Doorway States and Billiards
J. A. Franco-Villafañe, J. Flores, J. L. Mateos, R. A. Méndez-Sánchez, O. Novaro, T. H. Seligman
aa r X i v : . [ n li n . C D ] N ov Doorway States and Billiards
J. A. Franco-Villafañe ∗ , J. Flores †, ∗∗ , J. L. Mateos † ,R. A. Méndez-Sánchez ∗ , O. Novaro †,‡ and T. H. Seligman ∗ ,§ ∗ Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, P.O. Box 48-3, 62251Cuernavaca Mor., Mexico. † Instituto de Física, Universidad Nacional Autónoma de México, P.O. Box 20-364, 01000 México,D. F., Mexico. ∗∗ Centro Internacional de Ciencias, A. C., P.O. Box 6-101 C.P. 62131 Cuernavaca, Mor. Mexico. ‡ Member of El Colegio Nacional § Centro Internacional de Ciencias, A. C., P.O. Box 6-101 C.P. 62131 Cuernavaca, Mor., Mexico.
Abstract.
Whenever a distinct state is immersed in a sea of complicated and dense states, thestrength of the distinct state, which we refer to as a doorway, is distributed in their neighboringstates. We analyze this mechanism for 2-D billiards with different geometries. One of them issymmetric and integrable, another is symmetric but chaotic, and the third has a capricious form.The fact that the doorway-state mechanism is valid for such highly diverse cases, proves that it isrobust.
Keywords:
Doorway states, sedimentary valleys, seismic waves
PACS:
We dedicate this paper to the memory of Marcos Moshinsky.
INTRODUCTION
The doorway state mechanism, which was introduced in nuclear physics a long timeago [1], is effective when a “distinct” state is coupled to the scattering channel andalso to a sea of more dense and complicated states. Prime examples of doorway statesare isobaric analogue states and giant resonances in nuclei. The results given in Fig. 1for the photonuclear reaction Al ( p , g ) Si are well known for nuclear physicists andshow that the cross section acquires more structure as the energy average decreases. Inthe last few years, this mechanism has been found in many instances. Doorway stateshave been observed in many quantum systems: atoms and molecules [2], clusters [3],quantum dots [4], and in C60 fullerenes [5, 6]. Furthermore, they also appear in classicalwave systems: flat microwave cavities with a thin barrier inside [7], and even also in theseismic response of sedimentary valleys [8].The doorway state is not an eigenstate of the whole system. Therefore, the strength ofthe doorway state spreads among the eigenstates within some energy region around thedistinct state energy. This produces a strength function whose width is commonly knownas spreading width . In the simplest case, the strength function follows the Breit-Wignerform, a Lorentzian, as a function of energy [9].We therefore see that the doorway state concept is a unifying one in physics coveringa wide range of quantum and classical systems, established on scales ranging from
IGURE 1.
A very well known nuclear physics example of a giant resonance, intermediate and finestructure obtained with the nuclear reaction Al ( p , g ) Si. As the average interval diminishes the finestructures appears. fermis to tens of kilometers. In this paper we shall analyze this concept from anotherpoint of view, discussing its occurrence in two-dimensional billiards of different shapes:a rectangle, a stadium, and one that resembles the irregular shape of a sedimentarybasin [10]. The first is symmetric and integrable, the second is symmetric but chaotic,and the last one shows neither integrability nor symmetry. We will show numericallythat the doorway state mechanism is indeed robust.
II. A MODEL FOR THE ELASTIC DOORWAY STATE
In order that a system shows the phenomena described by the strength function, thepresence of a doorway state and a sea of complicate states must be assured. In theclassical systems analyzed up to now, this has been achieved in different ways. In thecase of microwave cavities, a doorway has been produced by introducing a thin barriernside a rectangular cavity. The barrier produces several states, called superscars, whichhave been detected experimentally and act as doorway states [7]. A different systemis a rigid parallelepiped cavity with an elastic membrane on one of its walls. This isan acoustic realization of the doorway states, provided now by the normal-mode statesof the elastic membrane coupled to the denser states of the fluid inside the otherwiserigid walls forming the cavity. In this case [11], the membrane states act literally asdoorways, since the energy of the scattering channel enters the fluid inside the cavitywhen the membrane excites the interior acoustic states. An elastic doorway state is alsopresent when the seismic response of sedimentary basins covered by a bounded regionof soft material is considered, as we shall now describe. The analysis of this responsewill serve us as a guide to introduce in billiards a model to study the strength-function.What is the doorway state in the seismic case? It has been known for a long time togeophysicists [12] that when a sedimentary layer is covered by a much softer material(such as happens, for example, on the ocean floor) a coupling occurs between evanescentSP waves in the soft layer and Rayleigh-type waves on the interface. The couplingcondition is [13] 0 . b < a = v , (1)where v is the phase velocity of the coupled mode and b , a are the S-wave velocityin the sediment and the P-wave velocity in the softer region, respectively. The couplingoccurs when the phase velocity of the dispersive Rayleigh waves is equal to the soundvelocity a of the very soft terrains. The coupled mode has many features in commonwith what is known as an Airy phase [12]. In