5

1 The direct lattice

1.1 Definition

Crystals are characterised by periodicity in three-dimensional space. This

notion of periodicity is embodied in the concept of a crystal lattice. The

lattice is a set of vectors v which characterise the translational symmetries

of a crystal. In other words, if a lattice translation is applied to the crystal

(considered to be an infinite object), the translated crystal superimposes with

the original one. The lattice is then viewed as an infinite set of vectors v that

can be expressed as linear combinations of some non-coplanar basis vectors

a, b, c with integer coefficients u, v,w.

v = ua + vb + wc u, v,w 2 Z (1)

The lattice representing the translational symmetry operations of a crystal

is also called the direct lattice in order to distinguish it from the reciprocal

lattice which will be defined later. A crystal can of course possess more

symmetry operations than just the translational symmetry (i.e. rotation

symmetry, inversion etc.). The full symmetry of a crystal is given by its

space group, of which the lattice is a subgroup. It is customary to define a

node in a lattice as representing the extremity (the tip) of a lattice vector v.

The lattice can then also be viewed as a set of nodes that are periodically

arranged in space. A node is labelled by the set of three integers u v w written

without commas. By convention, the horizontal bar is used as a symbol for

negative values of indices, e.g. ¯2 0 3 stands for the node at v = −2 a+0 b+3 c

It is important to realize that the lattice is an abstract notion and we

should resist any temptation to identify lattice nodes with e.g. atoms, ions

or molecules. The lattice should not be confonded with the true, physical,

1

crystal space and the two should not be superimposed on each other. Lattice

nodes, rows and planes (which will be defined below) do not exist in the

crystal !

For a given lattice, there exist an infinite number of possible choices for

the basis vectors a, b, c. Note that in general it might not be possible to

choose orthogonal basis vectors. The basis vectors must be non-coplanar (i.e.

linearly independent) and the set of all integer combinations of basis vectors

must generate the whole lattice. Such basis vectors are called primitive.

We then introduce the notion of unit cell - a concept which, unfortunately,

does not have a unique definition. In the context of the present chapter,

we designate by unit cell the parallelepiped that is generated by the basis

vectors a, b, c. It can be shown that the unit cell associated with a primitive

set of basis vectors does not contain any lattice nodes (apart from the 8 nodes

sitting on its vertices). Such a unit cell is therefore also called a primitive

cell. It can, however, be advantageous for reasons of symmetry to chose a

different set of basis vectors, called conventional, which may be non-primitive

(c.f. lectures on crystal symmetry). With a conventional set of basis vectors,

it may be necessary to include half or one-third integer coefficients u, v,w

to generate the whole lattice. Non-primitive unit cells contain lattice nodes

inside their volume or on their faces. For simplicity, we will assume that all

sets of basis vectors are primitive for the remainder of the discussion in this

section, but we will occasionally indicate implications that arise from the

choice of non-primitive basis vectors. The unit cell volume V is given by the

mixed product of the basis vectors.

V = |a · (b × c)| (2)

It is conventional to chose a right-handed set of basis vectors, i.e. a set of

vectors for which a · (b × c) > 0.

From a more mathematical point of view, the lattice (r) can be viewed

as a set of periodically spaced Dirac distributions, i.e. a Dirac (r − r◦)

distribution is located on each node of the lattice.

(r) =

+∞

X

u=−∞

+∞

X

v=−∞

+∞

X

w=−∞

(r − ua − vb − wc) (3)

The crystal structure C(r) can then be considered to arise from the con-

volution of a basis (or fundamental) domain1 B(r) with the crystal lattice

(r).

C(r) = B(r) ⊛ (r) (4)

1The basis domain is also sometimes called the motif. It is the smallest possible unit of

pattern which, by application of all translational symmetry, generates the whole crystal.

2

1.2 Lattice rows

A lattice row is a set of nodes that lie on a straight line that passes through

the origin. It can be seen from this definition that a given lattice node can

only belong to one lattice row (except the node at the origin, which belongs

to all lattice rows) . A lattice row is labelled by three integers, inserted into

square brackets ([. . . ]) without commas, which correspond to the first node

next to the origin on the row, i.e. if v = ua + vb + wc is the first node

next to the origin on the straight line, the lattice row is labelled [u v w]. As a

consequence, the integer indices u, v and w have no common integer divisor

different from one2. Also, [u v w] and [¯u ¯v ¯ w] denote the same lattice row.

A lattice row [u v w] also specifies a direction which is given by the vector

corresponding to the first node next to the origin on the row. Thus, the

direction symbols for the basis vectors are [1 0 0], [0 1 0] and [0 0 1], and often

these symbols are used in preference to a, b, c.

A lattice row can itself be considered as a one-dimensional periodic object,

i.e. the nodes on a lattice row present a repetitive pattern. If the first node

following the origin on the row is u v w, the following nodes will be 2u 2v 2w,

3u 3v 3w etc. Thus, the lattice row [u v w] contains all nodes nu nv nw with

n 2 Z.

lattice row : ruvw = n(ua + vb + wc) n 2 Z (5)

The distance between two consecutive nodes on a lattice row is called the

spacing of the lattice row3.

spacing : |ruvw| = |ua + vb + wc| (6)

1.3 Families of lattice planes

A lattice plane is a plane which passes through lattice nodes that do not all

lie on the same straight line. It can then be seen that a lattice plane contains

an infinity of nodes and the nodes on a lattice plane can be considered as a

two-dimensional periodic object, i.e. the nodes on a lattice plane present a

repetitive pattern.

A family of lattice planes is the set of all parallel lattice planes which,

taken together, contain all the nodes of the lattice. It follows from this

2When a non-primitive set of basis vectors is chosen, the first node next to the origin

on the lattice row may have half or one-third integer indices. It is then customary to

multiply all three indices by two or three so as to obtain integer indices for labelling the

lattice row.

3When a non-primitive set of basis vectors is chosen, the original (possibly non-integer)

indices must be used in this formula.

3

definition that each node belongs to all families of lattice planes. It can be

shown that neighbouring lattice planes in a family are at a constant distance

from each other. The distance between two neighbouring lattice planes in

a family is called the spacing d. Thus, a family of lattice planes is a set of

parallel and evenly spaced lattice planes.

Miller indices A family of lattice planes is labelled by three integers h, k

and l, inserted into parentheses without commas: (h k l). These indices are

also called Miller indices. Consider a family of lattice planes. We denote the

plane which passes through the origin by P◦. One of the planes in the family

will pass through the node located at a (i.e. the node 1 0 0). We denote

this plane by Pa. We define the integer h to be the number of lattice planes

which lie between P◦ and Pa plus one. In the special case where P◦ and Pa

are identical, h is set to 0. Similarly, we consider the planes Pb and Pc which

pass, respectively, through the nodes located at b and c. The integer k is

defined as the number of lattice planes which lie between P◦ and Pb plus one

and the integer l is defined as the number of lattice planes which lie between

P◦ and Pc plus one. Again, in the special cases where P◦ is identical to Pb or

Pc, the corresponding indices are set to zero. It is clear from this definition

that the Miller indices are integers.

