Reciprocal translations

Inductive storage L = inductance, /k = reciprocal translational or rotational stiffness, I = fluid inertance. 

Much of the wave nature of solids is carried out in reciprocal space because the momentum vector of photons, phonons, and particles such as electrons and neutrons can be expressed directly in terms of their vector k as fc = 2tt/A. The reciprocal lattice vectors A, B, and C are constructed such that they are perpendicular to the direct lattice vectors a, b, and c and their lengths are proportional to the inverse length of the direct lattice vectors. The lattice points in reciprocal space correspond to planes in direct space and a reciprocal translational vector is defined as Gm = hA -I- fcB - - C, where hk are fhe Miller indices of these planes. It is shown that Ghu is perpendicular to the hkl) plane and the distance of this plane from the origin is given by d M = 2-n/ Ghkt -... 

Synthesis of the transferrin receptor (TfR) and that of ferritin are reciprocally linked to cellular iron content. Specific untranslated sequences of the mRNAs for both proteins (named iron response elements) interact with a cytosolic protein sensitive to variations in levels of cellular iron (iron-responsive element-binding protein). When iron levels are high, cells use stored ferritin mRNA to synthesize ferritin, and the TfR mRNA is degraded. In contrast, when iron levels are low, the TfR mRNA is stabilized and increased synthesis of receptors occurs, while ferritin mRNA is apparently stored in an inactive form. This is an important example of control of expression of proteins at the translational level. 

Thus the point group part of the operation works on the momentum coordinates and the translation part gives rise to a phase factor. We notice that this phase factor reduces to 1 in the diagonal elements, or in general when the difference between the the two arguments of N(p,p ) is a reciprocal lattice vector. 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

For the reconstruction of the occupation number density (k) in the repeated zone scheme one uses the reciprocal form factor at lattice translation vectors R, as (k) can be written as [9]... 

The Y-intercept ( 3 /k Eq. 13) of the reciprocal correlation for the photolysis of diazirine 21-C1 in TME was 2.18, which translated into a 68% incursion of diazirine excited state in the genesis of the rearrangement products, 22-C1 and 23-C1 carbene 17-C1 only accounted for 32% of these products.28 A similar conclusion followed from the ratio of rearr/addn (68 32) at a high concentration (6.7 M) of TME in pentane, where carbene 17-C1 was almost completely diverted to the cyclopropane, and 22 and 23 were exclusively derived from the excited diazirine.28... 

Figure 6.8 summarizes the most important properties of the reciprocal lattice. It is important that the base vectors of the surface lattice form the smallest parallelogram from which the lattice may be constructed through translations. Figure 6.9 shows the five possible surface lattices and their corresponding reciprocal lattices, which are equivalent to the shape of the respective LEED patterns. The unit cells of both the real and the reciprocal lattices are indicated. Note that the actual dimensions of the reciprocal unit cell are irrelevant only the shape is important. 

The motion of atoms in the lattice can be depicted as a wave propagation (phonon). By dispersion we mean the variation in the wave frequency as reciprocal space is traversed. The propagation of sound waves is similar to the translation of all atoms of the unit cell in the same direction hence the set of translational modes is commonly defined as an acoustic branch. The remaining vibrational modes are defined as optical branches, because they are capable of interaction with light (see McMillan, 1985, and Tossell and Vaughan, 1992, for more exhaustive explanations). 

In view of the very poor fit of the data to this model, we decided to try a model involving a non-literal translation of hard and soft in which the parameters are reciprocal quantities ... 

All these methods exploit the fact that there is no need to recalculate the structure factors of the model each time it is rotated or translated it is sufficient to be able to sample the structure factors at the rotated Miller indices, with or without a phase shift coming from the translation, and this can be done quite effectively by interpolation in reciprocal space (see Protocol 7.4). 

Thus, the scattering of a periodic lattice occurs in discrete directions. The larger the translation vectors defining the lattice, the smaller a i=1 3, and the more closely spaced the diffracted beams. This inverse relationship is a characteristic property of the Fourier transform operation. The scattering vectors terminate at the points of the reciprocal lattice with basis vectors a i=1>3, defined by Eq. (1.21). 

Molecular entropies For a perfect monoatomic gas, there is only translational motion. According to quantum mechanics, the translational energy of molecules in a box is quantized and the size of the quantum is proportional to the reciprocal of the atomic weight. Heavier gases have smaller gaps and the number of states available and degeneracies are greater. 

At constant temperature, the observed widths of the spectral functions decrease with increasing mass of the collisional pair. This fact is a simple consequence of the mean translational energy of a pair, jm v = kT, which is the same for all pairs. The interaction time is roughly proportional to the reciprocal root mean square speed, and thus to the square root of the reduced mass. 

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