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Atomic Physics


Chapter 42 Atomic Physics Importance of the Hydrogen Atom The hydrogen atom is the only atomic system that can be solved exactly Much of what was learned in the ... – PowerPoint PPT presentation

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Title: Atomic Physics

Chapter 42
  • Atomic Physics

Importance of the Hydrogen Atom
  • The hydrogen atom is the only atomic system that
    can be solved exactly
  • Much of what was learned in the twentieth century
    about the hydrogen atom, with its single
    electron, can be extended to such single-electron
    ions as He and Li2

More Reasons the Hydrogen Atom is Important
  • The hydrogen atom is an ideal system for
    performing precision tests of theory against
  • Also for improving our understanding of atomic
  • The quantum numbers that are used to characterize
    the allowed states of hydrogen can also be used
    to investigate more complex atoms
  • This allows us to understand the periodic table

Final Reasons for the Importance of the Hydrogen
  • The basic ideas about atomic structure must be
    well understood before we attempt to deal with
    the complexities of molecular structures and the
    electronic structure of solids
  • The full mathematical solution of the Schrödinger
    equation applied to the hydrogen atom gives a
    complete and beautiful description of the atoms

Atomic Spectra
  • A discrete line spectrum is observed when a
    low-pressure gas is subjected to an electric
  • Observation and analysis of these spectral lines
    is called emission spectroscopy
  • The simplest line spectrum is that for atomic

Emission Spectra Examples
Uniqueness of Atomic Spectra
  • Other atoms exhibit completely different line
  • Because no two elements have the same line
    spectrum, the phenomena represents a practical
    and sensitive technique for identifying the
    elements present in unknown samples

Absorption Spectroscopy
  • An absorption spectrum is obtained by passing
    white light from a continuous source through a
    gas or a dilute solution of the element being
  • The absorption spectrum consists of a series of
    dark lines superimposed on the continuous
    spectrum of the light source

Absorption Spectrum, Example
  • A practical example is the continuous spectrum
    emitted by the sun
  • The radiation must pass through the cooler gases
    of the solar atmosphere and through the Earths

Balmer Series
  • In 1885, Johann Balmer found an empirical
    equation that correctly predicted the four
    visible emission lines of hydrogen
  • Ha is red, ? 656.3 nm
  • Hß is green, ? 486.1 nm
  • H? is blue, ? 434.1 nm
  • Hd is violet, ? 410.2 nm

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 3, 4, 5,
  • The spectral lines correspond to different values
    of n

Other Hydrogen Series
  • Other series were also discovered and their
    wavelengths can be calculated
  • Lyman series
  • Paschen series
  • Brackett series

Joseph John Thomson
  • 1856 1940
  • English physicist
  • Received Nobel Prize in 1906
  • Usually considered the discoverer of the electron
  • Worked with the deflection of cathode rays in an
    electric field
  • Opened up the field of subatomic particles

Early Models of the Atom, Thomsons
  • J. J. Thomson established the charge to mass
    ratio for electrons
  • His model of the atom
  • A volume of positive charge
  • Electrons embedded throughout the volume

Rutherfords Thin Foil Experiment
  • Experiments done in 1911
  • A beam of positively charged alpha particles hit
    and are scattered from a thin foil target
  • Large deflections could not be explained by
    Thomsons model

Early Models of the Atom, Rutherfords
  • Rutherford
  • 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

Difficulties with the Rutherford Model
  • Atoms emit certain discrete characteristic
    frequencies of electromagnetic radiation
  • The Rutherford model is unable to explain this
  • Rutherfords electrons are undergoing a
    centripetal acceleration
  • It 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
  • It doesnt

Niels Bohr
  • 1885 1962
  • Danish physicist
  • An active participant in the early development of
    quantum mechanics
  • Headed the Institute for Advanced Studies in
  • Awarded the 1922 Nobel Prize in physics
  • For structure of atoms and the radiation
    emanating from them

