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Teknillisen fysiikan ohjelmatyö

X-ray diffraction and structure analysis Introduction

Oleg Heczko

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Two Theta

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X-ray diffraction - structure studies and phase analysis Theoretical Introduction

Solid state physics is largely concerned with crystals and electrons in crystals. The study of

solid state physics could begin in the beginning of 20th century thanks (apart of others) the

discovery of the X-ray diffraction by crystals (Röngen, Laue, Bragg). The X-ray diffraction belongs

to the basic tools for condensed matter physics and materials science.

When crystal grows in a constant environment, the form develops as if identical building

blocks were added continuously. The building blocks are atoms or groups of atoms, so the crystal

is a solid three-dimensional periodic array of atoms.

Definition of crystal lattice

An ideal crystal is constructed by the repetition of identical structural unit in space. In the

simplest crystal the structural unit is a single atom, as in e.g. copper, silver, gold, iron. But the

smallest structural unit may comprise many atoms or molecules.

The structure of all crystals can be described in terms of lattice, with group of atoms

attached to every lattice point. The group of atoms is called the basis; when repeated in space, it

forms the crystal structure as demonstrated in Fig. 1(d).

Three fundamental translation vectors a1, a2, a3 define the lattice, such that the atomic

arrangement looks the same in every respect when viewed from the point

r' = r +u1a1 + u2a2 + u3a3 (1)

where ui are arbitrary integers. The set of the points r' defined by (1) for all ui defines a lattice.

A lattice is a regular periodic array of points in space. (The analogy in two dimensions is called

net.)

The lattice and the translation vectors ai are said to be primitive if any two points r, r' from

which the atomic arrangement looks the same, always satisfy (1) with suitable choice of the

integers ui. The primitive vectors, ai, form three adjacent edges of a parallelepiped called primitive

cell or unit cell. If there are lattice points only at the corners, then it is a primitive cell. It means

that there is just one lattice point per primitive cell. There is no cell of smaller volume than a

primitive cell that can serve as a building block for the crystal structure. There are many ways of

choosing the primitive cell for a given lattice. Examples of lattices in two dimensions are in Fig. 1.

The primitive translation vectors can be used to define the crystal axes. However, nonprimitive

crystal axes are used when they have a simpler relation to the symmetry of the structure (compare

e.g. Fig 1(d)).

A lattice is a mathematical abstraction; the crystal structure is formed when a basis of atoms is

attached identically to every lattice points. A basis of atoms is identical in composition,

arrangement, and orientation. The number of atoms on the basis can be one or more. The position

of the centre of an atom j of the basis relative to the associated lattice points is

rj = xja1 +yja2 +zja3 (2)

where 0 ≤ xj, yj, zj ≤ 1.

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Fig. 1. Examples of lattice in two dimensions.

Fig. 2. The cubic space lattices. The cells shown are conventional cells.

Fig. 3. A section of close packed spheres in face-centered cubic cell containing four atoms in unit cell (8 x 1/8 + 6

x 1/2 = 4).

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Index system for lattice (crystal) planes

The orientation of a plane is determined by three non-collinear points. For structure analysis

the orientation of the crystal plane is specified according following rules

- Find the intercept of the axes in term of the lattice constant a1, a2, a3. The axes may be those

of primitive or nonprimitive cell.

-Take the reciprocals of these numbers and then reduce to three integers having the same ratio,

usually the smallest of three integers. The result enclosed in parentheses (hkl) is called the index of

the plane or Miller indices of the plane.

For the plane whose intercept are 3, 2, 2, shown in Fig. 4, the reciprocal are 1/3, 1/2, 1/2; the

smallest three integers having the same ration are (233). For an intercept at infinity, the

corresponding index is zero. The indices (hkl) may denote a single plane or a set of parallel planes.

If plane cuts an axis on the negative side e.g. -k, the corresponding index is negative and it is

indicated by placing minus sign above the index (hk l). Planes equivalent by symmetry are

denoted by curly brackets around indices; set of cube faces is {100} which means a collection of

(100), (010), (001), (1 0 0), (01 0), (0 01) planes.

