Lattices and Crystals

8/2/2019 Lattices and Crystals

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Lattices and crystals

Real and reciprocal space

Definition

Base of direct space lattice a1, a2, a3 Base of reciprocal space b1, b2, b3, with ai bj= 2 lj.

The base vectors are often named as

Real space (a,b,c) Reciprocal space (a*,b*,c*)

The orthogonality conditions are then written as

a*a = a*c = b*a = b*c = c*a = c*b = 0

a*a = b*b = c*c = 1

Reciprocal base vectors are obtained as

a* = 2 bxc/V,

b*= 2 cxa/V,

c*= 2 axb/V,

where V is the volume of the unit cell V = abxc.

The volume of the reciprocal cell V* = 1/V.

How the lattice vectors are presented in the base a, b, and c?

Lattice vectors are R = p a + q b + r c, where r, q, and q are integers.

Reciprocal lattice vector are then G = ha* + kb* + lc*

The components h, k, and l are also integers.

1-dimensional

R = n a


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G = h a*Length a* = 2/a

2-dimensional

R = n a + m b G = h a* + kb*

Example. Two dimensional cubic structure

3-dimensional

R = p a + q b + r c

G = ha* + kb* + lc*

Miller indices

One can show that for the reciprocal lattice vector G the Laue conditions hold:

GRn= 2 (hq+kp+lr) = 2 integer.

Thus the diffraction maxima are obtained when

q=G.

This is equivalent to Braggs law

2d sin = .


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The integers h, k, l, the coordinates of lattice points in the reciprocal space, are called the Miller

indices.

If the Miller indices of a plane are (hkl), then the plane makes fractional intercepts of 1/h, 1/k, 1/l

with the axes, and, if the axial lengths are a, b, c, the plane makes actual intercepts of a/h, b/k,

c/l.

Example. Two-dimensional lattice planes 10 and 21


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Conventions in International tables of crystallography

Lattice planes (hkl)

Reflection hkl Direction [hkl]

Structure factor of the unit cell

Let the unit cell contain N atoms. The positions of the atoms are (xj, yj, zj), j = 1...N. The

structure factor may be computed as

Fhkl= j =1N fjexp(2 i(hxj + kyj + lzj)).

It gives the intensities for the reflections hkl.

The intensity is the square of the structure factor.

For correspondence to experimental data the intensity should be multiplied by polarization and

absorption factors.

http://www.ruppweb.org/Xray/comp/strufac.htm

Linear algebra properties

Reciprocal lattice vector Ghkl is perpendicular to the plane with Miller indices (hkl)

Consider two, not parallel vectors in the plane (hkl):

v1 = c/l - a/h, v2 = a/h - b/k.

Any vector v in the plane can be written as

v = e1v1+e2v2

Gv = (ha* + kb* + lc*)(e1/l c-e1/h a + e2/ha-e2/k b)

= (ha* + kb* + lc*)((e2-e1)/h a - e2/k b + e1/l c)

= 2(e2 - e1- e2 + e1) = 0
http://www.ruppweb.org/Xray/comp/strufac.htmhttp://www.ruppweb.org/Xray/comp/strufac.htmhttp://www.ruppweb.org/Xray/comp/strufac.htm


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Spacing of the lattice planes |Ghkl| = 2/dhkl

The plane spacing is the distance of the plane from the origin. Let n be a unit vector

perpendicular to the plane (hkl). The distance may be computed with of any vector pointing

from the origin to the plane, e.g. the vector a/h, as

dhkl = an/h

(projection). The vector n can be written as

n=Ghkl/|Ghkl|.

Thus dhkl = aGhkl /|Ghkl|/h = a(ha*+kb*+lc*)/|Ghkl| /h = 2/ |Ghkl|.

Metric matrices g and g*

The elements of the metric matrix g are

g11 = aa, g12 = ab, g13= ac

g21 = ba, g22 = bb, g23 = bc

g31 = ca, g32 = cb, g33 = cc

det(g) = V2

The matrix g* is defined analogously in terms of the reciprocal base (a*,b*,c*). The matrix g* is

the inverse of the matrix g:

g* = inv(g).

This is useful when computing e.g. distances of points presented in the lattice coordinate system.


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Angles between reciprocal base vectors

The angles may be computed using the dot product, for instance

b*c* = 2(cxa)(axb)/V2

Inserting the angles between the lattice vectors and using the formula (AxB)(CxD) =

(AC)(BD)-(AD)(BC) one obtains

cos(*) = (cos cos - cos)/(sin sin).

The length of the vector in base (a,b,c)

Vector r = za + yb + zc (column)

The length of the vector r is |r| = (xtg x)

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The square of the length of a vector r given in base (a,b,c) may also be written as

r2

= rtgr = x

2a

2+ y

2b

2+ z

2c

2 + 2xyabcos + 2xzaccos + 2yzbccos

Distance of two atoms at (x1,y1,z1) and (x2,y2,z2) in a unit cell

By taking the following new variables into use, A = a(x1-x2), B = b(y1-y2), C = c(z1-z2),

the length of the vector squared

r2

= A2

+ B2

+ C2+ 2ABcos + 2ACcos + 2BCcos


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The angle between two vectors given in the crystal coordinate system

Two vectors x1, x2 are given in the base (a,b,c). Their angle is

cos = x1tg

x2/(|x1||x2|).

Lattices and Crystals

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