Recall the a rigid bar with cavities, each containing a spring-mass
system with mass and complex spring constant (see
Figure 1).
The momentum of the bar () is related to its velocity ()
by the relation
where is the frequency and is the effective mass
which is given by
where is the mass of the rigid bar~.
Let us now consider a specific model for the springs. A simple model
is the one-dimensional Maxwell model shown in Figure 2.
In this case instead of a spring with a complex spring constant we have
an elastic spring (with real spring constant ) and a dashpot (with
a real viscosity ) in series. Let be the displacement of
the elastic spring and let be that of the dashpot under the action of a
force .
For the elastic spring, we have
For the dashpot, we have
Once again, assuming that the functions , , and can
be expressed as harmonic functions, we have
Plugging equations (3) into equations (1) and
(2), we get
Recall that the displacement of the sphere of mass inside the cavity
is related to the applied force by the relation
Now, . Hence,
or,
So, with the Maxwell model, the effective mass is
or,
This model is remarkably similar to a simple model for the frequency
dependent dielectric constant .
Consider the following simple model of an electron bound to an atom by
a harmonic force under the influence of a slowly varying electric
field (a detailed description can in found in Jackson75, Sec. 7.5).
A schematic of the situation is shown in Figure 3.
Let the electric field be and assume that it varies slowly in space
over the distance that the electron moves, i.e., . Let
be the mass of the electron
and let be its charge. Let be the frequency of the
harmonic force binding the electron to the atom and let be a
damping coefficient due to interactions with obstacles.
Then, the equation of motion of the electron is given by
Let
Plugging equations (7) into equation (6),
we get
or
Let be the charge density of the electron-atom system, i.e.,
where is the Dirac delta function and is the position of the
atom. Then, the dipole moment (the first moment of the charge) contributed by
the electron-atom system is given by
or,
Suppose now that instead of a binding frequency for all the
atoms and all the electrons in a volume, there are atoms per unit volume
with binding
frequencies and damping constants and. Then, we can
write the polarization as
Recall that the electric displacement is related to the electric field and
the polarization by
Therefore, from equations (8) and (9) we get
Comparing equation (5) with equation (10) we
observe that they have the same form which implies that the effective
permittivity is analogous to the effective mass. This also implies that
the electric field is analogous to the velocity. Similarly, the
polarization is analogous to the momentum.
More generally, we get a frequency dependent density if all the constituents
do not move in lock step. Lock step motion almost never occurs because
there are thermal vibrations at the microscale. There are also many
macroscopic situations in which lock step motion does not occur. Consider
the example of a porous rock containing some water
(see Figure 4). Both the rock grains and the water are
connected, though this is not obvious from the figure
In this case, the water will move with a different frequency that the
rock and the density of the composite will be dependent on the frequency.
At a molecular level, we can have a crystal with lead atoms attached by
single bonds to the structure (see Figure 5). Presumably,
the resonant frequency of such molecules is very high. so we will see
the frequency dependence of the mass only at very high frequencies.
In fact, Sheng et al. Sheng03 have shown in experiments that materials
indeed have frequency dependent masses. An example of such a material is
shown in Figure 6.
Consider the rigid body containing cavities shown in Figure~7.
This is just a two-dimensional extension of the model shown in
Figure~1. Here and are complex spring constants
in the and directions.
In this case, the effective mass along the -direction is given by
while that along the -direction is given by
In matrix form, we then have
Hence, the effective mass is clearly anisotropic. Note however that
from a macroscopic perspective it is not the average velocity in the
matrix which is important. In fact, such a quantity does not even make
sense because the velocity is not defined in the void phase. Rather it is
the velocity of the matrix that is the relevant quantity in this model.
One can generalize the model one step further by having the springs be
oriented at different angles to each other as shown in
Figure 8. Also let the springs in each cavity have
different spring constants and the masses in each cavity are different.
In this case, let be the rotation that is needed to orient each
set of springs with the and axes. Then, if is the
rotation matrix for cavity containing a mass , and if
and are the complex spring constants for that cavity, the effective
mass can be written as
Take any body with a rigid matrix. Suppose, instead of applying harmonically
varying velocities, we apply a time varying velocity
and observe what the momentum is. There will be some
linear constitutive relation
The kernel is second-order tensor valued and may possibly
be singular. Also, since both and are physical and real,
must be real. Causality implies that when
(or ) since the inertial force cannot depend
on velocities in the future.
Taking the Fourier transform of equation (13) and using the
convolution theorem, we get
where
The quantity can be shown to satisfy the Cauchy-Riemann
equations only if . This is a consequence of causality
and the fact that the integral in equation (15) only
converges in the upper half of the complex plane.
\footnote{ Note the positive sign in
the power of in equation (15). This occurs because
we have chosen the inverse Fourier transform to be of the form
The effect of such a choice is that the imaginary part of is
positive instead of negative.} Hence, is analytic in
when . The quantity is real.
Now, the complex conjugate of is given by
where denotes the complex conjugate of (for any complex
).
Also assume that, for large enough frequencies, the dynamic mass tends
toward the static mass, i.e.,
Equation (15) can be used to establish the Kramers-Kronig
equations for the material. To do that,
recall Cauchy's formula for a function which is analytic on a domain
that is enclosed in a piecewise smooth curve :
Since the function is analytic on the upper-half plane,
for any point in a closed contour in the upper-half plane, we
have
Let us now choose the contour such that it consists of the real
axis and a great semicircle at infinity in the upper half plane (see
Figure~9).
Also, from equation (17) we observe that
Hence, there is no contribution to the integral in equation (18)
due to the semicircular part of the contour and we just have to perform
an integration only over the real line:
Now, let us consider the integral
There is a pole at the point (in the figure, the contour is
shown as a semicircle of radius centered at the pole). In the
limit , this integral may be interpreted to mean the
Cauchy principal value (not the same as principal values in complex analysis)
From Figure~9 we observe that this integral can be
broken up into the integral over the path minus the sum of the
integrals over the paths and . From the Cauchy-Goursat theorem,
the integral over the closed path is zero. We have also seen that
since as ,
the integral over is zero. The integral over the path around the
pole is obtained from the Residue Theorem (where the value is divided
by two because the integral is over a semicircle), i.e.,
The negative sign arises because the curve is traversed in the
counter-clockwise direction.
Therefore,
or,
Letting and taking the limit as , we
get
Note that, in the above equation, both and are real.
Expanding equation (20) into real and imaginary parts and
collecting terms, we get the first form of the Kramers-Kronig relations
Therefore, the real part of the frequency dependent mass can be determined
if we know the imaginary part and vice versa.
We can also eliminate the negative frequencies from equations (21).
Recall from equation (16) that
Since, in equations (21), and are real, we have
This implies that
Consider the first of equations (21). We can write this relation
as
Similarly, the second of equations (21) may be written as
Therefore, the alternative form of the Kramers-Kronig relations is
The average rate of work done on the system in a cycle of oscillation will be
This quadratic form will be non-negative for all choices of if
and only if is positive semidefinite for all real
. Note that the quadratic form does not contain .
Since the work done in a cycle should be zero in the absence of
dissipation, this implies that the imaginary part of the mass is
connected to the energy dissipation (for instance, into heat).