Fig. 4. Miller indices of a plane.

The indices of a direction [uvw] in a crystal are the set of the smallest integers that have the

ratio of the components of a vector in the desired direction; referred to the axes e.g. a1 axis is the

[100] direction. In the cubic crystal the direction [hkl] is perpendicular to a plane (hkl) having the

same indices, but it is not generally true for other crystal systems.

Fundamental types of lattices and crystal structure

Crystal lattices can be carried or mapped into themselves by the lattice translation T

(translation symmetry) and by various other symmetry operations.

Lattice translation operation is defined as the displacement of a crystal by a crystal translation

vector

T = u1a1 +u2a2 +u3a3 (3)

Any two lattice points are connected by a vector of this form.

Other symmetry operations are

- Rotation about axis passing through a lattice point,

- Mirror reflection,

- Inversion.

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These are called point operations. There are one-, two-, three-, four-, and six-fold rotation,

which carry the lattice into itself, corresponding to rotations by 2π, 2π/2, 2π/3, 2π/4, 2π/6

radians. We have mirror reflections, m, about a plane through a lattice point. The inversion

operation is composed of a rotation of π, followed by a reflection in a plane normal to the

rotation axis; i.e. the effect is to replace r by -r. Finally, there may exist compound operations

made up of combined translation and point operations. Textbooks on crystallography are

largely devoted to the description of symmetry operations.

Table 1. Types of Bravais lattices and their parameters.

The point symmetry groups in three dimensions require the 14 different lattice types called

Bravais lattice listed in Table 1. The general lattice is triclinic, and there are 13 special lattices.

There are three lattices in the cubic system on which we will concentrate further: simple cubic (sc),

body-centered cubic (bcc) and the face centered cubic (fcc) lattices. The cubic cells are shown in

Fig. 2; only simple cubic (sc) is a primitive cell. It contains one lattice point. The bcc and fcc

contain two and four lattice points respectively and there are six, eight and twelve closest (nearest)

neighbours for sc, bcc, and fcc lattice respectively. This number is also called coordination

number. Example of fcc cell with close sphere packing is in Fig. 3. Often nonprimitive cell

(conventional cell) has more obvious relation with the point symmetry operation than the primitive

cell.

The position of a point in a cell is specified by (2) in terms of atomic coordinates x, y, z. Here

each coordinate is a fraction of the axial length ai in the direction of the coordinate axis with the

origin taken at one corner of the cell. Thus e.g. the coordinates of the lattice points in body

centered lattice are 0,0,0 and 1/2,1/2,1/2.

Examples of simple crystal structure

Many common metals crystallize in cubic structure (Fig. 2) with monoatomic base i.e. the

lattice point is occupied by a single atom; e.g. Cu, Ni, Pb, Al, Ag, Au, Pt, γ-Fe possesses fcc lattice,

the examples of bcc lattice are α-Fe, Cr, V, Nb, Mo, Ta, W etc.

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The sodium chloride, NaCl, structure is shown in Fig. 5. The lattice is face-centered cubic;

basis contains one Na atom and one Cl atom separated by one-half the body diagonal of a unit

cube. There are four units of NaCl in each unit cube. Each atom has a nearest neighbours six atoms

of the opposite kind.

The cesium chloride, CsCl, structure is shown in Fig. 6. There is one molecule per primitive

cell with atoms at the corners 000 and body-centered positions 1/2,1/2,1/2, of the simple cubic

space lattice. Each atom may be viewed as at the centre of a cube of atoms of the opposite kind, so

the number of the nearest neighbours or coordination number is eight. Examples of crystals having

this structure are BeCu, AlNi, CuZn (β-brass), CuPd, etc...)

