Chapter 28

Chapter 28

Atomic Physics

Plum Pudding Model of the Atom• J. J. Thomson’s “Plum Pudding” model of the atom:

• Electrons embedded throughout the a volume of positive charge

• A change from Newton’s model of the atom as a tiny, hard, indestructible sphere

Sir Joseph John Thomson1856 – 1940

Scattering Experiments

• The source was a naturally radioactive material that produced alpha particles

• Most of the alpha particles passed though the foil

• A few deflected from their original paths

• Some even reversed their direction of travel

Ernest Rutherford1871 – 1937

Planetary Model of the Atom• Based on results of thin foil scattering experiments,

Rutherford’s Planetary model of the atom:

• Positive charge is concentrated in the center of the atom, called the nucleus

• Electrons orbit the nucleus like planets orbit the sun

Chapter 28Problem 6

In a Rutherford scattering experiment, an α-particle (charge = +2e) heads directly toward a gold nucleus (charge = +79e). The α-particle had a kinetic energy of 5.0 MeV when very far (r → ∞) from the nucleus. Assuming the gold nucleus to be fixed in space, determine the distance of closest approach.

Difficulties with the Rutherford Model

• Atoms emit certain discrete characteristic frequencies of electromagnetic radiation but the Rutherford model is unable to explain this phenomena

• Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves of the same frequency

• The radius should steadily decrease as this radiation is given off

• The electron should eventually spiral into the nucleus, but it doesn’t

Emission Spectra• A gas at low pressure and a voltage applied to it emits

light characteristic of the gas

• When the emitted light is analyzed with a spectrometer, a series of discrete bright lines –emission spectrum – is observed

• Each line has a different wavelength and color

Emission Spectrum of Hydrogen• The wavelengths of hydrogen’s spectral lines can be

found from

• RH = 1.097 373 2 x 107 m-1 is the Rydberg constant and n is an integer, n = 3, 4, 5, …

• The spectral lines correspond to different values of n

• n = 3, λ = 656.3 nm

• n = 4, λ = 486.1 nm

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Johannes Robert Rydberg1854 – 1919

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

• Such 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

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 was an attempt to explain why the atom was stable and included both classical and non-classical ideas

Niels Henrik David Bohr1885 – 1962

The Bohr Theory of Hydrogen• The electron moves in circular orbits around the

proton under the influence of the Coulomb force of attraction, which produces the centripetal acceleration

• Only certain electron orbits are stable

• In these orbits electrons do not emit energy in the form of electromagnetic radiation

• Therefore, the energy of the atomremains constant and classicalmechanics can be used to describethe electron’s motion

The Bohr Theory of Hydrogen• Radiation is emitted when the electrons “jump” (not in

a classical sense) 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: Ei – Ef = h ƒ

• The size of the allowed electron orbits is determined by a quantization condition imposed on the electron’s orbital angular momentum:

me v r = n ħ where n = 1, 2, 3, …; ħ = h / 2 π

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• The radii of the Bohr orbits are quantized

• When n = 1, the orbit has the smallest radius, called the Bohr radius, ao = 0.0529 nm

• A general expression for the radius of any orbit in a hydrogen atom is rn = n2 ao

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• The lowest energy state (n = 1) is called the ground state, with energy of –13.6 eV

• The next energy level (n = 2) has an energy of –3.40 eV

• The energies can be compiled in an energy level diagram with the energy of any orbit of En = - 13.6 eV / n2

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• The value of RH from Bohr’s analysis is in excellent agreement with the experimental value of the Rydberg constant

• A more generalized equation can beused to find the wavelengths of anyspectral lines

Energy Level Diagram• The uppermost level corresponds to E = 0 and n

• The ionization energy: energy needed to completely remove the electron from the atom

• The ionization energy for hydrogenis 13.6 eV

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Chapter 28Problem 18

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Chapter 28Problem 24

Two hydrogen atoms collide head-on and end up with zero kinetic energy. Each then emits a 121.6-nm photon (n = 2 to n = 1 transition). At what speed were the atoms moving before the collision?