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
  • He applied Plancks ideas of quantized energy
    levels to orbiting electrons

Bohrs Theory, cont.
  • This model is now considered obsolete
  • It has been replaced by a probabilistic
    quantum-mechanical theory
  • The model can still be used to develop ideas of
    energy quantization and angular momentum
    quantization as applied to atomic-sized systems

Bohrs Assumptions for Hydrogen, 1
  • The electron moves in circular orbits around the
    proton under the electric force of attraction
  • The Coulomb force produces the centripetal

Bohrs Assumptions, 2
  • Only certain electron orbits are stable
  • These are the orbits in which the atom does not
    emit energy in the form of electromagnetic
  • Therefore, the energy of the atom remains
    constant and classical mechanics can be used to
    describe the electrons motion
  • This representation claims the centripetally
    accelerated electron does not emit energy and
    therefore does not eventually spiral into the

Bohrs Assumptions, 3
  • Radiation is emitted by the atom when the
    electron makes a transition from a more energetic
    initial state to a lower-energy orbit
  • The transition cannot be treated classically
  • The frequency emitted in the transition is
    related to the change in the atoms energy
  • The frequency is independent of frequency of the
    electrons orbital motion
  • The frequency of the emitted radiation is given
  • Ei Ef hƒ
  • If a photon is absorbed, the electron moves to a
    higher energy level

Bohrs Assumptions, 4
  • The size of the allowed electron orbits is
    determined by a condition imposed on the
    electrons orbital angular momentum
  • The allowed orbits are those for which the
    electrons orbital angular momentum about the
    nucleus is quantized and equal to an integral
    multiple of h

Mathematics of Bohrs Assumptions and Results
  • Electrons orbital angular momentum
  • mevr nh where n 1, 2, 3,
  • The total energy of the atom is
  • The total energy can also be expressed as
  • Note, the total energy is negative, indicating a
    bound electron-proton system

Bohr Radius
  • The radii of the Bohr orbits are quantized
  • This shows that the radii of the allowed orbits
    have discrete valuesthey are quantized
  • When n 1, the orbit has the smallest radius,
    called the Bohr radius, ao
  • ao 0.052 9 nm

Radii and Energy of Orbits
  • A general expression for the radius of any orbit
    in a hydrogen atom is
  • rn n2ao
  • The energy of any orbit is
  • This becomes
  • En - 13.606 eV / n2

Specific Energy Levels
  • Only energies satisfying the previous equation
    are allowed
  • The lowest energy state is called the ground
  • This corresponds to n 1 with E 13.606 eV
  • The ionization energy is the energy needed to
    completely remove the electron from the ground
    state in the atom
  • The ionization energy for hydrogen is 13.6 eV

Energy Level Diagram
  • Quantum numbers are given on the left and
    energies on the right
  • The uppermost level,
  • E 0, represents the state for which the
    electron is removed from the atom
  • Adding more energy than this amount ionizes the

Active Figures 42.7 and 42.8
  • Use the active figure to choose initial and final
    energy levels
  • Observe the transition in both figures

Frequency of Emitted Photons
  • The frequency of the photon emitted when the
    electron makes a transition from an outer orbit
    to an inner orbit is
  • It is convenient to look at the wavelength instead

Wavelength of Emitted Photons
  • The wavelengths are found by
  • The value of RH from Bohrs analysis is in
    excellent agreement with the experimental value

Extension to Other Atoms
  • Bohr extended his model for hydrogen to other
    elements in which all but one electron had been
  • Z is the atomic number of the element and is the
    number of protons in the nucleus

Difficulties with the Bohr Model
  • Improved spectroscopic techniques found that many
    of the spectral lines of hydrogen were not single
  • Each line was actually a group of lines spaced
    very close together
  • Certain single spectral lines split into three
    closely spaced lines when the atoms were placed
    in a magnetic field