The diamond structure shown in Fig. 7 is fcc. The primitive basis has two identical atoms at

000; 1/4,1/4,1/4 associated with each point of the fcc lattice. Thus the conventional unit cube

contains 4 x 2 = 8 atoms. There is no way to choose the primitive cell such that the basis of

diamond contains only one atom. It can be viewed as a structure of two interpenetrating face-

centered cubic lattices displaced along the body diagonal by one quarter the length of the diagonal.

The lattice is relatively empty, amount of closest neighbours are just four. Examples of this

structure are crystals of carbon (diamond), silicon, germanium and grey tin.

More than 70 percent of elements crystallise in one of the lattices: bcc, fcc, diamond and

hexagonal close packed (hcp). The remaining ones have a variety of crystal structures some of

them quite complex.

A

B

Fig. 5. Crystal structure of NaCl. We may construct the sodium chloride crystal structure by arranging Na+ and Cl

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ions alternately at the lattice points of simple cubic lattice. In the crystal each ion is surrounded by six nearest

neighbours of the opposite charge. b) Model of sodium chloride. The sodium atoms are smaller.

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Fig. 6. Crystal structure of CsCl

Fig. 7. Crystal structure of diamond showing

the tetrahedral bond arrangement.

Other (nonideal) structure of materials

The ideal crystal is formed by the periodic repetition of identical units in space. But no

general proof has been given that the ideal crystal is the state of minimum energy at absolute

zero temperature. At finite temperature there are always lattice defects present. Further there is

not always possible for structure to attain the equilibrium state in a reasonable time, which

leads to glasses or amorphous materials. In the amorphous materials the atoms are distributed

in a random pattern without regular lattice. Ordinary glass is a representative example of an

amorphous material. The glasses can be also considered as a frozen liquid.

Structures are known in which the stacking sequence of close packed planes is random.

This is known as random stacking and may be thought as a crystalline in two dimensions and

noncrystalline or glasslike in the third. Additionally many structures that occur in nature are

not entirely periodic but somehow quasiperiodic - quasicrystalline materials with five-fold

symmetry.

X-ray diffraction from crystal - Bragg law

The crystal structure is studied using the diffraction of photons, neutrons and electrons.

The diffraction depends on the crystal structure and on the wavelength. At optical wavelength

about 500 nm the superposition of the waves scattered elastically by the individual atoms of a

crystal results in ordinary optical refraction. When the wavelength is comparable or smaller

than the lattice constant, diffracted beams occurs in direction quite different from the incident

direction. W.L. Bragg presented a simple explanation of the diffracted beams from the crystal.

Suppose that the incident waves are reflected specularly (i.e. the angle of reflection equals the

angle of incidence) from parallel planes of atoms in the crystal, with each plane reflecting only

very small fraction of the radiation. The diffracted beams are found only when the reflections

from parallel planes of atoms interfere constructively. We consider only the elastic scattering,

in which the energy of the X-ray is not changed on reflection.

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Fig. 8. Bragg reflection from a particular family of lattice planes separated by distance d. Incident and

reflected rays are shown only for two neighbouring planes (in reality on each plane about 10-3 intensity is

reflected). The path difference is 2dsinθ.

Take parallel planes spaced d apart (as shown in Fig. 8). The path difference for rays

reflected from the adjacent planes is 2dsinθ, where θ is measured from the plane. Constructive

interference of the radiation from successive planes occurs when the path difference is an

integral number n of wavelengths λ, so that

2dsinθ = nλ (4)

This is Bragg law. Bragg reflection can occur only for wavelength λ ≤ 2d. This is why we

can use X-rays. It is important to notice that Bragg law is the consequence of the periodicity of

the lattice. The law does not refer to the composition of the basis associated with every lattice

points. This is purely geometrical law. However, the composition of the basis determines the

relative intensity of the various order of diffraction (denoted by n in eq. (4)) from the given set

of parallel planes.

Fig. 9. Two-dimensional lattice showing that lines of lowest indices have the greatest spacing and the

greatest density of lattice points.

