X-Ray Diffraction by Crystalline Materials

Abstract

Diffraction of X-rays by crystals offers the unique possibility to determine at atomic resolution the three-dimensional structure of solids. This opportunity is exploited by solid state chemists or by mineralogists to study the structure of crystals, by chemists to determine the structure of molecules and by molecular biologists to investigate structure and properties of living organisms molecules, like proteins, nucleic acids etc. In this chapter, after a brief introduction on crystals and crystal symmetries, the general principles of diffraction are introduced, then the reader is made familiar with the mathematical formalism of scattering and diffraction. The physical principles of diffraction by crystals are then introduced, and finally the phase problem in crystallography is presented and the main methods to solve it are shortly outlined. Few notes on the refinement process and on powder diffraction are also reported.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

In this chapter we will consider the diffraction of X-rays ¹ by crystalline solids. We will first give a very concise description of the main characters of the play: the crystal and the X-rays. For an exhaustive treatment the reader is referred to specific textbooks [1–6]. The mathematical tools needed for an elegant and concise treatment of X-ray diffraction ² are then introduced.

Finally we will give a more detailed account of what is the nature of the interaction of X-rays with crystals and on how we can exploit X-ray diffraction in order to obtain structural information at different levels.

The advent of very intense and highly collimated synchrotron X-ray sources has greatly widened the fields of application of diffraction techniques [7].

1.1 The Crystal

An ideal crystal (deviations from ideality are outside our scope) is formed by the periodic repeat, in three dimensions, of the same atomic or molecular motif. In Fig. 8.1 an example of the two-dimensional repeat of a motif is illustrated, and we can see that the repeating motif may be represented by a minimal repeating rectangular area, delimited by two vectors $\mathbf{a}$ and $\mathbf{b}$, which is called the unit cell . The periodic two-dimensional array formed by the repeat of the unit cell is the lattice and an infinite lattice may be represented by the set of vectors

$$\begin{aligned} \mathbf{r}_{u,v} = u \mathbf{a}+ v \mathbf{b}\end{aligned}$$

(8.1)

where $u$ and $v$ are integers going from $-\infty$ to $+\infty$.

In a similar way we may represent a crystal as a three-dimensional lattice, with a unit cell delimited by three vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ (Fig. 8.2). The lattice will be defined by the set of vectors

$$\begin{aligned} \mathbf{r}_{u,v,w} = u \mathbf{a}+ v \mathbf{b}+ w \mathbf{c}\end{aligned}$$

(8.2)

with $u$, $v$ and $w$ integers going from $-\infty$ to $+\infty$. The vectors $\mathbf{r}$ define the infinite set of lattice nodes. The vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ need not be orthogonal and their lengths $a$, $b$ and $c$, together with the three angles $\alpha = \mathbf{b}\ \times \ \mathbf{c}$, $\beta = \mathbf{a}\ \times \ \mathbf{c}$ and $\gamma = \mathbf{a}\ \times \ \mathbf{b}$, are the unit cell parameters . We may also define the lattice planes as those planes passing through the lattice nodes. Each family of planes with the same orientation with respect to the cell vectors is identified by three integers $(h,k,l)$, known as the Miller indices .

Fig. 8.1

Two-dimensional repeat of a motif and its representation as a lattice

Fig. 8.2

Three-dimensional lattice

1.2 Crystal Symmetries

The unit cell or crystal cell represents the minimum volume of the crystal that repeats itself in all three directions of space simply by translation. In the three-dimensional space only 7 types of cells are possible, and they give rise, according to the different possibilities of the cell of being primitive or centered, to 14 lattices, called Bravais lattices :

Crystal system	Cell lengths	Centering
Triclinic	$a\ne b \ne c, \alpha \ne \ \beta \ne \gamma$	P
Monoclinic	$a\ne b \ne c, \alpha = \gamma (or \beta ) = \pi /2$	P, C
Orthorhombic	$a \ne b \ne c, \alpha =\beta = \gamma = \pi /2$	P, C, I, F
Tetragonal	$a = b \ne c, \alpha =\beta =\gamma =\pi /2$	P, I
Trigonal	$a = b \ne c, \alpha =\beta = \pi /2, \gamma = 2\pi /3$	P
Hexagonal	$a = b \ne c, \alpha =\beta = \pi /2, \gamma = 2\pi /3$	P
Cubic	$a = b = c, \alpha =\beta = \gamma = \pi /2$	P, I, F

A crystal cell may contain more than one motif. If these motifs are all identical, they can be superimposed by applying appropriate transformations, called symmetry operations. Not all symmetry transformations are compatible with the crystal lattice. Those allowed, along with the symbols used to represent them, are listed below.

Centre of symmetry   $(\bar{1})$. It is a point in the unit cell such that, for any point defined by a vector $\mathbf{r}$, there is an equivalent point at position $-\mathbf{r}$.

Mirror plane $(m)$ . Also called plane of symmetry, is such that everything on one side of the plane is reflected on the other side.

Rotation axes  $(1, 2, 3, 4, 6)$. Rotation is necessarily about an axis. The numerical value of the rotation corresponds to $2\pi /n$, where $n$ in a crystal lattice can assume only the values $1, 2, 3, 4, 6$, corresponding to rotations of $2\pi$, $\pi$, $2\pi /3$, $\pi /2$, $\pi /3$, respectively.

Inversion axes  $(\bar{1}, \bar{2}, \bar{3}, \bar{4}, \bar{6})$. They are combinations of a rotation axis with the centre of symmetry. In practice, after a rotation of order $n$, the object is reflected through a point located on the axis.

The previous four types of symmetry elements do not include any translation. Their effect, after an integer number of applications, is a return to the starting point. They can be combined and all the combinations compatible with the properties of the lattice give rise to the 32 three-dimensional crystallographic point groups. They are reported, in association with each of the 7 crystal systems, in the Table below.

Crystal system	The 32 crystallographic point groups
Triclinic	$1, \bar{1}$
Monoclinic	$2, m, 2/m$
Orthorhombic	$222, mm2, mmm$
Tetragonal	$4, \bar{4}, 4/m, 422, 4mm, \bar{4}2m, 4/mmm$
Trigonal	$3, \bar{3}, 32, 3m, \bar{3}m$
Hexagonal	$6, \bar{6}, 6/m, 622, 6mm, \bar{6}2m, 6/mmm$
Cubic	$23, m\bar{3}, 432, \bar{4}3m, m\bar{3}m$

In addition to the previous ones, there are symmetry elements for which the periodic nature of the lattice plays a fundamental role. They include:

Glide planes  $(a, b, c, d, n)$. They represent a combination of a mirror plane and a translation. Their symbol indicates the direction of the translation: for example, $a$ represents a translation parallel to the unit cell vector $\mathbf{a}$.

Screw axes  $(n_m: 2_1, 3_1, 3_2, 4_1, 4_2, 4_3, 6_1, 6_2, 6_3, 6_4, 6_5)$. A combination of a rotation axis and a translation. The object is first rotated by an angle $2\pi /n$ and then displaced by $m/n$.

The possible combinations of the 32 point groups with the 14 Bravais lattices are 230 and are called the 230 space groups . They represent all the possible ways a motif can be arranged in a periodic way in the three-dimensional space.

The basic motif present in the unit cell is called the asymmetric unit , defined as the smallest portion of the crystal cell that does not contain crystallographic symmetries. In the case of molecular crystals, the asymmetric unit can correspond to a single molecule, or it can contain more than one molecule, or eventually less than it, if the molecule itself is symmetric.

A final aspect with relevant consequences must be remembered. The symmetry elements described above can be grouped in two classes: those, like rotation axes or screw axes, that keep unaltered the configuration of the object, and those, like mirror planes, centre of symmetry, inversion axes and glide planes, that invert it. Consequently, chiral molecules cannot crystallize in space groups containing second class symmetry elements, unless both enantiomers are present inside the crystal.

1.3 The X-Rays

Discovered in 1886 by Röntgen, it was only 10 years later that Sommerfeld measured their very short wavelength (around 0.4 Å) and identified their nature of electromagnetic waves. This was definitely confirmed in 1912 by M. von Laue who, inspired by an intuition of P. Ewald, suggested that crystal lattices could act as diffraction gratings for X-rays. This hypothesis was immediately confirmed by the successful experiment of Friedrich and Knipping.