An alternative but strictly equivalent definition is to consider the lattice

plane P1 in the family which is just next to the plane that passes through the

origin (i.e. there is no other lattice plane between P◦ and P1). This plane

cuts the [1 0 0], [0 1 0] and [0 0 1] directions at integer fractions of the basis

vectors: a/h, b/k, c/l. If the plane P1 is parallel to one of the basis vectors,

the corresponding index is set to zero. It is customary to denote the spacing

of a family of lattice planes (h k l) by dhkl.

It is important to remember that, by definition, a lattice plane must

contain lattice nodes. Similarly, all planes in a family of lattice planes must

contain lattice nodes. Therefore, the indices (2 2 0) are not valid indices for

a family of lattice planes, since the plane P1 (which cuts the basis vectors a

and b at 1

2 and which is parallel to c) would not contain any lattice nodes. It

is easy to show that valid Miller indices h, k and l have no common integer

divisor different from one4.

Normal direction and spacing Miller indices are one way to a charac-

terise a family of lattice planes. Another possibility is to specify the lattice

spacing d and the direction normal to the lattice planes by a unit vector n.

The specification of these two quantities uniquely defines a family of parallel

4This statement does not hold if non-primitive basis vectors have been chosen.

4

and evenly spaced planes. From simple geometric considerations, it can be

shown that the vector equation of a plane at a distance d from the origin

with normal direction n is given by

Plane: P = {p | n · r(p) = d} (7)

which reads: ”the plane P is the set of all point p such that n · r(p) = d,

where n and d are given parameters and r(p) is the position vector of point

p”.

If we now consider a family of parallel and evenly spaced planes of which

one plane P◦ passes through the origin, it is easy to see that the plane P1

(which is next to P◦) is at a distance d from the origin, the following plane

(P2) is at a distance 2d from the origin etc. The vector equation of such a

family of planes is therefore given by

Family of equidistant planes: = {Pm} = {p | n · r(p) = m · d} m 2 Z

(8)

Note that there are in fact two series of planes in such a family, one on each

side of P◦, corresponding respectively to positive and negative values of m.

It is convenient to group the two quantities n and d which define a family of

parallel and evenly spaced planes into a single quantity

N =

n

d

(9)

The vector equation of a family of parallel and evenly spaced planes is then

written

Family of equidistant planes: = {p |N· r(p) = m} m 2 Z (10)

It is important to note that this equation is a general equation for sets of

parallel and equidistant planes, which are not necessarily lattice planes. If

we want to restrict this equation to the particular case of families of lattice

planes, we have to impose certain restrictions on the values of d and on the

directions n. The question then arises as to which particular choices of N

do actually specify families of lattice planes ? It will be shown in the next

section that the set of all vectors N which specify families of lattice planes

do themselves generate a lattice, which is called the reciprocal lattice.

2 The reciprocal lattice

In the final paragraph of the previous section we have asserted the geometric

idea which defines the concept of a reciprocal lattice, namely that it is a

5

vector (or axial) representation of families of equidistant planes in the direct

lattice. In the present section, we will follow an apodeictic rather than as-

sertive approach where we adopt the relations between direct and reciprocal

basis vectors as a starting postulate. From this, we deduce the fundamental

geometric relation which exists between the direct and reciprocal lattices.

We then relate the reciprocal lattice to diffraction theory, thus emphasizing

the importance of this concept in understanding X-ray crystallography.

2.1 Definition of the reciprocal basis vectors

To every direct lattice we associate a reciprocal (or dual ) lattice ∗ which

is of the same dimension. The basis vectors for the reciprocal lattice, denoted

a∗, b∗, c∗ are defined by the following relations

a∗ · b = a∗ · c = 0 (11)

b∗ · a = b∗ · c = 0 (12)

c∗ · a = c∗ · b = 0 (13)

a∗ · a = b∗ · b = c∗ · c = 1 (14)

The first three of these equations specify that the reciprocal basis vector

a∗ is perpendicular to b and c, b∗ is perpendicular to a and c and c∗ is

perpendicular to a and b. The last set of equations specify the norms of the

reciprocal basis vectors as a function of the direct basis vectors.

Using elementary vector algebra, a strictly equivalent definition can be

formulated which involves vector products:

a∗ =

b × c

V

b∗ =

c × a

V

c∗ =

a × b

V

(15)

All the concepts that have been defined previously for the direct lattice

- such as nodes, unit cell, lattice rows, lattice planes - can be applied to the

reciprocal lattice. By convention, quantities which refer to the reciprocal

lattice are denoted with an ∗ superscript.

reciprocal lattice node: v∗ = ra∗ + sb∗ + tc∗ r, s, t 2 Z (16)

reciprocal lattice row: [r s t]∗ r∗

rst = n(ra∗ + sb∗ + tc∗) n 2 Z (17)

Since the vectors in the direct lattice have the dimension of a length, it

follows from equations (14) that the reciprocal lattice vectors have the di-

mension of the inverse of a length. This may at a first glance look odd, but

it must be emphasized that the two lattices exist in entirely different spaces.

Whereas the direct lattice vectors correspond to translation operations in

6

physical space, no such interpretation can be associated with reciprocal lat-

tice vectors. We will give a physical interpretation of the reciprocal lattice

vectors later in this section when we discuss its relation to diffraction. Suf-

fice it to say here that quantities, the dimension of which is the inverse of a

length, are not so uncommon in physics, think e.g. of the wavevectors k that

enter into wave equations of the type (x, t) = Acos(kx − !t). For similar

reasons, we should also resist any temptation to superimpose the direct and

reciprocal lattices on graphical representations. We should, however, remain

aware of the relation between reciprocal and direct lattice basis vectors. If

e.g. the direct lattice is rotated (which can physically correspond to the ro-

tation of the crystal), the reciprocal lattice is rotated in the same way, but

in its own space (we will later see what the physical implications of such a

rotation are).

The reciprocal unit cell is the parallelepiped generated by the basis vectors

a∗, b∗, c∗. Its volume is denoted by V ∗ and it is straightforward to show that

it is the inverse of the direct lattice unit cell volume V .