The various sets of planes in a lattice have various values of interplanar spacing. The

planes of large spacing have low indices and pass through a high density of lattice points,

whereas the reverse is true for planes with small spacing. It is illustrated in Fig. 9 for two-

dimensional lattice. The interplanar spacing dhkl, measured at right angles to the planes is a

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function both of the plane indices (hkl) and the lattice constants (a, b, c, α, β, γ). The exact

relation depends on the crystal system involved and for the cubic system it has simple form

(cubic) da

h k lhkl =

+ +2 2 2 (5)

It should be remembered that the Bragg law is derived for certain ideal conditions and the

diffraction is only a special kind of scattering. A single atom scatters an incident beam of X-

rays in all direction in space but a large number of atoms arranged in a perfectly periodic

lattice scatter (diffracts) X-ray in relatively few directions. It is so precisely because the

periodic arrangement of atoms causes destructive interference of the scattered rays in all

directions except those predicted by Bragg law where constructive interference

(reinforcement) occurs. If the crystal contains some imperfections the measurable diffraction

can occur at nonBragg angles because the crystal imperfection results in the partial absence of

one or more conditions necessary for perfect destructive interference at these angles. These

imperfection are generally slight compared to the over-all regularity of the lattice which cause

that the diffracted beams are confined to very narrow angular ranges centered on the angles

predicted by the Bragg law for ideal condition resulting in line broadening. The relation

between constructive interference and structural periodicity can be well illustrated by a

comparison of X-ray scattering by crystalline solids, liquids and gases (Fig. 10). The curve of

scattered intensity vs. 2θ for crystalline solid is nearly zero everywhere except at certain angle

where sharp maxima occur - diffracted beams. Both amorphous solids and liquids have

structures characterised by an almost complete lack of periodicity and tendency to order in the

sense that the atoms are fairly tightly packed together. The atoms in amorphous structure show

just statistical preference for a particular interatomic distance; resulting X-ray diffractogram

exhibits only one or two broad maxima. But in monoatomic gases there is no periodicity

whatever, the atoms are arranged perfectly random and their relative positions change all the

time. The corresponding scattering curve shows no maxima just a regular decrease of intensity

with increasing scattering angle (see also atomic factor).

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Fig. 10. Comparison of X-ray scattering curve for crystalline, amorphous structure and monoatomic gas

(schematic). The vertical scales are different.

Intensity of diffracted beam

The diffraction condition described by Bragg law gives the condition for constructive

interference of waves scattered from the lattice points. However, the scattering intensity from

crystal is determined by spatial distribution of electrons within each unit cell. Because a

crystal is invariant under any translation T (eq. 3), any local physical property of the crystal

such as electron number density, or magnetic moment density is also invariant under T. This is

very suitable for Fourier analysis. Fourier analysis of the electron concentration leads to the

concept of reciprocal lattice. The idea of reciprocal lattice is of paramount importance and

theoretical base in solid state physics but we will not pursue it here. We will take rather

crystallographic approach.

When monochromatic beam of X-rays strikes an atom, two scattering processes occur.

Coherent or unmodified scattering on tightly bound electrons and Compton (incoherent)

scattering on loosely bound electrons; both kind occurs simultaneously and in all directions.

The coherent scattering does not change the wavelength of X-ray beam. This coherent

scattering is important for X-ray diffraction. A quantity f, the atomic scattering factor, is used

to describe the magnitude of the scattering of a given atom in a given direction. It is defined as

a ratio of amplitudes

f = amplitude of the wave scattered by an atom/amplitude of the wave scattered by one

electron

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It is also called form factor because it reflects the distribution of electrons around nucleus.

In the forward direction f = Z (atomic number) and it decreases with the angle of scattering θ.

The atomic scattering factor also depends on the wavelength of the incident beam. The

calculated values for all elements are tabulated as a function of sinθ/λ. Since the intensity of

the wave is proportional to the square of its amplitude, a curve of scattered intensity from an

atom is the square of f. The resulting curve closely approximates the observed intensity per

atom of monoatomic gas (Fig. 10).

