Download Document

UNIT-III
ATOM WITH ONE ELECTRON
Bohr’s Model
Niels Bohr suggested that the problem about
hydrogen spectrum can be solved if we can make
some assumptions.
 According to classical theory, the frequency of the
electromagnetic waves emitted by a revolving
electron is equal to the frequency of revolution.
 As the electrons radiate energy, their angular
velocities would change continuously and they
would emit a continuous spectrum against line
spectrum actually observed.
 So, Bohr concluded that even if electromagnetic
theory successfully explained the macroscopic
phenomenon, it could not be applied to explain
microscopic phenomenon, that in atomic scale.

An unsatisfactory model
for the hydrogen atom
According to classical
physics, light should be
emitted as the electron
circles the nucleus. A
loss of energy would
cause the electron to be
drawn closer to the
nucleus and eventually
spiral into it.
Bohr's Postulates
Every atom consists of nucleus and suitable number of
electrons revolved around the nucleus in circular orbits.
 The force of attraction between the electron and the nucleus
provide necessary centripetal force for the circular motion.
 Electrons revolved only in certain non-radiating orbits called
stationery orbits for which the total angular momentum is an
integral multiple of h/2Л,
where h is planck's constant and L is the Angular momentum
of the revolving electrons
L=mvr=nh/2Л
 Radiation occurs when an electron
jumps from higher orbit to a lower
orbit
i.e., E2 - E1 = hf,
where f is frequency of radiation

Bohr’s Model of the Hydrogen Atom
Ze 2
v2
m
2
40 r
r
1
for L  mvr  n, n  1,2,3...
n 2
n 2 2
 Ze  40 mv r  40 mr (
)  40
mr
mr
n 2 2
 r  40
mZe2
n
1 Ze 2
 v

mr 40 n
2
2
Potential energy : V   

r
Ze 2
Ze 2
dr  
2
40 r
40 r
2
1
Ze
Kinetic energy : K  mv 2 
2
40 2r
Ze 2
mZ 2 e 4
1
Total energy : E  K  V  
 K  E  
2
( 40 ) 2r
( 40 ) 2 2 2 n 2
 
Ei  E f
h
1
2 4
mZ
e
1
1
2
(
)
( 2  2)
3
40
4
nf
ni
 
1



c
4
2
me
Z
1
1
2
(
)
( 2  2)
3
40
4 c n f
ni
1
1
1
 R Z ( 2  2 )
nf
ni
2
for R
4
me
(
)2
 RH
3
40
4 c
1
For Hydrogen atom, Z  1
r  n2
The stationary orbits are not equally spaced
On substituti ng various values
r  5.29 x 10 m
This is called the Bohr radius.
-11
 1 1 
Also,   RH  2  2 
 n1 n2 
 energy levels of the H atom  1/n 2
Hydrogen Atom Spectrum

Line spectra in other spectral regions also were observed:
Lyman series
ultraviolet
Balmer
visible
Paschen, Brackett, Pfund
infrared

wavelengths of emission is given by


1  R   1  1 
 m2

n2 



where m = 1, 2, 3,… and n = 2, 3, …(always at least m + 1)
Longest wavelength observed when n = m + 1.
Shortest wavelength observed when n = .
 1 1 
 RH  2  2 
n


 f ni 
For Lyman series, n f  1, ni  2,3, 4,...
1
For Balmer series, n f  2, ni  3, 4,5...
For Paschen series, n f  3, ni  4,5, 6...
For Brackett series, n f  4, ni  5, 6, 7...
For Pfund series, n f  5, ni 9 6, 7,8...
Correction for finite nuclear mass
mM
the reduced mass of the system :  
mM
 L  vr  n
1
1
2
   RM Z ( 2  2 )
n f ni
M

M
RM 
R  R , RM  R , as

mM
m
m
M
For hydrogen atom :
 1836
m
1
 RM 
R
2000
Bohr’s correspondence principle



The predictions of the quantum theory for the
behaviour of any physical system must
correspond to the prediction of classical physics
in the limit in which the quantum number
specifying the state of the system becomes very
large:
lim
n   quantum
theory = classical theory
At large n limit, the Bohr model must reduce to
a “classical atom” which obeys classical theory
The classical radiation frequency of  0 in Bohr orbit n is
v
1 2 me4 2
0 
(
)
3
2
2r
40 4 n
Bohr' s radiation theory for ni  n f  1
4
4
me
1
1
1
me
2n  1
2
2
 (
)
[
 2](
)
[
]
3
2
3
2 2
40 4 (n  1)
n
40 4 (n  1) n
1
4
me
2
2
n     (
)
 0
3
2
40 4 n
1
Bohr’s Quantum Condition

pq dq  nq h   Ld  L 
2
0
d  2L
nh
 n
2
nh
 L  mvr  pr  n 
2
 2L  nh  L 
h
h
for de Broglie wavelengt h     p 
p

h
nh
 r
 2r  n , n  1,2,3...

