Theory of diffraction on a crystal

The theory of diffraction on a crystal models the Interaction radiation-matter if the matter is organized in an ordered way (see also Cristallographie ).

These phenomena intervene primarily in the methods of analysis and of observation of the matter:

• electronic Microscopy in transmission (MET);
• Diffractometry of x-rays (DRX);
• Neutron diffraction.

One can have of it a purely simplified approach geometrical with the analogy with a Diffraction pattern and the Loi of Bragg.

On the whole, the analysis is independent of the nature of incidental radiation: electromagnetic Radiation (X-rays) or particle (electron S, Neutron S). However, the nature of the radiation intervenes for a finer analysis.

Diffusion by the atoms

The phenomenon at the base of diffraction by a crystal is the diffusion of the radiation by the atoms. One considers an elastic scattering exclusively (the radiation does not lose energy), it thus acts of Diffusion Rayleigh.

This diffusion is anisotropic; however, for a first approach, one can consider by approximation which this diffusion is isotropic, i.e. the intensity diffused by each atom is independent of the direction of space.

To simplify, a monochromatic radiation is considered. The radiation wavelength λ can be described by its Fonction of wave ψ in any point $\backslash \; vec\; \{X\}$ of space and at every moment T :

$\backslash \; psi\; (\backslash \; vec\; \{X\},\; T)\; =\; \backslash \; psi\_0\; \backslash \; cdot\; \backslash \; exp\; I\; \backslash \; left\; (\backslash \; Omega\; T\; -2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{X\}\; +\; \backslash \; varphi\_0\; \backslash \; right)$

where φ₀ is the phase in the beginning space and temporal, $\backslash \; vec\; \{K\}$ is the Vecteur of wave

$||\backslash \; vec\; \{K\}||\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; lambda\}$

and ω is the Pulsation

$\backslash \; Omega\; =\; \backslash \; frac\; \{2\; \backslash \; pi\; C\}\; \{\backslash \; lambda\}$

C being the Speed of light.

One chooses arbitrarily the origin such as φ₀ = 0.

A given mesh of the crystal is made up of N atoms. Each atom J placed in $\backslash \; vec\; \{R\}\; \_j$ diffuses the radiations in an elastic way. Let us consider the diffused wave having a vector of wave $\backslash \; vec\; \{K\}\; \text{'}$:

• $||\backslash \; vec\; \{K\}||\; =\; ||\backslash \; vec\; \{K\}\; \text{'}||$ since an elastic scattering is considered;
• the direction of $\backslash \; vec\; \{K\}\; \text{'}$ is the direction of the space in which the wave is diffused.

The function of the wave diffused by the atom J is ψ _J and is written:

$\backslash \; psi\_j\; (\backslash \; vec\; \{X\},\; T)\; =\; \backslash \; psi\_0\; \backslash \; cdot\; \backslash \; exp\; I\; \backslash \; left\; (\backslash \; Omega\; T\; +\; \backslash \; varphi\; (\backslash \; vec\; \{X\})\; \backslash \; right)\; \backslash \; cdot\; f\_j$

where φ is the dephasing of the wave in $\backslash \; vec\; \{X\}$ compared to the space origin and ƒ _J is the atomic Facteur of diffusion, which depends on the density of the electronic Nuage of the atom, therefore of its chemical nature.

Dephasing φ is the sum of two contributions:

• at the point $\backslash \; vec\; \{R\}\; \_j$ considered, dephasing φ of the incidental wave compared to the source placed at the origin is worth

$\backslash \; varphi\_1\; =\; -\; 2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_j$;

• dephasing φ₂ of the wave diffracted between its source (the atom in $\backslash \; vec\; \{R\}\; \_j$) and the point $\backslash \; vec\; \{X\}$ is worth

$\backslash \; varphi\_2\; =\; -\; 2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; “\backslash \; cdot\; (\backslash \; vec\; \{X\}\; -\; \backslash \; vec\; \{R\}\; \_j)$;

