Calculating D-Spacings in a Monoclinic Cell

[tappy]

TL;DR Summary

Monoclinc primitive crystal, why is (100) d-spacing not equal to lattice cell dimension a.

Why is it that the d-spacing of the (100) plane is not equal to the lattice cell dimension a.
I can calculate the (010) and (001) d-spacings and they are equal to b & c lattic cell dimensions which is what I expected.

Small bit of background.
My task was to write a small program that calculates the d-spacing for whatever plane the user inputs.
The user inputs the a,b,c & alpha, beta, gamma parameters and use an equation to calculate d-spacings for each of the planes.
The Monoclinc cell has dimensions a=0.52, b=0.54, c=56, alpha=gamma=90deg, beta = 99.18deg.
Any thoughts would be welcome.
Thanks

 

[anuttarasammyak]

d-spacing formula of orthogonal crystals
$$\frac{1}{d_{hkl}^2 }=\frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{l^2}{c^2}$$
tells (100) d-spacing is a.

Maybe I do not get your situation correctly.

 

[Lord Jestocost]

Summary:: Monoclinc primitive crystal, why is (100) d-spacing not equal to lattice cell dimension a.

...use an equation to calculate d-spacings for each of the planes.

Have a look at the $1/d^2$ equation for the monoclinic crystal system:
http://pd.chem.ucl.ac.uk/pdnn/unit1/unintro.htm

 

[tappy]

Thanks for your input. Indeed I used the calculation that you link to http://pd.chem.ucl.ac.uk/pdnn/unit1/unintro.htm
in my program.

I also checked my results with an online calculator at this link https://neutron.ornl.gov/user_data/hb3a/exp16/tools/Crystal%20Plane%20Spacings%20and%20Interplanar%20Angles.htm
so I am confident that the formula's and calculations are correct.

----
I still can't give an explanation in my own words for a reason that the (100) d-spacing is not equal to the lattice parameter for monoclinic whereas the (010) & (001) are equal to their respective lattic parameter.

I compare it to the case of the cubic crystal where it is easy to visualize that the (100) d-spacing is equal to the lattice parameter. However I don't see why this is different for the monoclinic crystal.

Thank you for your input.

 

[Lord Jestocost]

Maybe, try to compare your data with the results generated by Bernhard Rupp's Crystallographic Web Applet "Reciprocal Cell Spacing, N(ref)": http://www.ruppweb.org/new_comp/reciprocal_cell.htm

 

[crashcat]

The planes defined by the Miller indices are not normal to the crystal axes $\vec{a}$, $\vec{b}$, and $\vec{c}$. Instead, e.g. the plane defined by (100) is parallel to the plane defined by $\vec{b}$ and $\vec{c}$. This is because the "0" is the inverse of $\infty$; it means that the plane you're defining never intersects that axis. So the "d" spacing can be less than the vector length.

In the monoclinic lattice, there are two axes with some angle between them and then the third axis is normal to both of those. So only in the case of the third axis will you get that the d spacing is equal to the crystal axis length. In the other two cases, the d spacing will be slightly less. I've illustrated why it's less in a 2D diagram because I couldn't think of a succinct way to write it.