Building and Destroying Symmetry in 1-D Elastic Systems
J. Flores, G. Monsivais, P. Mora, A. Morales, R. A. Méndez-Sánchez, A. Díaz-de-Anda, L. Gutiérrez
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Building and Destroying Symmetry in 1-DElastic Systems
J. Flores ∗ , G. Monsivais ∗ , P. Mora ∗ , A. Morales † , R. A. Méndez-Sánchez † ,A. Díaz-de-Anda † and L. Gutiérrez † ∗ Instituto de Física, Universidad Nacional Autónoma de México, A. P. 20-364, 01000, México,D. F., México † Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, A.P. 48-3, 62251,Cuernavaca, Morelos, México
Abstract.
Locally periodic rods, which show approximate invariance with respect to translations,are constructed by joining N unit cells. The spectrum then shows a band spectrum. We then breakthe local periodicity by including one or more defects in the system. When the defects follow acertain definite prescription, an analog of the Wannier-Stark ladders is gotten; when the defects arerandom, an elastic rod showing Anderson localization is obtained. In all cases experimental valuesmatch the theoretical predictions. Keywords:
Vibrating rods, Anderson localization, Wannier-Stark ladders, Symmetry breaking
PACS:
We dedicate this paper to Marcos Moshinsky, who was always in love with symmetry
INTRODUCTION
Due to similarities between the stationary Schrödinger equation and the time indepen-dent wave equation, several classical analogs of quantum systems have been analyzedin the last few years. Among them, mechanical [1], microwave [2], optical [3, 4, 5, 6],and elastic systems [7] have been studied along these lines. This approach is sometimes,jokingly speaking, referred to as megascopic physics .In this paper we shall present a review of some analogs that we have recently discussedusing one-dimensional elastic rods. Normal-mode amplitudes and frequencies, whichare the classical analogs of quantum wave functions and eigenenergies, respectively, areobtained experimentally and compared with numerical calculations. The experimentalset up is shown in Fig. 1 where an electromagnetic acoustic transducer (EMAT), con-sisting of a permanent magnet and a coil, is used to excite and detect the vibrations ofthe rod. The transducer operates through the interaction of eddy currents in the metal-lic sample with a permanent magnetic field and an oscillating one. This EMAT is veryflexible and selective; it can excite or detect either compressional, torsional, or flexuralvibrations according to the relative position of the permanent magnet and the coil, asshown in Fig. 2. Experimental details are discussed in Refs. [7, 8].We shall first consider the building up of a rod which shows local translationalinvariance. Since we construct it with a finite number N of unit cells (see Fig. 3), werefer to it as a locally periodic rod [9]. A band spectrum emerges when N grows. This AMAC ExciterDetector coil MagnetEMATcoilDetectorAluminum rod OSCILLATORLOCK−INAMPLIFIERPREAMPLIFIER−+CursorMagnet POWER AMPLIFIER
FIGURE 1.
Experimental setup to measure the elastic vibrations of an aluminum rod. In the inset theEMAT is configured for torsional waves. The exciter should be configurated in the same way. (a)(c)(d) (b)(e) A l u m i n u m b a r CoilMagnet
FIGURE 2.
Different configurations of the permament magnet and of the coil, excite different modes:(a) y (b) compressional; (c) y (d) torsional; (e) flexural result is also valid for flexural vibrations, which obey instead a fourth order differentialequation [10, 11]. If the unit cell consists of two small different rods, an elastic analogof the diatomic chain results: optical and acoustical bands are formed as N grows.We then induce a symmetry breaking by changing the length l of one of the centralsmall rods that form the locally periodic system of Fig. 3. A normal-mode frequency isfound in the forbidden band and corresponds to a localized state. The number of rodswith different lengths is then increased until all of them are distinct. We proceed intwo different manners. If the lengths of the rods are altered following a systematic law,the analog of the Wannier-Stark ladders can be obtained. If the lengths are selected asrandom numbers, the elastic analog of Anderson localization is achieved. e e unit cell e z FIGURE 3.
Locally periodic rod. The unit cell consists of a small cylinder of length l and two smallercylinders of length e / In what follows we shall present our results with the aid of a set of figures, omittingthe details, which the interested reader can find in the references given to our publishedwork.
