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Chapter 6 Solid state physics

6.1Introduction

This section covers a few selected topics in solid state physics. There is no attempt to do more than scratch the surface of this vast ﬁeld, although the basics of many undergraduate texts on the subject are covered. In addition a period table of elements, together with some of their physical properties, is displayed on the next two pages.

Periodic table (overleaf) Data for the periodic table of elements are taken from Pure Appl. Chem., 71, 1593–1607 (1999), from the 16th edition of Kaye and Laby Tables of Physical and Chemical Constants (Longman, 1995) and from the 74th edition of the CRC Handbook of Chemistry and Physics (CRC Press, 1993). Note that melting and boiling points have been converted to kelvins by adding 273.15 to the Celsius values listed in Kaye and Laby. The standard atomic masses reﬂect the relative isotopic abundances in samples found naturally on Earth, and the number of signiﬁcant ﬁgures reﬂect the variations between samples. Elements with atomic masses shown in square brackets have no stable nuclides, and the values reﬂect the mass numbers of the longest-lived isotopes. Crystallographic data are based on the most common forms of the elements (the α-form, unless stated otherwise) stable under standard conditions. Densities are for the solid state. For full details and footnotes for each element, the reader is advised to consult the original texts.

Elements 110, 111, 112 and 114 are known to exist but their names are not yet permanent.

124	Solid state physics

6.2Periodic table

Hydrogen	name
1.007 94

1	1s1					atomic number											relative atomic mass (u)
1	1s1					atomic number											relative atomic mass (u)
	89 (β) 378											Titanium
	HEX	1.632		2	electron conﬁguration							47.867					symbol
	13.80	20.28		2	electron conﬁguration							22		Ti			symbol
	Lithium		Beryllium										[Ca]3d2
	6.941		9.012 182			density (kgm−3)					4 508			295			lattice constant, a (fm)
2	3	Li	4	Be							HEX			1.587
2	[He]2s1		[He]2s2				crystal type				1 943			3 563			c/a (angle in RHL,
	533 (β) 351		1 846	229													c/a	in ORC & MCL)
	533 (β) 351		1 846	229													c/a	in ORC & MCL)
	BCC		HEX	1.568													b/a
	453.65	1 613	1 560	2 745		melting point (K)											boiling point (K)
	Sodium		Magnesium			melting point (K)											boiling point (K)
	Sodium		Magnesium
	22.989 770		24.305 0
3	11	Na	12	Mg
3	[Ne]3s1		[Ne]3s2
	966	429	1 738	321
	BCC		HEX	1.624		3	4		5					6		7		8		9
	370.8	1 153	923	1 363		3	4		5					6		7		8		9
	Potassium		Calcium		Scandium		Titanium		Vanadium				Chromium			Manganese		Iron		Cobalt
	39.098 3		40.078		44.955 910		47.867		50.941 5					51.996 1		54.938 049		55.845		58.933 200
4	19	K	20	Ca	21	Sc	22	Ti	23	V			24		Cr	25	Mn	26	Fe	27	Co
4	[Ar]4s1		[Ar]4s2		[Ca]3d1		[Ca]3d2		[Ca]3d3					[Ar]3d54s1		[Ca]3d5		[Ca]3d6		[Ca]3d7
	862	532	1 530	559	2 992	331	4 508	295	6 090	302			7 194		388	7 473	891	7 873	287	8 800	( ) 251
	BCC		FCC		HEX	1.592	HEX	1.587	BCC				BCC			FCC		BCC		HEX	1.623
	336.5	1 033	1 113	1 757	1 813	3 103	1 943	3 563	2 193	3 673			2 180		2 943	1 523	2 333	1 813	3 133	1 768	3 203
	Rubidium		Strontium		Yttrium		Zirconium		Niobium				Molybdenum			Technetium		Ruthenium		Rhodium
	85.467 8		87.62		88.905 85		91.224		92.906 38					95.94		[98]		101.07		102.905 50
5	37	Rb	38	Sr	39	Y	40	Zr	41	Nb			42		Mo	43	Tc	44	Ru	45	Rh
5	[Kr]5s1		[Kr]5s2		[Sr]4d1		[Sr]4d2		[Kr]4d45s1					[Kr]4d55s1		[Sr]4d5		[Kr]4d75s1		[Kr]4d85s1
	1 533	571	2 583	608	4 475	365	6 507	323	8 578	330			10 222		315	11 496	274	12 360	270	12 420	380
	BCC		FCC		HEX	1.571	HEX	1.593	BCC				BCC			HEX	1.604	HEX	1.582	FCC
	312.4	963.1	1 050	1 653	1 798	3 613	2 123	4 673	2 750	4 973			2 896		4 913	2 433	4 533	2 603	4 423	2 236	3 973
	Caesium		Barium		Lanthanides		Hafnium		Tantalum					Tungsten		Rhenium		Osmium		Iridium
	132.905 45		137.327				178.49		180.947 9					183.84		186.207		190.23		192.217
6	55	Cs	56	Ba	57 – 71		72	Hf	73	Ta			74		W	75	Re	76	Os	77	Ir
6	[Xe]6s1		[Xe]6s2				[Yb]5d2		[Yb]5d3					[Yb]5d4		[Yb]5d5		[Yb]5d6		[Yb]5d7
	1 900	614	3 594	502			13 276	319	16 670	330			19 254		316	21 023	276	22 580	273	22 550	384
	BCC		BCC				HEX	1.581	BCC				BCC			HEX	1.615	HEX	1.606	FCC
	301.6	943.2	1 001	2 173			2 503	4 873	3 293	5 833			3 695		5 823	3 459	5 873	3 303	5 273	2 720	4 703
	Francium		Radium		Actinides		Rutherfordium		Dubnium				Seaborgium			Bohrium		Hassium		Meitnerium
	[223]		[226]				[261]		[262]					[263]		[264]		[265]		[268]
7	87	Fr	88	Ra	89 – 103		104	Rf	105	Db			106		Sg	107	Bh	108	Hs	109	Mt
7	[Rn]7s1		[Rn]7s2				[Ra]5f146d2		[Ra]5f146d3?				[Ra]5f146d4?			[Ra]5f146d5?		[Ra]5f146d6?		[Ra]5f146d7?
			5 000	515
			BCC
	300	923	973	1 773

