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A T O M I C & M O L E C U L A R S P E C T R O S C O P Y P H
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ATOMIC AND MOLECULAR
SPECTROSCOPY CONTENTS
CHAPTER DESCRIPTION PAGE
NO
PART A NOTES
1
QUANTUM
NUMBER
S
1.1 Principal quantum number 16
1.2 Orbital quantum number or azimuthal quantum number
16
1.3 Orbital magnetic quantum number 17
1.4 Magnetic spin quantum number 17
1.5 Space Quantization of an atom 19
2
MAGNETIC
DIPOLE
MOMENT
S, ELECTRO
N SPIN
AND
VECTOR
ATOM
2.1 Stern-Gerlach Experiment 23
2.2 Various types of coupling 27
2.2.1 L-S coupling 27
2.2.2 j-j coupling 29
2.3 Determination of spectral terms under L-S coupling
31
2.3.1 Atom with one optical
electrons
31
2.3.2 Atoms with two or more non-equivalent optical
electrons
32
2.3.3 Atoms with two or more
equivalent electrons
35
2.4 Calculation of spectroscopic term
for P2
36
2.5 The terms for various electrons configuration are given as
37
2.6 Order of terms and fine structure
levels
37
2.7 Spin-Orbit Interaction 38
2.8 Spectra of Alkali elements 39
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3 ZEENMAN
EFEFECT
3.1 Types of Zeeman effect 42
3.1.1 Normal Zeeman effect 42
3.1.2 Anomalous Zeeman effect 42
3.2 Explanation of Zeeman effect 42
3.3 Quantum theory of normal
Zeeman effect
43
3.4 Explanation of Anomalous Zeeman
effect
45
3.5 Spin-Orbit Correction 48
3.6 Few Example Of Zeeman Effect 48
4 STARK
EFFECT
4.1 Stark effect partially lift the
degeneracy
51
4.2 Hyperfine structure of spectral lines (hfs)
51
4.3 Vector model of atom for hfs 52
4.4 Interaction Energy 52
4.5 Problems 53
5 X-RAYS
5.1 X-rays 57
5.2 Types of X-rays 57
5.2.1 On the basis of energy 57
5.2.2 On the basis of spectrum
shown
57
5.3 Production of X-rays 58
5.3.1 Mechanism of production of
continuous X-rays
58
5.3.2 Mechanism of production of
characteristic X-rays
59
6 THE
BREADT
H OF
SPECTRAL LINES
6.1 Natural Breadth 63
6.2 Doppler effect 64
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7
MOLECULAR
SPECTRO
SCOPY
7.1 Molecular spectra 67
7.1.1 Electronic soectra 67
7.1.2 Vibrational-rotational spectra 67
7.1.3 Pure rotation spectra 67
8 PURE
ROTATIO
NAL
SPECTRA
(FOR I¬–R
REGION)
8.1 Salient features of Pure rotational
spectra
69
8.2The molecule as a rigid rotator 69
8.3 Problems 73
9 VIBRATION
AL
ROTATIO
NAL
SPECTRA
9.1 Salient features 81
9.2 Explanation of vibrational
rotational spectra
81
9.3 Energy in terms of wave number is 82
9.4 Molecule as Anharmonic oscillator 83
9.5 Fine structure of IR bands:
Molecule as vibrating rotator
84
9.5.1 R-Branch 85
9.5.2 P-Branch 86
9.5.3 Q-Branch 86
9.6 Nature of the Raman effect 95
9.7 Quantum theory of Raman effect 95
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10 RAMAN
EFFECT
10.1Nature of Raman Spectran:
Molecular spectra
101
10.2 Intensity of Rayleigh
line>Intensity of stokes
line>Intensity of anti-stokes Raman line
101
10.2.1 Vibrational Raman spectrum 101
10.2.2 Rotational Raman spectrum
selection rule are
102
10.3 Rule of Mutual Exclusion 103
10.4 Use of Raman effect. 105
11 ELECTRONIC
SPECTRA:
FRANCK-CONDONPRIN
CIPLE
11.1 Salient Features of Molecular Electronic spectra
107
11.2 Dissociation Energy and
Dissociation Products
108
11.3 Rotational Fine Structure of Electronic-Vibration Transitions
108
11.4 The Fortrat Diagram 109
11.5 Franck Condon principle 111
11.6 Temporal And Spatial Coherence 112
11.6.1 Coherence 112
11.6.2 Type of Coherence 112
12 LASERS
12.1 Absorption of the radiation 115
12.2 Spontaneous Emission 115
12.3 Stimulated Emission 115
12.4 Requirement of Lasing action 118
12.5 Mode Separation 120
12.6 Different types of lasers 121
12.7 NMR (Nuclear Magnetic
Resonance)
148
SOME IMPORTANT FORMULAE 149
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ATOMIC AND MOLECULAR SPECTROSCOPY
Some important constants:
1. Velocity of light
8
0
13 10 /sec
0c m
2. 0 (Permeability of free space) 74 10 /H m
0B H M
3. Permittivity of free space 12
0 8.86 10 /f m
4.
