P. J. Mjoberg, 5. 0. Ljunggren, and W. M. Ralowski The Rovol Institute of Technoloav S-10044 Stockholm 70, Sweden "
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Reciprocal Vectors in Function Spate
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The idea of reciprocal basis vectors is a very useful one in many branches of chemistry and physics. The best known application is probably that of the reciprocal lattice. This is fundamental to the theory of diffraction in crystal media (see below). However, it is seldom pointed out that reciprocal vectors actually do have a counterpart in function spaces. The main purpose of the present paper is to demonstrate the parallelism between reciprocal vectors and "reciprocal" functions, as well as to indicate some practical uses of the latter. As is well known, there exists a close formal analogy between function theory and vector algebra. Thus, most concepts of vector algebra have a counterpart in functional analysis (1, 2). The most fundamental of these concepts is that of a vector space itself (2-4). In this article, we shall examine the space Ea spanned by the usual three-dimensional vectors of analytical geometry and also a finite-dimensional metric function space. Important concepts common to different types of vector spaces are those of linear dependence and/or independence of vectors, linear combinations, dimensionality, basis and components, complete sets, linear manifolds, dual spaces, and isomorphism of vector spaces. Also of fundamental importance is the scalar or inner product defined _on a vsctor space. The scalar product of any two vectors a and b of the space is a real number denoted
The first notation is the one commonly used in analytical geometry and the second is the one used in quantum mechanics. In this paper the third notation will be used. Two vectors S and h are said to be orthogonal (or perpendicular) to one another if
From eqns. (2) and ( 5 ) , there is an obvious parallelism between spaces with ordinary vectors and spaces with real-valued functions as elements. Accordingly, we may denote the scalar product on a function space as
Reciprocal Vectors
In order to introduce the concept of reciprocal vectors, !et us start with three arbitrary non-coplanar vectors i, ez, and $ in ordinary space. Any vector a can then be written as a linear combination of these vectors
;I, &, and & are named basis vectors. In the majority of applications, the basis vectors are chosen as being mutually orthogonal and of unit length. However, in some cases, non-orthogonal basis vectors have proved to be a very valuable tool. For these vectors
In eqn. (8) thus
6ij
is the so-called Kronecker delta, defmed
In the case of non-orthogonal basis vectors, it is customary to define a set of reciprocal basis vectors by the following equations (5)
In the special case of a three-dimensional vector space, In order to illustrate these concepts, consider the ordinary space E3. It is common practice to choose the three basis vectors as being mutually orthogonal and of unit length. These orthogonal basis vectors are denoted *lo,6z0 and 630. The components a' of a vector1 H
are obtained by scalar multiplication of both sides of eqn. (3) by 6 0 . Due to the orthogonality of the basis vectors, the result is
From quantum mechanics, it is well known that two real-valued functions, hdx) and hz(x), are said to be orthogonal to each other on the interval a 5 x 5 b, if 1 In conformity with the rules of tensor analysis, all vector components are denoted by upper indices. These indices should not be confused with exponents.
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the reciprocal vectors can be written explicitly as follows
The physical implication of eqn. (10) is that the vector 3 is perpendicular to Gi (i.e., 3 l I t ) if 1 # i. Figure 1 shows an example of a non-orthogonal coordinate system with its corresponding reciprocal basis vectors. Using these relations we can easily calculate the compoBy scalar multiplication of nents a' of a given vector both sides of eqn. (7) by B we obtain
s.
Figure 2. Phase relations in scattering by a primitive lattice of point atoms.
one atom, the atoms and lattice points coincide. In this case, the position of each atom can he written as a linear combination-with integer coefficients of the three lattice vectors, ;I, ez, and ea, which form the edges of the unit cell. In diffraction theory, these vectors are customarily denoted a, b, and e. Thus the general formula for the position of atom number v in the lattice reads Figure i. Example of a nan-orthogonal coodinate system and the coneSponding recipracal basis vectors. The parallelepipeds formed by the two basis systems are also shown.
Like any other vector in three-dimensional space, 3 can
be expanded in the basis vectors Z,
It is easy to derive a mathematical expression for the coefficient &. By scalar multiplication of both sides of eqn. (13) by a,we obtain
Note the analogy of this relation to the corresponding relation of eqn. (8) for the original basis vectors. In the terminology of tensor analysis, g" are the doubly contravariant and gr, the doubly covariant components of what is known as the metric (or fundamental) tensor. Inserting eqn. (13) in eqn. (10) and using eqn. (a), we have
We now introduce the matrices G = (gi,) and G' = In matrix language, the last relationship can then he ten as G'G = E
where E is the unit matrix (6',). Thus G' = G-1 is the inverse matrix of G, which can easily be calculated using standard matrix methods. As was shown in eqn. (12), the coefficients in the expansion of a vector i according to eqn. (7) are then obtained as
This formula will thus generate all lattice points as the three coefficients nl assume all integer values. The incident wave on the crystal is mat&ematically described by its wave propagation vector k. This vector coincides with the direction of propagation of the wave and its length equals 2sJA. Similarly the scattered wave will he described by the wave propagation vector k' of the same length. By the relative phase of the wavelet scattered from atom u we mean the phase difference between the wavelet in question and a wave scattered from a reference atom denoted by 0 in Figure 2. From this figure, we can find the optical path difference between the waves scattered from atoms u and 0, respectively. It is
A 3
2-(kr'
-
- k1.l"
(19)
This corresponds to a relative phase (phase difference) of
Constructive interference is obtained when the phases of the wavelets scattered from different atoms are equal or differ by an integer multiple of 2r. The problem of determining the directions of constructive interference is easily dealt with by introducing the concept of the reciprocal lattice. This lattice is spanned by the reciprocal vectors 3, 52, and C3 defined according to eqn. (10). The usual crystallographic notation for the reciprocal vectors is a*, b*, and c*. Thus the coordinates of a lattice point in the reciprocal lattice are given by the formula
i=
(a)
,,-,-
where the coefficients mjare integers. It is now easy to see what happens when G , 27r times a vector of the reciprocal lattice. Then
I;. - C
3
=
- k equals
-
2r;h = 2*Zm,e'
(22)
1-1
The Utility of Reciprocal Vectors-Diffraction
Theory
As an example of the use of reciprocal vectors, let us briefly discuss the theory of diffraction (6) in a crystal lattice. In a crystal with a primitive unit cell containing only
By combining eqns. (la), (20), and (221, we obtain for the relative phase of the scattered wave from atom number u
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The coefficients c' in eqn. (24) are thus obtained by scalar multiplication off by @ where
is, of course, an integer. Thus, if eqn. (22) holds, the phase difference between waves scattered from different atoms in the lattice is always an integer multiple of 2r. Equation (22) can consequently be regarded as a criterion of constructive interference in the simple type of crystal cell we are considering. The type of crystal discussed above represents a highly idealized case. As a matter of fact, very few crystals are as simple as this. However, similar conclusions can he derived for more complex systems by using a more sophisticated mathematical approach. Reciprocal Functions
A problem often encountered in chemistry and physics is how to obtain the best possible coefficients in the expansion of an experimentally determined function f(x) in terms of a set of known functions @;(x), i.e.
Introducing the expansion (29) for dJ(x), we ohtain explicitly
In order to calculate the expansion coefficients, we thus have to apply the following procedure. First calculate all the integrals
and the integrals c V W + C2m2(z) + ... + c w x ) (24) In this expression, the functions @,(x) are, for instance, the uv-visible or ir molar extinction spectra of each of the components of a mixture, while f(x) is the extinction of the mixture itself. In such a case, x stands for the light wavelength or frequency. The coefficients c' then represent the molar concentrations to he calculated. If all the functions @i(x) are mutually orthogonal, the solution is obtained by the following well-known procedure (cf. eqns. (3) and (4)). Multiplying both sides of eqn. (24) by @,(x) and integrating, weobtain =
These integrals will generally have to be evaluated numerically unless the functional forms happen to he explicitly known. Secondly, invert the matrix S = (Sij) to obtain S' = S-1 = (S'j). The coefficients cj are finally calculated from eqn. (33). I t is easy to show that the values of c] calculated according to eqn. (33) are the same ones that minimize the integral
The condition that the integral should he stationary with respect to c' requires that
or in our shorthand notation
Owing to the orthogonality of the basis functions, all scalar products (C,$t) will vanish except the one for which 1 = i. Equation (26) then reduces to C' = (6, f)/(@,.~%) (n) This equation is analogous to the vector relation (4). In practice, the functions &(x) are, as a rule, not orthogonal to each other, i.e.
for all values of i between 1and n. This may be written (&f)
=
2C~(@,,b,)
(35)
1-1
Thus,
whence In this case, the argument leading up to eqn. (27) is no longer valid. The procedure discussed in connection with non-orthogonal vectors can be applied to the present problem without changes. We thus introduce "reciprocal" basis functions bJ(x) according to
The method of reciprocal vectors is thus equivalent to the method of least squares. Literature Cited 11) Taylor. A. E., "lnfmdudion to Functional Analysis." John Wiley & Sona. Inc.. New v-.,.
where the matrix S' = (811)is the inverse of the matrix S = (S,,). From the definition of &(x) (eqn. (29)) we derive the following orthogonality relations. (cf the analogous relation (10)) ,@,@,)
=
,~SN@,,@J =~ J W " $ J /-I
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Journal of Chemical Education
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(2) Stakgold. I., "Boundary Value Pmblems of Mathematical Phy~ies." Vol. I, The Macmillan Company. New York. 1968. (31 film-, P. R., "FiniV-Dimensional Vector Spacpr:' 2nd Ed., D. Van Nntrsnd Company, Inc.,Pnnmton, New Jersey, l a x . (41 Shilm, G. E., "An lntmduction to the Theory of Linear Spacpr," Pmntice-Hall. h e . . Englevood Cliffs, New J e w , 1964. 151 Wills. A. P.. "Vector b a h s i s with an lntmduction to Tensor Analvais." Dover . Publicatiom. he.,~ e w Y & k , 1958, p. 37. (61 Kittel. C.. "lntmduction to Solid S t a b Phyaim." 3rd Ed.. John Wilcy 8 Sona, loe.. New York, 1967. pp. 49-52.
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