electron diffraction patterns - tcd.ie · electron diffraction patterns • polycrystalline: ring...
TRANSCRIPT
Lecture 3
LRdhkl
Angles between g = Angles between the corresponding planes
• The diffraction planes are almost // the beam (angle~30’)
• Beam direction is approximately the zone axis
: The g is the plane normal )( plane hklghkl
hkl
hkld
g2
Diffraction Patterns (DPs): Lattices in the Intersecting plane of the Ewald Sphere and the reciprocal lattice - ZOLZ
Properties of Reciprocal space
Diffraction
DPs
R, angles between Rs
g, angles between g Planes, angles between planes
Beam direction
0 lwkvhu
Electron diffraction Patterns • Polycrystalline: Ring Pattern
– Identification of phase • extraction replicas • Fine grain size polycrystalline, CVD foils
• Single-crystal: – Spot
• The foil orientation • Accessibility of certain
orientations/diffracting vectors • Well-defined tilt axis • Identify: precipitates, twins, martensite
plates etc. • Orientation relationship between phases • Details of the fine structures
– Kikuchi line (Pairs of parallel bright/dark lines) • Reasonably thick: ~1/2 maximum usable
penetration • Low defect density
Selected-Area Diffraction Patterns
• Accuracy of operation of the TM is important – SADP: the image and
selector aperture coplanar (objective lens in focus)
– Spherical aberration – Sample height – Instability of lens current
and/or high tension
• Indexing? – Indices of diffraction
spots hkl • spacing between lattice
planes, lattice constants
– The orientation of the crystal and beam direction
Indexing: Ring Patterns
• Known materials – Measure n, (diameters of the rings) – Determine the ratios of n/ 1 (1
innermost ring) – Check the ratios against the table of
ratios of the inter-plannar spacing
• Unknown substance – Measure n, (diameters of the rings) – Convert the n into inter-plannar
spacing (camera constant) – Use ICDD index to identify the phase…
starting from the most likely… – Or… Determine the ratios of n/ 1 ..
(depending on the crystal structures) – If failed… try XRD and X-ray
microanalysis!
• Use of Ring Pattern – Calibrate the Camera Length – Identify phases
ICDD: The international Centre for Diffraction Data
1
Indexing: simple spot patterns
• Spots are in one zone axis – By inspection: Cubic… yes; Hexagonal…no (for particular c/a
patterns are similar for different zone axis)
• Indexing – Determine the {hkl} indices of the spot
• Camera constant: measure R – work out d • Ratios: measure dn/d1 – check low index planes
– Assign specific (hkl) to the individual spot • Closest spot (to the direct beam): arbitrarily indexed • The other two spots, check
– Vector addition – Angles between them
• All the others: vector addition with opposite spots (-h-k-l),(hkl)
