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Path Integral X-Ray Diffraction Theory

Path integral X-Ray diffraction (PIXRD) theory describes the interaction of photons with a regular atomic lattice. PIXRD utilizes the general concept of the path integral formulation of quantum mechanics to describe Bragg diffraction profiles. The principles of theory are first described for a single spin direction which is sufficient to describe the magnitude of X-Ray diffraction (XRD) profiles. The principle of theory is then described considering both spin directions.

Principles of theory (Positive spin direction)
Path integral X-Ray diffraction considers the many paths that may be taken by a photon at a particular diffraction angle. Each path is represented by a phasor which undergoes an amount of rotation dependent on the lattice spacing, photon energy (wavelength), and angle of diffraction. The phasors are summed in a tip to tail fashion on the complex plane resulting in a total real and imaginary value. The probability amplitude is equal to the sum of the squares of the total real and imaginary components.

Phasor
The photon is represented by a rotating phasor on the complex plane. The photon wavelength may be considered an interpretation of the phasor rotation rate (L) according to the following equation.


 * $$L = \frac{c} {\lambda} 2 \pi [rad/s]$$

where λ is the wavelength and c is the speed of light.

Photons travel at the speed of light undergoing light-like intervals between interactions. The phasors are only considered to undergo rotation within the diffracting material. The path difference between two planes (Δp) is equal to:


 * $$\Delta p = 2 d^{\{ hkl \}} \sin{\theta}$$

where d is the {hkl} plane spacing, and θ is the Bragg angle.

The phasor time difference between planes (Δt) is equal to:


 * $$\Delta t = \frac{\Delta p} {c} [s]$$

The total amount of rotation a phasor undergoes (Δψ) for a given wavelength, d spacing and Bragg angle can be determined by multiplying the phasor rotation rate (L) by the time difference (Δt):


 * $$\Delta \psi = L \Delta t = \frac{4 \pi d^{\{ hkl \}} \sin{\theta}} {\lambda} [rad]$$

The real and imaginary components of the phasor can be determined from the phase angle.


 * $$Re = \cos{\left (n \Delta \psi +2 \theta \right )}$$
 * $$Im = \sin{\left (n \Delta \psi +2 \theta \right )}$$

where n is an integer number of planes. An additional phase rotation of 2θ is added to bring the real and imaginary components into agreement with Thomson scattering polarization.

Phasor Summation
The phasors are summed for each plane with the following solutions:


 * $$Re = \sum_{n=1}^{P} \cos{\left (n \Delta \psi + 2 \theta \right )} = \csc{ \left ( \frac{\Delta \psi} {2} \right ) } \sin{ \left ( \frac{P \Delta \psi} {2} \right ) } \cos{ \left ( \frac{P \Delta \psi + \Delta \psi + 2 \theta} {2} \right ) }$$


 * $$Im = \sum_{n=1}^{P} \sin{\left (n \Delta \psi + 2 \theta \right )} = \csc{ \left ( \frac{\Delta \psi} {2} \right ) } \sin{ \left ( \frac{P \Delta \psi} {2} \right ) } \sin{ \left ( \frac{P \Delta \psi + \Delta \psi + 2 \theta} {2} \right ) }$$

where P is an integer number of planes.

The number of planes may be determined by the size of the ordered (crystalline) domain (τ):


 * $$P = \frac{\tau} {d^{ \{ hkl \} }}$$

Probability Amplitude
[[Image:PIXRD 001.PNG|frame|alt=alt text|FIG 1 Example PIXRD profile.

Wavelength = 0.22897 nm (Cr Ka1), d{hkl} = 0.1165 nm (Fe BCC {211}), P = 100 planes]]

The probability amplitude (|α|2) is equal to the sum of the squares of the real and imaginary components with the following solution:


 * $$k|\alpha|^2 = Re^2 + Im^2$$
 * $$k|\alpha|^2 = \csc^2{ \left ( \frac{\Delta \psi} {2} \right ) } \sin^2{ \left ( \frac{P \Delta \psi } {2} \right ) }$$

where k is the proportionality constant.

The maximum probability amplitude is equal to the square of the number of planes:


 * $$Max. |\alpha^2| = P^2$$

This probability amplitude is sufficient to describe an x-ray diffraction profile assuming the following conditions:
 * No blurring effects such as beam divergence.
 * Single particle size contribution.
 * Monochromatic incident x-ray photons.
 * All phasors are of a single spin direction such as those found in K-alpha emission lines.

Principles of theory (Negative spin direction)
The real and imaginary summations are the same as those for a positive spin direction with the acception that the imaginary summation is negative. (See priniciples of theory for both spin directions below). The probability amplitude remains the same as the imaginary component is squared.

