III Databook

7/31/2019 III Databook

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Natural Sciences Tripos Part III

MATERIALS SCIENCE

Data Book

2011-12

III


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Values of some Physical Constants

Avogadros number NA = 6.023 1023 mol1

Boltzmann constant k(kB) = 1.38 1023 J K1 = 8.617 105 eV K1

Gas constant R = 8.314 J K1mol1

Plancks constant h = 6.63 1034 J s

Plancks constant/2 = 1.054 1034 J s

Faraday constant F = 9.65 104 C mol-1

StefanBoltzmann constant W m2K4

Permeability of vacuum = 4 107H m1

Permittivity of vacuum = 8.854 1012 F m1

Velocity of light in vacuum c = 3 108

m s1

c

2 = 1

Electronic charge e = 1.602 1019 C

Rest mass of electron me = 9.110 1031 kg

Unified atomic mass constant mu = 1.66 1027 kg

Bohr magneton,em

e

2

B = 9.27 1024 J T1

Bohr radius, 22

04

eme

a0 = 5.29 1011

m

Bohr ground-state energy,22

0

4

8 h

eme

= 2.18 1018 J

em2

2 = 6 1039J m2

Flux quantum,e

h

2

= 2.07 1015 Wb

Electron volt 1 eV = 1.602 1019 J

Wavelength of 1 eV photon = 1.24 m

Frequency of l eV photon = 2.41 1014 Hz

kT (at T = 300 K) = 0.0259 eV

Energy of one photon of visible radiation = 1.7 eV 3.5 eV


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Mathematical Expressions

sin (A +B) = sinA cosB + cosA sinB cos (A +B) = cosA cosBsinA sinB

cos2 + sin2 = 1 cos +i sin = ei

ixix eei

x

2

1sin ixix eex

2

1cos

sinhx = )(21 xx

ee

coshx = xx

ee

21

lnN! NlnNN (Stirling's approximation)

Integration by parts: uvuvvu dd

The solution of the first order linear differential equation )()(d

dxQyxP

x

y is:

d)( CexexQe where xxP d)(

zx xez

0d2erf 2

2d

0

21

xex x

cn

xxx

nn

1d1

1

d

d nn nxxx

xxx

sincosd

d xx

xcossin

d

d

xfx

eex

xfxf

d

d

d

d e.g. xx ee

x

d

d RT

QRT

Q

eRT

Qe

T

2d

d

....!3

)2)(1(

!2

)1(11 32

x

nnnx

nnnxx

n

for n real and not a positive integer: 1x ;

for n a positive integer,x can have any value.

.....!3!2

132

xx

xex .....432

)1ln(432

xxx

xx 11 x

.....!5!3sin

53

xxxx

.....!6!4!21cos

642

xxxx

.....d

d

!3d

d

!2d

d

!1)()(

000

3

33

2

22

00 xxx

x

fh

x

fh

x

fhxfhx


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THE ERROR FUNCTION, erf (x)


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STRUCTURE TYPES

STRUCTURE TYPE FORMULA SYSTEM LATTICE LATTICE MOTIF

h.c.p. - hexagonal P 0,0,0 ; 2/3 ,1/3 ,1/2

diamond C cubic F C: 0,0,0 ; 1/4 ,1/4 ,1/4

caesium chloride CsCl cubic P Cl: 0,0,0 ; Cs: 1/2 ,1/2 ,1/2

sodium chloride NaCl cubic F Cl: 0,0,0 ; Na: 0,0,1/2

sphalerite ZnS cubic F S: 0,0,0 ; Zn: 1/4 ,1/4 ,1/4

(zinc blende)

wurtzite ZnS hexagonal P S: 0,0,0 ; 2/3 ,1/3 ,1/2

Zn: 0,0, 1/2+u; 2/3 ,1/3,u

(u 1/8 )

nickel arsenide NiAs hexagonal P As: 0,0,0 ; 2/3 ,1/3 ,1/2

Ni: 1/3

,2/3

,1/4

; 1/3

,2/3

,3/4

fluorite CaF2 cubic F Ca: 0,0,0 ; F: (1/4,

1/4,1/4)

rutile TiO2 tetragonal P Ti: 0,0,0 ;1/2 ,

1/2 ,1/2

O: (u,u,0); (1/2+u,1/2 -u,1/2)

(u 0.30)

perovskite BaTiO3 cubic P Ti: 0,0,0 ; Ba:1/2 ,

1/2 ,1/2

O: 1/2,0,0 ; 0,1/2,0 ; 0,0,1/2

Crystal structures of some metals at room temperature

Material Structure type Lattice parameters

Ag

Al

Au

Cu

Ni

Mg

Zn

Cr

-Fe

c.c.p.

