Closed orbits

Therefore it is shown that each closed orbit contains an element (i i, B2,0,0) such that both Bi,B2 are semisimple satisfying [ 1,52] = 0. Associating the orbit with the set of simultaneous eigenvalues of (Hi, i 2), we have the claimed isomorphism. ... 

Proof. First suppose is a complex torus T. Taking a basis of V, we may suppose we are in the above situation. (If some coordinates are 0, we replace P by a subspace.) We may also assume that Y is the unique closed orbit in the closure of x. The uniqueness follows from the existence of T -invariant polynomial which separates two disjoint T -invariant closed subsets (Theorem 3.3). 

It is easy to see a matrix has a closed orbit if and only if it is diagonalizable. Hence the set of closed orbits can be identified with the set of eigenvalues. On the other hand, a matrix B with [B, 5I] = 0 (i.e. a normal matrix) can be diagonalizable by a unitary matrix. Hence the quotient space is also identified with the set of eigenvalues. The identification can be seen directly in this example. 

Figure 22. The incident beam converges to a closed orbit.

We now can get further information on the electronic configuration by studying the magnetism of the compounds. Any electron revolving around a nucleus in a closed orbital is equivalent to a circular electric current, and thus produces a magnetic moment. Usually these orbital magnetic moments are directed in such a way that they just cancel each other, in which case the orbital moments are of no further interest for our present purpose. 

For such phase space representations with one variable, or indeed with any number of independent concentrations, there are a number of rules which the trajectories must obey. In particular, trajectories cannot cross themselves, except at singular points (the stationary states) or if they form closed orbits (such as limit cycles or some other forms we will introduce later). Also, the trajectories cannot pass over singular points. The first of these rules is perhaps most easily shown for two-dimensional systems, where we have a two-dimensional phase plane. Let us assume that the rate equations for the two independent concentrations (or concentration and temperature), x and y, can be written in the form... 

Curry, J. H., 1979, An algorithm for finding closed orbits. In Global Theory of Dynamical Systems (Edited by Nitecki, Z. and Robinson, C.), Lecture Notes in Math., Vol. 819, Springer, New York. 

Electron synchrotrons or storage rings use magnetic fields to bend the electrons onto a closed orbit. SR is produced at each of these bending magnets. The emitted... 

F2 F4 - 2F and F3 F4 - Fi, correspond to the forbidden orbits. From a quantum-mechanical point of view there is no semiclassical closed orbit to explain these frequencies. However, they can be understood in the frame of the quantum interference (QI) model [10] as two-arms Stark interferometers [11]. Within the QI model [10] the temperature damping of the oscillation amplitude is given by the energy derivative of the phase difference ((pi -cpj ) between two different routes i and j of a two-arms interferometer. This model states that 5(cpi - cpj) / de = ( /eB) <3Sk / de, where Sk is the reciprocal space area bounded between two arms. Since 3(difference between the effective masses of the two arms of the interferometer, the associated effective mass is given by m = mj - mj, where nij and mj are the partial effective masses of the routes i and j. In our case an interferometer connected with the frequency F3 consists of two routes, abcdaf and abef and another interferometer, connected with the frequency F2, includes two cyclotron orbits, abcdaf and abebef (see Fig 5). 

M 8] [P 7] The effect of switching between the flow fields (a) and (d) given in Figure 1.18 at various periods T was analyzed (see Figure 1.19) [28], For very high alterations, a simple superposition of the flow fields is achieved. Elliptic fixed points surrounded by closed orbits (tori) of various periods are found. 

The moment of inertia of a ring of particles, mrki was used as the criterion for stability to define a closed orbit that combines circular motion with simple harmonic displacements. A more general discussion that substantiates the derivation is given by Goldstein [11]. The frequency of revolution is obtained in the form of a square root, defined by a set of integers,... 

The closed orbit on the left represents a system in vibration (also called libration) and the open orbit on the right corresponds to rotation. Vibra-... 

