Post on 09-Apr-2016
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Lab 4 – Crystal Structure and
Diffraction Geometry
• Objectives Investigate the relationship between the geometry of
diffraction and the orientation of crystallographic planes in single crystals
Plot crystallographic planes and diffraction experiments in reciprocal space
• Experiment Diffraction of Si powder
Diffraction of single crystal Si (001)
Locate diffraction peaks from single crystal wafer
Copper X - ray source (Cu kα1)
40 KV, 30 mA for powder scan
40 KV, 30 mA for Si (001)
Powder Diffraction
The orientation of the powder is random
on the glass surface. Just by this random
distribution, any planes can end up such
that they are parallel to the surface. Thus
all possible reflections will show up in a
2θ-ω scan.
(110)
(200) (211)
Equations we already know
n λ = 2 dhkl sinθ
Bragg’s Law
D-Spacing Calculations
Si PDF
Single Crystal Diffraction
(004)
The orientation of the single crystal is
fixed on the surface. Thus only the
surface symmetric planes that are
possible reflections will show up in a
normal 2θ-ω scan.
10 20 30 40 50 60 70 80 90 100 11010
-2
10-1
100
101
102
103
104
105
106
107
Single Crystal Si
Powdered Si
Inte
nsity
(cp
s)
2 (degrees)
(004)
(220)
Powdered vs Single Crystal
Diffraction
Question: How can we
access other diffraction
peaks in a single
crystal?
(hkl) d (Å) φ G.E./G.I. 2Theta Omega Phi Intensity
111
111
220
202
202
113
113
004
331
331
224
224
115
115
333
333
404
404
???
What is the surface to plane angle for (110)?
(hkl) d (Å) φ G.E./G.I. 2Theta Omega Phi Intensity
111
111
220
202
202
113
113
004
331
331
224
224
115
115
333
333
404
404
Interplanar Angles
• Cubic
cos ϕ = (h1h2 + k1k2 + l1l2) / sqrt ((h12 + k1
2 + l12)(h2
2 + k22 + l2
2))
(004) and (110)
cos ϕ = ((0)(1)+(0)(1)+(4)(0))/ sqrt ((16)(2))
cos ϕ = 0
Φ = 90°
(hkl) d (Å) φ G.E./G.I. 2Theta Omega Phi Intensity
111
111
220
202
202
113
113
004
331
331
224
224
115
115
333
333
404
404
Is there a relationship between these three columns?
What is GI and GE?
Glancing Incidence
ω = θb - ϕ
Incident Diffracted
Glancing Exit
ω = θb + ϕ
Incident Diffracted
What difference can we expect
from G.E. and G.I.?
Sampling area
Outgoing intensity
Resolution
Reciprocal Space
• An easier way to represent diffraction
geometries, easier to visualize than real
space
Real space Reciprocal space
Reciprocal Space
• Interplanar spacing reciprocal value
• Angles preserved
• Crystal structure preserved
(001)
(110)
[001]
[110]
(000)
(001)
(002)
(003)
1 / d001
1 / d002
What is larger d001 or d002 ?
Reciprocal Space
[001]
[110]
What is larger d001 or d110 ?
Remember reciprocal space
is a 3d space, not all
reflections will be shown
here.
(220) (111) d111
d110
ϕ
Lab preparation
Si lattice parameter: 5.4309 Å
X-ray wavelength: 1.54056 Å
Single crystal Si wafer surface: (001)
Fill in blanks of the table on the handout.
Available diffraction spot:
h, k, l are all odd.
h, k, l are all even and h+k+l can be divided by 4.
(hkl) d (Å) φ G.E./G.I. 2Theta Omega Phi Intensity
111
111
220 1.92011 90 G.E. 47.302 113.651
202 1.92011 45 G.E. 47.302 68.651
202 1.92011 45 G.I. 47.302 -21.349
113
113
004 1.35773 0 Symmetric 69.129 34.564 Arbitrary
331
331
224
224
115
115
333
333
404
404
(hkl) d (Å) φ G.E./G.I. 2Theta Omega Phi Intensity
111 3.13553 54.736 G.E. 28.442 68.957 N/A
111 3.13553 54.736 G.I. 28.442 -40.515 N/A
220 1.92011 90.000 G.E. 47.302 113.651 N/A
202 1.92011 45.000 G.E. 47.302 68.651 N/A
202 1.92011 45.000 G.I. 47.302 -21.349 N/A
113 1.63748 25.239 G.E. 56.121 53.300 0
113 1.63748 25.239 G.I. 56.121 2.821 0
004 1.35773 0 Symmetric 69.129 34.564 Arbitrary
331 1.24593 76.737 G.E. 76.375 114.925 N/A
331 1.24593 76.737 G.I. 76.375 -38.550 N/A
224 1.10858 35.264 G.E. 88.028 79.279 0
224 1.10858 35.264 G.I. 88.028 8.750 0
115 1.04518 15.793 G.E 94.950 63.268 0
115 1.04518 15.793 G.I 94.950 31.682 0
333 1.04518 54.736 G.E. 94.950 102.211 N/A
333 1.04518 54.736 G.I 94.950 -7.260 N/A
404 0.95988 45.000 G.E. 106.706 98.353 45
404 0.95988 45.000 G.I. 106.706 8.353 45