Frequency stability

Accurate control of potential, stability, frequency response and uniform current distribution required the following low resistance of the cell and reference electrode small stray capacitances small working electrode area small solution resistance between specimen and point at which potential is measured and a symmetrical electrode arrangement. Their design appears to have eliminated the need for the usual Luggin capillary probe. 

Therefore, the so-called stability frequency is often used to describe the stability of a fluid system. In the case of a vertically stratified water column, the appropriate quantity is called the Bmnt-Vaisala frequency, N. It is defined by... 

Since it is assumed that the density gradient is not influenced by dissolved chemicals, the stability frequency, N, can be calculated from the temperature profiles alone ... 

Figure 2 The correlation between stability frequency, N, and vertical turbulent diffusivity, Ez, according to Eq. 22-32 yields q = 0.5.

Figure 22.7 Vertical turbulent diffu-sivity Ez versus square of stability frequency V2 in two Swiss lakes (see Eq. 22-32). (a) For Umersee (maximum depth 196 m), a basin of Lake Lucerne, the data refer to 10-100 m depth and indicate shear-produced turbulence. (b) For Zugersee (maximum depth 198 m) the values are calculated for an extreme storm of about two days duration. The data refer to the depth interval between 10 and 70 m they show a mixture between turbulence production by local shear and large-scale motion. (Fromlmboden and Wuest., 1995.)...

Abstract. We present a frequency comparison and an absolute frequency measurement of two independent -stabilized frequency-doubled Nd YAG lasers at 532 nm, one set up at the Institute of Laser Physics, Novosibirsk, Russia, the other at the Physikalisch-Technische Bundesanstalt, Braunschweig, Germany. The absolute frequency of the l2-stabilized lasers was determined using a CH4-stabilized He-Ne laser as a reference. This laser had been calibrated prior to the measurement by an atomic cesium fountain clock. The frequency chain linking phase-coherently the two frequencies made use of the frequency comb of a Kerr-lens mode-locked Ti sapphire femtosecond laser where the comb mode separation was controlled by a local cesium atomic clock. A new value for the R.(56)32-0 aio component, recommended by the Comite International des Poids et Mesures (CIPM) for the realization of the metre [1], was obtained with reduced uncertainty. Absolute frequencies of the R(56)32-0 and P(54)32-0 iodine absorp tion lines together with the hyperfine line separations were measured. 

In addition to the adiabatic frequency for the motion perpendicular to the RPO one can also determine the stability frequency or what is known as a characteristic eigenvalue (61)... 

Figure 4. Eddy diffusion coefficients (K) and the Vaisala stability frequency (N) plotted against depth.

When the density stratification is thermal in nature, the Vaisala stability frequency in Equation 19 may be written in terms of the temperature gradient... 

The constant coefficient 0.178 in Equation 22 does not imply that the eddy diffusion coefficient approaches the constant value of 0.178 cm2 sec"1 when the stability frequency N tends to zero. The factor 0.178... 

It is not to be expected that the straight-line Equation 21 would apply to the thermocline layers of other lakes. Both N and K were calculated from temperature profiles the shape of which depends in a complex manner on the climate of the area, thermal regime, depth, and volume of the lake. It seems, however, that by arguments presented earlier in this section, an inverse relationship between the stability frequency and eddy diffusion coefficient would, in general, hold in the pycnocline layers of lakes. If such a relationship is established, it would be possible to obtain estimates of K from the values of the stability frequency N, which are much easier to compute. 

The NMR of Ih in IhAs and IhSb powders was determined by means of an autod3nie detector with a stabilized frequency. The line shift was measured relative to the resonance line of In in an aqueous solution of In (0104)3. When both substances were in the same holder, the shift could be found directly from the calibrated separation between the lines. At low temperatures, the magnetic field was determined using one of the samples as the standard. The stability of the apparatus ensured that the error did not exceed 2 10 at frequencies starting from 200 Hz. 

Prestage, J.D. Chung, S. Le, T. Lim, L. Maleki, L. Liter sized ion clock with 10 stability. Frequency Control Symposium and Exposition, Proc. 2005 IEEE International Conference, Vancouver, BC, Canada, August 29-31 2005, 472-476. 

In principle, the numerical solution of Eq. (19) is not more difficult than the solution of Hamilton s equations of motion. All that one needs are second derivatives of the Hamiltonian evaluated at the pods. Moreover, it is often possible to evaluate the stability frequency of the pods analytically. For example in a symmetric exchange reaction the potential energy may be expanded in the vicinity of the saddle point, to third order such that... 

Given the adiabatic Hamiltonian (Eq. (26)) in the vicinity of the pods, one may define an adiabatic stability frequency (% for motion perpendicular to the orbit as... 

There is a close connection between the adiabatic frequency and the stability frequency (cf. Eq, 19) of a periodic orbit. Since at the pods the time average of is exactly d Em/du one finds that the adiabatic stability frequency is just the first order Magnus approximation to the exact stability frequency.From a practical point of view it is actually easier to compute the stability frequency. Finding the adiabatic frequency implies actual construction of the (u,v) coordinate system. 

The stability frequency may be computed by integrating along the pods for one period, but using the Cartesian set of coordinates. Note that the stability frequency is of course independent of the coordinate system used. 

From the properties of the Magnus approximation, one can show that if 3T<< than the adiabatic approximation is excellent. Thus, given a pods, one may easily compute its period and stability frequency and so assess the validity of the adiabatic approximation in the vicinity of the pods. 

The next simple estimate is to make use of the imaginary adiabatic or stability frequency of th e n-th adiabatic barrier to estimate the tunneling correction. Thus one replaces Eq. (29) with the well known expression ... 

There is though, one qualitative difference between the more accurate LSTH surface and the PK(II) surface. The former includes the van der Waals wells in the far entrance and exit valleys. Using pods, we We computed the adiabatic well depths for different H2 vibrational states. In addition we We computed the stability frequencies of the quantised pods. These frequencies are now of course real - the pods are truly stable. The results are summarised for Mu+DD in Table HI. Within... 

For Mu+DD we note that the n=2 and n=3 adiabatic surfaces support exactly one bound state each at E=0.9285 and 1.2705eV respectively. These energies should be compared with the exact results 0,9233 and 1.2646eV.A similar analysis may be carried out for the MU+H2 reaction,here the larger stability frequencies raise the resonance energy above the threshold energy so the resonance is, if at all, a virtual resonance, very broad and seemingly unobservable. 

To summarise, we have shown that periodic orbits are an extremely useful analytic tool. Many aspects of exact quantal computations were explained or even predicted, based on periodic orbit analysis. Here we should mention that the stability frequency of RPOW has been used to... 

As we have already seen in the previous section, finding adiabatic barriers and wells of the n-th quantal vibrational adiabatic potential surface for the y dependent Hamiltonian h(y) is equivalent semiclassica-lly to finding periodic orbits of h(y) with quantised action - (n+l/2)h if the periodic orbit is over a simple well potential. The time dependent coordinates and momenta of the (y dependent) periodic orbit are denoted r (t y), R (t y), Pr(t y), and PR(t y), and the period of the orbit is T (y). We thus find for each value of y a vibrationally adiabatic barrier or well at energy E (y), a stability frequency o)n(y) and effective mass M (uq) (cf. Eq. 27) for motion perpendicular to the... 

As noted earlier, the stability frequencies of periodic orbits are invariant under canonical transformation. Consider first the J=0 coplanar Hamiltonian... 

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