FORTH, E.N. Economou

FIXING by THINKING:FIXING by THINKING:The power of dimensional The power of dimensional

analysisanalysis

E.N. EconomouE.N. Economou

Dept. of Physics, U of CDept. of Physics, U of C

FORTHFORTH

March, March, 20020066

In the atomic idea “there is an enormous amount of information about the world, if just a little imagination and thinking are applied”.

R.P.Feynman

1. The atomic idea Everything is made of indestructible,

indivisible (α – τομα) microscopic particles which attract each other

This attraction must be counterbalanced for equilibrium to be established

HOW?

2. The wave – particle duality

comes to rescue, i.e., QM, which can be distilled in three basic principles:

(a) Heisenberg Principle:(a) Heisenberg Principle:

(b) Pauli Principle:(b) Pauli Principle:

(c) “Schrödinger” Principle:(c) “Schrödinger” Principle:

(a)+(b) (a)+(b) Equilibrium when P Equilibrium when Pforcesforces = P = Pkinetic energykinetic energy

(c) (c) Stability against small perturbations; changes still possible Stability against small perturbations; changes still possible

2 2 2

kin 2 2/3

x

kin 1/3

9E 4.87

2m 8 m r mVx p2 c c2

E c 3.123 V

p

pr

t

2 2/3

kin,t 2 /3

1/3

kin,t 1/3

NE 2.87N

V mVVN / 2 cN

E 2.32NV

2

1 GROUND 2/3mV

3. Equilibrium 3. Equilibrium minimization of Eminimization of Ett

t p KE E E

0 0t P KE E E

V V V

P KP P

Under constant Pext and Text

Equilibrium minimization of G

t ext extG E P V T S

Limits setting universal constants

Name/Symbol

Μass (ΜeV)

Electric charge

Name Strength Range

Electron

e -e

Gravitational

Proton

p +e

E/Μ

Neutron

n 0

Weak Nuclear

np 10-18 m

Netrino

ν

0 Strong Nuclear

ParticlesParticles ( (spinspin ½)½)

ForcesForces

3410 J s 8c 3 10 m/s

e1

m2

pme pm : m

1:1823

n pm m

9m 10v

1 2

12

Gm m,

r

212, but e / r¥

5w 10

s 15

2p

G

39

Gm

c

5.9 10

2e

c1/137

0

2r / r

se

137 ΄ er

0r c 1,41fm

For condensed matter only EM forces matter which are characterized by: e The kinetic energy is characterized by The three quantities define a system

of units: E.g.,the unit of length

More convenient to use (instead of )

2

2em e

aΒ

, ,em a Β , ,em e

, em, , ee m

determines the atomic radius

or, ,e am r

, 1 5a a ar f a f d d

Examples:

General Formula for any quantity A(a) so that (b) Replace by ; (c) A may depend on mα, T, P, c, etc.

Example: Ionization Potential

2 2

2 5, , , ,...o

o oe B e B e B B

EE P T

m a m a m a k

vo e BA m a oA A

, , , , ,...a

e

m T P cA A΄ f Z

m T΄ P΄ ΄

vBa

v va Bf a oA A΄

2

aa B

eI

f a , 0.5 1ad d

Shell & Subshell structure of the Shell & Subshell structure of the atomic orbitals based on their atomic orbitals based on their

angular dependenceangular dependence The latter is the same as that of (same degree)

polynomials satisfying Laplace equation.

Zero degree const no angular dependence s orbitals

1st degree x, y, z sinθcosφ, sinθsinφ, cosθ 3 p orbitals

2nd degree xy, yz, zx, x2-y2, y2-x2 … 5 d orbitals

3rd degree 7 f orbitals

etc..

xyz, z(x2-y2), x(y2-z2),y(z2-x2)y(x2 - 1/3 y2), z(y2 - 1/3 z2), x(z2 - 1/3 x2)

Molecules (diatomic)Molecules (diatomic)