Bohrs Correspondence Principle
  • Bohrs correspondence principle states that
    quantum physics agrees with classical physics
    when the differences between quantized levels
    become vanishingly small
  • Similar to having Newtonian mechanics be a
    special case of relativistic mechanics when v ltlt c

The Quantum Model of the Hydrogen Atom
  • The potential energy function for the hydrogen
    atom is
  • ke is the Coulomb constant
  • r is the radial distance from the proton to the
  • The proton is situated at r 0

Quantum Model, cont.
  • The formal procedure to solve the hydrogen atom
    is to substitute U(r) into the Schrödinger
    equation, find the appropriate solutions to the
    equations, and apply boundary conditions
  • Because it is a three-dimensional problem, it is
    easier to solve if the rectangular coordinates
    are converted to spherical polar coordinates

Quantum Model, final
  • ?(x, y, z) is converted to ?(r, ?, f)
  • Then, the space variables can be separated
  • ?(r, ?, f) R(r), ƒ(?), g(f)
  • When the full set of boundary conditions are
    applied, we are led to three different quantum
    numbers for each allowed state

Quantum Numbers, General
  • The three different quantum numbers are
    restricted to integer values
  • They correspond to three degrees of freedom
  • Three space dimensions

Principal Quantum Number
  • The first quantum number is associated with the
    radial function R(r)
  • It is called the principal quantum number
  • It is symbolized by n
  • The potential energy function depends only on the
    radial coordinate r
  • The energies of the allowed states in the
    hydrogen atom are the same En values found from
    the Bohr theory

Orbital and Orbital Magnetic Quantum Numbers
  • The orbital quantum number is symbolized by l
  • It is associated with the orbital angular
    momentum of the electron
  • It is an integer
  • The orbital magnetic quantum number is symbolized
    by ml
  • It is also associated with the angular orbital
    momentum of the electron and is an integer

Quantum Numbers, Summary of Allowed Values
  • The values of n can range from 1 to
  • The values of l can range from 0 to n - 1
  • The values of ml can range from l to l
  • Example
  • If n 1, then only l 0 and ml 0 are
  • If n 2, then l 0 or 1
  • If l 0 then ml 0
  • If l 1 then ml may be 1, 0, or 1

Quantum Numbers, Summary Table
  • Historically, all states having the same
    principle quantum number are said to form a shell
  • Shells are identified by letters K, L, M,
  • All states having the same values of n and l are
    said to form a subshell
  • The letters s, p, d, f, g, h, .. are used to
    designate the subshells for which l 0, 1, 2,

Shell and Subshell Notation, Summary Table
Wave Functions for Hydrogen
  • The simplest wave function for hydrogen is the
    one that describes the 1s state and is designated
  • As ?1s(r) approaches zero, r approaches and is
    normalized as presented
  • ?1s(r) is also spherically symmetric
  • This symmetry exists for all s states

Probability Density
  • The probability density for the 1s state is
  • The radial probability density function P(r) is
    the probability per unit radial length of finding
    the electron in a spherical shell of radius r and
    thickness dr

Radial Probability Density
  • A spherical shell of radius r and thickness dr
    has a volume of 4pr2 dr
  • The radial probability function is
  • P(r) 4pr2 ?2

P(r) for 1s State of Hydrogen
  • The radial probability density function for the
    hydrogen atom in its ground state is
  • The peak indicates the most probable location
  • The peak occurs at the Bohr radius

P(r) for 1s State of Hydrogen, cont.
  • The average value of r for the ground state of
    hydrogen is 3/2 ao
  • The graph shows asymmetry, with much more area to
    the right of the peak
  • According to quantum mechanics, the atom has no
    sharply defined boundary as suggested by the Bohr

Electron Clouds
  • The charge of the electron is extended throughout
    a diffuse region of space, commonly called an
    electron cloud
  • This shows the probability density as a function
    of position in the xy plane
  • The darkest area, r ao, corresponds to the most
    probable region

