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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,

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:

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.

*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:

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.

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}$$

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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