Atomic physics

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Atomic physics

• PHY232
• Remco Zegers
• zegers_at_nscl.msu.edu
• Room W109 cyclotron building
• http//www.nscl.msu.edu/zegers/phy232.html

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lon-capa 27,28,29
last homework set chapter 27 problems
1-10 chapter 28 problems 11-14 chapter 29
problems 15-17
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quiz (extra credit)

• If the photoelectric effect is observed in one
  metal, can you conclude that the effect will also
  be observed in another metal under otherwise the
  same conditions?
• a) Yes
• b) No

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models of the atom

• Newtonian era the atom is a solid sphere
• Thompson (1900) atom is a sphere of positive
  charge with negatively charged electrons in it
• Rutherford (1911) devised a planetary model
  with
• a positively charged nucleus in the core and
  electrons
• orbiting around it.
• where do atomic transitions come from
• continuous centripetal acceleration and thus
  emission of photons
• Bohr (1913) model for Hydrogen(-like) atoms
  (well look at it)
• Quantum-mechanical description of the atom

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atomic spectra
if an Hydrogen atom gets excited (for example
by heating it), light is emitted of certain
specific wavelengths following the equation
1/?RH(1/22-1/n2) RH1.097x107 m-1
n3,4,5,6... Light of these wavelengths gets
absorbed if white light (consisting of all wave
lengths) is shone on a Hydrogen gas.
Balmer series, after its discoverer Johann Balmer
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atomic spectra
Similar spectra are observed for other elements,
but the patterns are more complicated.
Nevertheless, measuring such spectra allows
one to identify which elements are present in a
sample
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Bohrs model for Hydrogen
proton
electron
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Bohrs theory for Hydrogen

• assumption 1
• the electron moves in circular orbits around the
  proton. The Coulomb force between the nucleus
  and the electron produces the centripetal
  acceleration. As a result, one can deduce the
  kinetic energy of the electron

proton
1
Note this is a pure classical reasoning
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Bohrs theory for Hydrogen

• assumption 2
• only certain orbits are stable, namely those in
  which no electromagnetic radiation is emitted in
  the absence of external forces. Hence, the energy
  of the atom is constant and the equations on the
  previous slide can be used.
• The size of the allowed orbits are those for
  which the orbital angular momentum of the
  electron is a multiple times h (h/(2?))
• This is derived (de Broglie) from the assumption
  that a fixed number of electron-wavelengths must
  fit in the orbit

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n1
n2
n3
11
n7
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Bohrs theory for Hydrogen

• assumption 3
• Radiation is emitted when an electron jumps from
  an outer orbit to an inner orbit. The energy of
  the radiation (and thus the frequency) is
  determined by the change in the atoms energy due
  to the jump
• radiation is absorbed if an electron jumps
• from an inner orbit to an outer orbit
• The energy of an orbit can be calculated with
• EEkineticEpotential

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3
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Bohrs theory for Hydrogen

• combining assumptions 1,2 3
• 1 2 give (solving for r, while eliminating v)
• combine with 3

3
1
2
5
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Bohrs theory for Hydrogen

• finally, by combining 4 and 5
• and using cf? and RH1.097x107 m-1

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5
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with nf and ni integers gt0
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The hydrogen spectrum
emission spectrum absorption spectrum
By measuring the wave length of the light, one
can determine the energy spectrum of Hydrogen
n1 ground state (energy is 13.6 eV) n?
electron is removed from atom the atom is
ionized. The n is usually referred to as a
shell the 1st shell, the 2nd shell etc
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question

• The quantum number n can increase without limit.
• Therefore, the frequency of the emitted light
  from state
• n to the ground state can also increase without
  limit.
• true
• false

answer b) if nf grows, f grows asymptotically
to cRH or if the frequency would grow without
limit, the energy involved would also increase
without limit, which doesnt make much sense
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example 1
How much energy does it take to ionize a Hydrogen
atom?
answer to ionize the atom, light must be
absorbed. The initial state has n1 (ground
state) The final state has n? E1-13.6 eV E?0
eV therefore, 13.6 eV must be absorbed by the
Hydrogen atom
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example 2
What is the wavelength of the light emitted if an
electron goes from the 5th shell to the 2nd shell
in a Hydrogen atom? What is the energy of the
photon?
answer in the initial state n5
E5-13.6/52-0.544 eV in the final state n2
E2-13.6/22-3.4 eV Ei-Ef-0.5543.42.846
eV 1/?RH(1/22-1/52)1.097x107(1/4-1/25)2.30x106
m-1 ?434 nm
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example 3
What is the wavelength of the light absorbed if a
hydrogen atom in its ground state is excited into
its n4 state? How much energy is absorbed (what
is the excitation energy)?
answer in the initial state n1
E1-13.6/12-13.6 eV in the final state n4
E4-13.6/42-0.85 eV Ei-Ef-13.60.85-12.75 eV
(note negative so absorbed) 1/?RH(1/12-1/42)1.0
97x107(1/1-1/16)1.03x107 m-1 ?97.2 nm note I
switched Ef and Ei in the equation for 1/?, since
that equation was derived for excitation and not
for absorption
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Heavier atoms
Bohrs equation also does well for heavier atoms
IF they have been ionized such that only one
electron remains in its orbit. For example for
Helium (2 protons (Z2) in the nucleus), 2
electrons in the orbits would make it neutral,
but only if one is missing can Bohrs equations
be applied. The equations need to be slightly
modified however, to take into account that the
Coulomb forces/energies are different. Change e2
into Ze2 everywhere where it occurs.
5a
6a
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question.

