Chapter 28:Atomic Physics

Atomic Spectra

Homework : Read and understand the lecture note.

Emission and absorption spectrum• Atoms emit and absorb light of specific wavelengths. Atoms from different elements have different wavelengths of emitted and absorbed light. These specific wave- lengths are the same for both emitted and absorbed light.• In 1885, Balmer found a formula that described these wavelengths for hydrogen atom.

constant Rydberg m 100973732.1

,...)5,4,3( 1

2

11

1-7

22

H

H

R

nn

R

(n=3,=656.3 nm) etc.

Basic assumptions of Bohr theory1. The electron moves in circular orbits about the proton under the influence of the Coulomb force of attraction.2. Only certain electron orbits are stable. These are orbits in which the hydrogen atom does not emit energy in form of EM radiation. Hence the total energy of the atom is constant and classical mechanics can be used to describe the electron’s motion. 3. Radiation is emitted by the hydrogen atom when electron jumps from a more energetic initial state to a less energetic state. The frequency f of the radiation emitted in the jump is related to the change in the atom’s energy and is independent of the frequency of the electron’s orbital motion:4. The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum:

Bohr Theory of Hydrogen (1913)

centripetal force

hfEE fi

,...)3,2,1( nnvrme

Consequences of Bohr theory

Bohr Theory of Hydrogen (1913)

• The electrical potential energy of the atom:

r

ek

r

eek

r

qqkPE eee

221 )(

• The total energy of the atom assuming that the nucleus is at rest:

r

ekvmPEKEE ee

22

2

1

• From Newton’s 2nd law applied to the electron:

r

vm

r

ek ee

2

2

2

r

ekvm ee 22

1 22 r

ekE e

2

2

• From the 4th assumption and this :

rm

ek

rm

nv

e

e

e

2

22

22

,...)3,2,1( 2

22

nekm

nr

een

radii of allowedelectron orbits

• Energy levels of electron :,...)3,2,1( eV

13.61

2 222

42

n

nn

ekmE een

mm 0529.02

2

0 emk

ae

,...)3,2,1( nm) (0.05292

02 nnanrn

Bohr’s radius

The negative signindicates the ele-ctron is bound tothe proton

Consequences of Bohr theory (cont’d)

Bohr Theory of Hydrogen (1913)

• Ionization energy :The upper most level corresponds toE=0 and n= and the energy neededto remove the electron completely fromthe atom (ionization energy = E1).

• Ground state :The lowest energy state (n=1) is calledthe ground state.

• Emitted/absorbed photon :

From the 3rd postulate, in the transition ofthe electron from an orbit with principalquantum number ni to another with nf, itemits a photon of frequency f given by:

223

42 11

4 if

eefi

nn

ekm

h

EEf

Consequences of Bohr theory (cont’d)

Bohr Theory of Hydrogen (1913)

• Wavelengths of emitted/absorbed photonsSince f=c,

22223

42 1111

4

1

ifH

if

ee

nnR

nnc

ekm

c

f

3

42

4 cekm

R eeH

• Named transitions :

- The Balmer series nf=2, ni=3,4,5,…

- The Lyman series nf=1, ni=2,3,4,…

- The Paschen series nf=3, ni=4,5,6,…

Bohr’s correspondence principle• Quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small.

Success of Bohr theory

Modification of Bohr Theory

• It explains the Balmer series and other series.• It predicts correctly a value for the Rydberg constant.• It gives an expression for the radius of the atom.• It predicts the energy levels of hydrogen.

This theory gives a model of what the atom looks like and how itbehaves. With some refinements and modifications, it can be usedas a model for other atoms than hydrogen atom.

Extension of Bohr theory to hydrogen-like atoms• A hydrogen-like atom contains only one electron : He

-, Li2-, Be

3- etc.• To extend the theory to hydrogen-like atoms, replace e2 with Ze2 where Z is the atomic number of the element.

,...)3,2,1( 1

2 22

422

nn

eZkmE een

223

422 11

4

1

if

ee

nnc

eZkm

Modification of Bohr Theory

Sommerfeld’s extention of Bohr theory • Sommerfeld extended the Bohr theory to include elliptical orbits. • Sommerfeld model introduced, in addition to the principal quantum number n, a new quantum number called orbital quantum number l , where the value of l ranges from 0 to n-1 in integer step.

For a given n, l =0,1,…,n-1 : n=2->l =0,1…. • An electron in any one of the allowed energy states of a hydrogen atom may move in any one of a number of orbits corresponding to different l.• All states with the same principal quantum number n are said to form a shell, which is identified by the letter K,L,M,…corresponding to n = 1,2,3,… • The letters s, p, d, f, g,… are used to designate the states with l = 0,1,2,3,4,…

Modification of Bohr Theory

Orbital magnetic quantum number ml

• Another quantum number was introduced when it was discovered that the spectral lines of a gas are split into several closely spaced lines when the gas is placed in a strong magnetic field (Zeeman effect).

