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The time dependent term

The derivatives are calculated from the basic equation for time-of-flight INS spectroscopy, where Wn is the neutron mass  [Pg.566]

Where Atm is the time width of the moderator pulse in microseconds. Its energy dependence is specific to a given neutron source and is described empirically, A is the moderator constant, (the units of in Eq. (A3.10) are quite exceptional, see Table A3.1). The time channel uncertainty, A/ch, in microseconds, is usually negligible in comparison to the moderator pulse width. [Pg.567]


In steady-state problems 6/S.l = 1 and the time-dependent term in the residual is eliminated. The steady-state scheme will hence be equivalent to the combination of Galerkin and least-squares methods. [Pg.132]

The next step is to introduce temperature by averaging out the bath operators appearing in the time dependent terms of Eq.(51) [137] over an adequate ensemble. To this end, the partial trace (or sum of the diagonal elements) over the surrounding subsystem has to be taken. For the system in interaction, the effective Hamiltonian of the solvent Hmeff must be defined in such a way that the sum of HSeff+Hmeff leads to the... [Pg.307]

The time-independent term on the right-hand side of Eq. (73) gives zero because the contour C may be closed in the upper half-plane where Y00(2) is everywhere analytic (see Section II-D) we are then left with the time-dependent term, which in the limit t — oo only contributes at its pole 2 = 0. From Eqs. (72) and (73), we then get ... [Pg.178]

Solution We first select the control volume as shown by the dotted line in the figure, assuming that, at the downstream end of the control volume, the velocity profile in the free jet is flat. Next, we apply the macroscopic momentum balance, Eq. 2.5-3, to the control volume. We need be concerned only with the x component, because this is the only momentum that crosses the control volume boundaries. The flow is steady, and therefore the time-dependent term vanishes, as do the forces, since there are none acting on the control volume. Thus the equation reduces to ... [Pg.36]

In order to carry out the transformation, one can take the following auxiliary integral, which accumulates the time-dependent terms with p=t—t/ of... [Pg.55]

In his model, the time-dependent term playing the role of [Oo(7 in the present situation, is obtained by solving a set of linear equations, which is Eq. (306), except the fact that the matrix elements are time independent. [Pg.368]

Nuclear relaxation in paramagnetic complexes occurs due to the time dependent terms in the nuclear spin Hamiltonian. The amount of relaxation effect is dependent on the intensity of electron-nuclear interaction and the rate at which this interaction is interrupted. Thus the relaxation rates of ligand nuclei are determined by the two factors, namely, molecular structure and molecular dynamics in solution. Thus the relaxation rates of ligand nuclei shed light on molecular structure and dynamics in solution. [Pg.794]

If we let (l>i = m ro represent the successive roots of this equation, the difference between consecutive values will be of the order of tt. Measured in units of (Tr/roY, thus increases very rapidly. The significance of this is that the exponents in the time-dependent terms of Eq. (XIV.6.10) increase very rapidly, and at times of the order of magnitude of (miD — kh) = iD/rl — [where rtii is the first admissible root of Eq. (XIV.6.13)] the first term is down to 1/e of its original value and the subsequent terms will be found to be negligible. This time can be taken as the half-life for the approach to the stationary state, and for reactions near the first explosion limit it is usually found to be of the order of seconds or less. At times greater than that, the solution becomes essentially the stationary-state solution, V(r,0 - P(r) [Eq. (XIV.6.12)] with the constant B determined by the continuity condition [Eq. (XIV.G.4)] ... [Pg.450]

Because we are concerned with intensities of the various waves under steady-state conditions, we can omit the time-dependent term, so Eq. (4.2) becomes... [Pg.90]

In the fast spinning limit, the time-dependent terms of Eq. (21) are av-... [Pg.149]

In the transformed form the T - and Fn-dependent terms are accompanied by a phase-dependent term, exp( 8< )). These terms are also accompanied by the time-dependent terms exp co — ffiijf], which oscillate with the difference of the laser frequencies. This shows that in any attempt to calculate phase-dependent effects, it is important to assume that the lasers have equal frequencies. Otherwise, for unequal frequencies the time-dependent terms rapidly oscillate in time and average out over a long period of the detection time. Furthermore, we note from Eq. (75) that in the case of q = 1 a phase dependence can be observed even in the absence of the vacuum induced quantum interference terms (I)2 = 0). Only for q = 0, that is, when each laser couples to only one of the transitions, the phase terms solely depend on the vacuum induced quantum interference. However, this condition can be achieved only for an imperfect interference (p =/= 1) between the atomic transitions that the dipole... [Pg.101]

