- #1

spf

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this is my first post on this forum and I hope that I am in the right category. My question concerns an independent study and not a homework assignment. However, it's a standard homework question in introductory solid state physics courses which makes me both the more frustrated that I cannot figure it out and the more surprised that I cannot find the answer on the web in spite of extensive googling.

## Homework Statement

The structure factor of graphite needs to be calculated.

The following article might be helpful to answer this question but is not necessary as I give all the values and formulas below: Chung, JMS 2002 (Journal of Materials Science), p.2, bottom right

http://www.springerlink.com/content/3u4j6xc32h49de1a/

More precisely, the problem lies in the calculation of the following sum in the structure factor

[itex]\sum_{j} e^{- i \vec{G}_m \vec{\rho}_j}[/itex]

with [itex]\vec{G}_m[/itex] being the reciprocal lattice vector and [itex]\vec{\rho}_j[/itex] being the coordinates of the four atoms of the graphite unit cell (assuming the standard ABAB stacking).

The correct result (according to Chung, JMS 2002) should be:

[itex](1 + e^{- i \frac{2\pi}{3}(2m_1-m_2)})(1 + e^{- i \pi m_3})[/itex]

which is the same as

[itex](1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(1 + e^{- i \pi m_3})[/itex]

because

[itex]1 = e^{i 2\pi} = e^{i \frac{2\pi}{3} \cdot 3 m_1}[/itex]

My problem is that I get

[itex](1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(e^{- i \frac{2\pi}{3}(m_1+m_2)} + e^{- i \pi m_3})[/itex]

The result from Chung JMS 2002 seems more reasonable as it explains the experimentally confirmed 001 Bragg spot extinctions which my result does not. But I cannot find any error in my calculations nor in the start values. Can anybody help me out?

## Homework Equations

The direct space basis vectors of the graphite lattice are

[itex]\vec{a}_1 = a (\sqrt{3}/2, - 1/2, 0)[/itex]

[itex]\vec{a}_2 = a (\sqrt{3}/2, 1/2, 0)[/itex]

[itex]\vec{a}_3 = c (0, 0, 1)[/itex]

a = 0.246 nm being the graphite in-plane lattice parameter, c = 0.671nm the graphite out-of-plane lattice constant (distance between two A planes in a ABAB stacking) and [itex]a/\sqrt{3}[/itex] being the next-neighbour carbon distance of 0.142nm.

A possible choice of the coordinates of the four atoms of the graphite unit cell (assuming the standard ABAB stacking) are

[itex]\vec{\rho}_A = (0, 0, 0)[/itex]

[itex]\vec{\rho}_B = (\frac{a}{\sqrt{3}}, 0, 0)[/itex]

[itex]\vec{\rho}_A' = (0, 0, \frac{c}{2})[/itex]

[itex]\vec{\rho}_B' = (- \frac{a}{\sqrt{3}}, 0, \frac{c}{2})[/itex]

One gets the reciprocal vectors with the following formula:

[itex]\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}[/itex]

[itex]\vec{b}_2 = 2\pi \frac{\vec{a}_3 \times \vec{a}_1}{\vec{a}_2 \cdot (\vec{a}_3 \times \vec{a}_1)}[/itex]

[itex]\vec{b}_3 = 2\pi \frac{\vec{a}_1 \times \vec{a}_2}{\vec{a}_3 \cdot (\vec{a}_1 \times \vec{a}_2)}[/itex]

The reciprocal lattice vector is then

[itex]\vec{G}_m = m_1 \vec{b}_1 + m_2 \vec{b}_2 + m_3 \vec{b}_3 [/itex]

The sum in the structure factor is then calculated by

[itex]\sum_{j} e^{- i \vec{G}_m \vec{\rho}_j}[/itex]

## The Attempt at a Solution

By using the formulas and values given above, one gets

[itex]\vec{b}_1 = \frac{4 \pi}{\sqrt{3} a} (+ 1/2, - \sqrt{3}/2, 0)[/itex]

[itex]\vec{b}_2 = \frac{4 \pi}{\sqrt{3} a} (+ 1/2, + \sqrt{3}/2, 0)[/itex]

[itex]\vec{b}_3 = \frac{2 \pi}{c} (0, 0, 1)[/itex]

and thus

[itex]\vec{G}_m = ( \frac{2 \pi}{\sqrt{3} a} (m_1 + m_2), \frac{2 \pi}{a} (m_2 - m_1), \frac{2 \pi}{c} m_3)[/itex]

and thus the following sum in the structure factor

[itex](1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(e^{- i \frac{2\pi}{3}(m_1+m_2)} + e^{- i \pi m_3})[/itex]

instead of

[itex](1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(1 + e^{- i \pi m_3})[/itex]

Where is the error? Any help would be appreciated.

Thanks,

spf