Dynamical Theory of X-Ray Diffraction

After Batterman & Cole

1.1 Introduction

The fundamental problem of the dynamical theory of x-ray diffraction as pointed out by von Laue is to solve Maxwell’s equations (classical electromagnetic wave theory) in a medium with a periodic complex dielectric constant (k). This constant is simply the multiplying factor by which capacitance is changed when dielectric materials (those that are capable of sustaining an electrical stress in which a steady electric field can be set up without causing an appreciable flow of current) are used between capacitor plates, it is given by k = 1+ c where c is the electric susceptibility of the material (c = e - 1, e is the relative permittivity measured in Farads m^-1 e = D / E).

The first step is to write down Maxwell’s equations using k to describe the medium, assuming wave solutions that are consistent with Bragg’s law, in order to obtain a set of homogeneous linear equations for the ratio of the field amplitudes. The next step is to write down a determinant whose value must be zero for nontrivial solutions of the equations to exist; this imposes certain conditions on the wave vectors. The loci of the tips of these permitted wave vectors in reciprocal space defines a construct called the dispersion surface. Using notions developed by Ewald the properties of this surface in reciprocal space can be understood and the results of the dynamical theory of diffraction can be generated.

On a fundamental level if the crystal can be represented by a periodic, anisotropic complex dielectric constant which is time dependent, then k as it is 3D periodic can be represented by a Fourier series over reciprocal space in much the same way as the charge density r(r) is. The Fourier coefficients of k are associated with those of r(r). A complex k is used to include the absorption, an anisotropic k allows waves propagating in different directions to have different refractive indices (this difference becomes the important factor in the problem).

1.2 The periodic, complex dielectric constant

The electron density at any point in a crystal r(r) can be written as a Fourier sum over the reciprocal lattice:

r(r) = (1 /V) S F_H exp (-2pi H.r) 1.1

^H

Where V is the volume of the unit cell, H is a reciprocal lattice vector (H = hx + ky + lz, where x, y, z are lattice vectors defining the unit cell in reciprocal space and h, k, l are the Miller indices of the reflection described by the reciprocal lattice point). This can also be written in terms of F_H the structure factor:

F_H = ∫_v r(r) exp (2pi H.r) dv 1.2

Assuming the atoms are rigid spheres and not vibrating thermally and the atomic scattering factor f_n can be introduced, then 1.2 becomes:

F_H = S f_n exp (+2pi H.r_n) 1.3

ⁿ

Where the sum is over n atoms in the unit cell. The connection between the Fourier series describing the electron density r(r) in equation 1.1 and that of the periodic dielectric constant k can be made by considering a dimensional analysis of the electric polarization.

Now the Polarization P (defined as the dipole moment per unit volume: charge density x displacement vector) can be written in terms of the electric field E (Nm^-1 or V^-1) as:

P = e_o c E 1.4

The susceptibility c is a property that is characteristic of each kind of matter and proportional to the ease of polarizing it. Now the electric displacement D can be written in terms of the dielectric constant:

D = k e_o E 1.5

Remembering that k = 1+ c

D = e_o E + P 1.6

Or alternatively

k = 1 + P /e_o E 1.7

Now consider a sinusoidal wave of amplitude E_o acting on a collection of electrons held by restoring forces such that they have a natural frequency w_o of oscillation then the amplitude of the induced electron motion is:

x = (e/m) E_o / (w_o² - w²) 1.8

This motion is simple harmonic i.e. d²x /dt² = -w² x, since d²x /dt² = (e/m) E _o (when eE/m = F/m = a = d²x /dt²). For S.H.M. the periodic time T = 2p/ w and remembering c = f l, then w = 2p c /l). If w is much greater than w_o, E ~ E_o (the polarization can be expressed in terms of the electron density P = rex) and k in 1.7 can then be rewritten, with x given by the expression1.8, as:

k(r) = 1 – [(e²/mc²) l² / 4p²e_o] r(r) 1.9

The quantity (e²/4pe_o mc²) is the classical electron radius r_e = 2.818 x 10^-15 m, we can write (irrespective of the system of units):

k(r) = 1 – r_e (l² / p) r(r) 1.20

If we define the symbol G:

G = r_el² / p V 1.21

Then from 1.1 for r(r)

k(r) = 1 - G S F_H exp (-2pi H.r) 1.22

^H

Of course if the x-ray frequency is near a resonant frequency for the electrons and absorption is present then the expression for polarization is more complicated than that for SHM given in equation 1.8. Usually k is taken to be complex including correction terms (Hönl) to the atomic scattering factor due to resonance and absorption:

F_H = S (f + Df ^¢ + iDf ^¢¢ )_n exp (+2pi H.r_n) 1.23

ⁿ

So that F_H can be written as the sum of a real part (f + Df ^¢) an imaginary part Df ^¢¢:

F_H = F_H^¢ + i F_H^¢¢ 1.24

Even so F_H^¢ and F_H^¢¢ may still be complex quantities due to the arrangement spatially of the atoms or the choice of origin of the unit cell. Considering values of hkl = 000 for which the term exp(-2pi H.r) Þ 0, in the Fourier series for k

k_o = 1 - G [F_o^¢ + i F_o^¢¢] 1.25

So that combining equations 1.22 and 1.24:

k_o = 1 - G S (f _o + Df _o^¢ )_n - iG S (Df _o^¢¢ )_n 1.26

_{n n}

For an x-ray beam traversing a slab of material without diffracting the linear absorption coefficient is related to the imaginary part of the dielectric constant:

m_o= (2p/l) G F_o^¢¢ 1.27

On dimensional grounds G º [(r_el² / p V) = (m³/ m³) dimensionless] and m_o \has the units m^-1. Using measured values of m_o and l a value can be estimated for G F_o^¢¢. For CuK_a radiation passing through Germanium m_o = 350 m^-1 , l = 1.54 Å So that

G F_o^¢¢ =0.86 x 10^-6 (a pure number). From these considerations, since k ~ (1- G H), that the dielectric constant differs only slightly from unity we can assume that k ~ 1 in the following argument.

1.3 Waves which satisfy Bragg’s Law and Maxwell’s Equations

For a start assume that the electrical conductivity s is zero (j = 0 since j = ds /dt ) at x-ray frequencies so that there is no resistive heat loss and that the crystal has the same magnetic behavior as empty space so that m = m_o then Maxwell’s equations can be written in the form:

Ñ x E = -¶B / ¶t = -m_o ¶ℋ / ¶t 1.28

Remembering that the magnetic flux density (Wb m^-2) B = m_o ℋ (magnetic field strength). The vector differential operator Ñ is defined as i ¶ / ¶x + j ¶ / ¶y + k ¶ / ¶z (the vector product Ñ x E = curl E while the scalar product Ñ . E = div E). Equation 1.28 is just an expression of Faraday’s law of induction V = - dF/dt. The next Maxwell equations are a consequence of the inverse square law of force applied to electric & magnetic fields (div E = (1/e_o)r & div B = 0). The fourth law results from the converse of the Faraday law that a varying electric field gives rise to a magnetic field, a general formalism for the production of electromagnetic waves by a displacement current:

Ñ x ℋ = (¶D / ¶t) + j 1.29

Substituting D = k e_o E from 1.5 and with j = 0, this reduces to:

Ñ x ℋ = e_o ¶ (kE) / ¶t 1.30

The fields E, D and ℋ can be expressed as sums of plane waves and if a wave with wave vector K_o ( |K _o| =1/l ) is scattered by the Fourier components of charge density with periodicity H ( |H | = reciprocal d spacing) then the wave vector of the scattered wave is

K _H = K _o + H 1.31

This relationship between the two wave vectors is a statement of Bragg’s law or thought of as the conservation of momentum for the wave scattering. Only waves coupled that satisfy 1.31 are of interest to us, and to include absorption a complex component is introduced:

K = K^¢ - i H^¢¢ 1.32

The substitution of plane wave expressions into Maxwell’s equations 1.28, 1.30 with use of the equation for the dielectric constant 1.22 and wave vector relation 1.31 leads to the fundamental set of equations for the wave fields inside a crystal. The field vectors can be written in the general form (where A can be replaced by any of the electrical quantities E, D, ℋ):

A = exp (2pint) SA_H exp (-2pi H_H.r) 1.33

^H

If we combine this expression with 1.30:

A = [ SA_H exp (-2pi H.r)] exp (-2pi K_o.r) exp (2pint) 1.34

^H

This wave expression (any wave can be written as: position term x time term) with a wave vector K_o has an amplitude which can be expressed as a Fourier series. Taking the curl of 1.32 & expressing it in terms of E:

Ñ x E = - (2pi) exp (2pint) S(K_H x E_H) x exp (-2pi K_H.r) 1.35

^H

Now taking the time derivative of 1.32, expressing the result in terms of the magnetic field strength H:

¶ℋ / ¶t = (2pi) n exp (2pint) Sℋ_H exp (-2pi K_H.r) 1.36

^H

Putting 1.35 & 1.36 into Maxwell’s equation 1.28 gives an equality for each term in the summation (Fourier series component)

K_H x E_H = nm_o H_H 1.37

Likewise for the other Maxwell equation 1.29 with the correct form:

K_H x H_H = - n D_H 1.38

Now H_H is perpendicular to K_H from eq. 1.37 and 1.38 implies that K_H , H_H and D_H form a mutually orthogonal set. We can now use the relation 1.5 for k D = k e_o E and the field vector relation 1.33 for the form for D & E:

S D_H exp (-2pi K_H.r) = e_o [1- G S F_H’ exp (-2pi H^¢.r)] S E_H exp (-2pi K_H.r) 1.39

^{H H’ H}

The index H^¢ is used to distinguish it from H. Changing indices and using the fact that K_H’ + H = K_H’+H from 1.31 the RHS of 1.39 can be rewritten:

SD_H exp(-2pi K_H.r) = e_oSE_H exp(-2pi K_H.r) - e_oGS(SF_H-P E_P) exp(-2pi K_H.r) 1.40

^{H P}

Accepting the infinite double sum. This must hold for each Fourier component, eliminating the exp(-2pi K_H.r) term from each side:

D_H = e_oE_H - e_oGSF_H-P E_P 1.41

^P

When P=H the sum S Þ F_o E_H for the first term:

D_H = e_o(1 - G F_o) E_H SF_H-P E_P 1.42

^P¹H

D_H is thus predominantly e_ok_oE_H since k_o = (1- G F_o), with small contributions from other Fourier components of the electric field. Taking the cross product of K_H with each side of 1.37 and substituting in 1.38:

K_H x (K_H x E_H) = nm_o (K_H x H_H) = nm_o (-n D_H) = -n²m_o D_H 1.43

Substituting for D_H from 1.42:

K_H x (K_H x E_H) = -n²m_oe_o(E_H- GSF_H-P E_P) 1.44

^P

Now using m_oe_o = 1 / c² and n² / c² = k² 1.44 can be written:

K_H x (K_H x E_H) + k² E_H - k² G(SF_H-P E_P) = 0 1.45

^P

Now using the vector identity for a triple crossproduct:

K_H x (K_H x E_H) = - (K_H.K_H) E_H + (K_H.E_H) K_H 1.46

And allowing 1.45 to be written like 1.42 this expression becomes:

[k²(1- G F_o) - (K_H.K_H)] E_H - k² G(SF_H-P E_P) + (K_H.E_H) K_H = 0 1.47

^P¹H

This is the fundamental set of equations describing the field inside the crystal. For each amplitude E_H there is a complex vector equation, these must hold for the real & imaginary parts & each component separately. We only consider the case where one reciprocal lattice point is near enough to the Ewald sphere to give any appreciable diffraction i.e. only allow the field amplitudes E_o & E_H with corresponding wave vectors K_o & K_H (these define the plane of incidence). Considering two states of polarization s normal & p parallel to the plane. Eq. 1.47 then becomes:

s case

[k²(1- G F_o) - (K_o.K_o)] E_o - k² G F_H E_H = 0 1.48

[k²(1- G F_o) - (K_H.K_H)] E_H - k² G F_H E_o = 0

Ignoring the term + (K_H.E_H) K_H

This pair of equations has a nontrivial solution for the ratio E_H / E_o only if its determinant is zero.

p case

The field vectors E_H & E_o each have two components in the plane of incidence: one along their respective wave vectors (longitudinal) and another at right angles. The longitudinal are negligible considering the following, D_H is dotted in 1.47 & 1.42 substituted :

[k²(1- G F_o) - (K_H.K_H)] E_H - k² G SF_H-P E_P = 0 1.49

^P¹H

Dividing by k² multiplying by e_o & collecting terms :

e_o [(1- G F_o) - G SF_H-P E_P ] = e_o E_H (K_H.K_H) / k² 1.50

^P¹H

D_H . D_H = e_o E_H . D_H (K_H.K_H) 1.51

k²

In 1.27 the dielectric constant (k = D / e_o E ) is almost 1 (differing by only ~ 10^-6) so that (K_H.K_H) / k² is about the same order of magnitude & likewise the component of e_o E_H not along D_H is of this order. Now dot K_H in 1.47:

K_H. E_H (1- G F_o) = G SF_H-P (K_H. E_P) 1.52

^P¹H

Again neglecting components of E_H along K_H. From the discussion above & in 1.3 since div E µ div D, Ñ.D = 0 rather than Ñ.D = r(r). The components of E then lie in the plane of incidence & are perpendicular to their respective wave vectors. Taking the angle between the directions of K_o and K_H to be 2q we can now write a pair of equations like 1.48 for the p components:

[k²(1- G F_o) - (K_o.K_o)] E_o - k² (cos 2q) G F_H E_H = 0 1.53

[k²(1- G F_o) - (K_H.K_H)] E_H - k² (cos 2q) G F_H E_o = 0

Letting P =1 or cos 2q

The determinant of 1.48 or 1.53 can be written as

| k²(1- G F_o) - K_o.K_o - k² P G F_H | 1.54

| - k² P G F_H k²(1- G F_o) - K_H.K_H |

Which when set to zero gives the permitted wave vectors.