University of Toronto
ADVANCED PHYSICS LABORATORY
ASL
Aperiodic Structures and
Anderson Localization
Revisions:
2011-2014 October: Minor Modifications: David Bailey <dbailey@physics.utoronto.ca >
2011 January:
Minor Corrections: Jason Harlow
2006 August:
Jason Harlow
2001:
Brian Statt
Please send any corrections, comments, or suggestions to the professor currently supervising this experiment,
the author of the most recent revision above, or the Advanced Physics Lab Coordinator.
Copyright © 2014 University of Toronto
This work is licensed under the Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 Unported License.
(http://creativecommons.org/licenses/by-nc-sa/3.0/)
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INTRODUCTION
Periodic systems have energy eigenvalues which are found in bands with accompanying band gaps
where no states exist. The eigenstates are extended states occupying the whole system. These systems
are typically modeled as coupled oscillators. Examples are crystal lattices, semiconductor
heterostructures and photonic bandgap structures. You may be familiar with the band gap in
semiconductor crystals where no electronic states exist. In a metallic crystal electrons propagate in
states, much like plane wave states, which are extended throughout the crystal. Nominally theses states
have the same amplitude throughout the crystal.
Adding disorder to periodic systems allows some of the states to become localized. This phenomenon,
first discussed by P.W. Anderson, is referred to as Anderson Localization. The energy eigenvalues of the
localized modes occur in the band gap of the system. These localization effects are strongest in onedimensional systems. An interesting example of Anderson Localization is to localize photons in a
photonic bandgap structure. One does not normally think of photons as standing still!
This experiment is based on a similar experiment by developed by He and Maynard described in
Physical Review Letters.
APPARATUS
Our one-dimensional system consists of masses on an Aluminum plate. There are 28 equal masses,
whose positions can be adjusted, on a 3 meter long plate. The wave equation for this system comes from
the Euler-Bernoulli equation (see “Wave Motion in Elastic Solids” by K. Graff 1975):
é Yd 2 ù ¶ 4 u
¶ 2u
+
[
r
+
V
(
x
)]
=0
ê
ú 4
¶t 2
ë 12 û ¶x
(1)
where ρ is the plate density, d is the plate thickness, Y is Young’s Modulus of the plate and V(x) is the
potential due to the masses. u(x,t) is the displacement of the plate parallel to the plate thickness, where t
is time and x is the distance along the length of the plate. Notice that this equation contains a fourth
order spatial derivative, resulting from the fact that the plate provides its own restoring stiffness,
proportional to its curvature, in marked contrast to the equation for a string, which requires externally
applied tension in order to support wave motion. The tension due to the fact that the plate is hanging
vertically is negligible for the waves we are studying, and so gravity is ignored in this analysis. The
potential can be modeled, assuming infinitesimal mass dimensions, as
V(x) =
M
d (x - xn )
wd
(2)
where M is the mass, w is the plate width and xn are the positions of the masses with n = 1-28. For a
periodic placement of the masses, xn = nl where l is the mean mass separation, 10 cm in our case. The
thickness of this aluminum plate is an eighth-inch: d = 0.3175 cm , its width is an inch: w = 2.54 cm and
the attached masses have M = 75g.
The sensors are piezoelectric crystals, “Piezo Ceramic Benders”, with the centre portion of the contact
pad removed. Thus they are most sensitive to flexing perpendicular to the cut. Their resonance
frequency is 6.4 KHz. How is this relevant to this experiment? Sensor numbers run from 0-7.
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EXPERIMENTAL
Safety Reminders
 Take great care when climbing the ladder to adjust the weights. Wear appropriate foot-wear, i.e. nonslip soles, not sandals or loose-fitting shoes. Do not overreach – move the ladder to the proper spot!
 Do not run the volume load enough to irritate other people in the room.
NOTE: This is not a complete list of every hazard you may encounter. We cannot warn against all
possible creative stupidities, e.g. juggling cryostats. Experimenters must use common sense to assess
and avoid risks, e.g. never open plugged-in electrical equipment, watch for sharp edges, don’t lift tooheavy objects, …. If you are unsure whether something is safe, ask the supervising professor, the lab
technologist, or the lab coordinator. When in doubt, ask! If an accident or incident happens, you must
let us know. More safety information is available at http://www.ehs.utoronto.ca/resources.htm.
The experiment involves investigating first a periodic lattice and then a lattice with disorder introduced
into the mass spacing. Use the Labview programs alspectra.vi to collect the frequency spectrum and
alchanelspectra.vi to collect the mode pattern at a fixed frequency. For the periodic lattice identify the
first two bands as well as the resulting band gap. How many modes are there in the bands and how many
do you expect? Estimate the fundamental frequency of this one-dimensional plate. Does it agree with
your findings for the lowest band? Find the mode patterns (eigenfunctions) for some of the resonances in
both bands. Comment on the extended nature of these modes.
Now introduce disorder into the lattice. The masses can be moved by loosening the wing nuts, moving
the mass, then tighten the wing nuts. Numerical simulations indicate that an rms displacement of 1.5 cm
will introduce detectable localized modes. Rather than trust yourself to randomly move the masses you
may want to use a random number generator to select the mass displacements.
Collect the frequency spectrum as before. Are there any new modes in the band gaps of the periodic
lattice? Compare the mode patterns of some resonances in the periodic band. Are they still extended
states? Now collect the mode pattern of the new states in the band gap. Are these states localized or
extended? You may have to try several different mass disorder configurations to find obvious localized
modes.
EXPERIMENTAL NOTE
Currently ( as of 20 September 2011) you must start running the LabView vi file and wait for the error
message saying that the function generator is not on, and only then turn on the function generator. If you
turn on the function generator first, you will not be able to run a scan or control the function generator;
you will only get an inaccurate error message.
Hints
Note that you can detect the mode pattern yourself by running your ear along the plate while it is being
excited.
An amplitude of 6Vp-p is sufficient for a useful signal to noise ratio. The sample delay time should be at
least 2 sec. to allow a steady state pattern to evolve. The apparatus is only capable of operating up to
4KHz. Sufficient signal averaging is required to collect good mode patterns.
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REFERENCES
1. Anderson, P.W. (1958): “Absence of diffusion in certain random lattices”, Phys. Rev. 109(5),
1492 -1505.
2. Graff, Karl F. (1975), “Wave Motion in Elastic Solids”, Oxford: Clarendon Press.
3. He, S. and Maynard, J.D. (1986), “Detailed measurements of inelastic scattering in Anderson
localization”, Phys. Rev. Lett. 57(25), 3171-3174.
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