Atomic Physics

1
Chapter 28

• Atomic Physics

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Introduction 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

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More Reasons the Hydrogen Atom is so Important

• The hydrogen atom is an ideal system for
  performing precise comparisons of theory and
  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

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28.1 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

The electrons are embedded inside the positive
charge
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Early Models of the Atom, 2

• 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

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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
• However, it doesnt

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28.2 Emission Spectra

• A gas at low pressure has a voltage applied to it
• Due to the voltage, the gas emits characteristic
  light
• 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

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Examples of Spectra
Emission
Absorption
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Emission Spectrum of Hydrogen Equation

• The wavelengths of hydrogens spectral lines can
  be found from
• RH is the Rydberg constant
• RH 1.0973732 x 107 m-1
• n is an integer, n 3, 4, 5
• The spectral lines correspond to different values
  of n

Balmer series
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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

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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 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

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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 (helios)

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28.3 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

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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

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Bohrs Quantum Conditions

• I. There are discrete stable tracks for the
  electrons. Along these tracks, the electrons move
  without energy loss.
• II. The electrons are able to jump between the
  tracks.

Ei-Efhf
In the Bohr model, a photon is emitted when the
electron drops from a higher orbit (Ei) to a
lower energy orbit (Ef).
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Bohrs Model Straightforward Approach

• Centripetal forceCoulomb force
• mv2/rkee2/r2 mv2kee2/r
• mv2/2kee2/(2r)
• Total energy of the atom E KEPE
• Emv2/2ke(-e)(e)/rkee2/(2r)-kee2/r

Kinetic energy
Electric potential energy of two point charges
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Bohrs Model Energy of the Atom
Orbit

• E -kee2/(2r)

Elementary charge
Coulomb constant
The negative sign indicates that the electron is
bound to the proton!
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Bohr Model Orbit Radius

• Bohr assumed that the angular momentum of the
  electron was quantized and could have only
  discrete values that were integral multiples of
  h/2?, where h is Planks constant
• mevrnh/(2p) n1, 2, 3,Quantum number
• vnh/(2p mer)

(or principal number)
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Bohr Model Orbit Radius, cont.

• It follows

Bohr orbit radius
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Orbital Radii and Energies (for the Hydrogen Atom)
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Orbital Radii and Energies, cont.

• rn0.0529?n2 (nm)
• En-13.6/n2 (eV)
• Energy difference between the levels
  DE13.6(1/nf2-1/ni2)

Initial State, ni
DE10.2 eV
Final state, nf
For example, between n1 and n2 (as drawn in the
picture) DE13.6(1/nf2-1/ni2)13.6(1/12-1/22)10.2
eV     
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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
• The ionization energy is the energy needed to
  completely remove the electron from the atom
• The ionization energy for hydrogen is 13.6 eV

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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

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Generalized Equation

• For the Balmer series, nf 2, ni3, 4, 5,
• For the Lyman series, nf 1, ni2, 3, 4,
• 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 ?

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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

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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 the Bohr
  equations
• Z is the atomic number of the element (number of
  protons)

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28.4 Modifications of the Bohr Theory
Elliptical Orbits

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

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Shell, Subshell Notation
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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

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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 an external
  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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28.5 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 (i.e., allowed) only if it contained an
  integral number of electron wavelengths

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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,

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de Broglie Waves in the Hydrogen Atom, cont.

• 2?r n ?
• ? h/(mev)
• 2?r nh/(mev)
• mevr nh/(2?) angular momentum of circular
  orbit
• This is precisely the quantization of angular
  momentum condition imposed by Bohr

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de Broglie Waves in the Hydrogen Atom, cont.

• This was the first convincing argument that the
  wave nature of matter was at the heart of the
  behavior of atomic systems
• Schrödingers wave equation was subsequently
  applied to atomic systems

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28.6 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

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Quantum Number, Summary

• 1. The principal quantum number n. As in the Bohr
  model, this number determines the total energy of
  the atom and can have only integer values, n1,
  2, 3,  
• 2. The orbital quantum number l.
• This number determines the angular momentum of
  the electron due to its orbital motion. The
  magnitude L of the angular momentum of the
  electron is L (ll1)1/2)h/(2p) l0, 1, 2,
  3,...(n-1)

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Quantum Number, Summary, final

• 3. The magnetic quantum number ml. The word
  magnetic is used here because an externally
  applied magnetic field influences the energy of
  the atom, and this quantum number is used in
  describing the effect. The values that ml can
  have depend on l 
• ml-l,-2, -1, 0, 1, 2,l 
• 4. The spin quantum number ms. This number is
  needed because the electron has an intrinsic
  property called spin angular momentum. Loosely
  speaking, we can view the electron as spinning
  while it orbits the nucleus, analogous to the way
  the earth spins as it orbits the sun. There are
  two possible values for the spin quantum number
  of the electron 
• ms1/2 or ms-1/2

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28.7 More about the 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 fine structure

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Example
(a)

• Determine the number of possible states for the
  hydrogen atom when the principal number is (a)
  n1 and (b) n2.

(b)