In the spectrum of electromagnetic waves, X-rays are placed in the region between $\gamma$-rays and ultraviolet radiation ($0.1 \le \lambda \le 100$ Å), but in X-ray crystallography only high energy radiation with $0.3 \le \lambda \le 3.0$ Å is usually employed.

Because of their high energy, the refractive index of X-rays does not deviate significantly from unity in all media. Therefore X-rays can not be focused by lenses as ordinary light or electrons [8]. Only in recent years improvements in X-ray focusing have been realized with the use of suitably shaped crystals acting as focusing mirrors [9].

Fig. 8.3

Representation of the Dirac $\delta$ function in one dimension

1.4 Some Useful Mathematical Concepts

• Dirac $\delta$ function: In one dimension is defined as

  $$\begin{aligned} \delta (x-x_o) \;\; \left[ \begin{array}{lll} = 0 &{} \mathrm{for} &{} x \ne x_o \\ = \infty &{} \mathrm{for} &{} x = x_o \end{array} \right. \end{aligned}$$

  (8.3)

  with

  $$\begin{aligned} \int _s \delta (x-x_o) dx = 1 \end{aligned}$$

  (8.4)

  It is therefore a normalized, infinitely narrow function (Fig. 8.3), which may be obtained as the limit for zero variance of a Gaussian function

  $$\begin{aligned} \delta (x-x_o) = \lim _{\sigma \rightarrow 0} \left[ \frac{1}{\sigma \sqrt{2\pi }} e^{-\frac{(x-x_o)^2}{2\sigma ^2} } \right] \end{aligned}$$

  (8.5)

  In three dimensions a point may be represented by a vector

  $$\begin{aligned} \mathbf{r}= x \mathbf{a}+ y \mathbf{b}+ z \mathbf{c}\end{aligned}$$

  (8.6)

  and the $\delta$ function is defined as in one dimension, as

  $$\begin{aligned} \delta (\mathbf{r}- \mathbf{r}_o) = \delta (x - x_o) \delta (y - y_o) \delta (z - z_o) \end{aligned}$$

  (8.7)

  We recall the following properties of the $\delta$ function

  $$\begin{aligned}&*\qquad \qquad&\delta (\mathbf{r}- \mathbf{r}_o) = \delta (\mathbf{r}_o - \mathbf{r}) \end{aligned}$$

  (8.8)
  $$\begin{aligned}&*\qquad \qquad&f(\mathbf{r}) \delta (\mathbf{r}- \mathbf{r}_o) \equiv f(\mathbf{r}_o) \delta (\mathbf{r}- \mathbf{r}_o)\end{aligned}$$

  (8.9)
  $$\begin{aligned}&*\qquad \qquad&\int _s f(\mathbf{r}) \delta (\mathbf{r}- \mathbf{r}_o) d\mathbf{r}= f(\mathbf{r}_o) \end{aligned}$$

  (8.10)
  $$\begin{aligned}&*\qquad \qquad&\int _s \delta (\mathbf{r}- \mathbf{r}_2) \delta (\mathbf{r}- \mathbf{r}_1) d\mathbf{r}= \delta (\mathbf{r}_2 - \mathbf{r}_1) \end{aligned}$$

  (8.11)

  which are an immediate consequence of its definition.

• Lattice function : A one-dimensional lattice is a sequence of equally spaced points, separated by a constant period $a$. It can be represented by a lattice function

  $$\begin{aligned} L (x) = \sum _{n=-\infty }^{\infty } \delta (x - x_n) \end{aligned}$$

  (8.12)

  where $x_n = n a$, $n$ is an integer and $a$ is the constant period; indeed $L (x) \ne 0$ for $x = na$ ($n = - \infty \rightarrow \infty$) (Fig. 8.4).

  A three-dimensional lattice defined by the vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ can then be represented by the function

  $$\begin{aligned} L (\mathbf{r}) = \sum _{u=-\infty }^{\infty } \sum _{v=-\infty }^{\infty } \sum _{w=-\infty }^{\infty }\delta (\mathbf{r}- \mathbf{r}_{u,v,w}) \end{aligned}$$

  (8.13)

  where, as we have seen

  $$\begin{aligned} \mathbf{r}_{u,v,w} = u \mathbf{a}+ v \mathbf{b}+ w \mathbf{c}\end{aligned}$$

  with $u, v, w$ integers, are the vectors defining the lattice nodes where $L(\mathbf{r}) \ne 0$.

• Fourier transform : Given a function $\rho (\mathbf{r})$, its Fourier transform is

  $$\begin{aligned} \mathbf{F}(\mathbf{S}) = \int _r \rho (\mathbf{r}) e^{2\pi \jmath \mathbf{S}\cdot \mathbf{r}} d \mathbf{r}\end{aligned}$$

  (8.14)

  where $\mathbf{S}$ is a vector in the space in which the transform is defined. It may be shown that the inverse relation applies, i.e.

  $$\begin{aligned} \rho (\mathbf{r}) = \int _{s} \mathbf{F}(\mathbf{S}) e^{- 2 \pi \jmath \mathbf{S}\cdot \mathbf{r}} d \mathbf{S}\end{aligned}$$

  (8.15)

  In short we can write

  $$\begin{aligned} \mathbf{F}(\mathbf{S}) = T [\rho (\mathbf{r})] \quad \text{ and } \quad \rho (\mathbf{r}) = T^{-1} [\mathbf{F}(\mathbf{S})] \end{aligned}$$

  In general $\mathbf{F}(\mathbf{S})$ is a complex function and may be expressed as

  $$\begin{aligned} \mathbf{F}(\mathbf{S}) = A (\mathbf{S}) + {\jmath }B (\mathbf{S}) \end{aligned}$$

  (8.16)

  with

  $$\begin{aligned} A (\mathbf{S}) = \int _r \rho (\mathbf{r}) \cos (2\pi \mathbf{S}\cdot \mathbf{r}) d\mathbf{r}\end{aligned}$$

  (8.17)
  $$\begin{aligned} B(\mathbf{S}) = \int _r \rho (\mathbf{r}) \sin (2\pi \mathbf{S}\cdot \mathbf{r}) d\mathbf{r}\end{aligned}$$

  (8.18)

  Some examples of Fourier transform

  • 1. For a Gaussian function

    $$\begin{aligned} \rho (x) = N (\sigma , 0) = \frac{1}{\sigma \sqrt{2\pi }} e^{- \frac{x^2}{2 \sigma ^2}} \end{aligned}$$

    (8.19)

    the transform is given by

    $$\begin{aligned} T [\rho (x)] = F(S) = e^ {- \pi ^2 \sigma ^2 {S}^2 } \end{aligned}$$

    (8.20)

    which is also a Gaussian type function, but now its width is inversely [and not directly as $N (\sigma , 0)$ in (8.19)] proportional to $\sigma ^2$ (Fig. 8.5).

  • 2. The Fourier transform of a $\delta$ function at the origin (Fig. 8.6) is

    $$\begin{aligned} F (S) = \int ^{\infty } _{- \infty } \delta (x) e^{2\pi \jmath {S} x} d x = 1 \end{aligned}$$

    (8.21)

    which is infinitely wide.

    For a $\delta$ function at a distance $a$ from the origin $\rho (x) = \delta (x-a)$ the transform is

    $$ F(S) = e^{2\pi \jmath a S} $$

  • 3. For a finite one dimensional lattice with $N = 2 p + 1$ nodes

    $$\begin{aligned} \rho (x) = \sum ^{p} _{n = - p} \delta ( x - n a) \end{aligned}$$

    (8.22)

    the Fourier transform is given by

    $$\begin{aligned} F (S) = \frac{\sin N \pi a S}{\sin \pi a S} \end{aligned}$$

    (8.23)

    which is a function with principal maxima or minima at $a S = h$ ($h$ integer) of height $\pm N$ and width $2/N$, separated by increasingly smaller ripples as $N$ increases (Fig. 8.7).

