ABSTRACT
In order to reach sufficiently large momentum transfers to cover several Brillouin zones, X-rays with energy of at least 10 keV need to be used. Considering that typical energies of phonons (E) are in the meV range with Ej ª Ei, and the momentum transfer is only dependent on the incident wave vector (energy) and the scattering angle 2q it follows
Q = 2ki sin(q). (6.5) The X-ray intensity of first-order TDS from a crystal containing N atoms in the unit cell at temperature T and momentum transfer Q can be written as follows: I k T
f
( ) coth (
Q q
q
Q
µ ( ) ( )Ê
ËÁ ˆ
¯˜ ◊
= Â 1 213 w w)exp( ( ) )( ( ))- + ◊ ◊ -
Q Q r Q q 1 21 2/ . (6.6)
Note that the above expression contains both eigenvalues wj (q) and eigenvectors sj (q) of the dynamical matrix D(q). This equation can be rewritten in the alternative form as follows: I k T
N( ) coth ( ) ( ) ( )Q Q Q q Q Qµ ◊ Ê ËÁ
ˆ ¯˜
◊ = Â 1 213 w w Z P Z* , (6.7)where Zaa (Q) = f[a/3] (Q)exp(–W)[a/3] (Q) + iQ ◊r[a/3])M[a/3]–1/2 and
the momentum transfer vector is redefined in 3N-dimensional space merging the 3D Q N times as QT = (Qx Qy Qz º º º Qx Qy Qz). Note that the matrix Z(Q) depends on the atomic coordinates and Debye-Waller factors only, and can therefore be obtained from a Bragg diffraction experiment. Since w2j are the eigenvalues of the (Hermitean) dynamical matrix D(q), the intensity reads thus: I(Q) µ QT ◊Z(q)S(q)Z* (Q) ◊ Q, (6.8)where S k T
k T k T
( ) ( ) ( ) ( )
q q
D q
D q I
= Ê
ËÁ ˆ
¯˜
= + Ê ËÁ
ˆ ¯˜
2 21121 2 D
coth 1720 4k TBÊËÁ ˆ¯˜ +ÊËÁÁ ˆ¯˜˜D q( ) ...
