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

		B.2 Characteristic Temperatures of the Elements					597
Atomic	Name of	Chemical	Melting	ΘD	Ordered	Transition
number	element	symbol	point (◦C)	(K)	phase	temperature
20	calcium	Ca	842	230
21	scandium	Sc	1541	360		Tc = 0.40 K
22	titanium	Ti	1668	420	S
23	vanadium	V	1910	380	S	Tc = 5.46 K
24	chromium	Cr	1907	630	AF	TN = 311 K
25	manganese	Mn	1246	410	AF	TN = 100 K
26	iron	Fe	1538	467	F	TC = 1043 K
27	cobalt	Co	1495	445	F	TC = 1388 K
28	nickel	Ni	1455	450	F	TC = 627 K
29	copper	Cu	1085	343		Tc = 0.86 K
30	zinc	Zn	420	327	S
31	gallium	Ga	30	320	S	Tc = 1.08 K
32	germanium	Ge	938	370
33	arsenic	As	817	282
34	selenium	Se	221	90
35	bromine	Br	−7
36	krypton	Kr	−157	72
37	rubidium	Rb	39	56
38	strontium	Sr	777	147
39	yttrium	Y	1522	280		Tc = 0.63 K
40	zirconium	Zr	1855	291	S
41	niobium	Nb	2477	275	S	Tc = 9.25 K
42	molybdenum	Mo	2623	450	S	Tc = 0.92 K
43	technetium	Tc	2157	351	S	Tc = 7.8 K
44	ruthenium	Ru	2334	600	S	Tc = 0.49 K
45	rhodium	Rh	1964	480	S	Tc = 0.035 mK
46	palladium	Pd	1555	274
47	silver	Ag	962	225		Tc = 0.52 K
48	cadmium	Cd	321	209	S
49	indium	In	157	108	S	Tc = 3.41 K
50	tin	Sn	232	199	S	Tc = 3.72 K
51	antimony	Sb	631	211
52	tellurium	Te	450	153
53	iodine	I	114	106
54	xenon	Xe	−112	64
55	cesium	Cs	28	38
56	barium	Ba	727	110		Tc = 5 K
57	lanthanum	La	920	142	S
58	cerium	Ce	798	146	AF	TN = 12.5 K
					Continued on the next page

598 B The Periodic Table of Elements
Atomic	Name of	Chemical	Melting	ΘD	Ordered	Transition
number	element	symbol	point (◦C)	(K)	phase	temperature
59	praseodymium	Pr	931	85	AF	TN = 0.03 K
60	neodymium	Nd	1016	159	AF	TN = 6 K
61	promethium	Pm	1042	158		TN = 14.0 K
62	samarium	Sm	1074	116	AF
63	europium	Eu	822	127	AF	TN = 90.4 K
64	gadolinium	Gd	1313	195	F	TC = 293 K
65	terbium	Tb	1356	150	F	TC = 220 K
66	dysprosium	Dy	1412	210	F	TC = 90 K
67	holmium	Ho	1474	114	F	TC = 20 K
68	erbium	Er	1529	134	F	TC = 18 K
69	thulium	Tm	1545	127	F	TC = 32 K
70	ytterbium	Yb	824	118		Tc = 0.1 K
71	lutetium	Lu	1663	210	S
72	hafnium	Hf	2233	252	S	Tc = 0.13 K
73	tantalum	Ta	3017	240	S	Tc = 4.47 K
74	tungsten	W	3422	400	S	Tc = 0.02 K
75	rhenium	Re	3186	430	S	Tc = 1.70 K
76	osmium	Os	3033	500	S	Tc = 0.66 K
77	iridium	Ir	2446	420	S	Tc = 0.11 K
78	platinum	Pt	1768	240
79	gold	Au	1064	165		Tc = 4.15 K
80	mercury	Hg	−39	72	S
81	thallium	Tl	304	78	S	Tc = 2.38 K
82	lead	Pb	327	105	S	Tc = 7.20 K
83	bismuth	Bi	271	119
84	polonium	Po	254	81
85	astatine	At	302
86	radon	Rn	−71
87	francium	Fr	27
88	radium	Ra	696	89
89	actinium	Ac	1051	124		Tc = 1.37 K
90	thorium	Th	1750	170	S
91	protactinium	Pa	1572	159	S	Tc = 1.4 K
92	uranium	U	1135	207	S	Tc = 0.68 K
93	neptunium	Np	644	121
94	plutonium	Pu	640	171		Tc = 0.60 K
95	americium	Am	1176		S
96	curium	Cm	1345

B.2 Characteristic Temperatures of the Elements

599

References

1.CRC Handbook of Chemistry and Physics, Editor-in-Chief D. R. Lide, 85th Edition, CRC Press, Boca Raton (2004).

