Chapter 28 Atomic Physics. General Physics What energy photon is needed to “see” a proton of radius 1 fm? 10 123456789 11121314151617181920 21222324252627282930.

Chapter 28Chapter 28

Atomic PhysicsAtomic Physics

General Physics

What energy photon is needed to What energy photon is needed to “see” a proton of radius 1 fm?“see” a proton of radius 1 fm?

1 2 3 4 5 6

17% 17% 17%17%17%17%

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

1.1. 1 eV (101 eV (1000 eV) eV)

2.2. 1 keV (101 keV (1033 eV) eV)

3.3. 1 MeV (101 MeV (1066 eV) eV)

4.4. 1 GeV (101 GeV (1099 eV) eV)

5.5. 1 TeV (101 TeV (101212 eV) eV)

6.6. 1 PeV (101 PeV (101515 eV) eV)

General Physics

Atom PhysicsAtom Physics

Sections 1–4Sections 1–4

General Physics

Emission Spectra When a high voltage is applied to a gas at low pressure, it

emits light characteristic of the gas When the emitted light is analyzed with a spectrometer, a

series of discrete bright lines is observed - emission spectrum Each line has a different wavelength and color

General Physics

Spectral Lines of Hydrogen

The Balmer Series has lines whose wavelengths are given by the preceding equation

Examples of spectral lines n = 3, λ = 656.3 nm n = 4, λ = 486.1 nm n = 5, λ = 434.1 nm n = 6, λ = 410.2 nm

General Physics

Emission Spectrum of Hydrogen – Equation

The wavelengths of hydrogen’s spectral lines experimentally agree with the equation

RH is the Rydberg constant RH = 1.0973732 x 107 m-1

n is an integer, n = 3, 4, 5, 6, … The spectral lines correspond to different values

of n

22H n

1

2

1R

1

General Physics

Absorption Spectra An element can also absorb light at specific wavelengths An absorption spectrum can be obtained by passing a

continuous radiation spectrum through a vapor of the gas The absorption spectrum consists of a series of dark lines

superimposed on the otherwise continuous spectrum The dark lines of the absorption spectrum coincide with the

bright lines of the emission spectrum

General Physics

Applications of Absorption Spectrum

The continuous spectrum emitted by the Sun passes through the cooler gases of the Sun’s atmosphere The various absorption lines can be used

to identify elements in the solar atmosphere

Led to the discovery of helium

General Physics

Importance of Hydrogen Atom

Hydrogen is the simplest atom Enables us to understand the periodic table Ideal system for performing precise

comparisons of theory with experiment Much of what we know about the hydrogen

atom can be extended to other single-electron ions For example, He+ and Li2+

General Physics

Sir Joseph John Thomson

“J. J.” Thomson 1856 - 1940 Developed model of the

atom Discovered the electron Did extensive work with

cathode ray deflections 1906 Nobel Prize for

discovery of electron

General Physics

Early Models of the Atom Newton’s model

tiny, hard, indestructible sphere

J.J. Thomson’s model A volume of positive charge Electrons embedded

throughout the volume Vibrational “modes”

responsible for spectral lines Didn’t work!

General Physics

Rutherford’s Scattering Experiments

The source was a naturally radioactive material that produced alpha particles (He++)

Most of the alpha particles passed though the gold foil

A few deflected from their original paths Some even reversed

their direction of travel

Active Figure: Rutherford Scattering

General Physics

Rutherford Model of the Atom

Rutherford, 1911 Planetary model Based on results of thin

foil experiments Positive charge is

concentrated in the center of the atom, called the nucleus

Electrons orbit the nucleus like planets orbit the sun

General Physics

Rutherford Model, Problems Atoms emit certain DISCRETE characteristic

frequencies of electromagnetic radiation The Rutherford model is unable to explain this phenomena

Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves at a frequency related to their orbital speed The radius should steadily decrease and the speed should

steadily increase as this radiation is given off The electron should eventually spiral into the nucleus, but it

doesn’t The radiation frequency should steadily increase – should

observe a continuous spectrum of radiation at progressively shorter and shorter wavelengths, but you don’t