  • 4. For a infinite one dimensional lattice with $N \rightarrow \infty$

    $$\begin{aligned} \rho (x) = L (x) = \sum ^{\infty }_{n = -\infty } \delta (x - n a) \end{aligned}$$

    (8.24)

    the Fourier transform is given by

    $$\begin{aligned} F (S) = \lim _{N \rightarrow \infty } \frac{\sin N \pi a S}{\sin \pi a S} \end{aligned}$$

    (8.25)

    which is a function with infinitely narrow maxima of infinite height at $S = h/a$ and zero value elsewhere; each maximum is a non-normalized $\delta$ function and it may be shown that the normalization factor is $1/a$. We can therefore write

    $$\begin{aligned} F(S) = \sum ^{\infty }_{h = - \infty } \delta (S - \frac{h}{a}) = \frac{1}{a} \sum ^{\infty }_{h = - \infty } \delta (a S - h) \end{aligned}$$

    (8.26)

    which is again a lattice function with nodes separated by an inverse period $1/a$.

  • 5. For a finite three dimensional lattice with $N_1 = 2p_1 +1$ nodes along $x$, $N_2 = 2p_2 +1$ nodes along $y$ and $N_3 = 2p_3 +1$ nodes along $z$

    $$\begin{aligned} \rho (\mathbf{r}) = \sum ^{p_1} _{u = - p_1} \sum ^{p_2} _{v = - p_2} \sum ^{p_3} _{w = -p_3} \delta (\mathbf{r}- \mathbf{r}_{u,v,w}) \end{aligned}$$

    (8.27)

    the Fourier transform is given by

    $$\begin{aligned} \mathbf{F}(\mathbf{S}) = \frac{\sin N_1 \pi \mathbf{a}\cdot \mathbf{S}}{\sin \pi \mathbf{a}\cdot \mathbf{S}} \cdot \frac{\sin N_2 \pi \mathbf{b}\cdot \mathbf{S}}{\sin \pi \mathbf{b}\cdot \mathbf{S}} \cdot \frac{\sin N_3 \pi \mathbf{c}\cdot \mathbf{S}}{\sin \pi \mathbf{c}\cdot \mathbf{S}} \end{aligned}$$

    (8.28)

    which, similarly to (8.23), will have maxima and minima at

    $$\begin{aligned} \mathbf{a}\cdot \mathbf{S}= h, \qquad \mathbf{b}\cdot \mathbf{S}= k, \qquad \mathbf{c}\cdot \mathbf{S}= l \end{aligned}$$

    (8.29)

    with $h, k, l$ integers.

    In order to identify these maxima and minima it is convenient to associate to the lattice defined by the base vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ (direct lattice ) an other lattice, the reciprocal lattice , with base vectors $\mathbf{a}^*$, $\mathbf{b}^*$, $\mathbf{c}^*$, uniquely defined by the relations

    $$\begin{aligned} \mathbf{a}^*\cdot \mathbf{a}= 1&\qquad \mathbf{a}^*\cdot \mathbf{b}= 0&\quad \mathbf{a}^*\cdot \mathbf{c}= 0 \nonumber \\ \mathbf{b}^*\cdot \mathbf{a}= 0&\qquad \mathbf{b}^*\cdot \mathbf{b}= 1&\quad \mathbf{b}^*\cdot \mathbf{c}= 0 \\ \mathbf{c}^*\cdot \mathbf{a}= 0&\qquad \mathbf{c}^*\cdot \mathbf{b}= 0&\quad \mathbf{c}^*\cdot \mathbf{c}= 1 \nonumber \end{aligned}$$

    (8.30)

    In fact, because of conditions (8.30), the maxima and minima of $\mathbf{F}(\mathbf{S})$ given in (8.29), will be at the end of vectors

    $$\begin{aligned} \mathbf{S}_{hkl} = h \mathbf{a}^*+ k \mathbf{b}^*+ l \mathbf{c}^*\end{aligned}$$

    (8.31)

    which are vectors in the reciprocal space.

  • 6. For a infinite three dimensional lattice with $(N_1, N_2, N_3) \rightarrow \infty$

    $$\begin{aligned} \rho (\mathbf{r}) = L (\mathbf{r}) = \sum ^{\infty }_{u = - \infty } \sum ^{\infty }_{v = - \infty } \sum ^{\infty }_{w = - \infty } \delta (\mathbf{r}- \mathbf{r}_{u,v,w}) \end{aligned}$$

    (8.32)

    Recalling (8.7) and the case of a one dimensional lattice, the Fourier transform will be given by

    $$\begin{aligned} \mathbf{F}(\mathbf{S}) = \frac{1}{V} \sum ^{\infty }_{h = - \infty } \sum ^{\infty }_{k = - \infty } \sum ^{\infty }_{l = - \infty } \delta (\mathbf{S}- \mathbf{S}_{hkl}) \end{aligned}$$

    (8.33)

    which is again a three-dimensional lattice defined by the infinite set of reciprocal vectors $S$ given in (8.31) with the integers $(h,k,l)$ going from $-\infty$ to $+\infty$.

• Convolution : Given two functions $\rho (\mathbf{r})$ and $g(\mathbf{r})$ their convolution is defined as

  $$\begin{aligned} C(\mathbf{u}) = \rho (\mathbf{r})\otimes g(\mathbf{r}) = \int _r \rho (\mathbf{r}) g(\mathbf{u}- \mathbf{r}) d\mathbf{r}. \end{aligned}$$

  (8.34)

  It is easy to show that for the convolution operation the commutative property

  $$\begin{aligned} \rho (\mathbf{r}) \otimes g(\mathbf{r}) = g(\mathbf{r}) \otimes \rho (\mathbf{r}) \end{aligned}$$

  (8.35)

  holds.

  An important theorem, relating Fourier transform and convolution, is the convolution theorem, stating that

  $$\begin{aligned} T[\rho (\mathbf{r})\otimes g(\mathbf{r})] = T[\rho (\mathbf{r})] T[g(\mathbf{r})] \end{aligned}$$

  (8.36)

  and conversely

  $$\begin{aligned} T[\rho (\mathbf{r}) g(\mathbf{r})] = T[\rho (\mathbf{r})] \otimes T[g(\mathbf{r})] \end{aligned}$$

  (8.37)

  Some examples of convolution

  • 1. The convolution of a function $\rho (\mathbf{r})$ with a $\delta$ function, because of property (8.10), is given by

    $$\begin{aligned} \delta (\mathbf{r}-\mathbf{r}_o)\otimes \rho (\mathbf{r}) = \rho (\mathbf{r}-\mathbf{r}_o) \end{aligned}$$

    (8.38)

    which corresponds to a translation of $\rho (\mathbf{r})$ by a vector $\mathbf{r}_o$ (Fig. 8.8a).

  • 2. The convolution of a function $f(x)$, defined in the range $0 \le x \le a$, with a one dimensional lattice function is given by

    $$\begin{aligned} L(x)\otimes f(x) = \sum ^{\infty }_{n=-\infty } f(x-na) = \rho (x) \end{aligned}$$

    (8.39)

    where $\rho (x)$ is now a periodic one-dimensional function. We obtain in this way the periodic repeat of the function $f(x)$ (Fig. 8.8b)

  • 3. The convolution of a function $f(\mathbf{r}) \equiv f(x,y,z)$, defined in the ranges $0 \le x \le a$, $0 \le y \le b$, $0 \le z \le c$, with a three dimensional lattice function is given by

    $$\begin{aligned} L(\mathbf{r})\otimes f(\mathbf{r}) = \sum ^{\infty }_{u,v,w=-\infty } f(\mathbf{r}-\mathbf{r}_{u,v,w}) = \rho (\mathbf{r}) \end{aligned}$$

    (8.40)

    which is a periodic three dimensional function. In Fig. 8.8c a two dimensional example is shown.

Fig. 8.4

Representation of the one dimensional lattice function defined in (8.12)

Fig. 8.5

Two Gaussian funtions with $\sigma = 1$ and $\sigma = 2$ and their Fourier transforms

Fig. 8.6

One dimensional $\delta$ function at the origin and its Fourier transform

Fig. 8.7

Fourier transforms of three one-dimensional finite lattices with $N = 6$, $N = 7$ and $N = 21$ nodes

Fig. 8.8

Convolution of a function $f(x)$ with: a a $\delta$ function; b a one dimensional lattice function; c convolution of a function $f(x,y)$ with a two dimensional lattice function

2 X-Ray Diffraction

When X-rays interact with matter, two types of scattering may occur:

• 1. Coherent (Thomson) scattering with a phase change of $\varDelta \phi = \pi$

• 2. Incoherent (Compton) scattering .