Another demonstration of the validity of these calculations is provided by BEDT-TTF-based salts. The calculated Fermi surface of these materials exhibit closed orbits characteristic of two-dimensional electronic interactions and this has been confirmed experimentally. For example, in the case of (BEDT-TTF)2I3, the calculated surface of these orbits (Fig. 21) [61] agrees well with the one measured by magnetic experiments [161]. However, the overall good agreement between calculation and experiment must not hide the fact that some qualitative discrepancies may arise in some cases. For example, (TMTTF)2X salts exhibit a resistivity minimum at a temperature at which no structural transition has yet been observed. The resistivity minimum is not explained by the one-electron band structure, and to account for this progressive electron localization, it is necessary to include in the calculations the effect of the electronic correlations [162]. Another difficulty has been met in the case of the semiconducting materials a -(BEDT-TTF)2X, for which the calculated band structure exhibits the characteristic features of a metal [93,97,100] and it is not yet understood... 

For a chemical reaction system, the characteristics of the periodic solutions are uniquely determined by the kinetic constants as well as by the concentrations of the reactants and final products. Starting from the neighborhood of steady state as an initial condition, the system asymptotically attains a closed orbit or limit cycle. Therefore, for long times, the concentrations sustain periodic undamped oscillations. The characteristics of these oscillations are independent of the initial conditions, and the system always approaches the same asymptotic trajectory. Generally, the further a system is in the unstable region, the faster it approaches the limit cycle. 

The Lotka-Volterra type of equations provides a model for sustained oscillations in chemical systems with an overall affinity approaching infinity. Perturbations at finite distances from the steady state are also periodic in time. Within the phase space (Xvs. Y), the system produces an infinite number of continuous closed orbits surrounding the steady state... 

Following Rutherford, Bohr depicted the hydrogen atom to be a positively charged nucleus of very small dimensions and an electron describing closed orbits around it. Bohr used the term... 

Closed orbits are of special interest for the semiclassical quantization of the helium atom. For a closed orbit F we have... 

The second effect, the rapid or fast oscillations, also briefly mentioned in Sect. 2.2.3, shows many similarities to the usual SdH effect and was, therefore, in the beginning interpreted as being due to closed orbits in the DW state. However, as pointed out already, the resistance oscillations, although periodic in 1/5, show a temperature and magnetic field dependence which is not understandable within the usual Lifshitz-Kosevich theory (3.6). 

All other a-phase salts (M = K, Rb, and Tl) showed AMRO periodic in tan 0, too. In contrast to the magnetoresistance shape in the former materials no resistance maxima as expected for a 2D warped cylinder but instead sharp resistance minima were observed [289, 309, 310, 311]. Figure 4.14a shows examples of this behavior for Q -(ET)2KHg(SCN)4 and a-(ET)2RbHg(SCN)4 at R = 11T and T — 1.4K [312]. Very prominent minima periodic in tan are visible in the magnetoresistance. These minima cannot be explained by the 2D warped FS model but are attributed to Lebed-like oscillations of the ID bands [313] (see also Sect. 3.3). From the calculated band structure shown in Fig. 2.20 one can see that the FS of the a phase consists of 2D closed orbits in the Brillouin zone corner and two ID open sheets perpendicular to the a direction. The maximum amplitude of the AMRO, however, was not observed for a rotation of the field parallel to these sheets but for angles 20° and 24° towards the a direction of the M = K and M = Rb crystals, respectively (see Fig. 4.14a). 

In the reconstructed FS the original quasi-lD bands disappear under the SDW potential but a new pair of open sheets tilted by against the original ones appear. Small lens-like closed orbits (Ai) remain which may be separated by only small energy barriers from the open orbits. Therefore, it is suggested... 

The main difference between the (3" structure compared to the / phase is the direction of the strong intermolecular interactions. Due to the smaller anion size the interaction directions are at 0°, 30°, and 60°, respectively, instead of face-to-face (90°) overlaps [335]. The more complicated interstack interaction results in a more anisotropic band structure with ID and 2D energy bands. There exists considerable disagreement between different band-structure calculations which might be caused by small differences in the transfer integral values [332, 335, 336]. One calculated FS based on the room temperature lattice parameters is shown in Fig. 4.27a [335]. Small 2D pockets occur around X and two ID open sheets run perpendicular to the a direction. In contrast, the calculation of [332] (not shown) revealed a rather large closed orbit around the F point. 

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