General Physics

Neils Bohr

1885 – 1962 Participated in the early

development of quantum mechanics

Headed Institute in Copenhagen

1922 Nobel Prize for structure of atoms and radiation from atoms

General Physics

The Bohr Theory of Hydrogen

In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory

His model includes both classical and non-classical ideas

His model included an attempt to explain why the atom was stable

General Physics

Bohr’s Assumptions for Hydrogen

The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction The Coulomb force

produces the centripetal acceleration

Only certain electron orbits are stable These are the orbits in

which the atom does not emit energy in the form of electromagnetic radiation

General Physics

Bohr’s Assumptions, cont Radiation is emitted by the atom when the electron

“jumps” from a more energetic initial state to a lower state The frequency emitted in the “jump” is related to the

change in the atom’s energy It is generally not the same as the frequency of the

electron’s orbital motion The size of the allowed electron orbits is

determined by a condition imposed on the electron’s orbital angular momentum Ln = me v r = n ħ where n = 1, 2, 3, …

General Physics

Bohr Radius

The radii of the Bohr orbits are quantized

This is based on the assumption that the electron can only exist in certain allowed orbits determined by the integer n

When n = 1, the orbit has the smallest radius, called the Bohr radius, ao

ao = 0.0529 nm

2 2

2 1, 2, 3,ne e

nr n

m k e

General Physics

Quantized Energies The total energy of the atom

Using the radius equation for the allowed Bohr orbits

When n = 1, the orbit has the lowest energy, called the ground state energy

E1 = -13.6 eV (the ionization energy)

r

ek

r

ekvmPEKEE e

ee 22

1 222

...3,2,11

2 22

42

n

n

ekmE ee

n

General Physics

Radii and Energy of Orbits A general expression for

the radius of any orbit in a hydrogen atom is rn = n2 ao

The energy of any orbit is En = - E0 / n2

Bohr radius and energy: ao = 0.529 Å

E0 = 13.6 eV

General Physics

Energy Level Diagram & Equation

Whenever a transition occurs from a state, ni to another state, nf (where ni > nf), a photon is emitted The photon is emitted with

energy hf = (Ei – Ef) with a wavelength λ given by

For the Paschen series, nf = 3 For the Balmer series, nf = 2 For the Lyman series, nf = 1

22

111

ifH nn

R

Active Figure: Bohr's Model of the Hydrogen Atom

General Physics

Bohr’s Correspondence Principle

Bohr’s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small Similar to having Newtonian Mechanics

be a special case of relativistic mechanics when v << c

General Physics

Successes of the Bohr Theory

Explained several features of the hydrogen spectrum Accounts for Balmer and other series Predicts a value for RH that agrees with experiment Gives an expression for the radius of the atom Predicts energy levels of hydrogen Model of what the atom looks like and how it

behaves Can be extended to “hydrogen-like” atoms

Those with one electron Ze2 needs to be substituted for e2 in equations

Z is the atomic number of the element

General Physics

Modifications of the Bohr Theory – Elliptical Orbits

Sommerfeld extended the results to include elliptical orbits Retained the principle quantum number, n

Determines the energy of the allowed states Added the orbital quantum number, ℓ

ℓ ranges from 0 to n-1 in integer steps All states with the same principle quantum

number are said to form a shell The states with given values of n and ℓ are said

to form a subshell

General Physics

Modifications of the Bohr Theory – Zeeman Effect

Another modification was needed to account for the Zeeman effect The Zeeman effect is the splitting of spectral

lines in a strong magnetic field This indicates that the energy of an electron is

slightly modified when the atom is immersed in a magnetic field

A new quantum number, m ℓ, called the orbital magnetic quantum number, had to be introduced

m ℓ can vary from - ℓ to + ℓ in integer steps

General Physics

Modifications of the Bohr Theory – Fine Structure

High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field This splitting is called fine structure Another quantum number, ms, called the spin

magnetic quantum number, was introduced to explain the fine structure

There are two directions for the spin Spin up, ms = ½ Spin down, ms = -½

General Physics

Spin Magnetic Quantum Spin Magnetic Quantum NumberNumber

It is convenient to think of the It is convenient to think of the electron as spinning on its axiselectron as spinning on its axis The electron is The electron is notnot physically physically

spinningspinning There are two directions for the There are two directions for the

spinspin Spin up, mSpin up, mss = ½ = ½ Spin down, mSpin down, mss = -½ = -½

There is a slight energy difference There is a slight energy difference between the two spins and this between the two spins and this accounts for the doublet in some accounts for the doublet in some lineslines

