Question

Derive an expression for Inter-Planar spacing of (hkl) planes of a cubic structure.

(ii) Explain line defects in crystal.​

Answer 1

Explanation:

For all orthogonal crystal systems (orthorhombic, tetragonal, cubic) the interplanar distance of a lattice plane with Miller indices is:

d=1./sqrt(h^2/a^2 + k^2/b^2 + l^2/c^2). This holds for each of the possible Bravais lattices in these systems. Thus all reflections h,k,l exist. If the lattice is primitive, all reflections may have non-zero intensity, if the lattice is body centered only reflections for which h+k+l=2n i.e. even may have non-zero intensity and for a face-centered lattice the reflection conditions are h+k and h+l and k+l must all be even. This means for a face centered lattice that Bragg reflections must have all even or all odd Miller indices.

I would not say that a face centered lattice has lower symmetry than a primitive one. As a matter of fact there are more symmetry operators in the group (the additional translations)!

In a body centered lattice there are always pairs of atoms in the structure. For each atom at a position x,y,z there is an identical atom at x, y, z + [1/2 , 1/2, 1/2]. You need to calculate the common contribution for both of these atoms to the scattered wave, the contribution to the structure factor F(hkl). This exercise will show you that the two contributions cancel each other (= add up to zero) for all Bragg reflections with h+k+l = 2n+1, while they add up to the sum of the two atomic form factors for h+k+l=2n.

Similar calculations can be done for all other Bravais lattices, and likewise for space groups that contain gliede planes and screw axes.

To understand these fundamental concepts of diffraction I strongly suggest to read the introduction to X-ray / neutron diffraction in any good text book, for example

Giacovazzo et al. "Fundamentals of Crystallography" ; Als-Nielsen "Elements of modern X-ray physics" etc etc etc. For a quick summary see slides 10 to 12 in the attached lecture notes.

At Merga, one should not use the wrong term "diamond lattice"; this is the diamond structure. A lattice is a mathematical object, where each point can be described as : p=n*a + m*b + p*c, with abc the base vectors, and nmp all whole integer numbers. The atom positions in the diamond structure do not fulfill this condition.

Answer 2

Explanation:

The interplanar distance of a lattice plane with Miller indices is: for all orthogonal crystal systems (orthorhombic, tetragonal, cubic):

$h2/a2 + k2/b2 + l2/c2) = 1./\sqrt{(h2/a2 + k2/b2 + l2/c2)}$. This is true for each of these systems' conceivable Bravais lattices. As a result, all h,k,l reflections exist. All reflections may have non-zero intensity if the lattice is primitive; only reflections for which $h+k+l=2n$ i.e. even may have non-zero intensity if the lattice is body centred; and for a face-centered lattice, the reflection conditions are $h+k$ and $h+l$ and $k+l$ must all be even if the lattice is face-centered. This means that Bragg reflections must have all even or all odd Miller indices for a face centred lattice.

A face centred lattice does not have less symmetry than a primitive one, in my opinion. In reality, there are more symmetry operators (the additional translations) in the group!

In a body centered lattice there are always pairs of atoms in the structure. For each atom at a position x,y,z there is an identical atom at x, y, z + [1/2 , 1/2, 1/2]. You need to calculate the common contribution for both of these atoms to the scattered wave, the contribution to the structure factor F(hkl). This exercise will show you that the two contributions cancel each other (= add up to zero) for all Bragg reflections with h+k+l = 2n+1, while they add up to the sum of the two atomic form factors for h+k+l=2n.

Similar calculations can be done for all other Bravais lattices, and likewise for space groups that contain gliede planes and screw axes.

To understand these fundamental concepts of diffraction I strongly suggest to read the introduction to X-ray / neutron diffraction in any good text book, for example

Giacovazzo et al. "Fundamentals of Crystallography" ; Als-Nielsen "Elements of modern X-ray physics" etc etc etc. For a quick summary see slides 10 to 12 in the attached lecture notes.

At Merga, the phrase "diamond lattice" is incorrect; this is the diamond structure. A lattice is a mathematical object in which each point is defined by the formula $p=n\times a + m\times b + p\times c$, where abc are the base vectors and nmp are all full integer numbers. This criterion is not met by the atom placements in the diamond structure.

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