Diffractometry of x-rays

The diffractometry of x-rays ( DRX , one also often uses the English abbreviation XRD for X-ray diffraction ) is a technique of analysis based on the Diffraction of the X-rays on the matter. Diffraction taking place only on the matter Crystal line, one also speaks about X-ray crystallography . For the materials not-crystalline lenses, one speaks about diffusion.

The measuring device is called a Diffractomètre. The collected data form the diagram of diffraction or diffractogram .

General presentation

; What is it?

the diffractometry of x-rays is a chemical method of analysis. It goes only on the matter Cristal smoothed (mineral, metals, Céramique S, organic materials crystallized), but not on the amorphous Matière (Liquide S, Polymère S, Verre S). On the other hand, it makes it possible to recognize products having the same rough chemical composition, but a form of different crystallization, for example to distinguish different the Silice S (which have all the same empirical formula SiO₂: quartz, Cristobalite…), different the Steel S (ferritic Steel, Austenite…) or different the Alumine S (which has all the same empirical formula Al₂O₃: Corundum /alumine α, γ, δ, θ…).

; How does it go?

One prepares the sample in the shape of a powder levelled in a cup, or in the shape of a solid plate punt. One envoit of x-rays on this sample, and a detector makes the turn of the sample to measure the intensity of x-rays according to the direction. For practical reasons, one makes turn the sample at the same time, or possibly one makes turn the tube producing x-rays.

; Who uses this technique?

the technique is used to characterize the matter. That concerns:

* research: when one creates a new material (often Céramique S), that one wants to know result of chemical reaction or physics (for example in Métallurgie, to recognize the products of Corrosion or to know which type of steel one manufactured), in Géologie (Géochimie) to recognize the rock taken at a place;

* for the follow-up of production in a factory (quality control of the product): in the Cement eries, the factories of Ceramic S…

* the Drug company:

** in research: the new molecules are crystallized, and the crystals are studied by diffractometry of x-rays;

** in production: that is in particular used to check that one did not manufacture another of the same molecule formulates, but of different form (in particular a enantiomer).

Interaction rays X-matter

X-rays, like all the electromagnetic waves, cause a displacement of the electronic Nuage compared to the core in the Atome S; these induced oscillations cause a réémission of electromagnetic waves of the same Fréquence; this phenomenon is called Diffusion Rayleigh.

Applications of the DRX

Identification of crystalline phases

Principles of the identification of the phases

The idea to use the diffraction of x-rays to identify a phase was developed at the beginning of the 20th century in an independent way by Albert Hull in 1919 on the one hand, and by Peter Debye and Paul Scherrer on the other hand. Because of the war, the publication and the diffusion of the scientific newspapers was difficult; chronologically, it is Hull which published the first its work, but the method carries the name of Debye and Scherrer.

A formed powder of a given crystalline phase always will give place to peaks of diffraction in the same directions, with about constant relative heights. This diagram of diffraction thus forms a true signature of the crystalline phase. It is thus possible to determine the nature of each crystalline phase within a mixture (mixture of powder or polyphase massive sample), in condition of having determined the signature of each phase before.

The determination of this signature can be done either in an experimental way (measurement of a pure product under ideal conditions), or by digital simulation starting from the known crystallographic structure - structure having been able itself to be given by X-rays diffraction (cf below). This signature is consigned in a card in the shape of a list of peaks; the position in 2θ is converted into distance interréticulaire D by the law of Bragg, so davoir a value independent the wavelength of x-rays (and thus of the type of source of x-rays used). The intensity I of each peak is expressed in Pourcent %, sometimes in Pourmille ‰, 100% (or 1 000 ‰) being the height of the most intense peak. This list of peaks is often indicated by the term “lists say ”. Databases thus are constituted, and the diagram measured on the unknown product is compared in a data-processing way with all the cards of the database. The most complete database at present (2004) is the Powder diffraction spins (pdf) of the ICDD (ex-JCPDS: Joint committee one powder diffraction standards , ex- E4 committee of ASTM), with more 150 000 cards (of which however of many redundancies).

The interest of this method is that it makes it possible to distinguish the various forms from crystallization of the same compound (for example for the Silice, to distinguish the quartz from the Cristobalite). However, it cannot generally make it possible to identify amorphous compounds. This technique is thus complementary to elementary Analyze.

