Atomic Physics

1
Chapter 28

• Atomic Physics

2
Importance of Hydrogen Atom

• Hydrogen is the simplest atom
• The quantum numbers used to characterize the
  allowed states of hydrogen can also be used to
  describe (approximately) the allowed states of
  more complex atoms
• This enables us to understand the periodic table

3
More Reasons the Hydrogen Atom is so Important

• The hydrogen atom is an ideal system for
  performing precise comparisons of theory with
  experiment
• Also for improving our understanding of atomic
  structure
• Much of what we know about the hydrogen atom can
  be extended to other single-electron ions
• For example, He and Li2

4
Sir Joseph John Thomson

• J. J. Thomson
• 1856 - 1940
• Discovered the electron
• Did extensive work with cathode ray deflections
• 1906 Nobel Prize for discovery of electron

5
Early Models of the Atom

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

6
Early Models of the Atom, 2

• 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

7
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

8
Difficulties with the Rutherford Model

• Atoms emit certain discrete characteristic
  frequencies of electromagnetic radiation
• 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

9
Emission Spectra

• A gas at low pressure has a voltage applied to it
• A gas emits light characteristic of the gas
• When the emitted light is analyzed with a
  spectrometer, a series of discrete bright lines
  is observed
• Each line has a different wavelength and color
• This series of lines is called an emission
  spectrum

10
Examples of Emission Spectra
11
Emission Spectrum of Hydrogen Equation

• The wavelengths of hydrogens spectral lines can
  be found from
• RH is the Rydberg constant
• RH 1.097 373 2 x 107 m-1
• n is an integer, n 1, 2, 3,
• The spectral lines correspond to different values
  of n

12
Spectral Lines of Hydrogen

• The Balmer Series has lines whose wavelengths are
  given by the preceding equation
• Examples of spectral lines
• n 3, ? 656.3 nm
• n 4, ? 486.1 nm

13
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

14
Applications of Absorption Spectrum

• The continuous spectrum emitted by the Sun passes
  through the cooler gases of the Suns atmosphere
• The various absorption lines can be used to
  identify elements in the solar atmosphere
• Led to the discovery of helium

15
Absorption Spectrum of Hydrogen
16
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

17
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

18
Bohrs 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

19
Bohrs Assumptions, cont

• Only certain electron orbits are stable
• These are the orbits in which the atom does 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
• Radiation is emitted by the atom when the
  electron jumps from a more energetic initial
  state to a lower state
• The jump cannot be treated classically

20
Bohrs Assumptions, final

• The electrons jump, continued
• The frequency emitted in the jump is related to
  the change in the atoms energy
• It is generally not the same as the frequency of
  the electrons orbital motion
• The frequency is given by Ei Ef h ƒ
• The size of the allowed electron orbits is
  determined by a condition imposed on the
  electrons orbital angular momentum

21
Mathematics of Bohrs Assumptions and Results

• Electrons orbital angular momentum
• me v r n h where n 1, 2, 3,
• The total energy of the atom
• The energy of the atom can also be expressed as

22
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 0.052 9 nm

23
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 - 13.6 eV/ n2

24
Specific Energy Levels

• The lowest energy state is called the ground
  state
• This corresponds to n 1
• Energy is 13.6 eV
• The next energy level has an energy of 3.40 eV
• The energies can be compiled in an energy level
  diagram

25
Specific Energy Levels, cont

• The ionization energy is the energy needed to
  completely remove the electron from the atom
• The ionization energy for hydrogen is 13.6 eV
• The uppermost level corresponds to E 0 and n ?
  ?

26
Energy Level Diagram

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

27
Generalized Equation

• For the Balmer series, nf 2
• For the Lyman series, nf 1
• Whenever an transition occurs between a state, ni
  to another state, nf (where ni gt nf), a photon is
  emitted
• The photon has a frequency f (Ei Ef)/h and
  wavelength ?

