Editing Electron diffraction (section)

=== Plane waves, wavevectors and reciprocal lattice ===
What is seen in an electron diffraction pattern depends upon the sample and also the energy of the electrons. The electrons need to be considered as waves, which involves describing the electron via a wavefunction, written in crystallographic notation (see notes{{efn|name=Pi}} and{{efn|name=RecP}}) as:<ref name="Form" /><math display="block">\psi (\mathbf r) = \exp(2\pi i \mathbf k \cdot \mathbf r)</math>for a position <math>\mathbf r</math>. This is a [[quantum mechanics]] description; one cannot use a classical approach. The vector <math>\mathbf k</math> is called the wavevector, has units of inverse nanometers, and the form above is called a [[plane wave]] as the term inside the exponential is constant on the surface of a plane. The vector <math>\mathbf k</math> is what is used when drawing ray diagrams,<ref name="Cowley95" />{{Rp|location=Chpt 3}} and in vacuum is parallel to the direction or, better, group velocity<ref name="Broglie" />{{Rp|location=Chpt 1-2}}<ref name=":21">{{Cite book |last=Schiff |first=Leonard I. |title=Quantum mechanics |date=1987 |publisher=McGraw-Hill |isbn=978-0-07-085643-1 |edition=3. ed., 24. print |series=International series in pure and applied physics |location=New York}}</ref>{{Rp|page=16}} or [[probability current]]<ref name=":21" />{{Rp|pages=27, 130}} of the plane wave. For most cases the electrons are travelling at a respectable fraction of the speed of light, so rigorously need to be considered using relativistic quantum mechanics via the [[Dirac equation]],<ref>{{Cite journal |last1=Watanabe |first1=K. |last2=Hara |first2=S. |last3=Hashimoto |first3=I. |date=1996 |title=A Relativistic n -Beam Dynamical Theory for Fast Electron Diffraction |url=https://scripts.iucr.org/cgi-bin/paper?S0108767395015893 |journal=Acta Crystallographica Section A |volume=52 |issue=3 |pages=379–384 |doi=10.1107/S0108767395015893 |bibcode=1996AcCrA..52..379W |issn=0108-7673|url-access=subscription }}</ref> which as spin does not normally matter can be reduced to the [[Klein–Gordon equation]]. Fortunately one can side-step many complications and use a non-relativistic approach based around the Schrödinger equation.<ref name="Schroedinger" /> Following Kunio Fujiwara<ref name="Fujiwara">{{Cite journal |last=Fujiwara |first=Kunio |date=1961 |title=Relativistic Dynamical Theory of Electron Diffraction |url=https://journals.jps.jp/doi/10.1143/JPSJ.16.2226 |journal=Journal of the Physical Society of Japan |language=en |volume=16 |issue=11 |pages=2226–2238 |doi=10.1143/JPSJ.16.2226 |bibcode=1961JPSJ...16.2226F |issn=0031-9015|url-access=subscription }}</ref> and [[Archibald Howie]],<ref name="AHDiss">{{Cite journal |last=Howie |first=A |date=1962 |title=Discussion of K. Fujiwara's paper by M. J. Whelan |journal=Journal of the Physical Society of Japan |volume=17(Supplement BII) |pages=118}}</ref> the relationship between the total energy of the electrons and the wavevector is written as:<math display="block">E = \frac{h^2 k^2}{2m^*}</math>with<math display="block">m^* = m_0 + \frac{E}{2c^2}</math>where <math>h</math> is the [[Planck constant]], <math>m^*</math> is a relativistic [[Effective mass (solid-state physics)|effective mass]] used to cancel out the relativistic terms for electrons of energy <math>E</math> with <math>c</math> the speed of light and <math>m_0</math> the rest mass of the electron. The concept of effective mass occurs throughout physics (see for instance [[Ashcroft and Mermin]]),<ref name=":7">{{Cite book |last1=Ashcroft |first1=Neil W. |title=Solid state physics |last2=Mermin |first2=N. David |date=2012 |publisher=Brooks/Cole Thomson Learning |isbn=978-0-03-083993-1 |edition=Repr |location=South Melbourne}}</ref>{{Rp|location=Chpt 12}} and comes up in the behavior of [[quasiparticles]]. A common one is the [[electron hole]], which acts as if it is a particle with a positive charge and a mass similar to that of an electron, although it can be several times lighter or heavier. For electron diffraction the electrons behave as if they are non-relativistic particles of mass <math>m^*</math> in terms of how they interact with the atoms.<ref name="Fujiwara" />