c.c.p.

c.c.p.

c.c.p.

c.c.p.

h.c.p.

h.c.p.

b.c.c.

b.c.c.

a = 4.09

a = 4.05

a = 4.08

a = 3.61

a = 3.52

a = 3.21 ; c = 5.21

a = 2.66 ; c = 4.95

a = 2.88

a = 2.87


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CRYSTAL SYSTEMS, LATTICES AND SYMMETRY

CRYSTAL SYSTEM TRICLINIC MONOCLINIC ORTHORHOMBIC TETRAGONAL

DEFINING SYMMETRY

(rotation or inversion)

Monad 1 diad 3 diads 1 tetrad

CONVENTIONAL UNIT

CELL

a b c

a b c

= 90

120

a b c

= 90

a = b c

== 90

CONVENTIONAL

LATTICE TYPES

P P, C P, C, I, F P, I

CRYSTAL SYSTEM TRIGONAL HEXAGONAL CUBIC

DEFINING SYMMETRY

(rotation or inversion)

1 triad 1 hexad 4 triads

CONVENTIONAL UNIT

CELL

a = b c

= 90

120

a = b c

= 90

120

a = b = c

= 90

CONVENTIONAL

LATTICE TYPES

P P,I,F

The symbol implies that equality is not required by symmetry


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T2 Electron Microscopy CD

Relativistic wavelength of electrons:

2212

cm

eVeVm

h

o

o

Gun brightness:2222

4

o

e

oo

e J

D

i

Expressions for probe size:

Source size22

4

Ido Spherical aberration

3

2

1ss Cd

Diffraction limit22.1dd Chromatic aberration c

oc C

EEd

Minimum current for a contrast level, C, in the SEM:2min

25

qC

feNI

p

Back scattered yield: = 0.0254 + 0.016Z 1.86 104Z2 + 8.30 107Z3

Minimum detectable concentration: tbpb

200

(wt %)

Structure factor: g.ri

i

ig efF 2

Extinction distance:g

Cg

F

V

cos

Intensity in diffracted beam: 2

22

sins

tstIg

g


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Dynamical correction:

2

2 1

g

ss

Intensity in a stacking fault image:

sztstss

I

g

g 2cos2

sin2

sin22

sin2

sin1 22

2

Intensity in a moir pattern:

tstsszszts

sI

g

g coscos2cos2coscos11

2

Phase shift of non-axial electron beams: 432 21 uCufu s

Scherzer defocus:2/1

Sch3

4

sCf

Beam spread in nm: 2/32/1

198.0 tAE

ZB

Concentration derived from EELS: tIIC

k

k

,,,

Poisson statistics for inelastic scattering:

/

!

1 tn

n et

nP


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The Energy and Associated Wavelengths of the Strongest K, L and M Lines of the Elements.

RelativeAtomic AtomicNumber Mass K1 L1 M1

Element Z ArE(keV) (nm) E(keV) (nm) E(keV) (nm)