The procedure of identification of the phases is done in two stages: a stage of research in a base ( search ), then a confrontation of the probable cards with what is possible chemically ( match ); one thus speaks frequently about search/match to indicate this procedure.

With final, it is the user who determines if a product is present or not: because of the possibilities of confusion (several very different products being able to have very close signatures), an automated algorithm cannot make only the decision. It is in last spring the competence of the user, his skill and its knowledge of the sample who intervene.

In certain fields, one wants simply to know if there are only the phases envisaged and not of other - in particular, problem of the enantiomer S) in the follow-up of the pharmaceutical production. In this context, it is enough to draw up a list of peaks on the diffractogram of the unknown product, which one compares with a list peaks established on the diffractogram of a product standard (i.e. whose chemical composition is controlled).

Encountered problems

In the case of a really unknown product and which one seeks to identify all the phases, one is confronted mainly with three problems:

the variation of the signature of a product compared to its theoretical or ideal signature:

the position in 2θ of the peaks of a phase can be shifted:

problem of alignment of the diffractometer,
problem of height of the surface of the sample;
problem of variation of the cell parameters crystalline, because of the Forced S or the solid Solution - nonpure product,

the relative heights of the peaks only is seldom respected:

preferential Orientation,
many crystallites insufficient to have good statistics,
superposition of peaks;

the mixture of the peaks is sometimes complex, with superpositions;
it is necessary to compare the diffractogram with several hundreds of thousands of cards of reference.

Manual algorithms of identification of the phases

Method of Hanawalt

The first algorithm was invented by Hanawalt in 1936. At the time, the cards of reference were in form paper. Hanawalt gathered the cards whose principal peak (known as “peak with 100%”) were at the same place (or more precisely in the same restricted zone 2θ), the categories thus create being classified by ascending order of position 2θ; then, in a category of cards, it gathered the cards of which the second most intense peak were at the same place, classifying in the same way the subcategories, and in a subcategory, it classified the cards by order of position of the third peaks most intense. To strip a diffractogram, it thus proceeded as follows:

one determined the three most intense peaks, and one sought in the list of Hanawalt the cards which can correspond;
the first product being identified, one eliminated the three peaks considered and one started again.

This method bears also the name of “method ASTM”.

However, it was as necessary to take into account the possible superpositions of peaks, therefore the possibility as a peak pertaining to an already identified phase also belongs to another phase. In fact, the identification became extremely complex beyond of a mixture of three phases, and was not very powerful to detect the phases present in small quantity, i.e. generating peaks low height.

Fink method

The Fink method was developed by W. Bigelow and J.V. Smith of ASTM with the beginning of the Années 1960, which gave him the name of William Fink, a referent of the JCPDS. The idea is to consider the four most intense peaks of a card, to apply all the possible permutations, then to classify all these solutions by order of D growing. At the time of a research, the operator takes the first value of D met on the diffractogram, then seeks in the index the cards to which this peak could belong. The other peaks of each card are then confronted with the diffractogram.

Data-processing algorithms of identification of the phases

Data processing made it possible to automate the manual procedures, in particular with algorithms of automatic search for peaks and comparisons with the cards in electronic forms. It also made it possible to improve the algorithm, by multiplying the possible comparisons instead of being satisfied with the three most intense peaks. It also made it possible to cross information on the peaks with information on the chemical composition (research known as “Boolean” because it uses logical operations type “and”, “not” and “or”).

The first programs appeared in the middle of the Années 1960, with limitations inherent in the quality of the diffractograms and the capacities of calculation of the computers: the programs were to consider important possibilities of error on the values of D and I .

Mr. C. Nichols thus adapts the algorithm of Hanawalt in 1966. G.G. Johnson Jr. and V. Vand adopt as for them a resolutely new approach in 1965: they compare in a systematic way all the cards of the base of data with the list of say extracted the diffractogram, and give a note to the card (FOM, figure off merit ). The cards of the database are thus classified by order of note of correspondence, then the “best pupils” are posted (typically, the 50 first are posted), classified according to the number of peaks common to the card and to the list of say extracted the diffractogram, then according to the note.

In 1982, manufacturing it Philips develops an algorithm owner (not published) based on the method of least squares: the note for each card is calculated according to the difference between the peaks of the card of reference and the list of says extracted the diffractogram.