9 2 2
0
19 10 coulomb / .
4m
5. Planck constant 346.6 10h J s
341.05 102
hJ s
6. 191.6 10l eV Joule
7. Charge on 191.6 10e coulomb
Charge on proton 191.6 10 coulomb
8. Rest mass of 319.1 10e kg
Rest mass of proton 271.6726 10 kg
Rest mass of neutron 271 6749 10. kg
Rest mass of neutron 271.6749 10 kg
9. Gravitational constant 11 2 26.67 10 /G Nm kg
10. Avogadro number 230 6.023 10N molecules/mole
11. Rydberg constant for infinite mass 71.09737 10 /R m
12. Bohr radius
2
00 2
0.53 ha A
me
13. Fine structure constant 1
137
14. 1836.15 1839p n
e e
m m
m m
15. Bohr magneton 249.27 10 /4
B
e
ehJ T
m
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16. Nuclear magneton 275.05 10 /4
N
N
ehJ T
m
Some important discoveries:
1. Atom : John Dalton
2. Proton : Goldstein
3. e : J.J. Thomson
4. Neutron : James Chadwick
5. Nucleus : Rutherford
6. e Positron : J-Anderson
Some known facts about the atom:
1. The most part of the atom is hollow p and n lies inside the nucleus while
the e revolve around nucleus.
2. The atom is neutral while the nucleus is vely charged.
3. Atom is generally represented by
A
z X
Where Z atomic and A mass number
In atom, number of protons number of electrons Z
And number of neutrons A Z
4. Nearly whole mass of the atom (more than 99.9%) is placed at the centre,
called nucleus.
5. Atom is taken to be spherical in shape.
Atomic Models:
1. Thomson’s atomic model.
2. Rutherford’s atomic model.
3. Bohr’s atomic model.
(Only for H-atom and one electron ions/atoms such as positronium atom,
,He Li etc ). According to this model, the electron revolve around the
nucleus in circular orbits.
This model fails to explain the fine structure of the H-atom spectrum and
in case of more than one electron systems.
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Sommerfeld’s atomic model:
Sommerfeld extended the Bohr’s model and make the following two corrections.
(a) He considered that the electron revolve around the nucleus in elliptical
orbits rather than circular orbit.
(b) He also taken into account the relativistic effect.
Bohr’s Atomic Model (For–H–atom):
According to it:
(i) Electron revolve around the nucleus in circular orbit.
(ii) Electron revolve around the nucleus only in those orbits in which its
angular momentum is an integral multiple of ,2
h
i.e. sin90n n n n nnL r p r p r p
2
n n
nhr p
2
n n
nhmv r
where 1,2,3,...n
(Number of orbits i.e. 1st and 2nd etc.)
(c) When electron gets sufficient energy from outside, it jumps to higher
energy state and after some time 810 sec , it jumps to lower energy state
and emit a photon of energy hv which is equal to the difference of energy of
two states i.e.
2 1hv E E E
Energy of photon 2 1hv E E E
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Bohr’s theory of H-atom:
(Energy levels in H-atom):
The required centripetal force to the electron is provided by coulomb force
of attraction between the electron and nucleus. Hence,
C.F. force of attraction (F).
2
2
0
1
4n n
Ze emVn
r r
2
2 2 2
0
1
4nn
m Vn r Ze r m
2
2
0
1
2 4n
nhZe r m
2 2
0
2... 1n
n hr
mZe
This expression show 2
nr n i.e. radius of second orbit is 4 time to that
of first orbit 2 14 .r r
Now,
2
01 2
hr
me [For H-atom z 1]
Bohr’s radius 0 10.53a A r 2 1 04 4r r a
Now, K.E. of electron is
2
21 1
2 2 2n
n
nhmV m
mr
2
2 2
2 2
1
2 4n
n hm
m r
Now putting 2nr from equation, (1) we get
K.E.
2 2 2
2 2 2 2
0
1
2 4
n h mZem
m n h
4 2
2 2
0
1. . ... 2
8
me zK E
h n
And
2
2 2 2
0 0
1 1. .
4 4 /n
Ze e ZeP E
r n h mZe
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4 2
2 2 2
0
1. ... 3
4
me zP E
h n
Now total energy nE K.E. P.E.