Hydrogen 1 1.0Helium 2 4.0Lithium 3 6.9 0.05Beryllium 4 9.0 0.11 11.40Boron 5 10.8 0.18 6.76Carbon 6 12.0 0.28 4.47Nitrogen 7 14.0 0.39 3.16Oxygen 8 16.0 0.52 2.36Fluorine 9 19.0 0.68 1.83Neon 10 20.2 0.85 1.46Sodium 11 23.0 1.04 1.19Magnesium 12 24.3 1.25 0.99Aluminium 13 27.0 1.49 0.93Silicon 14 28.1 1.74 0.71Phosphorus 15 31.0 2.01 0.61Sulphur 16 32.1 2.31 0.54Chlorine 17 35.5 2.62 0.47Argon 15 39.9 2.96 0.42Potassium 19 39.1 3.31 0.37Calcium 20 40.1 3.69 0.34 0.34 3.63Scandium 21 45.0 4.09 0.30 0.39 3.13Titanium 22 47.9 4.51 0.27 0.45 2.74Vanadium 23 50.9 4.95 0.25 0.51 2.42Chromium 24 52.0 5.41 0.23 0.57 2.16Manganese 25 54.9 5.90 0.21 0.64 1.94Iron 26 55.8 6.40 0.19 0.70 1.76Cobalt 27 58.9 6.93 0.18 0.77 1.60Nickel 28 59.7 7.48 0.17 0.85 1.46Copper 29 63.5 8.05 0.15 0.93 1.33Zinc 30 65.4 8.64 0.14 1.01 1.23Gallium 31 69.7 9.25 0.13 1.10 1.13Germanium 32 72.6 9.88 0.12 1.19 1.04Arsenic 33 74.9 10.54 0.12 1.28 0.97Selenium 34 79.0 11.22 0.11 1.38 0.90Bromine 35 79.9 11.92 0.10 1.48 0.84Krypton 36 83,8 12.65 0.10 1.59 0.78Rubidium 37 85.5 13.39 0.09 1.69 0.73Strontium 38 87.6 14.16 0.09 1.81 0.69Yttrium 39 88.9 14.96 0.08 1.92 0.64Zirconium 40 91.1 15.77 0.08 2.04 0.61Niobium 41 92.9 16.61 0.07 2.17 0.57Molybdenum 41 95.9 17.48 0.07 2.29 0.54Technetium 43 98.0 18.36 0.07 2.42 0.51Ruthenium 44 101.1 19.28 0.06 2.55 0.48Rhodium 45 102.9 20.21 0.06 2.70 0.46Palladium 46 106.4 21.17 0.06 2.70 0.44Silver 47 107.9 22.16 0.06 2.98 0.41Cadmium 48 112.4 23.17 0.05 3.13 0.39Indium 49 114.8 24.21 0.05 3.29 0.38Tin 50 118.7 25.27 0.05 3.44 0.36Antimony 51 121.7 26.36 0.05 3.60 0.34Tellurium 52 127.6 27.47 0.04 3.77 0.33Iodine 53 126.9 28.61 0.04 3.94 0.31Xenon 54 131.3 29.77 0.04 4.11 0.30Caesium 55 132.9 30.97 0.04 4.29 0.29Barium 56 137.3 32.19 0.04 4.46 0.28Lanthanum 57 138.9 33.44 0.04 4.65 0.27 0.83 1.49Hafnium 72 178.5 55.78 0.02 7.90 0.16 1.64 0.75Tantalum 73 181.0 57.52 0 02 8.14 0.15 1.71 0.73Tungsten 74 183.8 59.31 0.02 8.40 0.15 1.77 0.70Rhenium 75 186.2 61.13 0.02 8.65 0.14 1.84 0.67Osmium 76 190.2 62.99 0.02 8.91 0.14 1.91 0.65Iridium 77 192.2 64.88 0.02 9.17 0.14 1.98 0.63Platinum 78 195.1 66.82 0.02 9.44 0.13 2.05 0.60Gold 79 197.0 68.79 0.02 9.71 0.13 2.12 0.58Mercury 80 200.6 70.81 0.02 9.99 0.11 2.19 0.56Thallium 81 204.4 72.86 0.02 10.27 0.11 2.27 0.55Lead 82 207.2 74.96 0.02 10.55 0.12 2.34 0.53Bismuth 83 209.0 77.10 0.02 10.94 0.11 2.42 0.51Polonium 84 210.0 79.28 0.02 11.13 0.11 ? ?Astatine 85 210.0 81.50 0.02 11.43 0.11Radon 86 222.0 83.77 0.01 11.73 0.11Francium 87 223.0 96.09 0.01 12.03 0.10Radium 88 226.0 88.45 0.01 12.34 0.10Actinium 89 227.0 90.87 0.01 12.65 0.10Thorium 90 232.0 93.33 0.01 12.97 0.10 3.00 0.41Protactinium 91 231.0 95.85 0.01 13.29 0.09 3.08 0.40Uranium 92 238.0 98.42 0.01 13.61 0.09 3.17 0.30


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Threshold Energies of Ionisation Edges (in eV) observable by EELS

State 1s 2s 2p1/2 2p3/2 3p 3d 4p

Shell K L1 L2 L3 M23 M45 N23

2 He 24.6h

3 Li 55h4 Be 111h5 B 188h

6 C 284h7 N 400h8 O 532h9 F 685h

10 Ne 867h 18w

11 Na 1072h 32h12 Mg 1305h 52h13 Al 1560h 118h 73d14 Si 1839h 149h 100d

15 P 2149h 189h 135d

16 S 2472h 229h 165d17 Cl 2823 270h 200d18 Ar 3203 320h 246d19 K 3608 377h 294w20 Ca 4038 438h 350w 347w

21 Sc 4493 500h 406w 402w22 Ti 4965 564h 461w 455w 4723 V 5465 628h 520w 513w 4724 Cr 5989 695h 584w 575w 4825 Mn 6539 770h 652w 640w 51