V ∗ = |a∗ · (b∗ × c∗)| = V −1 (18)

2.2 The reciprocal basis and scalar products

We shall illustrate here that the reciprocal (or dual) basis is a useful concept

in geometry (and more generally in linear algebra) even if it is not used in

connection with a lattice. In particular, if a non-orthogonal basis a, b, c is

chosen to decompose vectors in a general three-dimensional vector space, the

expression for the scalar product of two vectors becomes rather cumbersome:

v1 = x1a + y1b + z1c

v2 = x2a + y2b + z2c

) v1 · v2 = x1x2|a|2 + y1y2|b|2 + z1z2|c|2

+ (x1y2 + y1x2)|a||b| cos(a, b)

+ (x1z2 + z1x2)|a||c| cos(a, c)

+ (y1z2 + z1y2)|b||c| cos(b, c)

Although correct, this expression is in asmuch unsatisfactory as it con-

tains terms which explicitly refer to the basis vectors, whereas in its very

nature, the scalar product of two vectors is independent of the choice of ba-

sis (i.e. v1 · v2 is the same, whatever basis vectors are chosen and it does

not change under a change of basis). Remember that if an orthonormal basis

7

is chosen, the scalar product reduces to the particularly simple expression

x1x2 + y1y2 + z1z2.

The trick here is to express one of the vectors in the direct basis

v1 = xa + yb + zc

and the other one in the reciprocal basis

v2 = a∗ + b∗ +

c∗

The scalar product then writes as

v1 · v2 = x + y + z

which is an expression that only contains the vector components.

2.3 The fundamental relation between the direct and

reciprocal lattices

Returning to the reciprocal lattice, we now state and prove the fundamental

geometric relation between the direct and reciprocal lattices - a relation which

deserves your most careful attention.

To every family of lattice planes (h k l) in the direct lattice corresponds a

lattice row [h k l]∗ of the same indices in the reciprocal lattice. The reciprocal

lattice row [h k l]∗ is perpendicular to the direct lattice planes (h k l) and its

spacing |r∗

hkl| is the inverse of the direct lattice plane spacing dhkl.

[h k l]∗ ? (h k l) and |r∗

hkl| = d−1

hkl (19)

To prove these assertions, we start with the general vector equation (10)

of a family of evenly spaced parallel planes (remember that this equation is

not restricted to families of lattice planes) and we decompose the vector N

in the reciprocal basis

N · r(p) = (a∗ + b∗ +

c∗) · r(p) = m (20)

Note that at this stage of the proof, the components of N (i.e. , ,

)

are not restricted to be integers. We now explicitly specify that the family

of planes under consideration is a family of lattice planes of indices (h k l).

What restrictions does this condition impose on the components , ,

? We

consider the plane P1, which is the plane that is next to the plane passing

through the origin. We saw earlier (in section 1.3) that this plane cuts the

direct basis vectors at the points a/h, b/k and c/l. These three points

8

therefore belong to plane P1 and must therefore satisfy equation (20) for

m = 1. Thus,

(a∗ + b∗ +

c∗) ·

a

h

= 1 (21)

)

h

= 1 or: = h (22)

similarly,

(a∗ + b∗ +

c∗) ·

b

k

= 1 (23)

)

k

= 1 or: = k (24)

and,

(a∗ + b∗ +

c∗) ·

c

l

= 1 (25)

)

l

= 1 or:

= l (26)

So we have proven that for a family of direct lattice planes (h k l), the com-

ponents of the vector N expressed in the reciprocal basis are h, k, and l. But

since these are integer coefficients, the vector N also defines a row in the

reciprocal lattice:

r∗

hkl = nN = n(ha∗ + kb∗ + lc∗) n 2 Z (27)

Using the definition of the vector N, equation (9), the spacing of this recip-

rocal lattice row is given by

|r∗

hkl| = |ha∗ + kb∗ + lc∗| = |N| = d−1

hkl (28)

which completes the proof.

To conclude then, the reciprocal lattice can be viewed as an ”axial” rep-

resentation of families of lattice planes in the direct lattice. Since a family of

direct lattice planes (h k l) defines a reciprocal lattice row [h k l]∗, the nodes

on this reciprocal lattice row are

nh nk nl n 2 Z.

It has become customary to actually label these nodes as (nh nk nl), i.e.

using parentheses, as for families of lattice planes. Although this is a rather

unfortunate convention, its usage is so widespread that we will also adopt it

here. It is, however, very important to keep in mind that a reciprocal lattice

node labelled (nh nk nl) does not correspond to a family of lattice planes in

the direct lattice. We have seen earlier that (nh nk nl) are not valid indices

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for families of lattice planes (except when |n| = 1). To be more precise then:

a single reciprocal lattice node does not correspond to a family of direct

lattice planes. It is only the reciprocal lattice row to which the node belongs

(and which contains other nodes) which establishes this correspondance. The

correspondance is between a family of direct lattice planes and a reciprocal

lattice row, not a single reciprocal lattice node.

2.4 The reciprocal lattice and diffraction

2.4.1 The reciprocal lattice and the Laue equations

We now return to the diffraction of X-rays from a periodic arrangement of

matter, re-stating the Laue equations:

a · S = n1 n1 2 Z

b · S = n2 n2 2 Z

c · S = n3 n3 2 Z

Decomposing the scattering vector in the reciprocal basis

S = a∗ + b∗ +

c∗

we obtain

= n1 2 Z

= n2 2 Z

= n3 2 Z

S = n1a∗ + n2b∗ + n3c∗ n1, n2, n3 2 Z (29)

Thus, the scattering vectors which satisfy the Laue equations have integer

components when decomposed in the reciprocal basis. Hence, they form a

lattice: the reciprocal lattice !

We will henceforth denote scattering vectors which satisfy the Laue equa-

tions as H. We have just shown that the diffraction conditions are only

satisfied if the scattering vector S coincides with a reciprocal lattice node.

Each reciprocal lattice node belongs to one reciprocal lattice row [h k l]∗.

Remember that this row contains all the nodes n(ha∗ + kb∗ + lc∗), n 2 Z.

Thus,

H = n(ha∗ + kb∗ + lc∗) (30)

The reciprocal lattice can therefore be viewed as being a representation of

all those scattering vectors H that satisfy the Laue equations i.e. that can

give rise to diffraction. To each scattering vector H corresponds a reciprocal

lattice node.

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2.4.2 Bragg’s law

We can now derive another important law which relates the direct lattice to

the geometry of diffraction. We recall from equation (9) that

|N| = d−1

hkl and |S| = |H| =

2 sin

Thus,

|H| =

2 sin

= nd−1

hkl n 2 Z (31)

or

2dhkl sin = n n 2 Z (32)

This is the celebrated Bragg’s law. It relates the geometry of a diffracted

beam, expressed in terms of its reflection angle , to a property of the direct

lattice: the spacing dhkl of the (h k l) family of lattice planes.

The standard interpretation of Bragg’s law is that diffraction occurs

through the reflection of an incident beam of radiation by all lattice planes

belonging to the same family (h k l) and where the beams reflected by each

plane in the family add up by constructive interferences to give rise to a series

of beams diffracted at different orders n

sin = n ·

1

2

d−1

hkl n = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

A derivation of Bragg’s law along these lines can be found in almost any

introductory text on X-ray crystallography. Although such a simple inter-

pretation is appealing, it presents some conceptual difficulties. In particular

the notion of X-rays that are reflected by lattice planes is rather problematic

since, as we have stressed before, there are no lattice planes in a crystal.

