Cristalografa de las fases en las muestras

LaB6: Grupo espacial Pm3m (221), coordenadas fraccionarias de los tomos

tomoSitioxyZ

La1a000

B6f0.1990.50.5

Parmetros de red: a = 4.16 A

CeO2: Grupo espacial Fm3m (225), coordenadas fraccionarias de los tomos

tomoSitioxyz

Ce4a000

O8c0.250.250.25

Parmetros de red:a = 5.41 A

Rutilo: Grupo espacial P42/mnm(136), coordenadas fraccionarias de los tomostomoSitioxyz

Ti2a000

O4fuu0

u = 0.3046Parmetros de red:a = 4.59556 A, c = 2.95996

Factor de Lorentz y polarizacin de los rayos XLa intensidad asociada a cada pico del patron de difraccin depende del tiempo que los rayos X pasan en los planos de difraccin (Factor de Lorentz) y de la polarizacin parcial del haz de rayos X en el monocromador y en la muestra.Esto aspectos influyen en la intensidad de los picos de difraccin, la cual depende del ngulo de difraccin de acuerdo a la siguiente relacin: ILP = I * (1+Cos2(M)*Cos2(2))/(Sin2()Cos())Mes el ngulo de difraccin en el monocromador is the scattering angle in the monochromatorPara una ptica sin monocromadorM = 0 grados.Par una radiacin totalmente polarizada (Radiacin de Sinchrotrn) M = 90 grados.Los valores de M para los monocromadores ms comunes empleados con radiacin de cobre se presentan a continuacin: Germanio : 27.3 gradosGrafito : 26.4 gradosQuarzo : 26.6 grados

Las siguientes notas se dejaron en el Idioma Ingls, tal como se encontraron en la literatura.Polarisation, PIt is fairly obvious that the direction of polarization of an X-ray photon can change as a result of scattering/diffraction. In fact there are two extreme cases to consider; when the change is maximal or when there is no change, depending on whether the initial polarization is or is not in the plane containing the pre- and post-scattered X-rays:

In the first case, the component of polarization resolved along the new (diffracted) direction is reduced by the cosine of the scattering angle (conventionally 2) and so the reduction in intensity would be by cos22 (since intensity is proportional to the square of the amplitude). In the second case, the component of polarization is clearly unaffected, and therefore unchanged, by the diffraction process so we would say, mathematically speaking, that the reduction factor is "1". These two extreme cases can in principle be realised by diffractometers on synchrotron sources due to the special polarised nature of the X-rays produced by synchrotron sources (see earlier section on Instrumentation: Synchrotron Sources and Methods). There is a third case, a mixture of the above two, which occurs with laboratory diffractometers since laboratory X-ray sources produce unpolarised X-rays, that is X-rays polarised equally in all possible directions; here the reduction factor will be the mean of the two previous extreme cases. We could summarise these three situations as follows: Case (1): Polarisation in plane of scattering,P = cos22

Case (2): Polarisation perpendicular to plane of scattering, P = 1

Case (3): Unpolarised X-rays,P = (1 + cos22 )/2

These three cases are shown graphically below (in red, dark blue, and magenta, respectively):

There are some obvious conclusions from all of this: Case (1): You would not use this arrangement out of choice since this has the worst reductions in intensity; in fact it is disastrously reduced to zero at 2=90; i.e. you would not be able to see any diffraction at all at this angle! However this case is used in a negative sense in spectroscopy measurements where diffraction effects are unwanted; for this reason spectroscopy measurements are often deliberately performed at 2=90 to remove the diffraction component. Case (2): This is obviously the best case of all since there is no loss of intensity anywhere due to polarisation. For this reason synchrotron diffractometers are usually set in this configuration. Case (3): The reduction centred at 90 is unavoidable with laboratory diffractometers and obviously must be corrected for.Now lets look at our case problem again, the 200 and 111 reflections in NaCl. To calculate the effect of polarisation here we need some additional information: that the measurements are done on a laboratory powder diffractometer using copper K X-rays. Armed with this we will calculate the correction for the 111 case, and again for easy reference we will structure the model answer: (1) Working out the angles , 2:First we remind you how to calculate the d spacing in cubic crystals: dhkl = a/(h2 + k2 + l2 )1/2

d111 = 5.638/(12 + 12 + 12 )1/2 = 5.638/3 = 3.255

For Copper K radiation the wavelength is about 1.54 , and so inserting into Bragg's law gives us: 1.54= 2 3.255 sin

sin = 0.237

= sin1(0.237) = 13.68

2= 27.37

(2) CalculatingP111:For the laboratory powder diffractometer we use the formula of Case (3): 2= 27.37

cos (2)= 0.888

cos2 (2)= 0.789

1 + cos2 (2)= 1.789

P111 = (1 + cos2 (2 ))/2= 0.894

So polarisation produces a reduction in intensity to 89% in the case of the 111 reflection. From the graphs, you should be able to see that it is reduced more for the 200 reflection, due to its smaller d spacing. Polarisation&MonochromatorsAn unpolarized laboratory X-ray source diffracted by monochromator crystal produces a partially polarized X-ray beam. As for a diffracting sample, the degree of polarisation depends upon the diffraction angle, 2M, of the monochromator. Assuming that the source, monochromator, sample, and detector are all in the same plane, then the polarisation factor in the presence of a monochromator is given by the expression: P = (1 + cos2 2M cos2 2) / (1 + cos2 2M) Given that 2M is small, the effect of the cos2 2M in this expression is small as shown in the figure below, where the solid line is the polarisation factor without a monochromator and the dotted curve shows the polarisation correction with 2M set to 28.44 (as for a Si and CuK1).

