In the usual geometric derivation of Bragg s Law one assumes that crystalline

Transcription

1 Diffraction Principles In the usual geometric erivation of ragg s Law one assumes that crystalline arrays of atoms iffract X-rays just as the regularly etche lines of a grating iffract light. While this quickly yiels the esire ragg expression, it presents an analogy that is inaequate to the task of explaining the results of an X-ray iffraction measurement. somewhat better physical analogy of X-ray iffraction by the atoms of crystals might go as follows: Suppose we ha a large, perfectly still, lake on which water waves were able to travel without the attenuation that we normally expect. Further suppose that at one ege of the lake we ha a large number of posts that emerge from the water to form perioic array (like a very large ock). The posts might be of various sizes, some as large as the pilings of a pier, some no bigger than broomsticks. ut whatever assortment of posts, they are arrange in an extene perioic array as they exten out into the lake. If a series of waves impinge on this array, there will be pattern of reflecte waves that emerge an propagate out into the lake. We can imagine that, by stuying the pattern of the reflecte waves, we coul work out the pattern of posts unerneath the ock. The situation escribe parallels the problem that faces a crystallographer when s/he hopes to etermine the structure of a crystal using iffraction techniques; the reaer may want to bear this analogy in min when thinking about iffraction from crystals. n even better physical analogy of the iffraction phenomenon is that offere by optical transforms, which we shall see emonstrate in class.

2 In hanling the elastic scattering of X-rays by matter, we will use a classical wave escription of the X-ray raiation. The incient electromagnetic raiation is characterize by specifying its wavelength,!, an its irection of propagation: { }; H = H 0 exp 2!i k "r # $t E = E 0 exp 2!i( k " r # $t) { ( )}, where E an H are the oscillating electric an magnetic fiels to which some point, r, is subject as the wave travels in the k irection. y x If the wave s electric fiel oscillates with frequency ν = c/λ in the x irection, the magnetic fiel oscillates in the y irection, then the wavevector, k, points in the z irection (E 0 = E 0^x ; H0 = H 0^y; k = k^z). In H E this expression, the magnitue of the z wavevector, k = k, is relate to the wavelength,!, by the relation k = 1/λ. When an X-ray is elastically scattere by an atom, it emerges with the same wavelength. We say that an X-ray with wavevector k is scattere to wavevector k, but k = k = 1/λ. However, the scattere photon is emitte as a kin of spherical wave but with an amplitue that ecreases to some extent as the angle by which it is iffracte increases. Representing the irection of propagation of the wavefronts by arrows, we will refer the angle with respect to the irection of the incoming iffracte wave as 2θ : 2! Every atom has a scattering power that scales with the number of electrons of the atom (or ion) an which falls off with scattering angle (2θ) in a characteristic manner as a function of wavelength, as illustrate in the figure below for the chlorine atom an ion:

3 In the absence of the angle epenence of the scattering, the scattere X-ray woul propagate outwar like a circular wave in water: If we inclue the angle epenence, there is stronger scattering in the forwar irection (in the irection of the incoming X-ray, 2θ = 0), the 2- imensional analog woul look something like the figure at right. X-ray iffraction for structure etermination is 2θ epenence of the atomic scattering represente in 2 imensions. The green-tippe, re arrow represents the incient X-ray propagation irection. concerne with interference effects that result from scattering by perioic, crystalline arrays of atoms. To see how this interference arises, first consier the interference effect cause by two scatterers, separate by a vector. In the immeiately following figure,

4 we see a pictorial representation of the wavefronts of an incient X-ray as they pass two scatterers at successively later times. In 1, the first wavefront has passe the first scatterer an the scattere wave is shown propagating outwar from that scatterer. In 2, we see the situation just as the first wavefront reaches the secon scatterer an scattere waves continue to emanate from the first scatterer as successive wavefronts pass its location. 1 2 k 3 4 k! k

5 In 3, time is move forwar by 1/ ν secons (=λ/c) so that the secon wavefront is just reaching the secon scatterer. Finally, in 4, we see the interference pattern that is being built up by the wave being scattere off the two scatterers. With the iagram below we analyze the interference phenomenon, where some vector algebra is use to specify the terms that concern us. k points in the irection of the incoming wave; k points in a irection for which we want to know the scattere intensity:! ˆk k 2" 2" ˆ k ˆk = k k k ˆ! = k! k! a unit vector in the k irection a unit vector in the k! irection path length ifference = (k ˆ k ˆ!) The amplitue of the scattere wave is f 1 (2θ) + f 2 (2θ) (phase shift factor), where f 1 (2θ) + f 2 (2θ) are just the scattering amplitues of atoms 1 an 2, respectively, an the phase shift factor epens on the path length ifference, (^ k ^ k ). For completely constructive interference in the scattere wave, the ifference in the length of the path that it traverses as it is scattere by the two scatterers is an integral multiple of the wavelength: ( ) = n$ where n = 1,2,3, ( ) = n since k = 1 $! ˆk " ˆk # or! k " k#

