1
4
Reciprocal lattice
Reciprocal vectors and the basis of the reciprocal vectors were first used by J. W. Gibbs. Round 1880
he made used of them in his lectures about the vector analysis ([1], pp. 10–11, 83). In structure analysis
the concept of the reciprocal lattice has been established by P. Ewald and M. v. Laue in 1913, at the
very begining of the discipline [2]. The reason was to facilitate calculations in analytic geometry of linear
forms in coordinate systems with non–orthonormal basis, which must inevitably be used when crystals
of lower symmetry (non–cubic) are investigated. Since that time the reciprocal lattice belongs to the
basic concepts of crystallography, solid state physics and other disciplines.
The above mentioned classicists and also authors of textbooks and reference books define the basis
vectors ~a +
s of the reciprocal lattice in an algebraic way (see relations 4.2(1) below or Appendix C). Then,
however, the textbooks and reference books declare the reciprocal lattice to be the Fourier transform of
a lattice. The proof of that statement is however almost always missing. (Probably the only exception
is the book by Guinier [3], pp. 85–87). The reason for that may lie with the fact that the proof is far
from being simple. The fact that the reciprocal lattice is the Fourier transform of a lattice is, however,
important not only for the study of cystalline solids but also for presentation of the Fourier series of
periodic functions of two and more variables that are not periodic in orthogonal directions, for the
formulation of the sampling theorem in dimensions N ≥ 2 etc.
In this chapter we first give that algebraic definition of the reciprocal lattice (section 4.2) and then
we submit the proof that reciprocal lattice is the Fourier transform of a lattice function (section 4.3). At
that, we obtain the expression for the Fourier series of general lattice functions (section 4.4), i.e. even
of such ones which characterize lattices that are periodic in non–ortogonal directions only.
4.1
The lattice function
The N –dimensional lattice with basis vectors ~ar , r = 1, 2, . . . , N , constituted by points is characterized
by the so called lattice function
X
X
f (~x) =
δ ~x − n1~a1 − n2~a2 − · · · − nN ~aN =
δ ~x − ~x~n ,
(1)
~
n ∈ inf
~
n ∈ inf
where
~x~n = n1~a1 + n2~a2 + · · · + nN ~aN
(2)
denotes a lattice vector and the symbol ~n ∈ inf implies that all the components n1 , n2 , . . . , nN of the
multiindex ~n acquire all integer values. Hence, the lattice function (1) is a N –multiple series of the Dirac
distributions of N variables.
Let us form the matrix
a11 a12 · · · a1N a21 a22 · · · a2N A = kars k = .
(3)
..
.. ,
..
..
.
.
.
aN 1 aN 2 · · · aN N the rows of which consist of the coordinates ars of basis vectors ~ar of the lattice in a system of coordinates
with orthogonal basis (~e1 , ~e2 , . . . , ~eN ). Let us consider the variable ~x and the multiindex ~n to be column
matrices formed by coordinates xr in that orthogonal basis and by integers nr , respectively. Then, we
may express the lattice function in the form
X
f (~x) =
δ ~x − AT ~n .
(4)
~
n ∈ inf
The determinant of the matrix A is called the outer product of the vectors ~ar , r = 1, 2, . . . , N , (see
e.g. [4], p. 95).
~a1 , ~a2 , . . . , ~aN = det A.
(5)
Its absolute value does not depend on the choice of the orthonormal basis and defines (cf. e.g. [9],
p. 216) the volume VU of the N –dimensional parallepiped (unit cell), the edges of which are the basis
vectors ~ar of the lattice:
2
4
|det A| = VU .
RECIPROCAL LATTICE
(6)
The independence of the absolute value of the outer product on the choice of the orthonormal basis
(~e1 , ~e2 , . . . , ~eN ) is obvious from the square of that product:
2
2 ~a1 , ~a2 , . . . , ~aN = det A = det A det AT = det AAT = det k~ar · ~as k = det G.
(7)
The symbol AT denotes the transposed
det G = matrix of the matrix A and
~a1 · ~a1 ~a1 · ~a2
~a2 · ~a1 ~a2 · ~a2
..
..
.
.
~aN · ~a1 ~aN · ~a2
···
···
..
.
···
~a1 · ~aN
~a2 · ~aN
..
.
~aN · ~aN
(8)
is the Gram determinant of the basis vectors of the lattice whose independence on the coordinate system
is evident. If the dimension N = 1, 2 and 3, the absolute value of the outer product has the meaning of
length, area, and volume,
respectively.
For N ≥ 4 it is also reasonable to consider the absolute value of
the outer product [~a1 , ~a2 , . . . , ~aN ] as a volume because it has properties attached to the volume ([4], p.
95, [9], pp. 216–217):
(i) [~a1 , ~a2 , . . . , ~aN ] = 0 if and only if the vectors forming the outer product are linearly dependent.
(ii) If one of the vectors involved in the outer product is multiplied by a number α then the outer
prouduct is multiplied by the number α.
(iii) [a1 , a2 , . . . , aN ] ≤ a1 a2 · · · aN , where ar means the length of the vector ~ar .
4.2
Algebraic definition of reciprocal lattice
In the middle of the last century the International Union of Crystallography recommended to define the
basis vectors ~a +
ar by scalar products
s of the lattice reciprocal to the lattice with the basis vectors ~
~ar · ~a +
s = K δrs ,
r, s = 1, 2, . . . , N,
where K is the so called reciprocal constant (see [6], p. 12), the value of which can be chosen as the
case may be. The recommendation of the Union has, however, not been accepted and in crystallography,
material science and related technological branches K = 1 (see e.g. [3], p. 386, [5], p. 63) is consistently
used, whereas in solid state physics, surface physics, etc. K = 2π is chosen (see e.g. [7], p. 62, [8], p.
87). (Emotional discussions have been carried on which of the two eventualities is the right one (cf. [10],
p. 62)). We will follow here the customs of crystallography, though it slightly complicates the formula
expressing the Fourier transform of the lattice function.
Thus, we define the basis vectors ~a +
s of the reciprocal lattice by scalar product
~ar · ~a +
s = δrs ,
r, s = 1, 2, . . . , N.
(1)
It is evident from the definition that the reciprocal lattice to the reciprocal lattice is the original lattice
and that the basis vector ~a +
ar of the
s of the reciprocal lattice is orthogonal to all the basis vectors ~
original lattice with exception of the vector ~as .
Definition (1) is an implicit definition of the basis vectors of the reciprocal lattice; it is not a formula
ar of the original lattice. To get such an explicit
expressing the vectors ~a +
s in terms of the basis vectors ~
expression we rewrite N 2 equations (1) in the matrix way
A A+
T
= I,
(2)
where
+
a 11
a+
21
A+ = a +
st = ...
a+
N1
a+
12
···
a+
22
..
.
···
..
.
···
a+
N2
a+
1N a+
2N .. . +
a NN (3)
4.2
Algebraic definition of reciprocal lattice
3
is the matrix the rows of which are coordinates of basis vectors ~a +
s of the reciprocal lattice in the
orthonormal system of coordinates and I is the unit matrix. It is evident from (2) that
A+ = A−1
T
,
i.e.
(−1)
a+
st = a ts
.
(4)
Expressed by words: If the rows of the matrix A are coordinates of the basis vectors of the original
(−1)
lattice, then the columns of the inverse matrix A−1 = a st are the coordinates of the basis vectors of
the original lattice:
~a +
s =
N
X
a+
et =
st ~
N
X
t=1
s+t
(−1)
t=1
det Ast
~et ,
det A
s = 1, 2, . . . N,
(5)
where det Ast is minor formed from det A by striking out the s–th row and t–th column. Expression (5)
can be written in a more simple way. Let us form the matrix ~As by replacing the s–th row of the matrix
A by vectors ~e1 , ~e2 , . . . , ~eN of the orthonormal basis. Then the sum
N
X
s+t
(−1)
det Ast ~et = det ~As
t=1
is the Laplace development of the det ~As along the elements of the s–th row and the expression (5) for
the basis vectors ~a +
s of the reciprocal lattice takes a particularly simple form
~a +
s =
det ~As
,
det A
s = 1, 2, . . . , N.
(6)
The disadvantage of these expressions is that they are related to an orthonormal system of coordinates.
In fact, the expression (6) provides the coordinates of the vector ~a +
s in terms of the coordinates art of
vectors ~ar in an orthonormal system of coordinates. It is true that this orthonormal system may be
arbitrary. In spite of that we will aim at an expression which does not require the use of any orthonormal
system of coordinates.