							Lanthanum		Cerium				Praseodymium			Neodymium		Promethium		Samarium
							138.905 5		140.116				140.907 65			144.24		[145]		150.36
				Lanthanides			57	La	58	Ce			59		Pr	60	Nd	61	Pm	62	Sm
				Lanthanides			[Ba]5d1		[Ba]4f15d1					[Ba]4f3		[Ba]4f4		[Ba]4f5		[Ba]4f6
							6 174	377	6 711	(γ) 516			6 779		367	7 000	366	7 220	365	7 536	363
							HEX	3.23	FCC				HEX		3.222	HEX	3.225	HEX	3.19	HEX	7.221
							1 193	3 733	1 073	3 693			1 204		3 783	1 289	3 343	1 415	3 573	1 443	2 063
							Actinium		Thorium				Protactinium			Uranium		Neptunium		Plutonium
							[227]		232.038 1				231.035 88			238.028 9		[237]		[244]
				Actinides			89	Ac	90	Th			91		Pa	92	U	93	Np	94	Pu
				Actinides			[Ra]6d1		[Ra]6d2				[Rn]5f26d17s2			[Rn]5f36d17s2		[Rn]5f46d17s2		[Rn]5f67s2
							10 060	531	11 725	508			15 370		392	19 050	285	20 450	666	19 816	618
							FCC		FCC				TET		0.825	ORC	1.736	ORC	0.733	MCL	1.773
							FCC		FCC				TET		0.825	ORC	2.056	ORC	0.709	MCL	0.780
							1 323	3 473	2 023	5 063			1 843		4 273	1 405.3	4 403	913	4 173	913	3 503