– Determine beam direction B • [uvw]: z = B = g1 g2
• Anticlockwise indexing rule: B upward normal from the pattern to the incident beam
Example: Pure Nickel
http://rruff.geo.arizona.edu/AMS/amcsd.php
American Mineralogist Crystal Structure database
FCC, a = 3.524 A
Cubic System
For cubic system: 22
222
2
1
a
N
a
lkh
d
1
2
1
2
2
1
N
N
R
R
d
d
hkl 100 110 111 200 210 211 220 300 221 310 311 222 320 321 400 410 322 411 330 331 420 421 332 422 500 430 510 431 511 333 520 432
h k l 1.0000 1.4142 1.7321 2.0000 2.2361 2.4495 2.8284 3.0000 3.0000 3.1623 3.3166 3.4641 3.6056 3.7417 4.0000 4.1231 4.1231 4.2426 4.2426 4.3589 4.4721 4.5826 4.6904 4.8990 5.0000 5.0000 5.0990 5.0990 5.1962 5.1962 5.3852 5.3852
1 0 0 1.0000 1.0000 1.4142 1.7321 2.0000 2.2361 2.4495 2.8284 3.0000 3.0000 3.1623 3.3166 3.4641 3.6056 3.7417 4.0000 4.1231 4.1231 4.2426 4.2426 4.3589 4.4721 4.5826 4.6904 4.8990 5.0000 5.0000 5.0990 5.0990 5.1962 5.1962 5.3852 5.3852
1 1 0 1.4142 0.7071 1.0000 1.2248 1.4142 1.5812 1.7321 2.0000 2.1213 2.1213 2.2361 2.3452 2.4495 2.5496 2.6458 2.8285 2.9155 2.9155 3.0000 3.0000 3.0822 3.1623 3.2404 3.3166 3.4641 3.5356 3.5356 3.6056 3.6056 3.6743 3.6743 3.8079 3.8079
1 1 1 1.7321 0.5773 0.8165 1.0000 1.1547 1.2910 1.4142 1.6329 1.7320 1.7320 1.8257 1.9148 1.9999 2.0816 2.1602 2.3093 2.3804 2.3804 2.4494 2.4494 2.5165 2.5819 2.6457 2.7079 2.8284 2.8867 2.8867 2.9438 2.9438 2.9999 2.9999 3.1091 3.1091
2 0 0 2.0000 0.5000 0.7071 0.8661 1.0000 1.1181 1.2248 1.4142 1.5000 1.5000 1.5812 1.6583 1.7321 1.8028 1.8709 2.0000 2.0616 2.0616 2.1213 2.1213 2.1795 2.2361 2.2913 2.3452 2.4495 2.5000 2.5000 2.5495 2.5495 2.5981 2.5981 2.6926 2.6926
2 1 0 2.2361 0.4472 0.6324 0.7746 0.8944 1.0000 1.0954 1.2649 1.3416 1.3416 1.4142 1.4832 1.5492 1.6125 1.6733 1.7888 1.8439 1.8439 1.8973 1.8973 1.9493 2.0000 2.0494 2.0976 2.1909 2.2360 2.2360 2.2803 2.2803 2.3238 2.3238 2.4083 2.4083
2 1 1 2.4495 0.4082 0.5773 0.7071 0.8165 0.9129 1.0000 1.1547 1.2247 1.2247 1.2910 1.3540 1.4142 1.4720 1.5275 1.6330 1.6832 1.6832 1.7320 1.7320 1.7795 1.8257 1.8708 1.9148 2.0000 2.0412 2.0412 2.0816 2.0816 2.1213 2.1213 2.1985 2.1985
2 2 0 2.8284 0.3536 0.5000 0.6124 0.7071 0.7906 0.8660 1.0000 1.0607 1.0607 1.1181 1.1726 1.2248 1.2748 1.3229 1.4142 1.4577 1.4577 1.5000 1.5000 1.5411 1.5811 1.6202 1.6583 1.7321 1.7678 1.7678 1.8028 1.8028 1.8372 1.8372 1.9040 1.9040
3 0 0 3.0000 0.3333 0.4714 0.5774 0.6667 0.7454 0.8165 0.9428 1.0000 1.0000 1.0541 1.1055 1.1547 1.2019 1.2472 1.3333 1.3744 1.3744 1.4142 1.4142 1.4530 1.4907 1.5275 1.5635 1.6330 1.6667 1.6667 1.6997 1.6997 1.7321 1.7321 1.7951 1.7951