The recent improvement most important took place in 1986, with the commercial program Eva (software continuation DIFFRAC-AT, then DIFFRAC^more) of the company Socabim, French SME working primarily for manufacturing Siemens. This algorithm owner (not published) takes again the logic of Johnson and Vand; however, it is not satisfied to extract a list from peaks with the diffractogram, but compares each card with the diffractogram itself to give a note to the card (the better the card corresponds to the diffractogram, plus the note is low). The cards of the database are thus classified by order of correspondence, then the “best pupils” are posted (typically, one posts the 50 first); the user superimposes then the cards (represented in the shape of sticks) with the diffractogram to determine the cards which it retains. Thus, the algorithm uses the totality of the measured points, and in particular the line basic, instead of being satisfied with a restricted list of top of peaks; it takes into account the superposition of the peaks (if the stick of a card is in a zone where the signal is above the basic line, it does not matter that it is alone or that there are other sticks) and makes it possible to detect the minority phases. Other companies developed similar algorithms thereafter.

Quantitative analysis

Method of surfaces of peak

The theory indicates that in a mixture, the clear surface of the peaks of a phase (known as also “integral intensity”) is propotionnelle with the concentration of the phase with the help of a term of absorption: x-rays are absorbed by the matter according to a Loi of Beer-Lambert, therefore 1% of a given material do not give the same signal according to the 99% remainder.

One can thus write a law of the form:

c_i = m_i \ cdot I_i \ cdot A

where

c_i is the concentration of the phase I ;
I_i is the integral intensity of a given peak of I ;
m_i is a coefficient of calibration, a constant of couple apparatus/phase;
has is the term of absorption, which is the same one for all the phases (since one works in monochromatic radiation).

The coefficient of calibration evolves/moves with the age of the apparatus, and in particular the ageing of the tube with x-rays.

One can abstract oneself from the absorption in two manners (methods of Chung):

by introducing a standard interns: if the sample is in the form of powder, one can mix a given and known quantity of a phase of stable R , and one works then in report/ratio of intensity and concentration:

\ frac {c_i} {c_r} = \ frac {m_i \ cdot I_i \ cdot has} {m_r \ cdot I_r \ cdot has} = \ frac {m_i} {m_r} \ cdot \ frac {I_i} {I_r}

by using an additional equation: if all the phases measurable and are measured), the sum of the concentrations is equal to 100%, one has as many equations then as unknown factors.

These two methods also make it possible to be abstracted from veillissement from the tube.

If the samples are almost identical, one can consider that the term of absorption is always the same one and to be satisfied to integrate this one in the coefficient of calibration. However, this method becomes erroneous if one leaves a range of concentrations restricted, and the calibration should be remade regularly to take into account vieillssement tube, or to determine the variation of intensity to correct it, method known as of “correction of drift” ( drift transistor correction ).

Report/ratio of intensity of reference

If one chooses a method of preparation of reference with a given internal standard, it is possible to establish a coefficient of calibration per defect; it is method known as RIR, for “report/ratio of intensity of reference” ( refers intensity ratio ).

The method which refers is the following one:

one prepares the sample in the form of powder;
one mixes it with 50% of Corindon (Alumine α-Al₂O₃) and one measures it;
one submits the relationship between the largest peak of the phase and the largest corundum peak.

This report/ratio of intensity is called I / I_cor , and constitutes the coefficient of calibration of reference.

So now one considers the equations above, one has, like c_i = c_cor :

\ frac {c_i} {c_ {horn}} = 1 = \ frac {m_i} {m_ {horn}} \ cdot \ frac {I_i} {I_ {horn}}

that is to say

\ frac {m_i} {m_ {horn}} = \ frac {1} {(I/I_ {horn}) _i}

In the general case c_i ≠ c_cor , one thus has:

\ frac {c_i} {c_ {horn}} = \ frac {1} {(I/I_ {horn}) _i} \ cdot \ frac {I_i} {I_ {horn}}

By submitting the report/ratio for two phases I and J , one obtains:

\ frac {c_i} {c_j} = \ frac {(I/I_ {horn}) _j} {(I/I_ {horn}) _i} \ cdot \ frac {I_i} {I_j}

It is seen whereas the concentration and intensity of corundum disappear from the formulas. One can thus measure the unknown sample without adding corundum and using all the same the coefficients of calibration established with corundum.

This method is known as “semi-quantitative” because it is not possible to define the error made to the measure. Indeed, as the reference samples do not have same nature that the unknown sample and do not have undergoes the same preparation, it is not possible to use the standard deviation obtained on the calibration to have an estimate of the error. In addition, the standard deviation on the calibration is in general not provided.