26 Fe 7113 846h 721w 708w 5727 Co 7709 926h 794w 779w 6228 Ni 8333 1008 872w 855w 6829 Cu 8979 1096 951h 931w 7430 Zn 9659 1194 1043 1020d 87

31 Ga 1298 1142 1115d 10532 Ge 1414 1248 1217d 125 3033 As 1527 1359 1323d 144 4134 Se 1654 1476 1436d 162 57h35 Br 1782 1596 1550d 182 70d

36 Kr 1921 1727 1675 214 89h37 Rh 2065 1846w 1804w 239 111d

38 Sr 2216 2007w 1940w 270 134d 20p39 Y 2373 2155w 2080w 300 160 28p40 Zr 2532 2307w 2222w 335 181 32p

41 Nb 2698 2465w 2371w 371 207h 35h42 Mo 2866 2625w 2520w 400 228h 37d44 Ru 3224 2967w 2838w 472 281h 42d


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State 3d3/2 3d5/2 4p 4d 4f 5p 5d

Shell M4 M5 N23 N45 N6, N7 O2, O3 O4, O5

45 Rh 312 308d 4846 Pd 340 335d 5047 Ag 373 367d 5948 Cd 411 404d 67

49 In 451 443d 77

50 Sn 494 485d 9051 Sb 537 528d 99 3252 Te 582 572h 110 4053 I 631 620h 123 5054 Xe 685 672h 147 64

55 Cs 740w 726w 7856 Ba 796w 781w 9357 La 849w 832w 9958 Ce 902w 884w 11059 Pr 951w 931w 114

60 Nd 1000w 978w 11862 Sm 1107w 1081w 13063 Eu 1161w 1131w 13464 Gd 1218w 1186w 14165 Tb 1276w 1242w 148

66 Dy 1332w 1295w 154 30, 2367 Ho 1391w 1351w 161 31, 2468 Er 1453w 1409w 168 31, 2569 Tm 1515 1468w 177 32, 2570 Yb 1576 1527w 184 33, 26

71 Lu 1640 1589w 195 35, 2772 Hf 1716 1662h 38, 30

73 Ta 1793 1735h 45, 3774 W 1872 1810h 37, 34 47, 3775 Re 1949 1883h 47, 45 46, 35

76 Os 2031 1960h 52, 50 58, 4677 Ir 2116 2041h 63, 60 63, 5178 Pt 2202 2122h 74, 70 66, 5179 Au 2291 2206h 87, 83 72, 5480 Hg 2385 2295h 81, 58

81 Tl 2485 2390h 14p82 Pb 2586 2284h 21p83 Bi 2688 2580h 27p90 Th 3491 3332 83p92 U 3728 3552 96p

N.B.

The most prominent edges (most suitable for analysis) are shown in italics.h denotes a hydrogenic edge with sawtooth profile.ddenotes a delayed maximum giving a rounded edge with a maximum at least 10eV above the threshold.w denotes a sharp white line peak at the edge threshold.

p denotes a low-energy edge that resembles a plasmon peak.


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T3 Optical, X-ray and neutron techniques HJS

Fourier transforms

Fourier transform: rrkexpr2

1=k

~

dif

Inverse Fourier transform: krkexpk

2

1=r

~

dif

Properties of Fourier transforms:

Property Function Fourier transform

Linearity xbgxaf kgbkfa~~

Scaling ax/ akfa~

Translation inx ax kfiak~

exp

Translation in k xfiaxexp ak~

Inversion x~

k

Fourier transform pairs

Fourier transform pairs

f x

f~

k

Exponential/ Lorentzian xexp 21

12

k

Gaussian/ Gaussian

2

exp2x

2

exp2k

Characteristic/ Sinc x k

k 2/sin2

Delta/ Complex exponential ax ikaexp2

1

Convolutions

dgxfg

Convolution theorem

2~~~

gfhgfh


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X-ray and neutron diffraction

Structure factor:

22 /sinexp2exp BlzkyhxifF mmmm

mhkl

Intensity from single crystal sample with X-rays

2sin2

2cos1

2

22

2

3

42

4

00 hklhkl Fcm

eAII

Intensity from polycrystalline sample with X-rays

sin2sin2

2cos1

32

22

22

3

42

4

00 hklhklhkl FpR

HW

cm

eAII

Intensity from polycrystalline sample with neutrons

2

2sinsinhklhkl

nhkl Fp

KI

Stress/ strain measurement

Strain measured in an arbitrary orientation making direction cosines l, m and n with the reference axes

