Fundamentals of the Physics of Solids / 16-back-matter

608 C Mathematical Formulas

A di erent normalization of Fourier transform can be chosen for functions defined on a lattice, too. The alternative

1				fˆ(k) eik·Ri ,
f (Ri) =	√
		N	k
				(C.1.49)
1
				f (Ri) e−ik·Ri ,
fˆ(k) = √N Ri

which is commonly used in the literature, is also extensively applied in the present book.

C.1.3 Fourier Transform of Some Simple Functions

Owing to the periodicity of sin ϕ, the function exp(iz sin ϕ) is also periodic with period 2π. Its Fourier decomposition is

	∞
eiz sin ϕ =	Jn(z) einϕ ,	(C.1.50)

n=−∞

where Jn(z) is the Bessel function of order n. Similarly,

	∞
eiz cos ϕ =
	inJn(z) einϕ .	(C.1.51)

n=−∞

The sawtooth wave is defined to take the value 2x/L over the interval −L/2 < x < L/2, which is then repeated with period L in both directions. Its Fourier representation is

f (x) =	2		∞ (−1)n sin 2πnx .
	−
		π	n=1		n					L
The Fourier transform of the Heaviside step function
θ(x) = 1					x = 0 ,
			1		x > 0 ,
			2
			0		x < 0
is	ˆ						i
θ(k) = πδ(k) −							k	.
To determine the Fourier transform of the top-hat function
Π(x) =	1/a −a/2 ≤ x ≤ a/2 ,
	0			x > a/2
			|		|

(C.1.52)

(C.1.53)

(C.1.54)

(C.1.55)

				C.1 Fourier Transforms		609
one has to exploit the relation
Π(x) =	1		[θ(x + a/2)	− θ(x − a/2)]	(C.1.56)
	a

and the property that the Fourier transform of θ(x − x0) di ers from that of θ(x) by a factor exp(−ikx0). The result is

ˆ	sin(ka/2)
Π(k) =		.
	ka/2

The Fourier transform of the Gaussian function

f (x) = e−ax2

is another Gaussian function,

ˆ π/a e−k2 /4a . f (k) =

The Fourier transform of the Lorentzian function

f (x) =

1 Γ

π (x − x0)2 + Γ 2

contains an exponentially decaying term,

ˆ −ikx0−Γ |k|

f (k) = e .

(C.1.57)

(C.1.58)

(C.1.59)

(C.1.60)

(C.1.61)

Because of its slow decay, the Coulomb potential 1/r does not have an unambiguous Fourier transform, but the Yukawa potential, which contains an additional exponentially decaying factor, does:

	e−αr	e−ik·r dr =			4π			(C.1.62)
							.
	r			k2 + α2
The relationship		r e−ik·r dr = k2						(C.1.63)
		1			4π

can be interpreted as a limit of the preceding formula. The inverse Fourier transform leads to

1	=	4π		1	eik·r .	(C.1.64)
r		V
			k	k2

Changing to a continuous variable,

r	= 2π2		k2 eik·r dk .		(C.1.65)
1	1		1

The following inverse Fourier transform is related to the Green function of particles with a k2 dispersion relation:

1		eik·r			=	cos ar	.	(C.1.66)
			−
V		k2		a2		4πr
	k

610 C Mathematical Formulas

C.2 Some Useful Integrals

Below we shall present some definite integrals of particular importance in solid-state physics. The special functions in terms of which certain integrals are expressed will be introduced and evaluated in the next section.

C.2.1 Integrals Containing Exponential Functions

Integrals that contain the product of an exponential and a power-law function can very generally be rewritten in the form

∞	e−μxxν−1 dx = μν Γ (ν) Re μ > 0 , Re ν > 0 .
0
	1

In the special case when the power is an integer or a half-integer,

∞
0	e−xxn−1 dx = Γ (n) ,
∞					√
						π
0		e−αx√x dx =						.
					2α3/2

When x2 appears in the exponent,

∞			x2n+1 dx =		2αn+1 ,
0	e−αx
		2			n!
	∞				(2 2n−+1
	0	e−αx		x2n dx =			/		α2n+1
			2		n 1)!!				π

for odd and even powers of x, while for n = 0

∞
0	e−αx	dx =	2	/		α	.
		2	1			π

When the argument of the exponent is imaginary,

∞
e−iαx		dx = e−iπ/4	/		α	.
	2				π

(C.2.1)