2.http://www.webelements.com

3.Springer Handbook of Condensed Matter and Materials Data, Editors: W. Martienssen and H. Warlimont, Springer-Verlag, Berlin (2005).

C

Mathematical Formulas

C.1 Fourier Transforms

When surface e ects are neglected and only bulk properties are examined in macroscopic crystalline samples, periodic boundary conditions are frequently applied, since the Fourier components that appear in the Fourier series or Fourier integral representation of the position-dependent quantities are often easier to determine. Nevertheless, owing to the invariance of crystals under discrete translations, functions that show the same periodicity as the crystal lattice and functions defined at the vertices of the crystal lattice are frequently encountered, too. It is often more convenient to specify them using the Fourier components associated with the vectors defined in the reciprocal lattice. The most important formulas of such functions are listed in the present section.

C.1.1 Fourier Transform of Continuous Functions

A periodic function of period L (f (x + L) = f (x)) can be expanded into a Fourier series as

f (x) = 21 a0	∞	an cos		2πn	x + bn sin		2πn	x .	(C.1.1)
	+ n=0			L			L

Making use of the orthogonality relation of trigonometric functions, it can be shown that the coe cients are given by the integrals

an = L

L/2

f (x) cos

L x dx,

bn = L

L/2

f (x) sin

L

x dx .

2

2πn

2

2πn

−L/2

−L/2

(C.1.2)

It is often more convenient to use exponential functions:

∞

f (x) =

fne2πinx/L ,

(C.1.3)

n=−∞

602 C Mathematical Formulas

where

fn = L	L/2	f (x)e−2πinx/L dx .	(C.1.4)
1

−L/2

This can be demonstrated directly by exploiting the completeness relation

∞

N

e2πinx/L = lim

N →∞

n=−∞

= lim

N →∞

	e2πinx/L		(C.1.5)
n=−N
	sin[π(2N + 1)x/L]		= Lδ(x)
	sin(πx/L)

and the orthogonality relation

	L/2	sin[π(n − n )]
	e−2πi(n−n )x/L dx =		= Lδ	n,n	.	(C.1.6)
		π(n − n )/L

−L/2

The coe cients fn of the Fourier series make up the Fourier spectrum of function f .

The Fourier series representation is straightforward to generalize to functions defined in d-dimensional space, provided they satisfy periodic boundary conditions on hypercubes (or even more generally, hyperparallelepipeds) of volume Ld – and are repeated periodically outside it. For simplicity, consider a function f defined inside a three-dimensional general parallelepiped of edges N1a1, N2a2, N3a3, and volume V that satisfies the periodic boundary conditions

f (r + Niai) = f (r) , i = 1, 2, 3

(C.1.7)

on and beyond the boundaries. The Fourier series can then be written as

	1
f (r) =	V	fˆ(k) eik·r .	(C.1.8)
		k

On account of the periodic boundary conditions, the allowed vectors k are most easily expressed in terms of the primitive vectors bi of the reciprocal

lattice:

3

k = mi bi , (C.1.9)

i=1 Ni

where the mi are arbitrary integers. Recall that the primitive vectors of the direct and reciprocal lattices are related by (5.2.13).

