X-RAY METHODS FOR ANALYSIS OF MATERIALS

Nikolay Petkov,

Tyndall, Block B,

E-mail: nikolay.petkov@tyndall.ie

Text Books- Instrumental Methods of Analysis- Atkins’ Physical Chemistry

Lecture notes, not really enough but with some background reading should be fine for the exam!

• Introduction• X-ray diffraction (XRD) •Crystal structure•X-ray fluorescence analysis (XRF, EDX)•X-ray photoelectron spectroscopy (XPS)

Topics

Traditional methods of analysis such as:Atomic Adsorption, Mass Spec, Chromatography, Infra-red spectroscopy, UV-Vis spectroscopy etc require dissolution into a fluid phase: destructive

Materials or solids analysis has been driven by the requirement to produce non-destructive methods.

They can be described in six categories:-Elemental – the atoms present – XRF, EDXStructural – define atomic arrangement - XRDChemical – define the chemical state - XPSImaging – what does the morphology look like? – Electron MicroscopySpectroscopic – energy level transitions – IR Thermal – effects of heating (sorption) – TGA, DSC

Problems:- Destructive Elemental No understanding of their form in the material

INTRODUCTION

Why to study solid state?

• Solid state includes most of the materials in that make modern technology possible

• Properties of the solid state differ significantly from the properties of isolated atoms or molecules

• The term ‘structure’ takes on a whole new meaning.

Example: Nanosized metal particles

Consider complete delocalized electrons in the metals, special effect called plasmons (quantized plasma oscillations, collective oscillations of the electron cloud) result is specific colour of metal particles with nanosized dimensions.

Challenges in analysis of solids– compared to molecular analysis

1. Solids have continuum energy states compared to discrete energy levels in molecular sates.

2. Very high absorptions so that not much energy gets out! Can lead to adsorption of signal of some elements within the matrix!

3. Saturation effects can not be simply diluted out.

4. Structural differences can often be difficult to resolve.

5. Many of the probes can not be used in simple laboratory environments.

6. Require complex detection equipment.

7. Standards can be quire difficult to prepare compared to simple dilution. Phase diagrams and structural change. Matrix effects.

Interaction of the radiation with the matter.

Elastic interactions - in which there is no lost in the energy Inelastic interactions – with lost of the exciting energy.

Examples of Methods and Interactions with solids

X-ray Diffraction – X-rays in and out but no loss of energy, elastic scattering of x-raysX-ray fluorescence – X-rays in and out but look at x-rays generated by secondary process, inelastic scattering of x-raysX-ray Absorption Spectroscopy - X-rays in and out but look for energy lossesX-ray Photoelectron Spectroscopy - X-rays in electrons out (secondary electrons)

Electron microscopy – electrons in and out analyse either transmitted (transmission EM) or secondary (secondary electrons, inelastic scattering of electrons)Low Energy Electron Energy Loss Spectroscopy - electrons in and out – energy analyse the electrons to look at energy loss to give vibrational information

Ultrasound - Sound in and out – use it to analyse for void formation in solids

NMR – radio frequency in and out analyse for energyloss No single techniques is capable of providing a complete characterization of a solid

Bulk vs Surface Analysis

Analysing solids is further complicated by the depth sensitivity. X-rays deeply penetrate matter so the analyte depth is high and the analysis is bulk sensitive rather than surface sensitive.

However, techniques involving electrons (even if they are excited by high energy techniques) are strongly absorbed and so originate from the surface region. Such are EDX and XPS.

This can be an advantage since surface chemistry is of fundamental importance. However, phase separation and segregation can give rise to problems.

Quantification

An ideal technique should be quantitative e.g.

Signal height is proportional to the number of atoms of element or chemical state present.

This is rare in a solid. In molecular analysis the intensity is usually directly proportional to the number of molecules in the analyte. This is because the systems are frequently dilute – there is little chance that there are multiple interactions of other interactions with primary or secondary radiation. In a solid exactly the opposite is true. Thus, solids suffer matrix effects.

Analyte

Matrix

Weak emitter in a strong adsorbing matrix. Analyte emission absorbed by matrix and no signal leaves sample.