2 2 1 3.0000 0.3333 0.4714 0.5774 0.6667 0.7454 0.8165 0.9428 1.0000 1.0000 1.0541 1.1055 1.1547 1.2019 1.2472 1.3333 1.3744 1.3744 1.4142 1.4142 1.4530 1.4907 1.5275 1.5635 1.6330 1.6667 1.6667 1.6997 1.6997 1.7321 1.7321 1.7951 1.7951
3 1 0 3.1623 0.3162 0.4472 0.5477 0.6325 0.7071 0.7746 0.8944 0.9487 0.9487 1.0000 1.0488 1.0954 1.1402 1.1832 1.2649 1.3038 1.3038 1.3416 1.3416 1.3784 1.4142 1.4491 1.4832 1.5492 1.5811 1.5811 1.6124 1.6124 1.6432 1.6432 1.7029 1.7029
3 1 1 3.3166 0.3015 0.4264 0.5223 0.6030 0.6742 0.7386 0.8528 0.9045 0.9045 0.9535 1.0000 1.0445 1.0871 1.1282 1.2061 1.2432 1.2432 1.2792 1.2792 1.3143 1.3484 1.3817 1.4142 1.4771 1.5076 1.5076 1.5374 1.5374 1.5667 1.5667 1.6237 1.6237
2 2 2 3.4641 0.2887 0.4082 0.5000 0.5774 0.6455 0.7071 0.8165 0.8660 0.8660 0.9129 0.9574 1.0000 1.0408 1.0801 1.1547 1.1902 1.1902 1.2247 1.2247 1.2583 1.2910 1.3229 1.3540 1.4142 1.4434 1.4434 1.4720 1.4720 1.5000 1.5000 1.5546 1.5546
3 2 0 3.6056 0.2773 0.3922 0.4804 0.5547 0.6202 0.6794 0.7844 0.8320 0.8320 0.8771 0.9198 0.9608 1.0000 1.0377 1.1094 1.1435 1.1435 1.1767 1.1767 1.2089 1.2403 1.2710 1.3009 1.3587 1.3867 1.3867 1.4142 1.4142 1.4411 1.4411 1.4936 1.4936
3 2 1 3.7417 0.2673 0.3780 0.4629 0.5345 0.5976 0.6546 0.7559 0.8018 0.8018 0.8452 0.8864 0.9258 0.9636 1.0000 1.0690 1.1019 1.1019 1.1339 1.1339 1.1650 1.1952 1.2247 1.2535 1.3093 1.3363 1.3363 1.3627 1.3627 1.3887 1.3887 1.4392 1.4392
4 0 0 4.0000 0.2500 0.3536 0.4330 0.5000 0.5590 0.6124 0.7071 0.7500 0.7500 0.7906 0.8292 0.8660 0.9014 0.9354 1.0000 1.0308 1.0308 1.0607 1.0607 1.0897 1.1180 1.1457 1.1726 1.2248 1.2500 1.2500 1.2748 1.2748 1.2991 1.2991 1.3463 1.3463
4 1 0 4.1231 0.2425 0.3430 0.4201 0.4851 0.5423 0.5941 0.6860 0.7276 0.7276 0.7670 0.8044 0.8402 0.8745 0.9075 0.9701 1.0000 1.0000 1.0290 1.0290 1.0572 1.0846 1.1114 1.1376 1.1882 1.2127 1.2127 1.2367 1.2367 1.2603 1.2603 1.3061 1.3061
3 2 2 4.1231 0.2425 0.3430 0.4201 0.4851 0.5423 0.5941 0.6860 0.7276 0.7276 0.7670 0.8044 0.8402 0.8745 0.9075 0.9701 1.0000 1.0000 1.0290 1.0290 1.0572 1.0846 1.1114 1.1376 1.1882 1.2127 1.2127 1.2367 1.2367 1.2603 1.2603 1.3061 1.3061
4 1 1 4.2426 0.2357 0.3333 0.4083 0.4714 0.5271 0.5774 0.6667 0.7071 0.7071 0.7454 0.7817 0.8165 0.8499 0.8819 0.9428 0.9718 0.9718 1.0000 1.0000 1.0274 1.0541 1.0801 1.1055 1.1547 1.1785 1.1785 1.2019 1.2019 1.2248 1.2248 1.2693 1.2693
3 3 0 4.2426 0.2357 0.3333 0.4083 0.4714 0.5271 0.5774 0.6667 0.7071 0.7071 0.7454 0.7817 0.8165 0.8499 0.8819 0.9428 0.9718 0.9718 1.0000 1.0000 1.0274 1.0541 1.0801 1.1055 1.1547 1.1785 1.1785 1.2019 1.2019 1.2248 1.2248 1.2693 1.2693
3 3 1 4.3589 0.2294 0.3244 0.3974 0.4588 0.5130 0.5620 0.6489 0.6882 0.6882 0.7255 0.7609 0.7947 0.8272 0.8584 0.9177 0.9459 0.9459 0.9733 0.9733 1.0000 1.0260 1.0513 1.0761 1.1239 1.1471 1.1471 1.1698 1.1698 1.1921 1.1921 1.2354 1.2354
