FORTH, E.N. Economou

CONDENSED MATTER PHYSICSwith “a little imagination and thinking”

E. N. EconomouFORTH

Dept. of Physics, U of CATHENS , July 16, 2014

Outline

1. Introduction: Three basic ideas and a powerful tool

2. Structures of matter:- The realm of electromagnetism

• Everything is made of indivisible (ά-τομα) microscopic particles which attract each other.

• The atomic idea implies that the equilibrium properties of matter at all scales depend on a few numberscharacterizing the elementary particles and their motion.

• For equilibrium to be established the attraction must be counterbalanced.

HOW?

1The atomic idea

It leads to the three basic principles of QM(to be remembered for ever):

= p k

The wave-particle duality comes to rescue:2

Every particle of energy and momentum p doesnot follow a trajectory but moves as a wave offrequency and wavevector/ω ε=

ε

k = p /

Every wave of frequency and wave vector kconsists of indivisible entities of energy andmomentum

ωε ω=

(a) Heisenberg Principle:

(b) Pauli Principle: Identical spin ½ particles do not “cohabit”

(c) “Schrödinger” Principle:Composite particles behave as elementary up to Δε

(a)+(b) provide for equilibrium when Pforces = Pkinetic energy (c) implies invariance under small perturbations; changes still possible.

/ 2xx p∆ × ∆ ≥ 2 2 2 2 2 2 2 2/3/ 2 ( / 2 ) / / /m p m m r m R mVκιν⇒ ≈ = ∆ ∝ ∆ ∝ ∝ε p

/ 2 / , / 2V V N V N N N N± + −⇒ → = = =

( )2 2/3 2/3 2 5/3 2, 2,87 / 1,105( / )E N N mV N m Rκιν Ν⇒ ≥ × =

2 2/31 /o mV∆ε ≡ ε − ε ∝

THE THREE PRINCIPLES OF QM, when particle confined within V

3

t p KE E E= +

0 0t P KP K

E E E P PV V V

∂ ∂ ∂= ⇒ + = ⇒ =

∂ ∂ ∂

Under constant Pext and Text

Equilibrium ⇔ minimization of G

t ext ext extG E P V T S H T S≡ + − ≡ −

Equilibrium ⇔ minimization of E

Attractive,due to interactions,

leads to collapse

Repulsive, due mostly to QM kinetic energy, counterbalances the attraction

4Dimensional Analysis

1 2 3 4 5 Χ depends In gener on Α ,Α , , .: .a , .l ,Α Α Α

4 50

40 50

( , ,...)f Α ΑΧ = Χ

Α Α

• Dimensions of • If X depends on , then

: [ ] a b cX X m t=

1 2 3, ,A A A 31 21 2 3

ss sX f A A A=

E.g. Frequency ω of oscillation of a pendulumIt depends on: 1) Physical constants, g

2) Physical quantities, ,m1

20, 0,aCg m a aβ γω γ β⇒ = ⇒ = + = =

numerical constant=1 /C gω⇒ =

m

Limits setting universal constants

Name/Symbol

Μass(ΜeV)

Electric charge

Name Strength Range

Electrone -e

Gravitational

Protonp +e

E/Μ

Neutronn 0

Weak Nuclearn to p, p to n

10-18 m

Neutrinoν

0 Strong Nuclear

Particles (spin ½) Interactions

3410 J s−≈ ⋅ 8c 3 10 m/s≈ ×

e1m2

≈

p em : m

1836≈

n pm m≈

9m 10v−≈ ?

