The missing factor of 2π in reciprocal lattice calculations

In summary, the conversation discusses the differences between the equations used in Kittel's "Introduction to Solid State Physics" and Hammond's "The Basics of Crystallography and Diffraction" for x-ray diffraction. Hammond's equation omits a factor of ##2\pi## in the relation between translation vectors in real and reciprocal space, causing discrepancies in later derivations. The source of this normalization is unclear and further clarification is needed.
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Wrichik Basu
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##\require{physics}## Recently, I wet my feet in X-ray diffraction a bit more than what is usually covered in standard solid state physics textbooks at the undergrad level, like Kittel. Two good books that I chanced upon included Christopher Hammond, The Basics of Crystallography and Diffraction and B. D. Cullity and S. R. Stock, Elements of X-Ray Diffraction. I will, however, stick to Kittel and Hammond in this thread.

In Kittel, the relation between the translation vector in real (or direct) space, ##\va{R},## and that in the reciprocal space, ##\va{G},## is given by $$\begin{align}
&\phantom{implies} \mathrm{e}^{i \va{G} \vdot \va{R}} = 1 \\
&\implies \va{G} \vdot \va{R} = 2\pi m; \quad m \in \mathbb{Z}.
\end{align}$$
If I write ##\va{G} = h \va{g}_1 + k \va{g}_2 + \ell \va{g}_3 ## and ##\va{R} = \sum\limits_{i=1}^3 n_i \va{a}_i## with ##n_i \in \mathbb{Z},## we can write $$\begin{equation}
\va{g}_i \vdot \va{a}_j = 2 \pi \delta_{ij} \label{eq:g_dot_a_kittel}
\end{equation}$$according to Kittel Chapter 2, and I can follow the reasoning fine.

In Hammond, however, this relation is written as $$\begin{equation}
\va{g}_i \vdot \va{a}_j = \delta_{ij},
\end{equation}$$omitting the ##2\pi.##

In all derivations throughout Chapter 8 in Hammond, this relation is used. Thus, ##\abs{ \va{G}_{hk\ell} }## becomes ##1/d_{hk\ell},## the radius of Ewald's sphere becomes ##1/\lambda,## and the vector form of Bragg's law, $$\begin{equation}
\dfrac{\vu{s} - \vu{s}_0}{\lambda} = \va{G}_{hk\ell}, \label{eq:bragg_law_vector}
\end{equation}$$when combined with the first Laue equation, becomes $$\begin{align}
&\phantom{\implies} \va{a} \vdot \qty( \vu{s} - \vu{s}_0 ) = n_x \lambda \nonumber \\
&\implies \va{a} \vdot \va{G}_{hk\ell} \cdot \lambda = n_x \lambda \nonumber \\
&\implies h = n_x,
\end{align}$$with a factor of ##2\pi## missing everywhere.

Things aggravate when Hammond derives the equation for the structure factor in Chapter 9, as follows:
structure_factor.png


The position vector ##\va{r}_1## is written as$$\begin{equation}
\va{r}_1 = \sum_{i=1}^3 x_i \va{a}_i
\end{equation}$$where ##\va{a}_i## are the lattice basis vectors, and ##x_i## are \textit{fractions} of the cell edge lengths.

The path difference,$$\begin{align}
\mathrm{P.D.} &= \mathrm{AB} - \mathrm{CD} \nonumber \\
&= \va{r}_1 \vdot \vu{S} - \va{r}_1 \vdot \vu{S}_0 \nonumber \\
&= \va{r}_1 \vdot ( \vu{S} - \vu{S}_0 ).
\end{align}$$
##\because## Bragg's Law is satisfied, we can write using ##\text{eqn.}~\eqref{eq:bragg_law_vector}##,$$\begin{align}
&\phantom{\implies} ( \vu{S} - \vu{S}_0 ) = \lambda \va{G}_{hk\ell} \nonumber\\
&\phantom{ \implies ( \vu{S} - \vu{S}_0 ) } = \lambda \qty( h \va{g}_1 + k \va{g}_2 + \ell \va{g}_3 ).\\
&\implies \mathrm{P.D.} = \lambda \qty( x_1 \va{a}_1 + x_2 \va{a}_2 + x_3 \va{a}_3 ) \vdot \qty( h \va{g}_1 + k \va{g}_2 + \ell \va{g}_3 ) \nonumber \\
&\implies \mathrm{P.D.} = \lambda \qty( h x_1 + k x_2 + \ell x_3 ). \label{eq:path_difference}
\end{align}$$where we have used ##\va{g}_i \vdot \va{a}_j = \delta_{ij}## to arrive at the last step.

The phase angle is given by$$\begin{align}
\phi &= \dfrac{2\pi}{\lambda} \mathrm{P.D.} \nonumber \\
&= 2 \pi \lambda \qty( h x_1 + k x_2 + \ell x_3 ).
\end{align}$$

Adding the scattering amplitudes ##f## with their respective phase angles ##\phi## in the complex plane,$$\begin{equation}
\boxed{F_{hk\ell} = \sum_{n = 1}^N f_n ~ \exp \qty[ 2\pi i \qty( h x_n + k y_n + \ell z_n ) ].}
\end{equation}$$
Note that if I simply put the factor of ##2\pi## in ##\text{eqn.}~\eqref{eq:path_difference},## the expression for the phase angle will have a factor of ##4\pi^2,## so will the expression for structure factor. Now, Wikipedia states that this expression for the structure factor is correct, so I just can't put in the ##2\pi## factor.

Can anyone please explain where the ##2\pi## factor is being normalized? And how do I derive the structure factor equation if I want to stick to the expression used in Kittel, i.e. ##\text{eqn.}~\eqref{eq:g_dot_a_kittel}?##

Interestingly, I just noticed that @ergospherical has also skipped the ##2\pi## factor in one of my previous thread on the Ewald sphere.
 
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  • #2
Does the equivalent of Eq. 1 in Hammond have a ##2\pi## in the exponent?
 
  • #3
Haborix said:
Does the equivalent of Eq. 1 in Hammond have a ##2\pi## in the exponent?
As far as I can see, Hammond has not written the equivalent of Eq. 1 anywhere, at least from Chapters 6 to 9. Chapter 6 deals with the reciprocal lattice, and there, he directly writes ##\va{g}_i \vdot \va{a}_j = \delta_{ij}## owing to the fact that ##\va{g}_1## is ##\perp## to both ##\va{a}_2## and ##\va{a}_3,## and so on. I didn't read through chapters 1-6 because I already studied them from Kittel.
 
  • #4
I don't have any of the textbooks at hand, but I would guess the two ##\vec{\mathbf{g}}## are related by a factor of ##2\pi##.
 
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  • #5
Haborix said:
I don't have any of the textbooks at hand, but I would guess the two ##\vec{\mathbf{g}}## are related by a factor of ##2\pi##.
It's the same as when choosing between f (frequency) and ω (angular frequency). Neither is 'correct' but the one which avoids a page full of π's is easier to write and interpret.
 
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