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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 Bohrs 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 .

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!

%& ()

!

" = 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

Highway 6563 to the US National Solar Observatory

in New Mexico

!

1/"# = RH122$1#

%

& '

(

) * => "# ~ 3646

Name Transitions Wavelength ()

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!2

1n2 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 = 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 fhc

=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 electrons reduced mass ():

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.

!

=mMm + M

!

RM =M

m + MR" =

mR"

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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 1.

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 en