Editing Electron diffraction (section)

=== Kinematical diffraction ===
In Kinematical theory an approximation is made that the electrons are only scattered once.<ref name="Cowley95" />{{Rp|location=Sec 2}}  For transmission electron diffraction it is common to assume a constant thickness <math>t</math>, and also what is called the Column Approximation (e.g. references<ref name="HirschEtAl" />{{Rp|location=Chpt 11}}<ref name="Tanaka">{{Citation |last=Tanaka |first=Nobuo |title=Column Approximation and Howie-Whelan's Method for Dynamical Electron Diffraction |date=2017 |url=http://dx.doi.org/10.1007/978-4-431-56502-4_27 |work=Electron Nano-Imaging |pages=293–296 |place=Tokyo |publisher=Springer Japan |doi=10.1007/978-4-431-56502-4_27 |isbn=978-4-431-56500-0 |access-date=2023-02-11|url-access=subscription }}</ref> and further reading). For a perfect crystal the intensity for each diffraction spot <math>\mathbf g</math> is then:<math display="block">I_{g} = \left|\phi(\mathbf k)\right|^2 \propto \left|F_{g}\frac{\sin(\pi t s_z)}{\pi s_z}\right|^2
</math>where <math>s_z</math> is the magnitude of the excitation error <math>|\mathbf s_z|</math> along z, the distance along the beam direction (z-axis by convention) from the diffraction spot to the [[Ewald's sphere|Ewald sphere]], and <math>F_{g}</math> is the [[structure factor]]:<ref name="Form" /><math display="block">F_{g} = \sum_{j=1}^N f_j \exp{(2 \pi i \mathbf g \cdot \mathbf r_j -T_j g^2)} </math>the sum being over all the atoms in the unit cell with <math>f_j</math> the form factors,<ref name="Form">{{Citation |last1=Colliex |first1=C. |title=Electron diffraction |date=2006 |url=https://xrpp.iucr.org/cgi-bin/itr?url_ver=Z39.88-2003&rft_dat=what%3Dchapter%26volid%3DCb%26chnumo%3D4o3%26chvers%3Dv0001 |work=International Tables for Crystallography |volume=C |pages=259–429 |editor-last=Prince |editor-first=E. |edition=1 |place=Chester, England |publisher=International Union of Crystallography |doi=10.1107/97809553602060000593 |isbn=978-1-4020-1900-5 |last2=Cowley |first2=J. M. |last3=Dudarev |first3=S. L. |last4=Fink |first4=M. |last5=Gjønnes |first5=J. |last6=Hilderbrandt |first6=R. |last7=Howie |first7=A. |last8=Lynch |first8=D. F. |last9=Peng |first9=L. M.|url-access=subscription }}</ref> <math>\mathbf g</math> the [[reciprocal lattice]] vector, <math>T_j</math> is a simplified form of the [[Debye–Waller factor]],<ref name="Form" /> and <math>\mathbf k</math> is the wavevector for the diffraction beam which is:<math display="block">\mathbf k = \mathbf k_0 + \mathbf g + \mathbf s_z</math>for an incident wavevector of <math>\mathbf k_0</math>, as in [[#Figure 6|Figure 6]] and [[Electron diffraction#Plane waves, wavevectors and reciprocal lattice|above]]. The excitation error comes in as the outgoing wavevector <math>\mathbf k</math> has to have the same modulus (i.e. energy) as the incoming wavevector <math>\mathbf k_0</math>. The intensity in transmission electron diffraction oscillates as a function of thickness, which can be confusing; there can similarly be intensity changes due to variations in orientation and also structural defects such as [[dislocations]].<ref>{{Cite journal | last1=Hirsch | first1=Peter | last2=Whelan | first2=Michael | date=1960 |title=A kinematical theory of diffraction contrast of electron transmission microscope images of dislocations and other defects |url=https://royalsocietypublishing.org/doi/10.1098/rsta.1960.0013 |journal=Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences |language=en |volume=252 |issue=1017 |pages=499–529 |doi=10.1098/rsta.1960.0013 | bibcode=1960RSPTA.252..499H | s2cid=123349515 |issn=0080-4614|url-access=subscription }}</ref> If a diffraction spot is strong it could be because it has a larger structure factor, or it could be because the combination of thickness and excitation error is "right". Similarly the observed intensity can be small, even though the structure factor is large. This can complicate interpretation of the intensities. By comparison, these effects are much smaller in [[x-ray diffraction]] or [[neutron diffraction]] because they interact with matter far less and often Bragg's law<ref name="Bragg" /> is adequate.

This form is a reasonable first approximation which is qualitatively correct in many cases, but more accurate forms including multiple scattering (dynamical diffraction) of the electrons are needed to properly understand the intensities.<ref name="Cowley95" />{{Rp|location=Sec 3}}<ref name="Peng" />{{Rp|location=Chpt 3-5}}