The wavelength of the electrons <math>\lambda</math> in vacuum is from the above equations<math display="block"> \lambda = \frac 1 k = \frac{h}{\sqrt{2m^* E}} = \frac{h c}{\sqrt{E(2 m_0 c^2 + E)}},</math>and can range from about {{val|0.1|ul=nm}}, roughly the size of an atom, down to a thousandth of that. Typically the energy of the electrons is written in [[electronvolt]]s (eV), the voltage used to accelerate the electrons; the actual energy of each electron is this voltage times the [[electron charge]]. For context, the typical energy of a [[chemical bond]] is a few eV;<ref>{{Cite web |date=2013-10-02 |title=Bond Energies |url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Chemical_Bonding/Fundamentals_of_Chemical_Bonding/Bond_Energies |access-date=2023-09-26 |website=Chemistry LibreTexts |language=en}}</ref> electron diffraction involves electrons up to {{val|5,000,000|u=eV}}.

The magnitude of the interaction of the electrons with a material scales as<ref name="Cowley95">{{Cite book |last=John M. |first=Cowley |url=http://worldcat.org/oclc/247191522 |title=Diffraction physics |date=1995 |publisher=Elsevier |isbn=0-444-82218-6 |oclc=247191522}}</ref>{{Rp|location=Chpt 4}}<math display="block">2 \pi \frac{m^*}{h^2 k} = 2\pi\frac{ m^* \lambda} {h^2} = \frac \pi {hc} \sqrt{\frac{2m_0 c^2}{E} + 1}.</math>While the wavevector increases as the energy increases, the change in the effective mass compensates this so even at the very high energies used in electron diffraction there are still significant interactions.<ref name="Fujiwara" />

The high-energy electrons interact with the Coulomb potential,<ref name="Bethe" /> which for a crystal can be considered in terms of a [[Fourier series]] (see for instance [[Ashcroft and Mermin]]),<ref name=":7" />{{Rp|location=Chpt 8}} that is<math display="block">V(\mathbf r) = \sum V_g \exp(2 \pi i \mathbf g \cdot \mathbf r)</math>with <math>\mathbf g</math> a [[reciprocal lattice]] vector and <math>V_g</math> the corresponding Fourier coefficient of the potential. The reciprocal lattice vector is often referred to in terms of [[Miller indices]] <math>(h k l)</math>, a sum of the individual reciprocal lattice vectors <math>\mathbf A,\mathbf B,\mathbf C</math> with integers <math>h, k, l</math> in the form:<ref name="Form" /><math display="block">\mathbf g = h \mathbf A + k \mathbf B + l \mathbf C</math>(Sometimes reciprocal lattice vectors are written as <math>\mathbf a^*</math>,  <math>\mathbf b^*</math>, <math>\mathbf c^*</math> and see note.{{efn|name=RecP}}) The contribution from the <math>V_g</math> needs to be combined with what is called the shape function (e.g.<ref>{{Citation |last=Vainstein |first=B.K. |title=Experimental Electron Diffraction Structure Investigations |date=1964 |url=http://dx.doi.org/10.1016/b978-0-08-010241-2.50010-9 |work=Structure Analysis by Electron Diffraction |pages=295–390 |publisher=Elsevier |doi=10.1016/b978-0-08-010241-2.50010-9 |isbn=9780080102412 |access-date=2023-02-11|url-access=subscription }}</ref><ref>{{Cite journal |last1=Rees |first1=A. L. G. |last2=Spink |first2=J. A. |date=1950 |title=The shape transform in electron diffraction by small crystals |journal=Acta Crystallographica |volume=3 |issue=4 |pages=316–317 |doi=10.1107/s0365110x50000823 |bibcode=1950AcCry...3..316R |issn=0365-110X|doi-access=free }}</ref><ref name="Cowley95" />{{Rp|location=Chpt 2}}), which is the [[Fourier transform]] of the shape of the object. If, for instance, the object is small in one dimension then the shape function extends far in that direction in the Fourier transform—a reciprocal relationship.<ref>{{Cite web |title=Kevin Cowtan's Book of Fourier, University of York, UK |url=http://www.ysbl.york.ac.uk/~cowtan/fourier/crys1.html |access-date=2023-09-26 |website=www.ysbl.york.ac.uk}}</ref>