Wave Function of the 2s state
  • The next-simplest wave function for the hydrogen
    atom is for the 2s state
  • n 2 l 0
  • The normalized wave function is
  • ?2s depends only on r and is spherically symmetric

Comparison of 1s and 2s States
  • The plot of the radial probability density for
    the 2s state has two peaks
  • The highest value of P corresponds to the most
    probable value
  • In this case, r 5ao

Active Figure 42.12
  • Use the active figure to choose values of r/ao
  • Find the probability that the electron is located
    between two values

Physical Interpretation of l
  • The magnitude of the angular momentum of an
    electron moving in a circle of radius r is
  • L mevr
  • The direction of is perpendicular to the plane
    of the circle
  • The direction is given by the right hand rule
  • In the Bohr model, the angular momentum of the
    electron is restricted to multiples of ?

Physical Interpretation of l, cont.
  • According to quantum mechanics, an atom in a
    state whose principle quantum number is n can
    take on the following discrete values of the
    magnitude of the orbital angular momentum
  • L can equal zero, which causes great difficulty
    when attempting to apply classical mechanics to
    this system

Physical Interpretation of ml
  • The atom possesses an orbital angular momentum
  • There is a sense of rotation of the electron
    around the nucleus, so that a magnetic moment is
    present due to this angular momentum
  • There are distinct directions allowed for the
    magnetic moment vector with respect to the
    magnetic field vector

Physical Interpretation of ml, 2
  • Because the magnetic moment of the atom can be
    related to the angular momentum vector, , the
    discrete direction of translates into the fact
    that the direction of is quantized
  • Therefore, Lz, the projection of along the z
    axis, can have only discrete values

Physical Interpretation of ml, 3
  • The orbital magnetic quantum number ml specifies
    the allowed values of the z component of orbital
    angular momentum
  • Lz ml?
  • The quantization of the possible orientations of
    with respect to an external magnetic field is
    often referred to as space quantization

Physical Interpretation of ml, 4
  • does not point in a specific direction
  • Even though its z-component is fixed
  • Knowing all the components is inconsistent with
    the uncertainty principle
  • Imagine that must lie anywhere on the surface
    of a cone that makes an angle ? with the z axis

Physical Interpretation of ml, final
  • ? is also quantized
  • Its values are specified through
  • ml is never greater than l, therefore ? can never
    be zero

Zeeman Effect
  • The Zeeman effect is the splitting of spectral
    lines in a strong magnetic field
  • In this case the upper level, with l 1, splits
    into three different levels corresponding to the
    three different directions of µ

Spin Quantum Number ms
  • Electron spin does not come from the Schrödinger
  • Additional quantum states can be explained by
    requiring a fourth quantum number for each state
  • This fourth quantum number is the spin magnetic
    quantum number ms

Electron Spins
  • Only two directions exist for electron spins
  • The electron can have spin up (a) or spin down
  • In the presence of a magnetic field, the energy
    of the electron is slightly different for the two
    spin directions and this produces doublets in
    spectra of certain gases

Electron Spins, cont.
  • The concept of a spinning electron is
    conceptually useful
  • The electron is a point particle, without any
    spatial extent
  • Therefore the electron cannot be considered to be
    actually spinning
  • The experimental evidence supports the electron
    having some intrinsic angular momentum that can
    be described by ms
  • Dirac showed this results from the relativistic
    properties of the electron

Spin Angular Momentum
  • The total angular momentum of a particular
    electron state contains both an orbital
    contribution and a spin contribution
  • Electron spin can be described by a single
    quantum number s, whose value can only be s ½
  • The spin angular momentum of the electron never

Spin Angular Momentum, cont
  • The magnitude of the spin angular momentum is
  • The spin angular momentum can have two
    orientations relative to a z axis, specified by
    the spin quantum number ms ½
  • ms ½ corresponds to the spin up case
  • ms - ½ corresponds to the spin down case

Spin Angular Momentum, final
  • The z component of spin angular momentum is Sz
    msh ½ h
  • Spin angular moment is quantized

Spin Magnetic Moment
  • The spin magnetic moment µspin is related to the
    spin angular momentum by
  • The z component of the spin magnetic moment can
    have values

Quantum States
  • There are eight quantum states corresponding to n
  • These states depend on the addition of the
    possible values of ms
  • Table 42.3 summarizes these states

Quantum Numbers for n 2 State of Hydrogen
Wolfgang Pauli
  • 1900 1958
  • Austrian physicist
  • Important review article on relativity
  • At age 21
  • Discovery of the exclusion principle
  • Explanation of the connection between particle
    spin and statistics
  • Relativistic quantum electrodynamics
  • Neutrino hypothesis
  • Hypotheses of nuclear spin

The Exclusion Principle
  • The four quantum numbers discussed so far can be
    used to describe all the electronic states of an
    atom regardless of the number of electrons in its
  • The exclusion principle states that no two
    electrons can ever be in the same quantum state
  • Therefore, no two electrons in the same atom can
    have the same set of quantum numbers
  • If the exclusion principle was not valid, an atom
    could radiate energy until every electron was in
    the lowest possible energy state and the chemical
    nature of the elements would be modified

Filling Subshells
  • The electronic structure of complex atoms can be
    viewed as a succession of filled levels
    increasing in energy
  • Once a subshell is filled, the next electron goes
    into the lowest-energy vacant state
  • If the atom were not in the lowest-energy state
    available to it, it would radiate energy until it
    reached this state

  • An orbital is defined as the atomic state
    characterized by the quantum numbers n, l and ml
  • From the exclusion principle, it can be seen that
    only two electrons can be present in any orbital
  • One electron will have spin up and one spin down
  • Each orbital is limited to two electrons, the
    number of electrons that can occupy the various
    shells is also limited

Allowed Quantum States, Example with n 3
  • In general, each shell can accommodate up to 2n2

Hunds Rule
  • Hunds Rule states that when an atom has orbitals
    of equal energy, the order in which they are
    filled by electrons is such that a maximum number
    of electrons have unpaired spins
  • Some exceptions to the rule occur in elements
    having subshells that are close to being filled
    or half-filled

Configuration of Some Electron States
Periodic Table
  • Dmitri Mendeleev made an early attempt at finding
    some order among the chemical elements
  • He arranged the elements according to their
    atomic masses and chemical similarities
  • The first table contained many blank spaces and
    he stated that the gaps were there only because
    the elements had not yet been discovered

Periodic Table, cont.
  • By noting the columns in which some missing
    elements should be located, he was able to make
    rough predictions about their chemical properties
  • Within 20 years of the predictions, most of the
    elements were discovered
  • The elements in the periodic table are arranged
    so that all those in a column have similar
    chemical properties

Periodic Table, Explained
  • The chemical behavior of an element depends on
    the outermost shell that contains electrons
  • For example, the inert gases (last column) have
    filled subshells and a wide energy gap occurs
    between the filled shell and the next available

Hydrogen Energy Level Diagram Revisited
  • The allowed values of l are separated
  • Transitions in which l does not change are very
    unlikely to occur and are called forbidden
  • Such transitions actually can occur, but their
    probability is very low compared to allowed

Selection Rules
  • The selection rules for allowed transitions are
  • ?l 1
  • ?ml 0, 1
  • The angular momentum of the atom-photon system
    must be conserved
  • Therefore, the photon involved in the process
    must carry angular momentum
  • The photon has angular momentum equivalent to
    that of a particle with spin 1
  • A photon has energy, linear momentum and angular

Multielectron Atoms
  • For multielectron atoms, the positive nuclear
    charge Ze is largely shielded by the negative
    charge of the inner shell electrons
  • The outer electrons interact with a net charge
    that is smaller than the nuclear charge
  • Allowed energies are
  • Zeff depends on n and l

