Determination of Crystal Structure (Chapt. 10)

Crystal Structure Diffraction PatternUnit Cell Line PositionsAtom Positions Line Intensities

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1. Use the angular positions of the diffraction lines to determine the shape and size of the unit cell. Assume either cubic, tetragonal, orthorhombic,

rhombohedral, hexagonal, monoclinic, or triclinic. Assign Miller indices to each reflection (“index the

pattern”). If a match is not obtained, the assumption of should be

changed and the pattern indexed again. Calculate the size of the unit cell based on the positions

and Miller indices of the diffraction lines.2. With the measured density of the material, the chemical

composition, and the size of the unit cell, calculate the number of atoms per unit cell.

3. Find the positions of the atoms in the unit cell by using the relative intensities of the diffraction lines.

Preliminary Treatment of Data

We want the values of sin2 for each diffraction line (in order to find the cell size and shape), however, there can be errors to what we measure, including extraneous lines in the diffraction pattern, and systematic errors (misalignment, film shrinkage, absorption, etc.)

Extraneous Lines1. X-ray beam with multiple wavelengths:

2. Contaminants or impurities in the sample, or the specimen mount!

Systematic Errors1. Film shrinkage (see Fig. 6-5 Cullity)2. Specimen is off-centered in Debye-Scherrer camera (see Fig. 11-3 Cullity)3. Absorption in the sample

Answer mix in a reference material and calibrate. (see Fig. 10-1 Cullity)

K

K

d

d

sin2

sin2

2

2

2

22

sinsin4

KKd

222

2

sinsinK

K

Cubic Crystals (Indexing the Patterns)

2222

22

4sin lkh

a

:222 lkh Simple cubic: 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, etc.Body-centered cubic: 2, 4, 6, 8, 10, 12, 14, 16, . . .Face-centered cubic: 3, 4, 8, 11, 12, 16, 19, 20, . . .Diamond cubic: 3, 8, 11, 16, 19, 24, 27, 32, . . .

Simple cubic bcc fccLinesin2(h2 + k2 + l2)

2

2

4a

(h2 + k2 + l2)2

2

4a

(h2 + k2 + l2)2

2

4a

a(Å)hkl

10.140 1 0.140 2 0.070 3 0.04673.5711120.185 2 0.093 4 0.046 4 0.04633.5920030.369 3 0.123 6 0.062 8 0.04613.5922040.503 4 0.126 8 0.063 11 0.04573.6131150.548 5 0.110 10 0.055 12 0.04573.6122260.726 6 0.121 12 0.061 16 0.04543.6240070.861 8 0.108 14 0.062 19 0.04533.6233180.905 9 0.101 16 0.057 20 0.04533.62420

Corrections for Systematic Errors

RS4

RR

SS

RS

ln4lnlnln

RR

SS

RS ,

2sin22 xONS

cossin4

2sin2Rx

Rx

SS

C

Absorption error can be lumped into this error.

cossin,,, Rx

RR

SS

ACRS

Correction for Systematic Error

For a cubic crystal:

Differentiation of Bragg’s Law: sin2d

sin2d

2

2

sin4

cossin22

sin4

cos2sin2

d

dddd

cotdd

222 lkhda dd

aa

Fractional error in a

(goes to zero as 90)

Correction for Systematic Error

sincos ,cossin , ,90

cossincossin

cossin

sincos

Rx

RR

SS

dd

At small (large ), this could be approximated as:

22 cossin KKdd

2cosKaaa

aa

dd

o

o

2cosKaaa oo

Correction for Systematic Error

0

0.5

1

1.5

2

2.5

3

3.5

4

-20 0 20 40 60 80 100

Mag

nitu

de

0.01*cos2

cos2

(p/2 - )*cos()/sin()

0.1*cos2

10*cos2

100*cos2(p/2 - )*cos()/sin()

+ cos2()

Nelson-Riley

0

0.5

1

1.5

2

2.5

3

3.5

4

-20 0 20 40 60 80 100

Mag

nitu

de

Nelson Rileyfunction

2*cos2

cos2

= (cos2/sin2) + (cos2/)(p/2 - )*cos()/sin()

Indexing Patterns for Non-Cubic Crystals

Tetragonal

2

222

22

2

2

22

2

11

ac

lkh

ac

l

a

kh

d

2

222loglog2log2

ac

lkhad

2

222

22

22

212

12

121 logloglog2log2

ac

lkh

ac

lkhdd

Depends on (c/a), but not on a.