6.2 Periodic table																	125

																18
																Helium
																4.002 602
																2	He
																1s2
BCC	body-centred cubic															1s2
BCC	body-centred cubic															120	356
CUB	simple cubic															120	356
CUB	simple cubic															HEX	1.631
DIA	diamond					13		14		15		16		17		HEX	1.631
DIA	diamond					13		14		15		16		17		3-5	4.22
FCC	face-centred cubic					Boron		Carbon		Nitrogen		Oxygen		Fluorine		Neon
HEX	hexagonal					Boron		Carbon		Nitrogen		Oxygen		Fluorine		Neon
HEX	hexagonal					10.811		12.0107		14.006 74		15.999 4		18.998 403 2		20.179 7
MCL	monoclinic					10.811		12.0107		14.006 74		15.999 4		18.998 403 2		20.179 7
MCL	monoclinic					5	B	6	C	7	N	8	O	9	F	10	Ne
ORC	orthorhombic					[Be]2p1		[Be]2p2		[Be]2p3		[Be]2p4		[Be]2p5		[Be]2p6
RHL	rhombohedral					2 466	1017	2 266	357	1 035 (β) 405		1 460	(γ) 683	1 140	550	1 442	446
TET	tetragonal					RHL	65◦ 7	DIA		HEX	1.631	CUB		MCL	1.32	FCC
TET	tetragonal					RHL	65◦ 7	DIA		HEX	1.631	CUB		MCL	0.61	FCC
(t-pt)	triple point					2 348	4 273	4 763	(t-pt)	63	77.35	54.36	90.19	53.55	85.05	24.56	27.07
						Aluminium		Silicon		Phosphorus		Sulfur		Chlorine		Argon
						Aluminium		Silicon		Phosphorus		Sulfur		Chlorine		Argon
						26.981 538		28.085 5		30.973 761		32.066		35.452 7		39.948
						13	Al	14	Si	15	P	16	S	17	Cl	18	Ar
						[Mg]3p1		[Mg]3p2		[Mg]3p3		[Mg]3p4		[Mg]3p5		[Mg]3p6
						2 698	405	2 329	543	1 820	331	2 086	1 046	2 030	624	1 656	532
						FCC		DIA		ORC	1.320	ORC	2.340	ORC	1.324	FCC
10		11		12							3.162		1.229		0.718
10		11		12		933.47	2 793	1 683	3 533	317.3	550	388.47 717.82		172	239.1	83.81	87.30
Nickel		Copper		Zinc		Gallium		Germanium		Arsenic		Selenium		Bromine		Krypton
58.693 4		63.546		65.39		69.723		72.61		74.921 60		78.96		79.904		83.80
28	Ni	29	Cu	30	Zn	31	Ga	32	Ge	33	As	34	Se	35	Br	36	Kr
[Ca]3d8		[Ar]3d104s1		[Ca]3d10		[Zn]4p1		[Zn]4p2		[Zn]4p3		[Zn]4p4		[Zn]4p5		[Zn]4p6
8 907	352	8 933	361	7 135	266	5 905	452	5 323	566	5 776	413	4 808	(γ) 436	3 120	668	3 000	581
FCC		FCC		HEX	1.856	ORC	1.001	DIA		RHL 54◦ 7		HEX	1.135	ORC	1.308	FCC
FCC		FCC		HEX	1.856	ORC	1.695	DIA		RHL 54◦ 7		HEX	1.135	ORC	0.672	FCC
1 728	3 263	1 357.8	2 833	692.68	1 183	302.9	2 473	1211	3103	883	(t-pt)	493	958	265.90	332.0	115.8	119.9
Palladium		Silver		Cadmium		Indium		Tin		Antimony		Tellurium		Iodine		Xenon		6
106.42		107.868 2		112.411		114.818		118.710		121.760		127.60		126.904 47		131.29
46	Pd	47	Ag	48	Cd	49	In	50	Sn	51	Sb	52	Te	53	I	54	Xe
[Kr]4d10		[Pd]5s1		[Pd]5s2		[Cd]5p1		[Cd]5p2		[Cd]5p3		[Cd]5p4		[Cd]5p5		[Cd]5p6
11 995	389	10 500	409	8 647	298	7 290	325	7 285 (β) 583		6 692	451	6 247	446	4 953	727	3 560	635
FCC		FCC		HEX	1.886	TET	1.521	TET	0.546	RHL 57◦ 7		HEX	1.33	ORC	1.347	FCC
FCC		FCC		HEX	1.886	TET	1.521	TET	0.546	RHL 57◦ 7		HEX	1.33	ORC	0.659	FCC
1 828	3 233	1 235	2 433	594.2	1 043	429.75	2 343	505.08	2 893	903.8	1 860	723	1 263	386.7	457	161.3	165.0
Platinum		Gold		Mercury		Thallium		Lead		Bismuth		Polonium		Astatine		Radon
195.078		196.966 55		200.59		204.383 3		207.2		208.980 38		[209]		[210]		[222]
78	Pt	79	Au	80	Hg	81	Tl	82	Pb	83	Bi	84	Po	85	At	86	Rn
[Xe]4f145d96s1		[Xe]4f145d106s1		[Yb]5d10		[Hg]6p1		[Hg]6p2		[Hg]6p3		[Hg]6p4		[Hg]6p5		[Hg]6p6
21 450	392	19 281	408	13 546	300	11 871	346	11 343	495	9 803	475	9 400	337			440
FCC		FCC		RHL 70◦ 32		HEX	1.598	FCC		RHL 57◦ 14		CUB
2 041	4 093	1 337.3	3 123	234.32	629.9	577	1743	600.7	2 023	544.59	1 833	527	1 233	573	623	202	211
Ununnilium		Unununium		Ununbium				Ununquadium
[271]		[272]		[285]				[289]
110 Uun		111 Uuu		112 Uub				114	Uuq