X-rays are scattered by the electrons in a crystal and these electrons are

certainly not located in planes (think of a crystal which is build up by com-

plicated molecules). Therefore, the picture of X-rays that are reflected by

planes is at best a useful optical analogy, but in no way does it correspond

the physical reality of diffraction. This does not, of course, invalidate Bragg’s

law, which can be directly derived from the Laue equations as we have shown

here.

2.4.3 An important comment regarding (h k l) indices

We saw earlier that it is customary to label reciprocal lattice nodes by their

indices, inserted into parentheses, and that these indices may have a common

divisor greater than one (contrary to Miller indices that are used to label

11

families of lattice planes). Thus, (6 3 3) is a valid label for a reciprocal lattice

node, although it does not correspond to a family of lattice planes. (We

should keep in mind that the reciprocal lattice node (6 3 3) is actually the 3rd

node on the lattice row [2 1 1]∗ and that this reciprocal lattice row corresponds

to the family of direct lattice planes (2 1 1)). Since each scattering vector H

is associated with a reciprocal lattice node, the scattering vectors are also

labelled by the indices of the nodes. Thus, in the jargon of crystallography,

we use e.g. the expression ”reflection (6 3 3)” to designate a beam diffracted

at H = 6 a∗ + 3 b∗ + 3 c∗. But, using the Bragg picture of diffraction, it

would be wrong to state that this reflection arises from the (6 3 3) lattice

planes. It is actually the 3rd order reflection (n = 3 in Bragg’s law) from

the (2 1 1) family of lattice planes. However, in the context of diffraction

it is generally irrelevant whether a reflection is fundamental (i.e. n = 1)

or of higher order so that they are all treated equally and there is then no

restriction on the (h k l) indices (except that they must be integers) that are

used to label ”reflections”.

With regard to Bragg’s law, treating all reflections on an equal footing

irrespective of their order actually amounts to incorporating the integer n

into the lattice spacing i.e. replacing

dhkl by

dhkl

n

Is then also possible to formally define the quantity

dhkl = |H|−1 = |ha∗ + kb∗ + lc∗|−1 (33)

irrespective of whether the indices h k l have a common divisor greater than

one or not. Bragg’s law then writes as

2dhkl sin = (34)

and we call the quantity dhkl defined by equation (33) simply the Bragg spac-

ing of reflection (h k l), remembering that it does not necessarily correspond

to the spacing of a family of lattice planes.

To summarise, if the first version of Bragg’s law, equation (32), is used,

the quantity dhkl corresponds to a true spacing of lattice planes that are

labelled by (h k l) indices which have no common divisor greater than one.

The order of the reflection is given by n. If the second version of Bragg’s

law, equation (34), is used, the quantity dhkl simply corresponds to the Bragg

spacing of the reflection (h k l) as defined by equation (33), where there is no

restriction on the indices.

12

2.5 The weighted reciprocal lattice

We stated at the end of section 1.1 that, from a more formal mathematical

point of view, the direct lattice (r) can be considered to consist of a set of

periodically spaced Dirac (r − r◦) distributions. It can be shown that the

reciprocal lattice ∗(S) is the Fourier cotransform (F¯T) of the direct lattice.

∗(S) = F¯T[] (35)

We also stated that the crystal structure C(r) can be considered to arise from

the convolution of a basis domain B(r) with the direct lattice (r). Using

the convolution theorem5, we can then see that the Fourier cotransform of

the crystal structure is the product of the Fourier cotransform of the basis

domain, denoted F(S) with the reciprocal lattice.

F¯T[C] = F¯T[B] · F¯T[] = F(S) · ∗(S) (36)

Thus, to each reciprocal lattice node (h k l), we can associate a complex

number that is given by the value of the transform of the basis domain F(H)

at the location H = ha∗ + kb∗ + lc∗ of the node. We say that each node is

”weighted” by a complex value and we call this the structure factor-weighted

reciprocal lattice.

From the point of view of diffraction, the related concept of intensity-

weighted reciprocal lattice is perhaps more important. The intensity-weighted

reciprocal lattice is obtained by associating with each reciprocal lattice node

H the square modulus of the structure factor |F(H)|2 at the location of the

node. The intensity of a beam diffracted at scattering vector H can be shown

to be proportional to |F(H)|2.

I(H) / |F(H)|2 (37)

The intensity-weighted reciprocal lattice is therefore accessible to experi-

mental investigations, since the vectors H and the intensities I of reciprocal-

lattice nodes are measurable quantities. The two categories of measurable

quantities are fairly different and contain complementary structural informa-

tion. The scattering vectors (i.e. a set of H) can be obtained from investi-

gating the geometry of diffracted beams (i.e. analysing the directions s◦ and

s corresponding respectively to the incident and diffracted beams). From

this purely geometric information, the characteristics of the reciprocal lat-

tice can be determined, hence also those of the direct lattice. We gain thus

5The convolution theorem states that the Fourier transform or cotransform of a con-

volution of two functions is the product of the Fourier transforms or cotransforms of each

function: F¯T[f ⊛ g] = F¯T[f] · F¯T[g]

13

information about the translational symmetries in the crystals and, more

generally, about the crystal packing. However, this information does not tell

us anything about the structure of the basis domain (which can e.g. be a

complicated molecular system). In order to obtain the genuine structural

information, it is necessary to measure the intensities I(H) of individual

diffracted beams, since these are related to the structure factors which are

Fourier cotransforms of the basis domain:

{I(H)}

Phase problem

−−−−−−−−−−−−−−! {F(H)}

Fourier transform

−−−−−−−−−−−−−−−−! B(r) (38)

The first step in this analysis is in asmuch problematic as it is impossible to

obtain a structure factor (which, in general, is a complex quantity) from its

modulus alone. Alas the diffracted intensities do not return any information

about the phase angles of the complex structure factors, so that this infor-

mation is not directly available from experiments. This is the famous phase

problem of X-ray crystallography. Many efforts have been devoted to invent

methods and techniques (both experimental and statistical) to overcome this

problem and recover the missing information. Another, somewhat less im-

portant problem arises from the fact that the Fourier transform involved

in the second step is, in principle, a summation over an infinite number of

terms. In practice, however, only a finite number of diffracted beams can

be measured. Truncation of the Fourier summation to a finite number of

terms ultimately puts a limit to the level of detail and precision to which the

structure B(r) can be determined.

A final word is in order concerning the symmetry of the weighted recip-

rocal lattice. An unweighted lattice possesses translational symmetry, since

all nodes are equivalent. However, if we put weights on each node (either

structure factor-weights or intensity-weights), the nodes are not equivalent

anymore. As a consequence, a weighted lattice has no translational symmetry

! It may however possess point symmetries (i.e. proper and improper rota-

tion symmetries) and the inspection of these symmetries yields information

about the symmetries of the crystal itself.