Weak emitter as a thin layer at the surface on strongly absorbing matrix.

Most of the electrons in the incident beam lose energy upon entering a material through inelastic scattering or collisions with other electrons of the material and form heat.

In such a collision the momentum transfer from the incident electron to an atomic electron can be expressed as dp = 2e2 / bv, where b is the distance of closest approach between the electrons, and v is the incident electron velocity.

The energy transferred by the collision is given by T = (dp)2 / 2m = e4 / Eb2, where m is the electron mass and E is the incident electron energy, given by E = (1 / 2)mv2.

By integrating over all values of T between the lowest binding energy, Eo, and the incident energy E, one obtains the result that the total cross section for collision is inversely proportional to the incident energy E.

How is heat produced ?

A second type of interaction in which the incident electron can lose its kinetic energy is an interaction with the nucleus of a target atom. In this type of interaction, the kinetic energy of the projectile electron is converted into electromagnetic energy.

Interaction of X-rays with matter

A single crystal, also called monocrystal, is a crystalline solid in which the crystal lattice of the entire sample is continuous and unbroken to the edges of the sample, with no grain boundaries.

The opposite of a single crystal sample is a polycrystalline sample, which is made up of a number of smaller crystals known as crystallites. Usually those crystallites are connected through a amorphous material to form extended solid.

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Symmetry Operations and ElementsA Symmetry operation is an operation that can be performed either physically or imaginatively that results in no change in the appearance of an object.   

There are 3 types of symmetry operations: rotation, reflection, and inversion. .

Rotational Symmetry As illustrated above, if an object can be rotated about an axis and repeats itself every 90o of rotation then it is said to have an axis of 4-fold rotational symmetry.  The axis along which the rotation is performed is an element of symmetry referred to as a rotation axis.   The following types of rotational symmetry axes are possible in crystals.

•1-Fold Rotation Axis - An object that requires rotation of a full 360o in order to restore it to its original appearance has no rotational symmetry.  Since it repeats itself 1 time every 360o it is said to have a 1-fold axis of rotational symmetry.

•2-fold Rotation Axis - If an object appears identical after a rotation of 180o, that is twice in a 360o rotation, then it is said to have a 2-fold rotation axis (360/180 = 2).  Note that in these examples the axes we are referring to are imaginary lines that extend toward you perpendicular to the page or blackboard.  A filled oval shape represents the point where the 2-fold rotation axis intersects the page.  

3-Fold Rotation Axis- Objects that repeat themselves upon rotation of 120o are said to have a 3-fold axis of rotational symmetry (360/120 =3), and they will repeat 3 times in a 360o rotation.  A filled triangle is used to symbolize the location of 3-fold rotation axis.

4-Fold Rotation Axis  - If an object repeats itself after 90o of rotation, it will repeat 4 times in a 360o rotation, as illustrated previously.  A filled square is used to symbolize the location of 4-fold axis of rotational symmetry.

6-Fold Rotation Axis - If rotation of 60o about an axis causes the object to repeat itself, then it has 6-fold axis of rotational symmetry (360/60=6).  A  filled hexagon is used as the symbol for a 6-fold rotation axis. 

Mirror Symmetry A mirror symmetry operation is an imaginary operation that can be performed to reproduce an object.  The operation is done by imagining that you cut the object in half, then place a mirror next to one of the halves of the object along the cut.  If the reflection in the mirror reproduces the other half of the object, then the object is said to have mirror symmetry.  The plane of the mirror is an element of symmetry referred to as a mirror plane, and is symbolized with the letter m.  As an example, the human body is an object that approximates mirror symmetry, with the mirror plane cutting through the center of the head, the center of nose and down to the groin.

Center of Symmetry  

Another operation that can be performed is inversion through a point.  In this operation lines are drawn from all points on the object through a point in the center of the object, called a symmetry center (symbolized with the letter "i"). The lines each have lengths that are equidistant from the original points.  When the ends of the lines are connected, the original object is reproduced inverted from its original appearance.  In the diagram shown here, only a few such lines are drawn for the small triangular face. The right hand diagram shows the object without the imaginary lines that reproduced the object.   