Europium		Gadolinium		Terbium		Dysprosium		Holmium		Erbium		Thulium		Ytterbium		Lutetium
151.964		157.25		158.925 34		162.50		164.930 32		167.26		168.934 21		173.04		174.967
63	Eu	64	Gd	65	Tb	66	Dy	67	Ho	68	Er	69	Tm	70	Yb	71	Lu
[Ba]4f7		[Ba]4f75d1		[Ba]4f9		[Ba]4f10		[Ba]4f11		[Ba]4f12		[Ba]4f13		[Ba]4f14		[Yb]5d1
5 248	458	7 870	363	8 267	361	8 531	359	8 797	358	9 044	356	9 325	354	6 966 (β) 549		9 842	351
BCC		HEX	1.591	HEX	1.580	HEX	1.573	HEX	1.570	HEX	1.570	HEX	1.570	FCC		HEX	1.583
1 095	1 873	1 587	3 533	1 633	3 493	1 683	2 833	1 743	2 973	1 803	3 133	1 823	2 223	1 097	1 473	1 933	3 663
Americium		Curium		Berkelium		Californium		Einsteinium		Fermium		Mendelevium		Nobelium		Lawrencium
[243]		[247]		[247]		[251]		[252]		[257]		[258]		[259]		[262]
95 Am		96	Cm	97	Bk	98	Cf	99	Es	100	Fm	101	Md	102	No	103	Lr
[Ra]5f7		[Rn]5f76d17s2		[Ra]5f9		[Ra]5f10		[Ra]5f11		[Ra]5f12		[Ra]5f13		[Ra]5f14		[Ra]5f147p1
13 670	347	13 510	350	14 780	342	15 100	338
HEX	3.24	HEX	3.24	HEX	3.24	HEX	3.24	HEX
1 449	2 873	1 618	3 383	1 323		1 173		1 133		1 803		1 103		1 103		1 903

126 Solid state physics

6.3 Crystalline structure Bravais lattices

Volume of

V = (a×b) · c

(6.1)

a,b,c

primitive base vectors

primitive cell

volume of primitive cell

a = 2πb

c/[(a

(6.2)

Reciprocal

b = 2πc

a/[(a

(6.3)

a ,b ,c reciprocal primitive base

primitive base

c = 2πa

b/[(a

(6.4)

vectors

vectorsa

a · a = b · b = c · c = 2π

(6.5)

a · b = a · c = 0

(etc.)

(6.6)

Lattice vector

Ruvw = ua + vb + wc

(6.7)

Ruvw

lattice vector [uvw]

u,v,w

integers

Reciprocal lattice

Ghkl = ha + kb + lc

(6.8)

Ghkl

reciprocal lattice vector [hkl]

vector

exp(iGhkl · Ruvw) = 1

(6.9)

i2 = −1

Weiss zone

hu+ kv + lw = 0

(6.10)

(hkl)

Miller indices of planec

equationb

Interplanar

dhkl =

2π

(6.11)

dhkl

distance between (hkl)

spacing (general)

Ghkl

planes

Interplanar

spacing

(6.12)

(orthogonal basis)

dhkl

aNote that this is 2π times the usual deﬁnition of a “reciprocal vector” (see page 20).

bCondition for lattice vector [uvw] to be parallel to lattice plane (hkl) in an arbitrary Bravais lattice.

cMiller indices are deﬁned so that Ghkl is the shortest reciprocal lattice vector normal to the (hkl) planes.

Weber symbols

	1				(2u− v)			(6.13)
	U =
			3
Converting	1			(2v − u)					U,V ,T ,W	Weber indices
	V =							(6.14)	u,v,w	zone axis indices
[uvw] to		3
[UV T W ]	1								[UV T W ]	Weber symbol
	T = −						(u+ v)	(6.15)	[uvw]	zone axis symbol
						3
	W = w							(6.16)

Converting	u = (U − T )							(6.17)
[UV T W ] to	v = (V − T )							(6.18)
[uvw]	w = W							(6.19)

Zone lawa	hU + kV + iT + lW = 0							(6.20)	(hkil)	Miller–Bravais indices

aFor trigonal and hexagonal systems.

