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2.3.2 Mean-square displacement surface next up previous
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

2.3.2 Mean-square displacement surface

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

  equation1547

with n referred to the unit vectors tex2html_wrap_inline2762 , tex2html_wrap_inline2764 , parallel to the reciprocal vectors tex2html_wrap_inline2766 , tex2html_wrap_inline2764 , respectively. The bases tex2html_wrap_inline2770 and tex2html_wrap_inline2772 are mutually reciprocal (Hirshfeld & Rabinovich, 1966). Note that whereas tex2html_wrap_inline2774 in eq. (2.3.59) is dimensionless, tex2html_wrap_inline2648 has dimension (length)2 . As n varies, the surface generated by tex2html_wrap_inline2648 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 tex2html_wrap_inline2782 between experimental tex2html_wrap_inline2756 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 up previous
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

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