Download o  Lecturer: Dr. Peter Gallagher Email:

o  Lecturer:
o  Dr. Peter Gallagher
o  Email:
o  [email protected]
o  Web:
o  www.physics.tcd.ie/people/peter.gallagher/
o  Office:
o  3.17A in SNIAM
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o 
Section I: Single-electron Atoms
o  Review of basic spectroscopy
o  Hydrogen energy levels
o  Fine structure
o  Spin-orbit coupling
o  Nuclear moments and hyperfine structure
o 
Section II: Multi-electron Atoms
o  Central-field and Hartree approximations
o  Angular momentum, LS and jj coupling
o  Alkali spectra
o  Helium atom
o  Complex atoms
o 
Section III: Molecules
o  Quantum mechanics of molecules
o  Vibrational transitions
o  Rotational transitions
o  Electronic transition
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o  Lecture notes will be posted at
http://www.tcd.ie/physics/people/peter.gallagher/
o  Recommended Books:
o  The Physics of Atoms and Quanta: Introduction to experiement and theory
Haken & Wolf (Springer)
o  Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles
Eisberg & Resnick (Wiley)
o  Quantum Mechanics
McMurry
o  Physical Chemistry
Atkins (OUP)
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o  Line spectra
o  Emission spectra
Bulb
Sun
o  Absorption spectra
Na
o  Hydrogen spectrum
Emission
spectra
o  Balmer Formula
o  Bohr’s Model
H
Hg
Cs
o  Chapter 4, Eisberg & Resnick
Chlorophyll
Diethylthiacarbocyaniodid
Absorption
spectra
Diethylthiadicarbocyaniodid
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o  Continuous spectrum: Produced by solids, liquids & dense gases produce - no
“gaps” in wavelength of light produced:
o  Emission spectrum: Produced by rarefied gases – emission only in narrow
wavelength regions:
o  Absorption spectrum: Gas atoms absorb the same wavelengths as they
usually emit and results in an absorption line spectrum:
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Gas cloud
3
1
2
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o  Electron transition between
energy levels result in emission
or absorption lines.
o  Different elements produce
different spectra due to differing
atomic structure.
o  Complexity of spectrum
increases rapidly with Z.
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o  Emission/absorption lines are due to radiative transitions:
1.  Radiative (or Stimulated) absorption:
Photon with energy (E! = h" = E2 - E1) excites electron from lower energy
level.
E! =h"
E2
E2
E1
E1
Can only occur if E! = h" = E2 - E1
2.  Radiative recombination/emission:
Electron makes transition to lower energy level and emits photon with energy
h"’ = E2 - E1.
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o 
Radiative recombination can be either:
a)  Spontaneous emission: Electron minimizes its total energy by emitting photon and
making transition from E2 to E1.
E2
E2
E1
E1
E!’ =h"’
Emitted photon has energy E!’ = h"’ = E2 - E1
b)  Stimulated emission: If photon is strongly coupled with electron, cause electron to
decay to lower energy level, releasing a photon of the same energy.
E! =h"
E2
E2
E!’ =h"’
E1
E1
E! =h"
Can only occur if E! = h" = E2 - E1 Also, h"’ = h"
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o  In ~1850s, hydrogen was found to emit lines at 6563, 4861 and 4340 Å.
H$
H#
6563
%& (Å)
4861
H!
4340
o  Lines fall closer and closer as wavelength decreases.
o  Line separation converges at a particular wavelength, called the series limit.
o  Balmer (1885) found that the wavelength of lines could be written
$ n2 '
" = 3646& 2
)
% n # 4(
where n is an integer >2, and RH is the Rydberg constant. Can also be written:
!
$1
1'
1/ " = R H & 2 # 2 )
%2
n (
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!
$1 1'
1/ " = R H & 2 # 2 ) => " = 6563 Å
%2
3 (
o  If n =3, =>
o  Called H# - first line of Balmer series.
! in Balmer series:
o  Other lines
Name
Transitions
Wavelength (Å)
H#
3-2
6562.8
H$
4-2
4861.3
H!
5-2
4340.5
Highway 6563 to the US
National Solar Observatory
in New Mexico
o  Balmer Series limit occurs when n '(.