Besides, X-rays are absorbed by matter also in two ways:

• (a) without photoemission

• (b) with photoemission

We will focus our attention on the kinematic theory of diffraction by ideal crystals, while for the more complex dynamic theory , capable of handling real crystals (with a finite size and defects in the lattice periodicity), the reader is referred to the textbooks cited in the introduction.

Fig. 8.9

Coherent scattering by a particle in O

2.1 X-Ray Scattering

Thomson scattering : The scattering, without loss of energy, of X-rays by a particle was studied by Thomson, who derived that the coherent scattered radiation has an intensity (Fig. 8.9) given by

$$\begin{aligned} I_{e(Th)} = I_i \left[ \frac{e^4}{m^2r^2c^4}\right] \frac{1+\cos ^2 2\theta }{2} \end{aligned}$$

(8.41)

where $I_i$ is the intensity of the incident beam, $e$ is the charge of the particle, $m$ is the mass of the particle, $2\theta$ is the scattering angle and $r$ is the distance from the particle to the point where the intensity is measured. Note that uncharged particles do not give coherent scattering and that, because of the inverse dependence from $m^2$, in practice only electrons (the smallest nucleus has a mass 1840 times that of electrons) contribute to the scattering in a significant way. The term $P_o = (1+\cos ^2 2\theta )/2$ is the form taken by the polarization factor when the incident beam is completely unpolarized, as in the case of the radiation from a conventional X-ray tube. Synchrotron sources give partially polarized radiation and the polarization factor changes accordingly.³

Compton scattering: Is the incoherent scattering in which the radiation loses part of its energy. The process may be described in terms of an inelastic collision between the X-ray photon and the electron. The difference in wavelength between the incident and the scattered beam is given by

$$\begin{aligned} \varDelta \lambda = 0.024 \; (1-\cos 2\theta ) \; \mathrm{\AA } \end{aligned}$$

(8.46)

and is maximum for $2\theta = 180^\circ$ (back-scattering).

2.2 Interference of Scattered Waves

Since we are interested in diffraction we have to consider the interference between waves scattered by different scattering particles. In Fig. 8.10, the incident beam, of wavelength $\lambda$, hits two point scatterers along a direction defined by the unit vector $\mathbf{s}_o$. Then the phase difference between the wave scattered by the particle in $O'$ in the direction indicated by the unit vector $\mathbf{s}$ and that scattered in the same direction by the particle in $O$, is given by

$$\begin{aligned} \delta = \frac{2\pi }{\lambda }(\mathbf{AO} + \mathbf{OB}) = \frac{2\pi }{\lambda } (\mathbf{s}- \mathbf{s}_o) \cdot \mathbf{r}= 2 \pi \mathbf{S}\cdot \mathbf{r}\end{aligned}$$

(8.47)

where

$$\begin{aligned} \mathbf{S}= \frac{\mathbf{s}- \mathbf{s}_o}{\lambda } \;\; \text{ with } \text{ modulus } \;\; |\mathbf{S}| = \frac{2 \sin \theta }{\lambda } \end{aligned}$$

(8.48)

where $2\theta$ is the angle between the incident and the scattered waves. If $\lambda \gg r$ then $\delta \rightarrow 0$ and interference becomes negligible.

Fig. 8.10

Scattering by two centers in $O$ and $O'$ and their interference

Then, if $A_o$ is the amplitude of the wave scattered by $O$, the amplitude of the wave scattered by $O'$ will be

$$\begin{aligned} A_o' e^ {\jmath \delta } = A_o' e^ {2\pi \jmath \mathbf{S}\cdot \mathbf{r}}. \end{aligned}$$

(8.49)

As we have seen only electrons contribute to the X-ray scattering and therefore $A_o$ and $A_o'$ will depend on the electron contents of $O$ and $O'$. Indicating with $A_{Th}$ the amplitude scattered by a free electron, we can define the scattering factor as $f = A/A_{Th}$, corresponding to the number of electrons in the scattering point.

If we now have $N$ scattering points at the end of the set of vectors {$\mathbf{r}_i$} ($i=1,N$), the resulting scattering amplitude, with respect to the amplitude scattered by a free electron at the origin, will be

$$\begin{aligned} \mathbf{F}(\mathbf{S}) = \sum ^{N}_{i=1} \frac{A_i}{A_{Th}} e^ {2\pi \jmath \mathbf{S}\cdot \mathbf{r}_i } = \sum ^{N}_{i=1} f_i e^ {2\pi \jmath \mathbf{S}\cdot \mathbf{r}_i }. \end{aligned}$$

(8.50)

A continuous scatterer may be described by a continuous electron density function $\rho (\mathbf{r})$ and a volume element $d\mathbf{r}$ at $\mathbf{r}$ will contain $\rho (\mathbf{r}) d(\mathbf{r})$ electrons, scattering with an amplitude

$$\begin{aligned} \rho (\mathbf{r}) d\mathbf{r}e^ {2\pi \jmath \mathbf{S}\cdot \mathbf{r}} \end{aligned}$$

(8.51)

The total scattered radiation will then be

$$\begin{aligned} \mathbf{F}(\mathbf{S}) = \int _r \rho (\mathbf{r}) e^ {2\pi \jmath \mathbf{S}\cdot \mathbf{r}} d\mathbf{r}= T [\rho (\mathbf{r})] \end{aligned}$$

(8.52)

which indicates that the scattered amplitude is the Fourier transform of the electron density function. Then $\rho (\mathbf{r})$ will be the inverse Fourier transform of $\mathbf{F}(\mathbf{S})$, i.e.

$$\begin{aligned} \rho (\mathbf{r}) = \int _{S} \mathbf{F}(\mathbf{S}) e^ {2\pi \jmath \mathbf{S}\cdot \mathbf{r}} d\mathbf{S}= T^{-1} [\mathbf{F}(\mathbf{S})] \end{aligned}$$

(8.53)

2.3 Scattering by Matter

We can now consider an atomic electron, for which the electron density may be derived from the wave function

$$\begin{aligned} \rho _e(\mathbf{r}) = |\psi (\mathbf{r})|^2 \end{aligned}$$

(8.54)

If we assume that $\rho _e(\mathbf{r})$ has a spherical symmetry,⁴ the radiation scattered by an atomic electron will have an amplitude

$$\begin{aligned} f_e(S) = \int \limits ^{\infty }_{0} U_e (r) \frac{\sin 2\pi r S}{2\pi r S} dr \end{aligned}$$

(8.55)

where $r = |\mathbf{r}|$ and $U_e (r) = 4\pi r^2 \rho _e (r)$ is the radial electron density function.

Fig. 8.11

Atomic scattering factors curves for $S$,$Na^+$ and $O$ (atoms at rest, $T = 0$). The dotted line shows the $O$ curve for a vibrating atom ($T > 0$)

We can then extend the same procedure to the scattering of the whole atom. The amplitude scattered by all $Z$ atomic electrons will be

$$\begin{aligned} f_a(S) = \int \limits ^{\infty }_{0} U_a (r) \; \frac{\sin 2\pi r S}{2\pi r S} dr = \sum ^{Z}_{j=1} f_{ei} \end{aligned}$$

(8.56)

where $U_a (r) = 4\pi r^2 \rho _a(r)$ is the atomic radial density function and $Z$ the atomic number. The trend of the atomic scattering factor $f_a(S)$ as a function of $\sin \theta /\lambda$ is shown in Fig. 8.11 where one can see its decrease as the scattering angle increases.

Atoms, also in the solid state, undergo thermal vibrations and are not fixed at a given position, but oscillate around it. If we assume that this oscillation is equal in all directions (isotropic displacement) we can derive that the atomic scattering factor at a temperature $T$ is given by

$$\begin{aligned} {^{T}f_a} (S) = \; {^{0}f_a} (S) e^ {-B \sin ^2 \theta /\lambda ^2 } \end{aligned}$$

(8.57)

where ${^{0}f_a} (S)$ is the scattering factor for the atom at rest ($T = 0$) and $B = 8 \pi ^2 U$, $U$ being the mean square displacement of the atom. In Fig. 8.11 the full lines represent the scattering factors for atoms at rest, while the dotted line is the scattering factor for $O$ at room temperature ($T = 298^\circ$). Note that the dotted curve shows a more rapid decrease as $\sin \theta /\lambda$ increases; a vibrating atom will give a negligible contribution at high scattering angles.