• Consider a hydrogen atom and a singly ionized
  helium atom. Which one has the lower ground state
  energy?
• a) hydrogen
• b) singly ionized helium
• c) the same

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More general description

• Bohrs classical approach breaks down if more
  than 1 electron is present in the atom.
• Instead, the problem has to be treated quantum
  mechanically by solving the Schrödinger equation
• The solutions give the distributions of electron
• clouds (so-called wave-functions) in the
  atom.
• The clouds describe the probability of
  finding
• an electron in a certain positions.
• The clouds are characterized by quantum
• numbers, which follow simple rules

the wave functions for Hydrogen
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atomic shells and quantum numbers

• The electrons are ordered according to 4 quantum
  numbers
• the principal quantum number n
• range 1,2,3.?
• Usually referred to as K(n1), L(n2), M(n3),
  N(n4) shells
• the orbital quantum number l
• range 0,1,2,n-1 (so there are n possibilities)
• usually referred to as s (l0),p (l1) d (l2) ,f
  (l3),g (l4),h,I
• the orbital magnetic quantum number ml
• range -l, -l1,-1,0,1l-1,l (there are 2l1
  possibilities)
• the spin magnetic quantum number ms
• range -1/2 (electron spin up) or ½ (electron
  spin down)
• in each state with given n,l,ml one can maximally
  place 2 electrons (ms-1/2 and ms1/2).
• Pauli exclusion principle
• no two electrons in an atom can have the same set
  of quantum numbers n,l,ml,ms

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example Hydrogen

• Hydrogen (not ionized) has only 1 electron.
• ground state quantum numbers
• n1
• l0,,n-10,0 so only l0
• ml-l,,l so only ml0
• ms-1/2 or 1/2
• one could maximally place 2 electrons in here
  (different ms)
• this level is referred to as 1s1 one electron in
  the level with quantum numbers n1, l0 (s)
• n2 states
• n2
• l0,n-10,1 so l0 or l1
• ml-l,l, so ml0 if l0 and ml-1,0,1 if l1
• for each ml, ms-1/2 ot 1/2
• these levels are referred to as
• 2s0 n2, l0 it is empty but I could put 2
  electrons in there
• 2p0 n2, l1 it is empty but I could put 6
  electrons in there, namely two each in n2,l1
  with ml-1,0,1

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more complicated example, Sodium Na

• Sodium has Z11 (11 protons), so if not ionized,
  it has 11 electrons.
• Atomic level will fill up according to lowest n,
  then lowest l
• there are exceptions to this (see also table in
  back of book and table 28.4)
• n1, l0, ml0, ms-1/2,1/2 1s2 2 electrons
• n2, l0, ml0, ms-1/2,1/2 2s2 2 electrons
• n2, l1, ml-1,0,1, ms-1/2,1/2 2p6 6
  electrons
• n3, l0, ml0, ms-1/2,1/2 3s1 1 electron (2
  possible, but only 1 needed to get to 11)
• SUM 11 electrons
• So, the ground state configuration can be
  described as
• 1s22s22p63s1
• Note that the n1 and n2 shells are filled with
  10 electrons (Neon)
• so this is sometimes referred to as Ne3s1

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energy levels (example for Li Z3)
electrons outside the last filled (sub)shell are
most important for the chemical properties of an
atoms.
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questions

• a) what is the electron configuration of Argon
  (Z18)
• b) what is the electron configuration of Titanium
  (Z22)

• start filling
• 1s2 n1, l0,ml0,ms-1/2 1/2 2 electrons 2
• 2s2 n2, l0, ml0,ms-1/2 1/2 2 electrons 4
• 2p6 n2, l1, ml-1,0,1,ms-1/2 1/2 6
  electrons 10
• 3s2 n3, l0, ml0,ms-1/2 1/2 2 electrons 12
• 3p6 n3, l1, ml-1,0,1,ms-1/2 1/2 6
  electrons 18
• The 3p sub-shell is just filled
• b) start with Argon and add 4 electrons
• Ar
• 3d4 n3, l2,ml-2,-1,0,1,2 ms-1/2 1/2 4
  electrons 22
• only 4 out of 10 are needed
• Note in reality the 4s shell gets filled before
  the 3d shell, so the real
• ground-state configuration is Ar4s23d2 (see
  table 28.4)

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For Titanium

• the 4s level has moved down and the 3d level
  moved up, so that the 4s level is lower in energy
  than the 3d level

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Periodic table of elements
see also back of book
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periodic table structure