6 More generally, the amplitue of the scattere wave, (2θ), is (2!) " f 1 (2!) + f 2 (2!) # exp(2$i( ˆk % ˆk &) ' ) = ( 1 when ( f 1 (2!) + f 2 (2!) # ˆk % ˆk &) ' = n, ) - *+ %1 when ( ˆk % ˆk &) ' = n The intensity of the scattere waves is given by the square of the amplitue, I(2θ) = (2θ) 2 = *. Note that if λ is large compare to there can be no completely constructive interference; in orer to get significant interference effect the wavelength of the incient raiation shoul be comparable to or smaller than the istances between the scattering atoms. The translational symmetry of a crystal allows us to quickly generalize these results. In a crystal, the position of every atom is relate to the position of an equivalent atom ( f 1 (2!) = f 2 (2! ) = f 3 (2!) =!) in another unit cell via translation by a lattice vector, R i. Thus, if we wish to specify the conitions on k an k so that all of a set of a crystal s equivalent atoms will scatter X-rays with constructive interference, we have: ( ) = n for all the lattice vectors R i R i! k " k # 2$i k" # or e ( k )!R i = 1 We have foun that the vectors k k are just the reciprocal lattice vectors. simple way to state the conition for constructive interference (iffraction) is to say that the X- ray will be iffracte by a reciprocal lattice vector: The importance of the reciprocal lattice in k! = k +. interpreting iffraction patterns is clear; the irection of every iffracte beam (the so-calle k reflections) can be etermine from the above equation. We will have more to say about the k geometrical relationships that relate the reflections to the lattice below. k

7 Of course, most interesting materials possess more than one atom per unit cell. To see how the effect of having more than one kin of scatterer, let us consier the case of crystal with two atoms, an, in each unit cell. For convenience we will choose the origin of the cell such that it is at the position of atom, an we let be the vector that joins an within the cell (see the figure below). In general, the atoms an will iffer in their ability to scatter X-rays an the atoms scattering amplitues will epen on the angle, 2!, through which the beam is scattere (2! is just the angle between k an k + ), the number of electrons on atoms an, an the istribution of electron ensity. These atomic scattering strengths are labele as f (2!) an f (2!) - these are calle the atomic form factors. The amplitue of the scattere wave, F hkl, associate with a given reciprocal lattice vector,, is F hkl! f (2") + f (2") # exp(2$i % ) If there are N atoms in the unit cell, this expression, which is known as the geometric structure factor, is generalize to

8 N F hkl!& f j (2") # exp(2$i % j ); j where f j (2θ) is the form factor for the j th atom an j specifies the position of j th atom. While this expression may seem intimiating, its use is straightforwar in practice. There exists a useful geometrical construction ai in unerstaning how one can preict when a crystal s orientation is such that the iffraction conition, = k k, is satisfie. The wavevector of the incient X-ray is place in the proper rotational orientation with respect to the reciprocal lattice (an with the proper length, etermine by constructing it such that 1/λ is properly scale relative to the reciprocal lattice imensions). The origin of k is place on the origin of the reciprocal lattice an a sphere of raius k (=1/λ) is constructe with its center at the tip of k, guaranteeing that the origin lies on the sphere s surface. When the crystal s orientation is change relative to k such that some other reciprocal lattice vector,, also lies on the sphere, the iffraction conition is satisfie (as illustrate). The tips of k an k touch in the center of what is calle the Ewal sphere. rotation of a crystal in the incient X-ray beam can, of course, be viewe as rotation of the angle of incience of the beam in the opposite irection, viewe from the frame of reference of the crystal. The illustration below shows the ac-plane of a monoclinic irect lattice an a*c*-plane of the associate reciprocal lattice. rotation of the irect lattice in the sense inicate will bring about a rotation of the reciprocal lattice through an equal angle. In the frame of reference of the crystal then, the Ewal sphere rotates in the opposite sense an as it rotates, successive reflections are brought in an out of the sphere of reflection.

