Atomic frequency

A pre-factor 1/r contains a time scale r or a frequency which for instance corresponds to the hard phonon or to an atomic frequency. The growth rate of the crystal is proportional to this rate (23). As will be shown later, the nucleus once formed expands in a time scale shorter than the one necessary for nucleation. If the process consists of a series of sequential subprocesses, the global velocity is governed by the slowest one. Therefore, this nucleation process determines the growth rate of a faceted surface. 

H. Ledbetter, Atomic Frequency and Elastic Constants, Zeit. Metallkunde, 82(11), 820 (1982). 

The signal assignments of the parent compounds 5 a- and 5 /i-androstane, 5 a- and 5 /i-pregnane and estrane [564] were performed by comparing the spectra with closely related derivatives, using substituent effects, off-resonance decoupled spectra and specifically deuterium-labeled analogues [564], The influence of different structural environments on the 13C chemical shift of the carbonyl carbon in keto steroids is illustrated by the values given in Table 5.10 [566]. The carbonyl carbon atom frequency in cyclopen-tanone moieties is shifted about 5 ppm downfield relative to that in cyclohexanone... 

ATOMIC FREQUENCY. The vibrational frequency of an atom, used particularly with respect to the solid state. 

It was known from experiment that all the spectral lines of an element could be represented as the differences of a relatively small number or terms. If these terms are arranged in a one-dimensional array 7 = Ti,T2,..., the atomic frequencies form a two-dimensinal array of elements u nm) = Tn—Tm,... 

It is now recognized that cold collision frequency shifts [32] is a crucial issue for every high precision atomic frequency standard, microwave or optical. For hydrogen at a density of 109 cm-3 the shift of the 1S-2S transition is about 0.4 Hz, [8], or a fractional shift of 1.7 x 10-16. For a rubidium hyperfine standard operating at the same density, the shift is about 6 xlO-14 [45,46]. 

This method provides an order of magnitude precision gain compared to measurements of the fine structure interval. Relativistic many-body calculations are used to reveal the dependence of atomic frequencies on a for a range of atomic species observed in quasar absorption spectra [1], It is convenient to present results for the transition frequencies as functions of a2 in the form... 

Another possibility is to use optical atomic frequency standards. Any evolution of a in time would lead to a frequency shift. To establish the connection between a and u>, relativistic calculations of the a dependence of the relevant frequencies for Cal, Srii, Bail, Ybn, Hgn, Inn, Tin and Rail have been performed [1]. The a dependence of the microwave frequency standards (Cs, Hg+) has also been accurately calculated. 

Ni-Ni distance. ) Binding energy per atom. Frequency of the totally symmetric vibration. 

Vq PB diluent concentration at the solubility limit under a standard state Vp Poisson s ratio of block copolymer composite Vp Atomic frequency factor in molecular chain scission Q Active craze front length per unit volume a Negative pressure (mean normal stress)... 

In Eq. (2), U is the energy for bond rupture, the macroscopic craze flow stress, X the craze fibril extension ratio accounting for the first order stress concentration, Q an activation volume of monomer dimensions, Vd a pre-exponential frequency factor of the order of atomic frequencies, and q a further stress concentration factor relatel to the fraction of the taut bundle of molecules in the fibril given by ... 

If the atomic frequency v has any real physical significance it should be possible to calculate it from various other physical properties. Thus Nemst supposed that the frequency of a body composed of electrically charged ions could be calculated from its optical properties. In agreement vdth this hypothesis he found that the frequencies of the ultra-red absorption bands of potassium and sodium chloride determined by Rubens were approximately equal to the values of v calculated from the specific heat. It is probable, however, that this agreement is purely accidental. 

Jump frequency of interstitial atoms Frequency of vibration of atoms in a lattice Average jump time (l/pt see pj above)... 

Figure 1. Mode density spectrum with a PBG. The atomic frequency u>at is inside a PBG, near the cutoff frequency u>o and defect mode frequency cOd-...

Figure 3. Left arrows sketch the periodic alternation between two values of the atomic frequency, starting from the atomic frequency lja, i.e., detuning Aa = u>u — wa (solid line arrows) oreOb, i.e., detuning A b (dashed line arrows). This frequency modulation occurs in the vicinity of a field in the discrete mode u>d, typically in a photonic crystal with defects, and of the photonic band gap edge u>u (right).

The parameters 8, are usually considered to be absorbed into the atomic frequencies ooi and 0)2, by redefining the frequencies >, = co, + 8, and are not explicitly included in the master equations. However, we are interested in the qualitative effects of the interactions between the atoms, and the role played by O12 in their dynamics. It is evident from Eq. (20) that the parameter fli2 does not appear as a shift of the energies, but rather as a coherent coupling between the atoms. Thus, the interaction with the vacuum field not only gives rise to the dissipative spontaneous emission but also leads to a coherent coupling between the atoms. 

Equations (120) contain time-dependent terms that oscillate at frequencies exp( /Af) and exp[ 2/((05 — ffiojt + <))]. If we tune the squeezed vacuum field to the middle of the frequency difference between the atomic frequencies, namely, v — oo0)t + < )] become stationary in time. None of the other time-dependent components is resonant with the frequency of the squeezed vacuum field. Consequently, for A > f, the time-dependent components oscillate rapidly in time and average to zero over long times. Therefore, we can formulate a secular approximation in which we ignore the rapidly oscillating terms, and find that Eqs. (120) give us the following steady-state solutions [64] ... 

When the distance between the two atoms is not too large (R < cfo) where is a typical atomic frequency), relativistic corrections do not have to be taken into account. One assumes that R is large enough, however, so that the overlap of the wavefimctions is small and one expands V in the small... 

Allan, D.W. Statistics of atomic frequency standards. Proc. IEEE, 1966, 54, 221-230. 

Major, EG. Microwave resonance of field-confined mercury ions for atomic frequency standard for atomic frequency standard. NASA Report, Goddard Space Flight Center, X-512-69-167, 1969. 

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