Fortunately in E3 the outer product is
det A = ~a1 · (~a2 × ~a3 )
and
det ~A1 = ~a2 × ~a3 ,
det ~A2 = ~a3 × ~a1 ,
det ~A3 = ~a1 × ~a2 .
Hence, the relations (6) are the well known expressions of the basis vectors of the reciprocal lattice
straight by the basis vectors of the lattice:
~a +
1 =
~a2 × ~a3
,
~a1 · (~a2 × ~a3 )
~a +
2 =
~a3 × ~a1
,
~a1 · (~a2 × ~a3 )
~a +
3 =
~a1 × ~a2
.
~a1 · (~a2 × ~a3 )
(7)
However, this happy disposition seems to be limited to N = 3 only. To be free from the necessity
to use an orthonormal system of coordinates also in the case of a general dimension N , we multiply the
numerator and denominator of (6) by the determinant det AT . In the denominator we get det A det AT =
det G and in the numerator det ~As det AT = det ~Gs , where det ~Gs is the determinant made up by replacing
the elements of the s–th row in the Gram determinant by the basis vectors ~a1 , ~a2 , . . . , ~aN of the lattice.
In this way we get the basis vectors ~a +
s of the reciprocal lattice expressed straight by the basis vectors
~ar of the lattice rather then by their coordinates:
det ~Gs
,
s = 1, 2, . . . , N.
(8)
det G
Note: The expressions (8) can be derived without any use of an orthonormal system of coordinates and
the matrix A: We resolve the vectors ~a +
s in the basis formed by the basis vectors of the lattice
~a +
s =
~a +
a1 + · · · + αsN ~aN ,
s = αs1~
s = 1, 2, . . . , N,
(9)
and multiply each of these decompositions successively by vectors ~ar , r = 1, 2, . . . , N . Taking into
account the definition (1) we obtain
4
4
αs1~a1 · ~ar + · · · + αsN ~aN · ~ar = δrs ,
RECIPROCAL LATTICE
r, s = 1, 2, . . . , N.
(10)
For a certain s, the system (10) represents N linear algebraic equations for N coefficients αs1 , αs2 , . . .,
αsN . The coefficient determinant of this system of linear algebraic equations is the Gram determinant
4.1(8). The right–hand side of the equations (10) is zero, if r 6= s, and one, if r = s. Cramer’s rule then
provides a very simple solution of the sytem of equations (10):
det Gst
,
t = 1, 2, . . . , N,
(11)
det G
is minor of the element ~as · ~at in the Gram determinant. On inserting solutions (11) in
s+t
αst = (−1)
where det Gst
(9), we get
~a +
s =
N
det ~Gs
1 X
s+t
(−1)
~at det Gst =
,
det G t=1
det G
in acordance with (8).
Let us use (8) to get expressions for basis vectors in E1 and E2 . In E1 we obtain
~a
,
a2
~a + =
~a =
~a +
.
(a + )2
(12)
In E2 it is
~a
~a2
1
~a2 · ~a1 a22
~a +
1 = 2
~a1 · ~a2
a1
~a2 · ~a1
a22
a2 ~a · ~a
1
2
1
~a1
~a2
~a +
2 = 2
~a1 · ~a2
a1
~a2 · ~a1
a22
,
,
i.e.
~a +
1 =
a22~a1 − (~a1 · ~a2 ) ~a2
2
a21 a22 − (~a1 · ~a2 )
~a +
2 =
,
a21~a2 − (~a1 · ~a2 ) ~a1
2
a21 a22 − (~a1 · ~a2 )
.
(13)
On the contrary
+
2 +
~a 1 − ~a +
a+
a2
a+
1 ·~
2 ~
2
~a1 =
,
+ 2
+ 2
+
+ 2
a1
a2 − ~a 1 · ~a 2
+
2 +
~a 2 − ~a +
a+
a1
a+
2
1 ·~
2 ~
~a2 =
.
+ 2
+ 2
+
+ 2
a1
a2 − ~a 1 · ~a 2
(14)
In E3 the traditional expressions (7) are much more simple than (8) and this simplicity invites us to
make use of them for two–dimensional lattices, too. However, in E2 the vector product is not defined.
This difficulty is usually bypassed by adding the unit vector ~n to the two basis vectors ~a1 , ~a2 of the
two–dimensional lattice in such a way that vectors ~a1 , ~a2 , ~n form the three–dimensional right–handed
basis. Then the basis vectors of the reciprocal lattice are calculated according to (7) (see e.g. [8], p. 87):
~a +
1 =
~a2 × ~n
,
|~a1 × ~a2 |
~a +
2 =
~n × ~a1
.
|~a1 × ~a2 |
(15)
Equivalence of expression (15) and (13) is easy to see, if we write the unit vector ~n as the fraction
~n =
~a1 × ~a2
,
|~a1 × ~a2 |
(16)
~ = (~a ·~c)(~b· d)−(~
~
~ ~b·~c).
insert it into (15) and use the identities ~a ×(~b×~c) = (~a ·~c)~b−(~a ·~b)~c, (~a ×~b)·(~c × d)
a · d)(
We get
~a +
1 =
~a2 × (~a1 × ~a2 )
(~a1 × ~a2 )2
=
a22~a1 − (~a1 · ~a2 )~a2
,
a21 a22 − (~a1 · ~a2 )2
~a +
2 =
(~a1 × ~a2 ) × ~a1
(~a1 × ~a2 )2
=
a21~a2 − (~a1 · ~a2 )~a1
,
a21 a22 − (~a1 · ~a2 )2
4.3
Reciprocal lattice and the Fourier transform of the lattice function
5
in accordance with (13). (As to the unimportant third basis vector of the reciprocal three–dimensional
basis is concerned, it is evident from (16) that ~a +
n.)
3 =~
4.2.1
Example: Reciprocal lattice to the primitive rectangular lattice in E2
Let us calculate the basis vectors of the reciprocal lattice to primitive rectangular lattice in E2 with
the ratio of the lengths of the basis vectors 1:2 and with the longer side of the unit cell in the vertical
direction. Let thelength of the basis vectors ~a1 , ~a2 be a1 = a, a2 = 2a (see Figure 1(a)). Then a21 = a2 ,
a22 = 4a2 , ~a1 · ~a2 = 0. and expressions (13) give the basis vectors of the reciprocal lattice in the form
~
a2
~
a1
a+
~a +
2 = (2a)2 . The basis vectors of the reciprocal lattice thus have the same direction as the
1 = a2 , ~
+
1
1
basis vectors of the original lattice and their magnitudes are a+
1 = a , a2 = 2a . The reciprocal lattice
is again primitive rectangular lattice, but the ratio of the primitive cell sides is 2:1, i.e. the longer side
has horizontal direction (see Figure 1(b)). If, for some reason, we do not choose orthogonal basis vectors
of our rectangular lattice but soume other ones we get, of course, the same reciprocal lattice. However,
the basis vectors specifying it are different. For example, let us choose the basis vectors ~a01 , ~a02 shown in
2
02
2
Figure 1(c). Obviously, a02
a01 · ~a02 = 6a2 , and expressions (13) give the following basis
1 = 8a , a2 = 5a , ~
0+
3 0
5 0
a1 + 2~a02 /a2 . Their magnitudes are
vectors of the reciprocal lattice ~a1 = 4 ~a1 − 23 ~a02 /a2 , ~a0+
2 = − 2~
√
√
0+
0+
a1 = 5/2a, a2 = 2/a (cf. Figure 1(d)).
4.2.2
Example: Reciprocal lattice to the centred rectangular lattice in E2
Now we find the reciprocal lattice to the centred rectangular lattice in E2 . Let the ratio of the sides of
the rectangular unit cell be 1:2 and let the longer side have the vertical direction. We choose the non–
orthogonal basis vectors ~a1 , ~a2 shown in Figure 2(a). Then a21 = a2 , a22 = 5a2 /4, ~a1 .~a2 = a2 /2. According
1
5
a1 − 12 ~a2 /a2 ,~a +
a1 + ~a2 /a2 . Their
to (13) the basis vectors of the reciprocal lattice are ~a +
2 = − 2~
1 = 4~
√
magnitudes are a+
5/2a, a+
1 =
2 = 1/a (cf.Figure 2(b)). Hence, the reciprocal lattice is again a centered
rectangular lattice, but with the side ratio 2:1, i.e. with the longer side in the horizontal direction.
4.3
Reciprocal lattice and the Fourier transform of the lattice function
We will calculate the Fourier transform of the lattice function 4.1(1) three times:
First in E1 (Section 4.3.1). In this case the calculation of the Fourier integrals gives the Fourier
transform having the shape of the Fourier series. This Fourier series is simultaneously a geometric series
whose ratio has the absolute value equal to one. The sum of the geometric series is the series of the
Dirac distributions which is proportional to the lattice functions of the reciprocal lattice.
The calculation in E2 (Section 4.3.2) takes advantage of the quoted properties of the one–dimensional
lattice function. Nevertheless, it requires also several artifices typical for deriving of the Fourier transform
of more–dimensional lattice function. The artifices are, however, geometrically illustrative.
The calculation in EN (Section 4.3.3) is formally identical with the calculation in E2 .
In all the cases we will find that the Fourier transform of the lattice function is proportional to the
lattice function of the reciprocal lattice with the reciprocal constant K = 2π/k.
4.3.1
The Fourier transform of the lattice function in E1
In one–dimmensional case the lattice function has the form
∞
X
f (x) =
δ(x − na)
(1)
n=−∞
and it represents a periodic distribution of the Dirac distributions of single variable with a period a (see
Figure 3(a)). Thus, we may formally expand it into the Fourier series
f (x) =
∞
X
h=−∞
x
ch exp i 2π h
.
a
It is noteworthy that in the case of lattice function (1) all the coefficients ch have the same value:
6
4
RECIPROCAL LATTICE
Figure 1: Two–dimensional primitive rectangular lattice (a), (c) whose basis vectors are chosen in
different ways and its reciprocal lattice (b), (d).
ch
=
1
|a|
Za/2
x
f (x) exp −i 2π h
dx =
a
−a/2
=
1
|a|
Za/2 X
∞
−a/2
=
1
|a|
Za/2
x
δ (x − na) exp −i 2π h
dx =
a
n=−∞
x
1
δ (x) exp −i 2π h
dx =
.
a
|a|
−a/2
Therefore, the Fourier series of the lattice function has the form
∞
X
n=−∞
δ (x − na) =
∞
∞
1 X
x
1 X
exp i 2π h
=
exp i 2π h a+ x ,
|a|
a
|a|
h=−∞
h=−∞
(2)
4.3
Reciprocal lattice and the Fourier transform of the lattice function
7
Figure 2: Two–dimensional centered rectangular lattice (a) with basis vectors ~a1 , ~a2 specifying a primitive lattice and its reciprocal lattice (b).
which is geometric series of the ratio equal to complex unity exp i 2π xa .
This is an important result which will be used when calculating the Fourier transform of more–
dimensional lattice function. Therefore, we prepare a needful formula and, according to (2), we give the
formula for the sum of geometric series of the ratio q = exp(ibx):
∞
X
∞
2π X
2π
δ x−n
.
|b| n=−∞
b
exp (ibhx) =
h=−∞
(3)
Now, we can calculate the Fourier transform of the one–dimensional lattice function either straight
from its definition (1) or from its Fourier series (2). We choose the first possibility.
By interchanging the order of integration and addition and using the sifting property of the Dirac
distribution we get the Fourier tranform of the lattice function (1) in the form of the Fourier series:
Z
F (X)
∞
∞
X
"
= A
−∞
= A
= A
#
δ x − na exp −ikXx dx =
n=−∞
∞ Z
X
n=−∞
∞
X
∞
δ x − na exp −ikXx dx =
−∞
∞
X
exp −ikXna = A
exp ikXna .
n=−∞
(4)
n=−∞
The Fourier series (4) of the Fourier transform of the lattice function (1) is geometric series of the
ratio exp(ikXa). According to (3) it is the Fourier series of the function
F (X) = A 2π
∞
X
h=−∞
∞
∞
X
2π X
2π h
2π h
ka = A
δ X−
.
δ kXa − 2πh = A 2π
δ
X−
k a
|ka|
k a
h=−∞
h=−∞
Thus, we conclude that the Fourier transform of the one–dimensional lattice function is proportional to
the one–dimensional lattice function with the period 2π
ka :
( ∞
)
∞
∞
X
X
X
1
2π h
1
2π +
FT
δ x − na
=
δ X−
=
δ X−
ha
.
(5)
B|a|
k a
B|a|
k
n=−∞
h=−∞
h=−∞
The lattice function (1) and its Fourier transform (5) are shown in Figure 3.
8
4
f(x)
F(X)
1
1
B |a |
-a
- 2π
ka
a 2a 3a x
0
2π
ka
0
2 2π
ka
(b)
(a)
RECIPROCAL LATTICE
32π
ka
X
Figure 3: Linear lattice 4.3.1(1) with the period a, (a) and its Fourier transform 4.3.1(5), (b).
4.3.2
The Fourier transform of the lattice function in E2
In two–dimensional space the lattice function has the form
f (~x) = f (x1 , x2 )
=
=
∞
X
∞
X
δ ~x − n1~a1 − n2~a2
n1 =−∞ n2 =−∞
∞
∞
X
X
=
δ x1 − n1 a11 − n2 a21 , x2 − n1 a12 − n2 a22 .
(6)
n1 =−∞ n2 =−∞
By direct calculation of the Fourier tranform of this function
~
F (X)
2
= A
= A2
= A2
Z∞
Z " X
∞
∞
X
#
δ ~x − n1~a1 − n2~a2
~ · ~x d2 ~x =
exp −ik X
−∞ n1 =−∞ n2 =−∞
∞
∞
X
X
~ · (n1~a1 + n2~a2 ) =
exp −ik X
n1 =−∞ n2 =−∞
∞
∞
X
X
~ · (n1~a1 + n2~a2 )
exp ik X
(7)
n1 =−∞ n2 =−∞
we obtain the Fourier tranform expressed again by the Fourier series, this time, of course, by the double
series. It is, however, evident that it may be rewritten as the product of two simple series:
~ = A2
F (X)
∞
X
∞
X
~ · ~a1 )n1
~ · ~a2 )n2 .
exp ik(X
exp ik(X
n1 =−∞
n2 =−∞
(8)
~ · ~a1 ) and exp(ik X
~ · ~a1 ), respectively. According
Each of them is geometric series with the ratio exp(ik X
to (3) they are the Fourier series of functions
2π
|k|
∞
X
h1 =−∞
~ · ~a1 − h1 2π
δ X
k
and
2π
|k|
∞
X
h2 =−∞
~ · ~a2 − h2 2π .
δ X
k
(9)
The Fourier transform (8) thus takes the form
~ = A2
F (X)
2π
k
2 X
∞
h1 =−∞
~ · ~a1 − h1 2π
δ X
k
X
∞
h2 =−∞
~ · ~a2 − h2 2π
δ X
k
.
(10)
Before going on with processing of (10), we will look up geometrical meaning of simple series in this
expression (see Figure 4). Evidently, the equations
~ · ~ai = 2π hi ,
X
k
i.e.
~ · ~ai = 2π hi ,
X
ai
kai
hi = 0, ±1, ±2, . . .
4.3
Reciprocal lattice and the Fourier transform of the lattice function
9
a2
a1
2π
ka 2
a2+
a2
a1
a1+
2π
ka 1
Figure 4: To the derivation of the Fourier transform of the lattice function in E2 . In the upper part of
the figure there is a lattice with basis vectors ~a1 , ~a2 . The lower part of the figure shows the corresponding
reciprocal lattice with basis vectors ~a +
a+
1,~
2.
2π
represent in E2 a system of stright–lines perpendicular to the vector ~ai and having the spacing ka
. The
i
product of the series in (10) then represents the system of the Dirac distribution of two variables with
~ specified by conditions
non–zero values at points X
~ · ~a1 = 2π h1 ,
X
a1
ka1
~ · ~a2 = 2π h2 ,
X
a2
ka2
h1 , h2 = 0, ±1, ±2, . . . ,
i.e. at points
~ = 2π h1~a + + h2~a +
X
1
2
k
constituting the two–dimensional reciprocal lattice with the reciprocal constant 2π
k .
Of course, we can get this result just by algebraic treatment of (10), without using the geometrical
interpretation. We will do it now and we will even complete the result. For this purpose we rewrite the
product of two simple series in (10) as the double series of the Dirac distributions of two variables. The
Fourier transform (10) takes then the form
~ = 1
F (X)
B2
∞
X
∞
X
h1 =−∞ h2 =−∞
2π
2π
~
~
δ X · ~a1 −
h1 , X · ~a2 −
h2 .
k
k
(11)
If we consider the argument of the Dirac distributions in (11) to be the row matrix we may process it
~ is taken as the row matrix the elements of which are coordinates X1 , X2 in an
further. The variable X
orthonormal basis and the multiindex ~h is the row matrix with integers h1 , h2 as elements.
10
4
RECIPROCAL LATTICE
2π ~ · ~a1 − 2π h1 , X
~ · ~a1 , X
~ · ~a2 ~ · ~a2 − 2π h2 = X
−
X
,
h
=
h
1
2
k
k
k
a11 a21 − 2π ,
h
= X1 , X2 h
1
2 =
a12 a22 k
~ AT − 2π ~h =
= X
k
−1 2π
~
~
h AT
=
X−
AT =
k
2π ~ +
~
hA
AT .
=
X−
k
(12)
The Fourier transform (11) with the argument of the Dirac distributions processed in this way can be
further adjusted and brought back from the matrix representation to the vectorial one which does not
depend on the choice of the orthonormal basis:
~
F (X)
=
1
B2
=
1
B2
=
1
B2
=
1
B2
∞
X
∞
X
2π ~ +
~
X−
δ
AT
=
hA
k
h1 =−∞ h2 =−∞
∞
∞
X
X
1
~ − 2π ~h A+ =
δ X
| det A|
k
h1 =−∞ h2 =−∞
1 X
2π 2π +
+
+
h1 a +
+
h
a
,
X
−
h
a
+
h
a
=
δ X1 −
2 21
2
1 12
2 22
11
VU
k
k
~
h ∈ inf
2π 1 X
+
+
~
δ X−
h1~a 1 + h2~a 2
.
VU
k
(13)
~
h ∈ inf
Here VU = | det A| is the area of the unit cell of the original lattice (cf. 4.1(6)). Hence, the Fourier
transform (13) of the lattice function (6) is proportional (with the proportionality factor 1/(B 2 VU )) to
the lattice function of the reciprocal lattice with the reciprocal constant K = 2π/k.
4.3.3
The Fourier transform of the lattice function in EN
The same result as in E2 is obtained also in a general case of the lattice function in EN . Algebraic and
analytic processing is the same as in E2 , only the geometric notion is missing.
We calculate the Fourier transform of the lattice function 4.1(1):
(
~
F (X)
=
)
X
FT
δ (~x − n1~a1 − n2~a2 − · · · − nN ~aN )
=
~
n ∈ inf
N
= A
Z
∞Z
···
−∞
= AN
X
X
~ · ~x dN ~x =
δ ~x − n1~a1 − n2~a2 − · · · − nN ~aN exp −ik X
~
n ∈ inf
~ · (n1~a1 + n2~a2 + · · · + nN ~aN ) =
exp −ik X
~
n ∈ inf
N
= A
X
~ · (n1~a1 + n2~a2 + · · · + nN ~aN ) .
exp ik X
(14)
~
n ∈ inf
Expression (14) represents the Fourier series of the N –dimensional lattice function in the form of N –
multiple Fourier series. The series can be factorized and — by the interchange of the order of summation
and multiplication — rewritten as the product of N simple geometric series:
4.4
The Fourier series of the lattice function in EN
~
F (X)
N
= A
N
X Y
11
h
i
~ · ~aj nj =
exp ik X
~
n ∈ inf j=1
= AN
N
∞
Y
X
h
i
~ · ~aj nj .
exp ik X
(15)
j=1 nj =−∞
Each of the simple geometric series in (15) can be expressed (according to (3)) as simple series of Dirac
distributions of single variable
~ = AN
F (X)
2π
k
N Y
N
∞
X
j=1 hj =−∞
2π
~
hj .
δ X · ~aj −
k
On the analogy of (11) the product of simple series of the Dirac distributions of single variable can be
expressed as the N –multiple series of the Dirac distributions of N variables:
X ~ · ~a1 + 2π h1 , . . . , X
~ · ~aN + 2π hN .
~ = 1
δ
X
F (X)
BN
k
k
~
h ∈ inf
The argument of these Dirac distributions can be considered to be the row matrix and the treatment
similar to (12) can be performed:
~
F (X)
=
1 X
2π ~
T
~
h =
δ XA −
BN
k
~
h ∈ inf
=
2π ~ T −1
1 X
~
δ
X−
h A
AT
=
BN
k
~
h ∈ inf
=
X 1
2π ~ +
1
~
δ X−
hA
.
