The number of diffraction peaks that will be observed in a diffraction experiment

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. For elastic scattering, $|\vec{k}|=|\vec{k}'|$ and diffraction can only occur for $2|\vec{k}| \gt |\vec{G}|$. Thus, there are only a finite number of diffraction peaks observable. The number of diffraction peaks can be estimated by dividing the volume of a sphere of radius $2|\vec{k}|$ by the volume of a primitive unit cell in reciprocal space. A more exact number can be obtained by testing if reciprocal lattice points lie inside the sphere. The form below calculates the primitive lattice vectors in reciprocal space from the primitive lattice vectors in real space and then determines the number of reciprocal lattice points that satisfy the diffraction condition.

Primitive reciprocal lattice vectors

$\vec{b}_1=2\pi\frac{\vec{a}_2\times\vec{a}_3}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [Å^-1]

$\vec{b}_2=2\pi\frac{\vec{a}_3\times\vec{a}_1}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [Å^-1]

$\vec{b}_3=2\pi\frac{\vec{a}_1\times\vec{a}_2}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}=$ $\hat{k}_x+$ $\hat{k}_y+$ $\hat{k}_z$ [Å^-1]

The approximate number of reciprocal lattice points where diffraction can be observed can be obtained by dividing the volume of a sphere of radius $2|\vec{k}|$ by the volume of a primitive unit cell in reciprocal space. This approximate number is .

The number of reciprocal lattice points that satisfy the diffraction condition is .

Show code

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The relevant part of the javascript code is shown below. It does not take advantage if the inversion symmetry of reciprocal space that for every $\vec{G}$ vector there is one at $-\vec{G}$.

<pre>

// calculate the lengths of the primitive lattice vectors
  b1 = Math.sqrt(b1x*b1x+b1y*b1y+b1z*b1z);
  b2 = Math.sqrt(b2x*b2x+b2y*b2y+b2z*b2z);
  b3 = Math.sqrt(b3x*b3x+b3y*b3y+b3z*b3z);

bmin = Math.min(b1,b2,b3); //find the shortest primitive lattice vector
 N = Math.ceil(2*k_vec/bmin)+1; //the number of unit cells from the origin to check
 var Glen; // the length of the reciprocal lattice vectors
 var count = 0;
 for (h=-N; h < N+1; h++) {
 for (k=-N; k < N+1; k++) {
 for (l=-N; l < N+1; l++) {
 Glen = Math.sqrt(Math.pow(h*b1x + k*b2x + l*b3x,2)+Math.pow(h*b1y + k*b2y + l*b3y,2)+ Math.pow(h*b1z + k*b2z + l*b3z,2));
 if (Glen < 2*k_vec) {count++;} //check if this reciprocal lattice vector is smaller than 2*k_vec.
 }
 }
 }
 document.getElementById('count').innerHTML=count;

</pre>

To see the full code, right click on the background of the webpage and select 'View Source'.
<a href="ewald.m">A Matlab version</a>