14

1 The direct lattice

1.1 Definition

Crystals are characterised by periodicity in three-dimensional space. This

notion of periodicity is embodied in the concept of a crystal lattice. The

lattice is a set of vectors v which characterise the translational symmetries

of a crystal. In other words, if a lattice translation is applied to the crystal

(considered to be an infinite object), the translated crystal superimposes with

the original one. The lattice is then viewed as an infinite set of vectors v that

can be expressed as linear combinations of some non-coplanar basis vectors

a, b, c with integer coefficients u, v,w.

v = ua + vb + wc u, v,w 2 Z (1)

The lattice representing the translational symmetry operations of a crystal

is also called the direct lattice in order to distinguish it from the reciprocal

lattice which will be defined later. A crystal can of course possess more

symmetry operations than just the translational symmetry (i.e. rotation

symmetry, inversion etc.). The full symmetry of a crystal is given by its

space group, of which the lattice is a subgroup. It is customary to define a

node in a lattice as representing the extremity (the tip) of a lattice vector v.

The lattice can then also be viewed as a set of nodes that are periodically

arranged in space. A node is labelled by the set of three integers u v w written

without commas. By convention, the horizontal bar is used as a symbol for

negative values of indices, e.g. ¯2 0 3 stands for the node at v = −2 a+0 b+3 c

It is important to realize that the lattice is an abstract notion and we

should resist any temptation to identify lattice nodes with e.g. atoms, ions

or molecules. The lattice should not be confonded with the true, physical,

1

crystal space and the two should not be superimposed on each other. Lattice

nodes, rows and planes (which will be defined below) do not exist in the

crystal !

For a given lattice, there exist an infinite number of possible choices for

the basis vectors a, b, c. Note that in general it might not be possible to

choose orthogonal basis vectors. The basis vectors must be non-coplanar (i.e.

linearly independent) and the set of all integer combinations of basis vectors

must generate the whole lattice. Such basis vectors are called primitive.

We then introduce the notion of unit cell - a concept which, unfortunately,

does not have a unique definition. In the context of the present chapter,

we designate by unit cell the parallelepiped that is generated by the basis

vectors a, b, c. It can be shown that the unit cell associated with a primitive

set of basis vectors does not contain any lattice nodes (apart from the 8 nodes

sitting on its vertices). Such a unit cell is therefore also called a primitive

cell. It can, however, be advantageous for reasons of symmetry to chose a

different set of basis vectors, called conventional, which may be non-primitive

(c.f. lectures on crystal symmetry). With a conventional set of basis vectors,

it may be necessary to include half or one-third integer coefficients u, v,w

to generate the whole lattice. Non-primitive unit cells contain lattice nodes

inside their volume or on their faces. For simplicity, we will assume that all

sets of basis vectors are primitive for the remainder of the discussion in this

section, but we will occasionally indicate implications that arise from the

choice of non-primitive basis vectors. The unit cell volume V is given by the

mixed product of the basis vectors.

V = |a · (b × c)| (2)

It is conventional to chose a right-handed set of basis vectors, i.e. a set of

vectors for which a · (b × c) > 0.

From a more mathematical point of view, the lattice (r) can be viewed

as a set of periodically spaced Dirac distributions, i.e. a Dirac (r − r◦)

distribution is located on each node of the lattice.

(r) =

+∞

X

u=−∞

+∞

X

v=−∞

+∞

X

w=−∞

(r − ua − vb − wc) (3)

The crystal structure C(r) can then be considered to arise from the con-

volution of a basis (or fundamental) domain1 B(r) with the crystal lattice

(r).

C(r) = B(r) ⊛ (r) (4)

1The basis domain is also sometimes called the motif. It is the smallest possible unit of

pattern which, by application of all translational symmetry, generates the whole crystal.

2

1.2 Lattice rows

A lattice row is a set of nodes that lie on a straight line that passes through

the origin. It can be seen from this definition that a given lattice node can

only belong to one lattice row (except the node at the origin, which belongs

to all lattice rows) . A lattice row is labelled by three integers, inserted into

square brackets ([. . . ]) without commas, which correspond to the first node

next to the origin on the row, i.e. if v = ua + vb + wc is the first node

next to the origin on the straight line, the lattice row is labelled [u v w]. As a

consequence, the integer indices u, v and w have no common integer divisor

different from one2. Also, [u v w] and [¯u ¯v ¯ w] denote the same lattice row.

A lattice row [u v w] also specifies a direction which is given by the vector

corresponding to the first node next to the origin on the row. Thus, the

direction symbols for the basis vectors are [1 0 0], [0 1 0] and [0 0 1], and often

these symbols are used in preference to a, b, c.

A lattice row can itself be considered as a one-dimensional periodic object,

i.e. the nodes on a lattice row present a repetitive pattern. If the first node

following the origin on the row is u v w, the following nodes will be 2u 2v 2w,

3u 3v 3w etc. Thus, the lattice row [u v w] contains all nodes nu nv nw with

n 2 Z.

lattice row : ruvw = n(ua + vb + wc) n 2 Z (5)

The distance between two consecutive nodes on a lattice row is called the

spacing of the lattice row3.

spacing : |ruvw| = |ua + vb + wc| (6)

1.3 Families of lattice planes

A lattice plane is a plane which passes through lattice nodes that do not all

lie on the same straight line. It can then be seen that a lattice plane contains

an infinity of nodes and the nodes on a lattice plane can be considered as a

two-dimensional periodic object, i.e. the nodes on a lattice plane present a

repetitive pattern.

A family of lattice planes is the set of all parallel lattice planes which,

taken together, contain all the nodes of the lattice. It follows from this

2When a non-primitive set of basis vectors is chosen, the first node next to the origin

on the lattice row may have half or one-third integer indices. It is then customary to

multiply all three indices by two or three so as to obtain integer indices for labelling the

lattice row.

3When a non-primitive set of basis vectors is chosen, the original (possibly non-integer)

indices must be used in this formula.

3

definition that each node belongs to all families of lattice planes. It can be

shown that neighbouring lattice planes in a family are at a constant distance

from each other. The distance between two neighbouring lattice planes in

a family is called the spacing d. Thus, a family of lattice planes is a set of

parallel and evenly spaced lattice planes.

Miller indices A family of lattice planes is labelled by three integers h, k

and l, inserted into parentheses without commas: (h k l). These indices are

also called Miller indices. Consider a family of lattice planes. We denote the

plane which passes through the origin by P◦. One of the planes in the family

will pass through the node located at a (i.e. the node 1 0 0). We denote

this plane by Pa. We define the integer h to be the number of lattice planes

which lie between P◦ and Pa plus one. In the special case where P◦ and Pa

are identical, h is set to 0. Similarly, we consider the planes Pb and Pc which

pass, respectively, through the nodes located at b and c. The integer k is

defined as the number of lattice planes which lie between P◦ and Pb plus one

and the integer l is defined as the number of lattice planes which lie between

P◦ and Pc plus one. Again, in the special cases where P◦ is identical to Pb or

Pc, the corresponding indices are set to zero. It is clear from this definition

that the Miller indices are integers.