particular, they are monochromatic and oflong duration. We have called this mode a PR mode [13].If the soft-clay terrain is bounded, as happens for example in the Mexico City basin aswell as in San Francisco, Kobe and many other cities around the world, once established,the PR mode reflects at the soft clay boundaries due to the large impedance contrastbetween the clays and the sediments surrounding them, and the very fact that theboundary layer on and near where it lives, terminates at this boundary. These surfacewaves have to be evanescent outside the interface, they imply horizontal compressionmovement in the soft clay above the interface, and thus strongly couple to several normalmodes in the soft clay bed located in the horizontal xy -plane. These modes provide thesea of complicated states [14].Since the source of the seismic waves is far away from the basin, we represent thePR mode by a plane wave exp ( i k · r ) , where the direction of k varies for different earth-quakes corresponding to different epicenters. Here r fixes a point within the bounded softterrain in the horizontal xy -plane. The magnitude of k for the PR mode is k = pn / a .What is the sea of dense and complicated states in the seismic case? The sea is formedby the normal-mode states with amplitudes f i and frequencies n i that correspond to theeigenfunctions of a 2-D Helmholtz equation (cid:209) f i + k i f i = k i = pn i / a , where a is the P-wave velocity inthe clays. We use Neumann boundary conditions ˆ n · (cid:209) f i =
0, where ˆ n is a vector normalto the boundary. Note that the Neumann conditions are somewhat arbitrary, but a similarcalculation with Dirichlet conditions yields qualitatively the same results.he spreading of the doorway state exp ( i k · r ) among the normal modes f i is thengiven by A ( n i ) = (cid:12)(cid:12)(cid:12)(cid:12) Z exp ( i k · r ) f i ( x , y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) (3)In a few words, our model for the doorway states in two-dimensional billiards is thefollowing: a monochromatic plane wave exp ( i k · r ) , with k = pn / a represents thedoorway state and the sea of complicated states is formed by the eigenstates f i of abilliard with a given boundary, phase velocity a and Newmann boundary conditions. III. SPREADING WIDTH IN BILLIARDS
In order to determine the parameters entering the calculations, we shall use data cor-responding to the seismic response of the Valley of Mexico. This is a basin formed byhard enough sedimentary deposits ( a = b = a = b = n = . A v e r a g e F ou r i e r S p ec t r a FIGURE 2.
Averaged Fourier spectrum obtained from seismograms of the earthquake measured inMexico City of magnitude 6.9, 1989 April 25. The continuous line corresponds to an average of stationslocated in the old lake bed; the dashed line to stations located in the rock zone.
We now present the resulting strength function A ( n i ) for three different billiards: arectangle, which is symmetric and integrable; a stadium, which has much the samesymmetry as the rectangle, but which is chaotic; and a third one, which is neither F ou r i e r s p ec t r u m m a gn it ud e ( c m / s ) FIGURE 3.
Fourier spectrum obtained from a seismogram measured in a station located on the lakebed clay in Mexico City during the earthquake of magnitude 7.1, 1997 January 11. symmetric nor integrable and has the shape of the lake. The linear dimensions ofthe three systems are comparable and a , n , which depend on the sediment or clayproperties, are taken to be equal to the appropriate values for the lake.The strength function for the first case can be obtained analytically and for the twoother cases it was obtained numerically, using a finite element method to obtain the waveamplitudes f i . The numerical results were calculated using the finite element methodwith linear polynomials. The region was discretized using 8272 points located in arectangular grid. The size of the grid is 118 m. We verified the accuracy of the programto be better than one percent using a region with a rectangular boundary, which can becomputed analytically.In Figs. 4 to 6 we show that a strength function A appears for the three systemsconsidered, it is plotted as a function of frequency. The peaks in this figure are relatedto the wave amplitudes shown as insets. In particular, we see from Fig. 3 that ourcalculation agrees qualitatively with what is observed in many seisms in Mexico City.In any case the strength function appears independently of the billiard symmetry, itsintegrability or lack of. It can therefore be deemed robust. CONCLUSIONS
As a conclusion, one can say doorway states appear in very diverse systems and circum-stances. They show up in many quantum and classical systems and also in billiards ofquite different shapes. Their existence has equivalent consequences on the response ofthe system not regarding whether this is integrable or chaotic, symmetric or arbitrary.
IGURE 4.
Strength function phenomenon for a rectangular billiard.
FIGURE 5.
Strength function phenomenon for a stadium.
Furthermore, they also appear covering many orders of magnitude in the characteristiclength of the system. As a matter of fact, the results of the seismic response of sedimen-tary valleys imply that these lengths range from fermis up to tens of kilometers.
IGURE 6.
Strength function. Some of the wave amplitudes of the lake, that correspond to the peaksof A , are shown. ACKNOWLEDGMENTS
This work was supported by DGAPA-UNAM under project PAPIIT-IN111308 and byCONACyT under project 79613.
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