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Solid State Physics 2. X-ray Diffraction
Diffraction using Light Diffraction Grating One Slit Two Slits http://physics.kenyon.edu/coolphys/FranklinMiller/protected/Diffdouble.html
Diffraction The diffraction pattern formed by an opaque disk consists of a small bright spot in the center of the dark shadow, circular bright fringes within the shadow, and concentric bright and dark fringes surrounding the shadow.
Diffraction for Crystals Photons Electrons Neutrons Diffraction techniques exploit the scattering of radiation from large numbers of sites. We will concentrate on scattering from atoms, groups of atoms and molecules, mainly in crystals. There are various diffraction techniques currently employed which result in diffraction patterns. These patterns are records of the diffracted beams produced.
Bragg Law William Lawrence Bragg 1980 - 1971
Mo 0.07 nm Cu 0.15 nm Co 0.18 nm Cr 0.23 nm
Bragg's Law can easily be derived by considering the conditions necessary to make the phases of the beams coincide when the incident angle is equal to the reflecting angle. The rays of the incident beam are always in phase and parallel up to the point at which the top beam strikes the top layer at atom z (Fig). The second beam continues to the next layer where it is scattered by atom B. The second beam must travel the extra distance AB + BC if the two beams are to continue traveling adjacent and parallel. This extra distance must be an integral (n) multiple of the wavelength (l) for the phases of the two beams to be the same: nl = AB +BC (2). Deriving Bragg’s Law
Recognizing d as the hypotenuse of the right triangle AbZ, we can use trigonometry to relate d and q to the distance (AB + BC). The distance AB is opposite q so, AB = d sinq (3). Because AB = BC and nl = AB +BC becomes, nl = 2AB (4) Substituting eq. (3) in eq. (4) we have, nl = 2 d sinq, (1) and Bragg's Law has been derived. The location of the surface does not change the derivation of Bragg's Law. Deriving Bragg’s Law
Bragg’s Law • There are a number of various setups for studying crystal structure using x-ray diffraction. In most cases, the wavelength is fixed, and the angle is varied to observe diffraction peaks corresponding to reflections from different crystallographic planes. • Using Bragg law, the distance between planes can be determined. Sharp x-ray diffraction peaks are observed only in the directions and wavelengths for which the x-rays scattered from all lattice points interferes constructively.
To find condition of constructive interference two scatterers separated by a lattice vector are considered. • X-rays are incident from infinity, along direction with wavelength and wavevector . • Assume the scattering is elastic i.e the x-rays are scattered in direction with same wavelength , so that the wavevector Diffraction Condition and Reciprocal Lattice . • The path difference between the x-ray scattered from the two atoms should be an integer number wavelengths. Therefore, the condition for constructive interference is . • Where m is an integer. Multiplying both sides of the above equation by 2π/λleads to a condition on the incident and scattered wave vectors
Diffraction Condition and Reciprocal Lattice Defining the scattering wave vector , the diffraction condition can be written as Where is, a vector where A set of vectors satisfying this condition form a reciprocal lattice. Vectors are called reciprocal lattice vectors. A reciprocal lattice is defined with reference to a particular Bravais lattice which is determined by a set of lattice vectors . The Bravais lattice that determines a particular reciprocal lattice is referred to as the direct lattice, when viewed in relation to its reciprocal. .
Scattered Wave Amplitude Fourier Analysis → where T → Define then mi integers bi is called the primitive vectors of the reciprocal lattice, and G a reciprocal lattice vector. i,j,k cyclic
Bragg’s Law using Reciprocal Lattice Vector In elastic scattering the photon energy is conserved, so the magnitudes of k and k’ are equal, therefore k2 =k’2. There from This another way of stating Bragg’s Law.
To Show: Reciprocal Lattice Vector is Orthogonal to the plane (hkl) Definition of Miller Indices If G is normal to the plane (hkl) Therefore G is orthogonal to the plane (hkl).
To show: The distance between two adjacent parallel planes of direct lattice is d = 2π/G Note: The nearest plane which is parallel to (hkl) goes through the origin of the Cartesian coordinates. Therefore the interplanar distance is given by the projection of one of the vectors xa1, ya2, za3 to the direction normal to the (hkl) plane. The direction is given by unit vector G/G, since G is normal to the plane. Therefore
To show: is equivalent to Bragg Law From : This angle is opposite to the angle between k and G. Therefore, k – same direction as X-ray beam G q Plane
To proof the relation between direct and reciprocal lattice is true To show that satisfies the equation
Reciprocal Lattices • Simple Cubic Lattice The reciprocal lattice is itself a simple cubic lattice with lattice constant 2/a.
Reciprocal Lattices • BCC Lattice The reciprocal lattice is represented by the primitive vectors of an FCC lattice.
Reciprocal Lattices • FCC Lattice The reciprocal lattice is represented by the primitive vectors of an BCC lattice.
Drawing Brillouin Zones Wigner–Seitz cell The BZ is the fundamental unit cell in the space defined by reciprocal lattice vectors.
Back to Diffraction Diffraction is related to the electron density. Therefore, we have a... Theorem The set of reciprocal lattice vectors determines the possible x-ray reflections.
For the diffracted wave, the phase difference is The difference in path length of the of the incident wave at the points O and r is The difference in phase angle is So, the total difference in phase angle is
Diffraction Conditions • Since the amplitude of the wave scattered from a volume element is proportional to the local electron density, the total amplitude in the direction k is
Diffraction Conditions G G G G G
Diffraction Conditions • When we introduce the Fourier components for the electron density as before, we get G G G Constructive Interference G G
Diffraction Conditions • For a crystal of N cells, we can write down G G SG=Structure factor= a measure of the scattering amplitude of a wave by a unit cell
Diffraction Conditions • The structure factor can now be written as integrals over s atoms of a cell. G G G G Atomic form factor Atomic form factor = a measure of the scattering amplitude of a wave by an isolated atom
Diffraction Conditions • Let • Then, for an given h k l reflection b1 b2 b3 G G
Diffraction Conditions • For a BCC lattice, the basis has identical atoms at and • The structure factor for this basis is • S is zero when the exponential is i × (odd integer) and S = 2f when h + k + l is even. • So, the diffraction pattern will not contain lines for (100), (300), (111), or (221).
Diffraction Conditions • For an FCC lattice, the basis has identical atoms at • The structure factor for this basis is • S= 4f when hkl are all even or all odd. • S= 0 when one of hkl is either even or odd.
KCl KBr
Structure Determination Simple Cubic When combined with the Bragg law:
X-ray powder pattern determined using Cu K radiation, = 1.542 Å
