lunes, 26 de julio de 2010

Diffraction and Quasi-Elastic Scattering Techniques


            The common electron-based diffraction techniques are LEED and RHEED. As with all surface diffraction techniques, the analysis is based in terms of the surface reciprocal lattice. An important aspect of diffraction from 2D structures is that k// is conserved to within a surface reciprocal lattice vector G//, whereas k^ is not, as shown in diagram 62 (Luth). This leads to the idea of reciprocal lattice rods, a concept which takes a little while to absorb. 

            The equipment for both diffraction techniques is simple, involving a fluorescent screen, with energy filtering in addition in the case of LEED, as indicated in diagram 63, to remove inelastically scattered electrons. There are three types of LEED apparatus in regular use. The 'Normal-view' arrangement has the LEED gun and screen mounted on a UHV flange, typically 8 inch (200 mm) across, and the pattern is viewed past the sample, which therefore has to be reasonably small, or it will obscure the view. Most new systems are of the 'Reverse-view' type, where the gun has been miniaturised, and the pattern is viewed through a transmission screen and a viewport. This enables larger sample holders to be used, which helps for such operations as heating, cooling, straining, etc. The third, and potentially most powerful technique is where, in addition to viewing the screen, the LEED beams can be scanned over a fine detector using electrostatic deflectors and focussing, in order to examine the spot profiles, which can be sensitive to surface steps and other forms of disorder at surfaces. This technique, which has been perfected by the Hannover group (M. Henzler et al.), is now known as SPA-LEED, emphasising the capability for Spot Profile Analysis. This would be a suitable topic for a seminar talk. 

            There are two aspects to diffraction techniques. The first, and simplest, is that the positions of the spots give the symmetry and size of the unit mesh, i.e the surface unit cell. This can be seen dramatically in diagram 64, which shows LEED patterns of Si(111) in the high T '1x1' phase (a), the 7x7 reconstruction at two different electron energies and exposures (b and c) and the 'root-3' structure associated with the ML phase of Ag/Si(111) (d). These patterns are strikingly different in a qualitative sense. The most common use of electron diffraction techniques is primarily, often solely, in this sense.

            The second, and much more subtle effect, is that the positions of atoms in the mesh is not determined by this qualitative pattern (see diagrams 28-30, and the discussion in Section 1.4), but requires a quantitative analysis of LEED intensities. The difference between LEED and X-ray structure analysis is that a kinematic diffraction theory has limited usefulness, because the scattering is very strong. The dynamic theory is quite complicated, and has attracted some very high-powered theorists over the years. Application of the dynamic theory has so far 'solved' some 400 surface structures. This is impressive, but it pales besides the number of bulk structures solved by X-rays, using (developments of) kinematic theory.

            We can go into the elements of dynamic theory, which is constantly developing, at a later stage. For now, realise that the intensities are typically collected in the form of so-called I-V or I(E) curves, where the size of the Ewald sphere is varied by varying the probe energy (say from 20-150 eV), and the intensity data is obtained by 'tracking' along an individual reciprocal lattice rod. Various computer-controlled, and frame-grabbing, schemes have been developed to do this.

            The data base of solved structures was referred to already in section 1.4 (P.R. Watson et al, Atlas of Surface Structures, vols 1A and 1B, ACS Publications). A description of experimental and theoretical methods is given by Clarke. Luth, Chap. 4. is a good starting point for LEED, and Chap. 4.4 gives an overall introduction to the physics of multiple scattering (brave attempt).


            Because ASU is well known for its work in Electron Microscopy and Diffraction, many of us have met dynamical theory at higher electron energies in the context of the interpretation of TEM images. The physics is very similar in LEED, but the language used is a bit different. The language of RHEED is very similar to that which may be familiar to you from THEED. The discussion can again be separated into Geometry and Intensities.

            As seen in diagram 67, the glancing angle geometry of RHEED means that the reciprocal lattice rods are closely parallel to the Ewald Sphere near the origin. This means that the low angle region often consists of streaks, rather than spots (diagram 68). This part of the pattern corresponds to the Zero order Laue Zone (ZOLZ) in THEED patterns. The higher angle parts of a RHEED pattern are then equivalent to Higher order (HOLZ) rings in THEED patterns.

            The apparatus for RHEED can consist of a simple 5-20kV electrostatically focussed gun, for example to monitor the surface crystallography in an MBE experiment, where the glancing angle geometry has many practical advantages over LEED, especially in the ease of access around the sample. Or it can utilise an electron gun approaching electron microscope quality, and produce finely focussed diffraction spots over a wide angular range. Several workers have perfected this technique in Japan, an example of Si(111)7x7 being shown in diagram 69, with more examples in the books (e.g. Luth, Panel 8, fig 5, page 207).

