Atomic Physics

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Chapter 28

• Atomic Physics

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Plum Pudding Model of the Atom

• J. J. Thomsons Plum Pudding model of the atom
• Electrons embedded throughout the a volume of
  positive charge
• A change from Newtons model of the atom as a
  tiny, hard, indestructible sphere

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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

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Planetary Model of the Atom

• Based on results of thin foil scattering
  experiments, Rutherfords 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

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Difficulties with the Rutherford Model

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

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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

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Emission Spectrum of Hydrogen

• The wavelengths of hydrogens 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 1, 2, 3,
• The spectral lines correspond to different values
  of n
• n 3, ? 656.3 nm
• n 4, ? 486.1 nm

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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

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Chapter 28Problem 6

• In a Rutherford scattering experiment, an
  a-particle (charge 2e) heads directly toward a
  gold nucleus (charge 79e). The a-particle had
  a kinetic energy of 5.0 MeV when very far (r ? 8)
  from the nucleus. Assuming the gold nucleus to be
  fixed in space, determine the distance of closest
  approach.

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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

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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 atom
• remains constant and classical
• mechanics can be used to describe
• the electrons motion

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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 atoms energy Ei Ef h ƒ
• The size of the allowed electron orbits is
  determined by a quantization condition imposed on
  the electrons orbital angular momentum
• me v r n h where n 1, 2, 3, h h / 2 p

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Radii and Energy of Orbits
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Radii and Energy of Orbits
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Radii and Energy of Orbits

• 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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Radii and Energy of Orbits

• 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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Energy Level Diagram
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Energy Level Diagram

• The value of RH from Bohrs analysis is in
  excellent agreement with the experimental value
  of the Rydberg constant
• A more generalized equation can be
• used to find the wavelengths of any
• spectral lines

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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 hydrogen
• is 13.6 eV

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Chapter 28Problem 18

• A particle of charge q and mass m, moving with a
  constant speed v, perpendicular to a constant
  magnetic field, B, follows a circular path. If in
  this case the angular momentum about the center
  of this circle is quantized so that mvr 2nh,
  show that the expression for the allowed radii
  for the particle are written in the corner, where
  n 1, 2, 3, . . .

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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?

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Modifications of the Bohr Theory Elliptical
Orbits

• Sommerfeld extended the results to include
  elliptical orbits
• Retained the principal 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 quantum
• number are said to form a shell, whereas the
• states with given values of n and l are said
• to form a subshell

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Modifications of the Bohr Theory Elliptical
Orbits
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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

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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

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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

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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

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de Broglie Waves in the Hydrogen Atom

• One of Bohrs 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 Bohrs equations as a
  natural consequence of standing wave patterns

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Quantum Mechanics and the Hydrogen Atom

• Schrödingers 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

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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

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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

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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

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Filling Shells

• As a general rule, the order that electrons fill
  an atoms 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(2l1)
  electrons

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Filling Shells
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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 Paulis
  Exclusion Principle

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The Periodic Table
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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?

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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

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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

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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

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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

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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

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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 little
• energy for electrons to jump
• from the partially filled to one of
• the nearby empty states

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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• Answers to Even Numbered Problems
• Chapter 28
• Problem 34
• 4
• 7

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Answers to Even Numbered Problems Chapter 28
Problem 36 Use the lecture notes
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Answers to Even Numbered Problems Chapter 28
Problem 44 137
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• Answers to Even Numbered Problems
• Chapter 28
• Problem 52
• 135 eV
• 10 times the magnitude of the ground state
  energy of hydrogen