Modifications of the Bohr Theory – Elliptical Orbits

• Sommerfeld extended the results to include elliptical orbits

• Retained the principle quantum number, n, which determines the energy of the allowed states

• Added the orbital quantum number, ℓ, ranging from 0 to n-1 in integer steps

• All states with the same principle quantumnumber are said to form a shell, whereas thestates with given values of n and ℓ are saidto form a subshell Arnold Johannes

Wilhelm Sommerfeld1868 – 1951

Modifications of the Bohr Theory – Elliptical Orbits

Arnold Johannes Wilhelm Sommerfeld

1868 – 1951

Pieter Zeeman1865 – 1943

Modifications of the Bohr Theory – Zeeman Effect

• Another modification was needed to account for the Zeeman effect: splitting of spectral lines in a strong magnetic field, indicating 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

Quantum Number Summary• The values of n can range from 1 to in integer steps

• The values of ℓ can range from 0 to n-1 in integer steps

• The values of m ℓ can range from -ℓ to ℓ in integer steps

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

Spin Magnetic Quantum Number• It is convenient to think of the electron as

spinning on its axis (the electron is not physically spinning)

• There are two directions for the spin: spin up, ms = ½; spin down, ms = - ½

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

• A classical description of electron spin is incorrect: the electron cannot be located precisely in space, thus it cannot be considered to be a spinning solid object

de Broglie Waves in the Hydrogen Atom• One of Bohr’s postulates was the angular momentum

of the electron is quantized, but there was no explanation why the restriction occurred

• de Broglie assumed that the electron orbit would be stable only if it contained an integral number of electron wavelengths

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de Broglie Waves in the Hydrogen Atom• This was the first convincing argument that the wave

nature of matter was at the heart of the behavior of atomic systems

• By applying wave theory to the electrons in an atom, de Broglie was able to explain the appearance of integers in Bohr’s equations as a natural consequence of standing wave patterns

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Quantum Mechanics and the Hydrogen Atom

• Schrödinger’s wave equation was subsequently applied to hydrogen and other atomic systems - one of the first great achievements of quantum mechanics

• The quantum numbers and the restrictions placed on their values arise directly from the mathematics and not from any assumptions made to make the theory agree with experiments

Electron Clouds• The graph shows the solution to

the wave equation for hydrogen in the ground state

• The curve peaks at the Bohr radius

• The electron is not confined to a particular orbital distance from the nucleus

• The probability of finding the electron at the Bohr radius is a maximum

Electron Clouds• The wave function for hydrogen

in the ground state is symmetric

• The electron can be found in a spherical region surrounding the nucleus

• The result is interpreted by viewing the electron as a cloud surrounding the nucleus

• The densest regions of the cloud represent the highest probability for finding the electron

The Pauli Exclusion Principle• No two electrons in an atom or in the same location can

ever have the same set of values of the quantum numbers n, ℓ, m ℓ, and ms

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

Wolfgang Ernst Pauli1900 – 1958

Filling Shells• As a general rule, the order that electrons fill an

atom’s subshell is:

• 1) Once one subshell is filled, the next electron goes into the vacant subshell that is lowest in energy

• 2) Otherwise, the electron would radiate energy until it reached the subshell with the lowest energy

• 3) A subshell is filled when it holds 2(2ℓ+1) electrons

Filling Shells

The Periodic Table• The outermost electrons are primarily responsible

for the chemical properties of the atom

• Mendeleev arranged the elements according to their atomic masses and chemical similarities

• The electronic configuration of the elements is explained by quantum numbers and Pauli’s Exclusion Principle

Dmitriy Ivanovich Mendeleyev1834 – 1907

The Periodic Table

Chapter 28Problem 28

(a) Construct an energy level diagram for the He+ ion, for which Z = 2. (b) What is the ionization energy for He+?