4 2 0 4.4721 0.2236 0.3162 0.3873 0.4472 0.5000 0.5477 0.6325 0.6708 0.6708 0.7071 0.7416 0.7746 0.8062 0.8367 0.8944 0.9220 0.9220 0.9487 0.9487 0.9747 1.0000 1.0247 1.0488 1.0955 1.1180 1.1180 1.1402 1.1402 1.1619 1.1619 1.2042 1.2042
4 2 1 4.5826 0.2182 0.3086 0.3780 0.4364 0.4880 0.5345 0.6172 0.6547 0.6547 0.6901 0.7237 0.7559 0.7868 0.8165 0.8729 0.8997 0.8997 0.9258 0.9258 0.9512 0.9759 1.0000 1.0235 1.0690 1.0911 1.0911 1.1127 1.1127 1.1339 1.1339 1.1751 1.1751
3 3 2 4.6904 0.2132 0.3015 0.3693 0.4264 0.4767 0.5222 0.6030 0.6396 0.6396 0.6742 0.7071 0.7386 0.7687 0.7977 0.8528 0.8791 0.8791 0.9045 0.9045 0.9293 0.9535 0.9770 1.0000 1.0445 1.0660 1.0660 1.0871 1.0871 1.1078 1.1078 1.1481 1.1481
4 2 2 4.8990 0.2041 0.2887 0.3536 0.4082 0.4564 0.5000 0.5773 0.6124 0.6124 0.6455 0.6770 0.7071 0.7360 0.7638 0.8165 0.8416 0.8416 0.8660 0.8660 0.8898 0.9129 0.9354 0.9574 1.0000 1.0206 1.0206 1.0408 1.0408 1.0607 1.0607 1.0992 1.0992
5 0 0 5.0000 0.2000 0.2828 0.3464 0.4000 0.4472 0.4899 0.5657 0.6000 0.6000 0.6325 0.6633 0.6928 0.7211 0.7483 0.8000 0.8246 0.8246 0.8485 0.8485 0.8718 0.8944 0.9165 0.9381 0.9798 1.0000 1.0000 1.0198 1.0198 1.0392 1.0392 1.0770 1.0770
4 3 0 5.0000 0.2000 0.2828 0.3464 0.4000 0.4472 0.4899 0.5657 0.6000 0.6000 0.6325 0.6633 0.6928 0.7211 0.7483 0.8000 0.8246 0.8246 0.8485 0.8485 0.8718 0.8944 0.9165 0.9381 0.9798 1.0000 1.0000 1.0198 1.0198 1.0392 1.0392 1.0770 1.0770
5 1 0 5.0990 0.1961 0.2773 0.3397 0.3922 0.4385 0.4804 0.5547 0.5884 0.5884 0.6202 0.6504 0.6794 0.7071 0.7338 0.7845 0.8086 0.8086 0.8320 0.8320 0.8549 0.8771 0.8987 0.9199 0.9608 0.9806 0.9806 1.0000 1.0000 1.0191 1.0191 1.0561 1.0561
4 3 1 5.0990 0.1961 0.2773 0.3397 0.3922 0.4385 0.4804 0.5547 0.5884 0.5884 0.6202 0.6504 0.6794 0.7071 0.7338 0.7845 0.8086 0.8086 0.8320 0.8320 0.8549 0.8771 0.8987 0.9199 0.9608 0.9806 0.9806 1.0000 1.0000 1.0191 1.0191 1.0561 1.0561
5 1 1 5.1962 0.1924 0.2722 0.3333 0.3849 0.4303 0.4714 0.5443 0.5773 0.5773 0.6086 0.6383 0.6667 0.6939 0.7201 0.7698 0.7935 0.7935 0.8165 0.8165 0.8389 0.8606 0.8819 0.9027 0.9428 0.9622 0.9622 0.9813 0.9813 1.0000 1.0000 1.0364 1.0364
3 3 3 5.1962 0.1924 0.2722 0.3333 0.3849 0.4303 0.4714 0.5443 0.5773 0.5773 0.6086 0.6383 0.6667 0.6939 0.7201 0.7698 0.7935 0.7935 0.8165 0.8165 0.8389 0.8606 0.8819 0.9027 0.9428 0.9622 0.9622 0.9813 0.9813 1.0000 1.0000 1.0364 1.0364
5 2 0 5.3852 0.1857 0.2626 0.3216 0.3714 0.4152 0.4549 0.5252 0.5571 0.5571 0.5872 0.6159 0.6433 0.6695 0.6948 0.7428 0.7656 0.7656 0.7878 0.7878 0.8094 0.8304 0.8510 0.8710 0.9097 0.9285 0.9285 0.9469 0.9469 0.9649 0.9649 1.0000 1.0000