Method of Rietveld

One can also make quantification by the Méthode of Rietveld: one starts from arbitrary concentrations, and one simulates the diffractogram which one would obtain, by using the theory of diffraction. Then, one adjusts the concentrations in order to bring closer the diffractogram simulated to the measured diffractogram (method of the Least squares).

The method of Rietveld is a method without standard, but it requires to acquire a diffractogram on a great angular beach with a good precision (thus a long measurement), whereas the method of the integral intensity makes it possible to measure only beaches of a few degrees around the interesting peaks. But the method of Rietveld is the only exploitable one if one cannot use peak isolated (problems of superposition of peaks).

To obtain the desired theoretical spectrum by means of computer, the experiementator can refine several parameters:

cell parameters
the rate of crystallinity
the shape of the peaks (Gaussian or Lorentzian) and adjusting the coefficient eta.
the shape of the foot of the peaks (Coefficients of Caglioti)
the background noise (polynomial of degree 5 in general)
the shift of origin
the scale factor

Stress measurement

If the crystal is compressed or stretched, the distances interréticulaires vary. This involves a variation of the position of the peaks.

By measuring the displacement of the peaks, one can deduce the deformation from it from the mesh, and thus, starting from the elastic Coefficients, the residual Contrainte in material.

While varying the orientation of the sample compared to the vector of diffraction (bisecting between incidental beam and detected beam), one can measure the variation of this constraint according to the orientation of the sample, and thus determine the Tenseur constraints.

See also: Determination of the tensor of the constraints by diffraction of x-rays

Measure texture

One of the assumptions of the diffraction of powder with geometry of Bragg-Brentano is that all the crystalline orientations must be respected. Indeed, as the vector of diffraction is always perpendicular to the surface of the sample, a plan ( HKL ) cannot give a peak that if there exist crystallites whose plan ( HKL ) is parallel to surface.

If the sample is not isotropic, then certain plans will give less low peaks, others higher than an isotropic powder. In addition, if one inclines the sample, the number of crystallites whose plan ( HKL ) diffracts will vary; thus, by measuring the height of two peaks for several orientations of the sample, one can determine the total orientation of crystallites, i.e. the texture.

See also: Measurement of texture by diffraction of x-rays

Determination of crystallographic structures

Starting from the diffracted intensities and opposite relation (reciprocal network - real network), it is possible, starting from a series of images of diffraction, to determine the three-dimensional arrangement of the atoms of a crystalline structure. This method took an importance considerable these last years for the determination of the structure of the biological Protéine S.

Using a software (for example Denzo), it is possible to determine the axes and centres of symmetry of a crystal and to propose the crystalline system most probable among seven what exists. It is then with the user to choose the group of the space (network of Faced) most suitable: the selected system is generally that which with highest symmetry in order to have the best resolution (it is generally at the end of the analysis, when all the atomic positions are given which can be specified the Groupe of space). Cell parameters are then proposed.

The Facteur of reliability R ( reliability ) makes it possible to calculate the degree of reliability of the mesh proposed compared to the real crystalline structure. When it reaches a sufficiently low value that means that the model of mesh is acceptable; one can then pass at the following stage i.e. the integration of the diffracted intensities and the refinement of the cell parameters.

The diffracted amplitudes are characteristic of the nature and the position of the atoms, in fact of the electronic Densité in any point of the mesh. More exactly, real space (of the crystalline structure) and reciprocal (of the directions of diffraction) are bound by Transformation of Fourier. Unfortunately, an important part of information is lost at the time of the collection of the images of diffraction, since only the standard of the intensities Complexe S is measurable by the detectors. The phases, which carry a very significant part of structural information, are lost and owe beings given (in experiments and/or by means of computer). It is necessary to integrate a great number of “tasks”, corresponding to the intensity of the reflections on the crystal lattice.

For the small compounds (meshs containing few atoms), of the procedures ab.initio was developed. On the other hand, for compounds of molar mass (or molecular weight) more important, one uses methods:

of derivation to the heavy atoms;
anomalous;
or of molecular replacement, when the structure (of the assymetric unit) is partially known.

By successive iterations, it is then possible to determine the missing phases, and consequently to refine the crystallographic structure of the compound.

See too

Random links: