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o  Lecturer: o  Dr. Peter Gallagher

o  Email: o  peter.gallagher@tcd.ie

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

o  Absorption spectra

o  Hydrogen spectrum

o  Balmer Formula

o  Bohr’s Model

o  Chapter 4, Eisberg & Resnick

Bulb

Sun

Na

H

Hg

Cs

Chlorophyll

Diethylthiacarbocyaniodid

Diethylthiadicarbocyaniodid

Absorption spectra

Emission spectra

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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

1 2

3

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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.

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.

E2

E1

E2

E1 E! =h"

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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.

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.

Can only occur if E! = h" = E2 - E1 Also, h"’ = h"

E2

E1

E2

E1

E2

E1

E2

E1

E!’ =h"’

E! =h" E!

’ =h"’

E! =h"

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o  In ~1850s, hydrogen was found to emit lines at 6563, 4861 and 4340 Å.

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

where n is an integer >2, and RH is the Rydberg constant. Can also be written:

!

1/" = RH122#1n2

$

% &

'

( )

6563 4861 4340

H# H$ H!

%& (Å)

!

" = 3646 n2

n2 # 4$

% &

'

( )

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o  If n =3, =>

o  Called H# - first line of Balmer series.

o  Other lines in Balmer series:

o  Balmer Series limit occurs when n '(.

!

1/" = RH122#132

$

% &

'

( ) => " = 6563 Å

Highway 6563 to the US National Solar Observatory

in New Mexico

!

1/"# = RH122$1#

%

& '

(

) * => "# ~ 3646Å

Name Transitions Wavelength (Å)

H# 3 - 2 6562.8

H$ 4 - 2 4861.3

H! 5 - 2 4340.5

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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

Series Limits

o  Other series of hydrogen:

o  Rydberg showed that all series above could be reproduced using

o  Series limit occurs when ni = !, nf = 1, 2, …

!

1/" = RH1n f2 #

1ni2

$

% & &

'

( ) )

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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., 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.

!

L = mvr = n!

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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.

o  As Coulomb force is a centripetal, can write (1)

o  As angular momentum is quantised (2nd postulate): n = 1, 2, 3, …

o  Solving for v and substituting into Eqn. 1 => (2)

o  The total mechanical energy is:

o  Therefore, quantization of AM leads to quantisation of total energy.

!

14"#0

Ze2

r2= m v 2

r

!

mvr = n!

!

E =1/2mv 2 +V

"En = #mZ 2e4

4$%0( )22!21n2 n = 1, 2, 3, …

!

r = 4"#0n2!2

mZe2

=> v =n!mr

=14"#0

Ze2

n!

(3)

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o  Substituting in for constants, Eqn. 3 can be written eV

and Eqn. 2 can be written 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

!

En ="13.6Z 2

n2

!

r =n2a0Z

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o  The wavelength of radiation emitted when an electron makes a transition, is (from 4th postulate):

or (4)

where

o  Theoretical derivation of Rydberg formula.

o  Essential predictions of Bohr model are contained in Eqns. 3 and 4.

!

1/" =Ei # E f

hc=

14$%0

&

' (

)

* +

2me4

4$!3cZ 2 1

n f2 #

1ni2

&

' ( (

)

* + +

!

1/" = R#Z2 1n f2 $

1ni2

%

& ' '

(

) * *

!

R" =14#$0

%

& '

(

) *

2me4

4#!3c

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):

o  As m only appears in R!, must replace by:

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.

!

µ =mMm + M

!

RM =M

m + MR" =

µmR"

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o  Consider 1H (hydrogen) and 2H (deuterium):

o  Using Eqn. 4, the wavelength difference is therefore:

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.

!

RH = R"1

1+ m /MH

=109677.584

RD = R"1

1+ m /MD

=109707.419

cm-1

cm-1

Balmer line of H and D

!

"# = #H $ #D = #H (1$ #D /#H )= #H (1$ RH /RD )

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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:

o  See Fig. 8.7 in Haken & Wolf. !

1/" = 4RHe132#1n2

$

% &

'

( )

0

20

40

60

80

100

120

Energy (eV)

1

n 2

1

2 3

Z=1 H

Z=2 He+

Z=3 Li2+

n n

1

2

3 4

13.6 eV

54.4 eV

122.5 eV

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o  Hydrogenic or hydrogen-like ions:

o  He+ (Z=2) o  Li2+ (Z=3) o  Be3+ (Z=4) o  …

o  From Bohr model, the ionization energy is:

E1 = -13.59 Z2 eV

o  Ionization potential therefore increases rapidly with Z.

Hydrogenic isoelectronic

sequences Energy (eV)

1

n 2

1

2 3

Z=1 H

Z=2 He+

Z=3 Li2+

n n

1

2

3 4

13.6 eV

54.4 eV

122.5 eV

0

20

40

60

80

100

120

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o  We also find that the orbital radius and velocity are quantised: and

o  Bohr radius (a0) and fine structure constant (#) are fundamental constants:

and

o  Constants are related by

o  With Rydberg constant, define gross atomic characteristics of the atom.

!

rn =n2

Zmµa0

!

a0 =4"#0!

2

me2

Rydberg energy RH 13.6 eV

Bohr radius a0 5.26x10-11 m

Fine structure constant ! 1/137.04

!

vn ="Znc

!

" =e2

4#$0!c

!

a0 =!

mc"

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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

o  However, know from Hiesenberg Uncertainty Principle, that

o  Comparing the two Eqns. above => p ~ n*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.

!

p = mv = m Ze2

4"#0n!$

% &

'

( ) =

n!r

!

"p ~ !"x

~ !r

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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