312312332

222

112 222,, nlmnlmnmlnml

In terms of the orientation angles

and

,

sincosl , sinsinm and cosn

Stress measurement with the

sin2 technique

d

E

dd x 2sin1

Stress/ strain relation

332211)21)(1(1

EEij

ijij


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X-ray atomic scattering factors and neutron bound coherent scattering factors

4

1

2/sinexp/sin

i

ii cba

Element X-rays Neutrons

a1 a2 a3 a4 c b1 b2 b3 b4 bc

H 0.49 0.32 0.14 0.04 0.00 10.51 26.13 3.14 57.80 0.374He 0.87 0.63 0.31 0.18 0.01 9.10 3.36 22.93 0.98 0.326Li 1.13 0.75 0.62 0.47 0.04 3.95 1.05 85.39 168.26 0.190Be 1.59 1.13 0.54 0.70 0.04 43.64 1.86 103.48 0.54 0.779B 2.05 1.33 1.10 0.71 0.19 23.22 1.02 60.35 0.14 0.53 + 0.02iC 2.31 1.02 1.59 0.87 0.22 20.84 10.21 0.57 51.65 0.665N 12.21 3.13 2.01 1.17 11.53 0.01 9.89 29.00 0.58 0.936O 3.05 2.29 1.55 0.87 0.25 13.28 5.70 0.32 32.91 0.580F 3.54 2.64 1.52 1.02 0.28 10.28 4.29 0.26 26.15 0.565Ne 3.96 3.11 1.45 1.13 0.35 8.40 3.43 0.23 21.72 0.457Na 4.76 3.17 1.27 1.11 0.68 3.29 8.84 0.31 129.42 0.363Mg 5.42 2.17 1.23 2.31 0.86 2.83 79.26 0.38 7.19 0.538Al 6.42 1.90 1.59 1.96 1.12 3.04 0.74 31.55 85.09 0.345Si 6.29 3.04 1.99 1.54 1.14 2.44 32.33 0.68 81.69 0.415P 6.43 4.18 1.78 1.49 1.11 1.91 27.16 0.53 68.16 0.513S 6.91 5.20 1.44 1.59 0.87 1.47 22.22 0.25 56.17 0.285Cl 11.46 7.20 6.26 1.65 9.56 0.01 1.17 18.52 47.78 0.958Ar 7.48 6.77 0.65 1.64 1.44 0.91 14.84 43.90 33.39 0.191K 8.22 7.44 1.05 0.87 1.42 12.79 0.77 213.19 41.68 0.367Ca 8.63 7.39 1.59 1.02 1.38 10.44 0.66 85.75 178.44 0.470Sc 9.19 7.37 1.64 1.47 1.33 9.02 0.57 136.11 51.35 1.229Ti 9.76 7.36 1.70 1.90 1.28 7.85 0.50 35.63 116.11 0.337

V 10.30 7.35 2.07 2.06 1.22 6.87 0.44 26.89 102.48 0.038Cr 10.64 7.35 3.32 1.49 1.18 6.10 0.39 20.26 98.74 0.364Mn 11.28 7.36 3.02 2.24 1.09 5.34 0.34 17.87 83.75 0.375Fe 11.77 7.36 3.52 2.30 1.04 4.76 0.31 15.35 76.88 0.954Co 12.28 7.34 4.00 2.35 1.01 4.28 0.28 13.54 71.17 0.249Ni 12.84 7.29 4.44 2.38 1.03 3.88 0.26 12.18 66.34 1.030Cu 13.34 7.17 5.62 1.67 1.19 3.58 0.25 11.40 64.81 0.772Zn 14.07 7.03 5.17 2.41 1.30 3.27 0.23 10.32 58.71 0.560Ga 15.24 6.70 4.36 2.96 1.72 3.07 0.24 10.78 61.41 0.729Ge 16.08 6.37 3.71 3.68 2.13 2.85 0.25 11.45 54.76 0.819As 16.67 6.07 3.43 4.28 2.53 2.63 0.26 12.95 47.80 0.658Se 17.00 5.82 3.97 4.35 2.84 2.41 0.27 15.24 43.82 0.797Br 17.18 5.24 5.64 3.99 2.96 2.17 16.58 0.26 41.43 0.680Kr 17.36 6.73 5.55 3.54 2.83 1.94 16.56 0.23 39.40 0.781Rb 17.18 9.64 5.14 1.53 3.49 1.79 17.32 0.27 164.93 0.709Sr 17.57 9.82 5.42 2.67 2.51 1.56 14.10 0.17 132.38 0.702Y 17.78 10.29 5.73 3.27 1.91 1.40 12.80 0.13 104.35 0.775Zr 17.88 10.95 5.42 3.66 2.07 1.28 11.92 0.12 87.66 0.716Nb 17.61 12.01 4.04 3.53 3.76 1.19 11.77 0.20 69.80 0.705Mo 3.70 17.24 12.89 3.74 4.39 0.28 1.10 11.00 61.66 0.672Tc 19.13 11.09 4.65 2.71 5.40 0.86 8.14 21.57 86.85Ru 19.27 12.92 4.86 1.57 5.38 0.81 8.43 24.80 94.29 0.703