Rotoinversion Combinations of rotation with a center of symmetry perform the symmetry operation of rotoinversion.  Objects that have rotoinversion symmetry have an element of symmetry called a rotoinversion axis.  A 1-fold rotoinversion axis is the same as a center of symmetr.

•3-fold Rotoinversion - This involves rotating the object by 120o (360/3 = 120), and inverting through a center.  A cube is good example of an object that possesses 3-fold rotoinversion axes.  A 3-fold rotoinversion axis is denoted as (pronounced "bar 3").  Note that there are actually four axes in a cube, one running through each of the corners of the cube. If one holds one of the axes vertical, then note that there are 3 faces on top, and 3 identical faces upside down on the bottom that are offset from the top faces by 120o.  

Combinations of Symmetry Operations As should be evident by now, in three dimensional objects, such as crystals, symmetry elements may be present in several different combinations.  In fact, in crystals there are 32 possible combinations of symmetry elements.  These 32 combinations define the 32 Point Groups.

Thus, this crystal has the following symmetry elements:•1  -  4-fold rotation axis (A4) •4 -  2-fold rotation axes (A2), 2 cutting the faces & 2 cutting the edges. •5 mirror planes (m), 2 cutting across the faces, 2 cutting through the edges, and one cutting horizontally through the center. •Note also that there is a center of symmetry (i). The symmetry content of this crystal is thus: i, 1A4, 4A2, 5m

Crystal familyCrystal system

Required symmetries of point group

point groups space groups bravais lattices Lattice system

Triclinic None 2 2 1 Triclinic

Monoclinic

1 twofold axis of rotation or 1 mirror plane

3 13 2 Monoclinic

Orthorhombic

3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.

3 59 4 Orthorhombic

Tetragonal1 fourfold axis of rotation

7 68 2 Tetragonal

HexagonalTrigonal

1 threefold axis of rotation

57 1 Rhombohedral18

1 HexagonalHexagonal

1 sixfold axis of rotation

7 27

Cubic4 threefold axes of rotation

5 36 3 Cubic

Total: 6 7 32 230 14 7

The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 lattice systems. This results in a space group being some combination of the translational symmetry of a unit cell including lattice centering, the point group symmetry operations of reflection, rotation and improper rotation (also called rotoinversion), and the screw axis and glide plane symmetry operations. The combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries

There are at least eight methods of naming space groups. Some of these methods can assign several different names to the same space group, so altogether there are many thousands of different names.

•International symbol or Hermann-Mauguin notation. describes the lattice and some generators for the group. It has a shortened form called the international short symbol, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, By way of example, the space group of quartz is P3121, showing that it exhibits primitive

centering of the motif (i.e., once per unit cell), with a threefold screw axis and a twofold rotation axis. Note that it does not explicitly contain the crystal system, although this is unique to each space group (in the case of P3121, it is trigonal).

Common crystal structures

Fd3m space group

Diamond structure, Si and Ge Rock salt structure, metals

Fm3m space group

Double Slit and Diffraction Grating.

a pattern of dark and bright fringes

At C all wavelengths arrive in phase and interfere constructively to produce a “central image”

At some other point P which is at a distance L from one slit and L + nλ from the other (λ is some specific wavelength present in the light beam; n is an integer) there is also constructive interferences and a bright fringe appears with the color pertaining to that specific wavelength. At intermediate points distant L and L + (2n + 1) (λ/2), destructive interference occurs for that wavelength λ.

X-ray diffraction

Bragg’s Law

When certain geometric requirements are met, X-rays scattered from a crystalline solid can constructively interfere, producing a diffracted beam. In 1912, W. L. Bragg and his son recognized a predictable relationship among several factors.

The path difference between two waves:2x = 2dsin(theta)

For constructive interference nλ = 2dsinθ

Another view of Bragg’s law

When an X-ray beam hits an atom, the electrons around the atom start to oscillate with the same frequency as the incoming beam. In almost all directions we will have destructive interference. However the atoms in a crystal are arranged in a regular pattern, and in a very few directions we will have constructive interference. Hence, a diffracted beam may be described as a beam composed of a large number of scattered rays mutually reinforcing one another.