BUILDING TRANSLATION SYMMETRY
In Fig. 4 we present the experimental spectrum of compressional waves of a rod as thenumber N of unit cells is increased. When N is large enough the spectrum consists ofallowed and forbidden bands, each consisting of N levels. The wave amplitudes are alsoobtained and compared with the ones obtained numerically using the transfer matrixmethod [7]. The waves are extended along the rod. This is consistent with Bloch’stheorem and what was found numerically for a Kronig-Penney model [12] or for anumber of other systems [9]. The same result holds for torsional vibrations [7] and,even more, for flexural vibrations [11]. This is interesting in itself since the latter obeyan equation which is not of the Helmholtz type, but rather a fourth order differentialequation, either the Bernoulli-Euler one or the more accurate one at higher modes, theso called Timoshenko equation [10]. The band spectrum for flexural vibration is shownin Fig. 5. DESTROYING THE SYMMETRY
We start by introducing in the locally periodic system depicted in Fig. 3 a single smallrod with a different length l = l + h . This is referred to in solid state physics as atopological defect, since it destroys the long range order of the locally periodic orsymmetrical rod. We have measured and calculated the spectrum for several values of h [13]. As given in Fig. 6, there appears a frequency in the first forbidden band of theordered rod. This is consistent with what one learns from the band theory of crystals.However, in the second forbidden gap two levels lie. In some of the higher gaps twofrequencies are also found. These state are localized, as seen is Fig. 7. The existence of _________________________ __________________________ ______________________________ __________________________________ ______________________________ ______________________________________ _____________________________________________ N fr e qu e n c y ( k H z ) FIGURE 4.
Spectrum of compressional waves of a locally periodic rod. As the number of unit cellsincreases, a band spectrum emerges. mode number fr e qu e n c y ( k H z ) mode number fr e qu e n c y ( k H z ) FIGURE 5.
Bloch theorem applies independently of the structure of the Lagrangian. Band spectra arealso observed for torsional waves and even for flexural vibrations. Here we show the result for flexuralvibrations. The circles correspond to the uniform rod, experiment empty circles, Timoshenko’s beamtheory is given by the filled circles. The squares show the results for the locally periodic rod, 12 cells,experiment empty squares, Timoshenko’s beam theory is given by the filled squares. uch localization was to be expected from the independent rod model because, if thereexists a normal-mode frequency proportional to ( l + h ) − , the neighboring rods of thedefect do not resonate. fr e qu e n c y ( k H z ) FIGURE 6.
One or more normal-mode frequencies appear in the forbidden gap. -1 A m p lit ud e ( a r b . un it s ) FIGURE 7.
The states in the forbidden band are localized at the topological defect. Mode 24 continuousline, and mode 25, dashed line. e can now go on introducing more defects in the locally periodic rod. We first do it inan ordered way to simulate, for example, the effect of an external constant static electricfield on a charged quantum particle that is moving in a periodic potential. This is thecase, indeed, with the Wannier-Stark ladders (WSL). In 1960, Wannier [14] suggestedthat a static external electric field would destroy the electron band spectrum and insteadproduce a picket fence one, in which the spacing between the energy levels is a constant,proportional to the electric field and the lattice constant. The WSL in solid state werefinally observed decades later using superlattices [15]. They have recently been found inoptics as well [3, 4]. As will be shown, by modifying appropriately the small rods in asystematic way one can obtain an elastic analog of WSL [11].Let us first use an independent rod model. The normal-mode torsional frequencies f ij of a rod of length l i and wave velocity c i are given by the well known expression [10] f ij = c i l i j , (1)where j is the number of nodes in the wave amplitude. We consider two options:either l i = l / ( + i g ) and c i = p G / r , with g an adimensional parameter, or l i = l and c i = c ( + i g ) . This happens for systems A or B, respectively, which are shown inFig. 8 [16]. Here r is the mass density and G the shear modulus of the rods and c thewave velocity. w l ee l i i h ’ A B
FIGURE 8.