(C.2.2)

(C.2.3)

(C.2.4)

(C.2.5)

(C.2.6)

(C.2.7)

0

C.2 Some Useful Integrals

611

C.2.2 Integrals Containing the Bose Function

When the chemical potential is zero, the definite integral over the interval (0, ∞) of the product of the Bose function 1/(ex −1) and an arbitrary positive power of x can be determined exactly:

∞	ν		−	1		∞	∞	xν−1
	ex				1 dx = k=1
	x
0		−					0

if ν > 1. When ν is an integer,

∞	1
e−kx dx =		Γ (ν) = Γ (ν)ζ(ν) (C.2.8)
k=1	kν

∞	n		∞	∞	∞ 1
	x		dx = k=1		xn e−kx dx = n! k=1		= n!ζ(n + 1) .	(C.2.9)
	ex	1				kn+1
0	−			0

For odd powers, the integral can be expressed in terms of Bernoulli numbers:

∞	x2n−1		(2π)2n		(2π)2n
0
		dx = (−1)n−1		B2n =		|B2n| .	(C.2.10)
	ex − 1		4n		4n

Integrals containing the derivative of the Bose function can also be evaluated:

∞

xn ex

(ex − 1)2 dx = n!

0

∞	1
		= n!ζ(n) .	(C.2.11)
	kn

k=1

When the spectrum is cut o at a finite frequency, the Debye function defined as

	n	x	tn
		0
Dn(x) =				dt	(C.2.12)
	xn		et − 1

appears. For small values of its argument

n

∞

nB2k

Dn(x) = 1 −

x +

x2k ,

2(n + 1)

(2k + n)(2k)!

k=1

while for large values of x

k2−

+ n(n −k3

Dn(x) = xn n!ζ(n + 1) −

e−kx

k

+

−

n

∞

xn

nxn

1

1)xn

2

k=1

Through integration by parts integrals of the form

x

n

t

t

Jn(x) =

e

dt ,

(et

− 1)2

(C.2.13)

+ . . . .

(C.2.14)

(C.2.15)

0

612 C Mathematical Formulas

which contain the derivative of the Bose function, can be rewritten as

	xn		x	tn−1
			0
Jn(x) = −		+ n
	ex − 1			et − 1
	xn		n

= −ex − 1 + n − 1 xn−1

dt

(C.2.16)

Dn−1(x) .

C.2.3 Integrals Containing the Fermi Function

The definite integral over the interval (0, ∞) of the product of the Fermi function 1/(ex + 1) – which specifies the occupation of states in fermionic systems – and a power α > −1 of x can be determined exactly:

∞

xν−1

ex + 1 dx =

0

=

if ν > 0. For ν = 1

∞ ∞

xν−1(−1)k−1e−kx dx

k=1 0

∞ (−1)k−1

kν

Γ (ν) = (1 − 21−ν )Γ (ν)ζ(ν)

k=1

∞

dx

ex + 1 = ln 2 .

0

(C.2.17)

(C.2.18)

For odd powers (α = 2n − 1) the integral can be expressed with the Bernoulli numbers:

∞	x2n−1		(2π)2n		(2π)2n
		dx = (−1)n−1(1 − 21−2n)		B2n = (1 − 21−2n)		|B2n| .
	ex + 1		4n		4n
0						(C.2.19)

Sometimes integrals containing the derivative of the Fermi function need to be evaluated:

∞

xnex

(ex + 1)2 dx

0

∞ ∞

= xn(−1)k−1ke−kx dx

k=1 0

∞ (−1)k−1

= n! = n!(1 − 21−n)ζ(n) . kn

k=1

When the chemical potential is nonzero, the Fermi integrals

	1	∞	xj dx
Fj (η) =
	j!		exp(x − η) + 1

(C.2.20)

(C.2.21)

0

C.2 Some Useful Integrals

613

appear, where the definition of j! is extended to noninteger values of j via the Γ function. For η = 0

Fj (0) = 1 − 2−j ζ(j + 1) .

