March 20, 2008

Fig. 1: Illustration of 2d ball and spring
model showing 1d surface. |

Problems with masses and springs are widespread in
physics literature. The reason for this, being their mathematical
simplicity, the clarity of their results and, most importantly, the fact
that they present excellent prototype models for many physical problems .
For these reasons we are to study a very simple two-dimensional model of
masses attached to springs. The masses are placed in a square lattice and
they are connected with their nearest and next nearest neighbors with
springs. We will investigate, in this very simple system, the response of
the masses at the surface (and we will compare them with that of the
masses of the interior) namely, the oscillations or waves that exist and
travel on the surface. Doing this we will discover the existence of the
*Rayleigh waves* (waves that they live and travel in the surface)
and we will gain some insight of their properties and dynamics.

The model that we are to use is a very simple and, at
the same time, very pedagogical. It is illustrated in Fig. 1. We will
consider a square lattice of bond length *b* which consists of masses
connected together with springs as shown in the figure above. There are
two types of springs in our model. The first type are springs that connect
nearest neighbor masses, having spring constant *k _{1}* which
are denoted by curly lines. The other type, denoted by broken lines, are
springs that connect the next nearest neighbors, having different spring
constant

Having all these thing in mind, we can easily write
down the equations of motion that this model obeys. But before we do this,
let’s make a couple of comments and assumptions that they will make this
task simpler. We will consider the lattice described above to be
semi-infinite in the y-direction - that is infinite in the negative
y-direction but has a bound in the positive one - so that our system has a
surface to investigate. Furthermore we will impose periodic boundary
conditions in the x-direction, which means that we are to close every
chain in x-axis to itself. This will simplify our problem considerably.
The periodic boundary conditions make the vibration of one mass to be the
same as that of its neighbor-mass (in the x-direction) times
*e ^{iqb}*,

These are the equations of motion for our system. The
matrix *D* (the so called dynamical matrix) gives the contribution to
the force due to the motion of the nth mass. The matrix
*D _{+}* gives the contribution due to the motion of the mass
n+1 and

Now Green's function is the one that it will give the information about the response of both the interior (the bulk as it is commonly called) and the surface of the system. That is why we will call, from now on, the imaginary part of the Green's function, response function. Now it is very easy to solve the Eq. (5) taking into account Eq. (6). So if we apply Eq. (6) to Eq. (5) we will get the results:

Eq. (8) being the response of the bulk while Eq. (9)
is the response of the surface. This is difference is, of course, due to
the fact that the masses at the surface are connected to fewer springs
that the ones in the bulk. This is the origin of the term *δ D
*, the change in the dynamical matrix that we must do to get the
response of the surface.

We have solved these equations by a little Fortran 77
program which you can find here. In order for us to solve
the problem we used *k _{1}=1* and

Fig. 2: Response in the bulk. |

With our program, mentioned above, we calculated the
response function for our system. First of all the response of the
interior (or the bulk as it is also known) of the system was calculated so
that we know the spectrum of the vibrations there. Then the response of
the surface was calculated in the way that we described above. These
results are plotted (for different values of *q*)

The response graphs can be understood in the following, very simple, way. Wherever we have non-zero value of the response function, we can have oscillations with this frequency. On the other hand, if we have zero value of the response function we cannot have any oscillation (or wave in general) with this frequency. On the light of these remarks we can now look, explain and interpret the results, starting from the bulk.

The very first thing that we observe in this spectrum
is the presence of a gap in the allowed vibration modes that is developing
with increasing *q*. This means that for a given non-zero *q*
there is a frequency below which we have no vibrations (this is often
referred to as cut-off frequency). So a wave or a collective oscillation
with frequency less than the cut-off frequency just cannot propagate in
the bulk. Another thing that we have to mention is that we have averaged
over the two different ways that the masses can oscillate i.e vertically
or horizontally. We did that via taking the trace of the response
function. Other than that there is nothing special about that spectrum.
It is pretty straight-forward and it does not have any strange features.
Now let's move to the, more interesting, response of the surface.

We can clearly see that there is a peak, for every
non-zero *q*, that appears inside the gap that we were talking about
previously, just next to the cut-off frequency. This is the so called
Rayleigh wave, a wave that can exist only in the surface. We can easily
understand why this is so. The Rayleigh peak is located inside the gap, of
the bulk response function. This means that it cannot propagate in the
bulk of our system and, therefore, it has to decay evanescently. Another
point, which cannot see it in the graph, is that this wave contains both
of the two different polarizations (transverse and longitudinal). This
cannot be seen in our graph because we have averaged over these two
polarizations, but it can be checked by plotting the diagonal components
of the response separately. The most important feature of these waves,
though, is that they are model-independent. This means that they will show
up in any model we might choose, provided that we will include
next-nearest neighbor interactions in it.

Fig. 3: Response at the surface. |

These kinds of waves are of outmost importance in several domains in science. First and foremost they are earthquake waves so their properties and dynamics have been studied extensively and they are very well understood from geologists and seismologists, because they are the most destructive waves in an earthquake. Furthermore they are very important to physics too, because they are a very important part of the surface dynamics of materials and because we can directly measure them from experiments. There are experiments like the one described in ref [3] of Helium atom scattering, in which the cross sections measured are directly proportional to the surface response function.

To sum up, we have investigated the behavior of a very simple model of harmonic oscillators and we arrived at the conclusion that this possesses a very interesting range of oscillations at the surface, the so called Rayleigh waves. Moreover we have seen here that even a very simple model, with very simple manipulations and calculations can yield to non-trivial results, which are of utmost importance.

© 2008 George Karakonstantakis. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial Rights are reserved to the author.

[1] P. C. Martin, *Measurements And Correlation
Functions* (Gordon and Breach, 1968).

[2] I. A. Viktorov, *Rayleigh and Lamb Waves*
(Plenum, 1967).

[3] J. P. Toennies *et al.*, Phys. Rev. B
**65**, 165427 (2002).