The expression is correct whether the monochromator is pre or post sample. However, there has been some debate as to whether cos2 2M or |cos 2M| should be used when the monochromator is a perfect crystal, e.g. Si (for similar reasons to the use of |F| versus F2 for intensity given in the earlier page on dynamic versus kinematic diffraction). If a crystal structure is refined from laboratory data collected with a monochromator and a monochromator polarization correction is not applied, then the thermal parameters will appear to be larger than expected so as to compensate for the lack of correction. An alternative scenario is the case of a pre-sample monochromator on a powder diffraction synchrotron beamline. Here the incident beam is 100% plane polarized. Assuming that the monochromator and diffractometer all function in the vertical plane, then the polarization remains unchanged from source to detector, so that no additional correction is required due to the monochromator.

Lorentz Factor, LIn previous discussions we have referred to ideas such as random crystallite orientations and the statistical likelihood of such crystallites being correctly oriented for diffraction. Bragg's law seems to imply that diffraction only occurs when the equation is satisfied exactly. In reality, the parameters to be fulfilled in Bragg's law (, d) will have a finite spread so that diffraction occurs in practice over a small range of angles around the mean, 2. Translated into the language of the reciprocal lattice used earlier, one would say that the reciprocal lattice points are not infinitesimal mathematical points of zero size, and similarly the Ewald sphere is not a vanishingly thin shell: the difference between the two is illustrated below:

In the left-hand "mathematical picture" a reciprocal lattice point is found to be slightly off the Ewald sphere which is equivalent to saying that diffraction will not occur unless the crystallite is re-oriented slightly to bring it onto the sphere; however in the "real" right-hand picture intersection (and therefore also diffraction) still occurs in spite of this slight mis-orientation. Given this latter more realistic view one might next ask what is the overall statistical probability, under the conditions pertaining to the powder diffraction experiment, that intersection/diffraction will occur? It turns out that the answer varies according to the experimental arrangement used. For standard powder diffraction arrangements the relationship is: L = c / (sin sin2) or, alternatively, L = c / (sin2cos) Both expressions appear in the literature and differ only by the definition of the constant c (since sin2=2sincos). The precise value of the constant, which can be set to unity, is unimportant since, in general we only calculate the relative intensities of reflections. The shape of the function is shown in the plot below:

It may be useful to consider the real-space processes that give rise to the Lorentz factor. Considering the Debye-Scherrer cones of scattered radiation from a powder sample, then a detector of finite aperture will have the potential to count more photons at low (or equally at high) angle, and less as the scattering angle approaches 90. The amount of the Debye-Scherrer cone measured is simply inversely proportional to the sine of the scattering angle 2, i.e. 1/sin2. The second term relates the probability of the crystallites having a plane in the correct orientation for Bragg scattering, and this is inversely proportional to the sine of the incident Bragg angle , i.e. 1/sin. Again taking the NaCl 111 reflection, and using the value for theta already obtained in the polarization calculation: = 13.68

sin = 0.237

sin 2= 0.460

sin sin 2= 0.1087

L = 1 / (sin sin 2)= 9.20

This is a relatively large value compared to, say, values of L for 2 values nearer to 90. You can see from the graph that the Lvalue for the 200 reflection will be much smaller. In addition, note that the powder patterns of materials with large dspacings are likely to have very intense reflections at low diffraction angles due to the Lorentz factor.

Medicin: Difraccin de rayos XMuestra:LaB6Fecha: 03/09/2012ptica del difractmetro empleado:

EquipoBruker D8 AdvanceGeometraBraggBrentanoConfiguracin Theta-Theta Dimetro de la geometra435 mm nodo del tubo de rayos XCobre Haz del tubo de rayos X Foco fino 0.04 x 12 mmVoltaje y corriente en el tubo de rayos X40kV, 35 mAMonocromadorNingunoColimador Soller en el haz incidente1.5 grados Apertura de divergencia0.5 grados Apertura de salidaNingunaCortador de fondo en el haz primario0.5 mm arriba de la muestra Apertura del haz difractado8.0 mm Colimador Soller en el haz difractado2.5 grados Filtro de Kbeta en el haz difractado Nquel 0.5 % AbsorbedorNingunoMonocromador secundarioNingunoApertura en el detector0.075 mmDetector Tira mltiple de Si (Lynxeye)rea activa mxima del detector14mm x 16mm Ventana activa del detector3.43 grados

Condiciones de medicin:Intervalos (en grados) de medicin en 2 theta 2-60 grados Modo de barrido del gonimetro Barrido contnuoIncremento en 2 theta0.0195 grados Tiempo de medicin (en segundos) por punto 72 segundos Tiempo total de la medicin por muestra 21:00 minutosIntensidadCuentasRotacin de la muestra NoPortamuestrasVidrioArchivos de datos generado con DIFFRAC PlusLaB6-4hrs-06-05-09.raw