Blake A.J.(ed.) Crystal Structure Analysis

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342 X-ray and neutron sources

References

Clegg, W. (2000). J. Chem. Soc., Dalton Trans. 3223–3232.

Helliwell, J. R. (1998). Acta Crystallogr. A54, 738–749.

Piccoli, P. M., Koetzle, T. F. and Schultz, A. J. (2007). Comments Inorg.

Chem. 28, 3–38.

Wilson, C. C. (2000). Single crystal neutron diffraction from molecular

materials, World Scientiﬁc: Singapore.

Appendix A: Useful

mathematics and

formulae

Peter Main

A.1 Introduction

To paraphrase Lord Kelvin, when you cannot express your observations

in numbers, your knowledge is of a meagre and unsatisfactory kind.

The use of scientiﬁc observation to add to our knowledge inevitably

means we need to express both observations and deductions mathe-

matically. The link between the two is mathematical also. We present

here some mathematics and a few formulae that are important in X-ray

crystallography.

Fig. A.1 A right-angled triangle for deﬁn-

ing trigonometric ratios.

A.2 Trigonometry

Trigonometry means ‘measurement of triangles’, but its use goes far

beyond what its name suggests. Many properties of triangles can be

summarized in terms of the ratios of thesides ofthe right-angledtriangle

in Fig. A.1, giving:

cos θ = a/c sin θ = b/c tan θ = b/a

so that tan θ = sin θ/cos θ.

The symmetry of the sine and cosine functions shows that cos(−θ) =

cos(θ) and sin(−θ) =−sin(θ).

Another relationship among these functions is obtained from

Pythagoras’ theorem:

+ b

= c

giving cos

θ +sin

θ = 1.

343

344 Useful mathematics and formulae

Also useful in crystallography are the multiple angle formulae, which

are given without derivation as:

cos(θ +φ) = cos θ cosφ − sin θ sin φ

and sin(θ + φ) = sin θ cosφ + cos θ sin φ.

These come into their own in the manipulation of the electron-density

equation for numerical calculation. For example, by putting θ = 2π(hx+

ky) and φ = 2πlz in the above expressions, cos2π(hx + ky + lz) can be

changed into:

cos 2π(hx + ky + lz) = cos2π(hx + ky) cos 2π lz

− sin 2π(hx + ky) sin 2πlz.

A similar operation gives

cos 2π(hx + ky) = cos(2πhx) cos(2π ky) − sin(2π hx) sin(2πky)

sin 2

π(hx + ky) = sin(2πhx) cos(2π ky) + cos(2πhx) sin(2πky),

so that

cos 2π(hx + ky + lz) = cos(2πhx) cos(2πky) cos(2π lz)

− sin(2π hx) sin(2π ky) cos(2πlz)

− sin(2π hx) cos(2πky) sin(2πlz)

− cos(2πhx) sin(2πky) sin(2πlz).

It looks

as if we have made things far more complicated by doing

this. However, these expressions usually simplify enormously in dif-

ferent ways according to space group symmetry and are useful in the

Beevers–Lipson factorization of the electron-density equation, which is

how many computer programs handle Fourier transform summations.

imaginary

real

Fig. A.2 The complex number a+ib plotted

on an Argand diagram.

A.3 Complex numbers

Much of the mathematics dealing with structure factors and discrete

Fourier transforms makes use of complex numbers. It is a pity these

numbers have the name they do, because it has the connotation of being

complicated. Complex numbers are simply numbers with two compo-

nents instead of the usual one. The components are called the real and

imaginary parts of the number and can be plotted on a two-dimensional

diagram, called an Argand diagram, as shown in Fig. A.2. The number

plotted has real and imaginary parts of a and b, respectively, and can be

written algebraically as a +ib where i

=−1. You may regard the imag-

inary constant i as a mathematical curiosity, but the important property

of its square given in the previous sentence enables complex numbers

to be multiplied and divided in a completely consistent way.

A.4 Waves and structure factors 345

An equivalent way to representa complex number is in polar form, i.e.

interms of r andθ inFig.A.2.These arecalled themodulusand argument

of the number respectively. A knowledge of trigonometry allows us

to write

a + ib = r cos θ + irsin θ = r(cos θ +i sin θ) = r e

iθ

The last relationship used in this equation is

cos θ + i sin θ = e

iθ

which is one of the most amazing relationships in the whole of mathe-

matics. Pythagoras’ theorem tells us that r

= a

+ b

and we also have

tan θ = b/a.

Some properties of complex numbers are important for the manip-

ulation of structure factors. A simple operation is to take the complex

conjugate, which means changing the sign of the imaginary part. Thus,

the complex conjugate of the complex number a+ib is written as (a+ib)

∗

and it is equal to a − ib. You should be able to conﬁrm that multiplying

a complex number by its complex conjugate gives a real number that is

the square of the modulus:

(a + ib)(a + ib)

∗

= (a + ib)(a − ib)

= a

− iab+iab− i

= a

+ b

= r

A.4 Waves and structure factors

X-rays are waves and we must be able to deal with them mathematically.

The obvious wavy functions are sines and cosines, so these are used in

the mathematical description of waves. It is an enormous convenience

to combine both sines and cosines into the single term exp(iθ) as seen

in the last equation but one above. This is the main reason why you

ﬁnd complex exponentials in the structure factor and electron-density

equations ((1.1) and (1.2), respectively, in Chapter 1).

Similarly, the structure factors F(hkl) are the mathematical represen-

tation of diffracted waves. When they are combined to form an image

of the electron density (which represents adding waves together), their

relative phases are important. The mathematical construction in Fig.A.2

allows both the amplitude of the wave, |F(hkl)|, and its relative phase,

φ(hkl), to be represented by the modulus and argument of a single com-

plex number. This leads us to write a structure factor in various ways

such as:

F(h) = A(h) + iB(h) =|F(h)|cos(φ(h)) + i|F(h)|sin(φ(h))

=|F(h)|exp(iφ(h)),

346 Useful mathematics and formulae

where the diffraction indices (hkl) are represented by the components of

the vector h.

The structure-factor equation, (1.1) in Chapter 1, shows that F(h) =

∗

(h), i.e. structure factors that are Friedel opposites are complex

conjugates of each other. This leads immediately to the relationship

F(h) × F(

h) =

F(h)

. In addition, we ﬁnd that the product of any

two structure factors can be written as:

F(h) × F(k) =

F(h)

iφ(h)

F(k)

iφ(k)

F(h)F(k)

i(φ(h)+φ(k))

showing that the structure factor magnitudes multiply and the phases

add. This is of importance when applying direct methods of phase

determination.

Fig. A.3 Addition of vectors: x

= x

A.5 Vectors

A vector is often described as a quantity that has magnitude and direc-

tion, as opposed to a scalar quantity that has only magnitude. This

deﬁnition is sufﬁcient for the present purpose and we shall see how

useful the directional properties of vectors are. One of the consequences

of this is that vectors can be added together as shown in Fig. A.3. The

vectors x

and x

are added together to give the resultant x

. This is

expressed algebraically as:

+ x

= x

Note that vectors are conventionally written in bold characters, as are

matrices when we come to them. If the vectors x

and x

give the posi-

tions of two atoms in the unit cell, they are known as position vectors;

is known as a displacement vector, giving the displacement of atom

2 relative to atom 1. A rearrangement of the above equation expresses

the displacement vector as x

= x

−x

and these displacement vectors

arise in the description of the Patterson function (see Chapter 9).

In the unit cell, the position vector x has components (x, y, z) such that

x=ax + by + cz,

where a, b and c are the lattice translation vectors (the edges of the unit

cell) and x, y and z are the fractional co-ordinates of the point. The vector

displacement of atom 2 from atom 1 can therefore be written as

= x

− x

= (ax

+ by

+ cz

) − (ax

+ by

+ cz

)

= a(x

− x

) + b(y

− y

) + c(z

− z

Similarly, the position of a point in reciprocal space is given by the vector

h, which has components (h,k,l) such that:

h = a*h + b*k +c*l,

A.6 Vectors 347

where a*, b* and c* are the reciprocal lattice translation vectors (the

edges of the reciprocal unit cell) and h, k and l are usually integers giving

the diffraction indices of the structure factor F(h) at that point in the

reciprocal lattice.