6.3 Crystalline structure	127

Cubic lattices

lattice

primitive (P)

body-centred (I)

face-centred (F)

lattice parameter

volume of conventional cell

lattice points per cell

1st nearest neighboursa

a√

a/√

1st n.n. distance

2nd nearest neighbours

a√

2nd n.n. distance

√

packing fractionb

π/8

π/6

reciprocal latticec

primitive base vectorsd

a1 = axˆ

a1 = a (yˆ + zˆ

xˆ )

a1 = a (yˆ + zˆ)

a2 = ayˆ

a2 = a

(zˆ + xˆ

− yˆ )

a2 = a (zˆ + xˆ )

−

a3 = azˆ

a3 = 2

(xˆ + yˆ − zˆ)

a3 = 2 (xˆ + yˆ )

aOr “coordination number.”

√

bFor close-packed spheres. The maximum possible packing fraction for spheres is

π/6.

cThe lattice parameters for the reciprocal lattices of P, I, and F are 2π/a, 4π/a, and 4π/a respectively. dxˆ , yˆ , and zˆ are unit vectors.

Crystal systemsa							6
system	symmetry			unit cellb		latticesc	6
triclinic	none			a =b =c;		P
triclinic	none			a =b =c;		P
				α =β =γ = 90◦
monoclinic	one diad	[010]		a =b =c;	= 90◦	P, C
				α = γ = 90◦, β	= 90◦
				a =b =c;
orthorhombic	three orthogonal diads			a =b =c;		P, C, I, F
				α = β = γ = 90◦
tetragonal	one tetrad		[001]	a = b =c;		P, I
				α = β = γ = 90◦
trigonald	one triad [111]			a = b = c;		P, R
trigonald	one triad [111]			α = β = γ < 120◦ = 90◦		P, R
hexagonal	one hexad		[001]	a = b =c;		P
				α = β = 90◦, γ = 120◦
cubic	four triads 111			a = b = c;		P, F, I
cubic	four triads 111			α = β = γ = 90◦		P, F, I

aThe symbol “ =” implies that equality is not required by the symmetry, but neither is it forbidden. bThe cell axes are a, b, and c with α, β, and γ the angles between b : c, c : a, and a : b respectively.

cThe lattice types are primitive (P), body-centred (I), all face-centred (F), side-centred (C), and rhombohedral primitive (R).

dA primitive hexagonal unit cell, with a triad [001], is generally preferred over this rhombohedral unit cell.

128	Solid state physics

Dislocations and cracks

Edge

unit vector line of

dislocation

l · b = 0

(6.21)

dislocation

b,b

Burgers vectora

Screw

dislocation energy per

(6.22)

dislocation

l · b = b

unit length

shear modulus

outer cuto for r

Screw

U =

µb

(6.23)

inner cuto for r

dislocation

4π

critical crack length

energy per

unit lengthb

µb

(6.24)

surface energy per unit

area

Critical crack

4αE

Young modulus

L =

(6.25)

Poisson ratio

lengthc

π(1

− σ

)p0

applied widening stress

aThe Burgers vector is a Bravais lattice vector characterising the total relative slip were the dislocation to travel throughout the crystal.

bOr “tension.” The energy per unit length of an edge dislocation is also µb2.

cFor a crack cavity (long L) within an isotropic medium. Under uniform stress p0, cracks ≥ L will grow and smaller cracks will shrink.

Crystal di raction

a,b,c

lattice parameters

a(cosα

−

cosα ) = hλ

(6.26)

,β

,γ

angles between lattice base

Laue

vectors and input wavevector

equations

b(cosβ1 − cosβ2) = kλ

(6.27)

α2,β2,γ2

angles between lattice base

c(cosγ1 − cosγ2) = lλ

(6.28)

vectors and output wavevector

h,k,l

integers (Laue indices)

2kin.G + |G|2 = 0

wavelength

Bragg’s lawa

(6.29)

kin

input wavevector

reciprocal lattice vector

f(G)

atomic form factor

Atomic form

iG r

position vector

factor

f(G) = vol e− · ρ(r) d

(6.30)

ρ(r)

atomic electron density

Structure

S(G)

structure factor

S(G) =

fj (G)e−iG·d j

(6.31)

number of atoms in basis

factorb

j=1

d j

position of jth atom within basis

change in wavevector

Scattered

I(K ) N

|S(K )|

(6.32)

(= kout − kin)

intensityc

I(K )

scattered intensity

number of lattice points

illuminated

intensity at temperature T

Debye–

intensity from a lattice with no

Waller

motion

IT = I0 exp − 3 u

|G|

(6.33)

factord

mean-squared thermal

displacement of atoms

aAlternatively, see Equation (8.32).

bThe summation is over the atoms in the basis, i.e., the atomic motif repeating with the Bravais lattice. cThe Bragg condition makes K a reciprocal lattice vector, with |kin| = |kout|.

dE ect of thermal vibrations.

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