E. K. A. ADVANCED PHYSICS LABORATORY PHYSICS 3081, 4051

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1 E. K. A. ADVANCED PHYSICS LABORATORY PHYSICS 38, 45 ELECTRON DIFFRACTION Introduction This experiment, using a cathode ray tube, permits the first hand observation that electrons, which are usually though of as particles, exhibit Fraunhofer-like diffraction from crystal lattices, a distinctly wave phenomenon. An included page shows the equipment and typical diffraction patterns obtained from the built-in randomly oriented polycrystalline aluminum (Debye-Scherrer Rings) or from a thin single (or few) crystals of pyrolytic graphite where the hexagonal lattice planes are nearly perpendicular to the electron beam direction. The pattern depends on the crystalline lattice structure and spacings, and thus provides a study of these aspects of the solid state lattice structure if the electron wave length is known. In 94 de Broglie proposed that the wavelength associated with particles of momentum p is λ = h/p, where h is Plancks constant. This is also true in optics where exp(ik r) is a plane wave of propagation vector k(radians/cm) for momentum p = hk, where h = h/π and k =π/λ. An electron of charge e accelerated through potential difference V from rest has a kinetic energy T = mv = ev 5 kev, the electron rest energy. This gives λ = h/ mev. The relation T = E m e = ev is relativistically and classically correct. T is however not mv relativistically. The correct relation between energy and momentum is E = T + m e = p + m, where we have set c= and measure masses and energies in ev, and momenta in ev/c. For V = kv, ev = kev, about % of the electron rest energy mc =5 kev, it is proper to include a small relativistic correction. From E = p + m we obtain p = (5 + ) 5 =.59 kev/c instead of: p = mt = 5 =.9 kev/c The relativistic correction increases p by about /% and decreases λ by the same amount. This correction should be included in calculations. In principle, electrons lose some energy (ionization energy loss) in the films, but this effect is not observable. Also, the electron energy loss in the aluminum (face centered cubic structure) does heat the foil above room temperature, causing some linear expansion. If your measured lattice size is slightly larger than the (room temperature) reference value, compute the implied temperature rise. Apparatus The Electron Diffraction equipment has:. An on-off switch. Ahigh voltage control and panel meter ( - kv). For greater precision the high voltage is read with a digital voltmeter and a precision voltage divider. The meter reads the high voltage on the volt scale where V corresponds to kv

2 EKALABORATORY Electron Diffraction 3. An intensity control which varies the electron beam current. This should initially be set to minimum before turning the power voltage on. There is (on top) a meter for reading the electron current. The current voltage shall never be allowedto exceed (kv) (µa). This precaution is absolutely necessary to extend the tube phosphor life. Keep the current always below µa!!!! See below. 4. Vertical and horizontal position adjust knobs and a focus control. The focus adjustment may change the electron current which should be checked frequently and readjusted as needed. During warm-up the current drifts upward and requires frequent adjustment. There are four small targets mounted internally on a plate at a distance D=78. mm from the phosphor. Examine the interior of the cathode ray tube assembly by looking through the transparent side window. When the electron beam is moved upward out of view, a sharp electron shadow pattern of the target structure is projected onto the CRT tube screen (for obscure reasons). This is useful for observing its structure. To obtain optimum patterns it is necessary to perform a search using the vertical and horizontal position controls. The pattern for the graphite usually shows contributions from more than one micro crystal, giving a complex pattern. Asearch should yield an essentially single crystal pattern, as shown in the illustration, giving a hexagonal dot array. In general seek a pattern having the rings or spots as bright as possible relative to the central spot, and try to have the pattern center near the center of the screen. In order to minimize the use of the (irreplaceable) tube at high intensities, the patterns are photographed with a Polaroid camera and measurements made on the photographs. The magnification of the camera can be determined by photographing a plastic ruler on white paper. Procedure. Taking Photographs of Ring Pattern from Aluminum. With the accelerating potential at kv, adjust the horizontal and vertical positions of the beam until you have a clear ring pattern with at least five or six rings. Cover the bright central spot with a black disk, and adjust the focus and intensity control until the outer rings are visible. Take a single photograph with the intensity control turned up, but never with a current greater then µaand this only for the few necessary seconds.. Photograph of Spot Pattern from Graphite. With the accelerating potential at kv, adjust the beam until you obtain a hexagonal spot pattern from a single graphite microcrystal. The graphite is deposited in such a way that the hexagonal-pattern layers of atoms are nearly perpendicular to the electron beam. For most positions of the electron beam, patterns from many microcrystals give a complex spot pattern on the screen; near the edge of the graphite, one may find a single microcrystal and obtain a simpler hexagonal spot pattern. As in part, take a single photograph with the intensity control is turned up for a brief time. 3. Verifying the Dependence of λ on the Accelerating Potential. With the accelerating potential V, as low as possible to detect a ring pattern from aluminum, measure the diameters of two or three prominent rings directly on the face of the tube. Then raise the accelerating potential in steps of kv until you reach the