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Rh 19.30 14.35 4.73 1.29 5.33 0.75 8.22 25.87 98.61 0.588Pd 19.33 15.50 5.30 0.61 5.27 0.70 7.99 25.21 76.90 0.591Ag 19.28 16.69 4.80 1.05 5.18 0.64 7.47 24.66 99.82 0.592Cd 19.22 17.64 4.46 1.60 5.07 0.59 6.91 24.70 87.48 0.487 0.07iIn 19.16 18.56 4.29 2.04 4.94 0.55 6.38 25.85 92.80 0.208 0.005iSn 19.19 19.10 4.46 2.47 4.78 5.83 0.50 26.89 83.96 0.623Sb 19.64 19.05 5.04 2.68 4.59 5.30 0.46 27.91 75.28 0.557Te 19.96 19.01 6.14 2.52 4.35 4.82 0.42 28.53 70.84 0.580I 20.15 18.99 7.51 2.27 4.07 4.35 0.38 27.77 66.88 0.528

Xe 20.29 19.03 8.98 1.99 3.71 3.93 0.34 26.47 64.27 0.492Cs 20.39 19.11 10.66 1.50 3.34 3.57 0.31 24.39 213.90 0.542Ba 20.34 19.30 10.89 2.70 2.77 3.22 0.28 20.21 167.20 0.507La 20.58 19.60 11.37 3.29 2.15 2.95 0.24 18.77 133.12 0.824Ce 21.17 19.77 11.85 3.33 1.86 2.81 0.23 17.61 127.11 0.484Pr 22.04 19.67 12.39 2.82 2.06 2.77 0.22 16.77 143.64 0.458Nd 22.68 19.68 12.77 2.85 1.98 2.66 0.21 15.89 137.90 0.769Pm 23.34 19.61 13.12 2.88 2.03 2.56 0.20 15.10 132.72Sm 24.00 19.43 13.44 2.90 2.21 2.47 0.20 14.40 128.01 0.08 0.165iEu 24.63 19.09 13.76 2.92 2.57 2.39 0.19 13.75 123.17 0.722 0.126iGd 25.07 19.08 13.85 3.55 2.42 2.25 0.18 12.93 101.40 0.65 1.382iTb 25.90 18.22 14.32 2.95 3.58 2.24 0.20 12.66 115.36 0.738Dy 26.51 17.64 14.56 2.97 4.30 2.18 0.20 12.19 111.87 1.69 0.0276iHo 26.90 17.29 14.56 3.64 4.57 2.07 0.20 11.44 92.66 0.801Er 27.66 16.43 14.98 2.98 5.92 2.07 0.22 11.36 105.70 0.779Tm 28.18 15.89 15.15 2.99 6.76 2.03 0.24 11.00 102.96 0.707Yb 28.66 15.43 15.31 2.99 7.57 1.99 0.26 10.66 100.42 1.243Lu 28.95 15.22 15.10 3.72 7.98 1.90 9.99 0.26 84.33 0.721Hf 29.14 15.17 14.76 4.30 8.58 1.83 9.60 0.28 72.03 0.777Ta 29.20 15.23 14.51 4.76 9.24 1.77 9.37 0.30 63.36 0.691W 29.08 15.43 14.43 5.12 9.89 1.72 9.23 0.32 57.06 0.486Re 28.76 15.72 14.56 5.44 10.47 1.67 9.09 0.35 52.09 0.920Os 28.19 16.16 14.93 5.68 11.00 1.63 8.98 0.38 48.16 1.070

