States of matter - Warsaw University of Technologyadam.mech.pw.edu.pl/~marzan/introduction_solids.pdf · Aristoteles (350 b.c) : Matter as continuous medium Democritus (400 b.c.)

JONIKA I FOTOWOLTAIKA MICHAŁ MARZANTOWICZ

States of matter

Solids- defined shape and volume- long range order- harmonic interactions

Liquids Gases

- poorly compressible- local order- create free surface

- compressible- molecules in constant motion- fill volume

Fluids Shear force (parallel to the surface) causes the fluid to flow.


Models of matter:history

Aristoteles (350 b.c) : Matter as continuous medium

Democritus (400 b.c.) : atom as indivisible particle. Matter is a combination of atoms.

John Dalton (1808)• Atom is uniform, invariable,and indivisible.• All atoms of a given element have identical chemical properties.• Atoms of one element are different than atoms of other element.• Chemical compounds are created by bonding of atoms in given constant proportions.


Electron

Charging bodies and flow of charge is performed by charge carriers –electrons and ions.

Thomson experiment(1897 r.)

q/m = 1.7·1011 C/kg

The mass of particlecoming from cathoderadiation is about 2000 lower than that of ionizedhydrogen (proton)


Properties of electron

Milikan experiment: estimation of electron charge

e = 1.602·10-19 C

m = 9.109·10-31 kg


Models of atoms – Thomson and Rutherford

Thomson model – „cake with raisins”

Rutherford experiment –„hollow matter”


Rutherford model

Mass and positive charge concentrated in the core

10-10 m10-15 m


Bohr model

Problems of Rutherford model:- radiation (electron „falls” onto the core)- emission spectra are not continuous


Bohr model

Balmer – lines in hydrogen spectrum

Rydberg RH =10 972 000 m−1

Lyman – ultraviolet spectrum

n=2,3,4...

Paschen, Brackett, Pfund, Humphrey series – infrared

n’=1,2,3... n>n’


Assumptions of Bohr model

1. Electron moves in circular orbit around the core. The energy of electron is constant (it does not radiate energy)

2. Only such orbits are allowed, for which the orbital angular momentum of electron is equal the multiple of h/2π

3. Radiation or absorption of energy is possible only when the electronjumps from one orbit to another. The frequency of the correspondingradiation is expressed by equation ∆E = hν

π2hnLn = n- quantum number


Hydrogen emission spectrum


Bohr model: energy of electron

20

22

4 nn

ne

rZe

rum

πε=

π2hnrumL nnen ==

)()()( nEnEnE kp +=

( ) 2220

42 124

)(n

eZmnE e

hπε−=

n=1 ground staten=∞ ionized state

⎟⎠⎞

⎜⎝⎛ −⎟⎟

⎠

⎞⎜⎜⎝

⎛= 223

42

0

1144

11mnc

eme

hππελ MmRR

e+=

1µRadiation length

RH


Basic physics: waves

Waves propagate in time and space.

Wave equation

⎟⎠⎞

⎜⎝⎛ −=Ψ

vxtf

2

22

2

2

xv

t ∂Ψ∂

=∂

Ψ∂Differential equation

Wavenumber

Frequency and angular frequency


Old and new quantum theory

Faults of Bohr model:- gives only the wavelength, and not the intensity- gives correct wavelengths only for hydrogen, and hydrogen-like atoms

The new theory: based on de Broglie’s hypothesis on matter waves

The light can be treated as a wave, and as a particle (photon).The light transmits momentum and energy.Hence, the matter particle that has momentum represents wavelength!


Explanation of Bohr model

The atom acts like a „resonance box” - electronorbit hosts a standing wave. The orbit length isdetermined by the electron wavelength.

π2hnrumL nnen == nλ=2πr


Matter waves

Davisson-Germer experiment:Wave properties of electrons

Thomson experiment: diffraction of electronson thin polycrystalline foil

Stern experiment: diffraction of hydrogen andhelium atoms on crystals of lithium fluorideand sodium chloride


Heisenberg uncertainty principle

Position and momentum cannot be estimated accurately at the same time.The particle can occupy a state with well defined energy for a long time. States with widespread energy are short-lived.

π4htE ≥∆⋅∆

π4hpx ≥∆⋅∆


Heisenberg uncertainty principle



Wave function

Wave function defines probability, that the particle can be found incertain space.

dxdxdxtxP 2*),( Ψ=ΨΨ= Probability density


Schrödinger equation

Wavefunction are a solution of Schrödinger equation

Electron in a electrical field (stationary condition)

Gradient of the field

Potential of the field

Electron energy

The function, and its derivative cannot change abruptly on borders of different potentials – there is no „jump” in probability, just „smooth” changes.Consequence: the electrons can penetrate „forbidden” areas – just with small probability!


Schrödinger equation – potential step

E<V0 I IIV0

V

0

Area I

Classical

QuantummEv 2

1 =Area I Electron does not penetrate area II

Area II

The electron penetrates area II...... but the probability decreases exponentially with distance.Eventually, the electron is always pushed away!


Potential step with finite length

The electron can pass through the barrier, despite „too low” energy.The probability decreases exponentially with barrier thickness.

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −−∝ l

EVmT

h

022exp


The model of atom: potential well

Inside the well:

The electron is a standing wave.

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States of matter - Warsaw University of Technologyadam.mech.pw.edu.pl/~marzan/introduction_solids.pdf · Aristoteles (350 b.c) : Matter as continuous medium Democritus (400 b.c.)