- #1
spf
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Hi,
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.
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
\sum_{j} e^{- i \vec{G}_m \vec{\rho}_j}
with \vec{G}_m being the reciprocal lattice vector and \vec{\rho}_j 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:
(1 + e^{- i \frac{2\pi}{3}(2m_1-m_2)})(1 + e^{- i \pi m_3})
which is the same as
(1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(1 + e^{- i \pi m_3})
because
1 = e^{i 2\pi} = e^{i \frac{2\pi}{3} \cdot 3 m_1}
My problem is that I get
(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})
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?
The direct space basis vectors of the graphite lattice are
\vec{a}_1 = a (\sqrt{3}/2, - 1/2, 0)
\vec{a}_2 = a (\sqrt{3}/2, 1/2, 0)
\vec{a}_3 = c (0, 0, 1)
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 a/\sqrt{3} 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
\vec{\rho}_A = (0, 0, 0)
\vec{\rho}_B = (\frac{a}{\sqrt{3}}, 0, 0)
\vec{\rho}_A' = (0, 0, \frac{c}{2})
\vec{\rho}_B' = (- \frac{a}{\sqrt{3}}, 0, \frac{c}{2})
One gets the reciprocal vectors with the following formula:
\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}
\vec{b}_2 = 2\pi \frac{\vec{a}_3 \times \vec{a}_1}{\vec{a}_2 \cdot (\vec{a}_3 \times \vec{a}_1)}
\vec{b}_3 = 2\pi \frac{\vec{a}_1 \times \vec{a}_2}{\vec{a}_3 \cdot (\vec{a}_1 \times \vec{a}_2)}
The reciprocal lattice vector is then
\vec{G}_m = m_1 \vec{b}_1 + m_2 \vec{b}_2 + m_3 \vec{b}_3
The sum in the structure factor is then calculated by
\sum_{j} e^{- i \vec{G}_m \vec{\rho}_j}
By using the formulas and values given above, one gets
\vec{b}_1 = \frac{4 \pi}{\sqrt{3} a} (+ 1/2, - \sqrt{3}/2, 0)
\vec{b}_2 = \frac{4 \pi}{\sqrt{3} a} (+ 1/2, + \sqrt{3}/2, 0)
\vec{b}_3 = \frac{2 \pi}{c} (0, 0, 1)
and thus
\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)
and thus the following sum in the structure factor
(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})
instead of
(1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(1 + e^{- i \pi m_3})
Where is the error? Any help would be appreciated.
Thanks,
spf
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
\sum_{j} e^{- i \vec{G}_m \vec{\rho}_j}
with \vec{G}_m being the reciprocal lattice vector and \vec{\rho}_j 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:
(1 + e^{- i \frac{2\pi}{3}(2m_1-m_2)})(1 + e^{- i \pi m_3})
which is the same as
(1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(1 + e^{- i \pi m_3})
because
1 = e^{i 2\pi} = e^{i \frac{2\pi}{3} \cdot 3 m_1}
My problem is that I get
(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})
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
\vec{a}_1 = a (\sqrt{3}/2, - 1/2, 0)
\vec{a}_2 = a (\sqrt{3}/2, 1/2, 0)
\vec{a}_3 = c (0, 0, 1)
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 a/\sqrt{3} 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
\vec{\rho}_A = (0, 0, 0)
\vec{\rho}_B = (\frac{a}{\sqrt{3}}, 0, 0)
\vec{\rho}_A' = (0, 0, \frac{c}{2})
\vec{\rho}_B' = (- \frac{a}{\sqrt{3}}, 0, \frac{c}{2})
One gets the reciprocal vectors with the following formula:
\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}
\vec{b}_2 = 2\pi \frac{\vec{a}_3 \times \vec{a}_1}{\vec{a}_2 \cdot (\vec{a}_3 \times \vec{a}_1)}
\vec{b}_3 = 2\pi \frac{\vec{a}_1 \times \vec{a}_2}{\vec{a}_3 \cdot (\vec{a}_1 \times \vec{a}_2)}
The reciprocal lattice vector is then
\vec{G}_m = m_1 \vec{b}_1 + m_2 \vec{b}_2 + m_3 \vec{b}_3
The sum in the structure factor is then calculated by
\sum_{j} e^{- i \vec{G}_m \vec{\rho}_j}
The Attempt at a Solution
By using the formulas and values given above, one gets
\vec{b}_1 = \frac{4 \pi}{\sqrt{3} a} (+ 1/2, - \sqrt{3}/2, 0)
\vec{b}_2 = \frac{4 \pi}{\sqrt{3} a} (+ 1/2, + \sqrt{3}/2, 0)
\vec{b}_3 = \frac{2 \pi}{c} (0, 0, 1)
and thus
\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)
and thus the following sum in the structure factor
(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})
instead of
(1 + e^{i \frac{2\pi}{3}(m_1+m_2)})(1 + e^{- i \pi m_3})
Where is the error? Any help would be appreciated.
Thanks,
spf