Lattices diffraction from

Electrons interact with solid surfaces by elastic and inelastic scattering, and these interactions are employed in electron spectroscopy. For example, electrons that elastically scatter will diffract from a single-crystal lattice. The diffraction pattern can be used as a means of stnictural detenuination, as in FEED. Electrons scatter inelastically by inducing electronic and vibrational excitations in the surface region. These losses fonu the basis of electron energy loss spectroscopy (EELS). An incident electron can also knock out an iimer-shell, or core, electron from an atom in the solid that will, in turn, initiate an Auger process. Electrons can also be used to induce stimulated desorption, as described in section Al.7.5.6. 

The following diagram shows two lattice planes from which two parallel x-rays are diffracted. If the two incoming x-rays are in phase, show that the Bragg equation 2d sin 6 = X is true when n is an integer. Refer to Major Technique 3 on x-ray diffraction, which follows this set of exercises. 

Figure 6. Conformation of cxjmpound 12, toxin C4, in the crystal lattice. Data from diffraction studies by S. D. Darling (13) computer graphics by T. Chambers.

The notion of a reciprocal lattice cirose from E vald who used a sphere to represent how the x-rays interact with any given lattice plane in three dimensioned space. He employed what is now called the Ewald Sphere to show how reciprocal space could be utilized to represent diffractions of x-rays by lattice planes. E vald originally rewrote the Bragg equation as ... 

Fig. 3.1. Left visible pump/X-ray probe scheme for femtosecond TRXRD experiments. Hard X-ray pulses are generated by shining intense femtosecond laser pulses on a metal target (laser plasma X-ray source). Right geometrical structure factor of bismuth as a function of inter-atomic distance for diffraction from (111) and (222) lattice planes. From [1] and [2]...

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

For crystals which have flat faces which extend for a fraction of 1 ym, a new type of phenomenon may be observed. Electrons incident at the edge of the crystal parallel to the surface may be channelled along the surface. The potential field of the crystal extending into the vacuum deflects the electrons so that they tend to enter the surface but they are scattered out of the crystal by the surface atoms or by diffraction from the crystal lattice planes parallel to the surface. If the scattering angle is less than the critical angle for total external reflection, the scattered electrons can not surmount the external potential barrier and are deflected back into the crystal (figure 4 (a)). 

Unlike simple inorganic compounds (e.g., NaCl or KC1), polymers do not have a perfectly ordered crystal lattice formation and are not completely crystalline. In fact, they contain both crystalline and amorphous regions. Hence, the X-ray diffractions from them are found to be a mixture of sharp as well as diffused patterns. 

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

The diffraction of visible light (with a wavelength X in the range 350 to 700 nm) by small holes, slits or lattices is a well-known phenomenon which results from destructive and constructive interferences of coherent waves. For a lattice, diffraction may occur when the wavelength is lower than the lattice repeat distances. 

Electrons diffract from a crystal under the Laue condition k — kg=G, with G = ha +kb +lc. Each diffracted beam is defined by a reciprocal lattice vector. Diffracted beams seen in an electron diffraction pattern are these close to the intersection of the Ewald sphere and the reciprocal lattice. A quantitative understanding of electron diffraction geometry can be obtained based on these two principles. 

The diSuse scatter arises because dislocations are defects which rotate the lattice locally in either direction. This gives rise to scatter, from near-core regions, which is not travelling in quite the same direction as the diffraction from the bulk of the crystal. This adds kinematically (i.e. in intensity not amplitude) and gives a broad, shallow peak that mnst be centred on the Bragg peak of the dislocated layer or substrate since all the local rotations are centred on the lattice itself. We can model the diffuse scatter quite well by a Gaussian or a Lorentzian function of the form ... 

Figure 7.8 A scattering map in reciprocal space. Equal intensity contours are shown schematically, and the Ewald sphere is represented as a plane near reciprocal lattice points 0 and h. The dynamical diffraction from the specimen is displaced slightly from the relp and from the centre of the diffuse scatter by the refractive index effect...

Fig. 4.3. Experimental intensity vs. voltage (energy) curves for electron diffraction from at Pt(l 11) surface. Beams are identified by different labels (h,k) representing reciprocal lattice vectors parallel to the surface. An incidence angle of 4° from the surface normal is used...

Fig. 5. Temperature variation of the hexagonal lattice parameters and of the volume of pure gadolinium measured by x-ray powder diffraction (this work). The values have been normalized to 300 K in order to show the relative changes. (The values at 300 K are a = 3.632 0.002 A, c = 5.782 0.002 A.) The lines represent the extrapolation of the lattice contribution from temperatures above Tq assuming a Debye temperature of 184 K (Bodnakov et al. 1998). The lowest part of the figure shows the magnetovolume effect, obtained by subtracting the lattice contribution from the volume expansion.

Fig. 14. Anisotropic thermal expansion of GdCuSn measured by x-ray powder diffraction (Gratz and Lindbaum 1998). The lines represent the extrapolation of the lattice contribution from the paramagnetic range by fitting...

The structures of several proteins have been solved by both NMR in solution and x-ray diffraction from the crystal, and the correspondence is excellent. Sometimes there are discrepancies at the contacts between neighboring molecules in the crystal lattice. Crystal packing can lead to such problems. 

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