Cryo-Crystallography: Diffraction at Low Temperature and More - Advanced X-Ray Crystallography - page 38

translational vectors. The U tensor affects the so-called Debye Waller factor T (S),
which is a kind of “damping factor” for the radiation scattered by an atom, as will be
discussed in Sect. 2.2 :

8p 2 u T i sin 2

exp

h #

T ð S Þ¼ : (2)

l 2

S is the scattering vector, u T is the atomic displacement parameter in this simplified
notation assumed to be isotropic, y is the scattering angle, and l the wavelength of the
incident radiation. The atomic displacement depends on the temperature, and hence so
does the Debye-Waller factor. If an atom is modeled by a classical oscillator, then the
atomic displacement would change linearly with temperature:

k B T

uhi¼

o 2 m :

(3)

m is the atomic reduced mass and k B is the Boltzmann constant. However, in a
quantum mechanical system, at low temperature the oscillator is temperature
independent:

h
2

u hi ¼

o m :

(4)

The typical behavior of an atomic displacement parameter is represented by the
curve plotted in Fig. 2 . This trend tells us that below the turn point (Y E /2) atomic
vibrations are not only smaller but also quite constant.
The important message from Einstein or Debye models is that vibrations of atoms
in a crystal contribute to Entropy S and to Heat Capacity C; therefore they affect the
thermodynamic equilibrium of a crystal by modifying both the Free energy F , which

Fig. 2 The mean square displacement of a single harmonic oscillator as a function of temperature.
Units are arbitrary because hu T i and T values depend on the frequency of the oscillator and the
mass of the particle. The plot shows that at low temperature the displacement is almost constant,
whereas at high temperature it varies linearly with T . The change of regime occurs approximately
at Y E /2