ˆ

The explicit form of the Fourier coe cient f (k) can be derived either using the generalization

eik·(r−r ) = V δ(r − r )

(C.1.10)

k

		C.1	Fourier Transforms	603
of the completeness relation (C.1.5), or the orthogonality relation
V	e−i(k−k )·r dr = V δk,k ,		(C.1.11)
leading to
fˆ(k) = V		f (r) e−ik·r dr .	(C.1.12)

It is easily seen that the convention used in the one-dimensional case is recovered if instead of (C.1.8) the Fourier series is defined as

f (r) =			(C.1.13)
		fˆ(k) eik·r ,
	k
and consequently the Fourier coe cients are given by
fˆ(k) = V	V	f (r) e−ik·r dr	(C.1.14)
1

instead of (C.1.12). The rationale behind choosing a di erent convention is that in su ciently large samples, where discrete sums are replaced by continuous integrals, the obtained formulas are independent of the sample volume in the V → ∞ limit. Since each vector k in the primitive cell of the reciprocal lattice is associated with a volume (2π)3/V , the sum over the k vectors can be replaced by an integral, using the formal substitution

		V		dk .	(C.1.15)
		→ (2π)3
	k

Then the Fourier integral representation of an arbitrary function f (r) defined on the whole space is, by definition,

	f (r) = (2π)3 fˆ(k) eik·r dk ,
				1
where		fˆ(k) =
as in the V → ∞ limit						f (r) e−ik·r dr ,
			(2π)3 eik·(r−r ) = δ(r − r ) ,
			dk
and	e−i(k−k )·r dr = (2π)3δ(k − k ) .

(C.1.16)

(C.1.17)

(C.1.18)

(C.1.19)

604 C Mathematical Formulas

ˆ

The function f (k) is the Fourier transform of f (r) and (C.1.16) defines the inverse Fourier transform.

More generally, the Fourier transform of a function defined in d-dimensional space is given in the space of the d-dimensional k vectors as

ˆ −ik·x d

f (k) = f (r) e d x ,

and the inverse Fourier transform is defined by

f (x) = (2π)d		fˆ(k) eik·x ddk .
1

The completeness and orthogonality relations then take the form

	ddk
	(2π)d eik·(x−x ) = δ(d)(x − x )

and

e−i(k−k )·xddx = (2π)dδ(d)(k − k ) .

(C.1.20)

(C.1.21)

(C.1.22)

(C.1.23)

In quantum mechanics, a common choice for the Fourier transform of the function f (r) is

fˆ(k) = (2π)d/2		∞	f (x)e−ik·x ddx ,
1
		−∞

and the inverse Fourier transform is then

	f (x) = (2π)d/2				∞
				1
					−∞
With this choice		f (x)		2ddx =
or more generally
	f (x)g(x)ddx =

fˆ(k)eik·x ddk .
fˆ(k) 2ddk ,

ˆ d

f (k)ˆg(k)d k ,

(C.1.24)

(C.1.25)

(C.1.26)

(C.1.27)

which indicates that the Fourier transform is a unitary transformation in the space of square integrable functions that preserves lengths and inner products.

Another convention is used for the time variable. The Fourier transform of an arbitrary time-dependent function f (t) is defined as

	∞
fˆ(ω) =		f (t) eiωt dt ,	(C.1.28)

−∞

				C.1 Fourier Transforms		605
	and the inverse transform as
		1	∞
	f (t) =			fˆ(ω) e−iωt dω .	(C.1.29)
		2π

−∞

Therefore the following formula is used for spaceand time-dependent functions that satisfy periodic boundary conditions at the boundaries of a sample

of volume V :

∞

fˆ(k, ω) =	dr dt f (r, t) e−i(k·r−ωt) ,	(C.1.30)
V	−∞

and

f (r, t) =

1 1

V k 2π

while for samples of infinite extent

∞

dω fˆ(k, ω) ei(k·r−ωt) ,

(C.1.31)

−∞

∞

	fˆ(k, ω) =		dr	dt f (r, t) e−i(k·r−ωt) ,	(C.1.32)
			−∞
	and		∞
	f (r, t) = (2π)4			dω fˆ(k, ω) ei(k·r−ωt) .	(C.1.33)
			dk
	1

−∞

These formulas can be applied to lattice-periodic functions, whose values inside the primitive cell of volume v spanned by the vectors a1, a2, a3 are repeated with the periodicity of the lattice – in other words, for each translation vector tn that can be written in the form (5.1.1),

f (r + tn) = f (r) .