% 1 1(
1/ "# = R H ' 2 $ * => "# ~ 3646 Å
& 2 #)
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!
o  Other series of hydrogen:
Lyman
UV
nf = 1, ni)2
Balmer
Visible/UV
nf = 2, ni)3
Paschen
IR
nf = 3, ni)4
Brackett
IR
nf = 4, ni)5
Pfund
IR
nf = 5, ni)6
o  Rydberg showed that all series above could be
reproduced using
$1
1'
1/ " = R H && 2 # 2 ))
%nf
Series Limits
ni (
o  Series limit occurs when ni = !, nf = 1, 2, …
!
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o  Term or Grotrian diagram for
hydrogen.
o  Spectral lines can be considered as
transition between terms.
o  A consequence of atomic energy levels,
is that transitions can only occur
between certain terms. Called a
selection rule. Selection rule for
hydrogen: *n = 1, 2, 3, …
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o 
Simplest atomic system, consisting of single electron-proton pair.
o 
First model put forward by Bohr in 1913. He postulated that:
1.  Electron moves in circular orbit about proton under Coulomb attraction.
2.  Only possible for electron to orbits for which angular momentum is quantised,
ie., L = mvr = n! n = 1, 2, 3, …
3.  Total energy (KE + V) of electron in orbit remains constant.
!
4.  Quantized radiation is emitted/absorbed if an electron moves changes its orbit.
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o  Consider atom consisting of a nucleus of charge +Ze and mass M, and an electron
on charge -e and mass m. Assume M>>m so nucleus remains at fixed position in
space.
1 Ze 2
v2
=
m
4 "#0 r 2
r
o  As Coulomb force is a centripetal, can write
(1)
mvr = n!
o  As angular momentum is quantised (2nd postulate):
n = 1, 2, 3, …
n 2!2
(2)
mZe 2
!
n!
1 Ze 2
=> v =
=
mr 4 "#0 n!
o  Solving for v and substituting into Eqn.!1 => r = 4"#0
o  The total mechanical energy is:
E = 1/2mv 2 + V
" En !
=#
mZ 2e 4
1
2
2
2
n
4
$%
2!
( 0)
n = 1, 2, 3, …
(3)
o  Therefore, quantization of AM leads to quantisation of total energy.
!
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o  Substituting in for constants, Eqn. 3 can be written
and Eqn. 2 can be written
r=
n 2 a0
Z
En =
"13.6Z 2
eV
n2
where a0 = 0.529 Å = “Bohr radius”.
!
o  Eqn. 3 gives a theoretical energy level structure for hydrogen (Z=1):
!
o  For Z = 1 and n = 1, the ground state of hydrogen is: E1 = -13.6 eV
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o  The wavelength of radiation emitted when an electron makes a transition, is (from
4th postulate):
4
&
)
&
)2
1/ " =
or
Ei # E f
1
me
1
1
=(
Z 2 (( 2 # 2 ++
+
3
hc
4
$%
4
$
!
c
n
n
'
0*
' f
i *
%1
1(
1/ " = R# Z 2 '' 2 $ 2 **
& n f ni )
!
(4)
% 1 ( 2 me 4
*
3
& 4 #$0 ) 4 #! c
where R" = '
!
o  Theoretical derivation of Rydberg formula.
!
o  Essential predictions of Bohr model
are contained in Eqns. 3 and 4.
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1.  Calculate the bind energy in eV of the Hydrogen atom using Eqn 3?
o  How does this compare with experiment?
2.  Calculate the velocity of the electron in the ground state of Hydrogen.
o  How does this compare with the speed of light?
o  Is a non-relativistic model justified?
3.  What is a Rydberg atom?
o  Calculate the radius of a electron in an n = 300 shell.
o  How does this compare with the ground state of Hydrogen?
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o  Spectroscopically measured RH does not agree exactly with theoretically derived
R!.
o  But, we assumed that M>>m => nucleus fixed. In reality, electron and proton
move about common centre of mass. Must use electron’s reduced mass (µ):
µ=
mM
m+ M
o  As m only appears in R!, must replace by:
!
RM =
M
µ
R" = R"
m+ M
m
o  It is found spectroscopically that RM = RH to within three parts in 100,000.
!
o  Therefore, different isotopes of same element have slightly different spectral lines.
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o  Consider 1H (hydrogen) and 2H (deuterium):
1
= 109677.584
1+ m / M H
1
RD = R"
= 109707.419
1+ m / M D
R H = R"
cm-1
cm-1
o  Using Eqn. 4, the wavelength difference is
therefore:
!
"# = # H $ #D = # H (1$ #D / # H )
= # H (1$ R H /RD )
o  Called an isotope shift.
!
o  H$ and D$ are separated by about 1Å.
o  Intensity of D line is proportional to fraction of
D in the sample.