Dull-Harvey Chart

Indexing Patterns for Non-Cubic Crystals

Tetragonal

2

2

2

222

22

2

2

22

2

sin411

ac

lkh

ac

l

a

kh

d

2

222

2

22 log

4logsinlog

ac

lkh

a

2

222

22

22

212

12

122

12 loglogsinlogsinlog

ac

lkh

ac

lkh

Depends on (c/a), but not on a.

2

222

2

22

4sin

ac

lkh

a

Scales for Left Side of Above Equations

1 102 3 4 65 7 8 9

0.010.11 0.5 0.05

d scale

sin2 scale

1 102 3 4 65 7 8 9

Hexagonal Hull-Davey Chart

Zinc Example (Cu K)

From table 10-2 (sin2)

0.010.11 0.5 0.05

0.010.11 0.5 0.05

10-7

0.010.11 0.5 0.05

10-5

0.010.11 0.5 0.05

10-8

Chapter 10 Example

sc fcc bcc diamond

line sin2 (h2 + k2 + l2) 2/4a2 (h2 + k2 + l2) 2/4a2 (h2 + k2 + l2) 2/4a2 (h2 + k2 + l2) 2/4a2

1 0.0462 1 0.0462 3 0.0154 2 0.0231 3 0.01542 0.1198 2 0.0599 4 0.02995 4 0.02995 8 0.0149753 0.1615 3 0.053833 8 0.020188 6 0.026917 11 0.0146824 0.179 4 0.04475 11 0.016273 8 0.022375 16 0.0111885 0.234 5 0.0468 12 0.0195 10 0.0234 19 0.0123166 0.275 6 0.045833 16 0.017188 12 0.022917 24 0.0114587 0.346 8 0.04325 19 0.018211 14 0.024714 27 0.0128158 0.391 9 0.043444 20 0.01955 16 0.024438 32 0.0122199 0.461 10 0.0461 24 0.019208 18 0.025611 35 0.01317110 0.504 11 0.045818 27 0.018667 20 0.0252 40 0.012611 0.575 12 0.047917 32 0.017969 22 0.026136 43 0.01337212 0.616 13 0.047385 35 0.0176 24 0.025667 48 0.01283313 0.688 14 0.049143 36 0.019111 26 0.026462 51 0.0134914 0.729 15 0.0486 40 0.018225 30 0.0243 56 0.01301815 0.799 16 0.049938 43 0.018581 32 0.024969 59 0.01354216 0.84 17 0.049412 44 0.019091 34 0.024706

Extended Hull-Davey Chart

Hull-Davey for Cubic

110100

Hull Davey cubic

(c/a

) =

1

dia.

bcc

dia.

fcc

dia.

sc

Sin2 Scale of Table 10-5

0.010.11 0.5 0.05

0.010.11 0.5 0.05

Chapter 10 Example

fcc

line sin2 (h2 + k2 + l2) 2/4a2

1 0.0462 3 0.01542 0.1198 8 0.0149753 0.1615 11 0.0146824 0.179 12 0.0149175 0.234 16 0.0146256 0.275 19 0.0144747 0.346 24 0.0144178 0.391 27 0.0144819 0.461 32 0.01440610 0.504 35 0.014411 0.575 40 0.01437512 0.616 43 0.01432613 0.688 48 0.01433314 0.729 51 0.01429415 0.799 56 0.01426816 0.84 59 0.014237