B N | det A|
k
(16)
~
h ∈ inf
The argument of the Dirac distributions can be expressed — similarly to (13) — in terms of basis vectors
of the reciprocal lattice and the use of 4.1(6) leads to the Fourier transform of the lattice function in the
form
~
F (X)
=
=
1 1 X
~ − 2π h1~a + + h2~a + + · · · + hN ~a +
δ
X
=
1
2
N
B N VU
k
~
h ∈ inf
1 1 X
2π ~
~
X~ ,
δ X−
B N VU
k h
(17)
~
h ∈ inf
where
~ ~ = h1~a + + h2~a + + · · · + hN ~a +
X
1
2
N
h
(18)
is the lattice vector of the reciprocal lattice.
The Fourier transform (17) of the lattice function in its most general shape is then proportional (with
proportionality factor 1/(B N VU )) to the lattice function of the reciprocal lattice with the reciprocal
constant K = 2π/k.
4.4
The Fourier series of the lattice function in EN
While deriving the Fourier transform of the lattice function we have got the transform in the form of
the Fourier series. In one–dimensional case it is the expression 4.3(4), in two–dimensional case 4.3(7)
and in N –dimensional case 4.3(14). In all these cases the exponents of the summation contain the scalar
12
REFERENCES
~ and of the lattice vector ~x~n of the lattice, not of the reciprocal lattice, as could
product of the variable X
be expected at the Fourier transform.
Now the question arises how does it look like the Fourier transform of the lattice function. For the
one–dimensional case we have found it and the expression 4.3(2) shows that in the exponent of the
summands there is the product of the variable x and of the reciprocal lattice vector ha+ . If we restrict
ourselves in the N –dimensional case to the lattice functions with mutually orthogonal basis vectors it
would be possible to factorize them, to calculate the Fourier series in each variable and then to put again
the individual factors together. Unfortunately it is not possible to proceed this way in the general case
when the vectors ~ar are not orthogonal. But, fortunately enough, the Fourier transform of the lattice
function 4.1(1) can be obtained in any case by the inverse Fourier transform of the Fourier transform of
the lattice function in the from 4.3(17):
X
δ ~x − n1~a1 − n2~a2 − · · · − nN ~aN
=
~
n ∈ inf
=



 1 1 X 2π
~ −
δ X
h1~a +
a+
a+
=
FT−1
1 + h2~
2 + · · · + hN ~
N

 B N VU
k
~
h ∈ inf
=
=
1
VU
Z
∞Z
···
−∞
X
~
h ∈ inf
~ − 2π h1~a + + h2~a + + · · · + hN ~a +
~ =
~ · ~x dN X
δ X
exp ik X
1
2
N
k
1 X
exp i 2π h1~a +
a+
a+
x .
1 + h2~
2 + · · · + hN ~
N ·~
VU
(1)
~
h ∈ inf
The Fourier series of the lattice function is frequently used for expressing the electron density in
crystalline solids ([5], p. 169, [6], pp. 353 and following), for deriving the sampling theorem of functions
of several variables when non–orthogonal sampling net is used [11], and elsewhere.
References
[1] Gibbs J. W.: Elements of Vector Analysis. Tuttle, Morehouse and Taylor, New Haven 1881–4.
[2] Ewald P.: Historisches und Systematisches zum Gebrauch des ”Reziproken Gitters” in der Kristallstrukturlehre. Zeitschrift für Kristallographie 93 (1936), 396–398.
[3] Guinier A.: X–Ray Diffraction In Crystals, Imperfect Crystals, and Amorphous Bodies. W. H.
Freeman and Co., San Francisco 1963.
[4] Čech E.: Základy analytické geometrie I. Přı́rodovědecké vydavatelstvı́, Praha 1951.
[5] Giacovazzo C. et al.: Fundamentals of Crystallography. International Union of Crystallography,
Oxford University Press 1992.
[6] Henry N. F. M., Lonsdale K. (eds.): International Tables for X-Ray Crystallography. Vol. 1. The
Kynoch Press, Birmingham 1952.
[7] Kittel Ch.: Úvod do fyziky pevných látek. Academia, Praha 1985.
[8] Lüth H.: Surfaces and Interfaces of Solid Materials. 3rd ed. Springer Verlag, Berlin 1995.
[9] Gantmacher F. R.: Těorija matric. 4. izd. Izdatěl’stvo Nauka, Moskva 1988.
[10] Kittel Ch.: Introduction to Solid State Physics. 4th ed., John Wiley, Inc., New York 1971.
[11] Petersen D. P., Middleton D.: Sampling and Reconstruction of Wave – Number – Limited Functions
in N –Dimensional Euclidean Spaces. Information and Control 5 (1962), 279–323.