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X-ray diffraction has been in use in two main areas:

1) Fingerprint characterization of crystalline materials – powder XRD

2) The determination of crystal structure e.g. identification of atomic position of crystals - single crystal XRD analysis.

But also…1) Strain and lattice mismatch in crystals2) Orientation of thin films – grazing incidence XRD, rocking

curves 3) Low angle X-ray scattering – form and shape of polymers,

meso-materials, colloidal particles.

Single Crystal Polycrystalline

Each crystalline solid has its unique characteristic X-ray powder pattern which may be used as a "fingerprint" for its identification. Once the material has been identified, X-ray crystallography may be used to determine its structure, i.e. how the atoms pack together in the crystalline state and what the interatomic distance and angle are etc. X-ray diffraction is one of the most important characterization tools used in solid state chemistry and materials science. Every lab in materials science has an XRD instrument.

Powder XRD

In any real solid we have a chance orientation. It would be almost impossible to study diffraction. However, 99% of all materials are polycrystalline or can be prepared (by grinding) so as to present many grains of material. In these some will always be at the correct alignment.

sample

source

detector

Bragg-Brentano Geometry (θ - 2θ)

θ θ

θ

Divergence slitAntiscatter slit

Monochromator

Detector slit

Tube

soller slit

•dimension of the elementary cell•content of the elementary cell•strain/crystallite size•quantitative phase amount

Which Information does a PowderPattern offer?

1) peak position – d-spacing2) peak intensity – structure factors3) peak broadening – size of a crystallite

Primary information

Secondary information - calculated

In the kinematical approximation for diffraction, the intensity of a diffracted beam is given by:

                             where             is the wavefunction of a beam scattered a vector          , and             is the so called structure factor which is given by:

                                                    Here, rj      is the position of an atom j in the unit cell, and fj is the scattering power of

the atom, also called the atomic form factor. The sum is over all atoms in the unit cell. It can be shown that in the ideal case, diffraction only occurs if the scattering vector           is equal to a reciprocal lattice vector      .

The structure factor describes the way in which an incident beam is scattered by the atoms of a crystal unit cell, taking into account the different scattering power of the elements through the term fj. Since the atoms are spatially distributed in the unit cell,

there will be a difference in phase when considering the scattered amplitude from two atoms. This phase shift is taken into account by the complex exponential term.

The atomic form factor, or scattering power, of an element depends on the type of radiation considered. Because electrons interact with matter though different processes than for example X-rays, the atomic form factors for the two cases are not the same.

When x-rays enter a solid they undergo refraction. For x-rays this is very small.

But the refraction angle differs from the incidence angle by only parts per thousand. But because of this the path length difference slowly varies from planes deeper into the material. The constructive interference slowly becomes destructive.Provided the sample is thick enough (if a crystallite is 10 μm there are 1 x 10-5/10-10 atom planes = 105) then all these slightly out of phase reflections will cancel. Leave just the perfect constructive interference feature. But if the crystallite is small , below 100 nm there are only 1000 planes and the effect of canceling is not so pronounced which will lead to broadening of the reflection.

X-ray line broadening

Scherrer equation

crystallite size = 0.9 λ/(B cos θ)B = = √(B2

obs – B2o)

where Bobs = FWHM of reflection Bo = instrumental FWHM minimum

Some types of unit cell give characteristic and easily recognizable pattern lines. For example, in a cubic primitive lattice of unit cell dimensions a the spacing is given by the equation:

sin θ = (h2 + k2 + l2)1/2 λ/2a

The reflections arte than predicted by substituting the values h, k, l: {h,k,l} {100} {110} {111} {200} {210} {211} {220}h2 + k2 + l2 1 2 3 4 5 6 8

The 7 (and 15…) is missing because the sum of the squares of three integers cannot be 7. Therefore the pattern has systematic absences that are characteristic of the cubic P lattice.

Systematic absences

For FCC, h,k,l all even or all odd are presentFor BCC, h+k+l = odd are absent