{{anchor|Figure 6}}[[File:EwaldS2.png|thumb|Figure 6: Ewald sphere construction for transmission electron diffraction, showing two of the Laue zones and the excitation error|alt=Illustration of how the wavevectors and diffraction from reciprocal lattice vectors is connected, called an Ewald sphere construction. This example is for transmission electron diffraction.]]

Around each reciprocal lattice point one has this shape function.<ref name="Cowley95" />{{Rp|location=Chpt 5-7}}<ref name="HirschEtAl">{{Cite book |last1=Hirsch |first1=P. B. | last2=Howie | first2=A. | last3=Nicholson| first3=R. B.| last4=Pashley | first4=D. W. | last5=Whelan | first5=M. J.|url=https://www.worldcat.org/oclc/2365578 |title=Electron microscopy of thin crystals |date=1965 |publisher=Butterworths |isbn=0-408-18550-3 |location=London |oclc=2365578}}</ref>{{Rp|location=Chpt 2}} How much intensity there will be in the diffraction pattern depends upon the intersection of the [[Ewald sphere]], that is energy conservation, and the shape function around each reciprocal lattice point—see [[#Figure 6|Figure 6]], [[#Figure 20|20]] and [[#Figure 22|22]]. The vector from a reciprocal lattice point to the Ewald sphere is called the excitation error <math>\mathbf s_g</math>.

For transmission electron diffraction the samples used are thin, so most of the shape function is along the direction of the electron beam. For both [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]]<ref name="LEEDB" /> and [[Electron diffraction#Reflection high-energy electron diffraction (RHEED)|RHEED]]<ref name="Ichimiya" /> the shape function is mainly normal to the surface of the sample. In [[#Low-energy electron diffraction|LEED]] this results in (a simplification) back-reflection of the electrons leading to spots, see [[#Figure 20|Figure 20]] and [[#Figure 21|21]] later, whereas in [[#Reflection high-energy electron diffraction|RHEED]] the electrons reflect off the surface at a small angle and typically yield diffraction patterns with streaks, see [[#Figure 22|Figure 22]] and [[#Figure 23|23]] later. By comparison, with both x-ray and neutron diffraction the scattering is significantly weaker,<ref name="Cowley95" />{{Rp|location=Chpt 4}} so typically requires much larger crystals, in which case the shape function shrinks to just around the reciprocal lattice points, leading to simpler Bragg's law diffraction.<ref name="Bragg">{{Cite journal |last1=Bragg |first1=W.H. |last2=Bragg |first2=W.L. |date=1913 |title=The reflection of X-rays by crystals |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1913.0040 |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=88 |issue=605 |pages=428–438 |doi=10.1098/rspa.1913.0040 |bibcode=1913RSPSA..88..428B |s2cid=13112732 |issn=0950-1207|url-access=subscription }}</ref>

