted inthis direction when the angle of incidentdiffers slightly from B!
B
B
2
1
2B
Inte
nsity
Diffraction Angle 2
Imax
Imax/2
2
IntegratedIntensity
B
2 1 2
A Ba
2
C D
Na
1N
B2
1
2
1
path difference for 11-22= AD – CB = acos2 - acos1
= a[cos(B-) - cos (B+)]= 2asin()sinB ~ 2a sinB.
2Na sinB = completelycancellation (1- N/2, 2- (N/2+1) …)
BNa sin2
Maximum angular range of the peak
(b) cos
1B
the width B increases as the thickness of the crystal decreases , as shown in the figure in the next page
t = m
d hkl
+
+
… …012
m
Constructive Interference
Bhkl
Bhkl
Bhkl
dd
d
sin)3(23sin)2(22
sin2
+ + .
);sin(2 Bhkld );sin()2(2)(2 Bhkld
);sin()3(2)(3 Bhkld …
Destructive interference:extra path difference (plane 0 and plane m/2): /2
);sin()2/(2))(2/( Bhkldmm 2/)2/( m
BBB cossincossin)sin( << 1s cos ~ 1 and sin ~ .
);cos)(sin2/(22/)2/( BBhkldmm
BhklBhkl mddmm cos2/sin)2/(2)2/( = t
BtB
cos2
Broadening (B): FWHMThickness dependent (just like slits)
2sin1
cos1
sin1
max BI
(iii-1) :factor due to the number of crystal counted in the powder method
cos
The number of crystal is not constant for aparticular B even through the crystal are oriented completely at random.
For the hkl reflection, the range of angle near the Bragg angle, over which reflection is appreciable, is . Assuming that N is the number of crystals located in the circular band of width r.
24
)2
sin(2
r
rr
NN B
2cos B
NN
# of grains satisfying the diffraction condition cosB. more grains satisfying the diffraction condition at higher B.
(iii-2) factor due to the segment factor in the Debye-Scherrer film
The film receives a greater proportion of a diffraction cone when the reflection is in the forward or backward direction than it does near 2/2
The relative intensity per unit length is proportional to
BR 2sin21
i.e. proportional toB2sin
1
Therefore, the Lorentz factor for the powder method is
cossin41
2sin1cos
2sin1
2
and for the rotating crystal method and the diffractometer method is
2sin1
(iv) Plot of polarization and Lorentz factors
0 20 40 60 80 100 120 140 160 180 2000
20
40
60
80
100
Lore
ntz-
Pola
rizat
ion
Fact
or
2 (Degrees)
cossin2cos1
2
2