Balmer line of H and D
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o  Bohr model works well for H and H-like atoms (e.g., 4He+, 7Li2+, 7Be3+, etc).
o  Spectrum of 4He+ is almost identical to H, but just offset by a factor of four (Z2).
o  For He+, Fowler found the following
in stellar spectra:
Z=1
H
0
$1
1'
1/ " = 4R He & 2 # 2 )
%3 n (
o  See Fig. 8.7 in Haken & Wolf.
!
13.6 eV
20
Z=2
He+
n
2
Z=3
Li2+
n
3
2
1
n
4
3
2
40
Energy
(eV)
60
54.4 eV
1
80
100
1
120
122.5 eV
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Z=1
H
o  Hydrogenic or hydrogen-like ions:
o 
o 
o 
o 
He+ (Z=2)
Li2+ (Z=3)
Be3+ (Z=4)
…
0
Hydrogenic
isoelectronic
sequences
20
13.6 eV
n
2
Z=2
He+
1
n
3
2
Z=3
Li2+
n
4
3
2
Energy
(eV)
o  From Bohr model, the ionization energy is:
40
60
54.4 eV
1
80
100
E1 = -13.59
Z2
eV
120
1
122.5 eV
o  Ionization potential therefore increases rapidly with Z.
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o  We also find that the orbital radius and velocity are quantised:
rn =
n2 m
a0
Z µ
and
vn = "
Z
c
n
o  Bohr radius (a0) and fine structure constant (#) are fundamental constants:
!
e2
4 "#0 ! 2 !
"
=
a0 =
and
4 #$0 !c
me 2
!
a0 =
o  Constants are related by
mc"
!
!
o  With Rydberg constant, define gross atomic characteristics of the atom.
! energy
Rydberg
RH
13.6 eV
Bohr radius
a0
5.26x10-11 m
Fine structure constant
!
1/137.04
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o 
Positronium
o  electron (e-) and positron (e+) enter a short-lived bound state, before they annihilate each
other with the emission of two !-rays (discovered in 1949).
o  Parapositronium (S=0) has a lifetime of ~1.25 x 10-10 s. Orthopositronium (S=1) has
lifetime of ~1.4 x 10-7 s.
o  Energy levels proportional to reduced mass => energy levels half of hydrogen.
o 
Muonium:
o  Replace proton in H atom with a µ meson (a “muon”).
o  Bound state has a lifetime of ~2.2 x 10-6 s.
o  According to Bohr’s theory (Eqn. 3), the binding energy is 13.5 eV.
o  From Eqn. 4, n = 1 to n = 2 transition produces a photon of 10.15 eV.
o 
Antihydrogen:
o  Consists of a positron bound to an antiproton - first observed in 1996 at CERN.
o  Antimatter should behave like ordinary matter according to QM.
o  Have not been investigated spectroscopically … yet.
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o  Bohr model was a major step toward understanding the quantum theory of the atom
- not in fact a correct description of the nature of electron orbits.
o  Some of the shortcomings of the model are:
1.  Fails describe why certain spectral lines are brighter than others => no
mechanism for calculating transition probabilities.
2.  Violates the uncertainty principal which dictates that position and momentum
cannot be simultaneously determined.
o  Bohr model gives a basic conceptual model of electrons orbits and energies. The
precise details can only be solved using the Schrödinger equation.
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o  From the Bohr model, the linear momentum of the electron is
$ Ze 2 ' n!
p = mv = m&
)=
% 4 "#0 n! ( r
o  However, know from Hiesenberg Uncertainty Principle, that
!
!
~
"x r
o  Comparing the two Eqns. above => p ~ n*p
!
"p ~
!
o  This shows that the magnitude
of p is undefined except when n is large.
o  Bohr model only valid when we approach the classical limit at large n.
o  Must therefore use full quantum mechanical treatment to model electron in H atom.
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o  Transitions actually depend on more than a
single quantum number (i.e., more than n).
o  Quantum mechanics leads to introduction on
four quntum numbers.
o  Principal quantum number: n
o  Azimuthal quantum number: l
o  Magnetic quantum number: ml
o  Spin quantum number: s
o  Selection rules must also be modified.
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Energy scale
Energy (eV)
Effects
Gross structure
1-10
electron-nuclear
attraction
Electron kinetic energy
Electron-electron
repulsion
Fine structure
0.001 - 0.01
Spin-orbit interaction
Relativistic corrections
Hyperfine structure
10-6 - 10-5
Nuclear interactions
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