2.4 Diffraction by a Crystal

We shall first consider the scattering by the molecules or atoms in the unit cell. Let us suppose that there are $N$ atoms in the unit cell (they may or may not be connected to form molecules) at positions defined by the set of vectors $\{ \mathbf{r}_i \}$ ($i = 1,N$). If we neglect the small fraction of bonding electron density, we can express the electron density of each atom as $\rho _i(\mathbf{r}- \mathbf{r}_i)$, and with this independent atom approximation, we can write

$$\begin{aligned} \rho _M(\mathbf{r}) = \sum ^{N}_{i=1} \rho _i (\mathbf{r}- \mathbf{r}_i) \end{aligned}$$

(8.58)

and the resulting scattered amplitude will be

$$\begin{aligned} \mathbf{F}_M(\mathbf{S}) =\sum ^{N}_{i=1} f_i (\mathbf{S}) e^ { 2\pi \jmath \mathbf{S}\cdot \mathbf{r}_i } \end{aligned}$$

(8.59)

An infinite crystal may be considered as the infinite periodic repeat of $\rho _M(\mathbf{r})$, i.e.

$$\begin{aligned} \rho _{\infty } (\mathbf{r}) = \rho _M(\mathbf{r}) \otimes L(\mathbf{r}) \end{aligned}$$

(8.60)

Because of the convolution theorem, we can obtain the scattered amplitude as

$$\begin{aligned} \mathbf{F}_{\infty }(\mathbf{S})&= T[\rho _M(\mathbf{r})] T[L(\mathbf{r})] = \mathbf{F}_M(\mathbf{S}) \frac{1}{V} \sum ^{\infty }_{h,k,l = -\infty } \delta (\mathbf{S}- \mathbf{S}_{hkl}) \nonumber \\&= \frac{1}{V} \mathbf{F}_M(\mathbf{S}) \sum ^{\infty }_{h,k,l = -\infty } \delta (\mathbf{S}-\mathbf{S}_{hkl}) \end{aligned}$$

(8.61)

with

$$\begin{aligned} \mathbf{S}_{hkl} = h \mathbf{a}^*+ k \mathbf{b}^*+ l \mathbf{c}^*\end{aligned}$$

(8.62)

$\mathbf{F}_{\infty }(\mathbf{S}) \ne 0$ only at the nodes of the reciprocal lattice, where the continuous function $\mathbf{F}_M(\mathbf{S})$ is sampled. The sampled values in (8.61) can be indicated by $\mathbf{F}_M(hkl)$, where $hkl$ is a row matrix of the three components of the vector $\mathbf{S}_{hkl}$ in the reciprocal space.

When $\mathbf{S}= (\mathbf{s}-\mathbf{s}_o)/\lambda$ is multiplied by $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$, the so called Laue diffraction conditions are obtained

$$\begin{aligned} \mathbf{a}\cdot (\mathbf{s}-\mathbf{s}_o)&= h\lambda \nonumber \\ \mathbf{b}\cdot (\mathbf{s}-\mathbf{s}_o)&= k\lambda \\ \mathbf{c}\cdot (\mathbf{s}-\mathbf{s}_o)&= l\lambda \nonumber \end{aligned}$$

(8.63)

which, for any given direction $\mathbf{s}_o$ of the incident beam, define the discrete values of $\mathbf{S}$ along which diffraction can occur.

Finally we will consider the case of a finite crystal of volume $\varOmega$. It is useful to define a form function

$$\begin{aligned} \varPhi (\mathbf{r}) \;\; \left[ \begin{array}{ll} = 1 &{} \text{ within } \text{ the } \text{ crystal } \\ = 0 &{} \text{ outside } \text{ the } \text{ crystal } \end{array} \right. \end{aligned}$$

(8.64)

We can then obtain the finite crystal as

$$\begin{aligned} \rho _{cry}(\mathbf{r}) = \rho _\infty (\mathbf{r}) \varPhi (\mathbf{r}) \end{aligned}$$

(8.65)

and, because of the convolution theorem, the scattered amplitude will be

$$\begin{aligned} \mathbf{F}(\mathbf{S})=T[\rho _\infty (\mathbf{r})] \otimes T[\varPhi (\mathbf{r})]=\mathbf{F}_\infty (\mathbf{S}) \otimes D(\mathbf{S}) \end{aligned}$$

(8.66)

where

$$\begin{aligned} D(\mathbf{S})=\int _\varOmega e^ {2\pi \jmath \mathbf{S}\cdot \mathbf{r}} d\mathbf{r}\end{aligned}$$

(8.67)

is the Fourier transform of the form function, and, recalling (8.61) and the properties of the $\delta$ function, we obtain

$$\begin{aligned} \mathbf{F}(\mathbf{S})=\frac{1}{V} \mathbf{F}_M(hkl)\sum ^{\infty }_{h,k,l=-\infty } D(\mathbf{S}- \mathbf{S}_{hkl}) \end{aligned}$$

(8.68)

Fig. 8.12

Crystal with the shape of a parallelepiped of edges $A_1$,$A_2$,$A_3$

If the crystal is a parallelepiped with edges $A_1$,$A_2$,$A_3$ (Fig. 8.12), it can be shown that

$$\begin{aligned} D(\mathbf{S})=\frac{\sin (\pi A_1 S_x)}{\pi S_x} \cdot \frac{\sin (\pi A_2 S_y)}{\pi S_y} \cdot \frac{\sin (\pi A_3 S_z)}{\pi S_z} \end{aligned}$$

(8.69)

which, similarly to (8.23) and (8.28), has maxima at $A_1 \cdot A_2 \cdot A_3 = \varOmega$, with amplitude $(A_1)^{-1} \cdot (A_2)^{-1} \cdot (A_3)^{-1}$. The nodes of the reciprocal lattice become domains of width $(A_i)^{-1}$, $i = 1,3$ along the three axes. Similarly for a spherical crystal of radius $R$ the reciprocal lattice nodes will be small spherical domains with radius $R^{-1}$.

$\mathbf{F}_M(hkl)$ in (8.61) and (8.68) is called structure factor. If we now indicate with ${\bar{\mathbf {x}}}_i \equiv (x_i,y_i,z_i)$ the row matrix with the three components of the position vector $\mathbf{r}_i$ in the direct space, the scalar product of the two vectors $\mathbf{S}_{hkl} \cdot \mathbf{r}_i$ is given by

$$\begin{aligned} \mathbf{S}_{hkl} \cdot \mathbf{r}_i = h x_i + k y_i + l z_i \end{aligned}$$

(8.70)

and, substituting in (8.58), the structure factor becomes

$$\begin{aligned} \mathbf{F}(hkl) =\sum ^{N}_{i=1} f_i e^ {2\pi \jmath (h x_i + k y_i + l z_i)} = A(hkl) + {\jmath }{B}(hkl) \end{aligned}$$

(8.71)

where

$$\begin{aligned} A(hkl) =\sum ^{N}_{j=1} f_i \cos [2\pi (h x_i + k y_i + l z_i)]\end{aligned}$$

(8.72)

$$\begin{aligned} B(hkl) =\sum ^{N}_{j=1} f_i \sin [2\pi (h x_i + k y_i + l z_i)] \end{aligned}$$

(8.73)

Fig. 8.13

Vector representation in the complex plane of the structure factor of a five atom structure

With reference to the vector representation of (8.71) in the complex plane, shown in Fig. 8.13, we can express the structure factor in terms of modulus and phase, i.e.

$$\begin{aligned} \mathbf{F}(hkl) = |\mathbf{F}(hkl)|e^{\jmath \phi (hkl)} \end{aligned}$$

(8.74)

where

$$\begin{aligned} |\mathbf{F}(hkl)| = (A(hkl)^2 + B(hkl)^2)^{\frac{1}{2}} \end{aligned}$$

(8.75)

and

$$\begin{aligned} \tan \phi (hkl) = \frac{B(hkl)}{A(hkl)} \end{aligned}$$

(8.76)

Finally, using (8.70), we can write the structure factor as

$$\begin{aligned} \mathbf{F}_{hkl} \equiv \mathbf{F}(hkl) =\sum ^{N}_{i=1} f_i e^ {2\pi \jmath (h x_i + k y_i + l z_i) } \end{aligned}$$

(8.77)

where we can immediately see that the diffracted amplitudes are a function of the coordinates of the atoms in the unit cell, i.e. of the crystal structure. We can therefore derive the crystal structure from the diffracted amplitudes, but, as we shall see, this is not a straightforward process. We just note here that relations such as (8.77) are far from being linear and this makes the process of deriving the atomic coordinates directly from these equations impossible in any practical case.