3 EKALABORATORY Electron Diffraction 3 maximum of kv, each time measuring the diameters of the same rings. Note that the beam current will rise as you increase the accelerating potential, so it is necessary to readjust the intensity control in order not to overload the tube. Theory of Crystalline Diffraction Aplane wave of momentum p = hk has propagation vector k radians/cm in the wave direction. The wave behavior (phase vs. r) is given by exp(ik r ) exp(iωt). The scattered wave in the k direction is exp(ik r). When one scattering atom is at the origin O and another displaced by r at P, the relative phase for scattering from P relative to scattering from O is (k k ) r, as indicated below. O O k r r ˆk r ˆk r P P k Incomming wave Outgoing wave The extra path lengths for point P with respect to O are ˆk r and ˆk r. The total phase difference is (k k ) r. One refers to the vector momentum change, in various notations as k = q = k k. If scattering atom j has scattering amplitude f j (q) for scattering through q, the combined total scattered amplitude through q is f(q) = j f j(q) exp(iq r j ), with k =π/λ. For elastic scattering through θ, simple geometry shows that q =ksin θ =(4π/λ) sin θ. θ k q θ k q = k sinθ if k = k

4 EKALABORATORY Electron Diffraction 4 For x-rays the scattering is due to the electrons, so f j (q) for an atom gives the Fourier transform of the electron distribution in the atom (in 3-dimension) and is used to measure that distribution. For electron scattering f(q) is mainly the coulomb (Rutherford) scattering by the nucleus, with an angular dependence as /θ (more precisely as /(θ +θ ) where θ comes from electron shielding of the nucleus for far collisions when λ nuclear size). For a lattice, if the f j (q) are all equal to a common f c (q), the expression j exp( iq r j) is the Fourier transform of the three dimensional lattice structure. This gives relatively large contributions for particular values of q (relative to the lattice orientation) and essential cancellation for other q values. It is the same as the theory for light scattering (diffraction) from a on- or two-dimensional grating. The lattice of points in q space is called the reciprocal lattice. Acrystalline lattice of atoms is a repeating periodic structure. The simplest case is a simple cubic lattice where the elemental cube spacing or nearest neighbor atoms is given by displacement vectors r = ±a, ±b or ±c, from a given atom. Here a, b, and c are mutually orthogonal lattice displacement vectors (all of equal length a in the cubic case). Ageneral lattice point is at r j = n a,j a + n b,j b + n c,j c, where the n s are integers. The diffraction condition equivalent to st, nd etc. order diffraction of light from a grating is that q r j is an integer number of cycles, so all atoms contribute in phase and the net amplitude is the sum of all the atoms in phase. This requires that q a =πh, q b =πl and q c = πm, where h, l and m are integers. The normal to q is a plane which is usually tilted with respect to the planes containing a and b, b and c, or c and a. The reciprocal lattice points are at q a = πh/a, q b = πl/b and q c = πm/c, where a = b = c in this case, and h, l, m assume all positive and negative integer values and zero. h, l, m are called Miller indices. The reciprocal lattice corresponding to a simple cubic lattice is itself an infinite cubic lattice having the same spatial axis directions as the crystal lattice. For a more complicated cubic lattice, such as the face centered cubic lattice (f.c.c.) of aluminum and many other materials, the unit cell is still most conveniently taken as a cube of sides a, i.e. r = a, b, c, a + b, a + c, b + c, a + b + c, which in addition to r= define the eight cube corners. The six face center sites are at r =(a + b)/, (a + c)/, (b + c)/, a + b/+c/, b + a/+c/ and c + a/+b/. A translation (shift) vector a, b or c shifts to a new origin from which the lattice looks the same. These shift vectors define the lattice. The crystal structure is composed of lattice plus basis. The basis consists of the subset of locations which must be specified so that all possible lattice shifts n a a + n b b + n c c sweep out all atoms with n a, n b, n c... In the simple cubic lattice, the basis consists of a single corner atom. For the f.c.c. lattice, the basis vectors are, a/+b/, a/+c/ and b/+c/ or four atoms per cell. When considering diffraction in this case we still require that q r =πn for all lattice shifts n a a + n b b + n c c. Thus all cells contribute identically to the net f(q). If there are N cells, each has the net f(q) =f cell (q), where f cell (q) = basis exp(iq r)