An alternative but strictly equivalent definition is to consider the lattice

plane P1 in the family which is just next to the plane that passes through the

origin (i.e. there is no other lattice plane between P◦ and P1). This plane

cuts the [1 0 0], [0 1 0] and [0 0 1] directions at integer fractions of the basis

vectors: a/h, b/k, c/l. If the plane P1 is parallel to one of the basis vectors,

the corresponding index is set to zero. It is customary to denote the spacing

of a family of lattice planes (h k l) by dhkl.

It is important to remember that, by definition, a lattice plane must

contain lattice nodes. Similarly, all planes in a family of lattice planes must

contain lattice nodes. Therefore, the indices (2 2 0) are not valid indices for

a family of lattice planes, since the plane P1 (which cuts the basis vectors a

and b at 1

2 and which is parallel to c) would not contain any lattice nodes. It

is easy to show that valid Miller indices h, k and l have no common integer

divisor different from one4.

Normal direction and spacing Miller indices are one way to a charac-

terise a family of lattice planes. Another possibility is to specify the lattice

spacing d and the direction normal to the lattice planes by a unit vector n.

The specification of these two quantities uniquely defines a family of parallel

4This statement does not hold if non-primitive basis vectors have been chosen.

4

and evenly spaced planes. From simple geometric considerations, it can be

shown that the vector equation of a plane at a distance d from the origin

with normal direction n is given by

Plane: P = {p | n · r(p) = d} (7)

which reads: ”the plane P is the set of all point p such that n · r(p) = d,

where n and d are given parameters and r(p) is the position vector of point

p”.

If we now consider a family of parallel and evenly spaced planes of which

one plane P◦ passes through the origin, it is easy to see that the plane P1

(which is next to P◦) is at a distance d from the origin, the following plane

(P2) is at a distance 2d from the origin etc. The vector equation of such a

family of planes is therefore given by

Family of equidistant planes: = {Pm} = {p | n · r(p) = m · d} m 2 Z

(8)

Note that there are in fact two series of planes in such a family, one on each

side of P◦, corresponding respectively to positive and negative values of m.

It is convenient to group the two quantities n and d which define a family of

parallel and evenly spaced planes into a single quantity

N =

n

d

(9)

The vector equation of a family of parallel and evenly spaced planes is then

written

Family of equidistant planes: = {p |N· r(p) = m} m 2 Z (10)

It is important to note that this equation is a general equation for sets of

parallel and equidistant planes, which are not necessarily lattice planes. If

we want to restrict this equation to the particular case of families of lattice

planes, we have to impose certain restrictions on the values of d and on the

directions n. The question then arises as to which particular choices of N

do actually specify families of lattice planes ? It will be shown in the next

section that the set of all vectors N which specify families of lattice planes

do themselves generate a lattice, which is called the reciprocal lattice.

2 The reciprocal lattice

In the final paragraph of the previous section we have asserted the geometric

idea which defines the concept of a reciprocal lattice, namely that it is a

5

vector (or axial) representation of families of equidistant planes in the direct

lattice. In the present section, we will follow an apodeictic rather than as-

sertive approach where we adopt the relations between direct and reciprocal

basis vectors as a starting postulate. From this, we deduce the fundamental

geometric relation which exists between the direct and reciprocal lattices.

We then relate the reciprocal lattice to diffraction theory, thus emphasizing

the importance of this concept in understanding X-ray crystallography.

2.1 Definition of the reciprocal basis vectors

To every direct lattice we associate a reciprocal (or dual ) lattice ∗ which

is of the same dimension. The basis vectors for the reciprocal lattice, denoted

a∗, b∗, c∗ are defined by the following relations

a∗ · b = a∗ · c = 0 (11)

b∗ · a = b∗ · c = 0 (12)

c∗ · a = c∗ · b = 0 (13)

a∗ · a = b∗ · b = c∗ · c = 1 (14)

The first three of these equations specify that the reciprocal basis vector

a∗ is perpendicular to b and c, b∗ is perpendicular to a and c and c∗ is

perpendicular to a and b. The last set of equations specify the norms of the

reciprocal basis vectors as a function of the direct basis vectors.

Using elementary vector algebra, a strictly equivalent definition can be

formulated which involves vector products:

a∗ =

b × c

V

b∗ =

c × a

V

c∗ =

a × b

V

(15)

All the concepts that have been defined previously for the direct lattice

- such as nodes, unit cell, lattice rows, lattice planes - can be applied to the

reciprocal lattice. By convention, quantities which refer to the reciprocal

lattice are denoted with an ∗ superscript.

reciprocal lattice node: v∗ = ra∗ + sb∗ + tc∗ r, s, t 2 Z (16)

reciprocal lattice row: [r s t]∗ r∗

rst = n(ra∗ + sb∗ + tc∗) n 2 Z (17)

Since the vectors in the direct lattice have the dimension of a length, it

follows from equations (14) that the reciprocal lattice vectors have the di-

mension of the inverse of a length. This may at a first glance look odd, but

it must be emphasized that the two lattices exist in entirely different spaces.

Whereas the direct lattice vectors correspond to translation operations in

6

physical space, no such interpretation can be associated with reciprocal lat-

tice vectors. We will give a physical interpretation of the reciprocal lattice

vectors later in this section when we discuss its relation to diffraction. Suf-

fice it to say here that quantities, the dimension of which is the inverse of a

length, are not so uncommon in physics, think e.g. of the wavevectors k that

enter into wave equations of the type (x, t) = Acos(kx − !t). For similar

reasons, we should also resist any temptation to superimpose the direct and

reciprocal lattices on graphical representations. We should, however, remain

aware of the relation between reciprocal and direct lattice basis vectors. If

e.g. the direct lattice is rotated (which can physically correspond to the ro-

tation of the crystal), the reciprocal lattice is rotated in the same way, but

in its own space (we will later see what the physical implications of such a

rotation are).

The reciprocal unit cell is the parallelepiped generated by the basis vectors

a∗, b∗, c∗. Its volume is denoted by V ∗ and it is straightforward to show that

it is the inverse of the direct lattice unit cell volume V .

V ∗ = |a∗ · (b∗ × c∗)| = V −1 (18)

2.2 The reciprocal basis and scalar products

We shall illustrate here that the reciprocal (or dual) basis is a useful concept

in geometry (and more generally in linear algebra) even if it is not used in

connection with a lattice. In particular, if a non-orthogonal basis a, b, c is

chosen to decompose vectors in a general three-dimensional vector space, the

expression for the scalar product of two vectors becomes rather cumbersome:

v1 = x1a + y1b + z1c

v2 = x2a + y2b + z2c

) v1 · v2 = x1x2|a|2 + y1y2|b|2 + z1z2|c|2

+ (x1y2 + y1x2)|a||b| cos(a, b)

+ (x1z2 + z1x2)|a||c| cos(a, c)

+ (y1z2 + z1y2)|b||c| cos(b, c)

Although correct, this expression is in asmuch unsatisfactory as it con-

tains terms which explicitly refer to the basis vectors, whereas in its very

nature, the scalar product of two vectors is independent of the choice of ba-

sis (i.e. v1 · v2 is the same, whatever basis vectors are chosen and it does

not change under a change of basis). Remember that if an orthonormal basis

7

is chosen, the scalar product reduces to the particularly simple expression

x1x2 + y1y2 + z1z2.

