University of TorontoADVANCED PHYSICS LABORATORY

ASL

Aperiodic Structures andLocalization

Revisions:2019 May-September: Natalia Krasnopolskaia2011-2014 October: Minor Modifications: David Bailey <dbailey@physics.utoronto.ca >2011 January: Minor Corrections: Jason Harlow 2006 August: Jason Harlow2001: 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 TorontoThis work is licensed under the Creative Commons

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INTRODUCTIONPeriodic 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.Crystalline periodic system with its regularly distributed potential in Cronig-Pennel model for a 1-dimensional case is shown below in Fig. 1a. Positions of ions are denoted as Z+.

Adding disorder to periodic systems allows some of the states to become localized. Localization due to the difference in the depth of potential well, first discussed by P.W. Anderson [1], is referred to as Anderson Localization (Fig.1b). The energy eigenvalues of the localized modes occur in the band gap of the system. These localization effects are strongest in one-dimensional 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! The other type of localization occurs if the disorder is caused by displacement of ions of the lattice from regularly distributed nodes. This is referred to as structure disorder (described in detail by I.M.Lifshitz [2]) and is shown in Fig.3.

The experiment in our lab has same goal as the experiment, developed by He and Maynard and described in Physical Review Letters [3], i.e. to get localized states of transverse vibrations in a one-dimensional mechanical system.

https://en.wikipedia.org/wiki/File:Periodic_square_potential_130707.png

(a)

https://slideplayer.com/slide/4932428/

(b)

https://slideplayer.com/slide/4932428/

FIG. 4: Truly random potential

FIG. 3: Structure disorder

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APPARATUSOur one-dimensional system consists of masses on a long and thin aluminum plate. There are 27 bars of almost equal masses of (72.7 ± 0.1) g, whose positions can be adjusted, on a 2.9-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 [4]):

[Yd2

12 ] ∂4 u∂ x4 +[ ρ+V ( x ) ] ∂

2 u∂ t2 =0 (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. We will consider the tension due to the fact that the plate is hanging vertically negligible for the waves we are studying. The potential can be modeled, assuming infinitesimal mass dimensions, as

(2)

where M is the mass of one bar, w is the plate width, d is the plate thickness and xn are the positions of the masses with n = 1-27. For a periodic placement of the masses, xn = nl where l is the mean mass separation. In our experiment, w = 2.54 cm; d = 0.3175 cm; l = 10 cm. The sensors are piezoelectric crystals, working as accelerometers. Their resonance frequency is above 20 KHz. Sensor numbers run from 0-13.Vibrations are actuated by a force generator FG-142, attached to the lower end of the plate. The other pieces of equipment are: an amplifier, a WaveTech signal generator, an oscilloscope and an interface NI-6259 connected to a computer. There are two options for studying and measuring frequencies of vibrations. Option 1 – making a sweep test of activated frequencies in the range of the driven frequency from 5 Hz to 2-3 kHz. A sweep test result on the PC display for 5 Hz – 1000 Hz is given in Fig. 5.

Run

FIG. 5.

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Option 1 requires all sensors to be connected to the interface input channels, and the force generator to be connected to the output channel of the interface NI-6259 via amplifier. To start the sweep test, activate the icon ASL Sweep Generator on the desktop. Set Start and Stop Frequency as required for your experiment. The apparatus is only capable of operating up to 3KHz. Set # Frequency Steps to the number you want to divide the range of frequencies into for recording. Frequency Step Duration should be 5 seconds. The colors of channels are chosen by a user to the right on the display and permit understanding which sensor registers which signal.Click command Run as shown in Fig. 5. The button OFF starts glowing in yellow. Click OFF to start data acquisition. After the spectrum is taken, and the system stops, slick on Export Data or make a right click on the graph to save the data to an Excel file for future processing. We recommend using Print Screen option to save the image.Option 2 – at a constant driving frequency from the signal generator and amplifier, watching vibrations at a position of a specific sensor with oscilloscope. This option may be used with the FFT function of the oscilloscope (Fig. 6). Warning! Fuse in the force generator will burn if you set both the signal generator and amplifier to maximum. -20dB on the signal generator is advised.

FIG. 6: Oscilloscope FFT function for the driving frequency 160 Hz.

EXPERIMENTAL

Safety Reminders Take great care when climbing the ladder to adjust the weights. Wear appropriate foot-wear, i.e. non-

slip 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 too-heavy 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.

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You may start with testing vibrations of the unloaded plate. Continue with investigating a periodic lattice and then a lattice with disorder introduced into the mass spacing. Like in the experiment, described in the source [3], varying a mass makes a negligible difference among spectra. On contrary, if you move just a couple of masses thus creating disorder in the one-dimensional system, the result will be much more significant.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 (eigenstates) 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.The Python codes were created by a SURF student Ms. Grace Li in summer 2019 to numerically solve the Eq.1 and predict the band structure and possible localized states under different disorder conditions. The codes are given in Appendix. Click the Start button on the computer desktop and click Spider. Wait for several seconds for loading the system. Allow access on request. Ignore the request for the Administrator PIN.In the folder C of the computer, open the folder Aperiodic Structures and open simulation_eigenvalue.py. This program simulate running the code for regularly distributed masses. After you click Run, the outcome in a row of dots identifying the locations of expected peaks will appear at the lower right quarter of the display. It would be more useful if you open the image in the Python editable console. Compare the structure of bands and band gaps with your experiment result.To simulate a disordered lattice, edit the Python file and save it under the new name. The example of the modified file with masses #5 and #6 been moved by 15 cm each from their regular positions find in Appendix.