5 EKALABORATORY Electron Diffraction 5 For the f.c.c. basis f cell (q) = + exp( i(l + m)π) + exp( i(h + l)π) + exp( i(h + m)π) For l + m, h + l or h + m odd, the corresponding exponential is.forevenitis+.a quick test shows that f cell (q) = unless h, l, m are either all odd, or all even (verify this!). For the resulting non-zero qlattice points, the magnitude of qis q π = sin θ λ = h + l + m a Case of non orthogonal a, b, c. The graphite crystal structure is as described in the figure below. The crystal has a layered structure. The layers contain attached hexagonal rings which are flat and weakly bound to adjacent layers. We are only concerned with diffraction associated with carbon spacings within layers and not with information associated with the layer spacing. The two dimensional unit cell contains two carbon atoms. The lattice displacement vectors a and b have 6 (or a multiple of 6 depending on arbitrary choice) between them and lengths of the spacing of next nearest neighbor in a hexagonal array. The second atom within the cell is at /3(a + b). We can consider c as a vector normal to the plane. In normal graphite it has two plane spacings, since adjacent planes have displaced symmetries while every other plane is generated by a plane displacement through c. We deal with only the case q c, so c is not important. Many layers can contribute in phase for the Laue spots.

6 EKALABORATORY Electron Diffraction 6 b a 3 ( a + b) The graphite structure, looking along the hexagonal axis. Heights of atoms are shown, as multiples of c. The atoms in the upper plane, at height c /, are shown with heavy lines, and are connected by a hexagonal framework, to make them easier to see. Note : a = a aˆ and b= a bˆ. When a, b, c (cell translation vectors) are not equal in length or mutually perpendicular, we still require that q a =πh, q b =πl and q c =πm. In this case we cannot simply choose q as made up of πhâ/a, πlˆb/b, πmĉ/c, since the components are no longer orthogonal. Instead we use a reciprocal vector set A, B, C chosen so A b = A c = B a = B c = C a = C b = A a B b C c

7 EKALABORATORY Electron Diffraction 7 Kittel sets the non-zero products equal to π using A =π b c V, B =π c a V, C =π a b V where V a b c = 3 a c is the lattice cell volume for graphite. (This is the formula for a hexagonal close packed structure. For the ideal hexagonal close packed structure c =.633 a. Magnesium is close to ideal with c =.63 a. Graphite is far from ideal with c =.7 a.) Then q = ha+lb+mc and each h, l, m set defines a point in the reciprocal lattice space. Note that m = or nearly zero so the phase spread of the different layers is π. Ais perpendicular to band cetc., A B, etc. are non-zero, so it is generally a complicated problem to evaluate q. Let α = angle between a and b c = A β = angle between a and c a = B γ = angle between a and a b = C Then A, B and C have lengths π a cos α, π π b cos β, c cos γ. For the graphite case, a = b = a, and cis along a b, therefore cos α = cos β = 3/ and cos γ =. The figure illustrates the geometry. We choose m =,soq = ha + lb and q (π) = 4 h + l + m 3 a. Since the reciprocal lattice is defined by all h, l sets of integers, it is also a twodimensional hexagonal lattice with hexagonal directions parallel to those of the space lattice. The fact that the reciprocal lattice is a hexagonal lattice of close packed equilateral triangles is seen from the diffraction pattern. Along a line of nearest neighbors in this pattern, the nearest neighbor spacing is constant as long as the small angle approximation sin θ = tan θ applies. To achieve accuracy, the spacing of points spaced along a line through the pattern center should be measured, where the line connecting them is along one of the three directions parallel to triangle bases passing through or near to the center point. The points should be four or more base lengths apart and equidistant from the center on opposite sides. Then a = 3 λd spacing Several measurements should be made to obtain accuracy and an uncertainty estimate for each triangle base spacing. Discussion of RelatedPhysics of Interest For forward scattering, h = l = m =, the contributions from all the atoms are in phase with each other, but usually 9 out of phase with the incident plane wave. The sum of the incident and forward scattered (or diffracted) component gives a phase advance or delay per cm. This is the index of refraction effect, familiar from optics. The index n is the ratio of the phase velocity in vacuum to that in the medium. Electrons with binding