4 4 2
2 2 2 2 3 2
0 0
1
8 8
me me hcz
h n ch n
2
2... 4n
R hcZE
n
Where
47
2 3
0
1.0974 10 /8
meR m
ch
For H-atom, Z 1,
2 2
13.6... 5n
R hc eVE
n n
Now, 1 13.6E eV
2 3.4 , 0E eV E eV
Therefore, The ionization energy of H-atom 1 13.6E E eV
& Ionization potential 13.6eV (There is only one I.P.
The Ist excitation energy 2 1 10.2E E eV
There are infinite number of excitation potential for H–atom.
Now, 1 2
2 2
2 2
1 2
,n n
R hcz R hczE E
n h
When electron jumps from higher energy state
2 1n n lower it emits energy hv .
2 1
2 2
2 2
2 1
n n
R hcz R hczhv E E
n n
h v R h
2
2 2
1 2
1 1cz
n nFor H-atom, Z 1
But c
2
1 22 2
1 2
1 1 1R z n n
n n
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For H-atom, Z 1 ... 6
2 2
1 2
1 1 1R
n n
Hydrogen spectrum:
(Series of spectrum):
1. Lyman series: This series is obtained when the transition of electron
occurs (observed in UV region) from 2 12,3,4... 1.n ton
eqn 7 gives
2 2
2
1 1 1
Lyman
Rl n
where 1 2n 1, n 2,3,4...
The minimum wavelength of lymann series is 912A and max. 1215A .
This is the only series observed in absorption spectra.
2. Balmer series: It is observed when the electron jump from
2 13,4,... 2.n ton
2 2
2
1 1 1
Balmer
RZ Z
where 2 3, 4,...n
Where 2n 3,the line observed is H
2n 4,the line observed is .H
Similarly, H ,H ,
Maximum wavelength is of 6563 .H line A
and min. wavelength 3646 .A
This is first series observed for H-atom and is found in visible region.
3. Paschen series: Observed in near IR (Infrared)
4. Brackett-series: Observed in far IR
5. Pfund series: Observed in far IR.
The above series are shown below
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(Fig. Hydrogen atom spectrum)
Ionization potential of H-atom:
2
13.6,nE eV
n 1 13.6 & 0E eV E eV
The ionization potential is defined as the potential which can remove the
electron from H-atom. As there is only one electron in H-atom. Hence there is
one I.P. for H-atom. Thus the I.P. energy of H-atom.
1E E 0 13.6eV 13.6eV
. 13.6I P eV
Excitation Potential: There are infinite number of discrete energy levels in
atom.(Franck- Hertz exp.) The excitation energy is defined as the energy
acquired to make the electron capable to jump to the higher energy state. The
potential. There may be infinite number of excitation potential. The first
excitation energy when electron is excited from
2 2
13.6 13.61 2
2 1
3.4 13.6
10.2
n ton
eV
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Fig. Different excitation of H-atom
Ist excitation potential for H-atom10.2V.
Variation of the Rydberg constant: R
The Rydberg constant for infinitely heavy nucleus is given as
4
2 3
0
...(1)8
meR
ch
And for nucleus of mass M,
4
M 2 3
0
eR ...(2)
8 ch
Where reduced mass,
1
mM m
mm M
M
4
2
08 3M
meR
ch
1
1m
M
increases
1M M
RR M R
m
M
Energy levels of positronium atom:
In positronium atom an electron revolve around positron
e ,antiparticleof electron
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2
2
positronium
n
R hczE
n
In this case,
2
1positronium
R RR
m
M
&
z 1 ...(2)
2 2
6.81 ,
2n
R hcEq gives E eV
n nexactly half of H-atom
Thus the I.P. for positronium atom is 6.8V.
PROBLEMS
1. The lowest series limit wavelength of the Balmer series in H-Spectrum is
3646A . Calculate the Rydberg constant.
Ans. 7 11.097 10R m
2. If the wavelength of the first line of the Lyman series of hydrogen is
6563A , calculate the wavelength of the second line of the series and the
series limit.
Ans. 1025 , 912A A
3. If the wavelength of the Ist line of Balmer series of hydrogen is 6563A ,
calculate the I.P. of the atom. 34 86.6 10 , 3 10 /h J s C m s
Ans. 13.6V.
4. Calculate the I.P. for ,He Li and Positronium atom.
Ans. 54.4 V, 122.4 V, 6.8 V)
5. How many revolutions does an electron in then n 2state of H-atom make
before dropping to then n 1state?
The average life time of an excited state is 810 sec .
3
.
/2 2int :
2 2n
n n
Classical ferqof revolution
nh mr R cvH f
r r n
Ans. 68.2 10