(C.2.22)

For negative values of η the Fermi integral can be expanded into a series as

1

∞

∞

elη

Fj (η) =

l

xj (−1)l−1elη e−lx dx =

l=1 (−1)l−1

.

(C.2.23)

j!

lj+1

0

For large negative values the leading-order term is

Fj (η) ≈ eη .

(C.2.24)

When η is positive, only an asymptotic form can be obtained:

Fj (η) = (jη+ 1)! &1 + r=1 a2r η−2r ' + R2s ,

(C.2.25)

j+1

s

where

a2r = 2(j + 1)j . . . (j − 2r + 2)(1 − 21−2r)ζ(2r) ,

(C.2.26)

and the remainder R2s is less than

1

(2s + 2)a2s+2ηj−2s−1 .

(C.2.27)

j!

For j = 1/2 and j = 3/2

η3/2

π2

7π4

F1/2(η) =

1 +

η−2 +

η−4 + R4 ,

(C.2.28-a)

(3/2)!

8

640

η5/2

π2

7π4

F3/2(η) =

1 +

5

η−2 −

η−4

+ R4 .

(C.2.28-b)

(5/2)!

8

384

C.2.4 Integrals over the Fermi Sphere

We shall repeatedly encounter sums and integrals over the Fermi sphere:

1

k <kF eik·r =

d3k

eik·r

V

(2π)3

|

|

|

k <kF

|

= (2π)3

kF

π

2π

(C.2.29)

0

k2 dk 0

sin θ dθ 0

dϕ eikr cos θ

1

=

kF3

j1(kFr)

=

kF3

sin kFr − kFr cos kFr

,

2π2

2π2

kFr

(kFr)3

614 C Mathematical Formulas

where j1(x) is the spherical Bessel function of the first kind of order n = 1.

1

1

d3k

1

=

(C.2.30)

V

k <kF

k

k

2

(2π)3

k

k

2

|

|

| − |

|k |<kF

| − |

= (2π)3

kF

π

2π

dϕ k2 + k 2

1

2kk cos θ .

k 2 dk

sin θ dθ

1

0

0

0

−

Using the substitution cos θ = x,

V

k <kF

k

1k 2

kF

1

k2 + k

2

2kk x

= (2π)2

k 2 dk

1

|

1

dx

|

|

−

|

0

kF

−1

−

= (2π)2

k

k

(C.2.31)

k

k dk ln

1

1

k + k

0

−

F

2

2

F

2

F

k

2π2

k

4kkF

kF

=

k

1

+

−

k

ln

k

+ k

.

−

The following integral can be determined along the same lines:

V

k <kF

k + q12

k2

= (2π)3

kF

π

sin θ dθ

2π

k2 dk

dϕ q2 + 2kq cos θ

1

|

1

1

|

|

|

−

0

0

0

1

kF

k2

q2 + 2kq

=

ln

dk

(2π)2

2kq

q2

−

2kq

0

F

2

2

q

F

2

F

(C.2.32)

8π2

8qkF

q

2kF

k

1

4k

−

q

+ 2k

=

+

ln

−

.

If (C.2.30) has to be summed with respect to the variable k on the Fermi sphere, then the introduction of the variable k = k + q gives

V12

k <kF k <kF

k k

2 =

(2π)3

(2π)3

k k

2

| |

|

1

d3k

d3k

1

|

| − |

|k|<kF

|k |<kF

| − |

=

d3k d3q

1

.

(C.2.33)

(2π)3

(2π)3

|q|2

|k|<kF |k+q|<kF

First the integration with respect to k (with q fixed) is performed. The region of integration allowed by the constraints is the intersection of two Fermi spheres with a relative displacement of q (see Fig. C.1).

