Reciprocal vector

Expressing (k) is complicated by the fact that k is not unique. In the Kronig-Penney model, if one replaced k by k + lTil a + b), the energy remained unchanged. In tluee dimensions k is known only to within a reciprocal lattice vector, G. One can define a set of reciprocal vectors, given by... 

The sum over symmetry operations in formula (16) can be rewritten by considering the effect of multiplying vector h7 by the rotation matrices The collection of distinct reciprocal vectors h7Rg is called the orbit of reflexion h7 [27] r7 is the set of symmetry operations in G whose rotation matrices are needed to generate the orbit ofh/ r, denotes the number of elements in the same orbit [50]. 

The projection of H on the plane formed by two reciprocal vectors a and b can be expressed in terms of real unit cell parameters a, b and y [1] ... 

Where flnm contains the origin and and the reciprocal vectors ... 

The wave function v /ex (r) of electrons at the exit face of the object can be considered as a planar source of spherical waves according to the Huygens principle. The amplitude of diffracted wave in the direction given by the reciprocal vector g is given by the Fourier transformation of the object function, i.e. 

If we consider a primitive Bravais lattice with cell edges defined by vectors a, 82, and aj, the corresponding reciprocal lattice is defined by reciprocal vectors bj, b2, and bj so that... 

D. Dynamical Reciprocal Vectors Drift Velocities and Diffusivities... 

The dynamical reciprocal basis vectors b, ..., bj( defined above are closely related to the modified reciprocal basis vectors defined in Eq. (16.3-6) of the monograph by BCAH [4] (see the table on p. 188 of BCAH for definitions), and to a set of corresponding basis vectors used by Ottinger, which he refers to by the notation 02,/0R introduced in Eq. (5.29) of his monograph, in which 2i refers in Ottinger s notation to one of the soft coordinates. The dynamical reciprocal vectors defined here are identical to another set of... 

In this section, we introduce generalized definitions of sets of reciprocal basis vectors, and of corresponding projection tensors, which include the dynamical reciprocal vectors and the dynamical projection tensor introduced in Section VI as special cases. These definitions play an essential role in the analysis of the constrained Langevin equation given in Section IX. 

A generalized set of reciprocal vectors for a constrained system is defined here to be any set off contravariant basis vectors b, ..., b- and K covariant basis... 

The biorthogonality and completeness relations presented above do not uniquely define the reciprocal basis vectors and mi a list of (3N) scalar components is required to specify the 3N components of these 3N reciprocal basis vectors, but only (3N) —fK equations involving the reciprocal vectors are provided by Eqs. (2.186-2.188), leaving/K more unknowns than equations. The source of the resulting arbitrariness may be understood by decomposing the reciprocal vectors into soft and hard components. The/ soft components of the / b vectors are completely determined by the equations of Eq. (2.186). Similarly, the hard components of the m vectors are determined by Eq. (2.187). These two restrictions leave undetermined both the fK hard components of the / b vectors and the Kf soft components of the K m vectors. Equation (2.188) provides another fK equations, but still leaves fK more equations than unknowns. Equation (2.189) does not involve the reciprocal vectors, and so is irrelevant for this purpose. We show below that a choice of reciprocal basis vectors may be uniquely specified by specifying arbitrary expressions for either the hard components of the b vectors or the soft components of m vectors (but not both). 

We now show, conversely, that for each projection tensor P j, there exists a unique set of corresponding reciprocal basis vectors that are related to P j, by Eq. (2.195). To show this, we show that the set of arbitrary numbers required to uniquely define such a projection tensor at a point on the constraint surface is linearly related to the set of fK arbitrary numbers required to uniquely specify a system of reciprocal vectors. A total of (3A) coefficients are required to specify a tensor P v- Equation (2.193) yields a set of 3NK scalar equations that require vanishing values of both the hard-hard components, which are given by the quantities n P = 0, and of the fK mixed hard-soft ... 

We now consider several possible ways of defining a system of reciprocal vectors and a corresponding projection tensor. 

This is perhaps the most obvious definition of a set of reciprocal vectors. 

Other definitions may be constructed by the following generalization of the relationship between the dynamical reciprocal vectors and the mobility tensor Given any invertible symmetric covariant Cartesian tensor S v with an inverse we may take... 

By repeating the reasoning applied in Section VI to the dynamical reciprocal vectors, we may confirm that any vectors so defined will satisfy Eqs. (2.186)-(2.189). It will hereafter be assumed that (except for pathological choices of S v) they also satisfy completness relation (2.190). A few choices for the tensors S v and T yield useful reciprocal vectors and projection tensors, for which we introduce special notation ... 

Dynamical reciprocal vectors and m) , which were introduced in Section VI, are defined by taking T = H in Eq. (2.208). The corresponding projection tensor is the dynamical projection tensor P v-... 

The pseudoforce associated with the dynamical projection tensor may be calculated by using dynamical reciprocal vectors to evaluate Eq. (2.205). In the simple case of a coordinate-independent mobility as in a free-draining model or a model with an equilibrium preaveraged mobility, we may use Eq. (A. 17) to express as a derivative... 

Note that the soft reciprocal vectors b are expanded in a basis of tangent vectors, and so are manifestly parallel to the constraint surface (as indicated by the use of a tilde), while the hard reciprocal vectors ihi are expanded in normal vectors, and so lie entirely normal to the constraint surface (as indicated by the use of a caret). These basis vectors may be used to construct a geometric projection tensor... 

The pseudoforce that is obtained by using geometric reciprocal vectors in Eq. (2.205) may be expressed, using Eq. (A.19), as a derivative... 

Inertial reciprocal vectors, which will be denoted by b and mf, are obtained by setting S v = nl v This yields... 

The inertial and geometrical projection tensors, and associated reciprocal vectors, are identical for models with equal masses for all beads, in which the mass tensor is proportional to the identity. 

Definition (2.144) for the dynamical reciprocal vector b may then be used to expand the second term in Eq. (2.384) as... 

The reciprocal vectors defined in Section VIII may be used to construct a more direct derivation. Let b and m refer to reciprocal basis vectors defined using Spv and in Eqs. (2.207) and (2.208), respectively. By substituting definitions (2.207) and (2.208) into completeness relation (2.190), we find that... 

Derivatives of the determinants S and f of the generic projected tensors Sah and Tij defined in Eqs. (2.20) and (2.24) may be expressed compactly in terms of the reciprocal vectors that are generated by applying Eqs. (2.207) and (2.208) to the corresponding Cartesian tensors S v and respectively. Using Eq. (A.14) to differentiate In 5 with respect to a soft variable gives... 

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