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

A

Physical Constants and Units

A.1 Physical Constants

The following table contains the values in SI units for some physical constants that are particularly important in solid-state physics.

Name	Symbol				Value
speed of light					299 792 458 m s−1
in vacuum	c
magnetic constant					4π × 10−7 N A−2
(permeability of free space)	μ0
electric constant	0 = 1/μ0c2				8.854 188 ×		10−12 F m−1
(permittivity of free space)
elementary charge	e				1.602 176 ×		10−19 C
Planck constant	h				6.626 069 ×		10−34 J s
in eV	h/{e}				4.135 667 ×		10−15 eV s
reduced Planck constant	= h/2π				1.054 572 ×		10−34 J s
in eV	/{e}				6.582 119 ×		10−16 eV s
fine-structure constant	α = e2/4π 0 c				7.297 353 ×		10−3
inverse of α	α−1				137.035 999
magnetic flux quantum	Φ0 = h/2e				2.067 834 ×		10−15 Wb
conductance quantum	G0 = 2e2/h				7.748 092 ×		10−5 S
inverse of G0	G0−1 = h/2e2				12 906.404 Ω
Josephson constant	K	J	= 2e/h		483 597.9	×	109 Hz V−1
				2
von Klitzing constant	RK = h/e				25 812.807 Ω
Boltzmann constant	kB				1.380 650 ×		10−23 J K−1
Avogadro constant	NA				6.022 142 ×		1023 mol−1

Continued on the next page

588 A Physical Constants and Units
	Name	Symbol	Value
	molar gas constant	R = NAkB	8.314 472 J mol−1 K−1
	Bohr magneton	μB = e /2me	9.274 009 × 10−24 J T−1
	nuclear magneton	μN = e /2mp	5.050 783 × 10−27 J T−1
	Bohr radius	a0 = 4π 0 2/mee2	0.529 177 × 10−10 m
	electron mass	me	9.109 382 × 10−31 kg
	electron magnetic	μe	−9.284 764 × 10−24 J T−1
	moment		−1.001 160 μB
	electron g-factor	ge = 2μe/μB	−2.002 319
	electron gyromagnetic	γe = 2|μe|/	1.760 860 × 1011 s−1 T−1
	ratio	γe/2π	28 024.9532 MHz T−1
	neutron mass	mn	1.674 927 × 10−27 kg
	neutron magnetic	μn	−0.966 236 × 10−26 J T−1
	moment		−1.913 043 μN
	neutron g-factor	gn = 2μn/μN	−3.826 085
	proton mass	mp	1.672 622 × 10−27 kg
	proton–electron		1836.152 667
	mass ratio	mp/me
	electron–proton		5.446 170 10−4
	mass ratio	me/mp
	proton magnetic	μp	1.410 607 × 10−26 J T−1
	moment		2.792 847 μN
	proton g-factor	gp = 2μp/μN	5.585 695
	proton gyromagnetic	γp = 2μp/	2.675 222 × 108 s−1 T−1
	ratio	γp/2π	42.577 481 MHz T−1
	Rydberg constant	R∞ = α2mec/2h	10 973 731.569 m−1
	Rydberg energy	Ry = R∞hc	2.179 872 × 10−18 J
	in electronvolts		13.605 692 eV
	Hartree energy	Eh = e2/4π 0a0	4.359 744 × 10−18 J
	in electronvolts		27.211 384 eV

A.2 Relationships Among Units

The fundamental units of the SI system are meter (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (Cd). In the next table derived units are expressed in terms of these fundamental ones – and other derived units.

	A.2 Relationships Among Units 589
1 coulomb (C) = 1 A s	1 pascal (Pa) = 1 N m−2
1 farad (F) = 1 C V−1 = 1 A2 s2 J−1	1 siemens (S) = 1 Ω−1 = 1 A V−1
1 henry (H) = 1 V s A−1 = 1 J A−2	1 tesla (T) = 1 Wb m−2
1 joule (J) = 1 N m = 1 kg m2 s−2	1 volt (V) = 1 W A−1
1 newton (N) = 1 kg m s−2	1 watt (W) = 1 J s−1
1 ohm (Ω) = 1 V A−1	1 weber (Wb) = 1 V s = 1 J A−1

It is immediately established that the unit of magnetic moment is 1 A m2 = 1 J T−1.