Ir 27.30 16.73 15.61 5.83 11.47 1.59 8.87 0.42 45.00 1.060Pt 27.01 17.76 15.71 5.78 11.69 1.51 8.81 0.42 38.61 0.960Au 16.88 18.59 25.56 5.86 12.07 0.46 8.62 1.48 36.40 0.763Hg 20.68 19.04 21.66 5.97 12.61 0.55 8.45 1.57 38.32 1.269Tl 27.54 19.16 15.54 5.53 13.17 0.66 8.71 1.96 45.81 0.878Pb 31.06 13.06 18.44 5.97 13.41 0.69 2.36 8.62 47.26 0.941Bi 33.37 12.95 16.59 6.47 13.58 0.70 2.92 8.79 48.01 0.853


17/27

16 02/12/2011

M1 Electrons and photons in solids CD

Dopant Ionisation Energy:2

0

4

1)4(2

*

r

emE

Fermi-Dirac Statistics:

kT

EEE

Fexp1

1)(

Density of States (3D): 2/12/322

2

2

1)( E

mEg

Number of Electrons in Conduction Band at Temperature T:

kT

EENn FCC

)(exp where

2/3

2

*22

h

kTmN nC

Number of Holes in Valence Band at Temperature T:

kT

EEN VFV)(exp where

2/3

2

*22

h

kTmN

pV

Semiconductor Equation: npni 2 Intrinsic Semiconductor Carrier Concentration:

kT

ENNn

gVCi

2exp

Fermi level:

Intrinsic semiconductor:

*

*

ln4

3

2 n

pg

V

i

FF m

mkTEEEE

Extrinsic semiconductor:

i

AD

n

pgVF

n

NNkT

m

mkTEEE

2sinhln

4

3

2

1

*

*

n-type (ND


18/27

17 02/12/2011


Hall Field (for holes):ep

Bj zxy E

Mobility:*n

nnn

m

e

E

v and

*p

ppp

m

e

E

v

Electron Current Density: dxdn

eDdiffJ nn )( and nn envdriftJ )(

Hole Current Density:dx

dpeDdiffJ pp )( and pp epvdriftJ

Einstein Relations: nn DkT

e

and pp DkT

e

Continuity Equation for Holes: ppp

RGdx

dJ

edt

dp

1

Continuity Equation for Electrons: nnn RG

dx

dJ

edt

dn

1

Diffusion Equations:

Holes:p

p

p

x

pD

t

p

2

2

Steady state:22

2

ppp L

p

D

p

dx

pd

Electrons:n

n

n

x

nD

t

n

2

2

Steady state:22

2

nnn L

n

D

n

dx

nd

Poissons Equation:

2

Specific Contact Resistance:

D

BnsnC

NmR

*

2exp~

p n Junction:

Depletion width: biDA

DAs VNN

NN

eW

2

Built-in Voltage: Wn

NN

e

kT

V mi

DA

bi 2

1

ln 2

Ideal Diode Equation:

1expkT

eVJJ S where

n

pn

p

npS

L

neD

L

peDJ

00


19/27

18 02/12/2011


JFET:

Channel resistance:)(2 WazNe

LR

Dn

Saturation Voltage: Gbis

DDsat VV

aeNV

2

2

Linear Region: DP

biG

P

PD V

V

VV

V

II

1

whereL

aNezI

s

DnP

322 and

s

DP

aeNV

2

2

Saturation Region:

2/3

3

2

3

1

P

biG

P

biGPDsat

V

VV

V

VVII

Cut-off Frequency:2

22

L

aeN

s

DnT

MESFET:

Channel resistance:)( WazNe

LR

Dn

Saturation Voltage: Gbis

DDsat VV

aeNV

2

2

Linear Region: DP

biG

P

PD V

V

VV

V

II

1

whereL

aNezI

s

DnP

2

322

ands

DP

aeNV

2

2

Saturation Region:

2/3

3

2

3

1

P

biG

P

biGPDsat

V

VV

V

VVII

Cut-off Frequency:2

2

2 L

aeN

s

DnT

MOS Diode:

i

ABS

n

N

e

kTinv ln

22.)( and

A

iAs

A

Bs

A

Ssm

Ne

nNkT

eNeN

invWW

2

)/ln(4)2(2.)(2


20/27

19 02/12/2011


MOSFET:

Linear Region: DTGonD VVVCL

zI )( and B

BAsT

C

eNV

2

)2(2

0

Saturation Region: 2)(2

TGon

Dsat VVL

CzI

Tunnel Diode:

kT

eVI

V

V

V

VII

PPP exp1exp 0

Ideal solar cell:

Open circuit voltage:

1lnS

Loc

I

I

e

kTV

Typical Semiconductor Properties (at room temperature):

Effective Mass (a) Mobility (b) Effective density of states (c) Energy gap (d)