To determine the intensity of the diffracted beam, we have to consider the coherent

scattering not from single atom but from all the atoms making up the crystal. When the atoms

are the part of regular periodic lattice, the coherent scattering undergoes the enhancement in

certain directions and cancellation in others, thus producing diffracted beams. The directions

of these beams are determined by Bragg law, which says in which direction the diffracted

beam may occur (but not necessarily will be). Since the crystal is a repetition of the

fundamental unit cells, it is enough to consider how the arrangement of atoms within a single

unit cell affects the diffracted intensity assuming the Bragg law is satisfied. In order to find

resulting wave from one unit cell, waves scattered by all the atoms of the unit cell, including

that in origin must be added together. The most convenient way of carrying this summation is

by expressing each wave as a complex exponential function. Any scattered wave with phase φ

can be expressed in the form

Ae fei i hu kv lwφ π= + +2 ( )

where indices (hkl) are indices of the plane of reflection, uvw are the indices of atom

position inside the cell. The resultant wave scattered by all the atoms of the unit cell is called

the structure factor, because it describes how the atoms arrangement, given by uvw for each

atom, affects the scattered beam. The structure factor, F, is obtained simply by adding together

all the waves scattered by the individual atoms. If the unit cell contains atoms 1,2, 3..., n with

fractional coordinates u1v1w1, u2v2w2,......, uNvNwN, and atomic factors f1, f2, ...fN then

structure factor is given by

F f ehkl n

i hu kv lwN

n n n= + +∑

2

1

π ( )

(6)

where the summation extent over all the N atoms of the unit cell. F is in general the

complex number and it expresses both the amplitude and phase of the resultant diffracted

wave. The intensity of the beam diffracted by all atoms in the direction predicted by the

Bragg law is proportional simply to F2, the square of the amplitude of the beam obtained by

multiplying the expression for F by its complex conjugate. The equation (6) is very important

relation in X-ray crystallography since it permits the calculation of the intensity of the

reflection of any (hkl) plane from the knowledge of the atomic positions.

Structure factor calculation - Example

1. The simplest case - unit cell contains only one atom in origin i.e. at 000.

Structure factor is F = f exp(2πi(0)) = f and the intensity is proportional to F2 = f2 .

2. Body centered cell (see Fig. 2) - unit cell contains two atoms of the same kind located

at 0 0 0, and 1/2 1/2 1/2.

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F = f exp(2πi(0)) + f exp(2πi(h/2+k/2+l/2)) = f [ 1 + exp(πi(h +k +l))]

F = 2f when (h +k +l) is even;

F = 0 when (h +k +l) is odd;

From the calculation it is clear that, the body-centered cell will not, for example, produce

001 reflection. This can be also concluded from the geometrical consideration (see Fig. 11).

However, a detailed examination of the geometry of all possible reflection would be quite

demanding process compared to the calculation of structural factor, which instantly yields a

set of rules for F2 or possible reflections for all combinations of plane indices. It should be

noted that the structure factor is independent of the shape and size of the unit cell. For

example any body-centered cell will have missing reflection for the planes where when (h +k

+l) is odd, whether the cell is cubic, tetragonal or orthorhombic.

Fig. 11. Geometrical explanation of absence of 100 line in body centered lattice. The phase difference

between successive planes is π, so the reflected amplitude from two adjacent planes is zero (1 + exp(-iπ) = 0).

Few general remarks on powder diffraction method and determination of crystal

structure

The powder diffraction method of X-ray diffraction was firs suggested by Debye and

Sherrer in 1916. It is the most useful of all diffraction methods, which can yield a great deal of

structural information about the material under investigation. Essentially the method employs

the diffraction of monochromatic X-rays by a powder specimen. In this sense the

monochromatic usually means filtered strong characteristic K radiation from X-ray tube

operated above the K excitation potential of the target material. "Powder" can mean either real

powder bound together or any specimen in polycrystalline form. This is a great advantage as

the polycrystalline materials can be examined without special preparation.