Orbital magnetic quantum number ml : -l =< ml =< l (2l+1 states)

Spin magnetic quantum number ms • Yet another quantum number was introduced when it was discovered that the spectral lines of a gas are actually split into two closely spaced lines (fine structure) even without a strong magnetic field due to spinning of electrons.

Spin magnetic quantum number ms : ms =-1/2,+1/2

Number of allowed states with n and l

2(2l+1) Example : p subshell has 2(2x1+1)=6 possible states

De Broglie Waves and Hydrogen Atom

Quantization of angular momentum and de Broglie waves• De Broglie found an interpretation of the Bohr’s angular momentum quantization in terms of his wave theory. An electron orbit would be stable (allowed) only if it contained an integral number of electron wavelengths.

1,2,3,...)(n 2 nrThe de Broglie wavelength of an electron is:

)/( vmh e

)/(2 vmnhr e

nvrme

This argument strengthened the wave theory of matter.

Hydrogen Atom Quantum numbers

Spin Magnetic Quantum Number

Electron Clouds

Wave function and electron clouds• The solution of the wave equation yields a wave function that depends on the quantum number n, l, and ml.

• The quantity |VP gives a probability of finding the electron in a small volume VP

around a point P.

• The maximum of the probability coincides with the Bohr radius.

Probability per unit length of findingthe electron at a distance r from thenucleus for 1s state of hydrogen atom

Quantum theory predicts that the electronis not located at a fixed point.

Exclusion Principle and Periodic Table

Pauli exclusion principle (1925)

No two electrons in an atom can ever have the same set of values for the set of quantum numbers n, l, ml, and ms.

• The Pauli exclusion principle explains the electronic structure of complex atoms as a succession of filled levels with different quantum numbers increasing in energy, where the outermost electrons are primarily responsible for the chemical properties of the element.

General rule for the order that electrons fill a subshell

• Once one subshell is filled, the next electron goes into the vacant subshell that is the lowest in energy.• If the atom were not in the lowest energy state available, it would radiate energy until it reached that state.• A subshell is filled when it contains 2(2l+1) electrons.

For hydrogen atom at the ground state : (n,l, ml, ms) = (1,0,0,1/2) or (1,0,0,-1/2)1s11 electron

Exclusion Principle and Periodic Table

General rule for the order that electrons fill a subshell (cont’d)For helium atom at the ground state : (n,l, ml, ms) = (1,0,0,1/2) and (1,0,0,-1/2)1s22 electrons

For lithium atom at the ground state : two electrons in 1s subshell one electron in 2s subshell1s22 electrons

( 2p subshell has higher energy)1 electron 2s1 1s22s1

Exclusion Principle and Periodic Table

Periodic table (Mendeleev 1871)

Characteristic X-Rays

Characteristic x-rays • The discrete lines in an x-ray spectrum are called characteristic x-rays.• When an electron beam bombards a metal to produce x-rays, knocked-off electrons leave vacancies at the energy levels at which they were located.• Then these vacancies are filled in by electrons that were at higher energy states. In such a transition, a photon is emitted with an energy corresponding to the difference in energy between the initial and the final energy of the electron that has filled a vacated state.

Atomic Transitions

Stimulated absorption process • When an atom is hit by a photon that carries an energy equal to the energy difference between two states, an electron at the lower energy state can move up to the higher energy state by absorbing the incident photon.• This process is called stimulated absorption process and the higher energy state is called an excited state.• When this happens, there is a probability that the electron that has moved up to the excited state goes back to a lower energy state by emitting a photon (spontaneous emission).

Atomic Transitions

Stimulated absorption process (cont’d) • When an atom is in the excited state E2 and a photon with energy hf=E2-E1 is incident on it, the incoming photon increases the probability that the excited atom will return to the ground state by emitting a second photon with the same energy (stimulated emission).

The emitted photon is exactly inphase with the incident photon.

Laser

Population inversion

• When an incident photon causes atomic transitions, stimulated absorption and stimulated emission happen with the equal probability.• When light is incident on a system of atoms, there is usually a net absorption. This is because when the system is in thermal equilibrium there are many more atoms in the ground state than in excited states.• If there are more atoms in excited states than in the ground state (population inversion), a net emission can result.• A mechanism exists to realize the population inversion: laser.

Laser

Laser (light amplification by simulated emission of radiation) • There are three conditions to achieve laser action:

- The system must be in a state of population inversion.

- The excited state of the system must be metastable state (longer lifetime than otherwise short liftime). Then stimulated emission occurs before spontaneous emission.

- The emitted photons must be confined within the system long enough to allow them to stimulate further emission from other excited atoms.