The following SI prefixes are used to denote the multiples and fractions of the units:

Name	Symbol	Corresponding power of 10
yotta-	Y	1024
zetta-	Z	1021
exa-	E	1018
peta-	P	1015
tera-	T	1012
giga-	G	109
mega-	M	106
kilo-	k	103
milli-	m	10−3
micro-	μ	10−6
nano-	n	10−9
pico-	p	10−12
femto-	f	10−15
atto-	a	10−18
zepto-	z	10−21
yocto-	y	10−24

Non-SI Units

Much of the solid-state physics literature continues to use CGS units. Some of the most frequently used units are:

1 Å = 10−10 m = 0.1 nm ,	1 erg = 10−7 J ,
1 atm = 101 325 Pa ,	1 bar = 105 Pa .

The CGS unit of magnetic field strength is the oersted (Oe), while that of magnetic induction (flux density) and magnetization is the gauss (G). In SI, magnetic induction is given in teslas, while magnetic field strength and magnetization in A/m. Therefore

1 G =@ 10−4 T

590 A Physical Constants and Units

when specifying the magnetic induction, however

1 G =@ 103 A/m

when it comes to magnetization. The relationship between the units of magnetic flux is

1 G cm2 =@ 10−8 Wb .

The units of magnetic field strength are related by

1 Oe =@ 103 A/m = 79.58 A/m .

4π

When CGS units are used, equations for electromagnetic quantities are usually written in their nonrationalized form. The table below shows the conversion factors used in the transformation of the rationalized equations into nonrationalized ones that apply to the Gaussian quantities (denoted by ); c2 = 1/ 0μ0.

Physical quantity	Conversion formula (CGS–SI)
electric charge	q = (4π 0)−1/2q
electric current	j = (4π 0)−1/2j
electric field	E = (4π 0)1/2E
electric displacement	D = (4π/ 0)1/2D
magnetic field	H = (4πμ0)1/2H
magnetic induction	B = (4π/μ0)1/2B
scalar potential	ϕ = (4π 0)1/2ϕ
vector potential	A = (4π/μ0)1/2A
resistivity	! = 4π 0!
conductivity	σ = (4π 0)−1σ
permittivity	= / 0
permeability	μ = μ/μ0
magnetic susceptibility	χ = χ/4π
magnetization	M = (μ0/4π)1/2M
magnetic flux	Φ = (4π/μ0)1/2Φ

For example, the expression ωc = eB/me for cyclotron frequency goes over

into

ωc = √ 0μ0e B /me = e B . mec

To convert electromagnetic SI units into CGS units, the next relationship (of mixed units) has to be used:

/					√		.
		μ0		= 0.1		dyn
		4π
						A

The nonrationalized form and CGS value of some electromagnetic constants are listed below.

A.2 Relationships Among Units

591

Name of physical constant

Symbol

Value

elementary charge

e

2

4.803 24 × 10−10 dyn1/2 cm

Bohr radius

a0 = 2

/mee

5.291 772 ×

10−9 cm

−1

Bohr magneton

μ

= e /2m

c

9.274 009

×

10−21

B

2

e

erg G

fine-structure constant

α = e

/ c

1/137.036

flux quantum

Φ

= hc/2e

2.067 834

×

10−7

G cm

2

0

Conversion Factors of Energy Equivalents

In solid-state physics energies are very often given in electronvolts: 1 eV = (e/C)J. Thermal energies are usually specified using the relation E = kBT , with the temperature given in kelvins, while magnetic field energies are commonly converted to field strengths given in teslas or gausses through E = μBB. In spectroscopy energy is often expressed in hertz units via E = hν or in inverse (centi)meters, via E = hc/λ. The energies corresponding to these units are

E(1 K) = (1 K)kB = 1.380 650 × 10−23 J ,

E(1 T) = (1 T)μB = 9.274 009 × 10−24 J ,

E(1 cm−1) = (1 cm−1)hc = 1.986 445 × 10−23 J ,

E(1 Hz) = (1 Hz)h = 6.626 069 × 10−34 J .

The rydberg (1 Ry = 1 R∞hc) and hartree (1 hartree = 2 Ry) units are widely used, too. The conversion factors relating energies given in the previous units are listed in the following table.