• total dephasing is worth

$\backslash \; varphi\; (\backslash \; vec\; \{X\})\; =\; \backslash \; varphi\_1\; +\; \backslash \; varphi\_2\; =\; -\; 2\; \backslash \; pi\; \backslash \; left\; (\backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_j\; +\; \backslash \; vec\; \{K\}”\; \backslash \; cdot\; (\backslash \; vec\; \{X\}\; -\; \backslash \; vec\; \{R\}\; \_j)\; \backslash \; right)\; =\; 2\; \backslash \; pi\; \backslash \; left\; ((\backslash \; vec\; \{K\}\; “-\; \backslash \; vec\; \{K\})\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_j\; -\; \backslash \; vec\; \{K\}”\; \backslash \; cdot\; \backslash \; vec\; \{X\}\; \backslash right)$;

If one defines the vector of diffraction $\backslash \; vec\; \{K\}$ as being

$\backslash \; vec\; \{K\}\; =\; \backslash \; vec\; \{K\}\; \text{'}-\; \backslash \; vec\; \{K\}$

one has then:

$\backslash \; psi\_j\; =\; \backslash \; psi\_0\; \backslash \; cdot\; e^\; \{I\; (\backslash \; Omega\; T\; -\; 2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; \text{'}\backslash \; cdot\; \backslash \; vec\; \{X\})\}\; \backslash \; cdot\; f\_j\; \backslash \; cdot\; \backslash \; exp\; \backslash \; left\; (I\; 2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_j\; \backslash \; right)$

---- ; Note

One considers only one direction of diffusion at the same time, the “direction of observation” (for example direction in which is the specific detector of radiation being used for measurement or site given of photographic film or the detector to space resolution), and thus that only one vector of diffraction; but the wave beautiful and is well diffused in all the directions simultaneously.

----

Influence organization of the matter

Factor of structure

One can now place oneself either on an atom scale, but on an unit cell scale. The wave ψ' diffracted by the mesh is the sum of the waves diffused by each one of its N atoms:

$\backslash \; psi\text{'}\; =\; \backslash \; sum\_\; \{J\; =\; 1\}\; ^n\; \backslash \; psi\_j\; =\; \backslash \; psi\_0\; \backslash \; cdot\; e^\; \{I\; (\backslash \; Omega\; T\; -\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \text{'}\backslash \; cdot\; \backslash \; vec\; \{X\})\}\; \backslash \; cdot\; \backslash \; sum\_\; \{J\; =\; 1\}\; ^n\; f\_j\; \backslash \; cdot\; \backslash \; exp\; \backslash \; left\; (I\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_j\; \backslash \; right)$

One defines the factor of structure F as being:

$F\; (\backslash \; vec\; \{K\})\; =\; \backslash \; sum\_\; \{J\; =\; 1\}\; ^n\; f\_j\; \backslash \; cdot\; \backslash \; exp\; \backslash \; left\; (I\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_j\; \backslash \; right)$

there is thus

$\backslash \; psi\text{'}\; (\backslash \; vec\; \{X\},\; T)\; =\; \backslash \; psi\_0\; \backslash \; cdot\; e^\; \{I\; (\backslash \; Omega\; T\; -\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \text{'}\backslash \; cdot\; \backslash \; vec\; \{X\})\}\; \backslash \; cdot\; F\; (\backslash \; vec\; \{K\})$

It was considered here that the wave was diffused by a specific atom. In any rigor, the wave is diffused by the electronic cloud, which is a continuous function of space. It is thus necessary to define in each point $\backslash \; vec\; \{R\}$ of the mesh a local factor of diffusion $f\; (\backslash \; vec\; \{R\})$, the then being written factor of structure:

$F\; (\backslash \; vec\; \{K\})\; =\; \backslash \; int\; \backslash \; int\; \backslash \; int\_\; \{\backslash \; rm\; mesh\}\; F\; (\backslash \; vec\; \{R\})\; \backslash \; cdot\; \backslash \; exp\; \backslash \; left\; (I\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \backslash \; right)\; \backslash \; cdot\; dv$

FD being the element of volume considered around the position $\backslash \; vec\; \{R\}$.