2.5 Other Useful Relations and Concepts

The most popular approach to the diffraction by crystals is that proposed in 1912 by W.L. Bragg, who showed that diffraction can be treated in terms of reflections by different families of crystal planes. In Fig. 8.14 the horizontal lines represent the traces of two adjacent planes of the family with Miller indices$(hkl)$. The incident ($\mathbf{s}_o$) and the reflected ($\mathbf{s}$) beams will form the same angle $\theta$ with the planes and beams reflected by the two planes will be in phase when their path difference $AB + BC = 2 d(hkl) \sin \theta$ is equal to an integer multiple of the wavelength. In this way we can derive Bragg’s law

$$\begin{aligned} 2 d(hkl) \sin \theta = \lambda \end{aligned}$$

(8.78)

Comparison with (8.48) shows that

$$\begin{aligned} |\mathbf{S}_{hkl}| = \frac{2\sin \theta }{\lambda } = \frac{1}{d_{hkl}} \end{aligned}$$

(8.79)

i.e. the reciprocal vector modulus is the inverse of the $d$-spacing.

Fig. 8.14

In phase reflection by two adjacent planes of the ($hkl$) family at an inter-planar distance $d(hkl)$

Fig. 8.15

Reflection sphere and limiting sphere

Now consider a sphere of radius $1/\lambda$ (Fig. 8.15) and imagine that the incident beam arrives at the origin of the reciprocal lattice in $O$ along the diameter IO. When the node $P$ of the reciprocal lattice, at the end of the vector OP $\equiv |\mathbf{S}_{hkl}|$, is on the surface of the sphere, then

$$\begin{aligned} OP = |\mathbf{S}_{hkl}| = \frac{1}{d_{hkl}} = IO \, \sin \theta = \frac{2\sin \theta }{\lambda } \end{aligned}$$

(8.80)

and Bragg’s law is fulfilled; the line IP, forming an angle $\theta$ with the incident beam, will be parallel to the trace of the ($hkl$) planes and the beam will be reflected in the AP direction forming an angle $2 \theta$ with IO. The sphere is known as the reflection or Ewald sphere . When OP $> 2/\lambda$ the node $P$ can never be on the surface of the reflection sphere and we can define a limiting sphere of radius $2/\lambda$ delimiting the set of reflections that can be measured with a given wavelength.

With a given wavelength we can therefore only measure a limited number of reflections, but in most cases this may not be an important limiting factor. Indeed, as we have seen the atomic scattering factors are decreasing functions of the scattering angle and the decrease is more pronounced when the thermal vibration increases. When a sufficiently short wavelength (less than 1.0 Å) is used, in most cases the reflections within the limiting sphere became so weak as $\sin \theta /\lambda$ increases, that beyond a certain value they are not detectable. We can then introduce the concept of resolution , which is the minimum $d$-spacing corresponding to the maximum useful $\sin \theta /\lambda$, i.e

$$\begin{aligned} \left( \frac{\sin \theta }{\lambda }\right) _{max} = \frac{1}{2 d_{min}} = \frac{1}{2 Res} \end{aligned}$$

(8.81)

For accurate structural determinations it is important to have high angle reflections, which correspond to large ($hkl$) indices and small $d$-spacings. Indeed, the exponential terms in the structure factor (8.77), of type $h x_i + k y_i + l z_i$, become more sensitive to errors in the atomic coordinates when $h$, $k$ and $l$ are large. For instance, with Cu $K_\alpha$ radiation ($\lambda = 1.5418$Å), at the limit imposed by the limiting sphere $\sin \theta _{max} = 1$ and $Res = \lambda /2 = 0.77$ Å. Since the minimum interatomic distance is around 1.0 Å, we shall say that this corresponds to atomic resolution. For a poorly diffracting crystal, the maximum angle at which intensities are still detectable may be quite small. Thus for protein crystals it is quite common that, with Cu $K_\alpha$ radiation, $\sin \theta _{max} = 0.3$, and then $Res = 2.6\,$Å, and in the electron density map linked atoms will not be resolved as separate maxima.

3 The Phase Problem

As indicated by (8.53), the electron density function is related to the scattered amplitude by the relation

$$\begin{aligned} \rho (\mathbf{r}) = \int _{s} \mathbf{F}(\mathbf{S}) e^ {-2\pi \jmath \mathbf{S}\cdot \mathbf{r}}d\mathbf{S}\end{aligned}$$

(8.82)

For a crystal we have seen that $\mathbf{F}(\mathbf{S})$ is sampled only at the nodes of the reciprocal lattice; in (8.82) it can be substituted by the discrete values of $F(hkl)$ and the space integral becomes a triple summation over the integer components $h$,$k$ and $l$ of the reciprocal lattice vectors

$$\begin{aligned} \rho (\mathbf{r})= \frac{1}{V}\sum ^{\infty }_{h,k,l=-\infty }\mathbf{F}_{hkl} e^ {-2\pi \jmath (hx + ky + lz)} \end{aligned}$$

(8.83)

The electron density is then expressed as a Fourier series. It is easy to see that, as expected, it is a real function, which, in terms of the real and imaginary parts of the structure factor [cf. (8.72)], is given by

$$\begin{aligned} \rho (x,y,z)=\frac{2}{V}\sum ^{\infty }_{h=0}\sum ^{\infty }_{k=-\infty }\sum ^{\infty }_{l=-\infty } [A_{hkl} \cos 2\pi (hx+ ky + lz) + B_{hkl} \sin 2\pi (hx + ky + lz)] \end{aligned}$$

(8.84)

or, in terms of modulus and phase [cf. (8.74)], by

$$\begin{aligned} \rho (x,y,z)=\frac{2}{V}\sum ^{\infty }_{h=0}\sum ^{\infty }_{k=-\infty }\sum ^{\infty }_{l=-\infty } |F_{hkl}|\cos [2\pi (hx+ky+lz)-\phi _{hkl}] \end{aligned}$$

(8.85)

We can thus see that it is possible to compute the electron density function within the unit cell, if the structure factors are known in modulus and phase. The positions of the maxima of $\rho (x,y,z)$ would then yield the atomic coordinates defining the crystal structure. Unfortunately the diffraction experiments only give the diffracted intensities, which are proportional to the square of the corresponding amplitude, i.e. $|F_{hkl}|\propto \sqrt{I_{hkl}}$, and the information about the phase $\phi _{hkl}$ is lost. In order to be able to use (8.85) we will have to overcome the phase problem.