1 eV = 1.602 176 ×

10−19 J

1 K = 1.380 650 ×

10−23 J

@

1.727 599

×

104

T

@

1.488 731 T

10−5 eV

1.160 451

×

104

K

8.617 343

×

6.333 631 ×

7.349 865 ×

10−2 Ry

10−6 Ry

8.065 544

×

14

6.950 356

×

10

103 cm−1

10−1 cm−1

2.417 989 ×

10

Hz

2.083 664 ×

10

Hz

1 T = 9.274 009 ×

10−24 J

1 Ry = 2.179 872 ×

10−18 J

@

0.671 713 K

10−5 eV

@

1.578 873 ×

105

K

5.788 382

×

13.605 692 eV

4.254 383 ×

10−6 Ry

2.350 517 ×

105 T

4.668 645

×

10

1.097 373

×

15

10−1 cm−1

105 cm−1

1.399 625 ×

10

Hz

3.289 842 ×

10

Hz

592 A Physical Constants and Units
1 cm−1 = 1.986 445		× 10−23 J	1 Hz = 6.626 069 ×			10−34	J
@	1.438 775 K		@	4.799 237	×	10−11	K
	1.239 842	× 10−4 eV		4.135 667	×	10−15 eV
	2.141 949 T			7.144 773 ×		10−11	T
	9.112 670	× 10−6 Ry		3.039 660	×	10−16	Ry
	2.997 925	× 1010 Hz		3.335 641	×	10−11 cm−1

Binding and cohesive energies in solids are commonly given per atom or per mole. In addition to eV, Ry, and J, older tables also contain data given in calories. Its usual value is 1 cal = 4.1868 J, however that of the thermochemical calory is 1 cal = 4.184 J. Consequently

1	eV		= 73.499	mRy		= 96.4853	kJ		= 23.05	kcal	.
	atom			atom			mol
										mol

Reference

1.P. J. Mohr and B. N. Taylor, 2002 CODATA recommended values of the fundamental physical constants, National Institute of Standards and Technology (2003). The data are available on the WWW at http://physics.nist.gov/cuu/Constants.

B

The Periodic Table of Elements

B.1 The Electron and Crystal Structures of Elements

The next table shows the electron structure for the elements of the periodic table as well as the Pearson symbol of their crystal structure that is stable at low temperatures and atmospheric pressure. When another crystal structure is stable at room temperature, its Pearson symbol is also listed.

Atomic	Name of	Chemical	Electron	Crystal structure
number	element	symbol	structure	at 0 K	at 300 K
1	hydrogen	H	1s1	cF4 (fcc)	gas
2	helium	He	1s2	hP2 (hcp)	gas
3	lithium	Li	[He] 2s1	hP2 (hcp)	cI2 (bcc)
4	beryllium	Be	[He] 2s2	hP2 (hcp)
5	boron	B	[He] 2s2 2p1	hR105
6	carbon	C	[He] 2s2 2p2	hP4
7	nitrogen	N	[He] 2s2 2p3	cP8	gas
8	oxygen	O	[He] 2s2 2p4	mC4	gas
9	fluorine	F	[He] 2s2 2p5	mC6	gas
10	neon	Ne	[He] 2s2 2p6	cF4 (fcc)	gas
11	sodium	Na	[Ne] 3s1	hP2 (hcp)	cI2 (bcc)
12	magnesium	Mg	[Ne] 3s2	hP2 (hcp)
13	aluminum	Al	[Ne] 3s2 3p1	cF4 (fcc)
14	silicon	Si	[Ne] 3s2 3p2	cF8
15	phosphorus	P	[Ne] 3s2 3p3	oC8
16	sulfur	S	[Ne] 3s2 3p4	oF128
17	chlorine	Cl	[Ne] 3s2 3p5	oC8	gas
18	argon	Ar	[Ne] 3s2 3p6	cF4 (fcc)	gas
19	potassium	K	[Ar] 4s1	cI2 (bcc)