4 3 2 5.3852 0.1857 0.2626 0.3216 0.3714 0.4152 0.4549 0.5252 0.5571 0.5571 0.5872 0.6159 0.6433 0.6695 0.6948 0.7428 0.7656 0.7656 0.7878 0.7878 0.8094 0.8304 0.8510 0.8710 0.9097 0.9285 0.9285 0.9469 0.9469 0.9649 0.9649 1.0000 1.0000
N1
N2 N1/N2
2
1
2
2
2
1
2
1
2
2
N
N
R
R
d
d
LRdhkl 222 lkhN
The Distances in Diffraction Pattern
R1
http://imagej.nih.gov/ij/
R1=503.337/6=83.89
Finite sizes of spots
The distances in diffraction pattern
R2
The distances in diffraction pattern
R3
518.21
3
3
1 R
R
d
d
497.21
2
2
1 R
R
d
d
d1 (largest spacing): {111} d2: {331} d3: {420}
d1 (largest spacing): {110} d2: {222} d3: {320}
Systematic absence: f.c.c: hkl mixed odd and even
• Work out the ratios
hkl 211 220 300 221 310 311 222 320 321 400 410 322 411 330 331 420 421 332
1 0 0 1.0000 2.4495 2.8284 3.0000 3.0000 3.1623 3.3166 3.4641 3.6056 3.7417 4.0000 4.1231 4.1231 4.2426 4.2426 4.3589 4.4721 4.5826 4.6904
1 1 0 1.4142 1.7321 2.0000 2.1213 2.1213 2.2361 2.3452 2.4495 2.5496 2.6458 2.8285 2.9155 2.9155 3.0000 3.0000 3.0822 3.1623 3.2404 3.3166
1 1 1 1.7321 1.4142 1.6329 1.7320 1.7320 1.8257 1.9148 1.9999 2.0816 2.1602 2.3093 2.3804 2.3804 2.4494 2.4494 2.5165 2.5819 2.6457 2.7079
Deciding indices
d1
d2
d3
d1 (largest spacing): {111} d2: {331} d3: {420}
• Arbitrarily deciding d1: (111)
Angle between (h1k1l1) and (h2k2l2)
2
2
2
2
2
2
222
2
2
2
2
2
2
2
1
2
1
2
1
212121
3
cos
lkh
lkh
lkhlkh
llkkhh
In the cubic system planes having the same indices regardless of order or sign are equivalent
(h1k1l1): (111)
Measuring the Angles
h k l cos
3 3 1 0.927173 22.01287
3 3 -1 0.662266 48.55168
-3 3 1 0.132453 82.43041
-3 3 -1 -0.13245 97.66089
-3 -3 1 -0.66227 131.5396
-3 -3 -1 -0.92717 158.0784
Note -331, 3-31, -313,.etc give the same result
2
2
2
2
2
2
222
3cos
lkh
lkh
Angles between the normal of {331} and {111}
Measuring the Angle
h k l cos
4 2 0 0.774597 39.25142
-4 2 0 -0.2582 105.0165
-4 -2 0 -0.7746 140.8399
Angle between (h1k1l1) and (h3k3l3)
Deciding indices:Vector Addition
-331 -313 3-31 31-3 1-33 13-3
-420 111 1-13 7-51 1-1-3 5-53 51-3
-402 13-1 111 1-3-1 71-5 5-31 53-5
2-40 -571 -553 111 15-3 -113 -17-3
20-4 -535 -517 1-35 111 -1-37 -131
0-42 -37-1 -351 31-1 35-5 111 17-5
02-4 -315 -3-17 3-55 3-11 1-57 111
d1
d2
d3
111
132 ddd
-d3
Possible pairs
(-420)-(-331) (-402)-(-313)
(2-40)-(3-31) (20-4)-(31-3)
(0-42)-(1-33) (02-4)-(13-3)
Deciding indices: the other spots
d1
d2
d3
111
9234.0)416(199
2343cos
)204()^313(
23)204()^313(
-331
-420
4-20
-1-1-1
Check Angle
Check Vector addition
(-4, 2, 0) = (-3,3,1) + (-1, -1, -1)
-242 (-420)-(-331)
• Use vector addition to index the other spots
Deciding Zone Axis
111
-331
-420
4-20
-1-1-1
3-3-1
-242
The g vectors belong to the ZOLZ
The zone axis [uvw] all the g vectors
21 ggB
cba
cba
cba
ggB
642
