Hydrogen Debye-Waller Factor
In order to fit neutron diffraction data on hydrogenated
GIC's (see Section 
), it is necessary to
have an estimate of the hydrogen Debye-Waller factor. The
C4KHx inelastic neutron scattering data
of Kamitakahara, Doll, and Eklund,[218] were employed here to make an
estimate of BH using a procedure suggested by
Kamitakahara.[122]
Kamitakahara, Doll, and Eklund[218] report that the hydrogen vibrational mode in C4KH0.8 has a frequency of 93 meV in-plane and 71 meV out-of-plane. The Debye-Waller factor corresponds approximately to the mean-square vibrational amplitude of the atom.[10] Semiclassically, a harmonic oscillator has energy
where m is the mass of the oscillator, Omega its characteristic frequency, and < x2 > its mean-square displacement, the quantity of interest. Plugging in the numbers for the out-of-plane vibration, the one relevant to (00l) diffraction patterns, gives < x2 > = 0.058Å2.
The Debye-Waller factor is actually 8 pi2 < x > 2 due to corrections that come into play in an exact calculation.[10] From the value of < x > 2, the final result is BH = 4.58 Å2.