α1 α2 α1 α2 Bd r r f f α exception: noble exception: noble gasesgases2

B me

Ed

e e0

α α

m m΄ E

m m 0,8 1,8d d,,

2e e

r r 02α αα1 α2

2 m mE

I m mf f

r0,5 1d d,,

2

0 2e

E 27.2 eVm α

SolidsSolids3

s s s4 V

r , r f3 N Ba

3B3

3 ss

m A2.68 g / cm

4 fr3

2

s s s2 2 2e s s s

27.2 eV 625 calE

atom molm r f f

s 1 ,,

211 2 11

s s5 5 5 2e s s s

294 180 NB c c 10 N / m 10

m r f f m

,, sc 0.6

e0 s

e s α s

m 82 Kmα

m r m sf

sα 1.6,,

SolidsSolids

3 2

s 5 5s e

2 10

100 f m α

oD 2

s B

23000K

f A

sB s0.1 β 1d d,,

max 2 2e

1ae s B

mc

mm f a

, 1c

13

2

30010 /

s w

rad sf A

Comparison with experimental dataComparison with experimental data

4,874,876,486,483,853,853,933,935,285,285,685,684,114,114,634,63 (Κ(Κm/s)m/s)

0,540,540,9980,9981,291,291,371,370,730,730,7220,7221,291,291,681,68B (B ( ))

2,692,694,634,633,823,823,493,493,043,043,393,393,823,824,284,28 (eV/atom(eV/atom))

2,362,362,332,339,019,018,968,962,732,732,792,797,927,927,867,86((gr/cmgr/cm33))

3,183,182,672,672,992,992,672,67

TheoryTheoryExpExp..TheoryTheoryExpExp..TheoTheoryry

Exp.Exp.TheoryTheoryExp.Exp.

SiSiCuCuAlAlFeFe

f

11 210 / mN

Μρ

δΕ

430430645645406406343343495495428428422422470470ΘD(K)

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 k 2 4

2 2 2

100.03m

s e s

T Kk f a m f

DC electrical resistivity ρe

2

/:

/

V E q E sR

I q s q

2/oR e

/ /e e eR S R

221.74eo

acm

e

[dimensions of time]

21.74 60e sf cm

DC electrical resistivity ρe

(2)Ti Pb Nb Bi Pr U Cs Mn Na Cu Al Ag43.1 21 14.5 116 67 25.7 20 139 4.75 1.7 2.74 1.61( )e cm

At T=295 K

at T2K ρ10-3 μΩ∙cm10-5 μΩ∙cm for pure Cu

ρ2x1023 μΩ∙cm for yellow sulfur

What went wrong?

Formula for ρeDepends on , , /e ee n m

2e

ee

m

e n

1,

/ F

2e F

e

m

e n

31/ 31/ 3

2 / 3

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

e s as

rr r

r

2 / 3

705 10 2 3

e

d

; 3e

If 6 710 10 0.5 !e d cm WAVE ?

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

Returning to my rented room one evening in early January, it was 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 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–paper and to treat the easiest case of a single electron in a one–dimensional periodic potential..”dimensional periodic potential..”

F. BlochF. Bloch

WAVE + PERIODICITYWAVE + PERIODICITY SYSTEMATIC SYSTEMATIC CONSTRUCTIVE CONSTRUCTIVE

INTERFERENCE, CANCELS SCATTERING INTERFERENCE, CANCELS SCATTERING FREE-LIKEFREE-LIKE

Metals2 2

3s v BV u E k T

31 1 1

33 1.67B

ο s ο sο o

k T Tc ρ f c ρ f c cm

E΄ T

Destructive interference Gaps Eg Semiconductor

/ 2g BE k Te

22g h p sV

Specific Heatph eC C C

3ph B aC k N3

2e B eC k N,

3 :ph B aC k N΄

Classically

;Bk T

a oN΄ d

2c QM:

3:

2e B eC k N΄3

2 ;2e B FN΄ k T Pauli 2b

F FE N

9

2e BF

TC Nk

T

max/ 0Bk

max/Bk t

0phC

3ph B aC k N

FLUIDS, FLUIDS, ΙΙ

[g, λ (or k = 2π/λ), d , ρΜ]

2g (g / k)f kd

f kd 1, kd 1 kd, kd 1

2 2 2 2g

k,

Velocity of sea waves

WINDWIND INDUCEDINDUCED

TSUNAMTSUNAMII

λ(λ(mm))

