Odd multiplicity

FT is essentially a mathematical treatment of harmonic signals that resolved the information gathered in the time domain into a representation of the measured material property in the frequency domain, as a spectrum of harmonic components. If the response of the material was strictly linear, then the torque signal would be a simple sinusoid and the torque spectrum reduced to a single peak at the applied frequency, for instance 1 Hz, in the case of the experiments displayed in the figure. A nonlinear response is thus characterized by a number of additional peaks at odd multiples of the... 

An inversion center is mentioned only if it is the only symmetry element present. The symbol then is 1. In other cases the presence or absence of an inversion center can be recognized as follows it is present and only present if there is either an inversion axis with odd multiplicity (N, with N odd) or a rotation axis with even multiplicity and a reflection plane perpendicular to it (N/m, with N even). 

A reflection plane that is perpendicular to a symmetry axis is designated by a slash, e.g. 2/m ( two over m ) = reflection plane perpendicular to a twofold rotation axis. However, reflection planes perpendicular to rotation axes with odd multiplicities are not usually designated in the form 3jm, but as inversion axes like 6 3jm and 6 express identical facts. 

The only information used in bisection is the sign of the function. The convergence is slow (of order 1), but never fails. Its disadvantages are the need for bracketing, which may be hard when two roots are very close, and the unability to find a root of odd multiplicity. 

The formula above for the index can be applied to fragment ions as well as to the molecular ion. When it is applied to even-electron (all electrons paired) ions, the result is always an odd multiple of 0.5. As an example, consider C7H5CK with an index of 5.5. A reasonable structure is... 

These two cases imply that two linearly polarized waves at right angles to each other, differing in amplitude and differing in phase by 0, ir or a multiple of vt, compound to give a resultant wave which is also linearly polarized but the plane of polarization lies at an angle tan-1 (b/a) or tan-1 ( b/a) to one of them, depending whether the phase difference is an even or an odd multiple of n. 

Figure 2.17 Plots of the trigonometric functions sine (dot-dash line), cos 0 (full line), and tane (dashed line) for -2n s, 0 < 2n. The principal branch of each function is shown by the thick lines. The dotted vertical lines at odd multiples of n 2 indicate the points of discontinuity in the tangent function at these values of e...

The tangent function, tanx = sinx/cosx, shown in Figure 3.3, is interesting because it exhibits infinite discontinuities whenever x passes through an odd multiple of f. 

The majority of nonlinear loads produce harmonics that are odd multiples of the fundamental frequency. Certain conditions need to exist for production of even... 

In another approach, ultrasonic acoustic flexural plate waves (or Lamb waves) were used to generate fluid mixing motion of microspheres in water in a Si device. A 10-pm-thick piezoelectric ZnO film was deposited on the back side of a Si wafer (see Figure 3.48). This thickness is an odd-multiple of the acoustic half-... 

We note an important feature of Eqs. (4.185)—(4.188)—they do not contain dispersion factors of even orders. This ensures that the frequency dependence of the full distribution function W incorporates only dispersion factors with odd multiples of the basic frequency. Qualitatively, this is the result of absence of the interwell mode for the statistical moments of even orders. Technically, it is due to vanishing of the products (qq fjf1) entering Eqs. (4.184)-(4.188) if the sum k + l is even. This mle follows immediately from combination of the oddity of cpj (see Section III.B.l) with the parity properties of the functions introduced in Section III.B.l. 

As one of these assumptions, we may suppose that the distribution of diffracting power conforms to the symmetry of the crystal This symmetry fixes the values of many, sometimes of all, of the 5 s as being either tt/2 or zero. For example, if the crystal has three mutually perpendicular planes of symmetry and if we take the intersections of these planes as axes of coordinates, the terms in the series can contain cosines only, for they must have the same values when we reverse the algebraic sign of either x> y, or 2. In this case, therefore, the 5 s must be odd multiples of 7r/2. What the multiples of tt/2 really are is immaterial, since the coefficient, A, is the square-root of a measured quantity and its algebraic sign is not determined by the diffraction data. 

In the last few years, first measurements have been performed determining the length of the attosecond soft X-ray bunches. Paul et al. measured 250 attosecond pulses in a two-color photoionization experiment [5]. Making use of the fact that in a train of soft X-ray bunches the frequency spectrum consists of odd multiples ( harmonics ) of the frequency of the driver laser, they studied the ionization of Ar atoms by simultaneous photo-absorption from the soft X-ray bunch and the fundamental color of the femtosecond laser. Measured... 

For large D we note that %C(D) converges to the surface transition %s, where 2r= a, while for small D the ordering occurs for even smaller values of %. For a>0, however, the thin film does not order at smaller values of % than the bulk would order (%=%Crit)- In fact, one finds that for odd multiples n %c(D)=%crit, there is no shift of the critical point in this case. However, for even multiples n Eq. (63) leads to... 

The sum in Equation 3.41 is a geometric series whose elements become unity, and add constructively, when kdl2 = mir, where m is an odd integer. This condition defines the relationship between SAW wavelength. A, and transducer periodicity, d, for coherent addition, as shown in Figure 3.19. The IDT excites odd harmonics at odd multiples of the synchronous frequency f = m/i. 

The concentration rapidly increases around 40 wt7. where the overlapping structure of a four-lobe clover at odd multiples of 45 and a small + type pattern in 0 and 90 directions appears. 

The gravimetric density of the crystals was not measured hut the content of the unit cell may be established by using Eq. 6.5 and the expectation that the reasonable value of p should be between 4 and 5 g/cm. The estimated density assuming NiMnOs composition has a reasonable value of 4.86 g/cm when Z = 4. The two closest numbers of formula units (Z = 3 or 5) are impossible due to the restrictions imposed by symmetry in a base-centered lattice, sites with odd multiplicities are impossible. The next two closest numbers (Z = 2 or 6) result in unrealistically low and high densities, respectively. Thus, we assume that there are 4 Mn, 4 Ni and 12 O atoms in the unit cell. 

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