General Physics

Wolfgang PauliWolfgang Pauli

1900 – 19581900 – 1958 Contributions includeContributions include

Major review of Major review of relativityrelativity

Exclusion PrincipleExclusion Principle Connect between Connect between

electron spin and electron spin and statisticsstatistics

Theories of relativistic Theories of relativistic quantum quantum electrodynamicselectrodynamics

Neutrino hypothesisNeutrino hypothesis Nuclear spin hypothesisNuclear spin hypothesis

General Physics

The Pauli Exclusion PrincipleThe Pauli Exclusion Principle

No two electrons in an atom can ever No two electrons in an atom can ever have the same set of values of the have the same set of values of the quantum numbers n, quantum numbers n, ℓ, ℓ, mm ℓℓ, and m, and mss

This explains the electronic structure This explains the electronic structure of complex atoms as a succession of of complex atoms as a succession of filled energy levels with different filled energy levels with different quantum numbersquantum numbers

General Physics

Number of Electrons in Filled Subshells and Shells (First Three Shells)

Shell - principle quantum number

Subshell - orbital

quantum number

orbital magnetic quantum number

spin magnetic quantum number

Electrons in Filled Subshell

Electrons in Filled

Shell

K ( n = 1 ) s ( ℓ = 0 ) mℓ = 0 ms = +½, −½ 2 2

L ( n = 2 )

s ( ℓ = 0 ) mℓ = 0 ms = +½, −½ 2

8

p ( ℓ = 1 ) mℓ = +1, 0, −1 ms = +½, −½ 6

M ( n = 3 )

s ( ℓ = 0 ) mℓ = 0 ms = +½, −½ 2

18p ( ℓ = 1 ) mℓ = +1, 0, −1 ms = +½, −½ 6

d ( ℓ = 2 ) mℓ = +2, +1, 0, −1, −2 ms = +½, −½ 10

General Physics

Atomic OrbitalsAtomic Orbitals

General Physics

Periodic TablePeriodic Table1s2; 2s2 2p6; 3s2 3p6; 4s2 3d10 4p6; 5s2 4d10 5p6; 6s2 4f14 5d10 6p6; 7s2 5f14 6d10

7p6

General Physics

General Physics

Atomic Transitions – Stimulated Absorption

When a photon with energy ΔE is absorbed, one electron jumps to a higher energy level These higher levels are

called excited states ΔE = hƒ = E2 – E1

In general, ΔE can be the difference between any two energy levels

General Physics

Atomic Transitions – Spontaneous Emission

Once an atom is in an excited state, there is a constant probability that it will jump back to a lower state by emitting a photon

This process is called spontaneous emission

General Physics

Atomic Transitions – Stimulated Emission

An atom is in an excited stated and a photon is incident on it

The incoming photon stimulates the excited atom to return to the ground state

There are two emitted photons, the incident one and the emitted one The emitted photon has the

same wavelength and is in phase with the incident photonActive Figure: Spontaneous and Stimulated Emission

General Physics

Population Inversion

When light is incident on a system of atoms, both stimulated absorption and stimulated emission are equally probable

Generally, a net absorption occurs since most atoms are in the ground state

If you can cause more atoms to be in excited states, a net emission of photons can result This situation is called a population inversion

General Physics

Lasers To achieve laser action, three conditions

must be met The system must be in a state of population inversion

More atoms in an excited state than the ground state The excited state of the system must be a

metastable state Its lifetime must be long compared to the normal

lifetime of an excited state The emitted photons must be confined in the system

long enough to allow them to stimulate further emission from other excited atoms

This is achieved by using reflecting mirrors

General Physics

Laser Beam – He Ne Example

The energy level diagram for Ne in a He-Ne laser

The mixture of helium and neon is confined to a glass tube sealed at the ends by mirrors

A high voltage applied causes electrons to sweep through the tube, producing excited states

When the electron falls to E2 from E*3 in Ne, a 632.8 nm photon is emitted

General Physics

Production of a Laser Beam

General Physics

Holography

Holography is the production of three-dimensional images of an object

Light from a laser is split at B One beam reflects off the

object and onto a photographic plate

The other beam is diverged by Lens 2 and reflected by the mirrors before striking the film

General Physics

Holography, cont The two beams form a complex interference

pattern on the photographic film It can be produced only if the phase relationship

of the two waves remains constant This is accomplished by using a laser

The hologram records the intensity of the light and the phase difference between the reference beam and the scattered beam

The image formed has a three-dimensional perspective