For all cases, when the reciprocal lattice points are close to the Ewald sphere (the excitation error is small) the intensity tends to be higher; when they are far away it tends to be smaller. The set of diffraction spots at right angles to the direction of the incident beam are called the zero-order Laue zone (ZOLZ) spots, as shown in [[#Figure 6|Figure 6]]. One can also have intensities further out from reciprocal lattice points which are in a higher layer. The first of these is called the first order Laue zone (FOLZ); the series is called by the generic name higher order Laue zone (HOLZ).<ref name="Reimer" />{{Rp|location=Chpt 7}}<ref>{{Cite web |title=higher-order Laue zone (HOLZ) reflection {{!}} Glossary {{!}} JEOL Ltd. |url=https://www.jeol.com/ |access-date=2023-10-02 |website=higher-order Laue zone (HOLZ) reflection {{!}} Glossary {{!}} JEOL Ltd. |language=en}}</ref>

The result is that the electron wave after it has been diffracted can be written as an integral over different plane waves:<ref name="Peng" />{{Rp|location=Chpt 1}}<math display="block"> \psi (\mathbf r) = \int \phi (\mathbf k) \exp(2 \pi i \mathbf k \cdot \mathbf r) d^3\mathbf k ,</math>that is a sum of plane waves going in different directions, each with a complex amplitude <math>\phi (\mathbf k)</math>. (This is a three dimensional integral, which is often written as <math>d\mathbf k</math> rather than <math>d^3\mathbf k</math>.) For a crystalline sample these wavevectors have to be of the same magnitude for elastic scattering (no change in energy), and are related to the incident direction <math>\mathbf k_0</math> by (see [[#Figure 6|Figure 6]])
<math display="block">\mathbf k = \mathbf k_0 + \mathbf g + \mathbf s_g.</math>
A diffraction pattern detects the intensities<math display="block"> I(\mathbf k) = \left| \phi(\mathbf k) \right| ^2 .</math>For a crystal these will be near the reciprocal lattice points typically forming a two dimensional grid. Different samples and modes of diffraction give different results, as do different approximations for the amplitudes <math>\phi (\mathbf k)</math>.<ref name="Cowley95" /><ref name="Reimer" /><ref name=":11" />

A typical electron diffraction pattern in TEM and [[Electron diffraction#Low-energy electron diffraction (LEED)|LEED]] is a grid of high intensity spots (white) on a dark background, approximating a projection of the reciprocal lattice vectors, see [[#Figure 1|Figure 1]], [[#Figure 9|9]], [[#Figure 10|10]], [[#Figure 11|11]], [[#Figure 14|14]] and [[#Figure 21|21]] later. There are also cases which will be mentioned later where diffraction patterns are [[#Aperiodic materials|not periodic]], see [[#Figure 15|Figure 15]], have additional [[#Diffuse scattering|diffuse]] structure as in [[#Figure 16|Figure 16]], or have rings as in [[#Figure 12|Figure 12]], [[#Figure 13|13]] and [[#Figure 24|24]]. With conical illumination as in [[#Convergent beam electron diffraction|CBED]] they can also be a grid of discs, see [[#Figure 7|Figure 7]], [[#Figure 9|9]] and [[#Figure 18|18]]. [[RHEED]] is slightly different,<ref name="Ichimiya" /> see [[#Figure 22|Figure 22]], [[#Figure 23|23]]. If the excitation errors <math>s_g</math> were zero for every reciprocal lattice vector, this grid would be at exactly the spacings of the reciprocal lattice vectors. This would be equivalent to a Bragg's law condition for all of them. In TEM the wavelength is small and this is close to correct, but not exact. In practice the deviation of the positions from a simple Bragg's law<ref name="Bragg" /> interpretation is often neglected, particularly if a column approximation is made (see below).<ref name="Peng" />{{Rp|page=64}}<ref name="HirschEtAl" />{{Rp|location=Chpt 11}}<ref name="Tanaka" />