Factor of form

The crystal is composed of m meshs. The function ψ' _L of the wave diffracted by a mesh L placed in $\backslash \; vec\; \{U\}\; \_l$ is written:

$\backslash \; psi\text{'}\_l\; (\backslash \; vec\; \{X\},\; T)\; =\; \backslash \; psi\_0\; \backslash \; cdot\; e^\; \{I\; (\backslash \; Omega\; T\; -\; 2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; \text{'}\backslash \; cdot\; \backslash \; vec\; \{X\})\}\; \backslash \; cdot\; F\; (\backslash \; vec\; \{K\})\; \backslash \; cdot\; \backslash \; exp\; \backslash \; left\; (I\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{U\}\; \_l\; \backslash \; right)$

(this is shown in a way similar to previously by considering dephasing between the source and the mesh, then between the mesh and the point $\backslash \; vec\; \{X\}$).

The wave ψ diffracted by the whole of the crystal is the sum of the waves diffracted by each mesh, that is to say:

$\backslash \; psi$ (\ vec {X}, T) = \ psi_0 \ cdot e^ {I (\ Omega T - 2 \ pi \ vec {K} '\ cdot \ vec {X})} \ cdot F (\ vec {K}) \ cdot \ sum_ {L = 1} ^m \ exp \ left (I 2 \ pi \ cdot \ vec {K} \ cdot \ vec {U} _l \ right)

One defines the factor of form $S\_\; \{\backslash \; vec\; \{K\}\}$ by:

$S\_\; \{\backslash \; vec\; \{K\}\}\; =\; \backslash \; sum\_\; \{L\; =\; 1\}\; ^m\; \backslash \; exp\; \backslash \; left\; (I\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{U\}\; \_l\; \backslash \; right)$

there is thus

$\backslash \; psi$ (\ vec {X}, T) = \ psi_0 \ cdot e^ {I (2 \ pi \ vec {K} '\ cdot \ vec {X} - \ Omega T)} \ cdot F (\ vec {K}) \ cdot S_ {\ vec {K}}

$S\_\; \{\backslash \; vec\; \{K\}\}$ depends on the shape of the crystal, from where its name. It is this factor which intervenes in the widening of the lines when the size of the Cristallite S is weak (lower than 1  μm).

See also: Formula of Scherrer

Diffracted intensity

The diffracted intensity I in a point of space $\backslash \; vec\; \{X\}$ is proportional to the square of the standard of the vector of the function of wave:

$I\; (\backslash \; vec\; \{X\})\; \backslash \; propto\; |\backslash \; psi$ (\ vec {X}, T)|^2

There is an effect of attenuation according to the distance which varies according to the reverse of the square of the distance: it is simply about the “distribution” of energy on a sphere (dimunution of the angular density). If this phenomenon is corrected, then the intensity depends only on the direction of space, that one can give by the vector of the diffracted wave $\backslash \; vec\; \{K\}\; “$:

$I\; (\backslash \; vec\; \{K\}”)\; \backslash \; propto\; |\backslash \; psi$ (\ vec {X}, T)|^2 with $||\backslash \; vec\; \{X\}||$ fixed arbitrarily,

that is to say

$I\; (\backslash \; vec\; \{K\}\; \text{'})\; \backslash \; propto\; |F\; (\backslash \; vec\; \{K\})|^2\; \backslash \; cdot\; |S\_\; \{\backslash \; vec\; \{K\}\}|^2$

Other factors intervene, in particular the geometry of the measuring device, optics. For example, the intensity can vary according to the slope of the detector by rarport to the sample.