In powder sample or fine polycrystalline sample there are many small crystallites

randomly oriented so there are always some crystals with planes favourably oriented to yield

the diffraction in particular angle determined by Bragg condition. By scanning the sample in

wide angle different planes of different crystals will satisfy the reflection condition and the

diffraction pattern or lines will occurs and can be recorded (see also the experimental set-up).

The diffraction intensity as a function of the measuring angle 2θ is called diffractogram. The

method assumes that the amount of the crystallites is large and truly randomly oriented. If not,

i.e. the sample has some kind of preferred orientation or the crystalline grains are large, the

observed intensity of the diffraction lines may radically differ from the calculated one. It is

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relatively easy to prepare powder specimens where ideal perfect randomness is closely

approached but virtually all solid polycrystalline materials either produced or natural will

exhibit some preferred orientation of the grains.

As it was shown previously the shape and size of the unit cell determines the angular

position of the diffraction lines. The arrangement of the atoms within the unit cell determines

the relative intensity of the lines (via structural factor). Since the structure determines the

diffraction pattern, it should be possible to go in the other direction and deduce the structure

from the diffraction pattern. However, there is not any direct way to achieve this. Given a

structure, the diffraction pattern can be easily calculated, but the reverse problem, directly

calculating the structure from the experimental diffraction pattern is in general impossible.

The procedure adopted is essentially the method of trial and error. On the basis of a reasonable

guess, a structure is assumed, the diffraction pattern calculated and the calculated pattern

compared with the observed one. If the two agree in all detail, the assume structure is correct;

if not the process is repeated to find a correct solution.

Calculation of powder pattern intensity

Any calculation of the intensity of diffraction must begin with the structure factor. The

rest of the calculation varies with the diffraction method involved.

There are six factors affecting the relative intensity of the diffraction lines in a powder

pattern.

1. Structural factor

2. Polarisation factor

3. Multiplicity factor

4. Lorenz factor

5. Absorption factor

6. Temperature factor

The polarisation factor is connected with the elastic scattering of X-ray by a single

electron and the fact that the incident X-ray radiation is unpolarised. Lorenz factor is

particular trigonometric factor influencing the intensity of the reflected beam for various

diffraction angles. The diffracted intensity is also affected by absorption, which takes place in

the specimen itself - absorption factor. The temperature factor is due to fact that the atoms are

not fixed in particular points but undergo the thermal vibration about that points. This is true

even for absolute zero and the amplitude of the vibration increases as the temperature

increases. This leads to decrease of the amplitude of the diffraction lines with increasing

temperature. The factor also depends on diffraction angle.

Multiplicity factor is connected with the fact that there are different planes with the same

spacing. Consider the 100 reflection from the cubic lattice. In the powder specimen, some of

the crystals will be oriented so that the reflection occurs from their (100) planes. Other crystals

of different orientations may be in a such position that reflection occurs from their (010) or

(001) planes. Since all the planes have the same spacing, the beams diffracted by all of them

form the part of the same line. This relative proportion of planes contributing to the same

reflection enters the intensity calculation. The multiplicity factor, p, can be defined as the

number of different planes having the same spacing. Parallel planes with different Miller

indices, such as (100) and (100) are counted separately. Thus for example the multiplicity

factor of the {100} planes of a cubic crystal is 6.

Taking into account all above-mentioned factors the relative intensity of the powder

pattern line in diffractometer measurement is

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I F p eM( ) (

cos

sin cos)θ

θθ θ

=+ −2

2

2

21 2 (7)

where term in brackets is Lorenz-polarisation term and e-2M is temperature factor. The

absorption factor does not enter the equation because the absorption is independent on the

angle θ for diffractometer measurement. For the comparison of the intensity of the adjacent

(neighbouring) lines on the pattern at the same temperature the temperature factor can be

omitted.

