Objectives
By the end of this section you should:• understand the concept of planes in crystals• know that planes are identified by their Miller
Index and their separation, d• be able to calculate Miller Indices for planes• know and be able to use the d-spacing
equation for orthogonal crystals• understand the concept of diffraction in crystals• be able to derive and use Bragg’s 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 identical• The planes are “imaginary”• The perpendicular distance between pairs of adjacent
planes is the d-spacing
Need to label planes to be able to identify them
Find 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) plane
General 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) plane
NB 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 1
• Miller indices define the orientation of the plane within the unit cell
• The 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 formula
For orthogonal crystal systems (i.e. ===90) :- 2
2
2
2
2
2
2 c
l
b
k
a
h
d
1
For cubic crystals (special case of orthogonal) a=b=c :- 2
222
2 a
lkh
d
1
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 Å, c=3.4 Å. Calculate the separation of the:
(1 0 0)
(0 0 1)
(1 1 1) planes
A cubic crystal has a=5.2 Å (=0.52nm). Calculate the d-spacing of the (1 1 0) plane
]ba[c
l
a
kh
d
12
2
2
22
2
Question 2 in handout:
If a = b = c = 8 Å, 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 Å, c = 9 Å.
Calculate the d-spacings for the same planes in a crystal with unit cell a = 7 Å, b = 8 Å, c = 9 Å.
X-ray Diffraction
Diffraction - an optical grating
XY
1
2
a
Coherent incident light Diffracted light
Path 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…..n
so a sin = n where n is the order of diffraction
Consequences: maximum value of for diffraction
sin = 1 a =
Realistically, sin <1 a >
So separation must be same order as, but greater than, wavelength of light.
Thus for diffraction from crystals:
Interatomic distances 0.1 - 2 Å
so = 0.1 - 2 Å
X-rays, electrons, neutrons suitable
Diffraction from crystals
XY
Z
d
Incident radiation “Reflected” radiation
Transmitted radiation
1
2
?
X-ray Tube
Detector
Beam 2 lags beam 1 by XYZ = 2d sin
so 2d sin = n Bragg’s Law
XY
Z
d
Incident radiation “Reflected” radiation
Transmitted radiation
1
2
We normally set n=1 and adjust Miller indices, to give
2dhkl sin =
2d sin = n
e.g. X-rays with wavelength 1.54Å are reflected from planes with d=1.2Å. Calculate the Bragg angle, , for constructive interference.
= 1.54 x 10-10 m, d = 1.2 x 10-10 m, =?
d2
nsin
nsind2
1
n=1 : = 39.9°
n=2 : X (n/2d)>1
Example of equivalence of the two forms of Bragg’s law:
Calculate for =1.54 Å, cubic crystal, a=5Å
2d sin = n
(1 0 0) reflection, d=5 Å
n=1, =8.86o
n=2, =17.93o
n=3, =27.52o
n=4, =38.02o
n=5, =50.35o
n=6, =67.52o
no reflection for n7
(2 0 0) reflection, d=2.5 Å
n=1, =17.93o
n=2, =38.02o
n=3, =67.52o
no reflection for n4
1d
ha
kb
lc2
2
2
2
2
2
2
Use Bragg’s law and the d-spacing equation to solve a wide variety of problems
2d sin = n
or
2dhkl sin =
X-rays with wavelength 1.54 Å are “reflected” from the (1 1 0) planes of a cubic crystal with unit cell a = 6 Å.
Calculate the Bragg angle, , for all orders of reflection, n.
Combining Bragg and d-spacing equation
056.06
0112
2
222
2 a
lkh
d
1
18d2 d = 4.24 Å
d = 4.24 Å
d2
nsin 1
n = 1 : = 10.46°
n = 2 : = 21.30°
n = 3 : = 33.01°
n = 4 : = 46.59°
n = 5 : = 65.23°
= (1 1 0)
= (2 2 0)
= (3 3 0)
= (4 4 0)
= (5 5 0)
2dhkl sin =
SummarySummary We 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 spacings
We model this diffraction by considering the “reflection” of radiation from planes - Bragg’s Law