3.1 Patterson Methods

The oldest method used to solve the phase problem is called the Patterson method from the name of its discoverer [10]. It simply consists in calculating the Fourier-transform of the scattered intensities, without phases:

$$\begin{aligned} P(\mathbf{U})=\frac{1}{V}\sum ^{\infty }_{h,k,l=-\infty }I(hkl) e^ {-2\pi \jmath \mathbf{U}\cdot \mathbf{S}} \end{aligned}$$

(8.86)

where $\mathbf{U}$ is a vector in the direct space defined as $\mathbf{U}=u\mathbf{a}+ v\mathbf{b}+ w\mathbf{c}$. The scattered intensity can be written as the product of the structure factor with its conjugate complex number

$$\begin{aligned} I(hkl)={\mathbf{F}(hkl) \mathbf{F}^*(hkl)}. \end{aligned}$$

(8.87)

Remembering the convolution theorem, stating that the Fourier transform of a product of two functions is the convolution of the Fourier transforms of each separated function, it is evident that (8.86) corresponds to the self-convolution of the electron density:

$$\begin{aligned} P(\mathbf{U})=\int _{V}\rho {(\mathbf{r})} \rho {(\mathbf{U}-\mathbf{r})}d\mathbf{r}. \end{aligned}$$

(8.88)

Application of (8.88) gives us the superposition of the electron density map of the crystal with itself, translated by a vector $\mathbf{U}$. In practice, the Patterson map will present peaks in correspondence to all the interatomic vectors, translated to the origin of the crystal cell. The deconvolution of the map, which is in principle possible, will allow the definition of the original positions of the atoms. The advantage of the Patterson function is that it can be easily calculated from the measured intensities. The disadvantage is that the number of peaks observed for a crystal containing $n$ atoms in the crystal cell is $n^2$. Since $n$ peaks will be located at the origin, the number of remaining peaks will be $n(n-1)$. We can easily guess that only very small molecules can be solved using this method, since even molecules of 5 or 10 atoms, assuming for example a simple space group with 4 molecules in the unit cell, will give rise to a map containing 380 or 1560 peaks in the crystal cell, respectively. Consider also that even at atomic resolution most of these peaks will, at least partially, overlap, giving rise to a large, flat background that makes the deconvolution impossible in most cases. Nevertheless, the Patterson method has found several applications in everyday practice. First, it can be used to find the position of one or few heavy-atom(s) in a molecule containing a large number of light atoms: in fact, the map will show large peaks corresponding to inter-atomic vectors between heavy-atoms and this reduces the problem to that of a molecule containing one or few atoms. In addition, Patterson techniques have found widespread use in macromolecular crystallography, in particular in the methods of single- or multiple-isomorphous replacement (SIR or MIR, respectively), described in Chap. 24 of this book.

3.2 Direct Methods

Nowadays the structure of small molecules, i.e. molecules containing less than about 100-200 atoms, can be routinely solved by the so called Direct Methods or, more generally, ab initio Methods. The term Direct Methods refers to methods that try to derive phases directly from the observed amplitudes through probabilistic relationships. They rely on simple assumptions, like that the electron density is everywhere positive and that crystals are composed of discrete atoms. Based on them, Karle and Hauptman [11], Sayre [12] and many others were able to devise formulas that allow to obtain phases that are approximate, but sufficient to calculate an electron density map that can be interpreted in terms of an atomic or molecular structure. Direct methods are mathematically very complex and they will not be discussed here. A general account of them for the solution of the phase problem may be found in the cited textbooks [13].

3.3 Other Methods Used to Solve the Phase Problem

Whilst the structure of small molecule crystals can be solved mainly using the direct methods described in the previous paragraph, the same is not true for macromolecular crystals, with few exceptions. The major reason of their failure is that a very large number of atoms is present in the asymmetric unit (more than 1000 atoms, but in general this number is much larger and can reach values of hundred of thousands in the case of viruses or large macromolecular complexes) and the fact that protein crystals often diffract to a relatively low resolution (generally a resolution better than 1.5 Å is considered quite good, and for very few protein crystals the resolution extends to more than 1 Å). For this reason other methods are used to solve the phase problem: (i) the isomorphous replacement, which can be distinguished in Single Isomorphous Replacement , SIR, and Multiple Isomorphous Replacement , MIR. They consist in diffusing a compound containing a heavy-atom inside the crystal cell, taking advantage of the presence of the solvent; (ii) molecular replacement , where an already known homologous structure already known is properly oriented in the crystal cell; (iii) anomalous scattering, that can assume the form of Single Anomalous Diffraction , SAD, and Multiple Anomalous Dispersion , MAD. These methods used to solved the phase problem in macromolecular crystallography will be described at length in Chap. 24 of this book.

In this paragraph we want to point out an important aspect of the anomalous scattering. It relies on the fact that, for some atoms and at some wavelength, the atomic scattering factor is not real and its Fourier transform, which is in any case a complex number, does not satisfy the Friedel law . As a consequence:

$$\begin{aligned} F(hkl) \ne {F(\bar{h}\bar{k}\bar{l})} \end{aligned}$$

(8.89)

This inequality not only can be used to determine the position of the anomalous scatterer(s) and consequently to solve the phase problem, but it also represents the only method able to assign the absolute configuration of a molecule. In fact, chiral molecules containing an anomalous scatterer will be able to differentiate between $F(hkl)$ and $F(\bar{h}\bar{k}\bar{l})$.

4 Refinement of the Crystal Structure

Once a preliminary or a partial model of the molecule properly positioned in the crystal cell is available, the structure must be refined, i.e. atomic coordinates and thermal parameters must be varied in order to improve the agreement between observed and calculated amplitudes. The most widely used method for the refinement of small molecule crystals is the least-squares method, where the differences between observed and calculated structure amplitudes are minimized:

$$\begin{aligned} \varDelta =(|\mathbf{F}_{obs}(hkl)| - |\mathbf{F}_{calc}(hkl)|)^2. \end{aligned}$$

(8.90)

The progress of refinement can be monitored through the calculation of different versions of the so-called crystallographic R factor, whose more common version is:

$$\begin{aligned} R = \frac{\sum |k|\mathbf{F}_{obs}(hkl)| - |\mathbf{F}_{calc}(hkl)||}{\sum k|\mathbf{F}_{obs}(hkl)|} \end{aligned}$$

(8.91)

where $k$ is the scale factor and the sum is extended over all observed reflections. Another important control parameter is called $R_{free}$, which is analogous to the $R$ factor, except that it is calculated using a small fraction of reflections, generally 5–10 % of the total, that were excluded from the refinement process. $F_{calc}(hkl)$ is calculated, according to (8.77) :

$$\begin{aligned} F_{hkl} =\sum ^{N}_{i=1} f_i(\mathbf{S}) e^ {2\pi \jmath (h x_i + k y_i + l z_i) } e^{ -B_i S^2 } \end{aligned}$$

(8.92)

where $B_i$ is the thermal parameter described in Sect. 8.2.3. It represents the isotropic approximation, which is usually not appropriate to describe the thermal motion of atoms. If enough observations are present, it is advisable to introduce an anisotropic description of the atomic thermal displacement using a $3\times 3$ tensor. The latter describes an ellipsoid, centered on the $i$th atom, in the reference crystal system.

The number of observations is a relevant aspect in the refinement process: their number, i.e. the number of independent reflections, must be at least 5–10 times the number of variables. The latter corresponds to 4–9 times the number of atoms present in the asymmetric unit of the crystal for the isotropic or anisotropic model, respectively. The number of observations, in turn, is strictly related to resolution: when the resolution extends to values close or lower than 1Å, the ratio observations/variables is largely sufficient. If this does not apply, as is generally the case for macromolecules, appropriate tools, based on other available information, must be applied.

The refinement process requires an iterative procedure, where automatic cycles of refinement are alternated with visual inspections of electron density maps. In the latter stage a very useful tool is represented by the Fourier-difference map, calculated with the formula:

$$\begin{aligned} \rho (\mathbf{r})= \frac{1}{V}\sum ^{\infty }_{h,k,l=-\infty } [kF_{obs}(hkl) - F_{calc}(hkl)] e^{j\phi } e^ {-2\pi \jmath (hx + ky + lz)} \end{aligned}$$

(8.93)

The map calculated with (8.93) will show positive maxima corresponding to atoms absent in the model and negative minima corresponding to atoms misplaced. These errors are hardly corrected by automatic procedures and often require manual intervention.

At the end of the refinement process it is expected that the crystallographic $R$ factor has a value as low as possible: in the case of small molecule crystals it can reach values from 0.02 to 0.06 (higher values, from 0.07 to 0.10 are looked suspiciously and must be justified, for example they can be due to the presence of disorder in some part of the molecule). For macromolecules, higher values are expected, as discussed in Chap. 24.

5 Diffraction by Polycrystalline Samples

As we have seen, for an accurate structural analysis, we must achieve the best possible representation of the electron density, $\rho (\mathbf{r})$, by measuring the intensity $I_{hkl}$ of as many reflections as possible. In practical terms we must perform an experiment in which as many as possible reciprocal lattice nodes cross the surface of the reflection sphere. This can be done in three different ways

• Using a single crystal and a monochromatic X-ray beam with a fixed direction $\mathbf{s}_o$, the crystal is moved so that most accessible reciprocal lattice nodes are brought to satisfy Bragg’s low. This method will be discussed in other lectures.

