ObjectivesBy the end of this section you should:understand the concept of planes in crystalsknow that planes are identified by their Miller Index and their separation, dbe able to calculate Miller Indices for planesknow and be able to use the d-spacing equation for orthogonal crystalsunderstand the concept of diffraction in crystalsbe able to derive and use Braggs law

Lattice Planes and Miller IndicesImagine representing a crystal structure on a grid (lattice) which is a 3D array of points (lattice points). Can imagine dividing the grid into sets of planes in different orientations

All planes in a set are identicalThe planes are imaginaryThe perpendicular distance between pairs of adjacent planes is the d-spacingNeed to label planes to be able to identify themFind intercepts on a,b,c: 1/4, 2/3, 1/2

Take reciprocals 4, 3/2, 2

Multiply up to integers: (8 3 4) [if necessary]

Exercise - What is the Miller index of the plane below?Find intercepts on a,b,c:

Take reciprocals

Multiply up to integers:

Plane perpendicular to y cuts at , 1, (0 1 0) planeGeneral label is (h k l) which intersects at a/h, b/k, c/l (hkl) is the MILLER INDEX of that plane (round brackets, no commas).This diagonal cuts at 1, 1, (1 1 0) planeNB an index 0 means that the plane is parallel to that axis

Using the same set of axes draw the planes with the following Miller indices:(0 0 1) (1 1 1)

Using the same set of axes draw the planes with the following Miller indices:(0 0 2) (2 2 2)NOW THINK!! What does this mean?

Planes - conclusions 1Miller indices define the orientation of the plane within the unit cellThe Miller Index defines a set of planes parallel to one another (remember the unit cell is a subset of the infinite crystal(002) planes are parallel to (001) planes, and so on

d-spacing formulaFor orthogonal crystal systems (i.e. ===90) :-

For cubic crystals (special case of orthogonal) a=b=c :-e.g. for(1 0 0)d = a(2 0 0)d = a/2(1 1 0)d = a/2 etc.

A tetragonal crystal has a=4.7 , c=3.4 . Calculate the separation of the:(1 0 0)(0 0 1)(1 1 1) planesA cubic crystal has a=5.2 (=0.52nm). Calculate the d-spacing of the (1 1 0) plane

Question 2 in handout:If a = b = c = 8 , find d-spacings for planes with Miller indices (1 2 3)

Calculate the d-spacings for the same planes in a crystal with unit cell a = b = 7 , c = 9 .

Calculate the d-spacings for the same planes in a crystal with unit cell a = 7 , b = 8 , c = 9 .

X-ray Diffraction

Diffraction - an optical gratingPath difference XY between diffracted beams 1 and 2:sin = XY/a XY = a sin For 1 and 2 to be in phase and give constructive interference, XY = , 2, 3, 4..nso a sin = n where n is the order of diffraction

Diffracted light

Coherent incident light

X

Y

a

(

(

1

2

Consequences: maximum value of for diffractionsin = 1 a = Realistically, sin So separation must be same order as, but greater than, wavelength of light.

Thus for diffraction from crystals:Interatomic distances 0.1 - 2 so = 0.1 - 2 X-rays, electrons, neutrons suitable

Diffraction from crystals?X-ray TubeDetector

2

1

X

Z

(

(

Transmitted radiation

Reflected radiation

Incident radiation

d

Y

Beam 2 lags beam 1 by XYZ = 2d sin so 2d sin = n Braggs Law

2

1

X

Z

(

(

Transmitted radiation

Reflected radiation

Incident radiation

d

Y

We normally set n=1 and adjust Miller indices, to give 2dhkl sin = 2d sin = ne.g. X-rays with wavelength 1.54 are reflected from planes with d=1.2. Calculate the Bragg angle, , for constructive interference. = 1.54 x 10-10 m, d = 1.2 x 10-10 m, =?n=1 : = 39.9n=2 :X (n/2d)>1

Example of equivalence of the two forms of Braggs law:Calculate for =1.54 , cubic crystal, a=52d sin = n(1 0 0) reflection, d=5 n=1, =8.86on=2,=17.93on=3,=27.52on=4,=38.02on=5,=50.35on=6,=67.52ono reflection for n7(2 0 0) reflection, d=2.5

n=1,=17.93o

n=2,=38.02o

n=3,=67.52ono reflection for n4

Use Braggs law and the d-spacing equation to solve a wide variety of problems2d sin = nor2dhkl sin =

X-rays with wavelength 1.54 are reflected from the (1 1 0) planes of a cubic crystal with unit cell a = 6 . Calculate the Bragg angle, , for all orders of reflection, n.Combining Bragg and d-spacing equationd = 4.24

d = 4.24 n = 1 : = 10.46n = 2 : = 21.30n = 3 : = 33.01n = 4 : = 46.59n = 5 : = 65.23= (1 1 0)= (2 2 0)= (3 3 0)= (4 4 0)= (5 5 0)2dhkl sin =

SummaryWe can imagine planes within a crystal Each set of planes is uniquely identified by its Miller index (h k l)We can calculate the separation, d, for each set of planes (h k l)Crystals diffract radiation of a similar order of wavelength to the interatomic spacingsWe model this diffraction by considering the reflection of radiation from planes - Braggs Law