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dielectric constants and β jn are dielectric impermabilities. Similarly, when linear piezoelectric effects exist
we can write linear relations between stress and electric fields such as σ ij = −g kij E k and E i = −e ijk σ jk ,
where g kij and e ijk are called piezoelectric constants. If there is a linear relation between strain and an
electric fields, this is another type of piezoelectric effect whereby e ij = d ijk E k and E k = −h ijk e jk ,where
d ijk and h ijk are another set of piezoelectric constants. Similarly, entropy changes can cause pyroelectric
effects. Piezooptical effects (photoelasticity) occurs when mechanical stresses change the optical properties of
the material. Electrical and heat effects can also change the optical properties of materials. Piezoresistivity
occurs when mechanical stresses change the electric resistivity of materials. Electric field changes can cause
variations in temperature, another pyroelectric effect. When temperature effects the entropy of a material
this is known as a heat capacity effect. When stresses effect the entropy in a material this is called a
piezocaloric effect. Some examples of the representation of these additional effects are as follows. The
piezoelectric effects are represented by equations of the form

σ ij = −h mij D m D i = d ijk σ jk e ij = g kij D k D i = e ijk e jk

where h mij , d ijk , g kij and e ijk are piezoelectric constants.
Knowledge of the material or electric interaction can be used to help modify the constitutive equations.
For example, the constitutive equations can be modified to included temperature effects by expressing the
constitutive equations in the form

σ ij = c ijkl e kl − β ij ∆T and e ij = s ijkl σ kl + α ij ∆T

where for isotropic materials the coefficients α ij and β ij are constants. As another example, if the strain is
modified by both temperature and an electric field, then the constitutive equations would take on the form

e ij = s ijkl σ kl + α ij ∆T + d mij E m .

Note that these additional effects are additive under conditions of small changes. That is, we may use the
principal of superposition to calculate these additive effects.
If the electric field and electric displacement are replaced by a magnetic field and magnetic flux, then
piezomagnetic relations can be found to exist between the variables involved. One should consult a handbook
to determine the order of magnitude of the various piezoelectric and piezomagnetic effects. For a large
majority of materials these effects are small and can be neglected when the field strengths are weak.

The Boltzmann Transport Equation
The modeling of the transport of particle beams through matter, such as the motion of energetic protons
or neutrons through bulk material, can be approached using ideas from the classical kinetic theory of gases.
Kinetic theory is widely used to explain phenomena in such areas as: statistical mechanics, fluids, plasma
physics, biological response to high-energy radiation, high-energy ion transport and various types of radiation
shielding. The problem is basically one of describing the behavior of a system of interacting particles and their
distribution in space, time and energy. The average particle behavior can be described by the Boltzmann
equation which is essentially a continuity equation in a six-dimensional phase space (x, y, z, V x ,V y ,V z ). We