1 2

12

Gm m,r

∞

212, but e / r∞ ±

5w 10−α ≈

s 1α ≈

2p

G

39

Gmc

5.9 10−

α ≡

= ×

2ec

1/137

α ≡

=

0

2r / re e

r−∝

0r 1.41fm cπ

≈ =

1. Introduction: Three basic ideas and a powerful tool

2 Structures of matter:- The realm of electromagnetism

Total kinetic energy of N identical spin ½particles with mass m, , anduniform concentration inside a volume

2 2/3 2 5/32

kin,t K2/32.87 1.105 , / 22N NE N p m

mmV R

ε≥ = =

343V Rπ=

total total 0= =P L

From ATOMS to ASTEROIDS EM reigns ; so

• Ep depends on

• EK depends on and mainly on

• Length: Bohr radius

• Instead of: use

• B a aa r r aΒ⇒ =

em2

B 2e

am e

≡

, , ee m , ,e Bm a

e

From atoms to asteroids:General formula for any quantity X

, ( )( , , , ,...)O Xae oo o

m T Pm T P

cX X f Z Z υ= ⋅

, , , etcB B o o o oa r r a X X T T ′′→ = → →

2 2 3 4 17/ / / / 2.42 10 so o e B B et E m a a e e m −= = = = = ×

/ /137 2188 km/s, / 1/137o e B om a c cυ α υ= = = ≡ ≈

2 5/ 294Mbar,o e BP m a= =

2 2 2 4 2/ / / 27.2eV, 315775KB o B e B ek E e a m a e mο οΤ Τ≡ = = = = =

; , , so that [ ] [ ]p qno e BX m a n p q οΧ Χ= =

The realm of electromagnetism

• Atoms• Molecules• Solids• Liquids

Solids3

s s s s4 V , , 33 N Br r r a rπ

≡ = ≈

3w33 ss

m A2.675 g / cm4 rr3

αΜρ = =

π

2

s s s2 2 2e s s s

27.2 eV 2625 kJE ,atom molm r r r

δ ≈ η ≈ η = η

s 1η ≈

211 2 11

s s5 5 5 2e s s s

294 180 NB c c 10 N / m 10 ,m r r r m

≈ = × ≈ ×

sc 0.6≈

e s0 s

e s α Ms w

m 51α km Bα ,m r m sr

′′υ ≈ ≈ =

ρΑ

sα 1.6 or 0.47′ ≈

Solids

3 2

s 5 5s e

1.2 10100 r m α

−

Β

µ ×τ ≈ ≈

osD 2

s w

7390 Kr A

aΘ ≈

sB ,µ ≈ β s0.1 β 1≤ ≤

smax 0 D s2 2

ek , or 1

m re

aB s

a m ama

ω ≈ υ ≈ ≈ π

132

97 10 rad / ss

s w

ar A

≈

Comparison with experimental data

1/3( / 0.74)s s sr r r f→ ≡

Fe Al Cu SiExp. Εst. Exp. Εst. Exp. Εst. Exp. Εst.

2.70 2.99 2.67 3.18

55.85 26.98 63.55 28.09

7.86 2.70 8.96 2.33

4.28 3.73 3.39 3.04 3.49 3.82 4.63 2.69

1.68 1.25 0.722 0.75 1.37 1.33 0.998 0.55

4.63 4.06 5.68 5.28 3.93 3.85 6.48 4.87

464 425 426 500 344 408 645 432

sr

wA3( / )M gr cmρ

(eV/atom)Eδ11 2B(10 / m )N

(km/s)υ

( )D KΘ

D2.454, 4.52, 1.98, 6.27, 568sr E B υ == δ = = Θ =

Better results, if ; f is volume fractionE. g. for the diamond structure of Si, f=0,34; hence

Solids Pb Be Mg Na

Exp. Est. Exp. Est. Exp. Est. Exp. Est.

3.66 2.35 3.34 3.93

(eV/atom) 2.03 2.03 3.32 3.93 1.51 2.44 1.113 1.76

0.43 0.27 1.003 2.47 0.354 0.426 0.068 0.189

2 1.57 7.88 11.62 4.57 4.98

105120

14401400

400422

158313

srEδ

11 2(10 / m )NB

(km/s)υ

(K)DΘ

The JELLIUM MODEL for &sr B2

2a s s e B

E aN r r m a

γ ′ ′= −

Kinetic Coulomb

energy energy

5/3 21.3 , 0.4 0.9ca rζ ηζ η′ = + ≤ ≤

4/3 20.56 0.9γ ζ ζ′ = +2

sar

γ′

⇒ =′

515.6 Mbars

aBr

′⇒ =

0s

Er

∂=

∂

r s /a

B

Specific Heatph eC C C= +

3ph B aC k N=32e B eC k N=,

3 :ph B aC k N ′=

Classically

( )/

;Bk T

a oN d

η′ = φ ω ω∫

( ), 2cε = ωQM:

3 :2e B eC k N ′= ( )2 ;e B FN k T ρ′ ≈ ( )Pauli 2b

92e B

F

TC NkT

≈

max/ 0Bk Τ ω →

max/ 1Bkη Τ ω ≥

0phC →

3ph B aC k N≈

F FE Nρ ≈

E

MELTING TEMPERATURE

s s m s mU PV T S U PV T S+ − = + −

sPV PV≈

0.03s sU U U− ≈

s aS S N kΒ− ≈

2 4

2 2 2

100.03 Kms e s

Tk r a m rΒΒ

≈ ≈

DC electrical resistivity ρe

2

/ :/

V E q E tRI q t q

⋅= = =

20/ 4108.236 ohm [ 2 , 4 ]o H o oR e R R Z Rα= = = π = π

[ ] [ ][ ][ ] [ / ] [ / ]e e eR S Rρ ρ ρ= = ⇒ =

2 21.74 cmeoae

Βρ = = µΩ⋅ [dimensions of time]

21.74 60 cme srρ = ≈ µΩ⋅

DC electrical resistivity ρe (2)

Ti Pb Nb Bi Pr U Cs Mn Na Cu Al Ag

43.1 21 14.5 116 67 25.7 20 139 4.75 1.7 2.74 1.61( cm)eρ µΩ⋅At T=295 K

at T=2K ρ=10-3 μΩ∙cm to 10-5 μΩ∙cm for pure Cu

ρ=2x1023 μΩ∙cm for yellow sulfur

What went wrong?

Formula for ρeDepends on , , /e ee n m τ

2 ,ee

e

me n

ρ = ητ

1η =

/ Fτ = υ1/3

20

(9 / 4),e F Fe

e e

me n r

ρυ υ π

= =υ

( ) 31/31/3

2 /3

4 / 3(9 / 4) 70 , / ,ee e s

e

rr r

rρ

ππ= ≈ = ζ

ζ

( )2/3

70 , (5 12) 2 5 ,e

dρ

≈ ≈ − Α ≈ −ζ

3eρ ≈

If 6 710 10 0.5 !e d cmρ −≈ ⇒ ≈ ≈ Is it possible ?

“The fact that periodicity of a crystal would be essential was somehow suggested to me by remembering a demonstration in elementary physics where many equal and equally coupled pendulums were hanging at constant spacing from a rod and the motion of one of them was seen to “migrate” along the rod from pendulum to pendulum.

Returning to my rented room one evening in early January, it was with such vague ideas in mind that I began to use pencil and paper and to treat the easiest case of a single electron in a one–dimensionalperiodic potential..”

F. BlochWAVE + PERIODICITY ⇒ SYSTEMATIC CONSTRUCTIVE INTERFERENCE, CANCELS SCATTERING ⇒FREE-LIKE

“The fact that periodicity of a crystal would be essential was somehow suggested to me by remembering a demonstration in elementary physics where many equal and equally coupled pendulums were hanging at constant spacing from a rod and the motion of one of them was seen to “migrate” along the rod from pendulum to pendulum.

Returning to my rented room one evening in early January, it was with such vague ideas in mind that I began to use pencil and paper and to treat the easiest case of a single electron in a one–dimensional periodic potential..”