9 (0,0,l) c k! k! k! a k k k k (h,0,0) The Ewal sphere concept helps us visualize several important aspects of iffraction. First, the raius of the sphere is inversely proportional to the wavelength of the raiation use in the iffraction experiment, while the spacing of the reciprocal lattice is inversely proportional to the unit cell imensions. Thus, more reflections are swept through the Ewal sphere when shorter wavelength raiation is use or when the unit cell imensions are larger. In neutron iffraction, if so-calle thermal neutrons are use, a typical wavelength woul be 1.45 Å (= λ n = h/p n = h [3m n k T] -1/2 ). In high-energy electron iffraction, electrons with 100 kev kinetic energies are commonly use. Electrons of this energy have eroglie wavelengths much shorter than typical wavelengths use in X-ray iffraction (λ Cu = Å, λ Mo = Å, λ e,100kev = h/p = h [2m e E] -1/2 = Å) an the Ewal sphere is very large in comparison with reciprocal lattice spacings. s a result, by rotating a crystal by only a few egrees, large portions of a reciprocal lattice plane nearly tangent to the Ewal sphere can be brought through it making the technique especially valuable for imaging.

10 PROPERTIES OF DIRECT LTTICES ND RECIPROCL LTTICES In Chem 673, we introuce the reciprocal lattice via group theory. In the above, we showe that the reciprocal lattice vectors are intimately relate the conition for constructive interference in iffraction, namely, the incoming x-ray is iffracte by a reciprocal lattice vector: k = k +. Now we will further evelop some geometrical relationships between the irect lattice that unerlies the translational perioicity of crystals an the reciprocal lattice that serves as such a useful geometrical construct for interpreting the iffraction patterns of crystals. First we note two important mathematical relations; the first relates the irect an reciprocal lattices an the secon merely recalls a bit of vector algebra: (1) The efinition of the reciprocal lattice basis vectors, (a*, b*, an c*), guarantees that the ot prouct between any irect lattice vector (DLV), R i = ua + vb + wc, an any reciprocal lattice vector (RLV), = ha* + kb* + lc*, will be an integer: R i K j = (ua + vb + wc) (ha* + kb* + lc*) = uh + vk + wl = integer!r i K j = m ; m = integer " e 2#i(R i K j ) = 1 (2) The projection of any vector on to an axis parallel to another vector is given by the quantity ( ) = Â = cos!, where ^ is a unit vector parallel to an θ is the angle between an. This relation can be unerstoo pictorially below: = cos! ˆ = cos!! Â cos!

11 The irect lattice can be partitione such that all the lattice points lie on planes. Crystallographers classify these planes using intercepts on the unit cell axes cut by the plane ajacent to the plane through the origin (see other hanouts for examples). The intercepts must be of the form (1 h,0,0), (0,1 k,0), an (0,0,1 l). These planes are normal to the RLV = ha* + kb* + lc*. c (hkl) plane (0,0,1/l) (0,1/k,0) -(1/h)a+(1/l)c (1/h,0,0) a b -(1/h)a+(1/k)b Proof: The two vectors!(1 / h)a + (1/ k)b an!(1 / h)a + (1 / l)c respectively connect the pairs of points (1 h,0,0) & (0,1 k,0) an (1 h,0,0) & (0,0,1 l) - see the rawing. s such, these two vectors must lie in the plane in question. Now, [!(1 h)a + (1 k)b] = [!(h h)a a * + (k k)b b * ] =!1+ 1 = 0 an [!(1 h)a + (1 l)c] = [!(h h)a a * + (l l)c c * ] =!1+ 1 = 0 So is normal to both vectors an therefore is normal to the plane in which they lie an therefore is normal to the entire family of mutually parallel planes. Since is perpenicular to the hkl family of planes, we can relate the istance between planes to the magnitue of as follows:

12 ssume that is the shortest RLV that points in the exact irection that it happens to point (i.e., = ha* + kb* + lc* where h, k, an l have no common factor). lattice planes b! a Kˆ hkl = - a unit vector (1/h,0,0) hkl = (1/h) a Kˆ hkl = (1/h) a K ˆ cos! The istance between planes, hkl, is given by (see the 2D example above): hkl = (a h) ˆ = (b k) ˆ = (c l) ˆ = (1 3)[(a h) + (b k) + (c l)] ( ) where K^ hkl is a unit vector parallel to. Substituting in the expression for we obtain: hkl = (1 3)[(h h)a a * + (k k)b b * + (k k)c c * ] (1 ) = (1 ) or = 1 hkl This is an important result that we will use later. Simply state, it says that reciprocal lattice vectors are inversely proportional to the spacing between irect lattice planes with which they are associate.