The trick here is to express one of the vectors in the direct basis

v1 = xa + yb + zc

and the other one in the reciprocal basis

v2 = a∗ + b∗ +

c∗

The scalar product then writes as

v1 · v2 = x + y + z

which is an expression that only contains the vector components.

2.3 The fundamental relation between the direct and

reciprocal lattices

Returning to the reciprocal lattice, we now state and prove the fundamental

geometric relation between the direct and reciprocal lattices - a relation which

deserves your most careful attention.

To every family of lattice planes (h k l) in the direct lattice corresponds a

lattice row [h k l]∗ of the same indices in the reciprocal lattice. The reciprocal

lattice row [h k l]∗ is perpendicular to the direct lattice planes (h k l) and its

spacing |r∗

hkl| is the inverse of the direct lattice plane spacing dhkl.

[h k l]∗ ? (h k l) and |r∗

hkl| = d−1

hkl (19)

To prove these assertions, we start with the general vector equation (10)

of a family of evenly spaced parallel planes (remember that this equation is

not restricted to families of lattice planes) and we decompose the vector N

in the reciprocal basis

N · r(p) = (a∗ + b∗ +

c∗) · r(p) = m (20)

Note that at this stage of the proof, the components of N (i.e. , ,

)

are not restricted to be integers. We now explicitly specify that the family

of planes under consideration is a family of lattice planes of indices (h k l).

What restrictions does this condition impose on the components , ,

? We

consider the plane P1, which is the plane that is next to the plane passing

through the origin. We saw earlier (in section 1.3) that this plane cuts the

direct basis vectors at the points a/h, b/k and c/l. These three points

8

therefore belong to plane P1 and must therefore satisfy equation (20) for

m = 1. Thus,

(a∗ + b∗ +

c∗) ·

a

h

= 1 (21)

)

h

= 1 or: = h (22)

similarly,

(a∗ + b∗ +

c∗) ·

b

k

= 1 (23)

)

k

= 1 or: = k (24)

and,

(a∗ + b∗ +

c∗) ·

c

l

= 1 (25)

)

l

= 1 or:

= l (26)

So we have proven that for a family of direct lattice planes (h k l), the com-

ponents of the vector N expressed in the reciprocal basis are h, k, and l. But

since these are integer coefficients, the vector N also defines a row in the

reciprocal lattice:

r∗

hkl = nN = n(ha∗ + kb∗ + lc∗) n 2 Z (27)

Using the definition of the vector N, equation (9), the spacing of this recip-

rocal lattice row is given by

|r∗

hkl| = |ha∗ + kb∗ + lc∗| = |N| = d−1

hkl (28)

which completes the proof.

To conclude then, the reciprocal lattice can be viewed as an ”axial” rep-

resentation of families of lattice planes in the direct lattice. Since a family of

direct lattice planes (h k l) defines a reciprocal lattice row [h k l]∗, the nodes

on this reciprocal lattice row are

nh nk nl n 2 Z.

It has become customary to actually label these nodes as (nh nk nl), i.e.

using parentheses, as for families of lattice planes. Although this is a rather

unfortunate convention, its usage is so widespread that we will also adopt it

here. It is, however, very important to keep in mind that a reciprocal lattice

node labelled (nh nk nl) does not correspond to a family of lattice planes in

the direct lattice. We have seen earlier that (nh nk nl) are not valid indices

9

for families of lattice planes (except when |n| = 1). To be more precise then:

a single reciprocal lattice node does not correspond to a family of direct

lattice planes. It is only the reciprocal lattice row to which the node belongs

(and which contains other nodes) which establishes this correspondance. The

correspondance is between a family of direct lattice planes and a reciprocal

lattice row, not a single reciprocal lattice node.

2.4 The reciprocal lattice and diffraction

2.4.1 The reciprocal lattice and the Laue equations

We now return to the diffraction of X-rays from a periodic arrangement of

matter, re-stating the Laue equations:

a · S = n1 n1 2 Z

b · S = n2 n2 2 Z

c · S = n3 n3 2 Z

Decomposing the scattering vector in the reciprocal basis

S = a∗ + b∗ +

c∗

we obtain

= n1 2 Z

= n2 2 Z

= n3 2 Z

S = n1a∗ + n2b∗ + n3c∗ n1, n2, n3 2 Z (29)

Thus, the scattering vectors which satisfy the Laue equations have integer

components when decomposed in the reciprocal basis. Hence, they form a

lattice: the reciprocal lattice !

We will henceforth denote scattering vectors which satisfy the Laue equa-

tions as H. We have just shown that the diffraction conditions are only

satisfied if the scattering vector S coincides with a reciprocal lattice node.

Each reciprocal lattice node belongs to one reciprocal lattice row [h k l]∗.

Remember that this row contains all the nodes n(ha∗ + kb∗ + lc∗), n 2 Z.

Thus,

H = n(ha∗ + kb∗ + lc∗) (30)

The reciprocal lattice can therefore be viewed as being a representation of

all those scattering vectors H that satisfy the Laue equations i.e. that can

give rise to diffraction. To each scattering vector H corresponds a reciprocal

lattice node.

10

2.4.2 Bragg’s law

We can now derive another important law which relates the direct lattice to

the geometry of diffraction. We recall from equation (9) that

|N| = d−1

hkl and |S| = |H| =

2 sin

Thus,

|H| =

2 sin

= nd−1

hkl n 2 Z (31)

or

2dhkl sin = n n 2 Z (32)

This is the celebrated Bragg’s law. It relates the geometry of a diffracted

beam, expressed in terms of its reflection angle , to a property of the direct

lattice: the spacing dhkl of the (h k l) family of lattice planes.

The standard interpretation of Bragg’s law is that diffraction occurs

through the reflection of an incident beam of radiation by all lattice planes

belonging to the same family (h k l) and where the beams reflected by each

plane in the family add up by constructive interferences to give rise to a series

of beams diffracted at different orders n

sin = n ·

1

2

d−1

hkl n = . . . ,−3,−2,−1, 0, 1, 2, 3, . . .

A derivation of Bragg’s law along these lines can be found in almost any

introductory text on X-ray crystallography. Although such a simple inter-

pretation is appealing, it presents some conceptual difficulties. In particular

the notion of X-rays that are reflected by lattice planes is rather problematic

since, as we have stressed before, there are no lattice planes in a crystal.