            Again the question of intensities is much more detailed, involving multiple scattering and inelastic processes, and there is a large amount of discussion/ assertions about whether streaks or spots constitute evidence for good (i.e. well prepared, flat) surfaces. We leave this to seminar material, but make some general remarks in the next section.

c) Elastic, Quasi-Elastic and Inelastic Scattering

            The theory of LEED and RHEED concentrates on elastic scattering, where the energy of the outgoing electron is the same as that of the incoming electron. But experimentally we cannot discriminate in energy very well in a typical diffraction apparatus. LEED grids/ screens are able to remove plasma loss electrons (~ 10-20 eV loss), but the intensities measured include phonon scattering (~ 25 meV losses and gains). This is thermal diffuse scattering, and is accounted for in the theories using a Debye-Waller factor, as in standard X-ray theory. At higher T, the intensities in the Bragg peaks fall off exponentially, as

            I/I0 = exp -(K2<u2>/2),  with <u2> = 3h2T/(4p2mkqd2),

qd being the Debye Temperature, and K the scattering vector (diagram 62a). Woodruff and Delchar give this formula, Chap 2, with Dk = K. This means that intensity measurements as a function of T measure <u2>, and several such studies have been done with LEED; examples are given by Prutton (Chap 5, pages 139-142, beware typographic errors in eqn 5.1). An interesting feature of such experiments is that the value of <u2> decreases towards the bulk value as the incident energy is increased, reflecting the increased sampling depth of the electrons.

            In a typical RHEED setup, there is no energy filtering, other than that caused by the fact that higher energy electrons produce more light from phosphor screens. Yet the geometry is such that plasmons, especially surface plasmons, will be produced very efficiently. Because plasmon excitation produces only a small angular deflection of the high energy beam, the diffraction pattern is not unduly degraded, but we should remember that they are present in the experiment, if not in the interpretation. A few groups have studied energy-filtered RHEED, but it has proven quite difficult to construct adequate filters which work over a large angular range.

            The basic reason for the surface sensitivity of LEED is the short inelastic mean free path for the excitation of plasmons (and other forms of electron-electron collision); this means that information from deeper in the crystal is effectively filtered out. One of the few calculations which we could usefully do in class (but there may not be time) is the pseudo-kinematic case, where one has single scattering and exponential attenuation. This calculation shows that the attenuation causes only a few layers at the surface to be sampled, which give rise to modulated reciprocal lattice rods, the width of the modulations being inversely proportional to the imfp. In the full dynamical LEED calculations, this attentuation effect is included by an imaginary potential, V0i. This is similar to the high energy case of RHEED and THEED, but the language is a little different. In particular, in TEM, we use imaginary potentials (V0i and Vgi) to describe contrast in images caused by inelastic scattering; but these are dependent on the aperture size used, and are typically due to the scattering of phonons and defects. In contrast to plasmons, these scattering effects cause a wide angular spread, and very little energy loss.

            LEED (especially SPA-LEED) and RHEED, and the corresponding microscopies (LEEM and REM) have been shown to be very sensitive to the presence of surface steps and other types of defects, including domain structures. Some of these sensitivities are due to the extra diffraction spots associated with particular domains; some of them are due to exploiting the difference between in-phase and out-of-phase scattering (see diagram 62b); some again depend on the small rotations produced by surface steps, and the increase in diffuse scattering. There are new techniques emerging, such as DLEED (Diffuse-LEED) and 'Electron Holgraphy'. An examination and discussion of some of these factors/ techniques would be a suitable seminar topic.

            The above discussion has concerned the effect of inelastic processes on the interpretation of elastic scattering processes. In the next two sections we are concerned with the understanding and use of the inelastic processes in their own right.
Arellano Wilson 17930016

HORIBA Analysis Techniques

Static Light Scattering

Also known as Low Angle Light Scattering (LALS), Fraunhofer diffraction, or Mie Scattering