Electronsmn*/m0

Holesmp*/m0

Electrons Holes NC NV

Si

GaAs

0.26

0.063

0.69

0.57

1450

9200

505

320

2.86 1019

4.7 1017

2.86 1019

7.0 1018

1.12

1.42

indirect

direct

(a) in terms of the rest mass of the electron (averaged to take account of anisotropy)

(b) in cm2 s1 V1

(c) in cm3

(d) in eV


21/27

M3 Extraction and recycling RVK


22/27

M3 Extraction and recycling RVK


23/27

22 02/12/2011

M5 Deformation kinetics WJC

Deformation data for different materials

Material: Copper 1Cr-Mo-V Alumina Olivine

Crystallographic and thermal data

Atomic volume ( 1029 m3) 1.18 1.18 4.25 4.92

Burgers vector ( 1010 m) 2.56 2.48 4.76 6.0

Melting temperature (K) 1356 1810 2320 2140

Modulus

Shear modulus at 300 K (GPa) 42.1 81 155 81.3

Temperature dependence of modulus,B

0.54 1.09 0.35 0.35

Lattice diffusion

Pre-exponential,Dov (m2 s1) 2 105 2 104 0.19 0.1

Activation energy, Qv (kJ mol1) 197 251 636 522

Boundary diffusion

Pre-exponential, Dob (m3 s1) 5 105 1.1 1012 1 1010

Activation energy, Qb (kJ mol1) 104 174 350

Power law creep

Exponent, n 4.8 6 3 3

Dorn constant,A 7.4 105 1.1 104 3.38 0.45

Data taken fromDeformation-Mechanism Maps by H.J. Frost and M.F. Ashby, Pergamon Press,1982, where the temperature dependent shear modulus is given by:

B

T

TGG

M

0

3001

Energy required to move unit length of edge dislocation against the lattice resistance, Up

)45.3(exp

18

12

p

b

w

Gb

U

where w = half-width of dislocation.


24/27

23 02/12/2011

M5 Deformation kinetics WJC

Peierls stress, Up

2

pp

2

bG

U

G

Friedel equation

kT

DGb

dt

d.

2 3


25/27

24 02/12/2011

M7 Electronic ceramics NDM

Electrical behaviour of isotropic ceramics

Relative dielectric constant (relative permittivity),

For a linear, isotropic, material

PEE 00

where E is the electric field, P is the electric polarisiation (electric dipole moment per unit volume),and

1

and EP 0

whereis the dielectric susceptibility.

Imperfect (lossy) dielectric

For a lossy (imperfect) dielectric the dielectric constant can be represented by

i

The imaginary part at a frequency is equivalent to a conductivity given by

0

The loss tangent or dissipation factor, tan , is defined through the equation

tan

Clausius-Mosotti relationship

0

0

3

2

1

n

where n0 is the number density of molecules of polarisability

For a single relaxation-type loss mechanism characterised by a relaxation time the values ofand

are given by

22

s

1

and

22

s

1

where s and are the values of the dielectric constant at zero and infinite frequency respectively.Note that these equations do not apply in the case ofresonance-type loss mechanisms.


26/27

25 02/12/2011


Electrical conductivity,

Evqn iii

wherej = current density, ni = number density of carriers of type i with charge qi and velocity vi.

Ionic conduction

kT

Ub

kT

nqexp

2

22

where= spacing between ionic sites separated by energy barrier U. The probability per unit time thatan ion will jump over the barrier is given by

kT

Ub exp

Piezoelectricity

Direct piezoelectric effect

jkijki dP

Converse piezoelectric effect

iijkjk Ed

where the dijkare the same as for the direct effect.Ferroelectric materials Landau theory

The free energy G of a ferroelectric is given as a function of polarisation P, temperature Tand appliedelectric fieldEby

664

42

06

1

4

1

2

1PgPgPTTEPG ...

For second order transitions g4 > 0, and terms O(P6) can be neglected.


27/27


Electro-optics

In general the equation of the optical indicatrix is a biaxial ellipsoid, the equation of which takes theform

1212112131331323223

2

333

2

222

2

111 xxbbxxbbxxbbxbxbxb

In the first order Pockels electro-optic effect, the electro-optic tensor rijkrelates the changes, bij, in thecoefficients specifying the indicatrix to the direction of applied fieldEk:

kijkij Er

where the rijkare the elements of the Pockels electro-optic tensor.

In the Kerr quadratic electro-optic effect, the electro-optic tensor isRijkland

lkijklij EER

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