The scalar (dot) product of the two vectors x and h is:

h.x = hx + ky + lz,

which is an expression to be found in both the structure factor and

electron-density equations. The vector (cross) product is used in the

relationships between the direct and reciprocal lattices:

∗

b × c

∗

c × a

∗

a × b

V = a.b ×c,

where V is the volume of the unit cell. It should be remembered that

a × b = ab sin γ n,

where γ is the angle between the vectors and n is a unit vector perpen-

dicular to both a and b, such that a, b and n are a right-handed set. It

should be clear from these relationships that a* is perpendicular to the

bc-plane; similarly, b* and c* are perpendicular to the ac- and ab-planes,

respectively. If you need convincing that vectors are the most convenient

way of expressing these relationships, here is the volume of the unit cell

without using vectors:

V = abc



1 − cos

α − cos

β − cos

γ + 2 cos α cos β cos γ .

The angles of thereciprocal lattice canbe obtained from the relationships

above but, to save you the trouble, they are:

cos α

∗

cos β cos γ − cos α

sin β sin γ

with corresponding expressions for cos β* and cos γ * obtained by cyclic

permutation of α, β, and γ .

The calculation of a Bragg angle is commonly required, for example

to calculate structure factors or the setting angles on a diffractometer. In

the triclinic system, the formula is:

4 sin

= h

∗2

+ k

∗2

+ l

∗2

+ 2hka

∗

cos γ

∗

+ 2klb

∗

cos α

∗

+ 2lhc

∗

cos β

∗

and this simpliﬁes enormously for other crystal systems.

348 Useful mathematics and formulae

A.6 Determinants

Determinants feature in inequality relationships among structure

factors, are needed in matrix inversion, and form a useful diagnostic

tool when your least-squares reﬁnement runs into trouble. A determi-

nant is a square array of numbers that has a single algebraic value. An

order two determinant is written and evaluated as:



= ad − bc,

and an order three determinant is:



abc

def

ghi



= aei + bfg + cdh − ceg −bdi − afh.

In general, a determinant can be expressed in terms of determinants

of order one less than the original. For an order n determinant, this is

expressed as

 =



i=1

(−1)

i+j



where a

is the ij element of  and 

is the determinant formed from

 by missing out the ith row and the jth column. The summation can

equally well be carried out over j instead of i and gives the same answer.

However, this is useful only for determinants of small order. Evalua-

tion of high-order determinants is best done using the process of Gauss

elimination (a standard mathematical procedure not discussed here) to

reduce the determinant to triangular form, then taking the product of

the diagonal elements.

A.7 Matrices

Matrices are used for a number of tasks in X-ray crystallography.

Typically, they represent symmetry operations, describe the orientation

of a crystal on a diffractometer, and are heavily used in the least-squares

reﬁnement of crystal structures. A brief refresher course will therefore

not be out of place. A matrix is a rectangular array of numbers or alge-

braic expressions and matrix algebra gives a very powerful way of

manipulating them.

One of the operations often required is to transpose a matrix. This

exchanges columns with rows so that, if A is the matrix

⎛

⎝

⎞

⎠

A.8 Matrices in symmetry 349

its transpose, A

,is



ace

bdf



If a square matrix is symmetric, it is equal to its own transpose.

Matrix multiplication is carried out by multiplying the elements in a

row of the ﬁrst matrix by the elements in a column of the second and

adding the products. This forms the element in the product matrix on

the same row and column as those used in its calculation:



abc

def



⎛

⎝

⎞

⎠



au + bv + cw ax + by + cz

du + ev + fw dx + ey + fz



Multiplication can only be carried out if the number of columns in the

ﬁrst matrix is thesame as the numberof rowsin the second. For example,

you may wish to verify that



−14



3 −21

45−3





18 11 −7

13 22 −13



Multiplication of a matrix by its own transpose always produces a

symmetric matrix.

A.8 Matrices in symmetry

Matrix multiplication is useful for representing symmetry operations.

For example, the operation of the 2

axis relating (x, y, z)to(

2+x,

2−y,

−z) may be written as:

⎛

⎝

10 0

0 −10

00−1

⎞

⎠

⎛

⎝

⎞

⎠

⎛

⎝

⎞

⎠

and this form of expression is used to represent symmetry operations

in a computer.