Conditions of diffraction

Condition of Laue

In a diagram of diffraction, a peak (or a point if it is a figure 2D) corresponds to a maximum of intensity, i.e. with a local maximum of F . Intuitively, F is maximum when the rays diffused by the atoms of the cell are all in phase. If one considers two atoms J and L , one must have

$2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_j\; \backslash \; equiv\; 2\; \backslash \; pi\; \backslash \; vec\; \{K\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\; \_l$

That is to say $(\backslash \; vec\; \{E\}\; \_1,\; \backslash \; vec\; \{E\}\; \_2,\; \backslash \; vec\; \{E\}\; \_3)$ the base of the network; the positions of the atoms are written

$\backslash \; vec\; \{R\}\; \_j\; =\; x\_j\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; \_1\; +\; y\_j\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; \_2\; +\; z\_j\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; \_3$

where x_j , y_j and z_j is positive numbers lower or equal to 1.

Let us consider the base of the reciprocal Espace $(\backslash \; vec\; \{E\}\; ^*\_1,\; \backslash \; vec\; \{E\}\; ^*\_2,\; \backslash \; vec\; \{E\}\; ^*\_3)$ with

$\backslash \; vec\; \{E\}\; ^*\_1\; =\; \backslash \; frac\; \{\backslash \; vec\; \{E\}\; \_2\; \backslash \; wedge\; \backslash \; vec\; \{E\}\; \_3\}\; \{V\}$

$\backslash \; vec\; \{E\}\; ^*\_2\; =\; \backslash \; frac\; \{\backslash \; vec\; \{E\}\; \_3\; \backslash \; wedge\; \backslash \; vec\; \{E\}\; \_1\}\; \{V\}$

$\backslash \; vec\; \{E\}\; ^*\_3\; =\; \backslash \; frac\; \{\backslash \; vec\; \{E\}\; \_1\; \backslash \; wedge\; \backslash \; vec\; \{E\}\; \_2\}\; \{V\}$

V being the volume of the mesh

$V\; =\; \backslash \; vec\; \{E\}\; \_1\; \backslash \; cdot\; (\backslash \; vec\; \{E\}\; \_2\; \backslash \; wedge\; \backslash \; vec\; \{E\}\; \_3)$

The condition applied to all the atoms of the mesh two to two cost then to say that the vector of diffraction must be a linear combination with whole factors of the vectors of the reciprocal base:

$\backslash \; vec\; \{K\}\; =\; H\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^*\_1\; +\; K\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^*\_2\; +\; L\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; ^*\_3$

H , K and L being entireties, they are the Indices of Miller. The equation above is the condition of diffraction of Laue . One can show that it is equivalent to the condition of Bragg.

One thus can indicer the vectors of wave giving of the peaks/points by the indices of Miller and to write $\backslash \; vec\; \{K\}\; \_\; \{HKL\}$. One also can indicer the corresponding factors of structure:

$F\_\; \{HKL\}\; =\; F\; (\backslash \; vec\; \{K\}\; \_\; \{HKL\})$

The places of the ends of the $\backslash \; vec\; \{K\}\; \_\; \{HKL\}$ form a network in reciprocal space, called reciprocal Réseau. One can in makes associate each point of the reciprocal network (i.e. each vector $\backslash \; vec\; \{K\}\; \_\; \{HKL\}$) with the direction of a crystallographic Plan.

Vector of diffraction and reciprocal network

Thus according to the condition of Laue, there is diffraction if

$\backslash \; exists\; (H,\; K,\; L)\; \backslash \; in\; \{\backslash \; mathbb\; NR\}/\backslash \; vec\; \{K\}\; =\; \backslash \; vec\; \{K\_\; \{HKL\}\}$

thus if $\backslash \; vec\; \{K\}$ is a vector of the reciprocal network of one of crystallites enlightened.