)3(31331
133
111
21
[uvw] =[-1 -2 3]
Is this confusing?
Uniqueness in indexing DP: other possibilities
cba
cba
cba
ggB
642
331331
133
111
21
[uvw] =[-1- 23]
• 180o: g and -g - diffraction from both sides of the reflecting plane (From one zone)
• Coincidence ambiguity: {hkl} the same inter-planar spacing
• Solved by tilt sample
-420
-33-1
-242
111
-1-1-1
3-31 4-20
[uvw] =[-1- 23]
-1-1-1
3-31 4-20
-420
111
-33-1
2-4-2
[uvw] =[-1 -2 3]
The same beam direction
Beam direction: Source of Errors
• Uniform intensity: B < 0.5o
• Rings of bright spots: B>1o
• Nonuniform intensity: B~ 15o
• Kikuchi lines: B~0.1o
• For high accuracy – Cs – Errors in lattice parameters/beam
energy – Errors in measurement of SADP
• B: average for the selected region – Bent, defective – K lines not for deformed/phase
transformations
Sketch Zero Laue Zone FCC (421)
In the reciprocal space, (421) Intercepts are:
4 (1/4, ½, 1) = 1,2, 4
Consider the plane passing through the origin: lets move (1,0,0) to the origin
100, 020, 004
1,0,0; 0,2,0; 0,0,4 0,0,0; -1,2,0;-1,0,4 -1,0,0
For FCC: must be all even/odd numbers
0,0,0; -2,4,0;-2,0,8
The angle between (-2,4,0) and (-2,0,8) is 83o49’
Length ratio is 844.1
1
68
20
082
402 R
R Simple pattern low zone axis Weiss zone law
0111 wlvkuh
ZOLZ of the example Zone axis [-1-23] The plane in the reciprocal space (-1-23) Intercepts of the plane: (-1,0,0) (0,-1/2,0) (0,0,1/3) (-6,0,0) (0,-3, 0) (0,0,2)
Move this plane to the origin: (0,0,0) (6,-3, 0) (6,0,2)
fcc requires: (0,0,0) (12,-6, 0) (6,0,2) Angle : 32o
The length ratio: 2.1
(6,0,2)
(12,-6, 0)
(-6,0,-2)
(6,-6,-2)
(3,-3,-1)
(-3,3,1) (3,3,3)
(1,1,1) (2,2,2)
(-1,-1,-1)
(-2,-2,-2)
(4,-2,0)
Kikuchi Lines • 1928 Kikuchi • More accurate than spot for
– Specimen orientation – Accessibility of certain
orientation/diffracting vectors – A well-defined tilt axis – Orientation relationship
• Kikuchi map: distribution of K lines – A unit triangle of the stereogram – Identify K pattern rapidly, determine B – Tilt sample in a self-consistent manner
• Determine the magnitude and sign of the deviation s
• Define the sense of tilt (moves in the same sense as the crystal)
• Small angle tilt can be estimated • Large angular tilts can be measured • Determine crystal symmetry – real
crystal symmetry
Kikuchi Lines • The diffraction of electrons which
have been previously inelastically scattered – Inelastically scattered
• Small Energy loss < 50 eV and change direction
• A diffuse halo (small angle)/overall faint background
– The Bragg law (hkl) • Not satisfied for the incident beam • Satisfied for the inelastically scattered
beam – Removed from the T beam: local
reduction in background intensity – Fewer will be diffracted back to the