Continued on the next page

594	B The Periodic Table of Elements
Atomic	Name of	Chemical	Electron	Crystal structure
number	element	symbol	structure	at 0 K	at 300 K
20	calcium	Ca	[Ar] 4s2	cF4 (fcc)
21	scandium	Sc	[Ar] 4s2 3d1	hP2 (hcp)
22	titanium	Ti	[Ar] 4s2 3d2	hP2 (hcp)
23	vanadium	V	[Ar] 4s2 3d3	cI2 (bcc)
24	chromium	Cr	[Ar] 4s1 3d5	cI2 (bcc)
25	manganese	Mn	[Ar] 4s2 3d5	cI58
26	iron	Fe	[Ar] 4s2 3d6	cI2 (bcc)
27	cobalt	Co	[Ar] 4s2 3d7	hP2 (hcp)
28	nickel	Ni	[Ar] 4s2 3d8	cF4 (fcc)
29	copper	Cu	[Ar] 4s1 3d10	cF4 (fcc)
30	zinc	Zn	[Ar] 4s2 3d10	hP2 (hcp)
31	gallium	Ga	[Ar] 4s2 3d10 4p1	oC8
32	germanium	Ge	[Ar] 4s2 3d10 4p2	cF8
33	arsenic	As	[Ar] 4s2 3d10 4p3	hR2
34	selenium	Se	[Ar] 4s2 3d10 4p4	hP3
35	bromine	Br	[Ar] 4s2 3d10 4p5	oC8	liquid
36	krypton	Kr	[Ar] 4s2 3d10 4p6	cF4 (fcc)	gas
37	rubidium	Rb	[Kr] 5s1	cI2 (bcc)
38	strontium	Sr	[Kr] 5s2	cF4 (fcc)
39	yttrium	Y	[Kr] 5s2 4d1	hP2 (hcp)
40	zirconium	Zr	[Kr] 5s2 4d2	hP2 (hcp)
41	niobium	Nb	[Kr] 5s1 4d4	cI2 (bcc)
42	molybdenum	Mo	[Kr] 5s1 4d5	cI2 (bcc)
43	technetium	Tc	[Kr] 5s2 4d5	hP2 (hcp)
44	ruthenium	Ru	[Kr] 5s1 4d7	hP2 (hcp)
45	rhodium	Rh	[Kr] 5s1 4d8	cF4 (fcc)
46	palladium	Pd	[Kr] 4d10	cF4 (fcc)
47	silver	Ag	[Kr] 5s1 4d10	cF4 (fcc)
48	cadmium	Cd	[Kr] 5s2 4d10	hP2 (hcp)
49	indium	In	[Kr] 5s2 4d10 5p1	tI2
50	tin	Sn	[Kr] 5s2 4d10 5p2	cF8	tI4
51	antimony	Sb	[Kr] 5s2 4d10 5p3	hR2
52	tellurium	Te	[Kr] 5s2 4d10 5p4	hP3
53	iodine	I	[Kr] 5s2 4d10 5p5	oC8
54	xenon	Xe	[Kr] 5s2 4d10 5p6	cF4 (fcc)	gas
55	cesium	Cs	[Xe] 6s1	cI2 (bcc)
56	barium	Ba	[Xe] 6s2	cI2 (bcc)
57	lanthanum	La	[Xe] 6s2 5d1	hP4 (dhcp)
58	cerium	Ce	[Xe] 6s2 4f2	cF4 (fcc)	hP4 (dhcp)
59	praseodymium	Pr	[Xe] 6s2 4f3	hP4 (dhcp)
60	neodymium	Nd	[Xe] 6s2 4f4	hP4 (dhcp)
61	promethium	Pm	[Xe] 6s2 4f5	hP4 (dhcp)
62	samarium	Sm	[Xe] 6s2 4f6	hR3