1010-3-3

101000

101011

101022

101033

101044

101055

1010-2-2

1010-1-1

1010-1-1

101000

101022

101011

ph

ωυ = m/s

k

0.2320.232

0.84 0.84 km/hkm/h

1.7 cm1.7 cm

σk

ρg

k gd

FLUIDSFLUIDS

Drag forceDrag force , , , 2

α 1F c S ,,2S , LARGE BODIES, HIGH , LARGE BODIES, HIGH

SPEEDSPEEDη 2F c ,, 2c 6 R , SMALL BODIES, LOW , SMALL BODIES, LOW

SPEEDSPEEDαF

ReF /

Reynolds Reynolds numbernumber

watwaterer

0.010.01 0.010.01

airair 0.000180.00018 0.150.15

1 1gcm s 2 1/ cm s

Black Body RadiationBlack Body Radiation BS, c, , k T

422 2 SB S

SUN s s2 2 3

B B

k Tk TI 4 R 4 R

60 cck T k T

422 E

E s 2 3

k TI 4 R

60 c

Why Earth is round?Why Earth is round? Why are there mountains?Why are there mountains? What is the largest possible height of a mountain in a planet?What is the largest possible height of a mountain in a planet?

When the shear stress exceeds the critical valueWhen the shear stress exceeds the critical value

c

S

B V g 1

V SH3

3 3

3 m

4 f

2g GM / R34

M R3

23

c 5 5e

2 10m f

322 2

B G

RH 0,025f0,3 10

α A

11 2RH 10 m

,

STARSSTARS MINIMUM NUMBER OF NUCLEONS:MINIMUM NUMBER OF NUCLEONS:

MAXIMUM NUMBER OF NUCLEONS:MAXIMUM NUMBER OF NUCLEONS:

NUMBER OF NUCLEONS IN OUR SUN:NUMBER OF NUCLEONS IN OUR SUN:

MAXIMUM NUMBER OF NUCLEONS IN A WHITE MAXIMUM NUMBER OF NUCLEONS IN A WHITE DWARF:DWARF:

3/ 4 3/ 257u

,mine G

mN 0.6 0.23 10

m

ν

3/ 259s

,maxG

N 1.3 10

ν ,, s 15

571,2 10

3/ 257

G

11.775 1.71 10

UniverseUniverse Homogeneous & IsotropicHomogeneous & Isotropic

Expanding according to Hubble’s law:Expanding according to Hubble’s law:

Hubble’ s constant (indipendent of R):Hubble’ s constant (indipendent of R):

, dimensionally; , dimensionally;

dR= R = H t R

dt

R

m 22

2

2

1

2

8 8

3 3

GM m GM mF mR

R R

R GG

R c

R

H tR

GH t2c

2

C

R

2

22

R 8πGH = ε

R 3c

Euclidean geometry

the equivalent of 5.51 protons perthe equivalent of 5.51 protons per m3

This density equals to the critical one with an uncertainty This density equals to the critical one with an uncertainty of 2%of 2%

protons per protons per mm33, i.e. about , i.e. about 4.2%4.2% of of εε

The Rest? DARK MATTER

DARK ENERGY

272 3

Kg9.47 10

c m

nucleons2

1

4c

ph2

0c

21.5%

c

v d

21 4%

73 4%

Einstein’ s cosmological constant

nucleons 3

const

R

4 3 3ph ph ph4

const: , S R T const

R

v 3

const

R

DM 3

const

R

DE const

Cosmic EpochsCosmic Epochs

1. Inflation

2. Early t 10-4 s

3. Photon domination 1ms t 50 ky

4. Matter domination 200 ky t 5 ky

5. Dark energy domination 20 Gy t

Photon domination

Matter/ Dark Energy DominationMatter/ Dark Energy Domination

1/ 22

1/ 2

2

R 1RR const R t

R R

T t

t

1/ 221

2c

R c RR

1/ 2 2/3

12

2

c 3R t sinh c t

c 2

2 / 3TodayR t 0.7 R sinh 0.0936 t , t in Gy

R 0 for t 7Gy t 8Gy

R t 379 ky 1 1

R 1310 1090

BB

t

q→B P,n→nuclei

εph→εnucl decoupling

Protostars

Galaxies

R 0 Today10-

4s1s

70 ky

380 ky

180 My

8 Gy

13.7 Gy

500 My

Acoustic waves in the infant universe of a basic frequency and its overtones

The seeds of the future galaxies

This cosmic “music” must have been inprinted in the cosmic photons

(cosmic background radiation)

Is it there? Can we listen to it?

OBSERVATIONSOBSERVATIONS

WMAP WMAP Wilkinson Microwave Anisotropy ProbeWilkinson Microwave Anisotropy Probe