Phasor
The phasor may spin in one of two directions, clockwise (CW) or counterclockwise (CCW). The phase rotation is shown with a plus or minus sign to indicate direction:
 * $$CW = + \Delta \psi$$
 * $$CCW = - \Delta \psi$$

A phase offset value is assigned to each spin direction. The real and imaginary equations now become:
 * $$Re^+ = \cos{\left (+ \Delta \psi + 2 \theta \right )}$$
 * $$Im^+ = \sin{\left (+ \Delta \psi + 2 \theta \right )}$$
 * $$Re^- = \cos{\left (- \Delta \psi - 2 \theta \right )}$$
 * $$Im^- = \sin{\left (- \Delta \psi - 2 \theta \right )}$$

Phasor Summation
The phasor summations for each plane now become:
 * $$Re^+=\sum_{n=1}^P U \cos{\left (+n \Delta \psi + 2 \theta \right )} = U \csc{\left ( \frac{\Delta \psi} {2} \right )} \sin{\left ( \frac{P \Delta \psi } {2} \right )} \cos{\left ( \frac{P \Delta \psi + \Delta \psi + 2 \theta} {2} \right ) }$$
 * $$Im^+=\sum_{n=1}^P U \sin{\left (+n \Delta \psi + 2 \theta \right )} = U \csc{\left ( \frac{\Delta \psi} {2} \right )} \sin{\left ( \frac{P \Delta \psi } {2} \right )} \sin{\left ( \frac{P \Delta \psi + \Delta \psi + 2 \theta} {2} \right ) }$$
 * $$Re^-=\sum_{n=1}^P V \cos{\left (-n \Delta \psi - 2 \theta \right )} = V \csc{\left ( \frac{\Delta \psi} {2} \right )} \sin{\left ( \frac{P \Delta \psi } {2} \right )} \cos{\left ( \frac{P \Delta \psi + \Delta \psi + 2 \theta} {2} \right ) }$$
 * $$Im^-=\sum_{n=1}^P V \sin{\left (-n \Delta \psi + 2 \theta \right )} = -V \csc{\left ( \frac{\Delta \psi} {2} \right )} \sin{\left ( \frac{P \Delta \psi } {2} \right )} \sin{\left ( \frac{P \Delta \psi + \Delta \psi + 2 \theta} {2} \right ) }$$

where U:V is the CW:CCW intensity ratio ranging between 1:0 and 0:1.

Probability Amplitude
The probability amplutide (|α|2) is the sum of the squares of the real and imaginary components.


 * $$k|a|^2 = \csc^2{\left ( \frac{\Delta \psi} {2} \right )} \sin^2{\left ( \frac{P \Delta \psi} {2} \right )} (U^2 + V^2 + 2UV \cos{\left ( -P \Delta \psi - \Delta \psi \right )})$$

where k is the proportionality constant.

Electron Density Influence
The crystallographic planes exist as an electron probability distribution f(r), and not just a set of discrete planes. The phasors can be summed over an infinite number of locations between planes.


 * $$\lim_{\delta r\rightarrow 0}\sum_{r=-d^{hkl}/2}^{d^{hkl}/2} f(r)=\int\limits_{-d^{hkl}/2}^{d^{hkl}/2} f(r), dr$$

Assuming each plane has the same electron probability distribution, the phasor summations become:


 * $$Re^+ =\sum_{n=1}^P \left [ \int\limits_{-d^{hkl}/2}^{d^{hkl}/2} f(r) U \cos{\left (\Delta \psi(n+r/d^{hkl}+1/2) + 2 \theta \right )}, dr \right ]$$


 * $$Im^+ =\sum_{n=1}^P \left [ \int\limits_{-d^{hkl}/2}^{d^{hkl}/2} f(r) U \sin{\left (\Delta \psi(n+r/d^{hkl}+1/2) + 2 \theta \right )}, dr \right ]$$


 * $$Re^- =\sum_{n=1}^P \left [ \int\limits_{-d^{hkl}/2}^{d^{hkl}/2} f(r) V \cos{\left (-\Delta \psi(n+r/d^{hkl}+1/2) + 2 \theta \right )}, dr \right ]$$


 * $$Im^- =\sum_{n=1}^P \left [ \int\limits_{-d^{hkl}/2}^{d^{hkl}/2} f(r) V \sin{\left (-\Delta \psi(n+r/d^{hkl}+1/2) + 2 \theta \right )}, dr \right ]$$

Divergence
Bla...

Outcomes
PIXRD accurately describes the following:
 * Diffraction angle predicted by the Bragg equation.
 * Full width at half maximum (FWHM) value predicted by the Schrerrer equation.
 * Darwin plateau.
 * Thomson Scattering.
 * Polarization of diffracted profile.