The missing factor of 2π in reciprocal lattice calculations

##\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:

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.

Does the equivalent of Eq. 1 in Hammond have a ##2\pi## in the exponent?

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.

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##.

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.