In any diffraction technique **Bragg’s law** is one of the fundamentals:

n\cdot\lambda = 2\cdot d_{(hkl)}\cdot\sin\theta_{(hkl)}

For the determination of *d*_{(hkl)} which is used to describe the lattice parameters, the **Bragg angle** has to be **measured** and the **wavelength must be known**.

Electron wavelength

The wavelength of electron λ_{e} is a function of the acceleration voltage *E*_{0} used to controll the electron energy.

The **classic approach**

\lambda_e [\textnormal{\AA}]= \dfrac{h}{\sqrt{2m_0eE_0\rule{0em}{1.8ex}}}=\dfrac{12.2653} {\sqrt{E_0\rule{0em}{1.8ex}} \;[\textrm{V}] }

of deBroglie is often assumed as sufficient for applications in SEM.

For higher acceleration energies – commonly used in TEM – the **relativistic approach**

\begin{align*} \lambda_e [\textnormal{\AA}]&= \dfrac{h}{\sqrt{2m_0eE_0\left(1+\dfrac{e}{2m_0c^2}E_0\right)}}\\ & = \dfrac{12.2653}{\sqrt{E_0+\rule{0em}{2.25ex}0.97845\cdot 10^{-6}{E_0}^2}} \end{align*}

has been developed. The left diagram (below) shows that the differences between the two approaches are quite small. However, the relevant relative difference (right diagram) proves a practically linearly increasing offset, which is 1% at 20kV and 1.5% at 30kV. This would not be negligible for the desired accuracy in CALM so that it uses the relativistic approach.

Bragg angle

For 20kV the wavelength λ_{e} is about 18 times shorter than the commonly used Cu Kα-radiation in X-ray diffraction.

Therefore, the Bragg angles in EBSD and TKD patterns is very small compared to XRD. A reasonably experienced XRD expert can name offhand some characteristic Bragg angles for commonly occurring phases. In contrast, it is almost impossible to find an EBSD user who has approximate ideas of typical Kikuchi bandwidths. Therefore, in the following diagram Bragg angles for Ferrite (α-Fe) are shown considering the most intense ten interferences. Whereas the θ position of each bar indicates the geometrically predicted Bragg angle position for 20keV electrons, their height scales with the relative intensity derived from the kinematic theory. It is directly proportional to the structure amplitude: I_{hkl} ~ |F_{hkl}|.

*Bragg angles (vertical bars) for Ferrite using 20kV electrons. The superimposed curve displays the deviation between the Bragg angle and its sine which is used in Bragg’s law. Even for wider bands the difference (θ-sinθ) is sma*ller than 0.1%.

The small Bragg angles are the reason why the determination of lattice parameters by EBSD is widely considered a waste of time. After all, everyone knows that lattice parameters with XRD are preferably done only at the largest possible Bragg angles. Small lattice parameter chances shift peaks more the higher θ.

Please note: *Apart from the inadmissible generalization, this is a major reason why phase separation by means of EBSD can never be better than that by means of XRD in the standard case. EBSD should only be used if XRD fails for other reasons. But, and this should be clear, the correctness of EBSD will never reach that of XRD.*

Bragg’s law for EBSD

In the upper graph a curve is overlaid which displays the difference between angle and its sine.

Because of this small difference, for EBSD **Bragg’s law** can be simplified:

\color{red}\theta_{hkl} \cdot d_{hkl} \approxeq \dfrac{\lambda_e}{2}\color{black}= \textrm{const.}

This means, that the **Bragg angle** is** inversely proportional to** the interplanar spacing **d _{hkl}**.

It also means: We can

**directly measure**the

**interplanar spacing**

**from**the

**angular width**of a Kikuchi band.

*The really big problems are of a practical nature: Where is the Bragg angle? And how exactly can the Bragg angle indicated by the edge of the Kikuchi band be determined?*

From the periodicity of a crystal and reciprocal lattice follows, that all bands need to describe the phase-specific lattice. **Lattice plane and zone axis** describe the **crystal lattice** whereas the **normal direction** **of** **the** diffracting lattice **plane **as trace of a Kikuchi band **indicates** a **reciprocal lattice direction**.

The **bandwidth** tells us the **distance to a reciprocal lattice point** from the origin.