HintsNote that you can detect the mode pattern yourself by running your ear along the plate while it is being excited.

References1. Anderson, P.W. (1958): “Absence of diffusion in certain random lattices”, Phys. Rev. 109(5),

1492 -1505.2. Lifshitz, I. M. “Energy spectrum structure and quantum states of disordered condensed systems”,

Sov. Phys. Usp. 7 549-573 (1965).3. He, S. and Maynard, J.D. (1986), “Detailed measurements of inelastic scattering in Anderson

localization”, Phys. Rev. Lett. 57(25), 3171-3174.4. Graff, Karl F. (1975), “Wave Motion in Elastic Solids”, Oxford: Clarendon Press.

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Appendix. Python codes for simulating eigenvalues.1. Regularly distributed masses: simulation_eigenvalue.py.

from scipy.linalg import eigimport numpy as npfrom matplotlib import pyplot as plt

n = 290 # number of elementsY = 69*10**9 # Young's modulus for aluminiumd = 0.00314 # thicknesswid = 0.0254 # widthL = 2.904 # length of the aluminium platerho = 2700 # density of aluminiumdelta_x = L/n # the length between each element (1cm)rho_mass = 0.0726/(d*wid*delta_x)rho_motor = 0.295/(d*wid*delta_x)const = (Y*(d**2)/12/rho)**0.5/(delta_x**2) # time scaling factor

S = np.zeros((n,n)) # set up an empty nxn matrix

# fixed endS[0][0] = 7*rho/(rho+rho_motor)S[0][1] = -4*rho/(rho+rho_motor)S[0][2] = 1*rho/(rho+rho_motor)S[1][0] = -4S[1][1] = 6S[1][2] = -4S[1][3] = 1num_m = 0

# constructing matrix Sfor i in range(2,n-2): if (i+1-12)%10 == 0 and num_m < 27: # add a mass S[i][i-2] = 1*rho/(rho+rho_mass) S[i][i-1] = -4*rho/(rho+rho_mass) S[i][i] = 6*rho/(rho+rho_mass) S[i][i+1] = -4*rho/(rho+rho_mass) S[i][i+2] = 1*rho/(rho+rho_mass) num_m += 1 else: S[i][i-2] = 1 S[i][i-1] = -4 S[i][i] = 6 S[i][i+1] = -4 S[i][i+2] = 1 # fixed endS[-2][-4] = 1

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S[-2][-3] = -4S[-2][-2] = 6S[-2][-1] = -4S[-1][-3] = 1S[-1][-2] = -4S[-1][-1] = 7

evals,evecs = eig(S) # solve for eigenvalues of matrix Sf = (abs(np.real(evals))*const**2)**0.5/2/np.pi # convert to frequency (Hz)plt.plot(f, [0]*290, 'o')plt.xlabel('Frequency (Hz)')

2. Disorder in spacing: aperiodic_simulation_eigenvaluefrom scipy.linalg import eigimport numpy as npfrom matplotlib import pyplot as plt

n = 290 # number of elementsY = 69*10**9 # Young's modulus for aluminiumd = 0.00314 # thicknesswid = 0.0254 # widthL = 2.904 # length of the aluminium platerho = 2700 # density of aluminiumdelta_x = L/n # the length between each element (1cm)rho_mass = 0.0726/(d*wid*delta_x)rho_motor = 0.295/(d*wid*delta_x)const = (Y*(d**2)/12/rho)**0.5/(delta_x**2) # time scaling factor

S = np.zeros((n,n)) # set up an empty nxn matrix

# fixed endS[0][0] = 7*rho/(rho+rho_motor)S[0][1] = -4*rho/(rho+rho_motor)S[0][2] = 1*rho/(rho+rho_motor)S[1][0] = -4S[1][1] = 6S[1][2] = -4S[1][3] = 1num_m = 0

# constructing matrix Sfor i in range(2,n-2): # moving m5 15cm down, m6 15cm up: if ((i+1-12)%10 == 0 and num_m < 27 and (i+1-12)/10 != 5 and (i+1-12)/10 != 6)\ or (i+1-12) == 3*10+5 or (i+1-12 == 7*10+5): # add a mass S[i][i-2] = 1*rho/(rho+rho_mass) S[i][i-1] = -4*rho/(rho+rho_mass) S[i][i] = 6*rho/(rho+rho_mass) S[i][i+1] = -4*rho/(rho+rho_mass)

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S[i][i+2] = 1*rho/(rho+rho_mass) num_m += 1 else: S[i][i-2] = 1 S[i][i-1] = -4 S[i][i] = 6 S[i][i+1] = -4 S[i][i+2] = 1 # fixed endS[-2][-4] = 1S[-2][-3] = -4S[-2][-2] = 6S[-2][-1] = -4S[-1][-3] = 1S[-1][-2] = -4S[-1][-1] = 7 evals,evecs = eig(S) # solve for eigenvalues of matrix Sf = (abs(np.real(evals))*const**2)**0.5/2/np.pi # convert to frequency (Hz)plt.plot(f, [0]*290, 'o')plt.xlabel('Frequency (Hz)')

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