Expansion period

The foam density at any time during the expansion period results from... 

Fig. 4 Plugging period of Fig. 5 Expansion period of the solids. the solids.

In discussing experimental techniques, Dunn and Taylor [134] mention that while dilatometric studies with degassed materials eliminated the induction period, the start of the reaction was difficult to observe since the initiation of the polymerization takes place during the thermal expansion period as the equipment was thermostated. 

There sure are. While the Big Bang theory is probably the most widely known and accepted theory surrounding the origins of the current universe, there are still other theories being explored and proposed. Some of these are much more scientifically feasible than others take a look around the Web and you can find lots of different ideas out there. One alternative that has received notable attention describes the universe as a continuous cycle of expansion and rebirth the expansion period, similar to that de-... 

Schematic drawing illustrating these aspects in case of NbsGe is presented in Fig. 27.6. The Fu phonon mode covers out-of phase stretching vibration of two perpendicular Nb chains in two planes - see Fig. 27. Id. For simplicity, drawing of only a single chain of Nb atoms in a plane (e.g. b-c plane) is sketched in Fig. 27.6. For equilibrium high-symmetry structure (Req) on the crude-adiabatic level, the highest electron density is localized at equilibrium position of Nb atoms in a chain - Fig. 27.6a. For distorted nuclear geometry (Rd,cr) in the Fn mode, electron density is polarized and the highest value is shifted into the inter-site positions-bipolarons are formed. The Fig. 27.6b corresponds to compression period in stretching vibration of Nbl-Nb2 which induces increase of Nbl-Nb2 inter-site electron density and decreases of Nb2-Nb3 electron density. For an expansion period. Fig. 27.6c, situation is opposite. Inter-site electron density is decreased for Nbl-Nb2 and increased for Nb2-Nb3. On the lattice scale, increase and decrease of electron density is periodic. On the adiabatic level, alternation of electron density is bound to vibrations at equilibrium nuclear positions (Fig. 27.6a-c).

Fig. 27.6 Schematic drawing of vibration periods and electron density in Nb-chain of A15 superconductors (NbsGe, Nb3Al,) Equilibrium geometry characterizes the line (a). Lines (b) and (c) represent compression and expansion period in vibration mode. The circles depict circumferences of flux-circles of degenerate antiadiabatic ground state. Lines (d), (e) represents cooperative transversal positions of Nb atoms at circumferential motion in antiadiabatic state -see text...

Flux-circle position 90" d Expansion period in vibr.-180°... 

This equation may be solved by the same methods as used with the nonreactive coupled-channel equations (discussed later in section A3.11.4.2). Flowever, because F(p, p) changes rapidly with p, it is desirable to periodically change the expansion basis set ip. To do this we divide the range of p to be integrated into sectors and within each sector choose a (usually the midpoint) to define local eigenfimctions. The coiipled-chaimel equations just given then apply withm each sector, but at sector boundaries we change basis sets. Let y and 2 be the associated with adjacent sectors. Then, at the sector boundary p we require... 

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

Each logarithm in the last temi can now be expanded and the (—n)th Fourier coefficient arising fi om each logarithm is — jn) zk-Y- To this must be added the n = 0 Fourier coefficient coming from the first, f-independent term and that arising from the expansion of second term as a periodic function, namely. 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

The speed of the method comes from two sources. First, all of the macroscopic cells of the same size have exactly the same internal structure, as they are simply formed of tessellated copies of the original cell, thus each has exactly the same multipole expansion. We need compute a new multipole expansion only once for each level of macroscopic agglomeration. Second, the structure of the periodic copies is fixed we can precompute a single transfer... 

Hence we see that this simple periodic function has just two terms in its Fourier series. In terms of the Sine and Cosine expansion, one finds for this same f(t)=Sin3t that an = 0, bn =... 

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. 

Strong growth in demand during the period 1985—95 from existing and new appHcations led to firm pricing and expansion of world capacity. Total world demand in 1995 was estimated to be approximately 60,000 metric tons. Balance of capacity and demand at that time resulted in a selling price of 16.00/kg. More recently (ca 1997), prices have dropped to less than 10/kg. 

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