The interference function calculates the scattering effect o f the array o f unit cells that form a crystal. For example consider a one-dimensional array o f cubic unit cells. If there are N such unit cells and the phase difference between the waves diffracted by adjacent cells is F, then the scattered amplitude for entire array of cells is:
A = F + Fe'^ + Fe*2^ + Fe*2^ ... + Fe^N-l)^ (5)
reducing this to the geometrical series produces:
A = F (1 - e iN<t>)/(l - e ^ ) (6)
the actual scattered intensity is I = A*A, where A* is the complex conjugate o f A. The formula for the intensity is thus:
/ = i f f
( l - e w» ) ( l - e ' w*)( l - e * ) ( l —e- ")
(7)
replacing this with the equivalent trigonometric terms this reduces to:
/ = i f f ^1 - cos(/V</>)] [ l - c o s ( 0 ) j
and using the relation 2sin2x = l-cos(2x) this becomes
j _ .^ 2 sin2(A/r0 / 2 ) sin2(0 / 2)
(8)
(9)
when (() is near a multiple o f 2tc, which w ill occur near a Bragg reflection, then <J)=27c(n+l) can be substituted:
/ = |F|2 shr(AW)
A^sin“(7r/) (10)
The trigonometric part o f equation 10 is the interference function for the (001) series. A factor N has been added to the denominator to normalise the peak area to a constant value.
The complete formula for intensity requires three interference functions, one for h, k and 1:
_ Lp(0)\F(hkl)\2 sin2{Nnh) sin\ N n k ) s m fN n l) N\ N2 N2 sin2(/r/z) sin2(/r&) sin2(/r/)
(11)
This equation contains a new factor: the Lorentz polarization term Lp, which includes the polarization effect, a result of the difference in the direction of the electric vector in the incoming x-rays and the direction of scattering, and the Lorentz or angular-velocity factor. Here, the entire Lp correction is:
1 + cos2 26
sin # sin 2 0 (12)
A post diffraction monochromator, employed in the experimetntal arrangement used, adds another polarization term (6) o f the form p = (l+ k C o s22 0 )/(l+ k ), where k = cos220m for a mosaic monochromator with set reflection angle 20m.
0 1 2 3 4 5 6
I
Figur e 2.6: Diagram showing the effect o f crystallite size N on the 1-D interference <t> as a function o f l. As N increases, the peaks become much higher and narrower. (After Reynolds
Figure 2.6 plots the ID interference function O from equation (10) as a function of /, for N=3, 10 and 20. As N increases, the peaks of the function become narrower and more intense, so that at very high values of N the effect of the interference function on the peak shape is negligible, and the shape of diffracted peaks is dominated by instrumental effects. At low values of N, the function produces broad peaks with significant intensity
at non-integral values of /; at N =l, representing an isolated unit cell, the function O = 1 at all values of /, and Bragg diffraction will not take place. As explained above, the ab-plane structures studied in this work are either uncorrelated or weakly correlated along the c- axis; effectively these structures are only one layer thick and thus N3= l for these structures. To simulate the diffraction pattern requires that the calculation of the interference function with non-integral values of h, k and /. The approach used here, taken from Reynolds (1989), uses a reciprocal space computation.
2 . 5 Diffraction Intensities
The scattering of x-rays by atoms is a function of several processes, primarily coherent (Rayleigh) scattering and incoherent (Compton) scattering as well as thermal, absorption and dispersion effects. Coherent scattering is the result of the interaction of an x-ray with a tightly bound electron, such that the scattering is an elastic process. The intensity of coherent scattering from a single electron of an unpolarized incident beam of photons in a direction 0 relative to the incident beam is:
K
=' X
1 + cos2 20
(13)
where re is the classical radius of the electron and IQ is the intensity of the incident b e a n r 7\ Coherent scattering induces a phase shift of K in the scattered photon. In incoherent scattering the electron is considered to be effectively free, with the result that there is momentum transferred between the photon and the electron, and change in wavelength for the scattered photon, of the form:
a t 2 /i . 2
A/l = — sin“ 0
me (14)
The formula for incoherent scattering from a single electron is that same as ( 1 3 )^ , but with additional correction factor [A/(A,+A?i)P for proportional counters or [A/(A,+AA,)]2 for scintillation counters^.
Generalised to the scattering from an atom of atomic number Z, the coherent and incoherent scattering intensities are:
^coh ^e
h,
7 incoh ^e J k
j ’k
m
where fj =
J
p(rj) exp(iQ rj) drj, fjk = 1 Vj*Vk ex p(iQ rj) drj, Q= ko-k(7).If all the electrons in an atom were concentrated at a point the scattering factor would be the Fourier transform of a delta function, and would thus be a constant and independent of 0. This is the case for thermal neutron scattering by a nucleus, but the electrons of an atom are distributed over space, and produce a scattering factor which is a function of 0. The scattering intensities of the neutral atoms and the more chemically important ions have been calculated by several techniques and are tabulated in the International Tables for Crystallography*7^. Interpolation to these calculated intensities allows an analytical approximation to be used:
f(sinO / A) = ^ a i exp(~bi sin2 6 / X2) + c
/ = i (17)
where the coefficients aj, bj and c are again collected in the International Tables. This analytical approximation was used to calculate the scattering intensities in the simulations that follow. Though they are known to provide a poor fit to the atomic scattering curves outside the range 0 < sin0/A < 2 Ä '1, the diffraction data was taken well within this range.
35 H
o 25 -
20 (Degrees)
Figure 2.7: the calculated scattering factors using the analytic approximation^^ fo r the elements that compose the compounds studied in this work.
The GICs that were studied in this work are composed of covalently bonded carbon atoms, and rubidium and potassium ions, while the physisorbed ternary species contained argon, nitrogen, carbon and hydrogen atoms. The calculated scattering intensities (in units of the scattering factor of a single electron) of these atoms and ions, over the same range in 20 that the experimental data covered, is shown in Figure 2.7. These curves are the atomic form factors and reflect not only the number of electrons but the atomic size. The form factors are the Fourier transform of the atomic or ionic electron distribution.
In addition to coherent and incoherent scattering, the scattering intensity of an atom is modified by other terms such as the displacement factor and dispersion correction. While the dispersion correction, which adds an imaginary component to the atomic form factors, is sufficiently small that it can be ignored in these simulations, a displacement or temperature correction must be used. The tem perature correction accounts for the vibration of an atom around its mean position; ideally six variables are needed to account for anisotropic harmonic thermal vibrations in three dimensions, but here a simple isotropic Debye-Waller factor of the form:
f = fo e x p (-ß s in 2 6 / A2) (18)
is included, where B is a single refinable variable of dimensions Ä2. Figure 2.8 shows the effect of a temperature factor B = 2 Ä2 on the scattering factors calculated for Figure 2.7. For a simple Bravais lattice of atoms the diffraction intensities will be proportional to the squares of the scattering factors at the allowed 20 positions for Bragg diffraction.
35 H
ü) 30
20 (Degrees)
Figure 2.8: The scattering factors shown in Figure 2.7 with a temperature correction B -2 .0 A 2 applied to the calculations. Note the large effect at high 26.