F. Bloch

Let us use also pencil and paper to treat this 1D case

2 2 2(2 / )(1 cos ), /m ka g lο οω ω κ ω = + − =

H Eψ ψ=n nn

cψ ϕ= ∑( ) 0, 0, 1, 2, ...mn mn nnH E c m− δ = = ± ±∑

1 2, ,mm mmH H Vοε ±= =

2 1 1( ), 0, 1, 2, ...n n n nEc c V c c nοε − +⇒ = + + = ± ±

2 2

2

2n o

n

u m mc Vο

ω ω κ κ Ε ε

+−

2 2n 1 n 1 n) )n o n nm u m u u u u uω ω κ κ− +⇒ − = − + ( − + ( −

Re exp( ),n n k nu A i kx i t x naω= − =

22 cosk oE V kaε= +

Metals2 2 3s v BV u E k Tρ σ∝ ∝ ∝ ∝ ≈

31 1 1

3 3 1.67 cmBο s ο s

ο o

k T Tc ρ r c ρ r cE΄ T

ρ = = ≈ µΩ⋅

SemiconductorsDestructive interference ⇒ Gaps, Eg

1 /2g Be

E k Tn eρ −∝ ∝

( ) 2 22 22 , 3.22 /g h p s h eV V m dΕ ≈ − ε − ε =

2 22 32 ( )g h h p sE V V ε ε≈ + − −

(a)

(b)

(α)

(β)

1,42Ad = 3 2,46Aa d= =(α) (β)

,An z Bn zA Bn nn n

c p c pψ∞ ∞

=−∞ =−∞

= +∑ ∑

[ ]2 2 3A p A B B BEc c V c c cε= + + +

2 2exp( )B Bc c i= ⋅k a

3 1exp( )B Bc c i= − ⋅k a

2 (1)A p A BEc c V f cε= +

1 21 exp( ) exp( )f i a i= + − ⋅ + ⋅k k a*

2 (2)B p B AEc c V f cε= +

2pE V fε± = ±

3 ,2 Pdf δ δ = −k k k k

The realm of electromagnetism

• Atoms• Molecules• Solids• Liquids

LIDUIDS, Ι

[g, λ (or k = 2π/λ), d , ρΜ]

( )2g (g / k)f kdυ =

( )f kd 1, kd 1→ >>kd, kd 1→ <<

2 2 2 2g

k, σ σσ

υ = υ + υ υ =ρ

Velocity of sea waves

( )f kd tanh kd=

( )nn nn

2

1 A΄2

2 r

Α − εσ ≈

× π

wr r αΒ=

nn1 0.45 eV / molecule2

Α ε ≈

nn 8Α ≈

nn΄ 5Α ≈

wr = 3.64

2J0.1

mσ ≈ exp: 0.073 J/m2

WIND INDUCED

TSUNAMI

λ(m)10-3 100 101 102 103 104 10510-2 10-1

10-1

100

102

101

( )phωυ = m/sk

0.232

0.84 km/h

1.7 cm

σkρ

gk gd

FLUIDS

Drag force [ ], , ,υ ρ η 2

α 1F c S= ρ υ , 2S≈ , LARGE BODIES, HIGH SPEED

η 2F c= η υ , 2c 6 R= π , SMALL BODIES, LOW SPEED

αFReF /η

υ= ≈

η ρ Reynolds number

water 0.01 0.01air 0.00018 0.15

( )1 1gcm s− −η ( )2 1/ cm s−η ρ

[ ] [ ] [ ]Pr essure timeη = ×

12c π

η = µω

W

16e

2 22 2 2αe Β w

m 4.13 10 1c cm 18 1823m α r r

×ω = =

×

132c 1.72 10 rad / sω = ×

931 1

132 2

c 2.44 10 kg c kg2 0.89 10c ms c ms1.72 10

−×η = π = ×

×

exp: 3 kg10ms

−

THREE GRAND FRONTS:The minute, the great, and the complex

I. The journey towards the ultimate small continues(Other particles? Strings? Branes?)

II. Exploring the Universe(s ?) continues to astonish us and to widen our horizons

III. Exploring and integrating light and matter at the nanoscale goes towards the merging with the living structures, the ultimate of complexity

The snake bites its tail

Pps 3-45, 83-106, 149-175, 179-208