X-Ray Spectra
  • These x-rays are a result of the slowing down of
    high energy electrons as they strike a metal
  • The kinetic energy lost can be anywhere from 0 to
    all of the kinetic energy of the electron
  • The continuous spectrum is called bremsstrahlung,
    the German word for braking radiation

X-Ray Spectra, cont.
  • The discrete lines are called characteristic
  • These are created when
  • A bombarding electron collides with a target atom
  • The electron removes an inner-shell electron from
  • An electron from a higher orbit drops down to
    fill the vacancy

X-Ray Spectra, final
  • The photon emitted during this transition has an
    energy equal to the energy difference between the
  • Typically, the energy is greater than 1000 eV
  • The emitted photons have wavelengths in the range
    of 0.01 nm to 1 nm

Moseley Plot
  • Henry G. J. Moseley plotted the values of atoms
    as shown
  • ? is the wavelength of the Ka line of each
  • The Ka line refers to the photon emitted when an
    electron falls from the L to the K shell
  • From this plot, Moseley developed a periodic
    table in agreement with the one based on chemical

Stimulated Absorption
  • When a photon has energy hƒ equal to the
    difference in energy levels, it can be absorbed
    by the atom
  • This is called stimulated absorption because the
    photon stimulates the atom to make the upward

Active Figure 42.25
  • Use the active figure to adjust the energy
    difference between the states
  • Observe stimulated absorption

Spontaneous Emission
  • Once an atom is in an excited state, the excited
    atom can make a transition to a lower energy
  • Because this process happens naturally, it is
    known as spontaneous emission

Stimulated Emission
  • In addition to spontaneous emission, stimulated
    emission occurs
  • Stimulated emission may occur when the excited
    state is a metastable state

Stimulated Emission, cont.
  • A metastable state is a state whose lifetime is
    much longer than the typical 10-8 s
  • An incident photon can cause the atom to return
    to the ground state without being absorbed
  • Therefore, you have two photons with identical
    energy, the emitted photon and the incident
  • They both are in phase and travel in the same

Active Figure 42.27
  • Use the active figure to adjust the energy
    difference between states
  • Observe the stimulated emission

Lasers Properties of Laser Light
  • Laser light is coherent
  • The individual rays in a laser beam maintain a
    fixed phase relationship with each other
  • Laser light is monochromatic
  • The light has a very narrow range of wavelengths
  • Laser light has a small angle of divergence
  • The beam spreads out very little, even over long

Lasers Operation
  • It is equally probable that an incident photon
    would cause atomic transitions upward or downward
  • Stimulated absorption or stimulated emission
  • If a situation can be caused where there are more
    electrons in excited states than in the ground
    state, a net emission of photons can result
  • This condition is called population inversion

Lasers Operation, cont.
  • The photons can stimulate other atoms to emit
    photons in a chain of similar processes
  • The many photons produced in this manner are the
    source of the intense, coherent light in a laser

Conditions for Build-Up of Photons
  • The system must be in a state of population
  • The excited state of the system must be a
    metastable state
  • In this case, the population inversion can be
    established and stimulated emission is likely to
    occur before spontaneous emission
  • The emitted photons must be confined in the
    system long enough to enable them to stimulate
    further emission
  • This is achieved by using reflecting mirrors

Laser Design Schematic
  • The tube contains the atoms that are the active
  • An external source of energy pumps the atoms to
    excited states
  • The mirrors confine the photons to the tube
  • Mirror 2 is only partially reflective

Energy-Level Diagram for Neon in a Helium-Neon
  • The atoms emit 632.8-nm photons through
    stimulated emission
  • The transition is E3 to E2
  • indicates a metastable state

Laser Applications
  • Applications include
  • Medical and surgical procedures
  • Precision surveying and length measurements
  • Precision cutting of metals and other materials
  • Telephone communications