Editing Electron diffraction (section)

== Notes ==
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{{efn|name=Diff|Sometimes electron diffraction is defined similar to light or water wave diffraction, that is interference or bending of (electron) waves around the corners of an obstacle or through an aperture. With this definition the electrons are behaving as waves in a general sense, corresponding to a type of Fresnel diffraction. However, in every case where electron diffraction is used in practice the obstacles of relevance are atoms, so the general definition is not used herein.}}
{{efn|name=Wlength|In their first, shorter paper in Nature Davisson and Germer stated that their results were consistent with the de Broglie wavelength. Similarly Thomson and Reid used the de Broglie wavelength to explain their results. However, in their subsequently, more detailed papers Davisson and Germer specifically stated that their work was consistent with ''undulatory mechanics'', and not consistent with the de Broglie wavelength. More importantly, the (non-relativistic) wavelength comes automatically from the Schrödinger equation, as do the equations for the amplitudes of electron diffraction; these cannot be derived from the de Broglie wavelength. As cited in the main text, Davisson and Germer were able to demonstrate that the diffraction angles were different from those of [[Bragg's Law]], needing a proper treatment which includes the average potential inside the material. Since all theoretical models start from the Schrödinger equation (with relativistic terms included) this is really the key to electron diffraction, not the ''de Broglie wavelength''. See [[matter waves]] for more discussion.}}
{{efn|name=Pi|Herein crystallographic conventions are used. Often in physics a plane wave is defined as <math>\exp(i \mathbf k \cdot \mathbf r)</math>. This changes some of the equations by a factor of <math>2 \pi</math>, for instance <math>\hbar</math> appears instead of <math>h</math>, but nothing significant.}}
{{efn|name=RecP|Notations differ depending upon whether the source is crystallography, physics or other. In addition to <math>\mathbf A, \mathbf B, \mathbf C</math> for the reciprocal lattice vectors as used herein, sometimes <math>\mathbf a^*, \mathbf b^*, \mathbf c^*</math> are used. Less common, but still sometimes used, are <math>\mathbf a_1 ,\mathbf a_2,\mathbf a_3</math> for real space, and <math>\mathbf b_1, \mathbf b_2,\mathbf b_3</math> for reciprocal space. Also, sometimes reciprocal lattice vectors are written with capitals as <math>G</math> not <math>g</math>, and the length can differ by a factor of <math>2 \pi</math> as mentioned above if <math>\exp(i\mathbf k \cdot \mathbf r)</math> is used for plane waves. (Different notations also exist for the wavevectors <math>\mathbf k</math>, <math>\mathbf \chi</math> or <math>\mathbf q</math>.) Similar notation differences can occur with aperiodic materials and superstructures. Furthermore, when dealing with surfaces as in [[#Low-energy electron diffraction|LEED]], normally two-dimensional real and reciprocal lattice vectors in the surface are used, defined in terms of a matrix multiplier of the simple surface unit cell when there are reconstructions.  To make things slightly more complicated, frequently four [[Miller indices]] are used for hexagonal systems even though only three are needed.}}
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