It is sometimes useful to be able to deal with symmetry operations

in reciprocal space also. The operation above can be written in terms of

matrix algebra as



= Cx + d,

where C is the 3 ×3 matrix and d the translation vector. If a space group

symmetry operation is carried out on the whole crystal, by deﬁnition

350 Useful mathematics and formulae

the X-rays see exactly the same structure. The structure-factor equation

may then be written as

F(h) =



j=1

exp(2πih.(Cx

+ d) =



j=1

exp(2πih

) × exp(2π ih.d)

= F(h

C) exp(2π ih.d).

That is, the two reﬂections F(h) and F(h

C) are symmetry related. Their

magnitudes are the same and there is a phase difference between them

of 2πh.d.

This is easier to understand if we continue with the example above.

The 2

axis is one of those that occur in the space group P2

. The

symmetry-related reﬂections that it produces are given by

C =



hkl



⎛

⎝

10 0

0 −10

00−1

⎞

⎠



k l



That is, F(hkl) is related by symmetry to F(h

kl). Their magnitudes must

be the same and there is a phase shift between them of

2πh.d = 2π(hkl).(

2 0) = π(h + k).

Putting this all together gives the relationships |F(hkl)|=|F(h

kl)| and

φ(hkl) = φ(hkl) + π(h + k). Thus, the phase is the same if h + k is even,

but shifted by π if h +k is odd.

Even with anomalous scattering, these relationships are strictly true.

It is only when structure factors are related by a complex conjugate that

they are affected differently by anomalous scattering. For example, in

, we have already seen that |F(hkl)| and |F(hkl)| are always the

and |F(hkl)|.

A.9 Matrix inversion

The inverse of the square matrix A is the matrix A

−1

that has the

property that

−1

= A

−1

A = I,

where I is the identity matrix (1s down the diagonal and 0s every-

where else).

Operations performed by multiplying by a matrix, A, can be undone

by multiplying by the inverse of the matrix, A

−1

. For an order 2 square

A.10 Convolution 351

matrix, the recipe for inversion is:

if A =





then A

−1

det(A)



d −b

−ca



where det(A) is the determinant of the matrix A.

Inversion of an order three matrix is achieved by the following recipe:

if A =

⎛

⎝

⎞

⎠

, then form C =

⎛

⎝

⎞

⎠

, where c

is the

determinant obtained from A by removing the ith row and jth column

and multiplying by (−1)

i+j

. We then have:

−1

det(A)

This recipe will work for any order of matrix, but it is extremely inef-

ﬁcient for orders higher than three. Larger matrices are best inverted

using Gauss elimination, as mentioned earlier. It is a commonly believed

fallacy that matrix inversion is necessary for solving systems of linear

simultaneous equations. Since it is quicker to solve equations than to

calculate an inverse matrix, the inverse should be calculated only if it is

speciﬁcally required, for example, to estimate standard uncertainties of

parameters determined by the equations.

A.10 Convolution

Convolution is an operationthat affectsthe lives of allscientists. Since no

measuring or recording instrument is perfect, it will affect the quantity

that is detected before the recording takes place. For example, loud-

speakers change the signal that is fed to them from an ampliﬁer, thus

altering (hopefully slightly) the sound that you hear. The mathematical

description of this is called convolution. It also appears in the mathemat-

ics of crystallography, although many people function quite adequately

as crystallographers without knowing much about it.

The simplest example of convolution is in the description of a crystal.

The convolution of a lattice point with anything at all, e.g. a single unit

cell, leaves that object unchanged. However, the convolution of two

lattice points with a unit cell gives two unit cells, one at the position

of each lattice point. A complete crystal, therefore, can be described as

the convolution of a single unit cell with the whole crystal lattice. This

would seem to be an unnecessary complication except for the intimate

association of convolution with Fourier transforms.

The convolution theorem in mathematics states that: “the Fourier

transform of a product of two functions is given by the convolution

of their respective Fourier transforms.” That is, if c(x), f(x) and g(x) are

Fourier transforms of C(S), F(S) and G(S), respectively, the theorem may