8 EKALABORATORY Electron Diffraction 8 energies smaller then the photon energy, give n< contributions, while large binding energy electrons give n> contributions (the usual case for visible light in transparent materials). For x-rays the net n tends to be smaller than, which makes possible the phenomenon of total external reflection of x-rays near grazing incidence from the (air) side. Similar n< effects occur for slow neutrons in most materials. When diffraction is considered, the value of λ should be taken as that in the medium. Usually, for x-rays and slow neutrons n. Analysis of the Aluminum Data It is assumed that the electron beam is moving perpendicular to the screen before scattering through an angle of θ to form a ring of diameter d. Since the theoretical interpretation involves sin θ, it is convenient to convert the measured diameter, d, of each ring in the aluminum data to the corresponding value of sin θ: [ sin θ = sin d ] tan D The first ring has h = l = m =, and succeeding rings have only all even or all odd values of h, l and m, since aluminum is a face centered cubic crystal. Since sin θ is proportional to h + l + m, you should check the assignment of h, l, m to each measured ring by calculating sin θ/ h + l + m. (In practice, some of the outer rings are not measurable and some are double within the resolution.) If there is a question about assignment, rely on the first three rings ( 3, 4, 8, ) to establish the calculated ratio, and assign indices to the outer rings for the best agreement. From the above data, calculate the unit cell size, a, for aluminum. Compare this result with that which you can calculate from handbook values of the atomic weight, the metal density, and Avogadro s number. Analysis of the Graphite Data For the graphite pattern array of equilateral triangles, the most precise measurement of the common triangle side length can be made by measuring the distance between spots separated by several base lengths. Use the result of several such measurements to calculate the unit cell size for graphite. Analysis of the Dependence of λ of V From the data taken on the aluminum rings at different values of the accelerating potential V, verify the dependence of d, and thus λ, onv. Calculation of the lattice constant a, for aluminum

9 EKALABORATORY Electron Diffraction 9 a The number of atoms, n, associated with a unit cell for a face-centered cubic lattice (the case of aluminum), can be found from the following reasoning. Each of the atoms at a corner as belongs to the 8 unit cells for which it is a common corner point. We can say, therefore, that /8 of it belongs to our unit cell Since there are 8 of these corner points, together they furnish atom to our unit cell. If of the remaining atoms, as in the center of a face, each belongs to two unit cells, and thus / of each belongs to our unit cell. Since there are 6 faces, these atoms furnish, in all, 3 atoms to our unit cell, thus n =4. The mass of the material in a unit cell can be expressed in two ways. ) m = a 3 ρ and ) m = n A N where ρ is the density of aluminum A is the atomic weight of aluminum and N is Avogadro s number. (See Semat, 4th Ed., pp 53 and 577.) Equating the two relations gives a. For a check, see Semat, p.

Electron Diffraction

$Electron Diffraction$ Exp-3-Electron Diffraction.doc (TJR) Physics Department, University of Windsor Introduction 64-311 Laboratory Experiment 3 Electron Diffraction In 1924 de Broglie predicted that the wavelength of matter