Geometry of Bragg-Brentano

Let us study only the case where the vector of diffraction always keeps the same orientation compared to crystallite (the bisectrix between the incidental beam and the direction of observation is always on the same line); that means that the vectors of the incidental wave and the diffused wave are always symmetrical compared to this direction, in real space as in reciprocal space. That corresponds to the geometry of Bragg-Brentano , one places the detector of manner symmetrical at the normal at the sample passing by the center of this one.

We place a monocrystal in the case of. One sees that according to the deviation of the beam, i.e. the angle which the incidental beam with the direction of observation forms, one is in condition of diffraction or not.

Let us suppose now that one makes turn crystallite in all the directions during measurement, or, which is equivalent, that the sample consists of a crystallite multitude directed in all the directions (powder). Then, it is necessary to superimpose all the reciprocal networks to know the deviations giving a peak/not diffraction. That gives concentric spheres; there is diffraction if the vector of diffraction meets a sphere.

” (or “sphere of reflection”), and it contains the origin O reciprocal network.

The directions in which one will have diffraction are thus given by the intersection of the sphere of Ewald with the spheres of the $\backslash \; vec\; \{K\}\; \_\; \{HKL\}$. The intersection of two nonconcentric spheres, when it exists, is a circle. One from of deduced that the ends of the vectors of diffraction for which there is diffraction form a circle, therefore that the ends of the vectors of diffused wave for which there is diffraction describe a circle, i.e.: the diffracted rays form cones.

Let us consider now that one keeps the motionless reciprocal network (monocrystal), but which one makes turn the sphere of Ewald. It is seen that the sphere of Ewald will sweep a ball of center O and whose ray is the diameter of the sphere of Ewald. The points contained in this “supersphère” correspond to the various possible conditions of diffraction; the points outside cannot, under the conditions of measurement given (i.e. for the wavelength λ given), to give diffraction. This “supersphère” is called “ sphere of resolution ”, it has a ray of 2/λ.

If λ is too large, the sphere of resolution contains only the center of the reciprocal network, diffraction is thus not possible. This is why it is necessary to resort to radiation sufficiently small wavelength (x-rays or particles having a sufficiently raised speed) to be able to characterize a crystal lattice.

If one recovers in a geometry from Bragg-Brentano (direction of the vector of fixed diffraction), the vector of diffraction is obtained by taking the intersection of the sphere with the axis of the imposed direction.

Factor of form and reciprocal network

For the conditions of diffraction, we considered up to now only the factor of structure. The conditions of diffraction for a monocrystal are represented like a specific network in reciprocal space.

This would be true only for one monocrystal of “infinite” size. For a crystallite of finished size, there is a diffraction with the direction diffraction of Fraunhofer; on a photographic film, the trace of diffraction is thus not a whole of points infinitely smalls, but of the spots of Airy.

See the detailed article Theory of diffraction.

In reciprocal space, the condition of diffraction is not a network of points, but a network of three-dimensional spots.

The shape of these spots in reciprocal space is described by the factor of form. In a traditional way as regards diffraction, the spot of the reciprocal network is extended in the direction perpendicular to the narrowest dimension of crystallite.

If crystallite is spherical but of small size (lower than the Micromètre), the spot in reciprocal space will be of spherical symmetry, the density decreasing with the ray (diffracted intensity being proportional to this density).

If crystallite is a disc (cylinder flattened in its axis), the spot of diffraction will be a needle (cylinder of weak ray but stretched according to its axis).

Kinetic theory and dynamic theory

We above mentioned the theory known as “kinetic” of diffraction. In the kinetic theory, one considers that the wave diffused by the nodes does not diffract itself. This assumption is valid when the diffracted intensity is low in front of the incidental intensity, which is the case with x-rays and the neutrons.

This assumption is not in general valid any more with the electrons, except in the case of diffraction by a thin blade (in a Electron microscope in transmission). One then has recourse to the theory known as “dynamic”.

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