Continued on the next page

	B.1	The Electron and Crystal Structures of Elements				595
Atomic	Name of	Chemical	Electron	Crystal structure
number	element	symbol	structure	at 0 K	at 300 K
63	europium	Eu	[Xe] 6s2 4f7	cI2 (bcc)
64	gadolinium	Gd	[Xe] 6s2 4f7 5d1	hP2 (hcp)
65	terbium	Tb	[Xe] 6s2 4f9	hP2 (hcp)
66	dysprosium	Dy	[Xe] 6s2 4f10	oC4	hP2 (hcp)
67	holmium	Ho	[Xe] 6s2 4f11	hP2 (hcp)
68	erbium	Er	[Xe] 6s2 4f12	hP2 (hcp)
69	thulium	Tm	[Xe] 6s2 4f13	hP2 (hcp)
70	ytterbium	Yb	[Xe] 6s2 4f14	hP2 (hcp)	cF4 (fcc)
71	lutetium	Lu	[Xe] 6s2 4f14 5d1	hP2 (hcp)
72	hafnium	Hf	[Xe] 6s2 4f14 5d2	hP2 (hcp)
73	tantalum	Ta	[Xe] 6s2 4f14 5d3	cI2 (bcc)
74	tungsten	W	[Xe] 6s2 4f14 5d4	cI2 (bcc)
75	rhenium	Re	[Xe] 6s2 4f14 5d5	hP2 (hcp)
76	osmium	Os	[Xe] 6s2 4f14 5d6	hP2 (hcp)
77	iridium	Ir	[Xe] 6s2 4f14 5d7	cF4 (fcc)
78	platinum	Pt	[Xe] 6s1 4f14 5d9	cF4 (fcc)
79	gold	Au	[Xe] 6s1 4f14 5d10	cF4 (fcc)
80	mercury	Hg	[Xe] 6s2 4f14 5d10	hR1		liquid
81	thallium	Tl	[Xe] 6s2 4f14 5d10 6p1	hP2 (hcp)
82	lead	Pb	[Xe] 6s2 4f14 5d10 6p2	cF4 (fcc)
83	bismuth	Bi	[Xe] 6s2 4f14 5d10 6p3	hR2
84	polonium	Po	[Xe] 6s2 4f14 5d10 6p4	cP1 (sc)
85	astatine	At	[Xe] 6s2 4f14 5d10 6p5
86	radon	Rn	[Xe] 6s2 4f14 5d10 6p6			gas
87	francium	Fr	[Rn] 7s1
88	radium	Ra	[Rn] 7s2	cI2 (bcc)
89	actinium	Ac	[Rn] 7s2 6d1	cF4 (fcc)
90	thorium	Th	[Rn] 7s2 6d2	cF4 (fcc)
91	protactinium	Pa	[Rn] 7s2 5f2 6d1	tI2
92	uranium	U	[Rn] 7s2 5f3 6d1	oC4
93	neptunium	Np	[Rn] 7s2 5f4 6d1	oP8
94	plutonium	Pu	[Rn] 7s2 5f6	mP16
95	americium	Am	[Rn] 7s2 5f7	hP4 (dhcp)
96	curium	Cm	[Rn] 7s2 5f7 6d1	hP4 (dhcp)
97	berkelium	Bk	[Rn] 7s2 5f9	hP4 (dhcp)
98	californium	Cf	[Rn] 7s2 5f10	hP4 (dhcp)
99	einsteinium	Es	[Rn] 7s2 5f11	hP4 (dhcp)
100	fermium	Fm	[Rn] 7s2 5f12
101	mendelevium	Md	[Rn] 7s2 5f13
102	nobelium	No	[Rn] 7s2 5f14
103	lawrencium	Lr	[Rn] 7s2 5f14 6d1
104	rutherfordium	Rf	[Rn] 7s2 5f14 6d2
105	dubnium	Db	[Rn] 7s2 5f14 6d3

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596	B The Periodic Table of Elements
Atomic	Name of	Chemical	Electron		Crystal structure
number	element	symbol	structure		at 0 K at 300 K
106	seaborgium	Sg	[Rn] 7s2 5f14		6d4
107	bohrium	Bh	[Rn] 7s2 5f14		6d5
108	hassium	Hs	[Rn] 7s2 5f14		6d6
109	meitnerium	Mt	[Rn] 7s2 5f14		6d7
110	darmstadtium	Ds	[Rn] 7s1	5f14	6d9
111	roentgenium	Rg	[Rn] 7s1	5f14	6d10

B.2 Characteristic Temperatures of the Elements

The following table contains the melting point (in Celsius degrees), the Debye temperature (in kelvins), and the critical temperature of the superconducting or magnetic phase transition (in kelvins) for the elements of the periodic table. The symbols S, F, and AF denote superconducting, ferromagnetic, and antiferromagnetic phases, respectively. When more than one magnetic phases are possible, only the transition temperature to the ground-state structure is given. The symbol F does not mean that moments are rigorously parallel, only that the material possesses a finite net magnetization.

Atomic	Name of	Chemical	Melting	ΘD	Ordered	Transition
number	element	symbol	point (◦C)	(K)	phase	temperature
1	hydrogen	H	−259	105
2	helium	He		26
3	lithium	Li	180	344		Tc = 0.026 K
4	beryllium	Be	1287	1440	S
5	boron	B	2075	1315
6	carbon	C	3825	420
7	nitrogen	N	−210	68
8	oxygen	O	−218	91
9	fluorine	F	−220
10	neon	Ne	−249	75
11	sodium	Na	98	158
12	magnesium	Mg	650	400		Tc = 1.175 K
13	aluminum	Al	660	428	S
14	silicon	Si	1414	640
15	phosphorus	P	44	193
16	sulfur	S	115	250
17	chlorine	Cl	−101	115
18	argon	Ar	−189	93
19	potassium	K	63	91
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