References: 1. Ashcroft N., Mermin N. D., Solid State Physics, international edition, 1976

2. B.C. Cullity Elements of X-ray diffraction, second edition, Addison-Wesley Publ., 1978

3. W.P. Pearson, Handbook of Lattice Spacing and Structures of Metals, Pergamon Press, Oxford 1967

4. International Tables for X-ray Crystallography, four volumes, Kynoch Press, 1952-1974

5. H. P. Klug and L E. Alexander, X-ray Diffraction Procedures, John Wiley & Sons, N.Y., 1956

6. J.M. Ziman Models of disorders, Cambridge University Press, 1979, Cambridge

Questions:

Why is necessary and possible to use X-rays for structure studies? Can we use other

radiation?

What are the basic Bravais lattices, list at least five.

How many cubic lattices crystallographers recognize and which one is primitive?

What are basic symmetry operations in the lattice? Why it is not possible to have lattice

with five-fold symmetry?

What are the Miller indices of the plane? Determine them for given simple planes.

The face-centered cubic is the densest and the simple cubic is the least dense of the three

cubic Bravais lattice. The diamond structure is even less dense than any of these. One measure

of this is coordination number. What is the coordination number and what is the number for

fcc, bcc, sc and diamond structure?

What is multiplicity?

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Diffraction method - Experiment

X-ray diffraction is a non-destructive method for identification and quantitative

determination of the various crystalline forms known as “phases” of compounds present in

powdered and solid samples.

Identification is achieved by comparing the X-ray diffraction pattern- or “diffractogram”-

obtained from an unknown sample with an internationally recognized database containing

reference patterns. In our case the commercial database contains more than 70 000 of the most

common phases.

When a monochromatic X-ray beam with wavelength λ is incident on lattice plane in a

crystal at an angle θ, diffraction occurs only when the distance traveled by the rays reflected

from successive planes differs by a complete number n of wavelengths. Bragg law states the

essential condition that must be met if diffraction is to occur

n λ = 2dsin θ

By varying the angle θ, the condition of Bragg law is satisfied by different d-spacing in

polycrystalline materials. Plotting the angular position and intensities of the resultant

diffraction peaks produces a pattern that is characteristic of the sample - diffractogram. Where

a mixture of different phases is present the diffraction is formed by the superposition of the

individual patterns.

X-ray diffractometer

Diffractometer is an instrument for studying crystalline (and non crystalline) materials,

which gives possibility to get “diffractogram” (diffraction pattern) from a sample. The

monochromatic X-ray of known wavelength is used. The essential features of diffractometer

are presented schematically in Fig 1.

Fig. 1 X-ray diffractometer (schematic).

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A specimen C in a form of flat plate is supported on the table H, which can be rotated

about an axis O perpendicular to the plane of the drawing (diffractometer axis). The X-ray

source is S, the focal sport on the target of the X-ray tube. X-rays diverge from this source and

are diffracted by the specimen to form a convergent diffracted beam, which come to a focus at

the slit F and then enters the electronic counter G. A and B are special slits which define and

collimate the incident and diffracted beam. The support of table H and carriage of the counter

E, whose angular position 2θ may be read on the graduated scale K, are mechanically coupled

so that a rotation of the counter through 2θ degrees is automatically accompanied by rotation

of the specimen through θ degrees. This coupling ensures that the angles of incidence on, and

reflection from, the flat specimen will always be equal to one another and equal to half the

total angle of diffraction, an arrangement necessary to preserve focusing conditions.

Used configurations

X-ray optics used for measurements defines required configuration. Configuration usually

used for phase analysis is presented in Fig .2 (focusing arrangement). This configuration is

preferable if resolution is more important.

Fig.2 Phase analysis configuration.

Optionally, configuration presented in Fig.3 can be mounted. Parallel beam optics (X-ray

lens and parallel plate collimator) is used if more intensity is required e.g. for texture studies

and crystal orientation.

