2.3.2 Mean-square displacement surface
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### 2.3.2 Mean-square displacement surface

The mean-square displacement amplitude (MSDA) in a direction defined by a unit vector n is

with n referred to the unit vectors , , parallel to the reciprocal vectors , , respectively. The bases and are mutually reciprocal (Hirshfeld & Rabinovich, 1966). Note that whereas in eq. (2.3.59) is dimensionless, has dimension (length)2 . As n varies, the surface generated by is not an ellipsoid; it is usually peanut-shaped.

Such surfaces can be constructed even for non-positive-definite tensors and they are therefore particularly useful for inspecting difference tensors between experimental tensors and those obtained from kinematic or dynamic models of atomic and molecular motion (Hummel, Raselli & B?rgi, 1990).

The distinction between the surfaces defined by eqs. (2.3.2) and (2.3.3) has often proved puzzling. Note that the right-hand side of (2.3.2) is a constant, the (arbitrarily chosen) equi-probability level for defining the ORTEP ellipsoids. When the matrix of the mean-square displacement tensor is non-positive definite, the quadratic surface defined by (2.3.2) is no longer closed, and no ellipsoid can be plotted. In contrast, the right-hand side of (2.3.3) is the mean-square displacement amplitude (MSDA) in a given direction, and varies as n varies. Only positive values of the quantity defined in (2.3.3) are meaningful for an individual atom, but negative values can be meaningful when differences in MSDA values are calculated. The MSDA surfaces can be plotted with the aid of the program PEANUT of Hummel et al. (1990); negative values are plotted as dashed contours.

Next: 3. Beyond the Gaussian Approximation Up: 2.3 Graphical representations of the Gaussian mean-square displacement matrix Previous: 2.3.1 Ellipsoids of constant probability