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.