• A monochromatic X-ray beam with a fixed direction $\mathbf{s}_o$ hits a large number of randomly oriented crystals. This is the case of the so called powder diffraction technique, which we will treat in more details in this section.

• Using a single crystal and a polychromatic radiation each reciprocal lattice node will have a chance of being on the surface of one of the infinite reflection spheres (one for each wavelength of the polychromatic radiation) available. This was the technique used in the first experiment of Friedrich and Knipping and is known as the Laue method. In recent years intense white synchrotron radiation sources have been used to obtain Laue diffraction patterns with very short exposure times in order to carry out time resolved experiments on materials and biological molecules [14, 15].

In order to measure their intensity the reflections are collected on a detector, which can be a film, a counter or an area detector (image plate or CCD). The intensities are proportional to the square of the structure factors

$$\begin{aligned} I_{hkl} = K|F_{hkl}|^2 \end{aligned}$$

(8.94)

where $K$ includes several terms, among which the Lorentz (taking into account the time needed by the little volume around the reciprocal lattice node to cross the surface of the reflection sphere) and polarization factors, the incident beam intensity $I_o$, $\lambda ^3$ and the volume of the sample.

We will now give a brief overview of the powder diffraction technique. For a more detailed description the reader is referred to specific publications [16–18].

An ideal crystalline powder is formed by an infinite number of randomly oriented small crystals (crystallites or grains). The Fourier transform of this set of randomly oriented lattices will be an infinite number of randomly oriented identical reciprocal lattices all with a common origin. Each reciprocal lattice vector $\mathbf{S}_{hkl}$ will then assume all possible orientations (Fig. 8.16) and the corresponding reciprocal lattice node will becomes the surface of a sphere of radius $|\mathbf{S}_{hkl}|$. This sphere will intersect the reflection sphere on a circle and the reflected beams will be on a cone with vertex in A and aperture $4 \theta$. Of course in a real sample the number of grains will be finite and not necessarily the crystallites will be randomly oriented. In order to approach the ideal case we will have to grind our sample to increase the number of grains and to avoid as much as possible preferred orientations of the crystallites.

Fig. 8.16

Reflection sphere and sphere generated by a randomly oriented reciprocal lattice vector. Cone of reflected beams

5.1 Experimental Techniques for Obtaining Powder Diffraction Patterns

The most common setting of powder diffractometers is the so called Bragg-Brentano geometry, shown in Fig. 8.17.⁵ The X-rays hit the flat sample CP (powder pressed in a flat plate) and with this geometry the reflected beams are focalized on the counter D.

Fig. 8.17

Bragg-Brentano geometry

Fig. 8.18

Debye-Scherrer geometry

An other common setting is the so called Debye-Scherrer geometry, in which the sample is in a cylindrical capillary and is usually rotated to reduce the effects of preferred orientations. In the early days this geometry was used with film detectors as shown in Fig. 8.18. In modern instruments the film is replaced by a counter or a position sensitive area detector (like on a film several reflections are recorded at the same time).

Fig. 8.19

Powder diffraction pattern of Silicon (NBS 640b)

X-rays are monochromatized by a graphite or a silicon monochromator and collimated by different types of slits. Focusing mirrors are always used with synchrotron radiation and often employed with conventional X-ray sources. The recorded diffraction pattern is then digitized and stored in a file with a suitable format. The plot of the pattern of a Silicon sample, used as standard, is shown in Fig. 8.19. From the $2\theta$ value we obtain $\sin \theta$ and for a given wavelength [recalling (8.62)] we have

$$\begin{aligned} |\mathbf{S}_\mathbf{H}| = \frac{2\sin \theta }{\lambda } = |h\mathbf{a}^*+ k\mathbf{b}^*+ l\mathbf{c}^*| \end{aligned}$$

(8.95)

Since $d_\mathbf{H}= 1/|S_\mathbf{H}|$ we can write

$$\begin{aligned} \frac{\sin ^2\theta }{\lambda ^2} = \frac{1}{4 d_\mathbf{H}^2}&= \frac{1}{4} (h^2 {a^*}^2 + k^2 {b^*}^2 + l^2 {c^*}^2 + 2 \ h \ k \ a^*b^*\cos \gamma ^*\nonumber \\&\;\;\;\;\; + \; 2 \ h \ l \ a^*c^*\cos \beta ^*+ 2 \ k \ l \ b^*c^*\cos \alpha ^*) \end{aligned}$$

(8.96)

We will have a relation of this type for each reflection and from this set of relations it is possible to derive the reciprocal cell parameters and the indices of the different reflections. This is not a straightforward procedure for low symmetry crystals and becomes usually rather simple for cubic crystals.

The most serious limitation inherent to powder diffraction is reflection overlap. In fact, especially at high reflection angles, different families of planes with different ($hkl$) indices (i.e. with different orientations with respect to the reference axes) may have very close $d_\mathbf{H}$ values and the orientational averaging arising from the random orientation of the crystallites will cause the overlap of the corresponding reflections. If a peak is the overlap of two or more reflections it will be impossible to decide a-priori what fraction of the total intensity should be assigned to the individual reflections. Synchrotron radiation allows a much higher angular resolution and peaks which overlap in an experiment with a conventional source may often be resolved.

5.2 Analysis of Powder Diffraction Patterns

We now give a very brief account of the information which can be attained from the analysis of powder diffraction patterns. A more detailed description is reported in Chap. 10.

• 1. Phase analysis: Each crystalline phase has its typical diffraction pattern which is characterized by the $d$-spacings $d_\mathbf{H}$ and the relative intensities $I_\mathbf{h}$ of all reflections. If a data base of powder diffraction data of the pure compounds of interest is available (a general database can be purchased [19]), by comparing the set $\{d_\mathbf{H}, I_\mathbf{h}\}$ of experimental values of an unknown sample with those reported in the database we can identify the crystalline phases present in the sample.

• 2. Phase transitions : When a crystalline compound undergoes a phase transition, its diffraction pattern changes as a consequence of the modification of the lattice. It is possible to record diffraction patterns at continuously increasing temperatures and a transition point can be identified by the sudden change in the pattern.

• 3. Quantitative phase analysis : From the ratios of the intensities of the reflections of the different phases it is possible to obtain an estimate of the relative quantity of each phase. This can be done using the Internal Standard method or an extension [17] of the Rietveld refinement technique (see Chap. 10).

• 4. Structure determination and refinement: As we have seen, the determination of a crystal structure requires the knowledge of the structure factors in modulus and phase. The first may be derived from the intensities, but reflection overlap will make this process much more difficult. Patterson and direct methods can be applied to achieve phase information, but the uncertainty on the moduli makes crystal structure solution from powder data more problematic. Recent progresses in the algorithms [20, 21] and the use of high resolution synchrotron diffraction patterns have allowed the solution of increasingly complex structures with up to 30–40 independent atoms. The refinement of the structure is usually performed by the Rietveld method.

• 5. Crystallite dimensions and deformations: (8.68) and (8.69) indicate that finite crystals give wider diffraction maxima. It is therefore possible to derive the average size $D$ of the grains from the analysis of the profiles of the diffraction maxima. The simplest (but less accurate) relation between $D$ and the full width at half maximum (FWHM) $\varDelta (2\theta )$ is

  $$\begin{aligned} D = 0.9 \frac{\lambda }{\cos \theta \varDelta (2\theta )} \end{aligned}$$

  (8.97)

  More complex and accurate relations have been derived [22] and a more detailed description is given in Chap. 10.

  Deformations of the sample by mechanical, thermal or other treatments cause variations in the lattice cell parameters and consequently the positions of the diffraction maxima will change. It is possible to obtain an estimate of the strain and stress tensors from diffraction experiments [2].

• 6. Preferred orientations : As mentioned before, it may happen that in the powder sample the crystallites are not randomly oriented. Indeed acicular or plate crystals will tend to align or pile in a preferred way and sometimes mechanical and/or thermal treatments can orient the grains. We will then have textured samples and in this cases the intensity of the diffracted beams along the cones of Fig. 8.16 will not have a uniform distribution. An analysis of these distributions for some reflections will allow the construction of the so called polar figures [22, 23], from which the orientation distribution functions (ODF) may be derived.

  A very promising method for exploiting preferred orientation to obtain single-crystal-like diffraction data from textured polycrystalline samples has been recently proposed [24].