(C.1.34)

Since condition (C.1.7) is now met by the choice N1 = N2 = N3 = 1, the vectors k appearing in the Fourier representation are the same as the vectors G of the reciprocal lattice, hence the Fourier transform of f (r) is

	fˆ(G) = v		f (r) e−iG·r dr ,	(C.1.35)
	while the inverse transform is
		1
	f (r) =		fˆ(G) eiG·r .	(C.1.36)
		v
			G

606 C Mathematical Formulas

C.1.2 Fourier Transform of Functions Defined at Lattice Points

Functions defined at the vertices of a discrete lattice are frequently used in solid-state physics. Consider a discrete lattice of volume V , with N lattice points. When the function f (Ri) is subject to periodic boundary conditions, it can be represented as

			1
	f (Ri) =		N		fˆ(k) eik·Ri	,	(C.1.37)
					k
	ˆ
and the Fourier transform f (k) can be written as
	fˆ(k) =				f (Ri) e−ik·Ri .		(C.1.38)
		Ri
ˆ	ˆ

Since f (k) = f (k + G) for any vector G of the reciprocal lattice, it is su cient

to consider one vector k in each set of equivalent vectors (which di er by a reciprocal-lattice vector) – in other words, the vectors k given in (C.1.9) are defined within the primitive cell or Brillouin zone of the reciprocal lattice. The number of allowed vectors k is just N .

To justify the previous formulas, we shall demonstrate that in the limit

where the number of lattice points is large,
1			e−i(k−k )·Ri = δk k ,G .		(C.1.39)
				−
	N	Ri	G

The equality obviously holds when k reciprocal lattice, since each of the N

− k is the same as a vector G of the terms in the sum

	(C.1.40)
e−i(k−k )·Ri

Ri

is then unity. Otherwise the phase factors cancel out to a good approximation. This cancellation can be most easily demonstrated in the case where the crystal contains odd numbers of lattice points along the direction of each primitive vector (a1, a2, a3). Expressed in terms of the primitive vectors b1, b2, b3 of the reciprocal lattice, k − k is

				k − k
					= (ki − ki)bi ,	(C.1.41)
					i
and so the sum in question reads
1		N1	N2	N3
				exp{−2πi[(k1 −k1)n1 + (k2 −k2)n2 + (k3 −k3)n3]} .
	N
		n1=−N1 n2=−N2 n3=−N3

(C.1.42)

C.1 Fourier Transforms

607

Performing the sum separately along the three directions,

1		Ni
		exp[−2πi(ki − ki)ni]
2Ni + 1
	ni =−Ni
				2Ni
=		exp[2πi(ki − ki)Ni]				exp[		2πi(k	k )n	]
							−
		2Ni + 1
				ni =0
=		exp[2πi(ki − ki)Ni]			exp[−2πi(ki − ki)(2Ni + 1)] − 1
		2Ni + 1				exp[−2πi(ki − ki)] − 1
=		sin[π(ki − ki)(2Ni + 1)]				.					(C.1.43)
		(2Ni + 1) sin[π(ki − ki)]

Wherever the denominator is zero, i.e., when ki −ki = 0, ±1, ±2, . . . (in other words, k − k is a vector of the reciprocal lattice), the expression tends to 1 in the limit N → ∞. Everywhere else it vanishes in the same limit.

Since the allowed vectors k are distributed continuously in the N → ∞

limit, the relation (C.1.39) can be recast in the equivalent form

1

)

(2π)3

δ(k − k − G) .

N

e−i(k−k

·Ri

=

V

(C.1.44)

Ri

G

In one dimension this can be rewritten as

1

) R

2π

e−i(k−k

·

n =

δ(k − k − Gh) ,

(C.1.45)

N

L

Rn

h

where Rn = na and Gh = (2π/a)h. Owing to the properties of the delta function, this is equivalent to

∞∞

			h) .
e−2πinx =	δ(x	−		(C.1.46)
n=−∞	h=−∞

It can be shown along the same lines that for the sum over the discrete vectors k in the Brillouin zone

1
	eik·(Ri −Rj ) = δRi ,Rj .	(C.1.47)
N k BZ

In the N → ∞ limit, where the sum over the reciprocal lattice can be replaced by an integral,

N (2π)3			dk eik·(Ri −Rj ) = δRi ,Rj .	(C.1.48)
V	1

k BZ

Naturally, integration is once again over the primitive cell or Brillouin zone of the reciprocal lattice.