Fraunhofer diffraction is the simplest method of determining particle size from light scattering measurements. It applies to particles larger than approximately one micron.
For particles larger than the wavelength of light, the light scatters from the edge of the particle at an angle which is dependent on the size of the particle. Larger particles scatter light at relatively smaller angles than light scattered from smaller particles. From observing the intensity of light scattered at different angles, we can determine the relative amounts of different size particles.
As the particles get close to or smaller than the wavelength of light, more of the light intensity is scattered to higher angles and back-scattered. The Mie Scattering Theory accounts for this different behavior and requires that we input information about the optical properties of the particles, such as refractive index. In order to make particle size measurements, the light intensity pattern must be measured over the full angular range. The Mie Theory applies to all sizes. When the particle size is larger that the wavelength of the incident light, the Mie equation reduces to the Fraunhofer equation. This allows one algorithm to cover the entire size range.
HORIBA incorporates the full Mie Scattering Theory over the entire size range of interest. An array of detectors. including high-angle and back-scatter detectors, and multiple light sources of different wavelengths are employed to provide an instrument that allows measurement of the full size range in one analysis. There is no need to combine results from two optical systems or analysis techniques, along with the problems that entails.
HORIBA offers a range of static light scattering particle size analyzers with different size ranges and with a range of capabilities.

Dynamic Light Scattering

Also known as Photon Correlation Spectroscopy (PCS).

Dynamic light scattering is a general term for the measurement of smaller particles (less than a few microns) by observing the Doppler shift of the incident light due to the Brownian motion of the suspended particulates. LB-550 Optics Layout Particles suspended in a fluid exhibit Brownian motion. This is random movement of the particles caused by the fluid molecules hitting the particles. Smaller particles with more more rapidly that larger particles due to their low inertia. When a coherent light source shines on these particles, light will be scattered from the particles, but the frequency will be shifted because the particles are in motion (Doppler shift). The speed of the particles determines how much the frequency is shifted.
By knowing the incident light frequency and measuring the scattered light frequency to determine the shift, we can calculate particle size.Dynamic Light Scattering Diagram 1 Early particle size analyzers using the PCS method, placed the detector at 90 degrees to the incident light. This required a very dilute suspension to prevent multiple scattering and over-attenuation of the light beam. This required a high powered laser and was sensitive to dust contamination of the fluid. The correlation technique limited the resolution capabilities of the algorithm, so it was often not capable of independently determining the presence of multiple modes in a sample.
More recent DLS instruments place the detector at or near 180 degrees to the incident light beam. This allows the measurement of higher concentration suspensions, eliminating the need for dilution and worries about dust contamination.
HORIBA's LB-550 places the detector at 180 degrees, allowing the measurement of samples up to 20% solids (by weight). The algorithm used to determine particle size from the light signal incorporates the Fourier-transformed power spectrum and iterative deconvolution of the relative contribution of various size particles in the mixture. This allows for and accurate accounting of not only median size, but also the distribution shape or multiple modes if they are present, without input from the operator.
HORIBA offers the LB-550 Dynamic Light Scattering Particle Size Analyzer, covering a range of 3nm to 6µm.
Arellano Wilson 1793016