X-rays are scattered by the electrons in a crystal and these electrons are

certainly not located in planes (think of a crystal which is build up by com-

plicated molecules). Therefore, the picture of X-rays that are reflected by

planes is at best a useful optical analogy, but in no way does it correspond

the physical reality of diffraction. This does not, of course, invalidate Bragg’s

law, which can be directly derived from the Laue equations as we have shown

here.

2.4.3 An important comment regarding (h k l) indices

We saw earlier that it is customary to label reciprocal lattice nodes by their

indices, inserted into parentheses, and that these indices may have a common

divisor greater than one (contrary to Miller indices that are used to label

11

families of lattice planes). Thus, (6 3 3) is a valid label for a reciprocal lattice

node, although it does not correspond to a family of lattice planes. (We

should keep in mind that the reciprocal lattice node (6 3 3) is actually the 3rd

node on the lattice row [2 1 1]∗ and that this reciprocal lattice row corresponds

to the family of direct lattice planes (2 1 1)). Since each scattering vector H

is associated with a reciprocal lattice node, the scattering vectors are also

labelled by the indices of the nodes. Thus, in the jargon of crystallography,

we use e.g. the expression ”reflection (6 3 3)” to designate a beam diffracted

at H = 6 a∗ + 3 b∗ + 3 c∗. But, using the Bragg picture of diffraction, it

would be wrong to state that this reflection arises from the (6 3 3) lattice

planes. It is actually the 3rd order reflection (n = 3 in Bragg’s law) from

the (2 1 1) family of lattice planes. However, in the context of diffraction

it is generally irrelevant whether a reflection is fundamental (i.e. n = 1)

or of higher order so that they are all treated equally and there is then no

restriction on the (h k l) indices (except that they must be integers) that are

used to label ”reflections”.

With regard to Bragg’s law, treating all reflections on an equal footing

irrespective of their order actually amounts to incorporating the integer n

into the lattice spacing i.e. replacing

dhkl by

dhkl

n

Is then also possible to formally define the quantity

dhkl = |H|−1 = |ha∗ + kb∗ + lc∗|−1 (33)

irrespective of whether the indices h k l have a common divisor greater than

one or not. Bragg’s law then writes as

2dhkl sin = (34)

and we call the quantity dhkl defined by equation (33) simply the Bragg spac-

ing of reflection (h k l), remembering that it does not necessarily correspond

to the spacing of a family of lattice planes.

To summarise, if the first version of Bragg’s law, equation (32), is used,

the quantity dhkl corresponds to a true spacing of lattice planes that are

labelled by (h k l) indices which have no common divisor greater than one.

The order of the reflection is given by n. If the second version of Bragg’s

law, equation (34), is used, the quantity dhkl simply corresponds to the Bragg

spacing of the reflection (h k l) as defined by equation (33), where there is no

restriction on the indices.

12

2.5 The weighted reciprocal lattice

We stated at the end of section 1.1 that, from a more formal mathematical

point of view, the direct lattice (r) can be considered to consist of a set of

periodically spaced Dirac (r − r◦) distributions. It can be shown that the

reciprocal lattice ∗(S) is the Fourier cotransform (F¯T) of the direct lattice.

∗(S) = F¯T[] (35)

We also stated that the crystal structure C(r) can be considered to arise from

the convolution of a basis domain B(r) with the direct lattice (r). Using

the convolution theorem5, we can then see that the Fourier cotransform of

the crystal structure is the product of the Fourier cotransform of the basis

domain, denoted F(S) with the reciprocal lattice.

F¯T[C] = F¯T[B] · F¯T[] = F(S) · ∗(S) (36)

Thus, to each reciprocal lattice node (h k l), we can associate a complex

number that is given by the value of the transform of the basis domain F(H)

at the location H = ha∗ + kb∗ + lc∗ of the node. We say that each node is

”weighted” by a complex value and we call this the structure factor-weighted

reciprocal lattice.

From the point of view of diffraction, the related concept of intensity-

weighted reciprocal lattice is perhaps more important. The intensity-weighted

reciprocal lattice is obtained by associating with each reciprocal lattice node

H the square modulus of the structure factor |F(H)|2 at the location of the

node. The intensity of a beam diffracted at scattering vector H can be shown

to be proportional to |F(H)|2.

I(H) / |F(H)|2 (37)

The intensity-weighted reciprocal lattice is therefore accessible to experi-

mental investigations, since the vectors H and the intensities I of reciprocal-

lattice nodes are measurable quantities. The two categories of measurable

quantities are fairly different and contain complementary structural informa-

tion. The scattering vectors (i.e. a set of H) can be obtained from investi-

gating the geometry of diffracted beams (i.e. analysing the directions s◦ and

s corresponding respectively to the incident and diffracted beams). From

this purely geometric information, the characteristics of the reciprocal lat-

tice can be determined, hence also those of the direct lattice. We gain thus

5The convolution theorem states that the Fourier transform or cotransform of a con-

volution of two functions is the product of the Fourier transforms or cotransforms of each

function: F¯T[f ⊛ g] = F¯T[f] · F¯T[g]

13

information about the translational symmetries in the crystals and, more

generally, about the crystal packing. However, this information does not tell

us anything about the structure of the basis domain (which can e.g. be a

complicated molecular system). In order to obtain the genuine structural

information, it is necessary to measure the intensities I(H) of individual

diffracted beams, since these are related to the structure factors which are

Fourier cotransforms of the basis domain:

{I(H)}

Phase problem

−−−−−−−−−−−−−−! {F(H)}

Fourier transform

−−−−−−−−−−−−−−−−! B(r) (38)

The first step in this analysis is in asmuch problematic as it is impossible to

obtain a structure factor (which, in general, is a complex quantity) from its

modulus alone. Alas the diffracted intensities do not return any information

about the phase angles of the complex structure factors, so that this infor-

mation is not directly available from experiments. This is the famous phase

problem of X-ray crystallography. Many efforts have been devoted to invent

methods and techniques (both experimental and statistical) to overcome this

problem and recover the missing information. Another, somewhat less im-

portant problem arises from the fact that the Fourier transform involved

in the second step is, in principle, a summation over an infinite number of

terms. In practice, however, only a finite number of diffracted beams can

be measured. Truncation of the Fourier summation to a finite number of

terms ultimately puts a limit to the level of detail and precision to which the

structure B(r) can be determined.

A final word is in order concerning the symmetry of the weighted recip-

rocal lattice. An unweighted lattice possesses translational symmetry, since

all nodes are equivalent. However, if we put weights on each node (either

structure factor-weights or intensity-weights), the nodes are not equivalent

anymore. As a consequence, a weighted lattice has no translational symmetry

! It may however possess point symmetries (i.e. proper and improper rota-

tion symmetries) and the inspection of these symmetries yields information

about the symmetries of the crystal itself.

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