PHY.K02UF Molecular and Solid State Physics

Ewald sphere

Diffraction can occur whenever the diffraction condition, $\vec{k}' -\vec{k} =\vec{G}$, is satisfied. Here $\vec{k}$ is the wave vector of the incoming waves, $\vec{k}'$ is the wave vector of the scattered wave, and $\vec{G}$ is a reciprocal lattice vector. The diffraction condition can be rewritten as $\hbar\vec{k}' -\hbar\vec{k} =\hbar\vec{G}$. This is a statement of the conservation of momentum. For elastic scattering, energy is conserved so the wavelength of the scattered photon is the same as the incoming photon, $|\vec{k}|=|\vec{k}'|$. Diffraction can only occur for $2|\vec{k}| \gt |\vec{G}|$ so there are only a finite number of diffraction peaks observable. The diffraction condition can be visualized with an Ewald construction as illustrated below. You draw the incoming wavevector $\vec{k}$ so that it ends at the origin of reciprocal space and draw a sphere around the beginning of the $\vec{k}$ vector with a radius $|\vec{k}|$. When you rotate the crystal it will rotate the points in reciprocal space and whenever a reciprocal lattice point lies on the sphere, the diffraction condition will be satisfied. In the simulation below, the crystal is aligned so that the $(hkl)$ plane is parallel to the beam. The crystal is then rotated around the normal to the $(hkl)$ plane. Since $\vec{G}_{hkl}$ is normal to the $(hkl)$ plane, the reciprocal lattice points are rotated around $\vec{G}_{hkl}$. Every time the crystal is rotated a little, the detector is scanned around the $(hkl)$ plane to search for diffraction peaks. When diffraction occurs, both $\vec{k}$ and $\vec{k}'$ will lie on the Ewald sphere. The angle between $\vec{k}$ and $\vec{k}'$ is $2\theta$. Press the and buttons to rotate the crystal.

Primitive lattice vectors:

 $\vec{a}_1=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å] 
 $\vec{a}_2=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å]
 $\vec{a}_3=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å]

 The x-rays scan the $hkl$ plane.
$h=$ $k=$ $l=$

X-ray wavelength λ = [Å]
= 1.540598 Å
= 2.28975 Å
= 0.7093165 Å

Reciprocal lattice points in the in the ( ) plane

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Primitive reciprocal lattice vectors

$\vec{b}_1=$ [ , , ] [Å-1]
$\vec{b}_2=$ [ , , ] [Å-1]
$\vec{b}_3=$ [ , , ] [Å-1]