Miller indices vs Laue indices?

Question

I am looking at the difference between the Miller indices and Laue indices in crystal structures. I will denote the former by $hkl$ and the latter by $HKL$.

I understand that $hkl$ must be coprime integers, however, I have also heard that $hkl$ represent families of lattice planes. For the case of e.g. the bcc lattice these seem contradictory since $(200)$ rather then $(100)$ forms faimiles of lattice planes.

For $HKL$ it appears that they to must be integers (but not necessarily coprime). These are the indices used to determine if we have a systematic absence or not. But this seems to be assuming we only get diffraction peaks from families of planes which are families of lattice planes - yet as far as I can tell you will get a diffraction peak from the planes $(0.5 \ \ 0.5\ \ 0.5)$ .

Please can someone therefore explain to me the difference, what values they can take and how they can be defined?

Answer 1

The distinction between the two indices is quite subtle.Perhaps it is better to start from the laue condition. Let's say I have rows of atoms with spacing $a$. We are in the fraunhofer limit so when two x-ray beams come in and scatter off neighboring atoms they come in parallel and go out parallel. We can then consider the path length difference between the incoming beams and out going beams i.e $ \Delta^1 = \Delta_1 - \Delta_2= a \cos \alpha_1 - a \cos \alpha_2 $ where $\alpha_1$ is the angle between the incoming beam and the row of atoms and $\alpha_2$ is the angle between the outgoing beam and the row of atoms.The condition for constructive interference is that $\Delta^1 = h \lambda$ where $\lambda$ is the wavelength and h is an integer. This analysis can be done in all three directions and then the condition has to be true in all three directions giving $\Delta^2 = k\lambda \text{ and } \Delta^3 = l\lambda $.(There will be corresponding angles for the other path length differences).

So we can define incoming unit vectors $s_1 = (\cos \alpha_1,\cos \beta_1, \cos \gamma_1)$ and outgoing $s_2 = (\cos \alpha_2, \cos \beta_2, \cos \gamma_2)$. From this we consider the scatter wave $s_1-s_2= G \lambda $ with $G = \frac{1}{a}(h,k,l)$ since $s_1 ,s_2$ are unit vectors $G$ will be perpendicular to the vector that bisects the angle between $s_1 $ and $s_2$.

Now comes the first crucial point, draw planes that are parallel to the plane that bisects the angle between the incoming and outgoing beams. These are the lattice planes and they define the miller indices.Laue indices will be defined by another set of planes. Back to the lattice planes , call the distance between each successive plane $d_{hkl}$ this is the distance that appears in Bragg's law.When we pick a primitive cell then (hkl)are our miller indices.

What about laue indices? Well we look at the bragg's law $n \lambda = 2 d_{hkl} \sin \theta_n $ . Now comes the second crucial point, note that the $m^{th}$ order reflection off the (hkl) plane can be regarded as $n^{th}$ order if I $ \textit{define}$ new planes at distance $\frac{d_{hkl}n}{m}$ i.e $ m \frac{n}{m}\lambda=2 (d_{hkl}\frac{n}{m})\sin \theta_m $ This family of planes is then given by the notation $ \frac{m h}{n} \frac{m k}{n} \frac{m l}{n}$ without parantheses. These are the laue indices, note how they can have common factors even if we started off with a primitive cell.

Answer 2

From what I have read and Amara's answer, this is what I have found.

Miller Indices

A set of 3 indices $h,k,l$ which specify a family of lattice planes as well as the shortest reciprocal lattice vector in a given direction.

Laue Indices

aka Reflection Indices¹.

A set of 3 indices $H$,$K$,$L$, which specify a family of planes (note no 'lattice') from which Bragg reflection peaks form.

Restriction on Values

It is a common statement that Miller indices have to be coprime (i.e. have no common factors) yet this simply is not true. Miller indices take take any value they want as long as they correspond to a family of lattice planes (which restricts their values to small integers). Laue indices can take any value which is an integer multiple of a valid set of Miller indices.

Examples

For an simple cubic cell (111) is a set of miller indices (222) is not but (222) is a valid set of Laue indices.

For a fcc with a conventional unit cell (100) is not a valid set of miller indices (it does not correspond to a set of lattice planes) (200) is however. (300) is not a valid set of Laue indices but (200) and (400) etc are.

Warning

I have seen books where all sets of indices $hkl$ are simply called Miller indices without any restriction or the concept of Laue indices been introduced.

References

[1] Hammond, C. 2009. The basics of crystallography and diffraction (Vol. 12). Oxford: Oxford University Press. (~ pg 111)

[2] Fredriksson, H. and Åkerlind, U., 2008. Physics of functional materials. John Wiley & Sons. (pg 134)

[3] Clegg, W. 2015. X-ray Crystallography. Oxford University Press (pg 109)

_{¹Reflection Indices is a much more lucrative search term.}