Difracción de rayos X

Röentgen en 1895 descubre los rayos X: comienzan a utilizarse sin saber muy bien lo que eran (radiografías de cuerpos opácos). En 1912 Von Laue descubre que los cristales difractan los rayos X y según la forma de difractar permitía identificar la estructura del cristal.
Los rayos X son radiaciones electromagnéticas de  entre 0,5 - 2,5 Å aunque puede abarcar desde 0,1 a 10 Å, ( luz visible 4000 - 8000 Å ). Los rayos X se producen cuando una partícula cargada con suficiente energía cinética, se desacelera rapidamente, se emplean electrones como partículas cargadas. Un tubo de rayos X tienen una fuente de electrones y dos electrodos metálicos, se establce un voltaje alto entre los dos electrodos (decenas de miles de voltios), los electrodos van al ánodo chocando a gran velocidad. En el punto de impacto se producen los rayos X que radian en todas direcciones. . Donde "e" es la carga del electrón Carga de un electrón, "v" es el voltaje, "m" la masa del electrón Masa de un electron y "v" es la velocidad justo antes del impacto. Si se aplican 30.000 V => V = 1/3 velocidad de la luz. La mayor parte de la energía cinética se convierte en calor, solo menos de un 1 % se transforma en rayos X, nalizados estos rayos X se comprueba que son una mezcla de de diferentes  y la variación de la intensidad con la  depende del voltaje del tubo. la radiación continua está constituida por rayos de muchas , esto es debido al desaceleración que sufren los electrones al golpear el ánodo, no todos los electrones sufren la misma desaceleración, unos paran del todo, otros son desviados, los que son completamente desacelerados comunicarán la máxima energía. Los electrones desviados tienen menor energía y mayor . La totalidad de esas  por encima de la SWL (short wavelength limit) constituyen el espectro continuo. A mayo V, las curvas son más altas y desplazadas hacia la izquierda - ver figura - (mayor número de fotones por segundo y mayor energía por fotón). El total de rayos X emitidos depende también del Z del ánodo y de la intensidad de la corriente en el tubo.
La intensidad total de rayos X es , donde A es la cosntante de proporcionalidad y m es constante de valor 2. Si se quiere mucha radiación blanca, se usará un metal de Z alto, el material del ánodo afecta a la intensidad de la  pero no a la distribución de la  del espectro continuo.
Cuando se sube el voltaje de un tubo de rayos X por encima de un valor crítico (propio del metal del ánodo) aparecen máximos de intensidad a ciertas  superpuestos al espectro continuo, por ser característicos del metal se llaman lineas características. Estas lineas caen en varios grupos a las que se denominan K, L, M, en orden creciente de , todas ellas forman el espectro caracteristico del metal usado como ánodo. Para el Molibdeno las lineas K tienen una  de 0,7 Å las L de 5 Å y las M mayores. Normalmente solo las lineas K son útiles en los rayos X. Hay varias lineas K pero solo las tres mas fuertes son observadas en la difracción normal. Son la  . Para el caso del Molibdeno las  serían:  = 0,70926 Å;  = 0.71354; y para  = 0.63225 Å. Los componentes  y  tienen  tan próximas que no siempre se resuelven en lineas separadas si lo hacen se suele llamar linea , igualmente se suele llamar linea  es siempre aproximdamente dos veces mayor que  mientras que el cociente  depende de Z pero suele ser un promedio de 5/1. Puesto que el voltaje de excitación K, para el Molibdeno es de 20,01 kv, las lineas K no aparecen por debajo de este valor. Un aumento del voltaje sobre el valor crítico aumenta la intensidad de la linea caracteristica del espectro continuo pero no cambia sus .

Espectro del Molibdeno a 35 kv:
El aumento de voltaje ha llevado el espectro continuo a menores  y ha aumentado la intensidad de las lineas K pero no variado las  del espectro caracteristico. La intensidad de una linea K viene dada por  donde B es la constante de proporcionalidad,  es el voltaje de excitación de K y n la constante = 1,5 . La intensidad de una linea caracteristica puede ser muy grande, así un anodo de cobre a 30 kv tiene la linea  con una intensidad 90 veces mayor que las  adyacentes del espectro continuo. A pesar de ser tan intensas las lineas son muy estrechas (<0,001 Å) en la mitad de su I max. La existencia de esta linea , fuerte y nitida explica el papel importante de los rayos X pues muchos experimentos requieren el empleo de radiación monocromática.
Las lineas caracteristicas de los rayos X fueron descubiertas por Bragg y concretadas por Moseley, este encontró que las  de una linea determinada disminuyen al aumentar el Z del emisor. En concreto encontro que  donde C y sigma son constantes. Veamos en la figura siguiente esta relación para las lineas  y , esta última la linea mas fuerte en la serie L.
Muestran que las lineas L, no son simpre de  larga: la linea  de un metal pesado como el wolframio tiene casi la misma  que la linea  del cobre, sobre 1,5 Å. Se han medido con precisión las  de las lineas caracteristicas de rayos X de casi todos los elementos conocidos y están tabulados. Mientras el espectro continuo se debe a la desaceleración de los electrones, el espectro caracteristico depende del anodo, consideremos el átomo como un nucleo rodeado de electrones en varias capas. Un electrón que bombardee el nucleo con la suficiente energía cinética, puede llevarse un electrón de la capa K dejando al átomo en un estado activado. Uno de los electrones exteriores cae inmediatamente en la vacante de la capa K emitiendo energía y alcanzando el átomo su estado normal. Esta energía emitida en forma de radiación tiene unadefinida y es, de hecho la radiación caracteristica K. La capa K se puede llenar con electrones de la L, M, etc. dando una serie de lineas K, de igual manera que mientras en un atomo cambie un electron de la capa L, en otro lo haga de la capa M, también es imposible excitar una linea K sin excitar también las otras. Lo mismo podría decirse de las lineas L. La radiación no se dará hasta que el voltaje del tubo sea tal que el bombardeo tenga la energía necesaria para golpear un electrón K y lo saque. Llamando  al trabajo requerido para quitar un electrón K, la energía cinética necesaria sera de .
Arellano Wilson 17930016