28
Bohrs Correspondence Principle

• Bohrs 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 ltlt
  c

29
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 the
  experimental value
• Gives an expression for the radius of the atom
• Predicts energy levels of hydrogen
• Gives a 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

30
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

31
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

32
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

33
de Broglie Waves

• 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

34
de Broglie Waves in the Hydrogen Atom

• In this example, three complete wavelengths are
  contained in the circumference of the orbit
• In general, the circumference must equal some
  integer number of wavelengths
• 2 ? r n ? n 1, 2,

35
de Broglie Waves in the Hydrogen Atom, cont

• The expression for the de Broglie wavelength can
  be included in the circumference calculation
• me v r n h
• This is the same quantization of angular momentum
  that Bohr imposed in his original theory
• This was the first convincing argument that the
  wave nature of matter was at the heart of the
  behavior of atomic systems

36
de Broglie Waves, cont.

• 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
• Schrödingers wave equation was subsequently
  applied to atomic systems

37
Quantum Mechanics and the Hydrogen Atom

• One of the first great achievements of quantum
  mechanics was the solution of the wave equation
  for the hydrogen atom
• The significance of quantum mechanics is that 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

38
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
• Also see Table 28.2

39
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

40
Spin Notes

• A classical description of electron spin is
  incorrect
• Since the electron cannot be located precisely in
  space, it cannot be considered to be a spinning
  solid object
• P. A. M. Dirac developed a relativistic quantum
  theory in which spin appears naturally

41
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

42
Electron Clouds, cont

• 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

43
Wolfgang Pauli

• 1900 1958
• Contributions include
• Major review of relativity
• Exclusion Principle
• Connect between electron spin and statistics
• Theories of relativistic quantum electrodynamics
• Neutrino hypothesis
• Nuclear spin hypothesis

44
The Pauli Exclusion Principle

• No two electrons in an atom 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

45
Filling Shells

• As a general rule, the order that electrons fill
  an atoms subshell is
• Once one subshell is filled, the next electron
  goes into the vacant subshell that is lowest in
  energy
• Otherwise, the electron would radiate energy
  until it reached the subshell with the lowest
  energy
• A subshell is filled when it holds 2(2l1)
  electrons
• See table 28.3

46
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
  explained by quantum numbers and Paulis
  Exclusion Principle explains the configuration

47
Characteristic X-Rays

• When a metal target is bombarded by high-energy
  electrons, x-rays are emitted
• The x-ray spectrum typically consists of a broad
  continuous spectrum and a series of sharp lines
• The lines are dependent on the metal of the
  target
• The lines are called characteristic x-rays

48
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
• 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

49
Moseley Plot

• ? is the wavelength of the K? line
• K? is the line that is produced by an electron
  falling from the L shell to the K shell
• From this plot, Moseley was able to determine the
  Z values of other elements and produce a periodic
  chart in excellent agreement with the known
  chemical properties of the elements

50
Atomic Transitions Energy Levels

• An atom may have many possible energy levels
• At ordinary temperatures, most of the atoms in a
  sample are in the ground state
• Only photons with energies corresponding to
  differences between energy levels can be absorbed

51
Atomic Transitions Stimulated Absorption

• The blue dots represent electrons
• When a photon with energy ?E is absorbed, one
  electron jumps to a higher energy level
• These higher levels are called excited states
• ?E hƒ E2 E1
• In general, ?E can be the difference between any
  two energy levels

52
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

53
Atomic Transitions Stimulated Emission

• An atom is in an excited stated and a photon is
  incident on it
• The incoming photon increases the probability
  that the excited atom will return to the ground
  state
• There are two emitted photons, the incident one
  and the emitted one
• The emitted photon is in exactly in phase with
  the incident photon

54
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

55
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

56
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 E3 in Ne, a
  632.8 nm photon is emitted

57
Production of a Laser Beam
58
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

59
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

60
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

61
Energy Bands, Detail

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

62
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

63
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

64
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

65
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 white area in the valence band represents
  holes

66
Semiconductors, cont

• Holes are empty states in the valence band
  created by electrons that have jumped to the
  conduction band
• 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

67
Movement of Charges in 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

68
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

69
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

70
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

71
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
• A depletion region

72
The Depletion Region

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

73
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

74
Applications of Semiconductor Diodes

• Rectifiers
• Change AC voltage to DC voltage
• A half-wave rectifier allows current to flow
  during half the AC cycle
• A full-wave rectifier rectifies both halves of
  the AC cycle
• Transistors
• May be used to amplify small signals
• Integrated circuit
• A collection of interconnected transistors,
  diodes, resistors and capacitors fabricated on a
  single piece of silicon