Explanation of Characteristic X-Rays

• The details of atomic structure can be used to explain characteristic x-rays

• A bombarding electron collides with an electron in the target metal that is in an inner shell

• If there is sufficient energy, the electron is removed from the target atom

Explanation of Characteristic X-Rays

• The vacancy created by the lost electron is filled by an electron falling to the vacancy from a higher energy level

• The transition is accompanied by the emission of a photon whose energy is equal to the difference between the two levels

Energy Bands in Solids• In solids, the discrete energy levels of isolated atoms

broaden into allowed energy bands separated by forbidden gaps

• The separation and the electron population of the highest bands determine whether the solid is a conductor, an insulator, or a semiconductor

Energy Bands in Solids• Sodium example

• Blue represents energy bands occupied by the sodium electrons when the atoms are in their ground states, gold represents energy bands that are empty, and white represents energy gaps

• Electrons can have any energy within the allowed bands and cannot have energies in the gaps

Energy Level Definitions

• The valence band is the highest filled band

• The conduction band is the next higher empty band

• The energy gap has an energy, Eg, equal to the difference in energy between the top of the valence band and the bottom of the conduction band

Conductors

• When a voltage is applied to a conductor, the electrons accelerate and gain energy

• In quantum terms, electron energies increase if there are a high number of unoccupied energy levels for the electron to jump to

• For example, it takes very littleenergy for electrons to jump from the partially filled to one of the nearby empty states

Insulators• The valence band is completely

full of electrons

• A large band gap separates the valence and conduction bands

• A large amount of energy is needed for an electron to be able to jump from the valence to the conduction band

• The minimum required energy is Eg

Semiconductors• A semiconductor has a small energy

gap

• Thermally excited electrons have enough energy to cross the band gap

• The resistivity of semiconductors decreases with increases in temperature

• The light-color area in the valence band represents holes – empty states in the valence band created by electrons that have jumped to the conduction band

Semiconductors• Some electrons in the valence

band move to fill the holes and therefore also carry current

• The valence electrons that fill the holes leave behind other holes

• It is common to view the conduction process in the valence band as a flow of positive holes toward the negative electrode applied to the semiconductor

Semiconductors• An external voltage is supplied

• Electrons move toward the positive electrode

• Holes move toward the negative electrode

• There is a symmetrical current process in a semiconductor

Doping in Semiconductors

• Doping is the adding of impurities to a semiconductor (generally about 1 impurity atom per 107 semiconductor atoms)

• Doping results in both the band structure and the resistivity being changed

n-type Semiconductors• Donor atoms are doping materials that contain one

more electron than the semiconductor material

• This creates an essentially free electron with an energy level in the energy gap, just below the conduction band

• Only a small amount of thermal energy is needed to cause this electron to move into the conduction band

p-type Semiconductors• Acceptor atoms are doping materials that contain one

less electron than the semiconductor material

• A hole is left where the missing electron would be

• The energy level of the hole lies in the energy gap, just above the valence band

• An electron from the valence band has enough thermal energy to fill this impurity level, leaving behind a hole in the valence band

A p-n Junction

• A p-n junction is formed when a p-type semiconductor is joined to an n-type

• Three distinct regions exist: a p region, an n region, and a depletion region

• Mobile donor electrons from the n side nearest the junction diffuse to the p side, leaving behind immobile positive ions

A p-n Junction

• At the same time, holes from the p side nearest the junction diffuse to the n side and leave behind a region of fixed negative ions

• The resulting depletion region is depleted of mobile charge carriers

• There is also an electric field in this region that sweeps out mobile charge carriers to keep the region truly depleted

Diode Action• The p-n junction has the

ability to pass current in only one direction

• When the p-side is connected to a positive terminal, the device is forward biased and current flows

• When the n-side is connected to the positive terminal, the device is reverse biased and a very small reverse current results

Answers to Even Numbered Problems

Chapter 28:

Problem 34

(a) 4(b) 7

Answers to Even Numbered Problems

Chapter 28:

Problem 36

Use the lecture notes

Answers to Even Numbered Problems

Chapter 28:

Problem 44

137

Answers to Even Numbered Problems

Chapter 28:

Problem 52

(a) 135 eV(b) ~ 10 times the magnitude of the

ground state energy of hydrogen