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Fig. 3. Optional configuration.

Hardware setup (configuration mounting)

1. Install necessary focus of X-ray tube (point focus).

2. Install necessary configuration (in this case- see configuration at Fig.3). Use X-ray

lens as the primary X-ray optics and thin film collimator as the secondary.

3. Set the crossed slits of 3mm width and of 3 mm height for the incident beam.

4.Set the iron β-filter for the incident beam.

4. Set the mask of 20mm height for diffracted beam optics.

Preparation of the system

1. Open the water tap of the cooling system and switch on the cooling system.

2. Switch the MRD system on by pressing the “Power on “ button. When the display shows

15 kV and 5 mA, the system is ready for use.

User setup

1.Open the ORGANIZER program.

2.Enter your user name and password.

3. Select the USER&PROJECTS menu.

4.Select SELECT PROJECT to select an existing project or EDIT PROJECT to create a

new project.

5.Select MODULES\X PERT DATA COLLECTOR or the Data Collector button on the

tool bar to open the Data Collector Program.

6.Select CONTROL in the Data Collector. The Go On Line box will appears.

7.Select the configuration that defines the selected optics setup. Then press the OK

button. CONTROL window will appear.

Sample mounting

1.Mount a sample as flat as possible on the stage.

2.Mount the micrometer on the stage and close the enclosure doors.

3. Change X, Y according to the chosen place on the sample.

3. Move the Z position of the sample stage until the micrometer reads 1.0 and the large

outer dial should point to the 0 at the top of the micrometer.

4.Remove the micrometer and close the doors.

Optics Setup

1.Select the incident beam optics tab. Change the optics to reflect the current instrument

setup.

2.Select the diffracted beam optics tab. Set the optic to resemble the chosen diffracted

beam optics.

Select the GONIOMETER&SAMPLE STAGE tab. Set the generator to 35 kV and 50

mA. Then press OK.

18

Measurement program

1. Select FILE\NEW PROGRAM\ABSOLUTE SCAN or FILE\OPEN

PROGRAM\ABSOLUTE SCAN if the program already exists.

2. Enter the information in the field of PREPARE ABSOLUTE SCAN window. Chose

the continuous scan mode. Close the window.

3. Start the measurement program by selecting MEASURE\PROGRAM. The scan data will

be automatically saved.

When the measurement is finished.

4. Set the generator to 15 kV and 5 mA and close X' PERT DATA COLLECTOR.

5. Switch of the system. Press on “Stand by” button. Switch off the cooling system. Close

the water tap.

Data analysis

1. Use X'PERT GRAFICS&IDENTIFY to identify the compounds.

2. Open NEW GRAF. Select your graph and press OK.

3. Select CUSTOMISE –REFERENCE DATABASE. Check the “Use” radio button for the

chosen ICDD database.

4. Select ANALYSE – Kα2 strip. Perform Kα2 strip for the scan data, using “Philips default”

parameters.

5. Select EDIT-PEAK SEARCH PARAMETERS and then press |”Pre-defined”.

6. Select “Default peak search parameters. Stored as fixed intensities” and press OK.

7. Press OK to save the peak search parameters.

8. Select ANALISE – PEAK SEARCH, then “Fixed slit intensities”. Press OK.

The search is performed. Accept the “Philips default” and press OK. A list of the

peaks found in the scan pattern is now displayed.

9. Select EDIT- SEARCH- MATCH…and in the “Edit Search-Match Parameters” window

press “Pre-defined”.

10. Select “Normal search-match” and press OK.

11. Tick the ”C:identdb” and press OK.

12. Select ANALISE - SEARCH-MATCH. Select “Normal” and press OK.

The MATCH SCORE LIST will be displayed.

13. Tile the graph and the list by selecting WINDOW-TILE HORIZONTALLY.

14. Compare diffractogram with reference patterns and identify the phases in the specimen.

15. Perform indexing the pattern (diffractogram). Determine structure and lattice parameters

of the phases.