Elastic rods showing the Wannier-stark ladders
In the first case, which is the elastic analog of the optical system discussed in Ref. [3],we have f ij = l s G r ( + i g ) j (2)and the difference D f ij = f i + j − f ij is equal to D f ij = n L s G r j g = D f j = j D (3)which is independent of the index i .e should remark that in both cases D ji , is independent of i , a fact which willlead to a WSL as we will now show. In Fig. 9 the WSL measured for system A arecompared with the numerical values obtained using the transfer matrix method. Thenormal-mode amplitudes are localized as one would expect from the independent rodmodel. This is shown in Fig. 10 for several modes in a given ladder; experimentalwave amplitudes fit very well the theoretical ones, as exemplified in Fig. 11. The sameconclusions can be obtained by analyzing system B; results for system B can be foundin References [16, 17]. ______________ ______________ _____ _______________ _____________ _____ ___ __ __ __ __ __ __ __ __ _ __ _ __ _ __ _ __ _ _ _ j = 1 j = 2 j = 3 j = 4 Wannier-Stark ladder number fr e qu e n c y ( k H z ) FIGURE 9.
Wannier-Stark ladder spectrum for system A
A very different effect appears if one now introduces random defects: a disorderedrod is built. This is reminiscent of the well known Anderson localization. The quantummechanical states in a disordered chain of potentials show always a localization length x . Let us now consider what happens in the elastic case, when we analyze a 1-D systemformed by N small rods each of length l i , with l i = l + e i and e i an independent randomvariable distributed according to P ( e i ) = / − ≤ e i ≤ P ( e i ) = P b ( z ) = Az b exp ( − a z b + ) ; z = s / d (4) | |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| ||||f = 23.8190506 kHzf = 25.1494993 kHz|||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||||||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||||||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| ||||f = 26.4663602 kHz|||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| |||| ||||f = 27.7694067 kHzf = 27.7694067 kHz FIGURE 10.
Wannier-Stark ladder wave amplitudes obtained with the transfer matrix method. |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| position (mm) | a m p lit ud e | FIGURE 11.
Experimental wave amplitude of the Wannier-Stark ladder and its comparison with thetheory. with A = b + a = h G (cid:16) b + b + (cid:17)i b + , s the space between frequencies and d themean level spacing. When b = b = position (m) -505 a m p lit ud e ( a r b it r a r y un it s ) FIGURE 12.
Localized wave function calculated with the method of the Poincaré map of a disorderedrod. position (m) -0.4-0.200.20.4 a m p lit ud e ( a r b it r a r y un it s ) FIGURE 13.
Measured localized wave amplitude of a disordered rod. (GOE) corresponds to the value b = . b is a function of frequency, showing the rich-ness of elastic systems; this is shown in Fig. 15. If one eliminates the frequency, Fig. 16 frequency (Hz) -1 l o ca li za ti on l e ng t h ( m ) FIGURE 14.
Localization length as a function of frequency frequency (Hz) -1 B r ody r e pu l s i on p a r a m e t e r FIGURE 15.
Repulsion parameter as a function of frequency is obtained, showing that the repulsion parameter is to a good approximation a linearfunction of the localization length. A similar result was obtained numerically using atight-binding Hamiltonian with nearest neighbor interaction and diagonal disorder [20]. localization length (m) B r ody r e pu l s i on p a r a m e t e r FIGURE 16.
Repulsion parameter vs localization length
CONCLUSIONS
We have discussed, both from experimental and numerical points of view, differentconfigurations of elastic rods. They serve as classical analogs of 1-D quantum systems.We first build up a locally periodic rod and a band spectrum emerges. We then destroy thelocal translational symmetry by introducing defects in the rod. If one does it followingspecific rules, the analog of Wannier-Stark ladders can be obtained. If the defects haverandom properties, features of the Anderson transition take place. In all cases, theexperimental and theoretical results coincide. The elastic analogs are useful to betterunderstand several features of one-dimensional models dealt with in the quantum theoryof solids. In particular, the wave amplitudes can be measured, which is only partiallypossible for microscopic systems.
ACKNOWLEDGMENTS
This work was supported by DGAPA-UNAM under projects IN111308, IN119509 andby CONACyT under projects 79613 and 82474.
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