background (angle distribution)
• Cones of radiation – Centre: the sample – Cone I: high intensity; Cone II: low
intensity – Cones attached to the crystal
» Move about a radius: 1/ » Spots move about : g 1/25
1/ – Cones intersect Ewald sphere (’
): hyperbolae
Summary
• Indexing simple patterns
– Diffraction spots
– Beam direction
– ZOLZ
• Kikuchi Lines
Lecture 5
• Dynamical Diffraction Theory
• Diffraction Contrast
SUPPLEMENTARY
Indexing For cubic system:
22
222
2
1
a
N
a
lkh
d
N: Sum of squares of three integers 784 mN n
bcc: N = 2m; fcc: N=4n; 8n+3
Assume two spots: R1 and R2
Camera Length:
LRdhkl
For Tetragonal system: 2
2
22
2
2
22
2
1
c
l
a
A
c
l
a
kh
d
For l=0: A, 2A must in the series
For Hexagonal system: 2
2
22
2
2
22
2 3
4
3
41
c
l
a
H
c
l
a
khkh
d
For l=0: H, 3H must in the series
2
1
2
2
2
1
2
1
2
2
N
N
R
R
d
d
ZnO (Znic Blende structure) cubic reciprocal lattice layers along [111] direction
0-level
l = +1 level
l = -1 level
Note: All diffraction patters are Centro-symmetric even if the crystal is not (Friedel’s law)
Diffraction Sysmtry
Some ZOLZ patterns thus exhibit Higher rotational symmetry than the structure has
Indexing: Kikuchi Patterns • B = low zone axis
– a high degree of symmetry
– Direct comparison: standard patterns
• Two features
– Kikuchi line spacing Dd = L: no error because of s
– Angles between K lines – angles between planes
• Simple patterns: in the same zone
– Identical for spot patterns
– B is close to zone axis: T beam away from the centre of the K pattern
– 180o ambiguity
Kikuchi Lines: Complex pattern
• Three non-parallel line pairs
Kikuchi Maps
• Help to eliminate the trial and error stage of indexing K patterns
• Rapidly obtain: B with 2o
Use of diffraction patterns: Basic
• Specimen tilting
• Orientation relationship determination
• Twining
• Second phases
• Crystallographic information
USE: The fine structures
• Extra spots
• Spot splitting and satellite spots
• Streaks
• Diffuse scattering Effects
Indexing: Complicated patterns • Large lattice para
– Thin sample: high-order laue zero – at the edge of the pattern
– Thin sample: far away from low zone axis
• Fragmentary arrays of spots – avoid this!
• Laue zones: hu+kv+lw = n
Indexing: Imperfect patterns • Spots: from different zero-order Laue zones
– Remove ambiguity in indexing
• Indexing: Three vectors g1, g2, g3, form a circle – T is inside the circle:
• O lying below the plane • Labelled anticlockwise
– T is outside • O lying above the plane • Labelled clockwise
– Evaluate {hkl}: use camera length – Arbitrarily index on (hkl), label g2, g3 according to the
above rule – Zone axis = g1Xg2 – All zone axes: close – Check g1.(g2Xg3)>0 – Work out B: 21
2
313
2
232
2
1 gggggggggB