C.3 Special Functions

615

Fig. C.1. Domain of integration in the space of variables k and k = k + q

With q fixed, the k-integration is most easily done using cylindrical coordinates; the component along q is then denoted by kz . Since |q| may vary between 0 and 2kF, and as the regions −kF < kz < −q/2 and −q/2 < kz < kF −q contribute equally,

V 2

k <kF k <kF

k k

2 =

(2π)6

2kF

4π dq

−q/2

dkz 2π[kF2 − kz2]

1

| |

|

1

1

|

| − |

0

−kF

=

2π

4

(C.2.34)

.

kF

C.2.5 d-Dimensional Integrals

The d-dimensional integral of a function that depends only on the magnitude of the vector k is given by

	(2π)d					∞			(C.2.35)
			f (k2) ddk = Kd 0				f (k2)kd−1 dk ,
1
where				1			1
			Kd =					.	(C.2.36)
				πd/22d−1		Γ (d/2)

If the integrand depends not only on the magnitude of k but also on its component along the direction of a given vector k1,

∞π

(2π)d		f (k2, k·k1) ddk = 2π Kd−1			0	dk 0	dθkd−1(sin θ)d−2f (k2, kk1 cos θ) .
1		1
							(C.2.37)

C.3 Special Functions

C.3.1 The Dirac Delta Function

By definition, the Dirac delta function (or simply delta function) vanishes everywhere except for x = 0 where it takes an infinitely large value, in such

616 C Mathematical Formulas

a way that its integral over the entire number line is unity. Some common representations are

1

2

2

δ(x) = lim

e−x

/ε

√

ε 0

→

πε

= lim

1 sin(x/ε)

π

x

ε→0

= lim

1

ε

π x2 + ε2

ε→0

= 2π

∞

dk eikx.

1

−∞

The d-dimensional generalization of the last formula is

(C.3.1-a)

(C.3.1-b)

(C.3.1-c)

(C.3.1-d)

δ(d)(r) = (2π)d		dk eik·r .	(C.3.2)
1

The imaginary part of a function that has a simple pole close to the real axis can be expressed in terms of the Dirac delta function as

	1	1
			= P		iπδ(x) ,	(C.3.3)
x	±	iα		x

where α is infinitesimally small, and P stands for the principal value. This expression appears in the result of the integral

	∞		i
	0	eiωt dt =		(C.3.4)
			ω + iα

when an infinitesimally small imaginary part is added to the variable ω to render the integral convergent at the upper limit.

The Heaviside step function can be defined as the integral of the Dirac delta function:

				x
θ(x) =					δ(t) dt ,				(C.3.5)
that is		−∞
				d
δ(x) =						θ(x) .			(C.3.6)
				dx
An alternative representation is
		i		∞
θ(t) = α→0							e−iωt		(C.3.7)
	2π						ω + iα dω ,
lim

−∞

which can be obtained by taking the inverse Fourier transform of (C.3.4).

C.3 Special Functions

617

The Dirac delta function also appears in the frequently encountered formula

1		1
		= 2		= −4πδ(r) .	(C.3.8)
	r		r

To demonstrate this, consider a sphere of radius R and volume V , centered at the origin, and integrate the left-hand side of the previous formula over this sphere. Using Gauss’s law, the volume integral can be converted into an integral over the surface S of the sphere:

V	2 r dr = S		r dS = − S		r2 = −4π ,
	1		1		dS

which is just the integral of the right-hand side.

C.3.2 Zeta and Gamma Functions

The Riemann Zeta Function

The Riemann zeta function is defined by the sum

∞

ζ(z) = k−z ,

k=1

(C.3.9)

(C.3.10)

which is convergent for Re z > 1. Alternatively, it may be represented by the integral

∞

z

−

1

∞

∞

∞

xz−1 dx .

(C.3.11)

ζ(z) = Γ (z)

ex

1 dx =

Γ (z) k=1

e−kx

1

x

1

0

−

0

0

It can be analytically continued to the entire complex plane except for the point z = 1.

For the first few integer values of the argument
ζ(2) = 1 +	1	+	1	+ · · · =	π2
						,
	22		32		6
ζ(3) = 1 +	1	+	1	+ · · · = 1.202 ,
	23		33
ζ(4) = 1 +	1	+	1	+ · · · =	π4			(C.3.12)
						,
	24		34		90
ζ(5) = 1 +	1	+	1	+ · · · = 1.037 ,
	25		35
ζ(6) = 1 +	1	+	1	+ · · · =	π6
							,
	26		36		945

while for a couple of half-integer values