The fact that we call the exponent of the time-dependent term —iaxct instead of just a t is just for convenience and does not change the actual problem at all from the forms analyzed in the chapter. You should find... [Pg.887]

Since HB stars are in a long-lived phase of evolution, the time- dependent terms in the stellar structure equations are negligible and equilibrium solutions... [Pg.75]

All the time-dependent terms in Equation 50 assembled between the braces are independent of the oxygen consumption rate, v. This means that the rate at which the concentration C at any depth approaches a steady-state value (dC/dt) is independent of the zero order reaction rate constant. The time-dependent terms contain, however, the eddy diffusion coefficient and advection velocity, and the rate of approach to steady-state is therefore dependent on these two physical characteristics of the environment. [Pg.69]

In Equation 58, the time-dependent terms between the braces contain the decay constant A. Therefore, the rate of change in Ra-226 concentration at any depth (dC/dt) depends on the decay rate constant. Thus, in the case of a first-order reaction (radioactive decay), the rate of change in concentration depends on the reaction rate constant, whereas it has been shown in the preceding section that for a zero-order reaction (oxygen consumption), the rate of change in concentration (dC/dt) is independent of its rate constant. [Pg.73]

A new steady-state concentration (Ct = x>) will be attained when t tends to infinity in the time-dependent terms of Equation 58. [Pg.73]

Clearly, the time dependence of Eq. (5.130) is simpler than its length dependence. To eliminate [T], we pick any one of the time dependent terms on the righthand side, V or p/ix. Since the final dimensionless numbers will ultimately involve all quantities describing Eq. (5.129), and since we have already manipulated with p, let us pick V this time and combine it with Fjp and p/p in such a way that the time dependence disappears. Thus... [Pg.271]

This timescale can also be obtained from the complete solution of the problem in (12.8). Note that the time-dependent term approaches zero as the upper limit of the integration approaches zero or when l /Dgi r - Rp or equivalently when (r - RP)2/4 Dg. For a point close to the particle surface (r - Rp f< Rp and... [Pg.551]

Note that for a point where r Rp, the time-dependent term is practically zero because of the existence of the term (Rp/r) in front of the integral. One should not be bothered by the different numerical factor in the two timescales in (12.48) and (12.49). These timescales are, by definition, order-of-magnitude estimates and as such either value is sufficient for estimation purposes. [Pg.551]

By quasi-steady motion here, we mean an unsteady motion for which the time-dependent terms in the equations of motion and boundary conditions are of higher order in R than 0(.R In R). [Pg.368]

The system-bath coupling is described by the time-dependent term... [Pg.103]

The first-order equation to be satisfied is similar to the time-independent case but, of course, has a non-zero right-hand-side arising from the time-dependent term in the TDHF equation. Here is eqn ( 26.7), slightly rearranged... [Pg.324]

The cycle for the chosen example is the rotation period. Hence, to calculate the lowest-order average Hamiltonian one has to average over one rotation period. The time-dependent terms in Eq. (36) are exp(i(m n)uv t) the average of these over a rotation period is given by... [Pg.193]

The boundary conditions used in conjunction with the above equations can vary and are to some degree simulation dependent. Normally, the current density, water flux, reference potential, and water chemical potential are specified but two water chemical potentials or the potential drop in the membrane can also be used. If modeling more regions than just a membrane, additional mass balances and internal boundary conditions must be specified. In addition, for modeling the membrane in the catalyst layers, rate equations are required for kinetics and water transfer among its various phases [40]. The above equations are also valid only for the steady-state case (the time-dependent terms have been ignored). [Pg.166]


See other pages where The time dependent term is mentioned: [Pg.8]    [Pg.411]    [Pg.411]    [Pg.51]    [Pg.64]    [Pg.223]    [Pg.106]    [Pg.13]    [Pg.137]    [Pg.13]    [Pg.359]    [Pg.6146]    [Pg.293]    [Pg.132]    [Pg.128]    [Pg.566]    [Pg.79]    [Pg.208]    [Pg.660]    [Pg.844]    [Pg.6145]    [Pg.51]    [Pg.64]    [Pg.223]   


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Term dependence

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