2
Modification of the Bohr Model

Successes of the
Bohr Theory
◦ Model of the Atom
◦ Explained Atomic
Spectra
◦ Predicted Rydberg
Constant
◦ Gave expression for the
radius of an atom
◦ Predicted the energy
levels of the hydrogen
atom

Modifications to the
Bohr Model
◦ Elliptical orbits
◦ Orbital quantum
number
◦ Orbital magnetic
quantum number
◦ Spin magnetic
quantum number
Sommerfeld’s model

Fine structure: a splitting
of spectral lines due to
spin-orbit interaction
Sommerfeld’s explanation
for an elliptical orbit:
 Ld  n h  L2  n h  L  n / , n  1,2,3..
 p dr  n h  L(a / b  1)  n h, n  0,1,2,...
r
r
r
r
40 n 2  2
n
1 2 Z 2 e 4
a
,b  a
 E  (
)
2
Ze
n
40 2n 2  2
 : reduced mass
nr : radial quantum number
n : azimuthal quantum number
n  nr  n principal quantum number
(1) n  n circular orbit
(2) n  nr elliptical orbit
 For the same n, but different nr and n energy is degenerate .
Frank-Hertz experiment



Shows the excitation of atoms to discrete energy levels
Mercury vapour is bombarded with electron accelerated
under the potential V (between the grid and the filament)
A small potential V0 between the grid and collecting plate
prevents electrons having energies less than a certain
minimum from contributing to the current measured by
ammeter
The electrons that arrive at the anode
peaks at equal voltage intervals of 4.9 V
As V increases,
the current
measured also
increases
 The measured
current drops at
multiples of a
critical potential
 V = 4.9 V, 9.8V,
14.7V

Interpretation






As a result of inelastic collisions between the accelerated
electrons of KE 4.9 eV with the the Hg atom, the Hg atoms
are excited to an energy level above its ground state
At this critical point, the energy of the accelerating electron
equals to that of the energy gap between the ground state and
the excited state
This is a resonance phenomena, hence current increases
abruptly
After inelastically exciting the atom, the original (the
bombarding) electron move off with too little energy to
overcome the small retarding potential and reach the plate
As the accelerating potential is raised further, the plate
current again increases, since the electrons now have enough
energy to reach the plate
Eventually another sharp drop (at 9.8 V) in the current
occurs because, again, the electron has collected just the
same energy to excite the same energy level in the other
atoms
If bombared by electron with Ke = 4.9 eV excitation
of the Hg atom will occur. This is a resonance
phenomena
First excitation
Hg
energy of Hg
atom DE1 = 4.9eV
Third
resonance
initiated
Ke= 4.9eV
Ke= 0 after first
resonance
Ke= 0
Ke reaches 4.9 eV again
here
Hg
Ke reaches 4.9 eV again
second resonance
initiated
K = 0 after second
Electron continue to
be accelerated by the
First
external potential until
resonance
Plate C
the second resonance
at 4.9 eV occurs
electron is
accelerated
under the
external
potential
V = 14.7V
e
resonance
Hg
Hg
Electron continue to Plate
be accelerated by the
external potential until
the next (third)
resonance occurs
P
The higher critical potentials result from
two or more inelastic collisions and are
multiple of the lowest (4.9 V)
 The excited mercury atom will then deexcite by radiating out a photon of
exactly the energy (4.9 eV) which is also
detected in the Frank-Hertz experiment
 The critical potential verifies the
existence of atomic levels

Quantum Numbers
Schrödinger’s equation requires 3 quantum numbers:
◦ Principal Quantum Number, n. This is the
same as Bohr’s n. As n becomes larger, the atom
becomes larger and the electron is further from
the nucleus. ( n = 1 , 2 , 3 , 4 , …. )
◦ Angular Momentum Quantum Number, l.
This quantum number depends on the value of n.
The values of l begin at 0 and increase to (n - 1).
We usually use letters for l (s, p, d and f for l = 0,
1, 2, and 3). Usually we refer to the s, p, d and forbitals.
◦ Magnetic Quantum Number, ml. This
quantum number depends on l . The magnetic
quantum number has integral values between -l
and +l . Magnetic quantum numbers give the 3D
orientation of each orbital.
Quantum
Number
Symbol
Values
Description
Principal
n
1,2,3,4,…
Size & Energy of orbital
Angular
Momentu
m
l
0,1,2,…(n-1)
for each n
Shape of orbital
Magnetic
ml
-l…,0,…+ l
for each l
Relative orientation of orbitals within
same l
Spin
ms
+1/2 or –1/2
Spin up or Spin down
Angular Momentum Quantum
Number
Name of Orbital
0
s (sharp)
1
p (principal)
2
d (diffuse)
3
f (fundamental)
4
g
Orbital Magnetic Moment
gl  B L
l  

e
where  B 
 0.927 10  23 Am  2 (Bohr Magneton)
2m
and g l  1 (orbital factor )
Electron Spin
1925: S.A Goutsmit and G.E. Uhlenbeck suggested that an
electron has an intrinsic angular momentum (i.e.
magnetic moment) called its spin.
 The extra magnetic moment μs associated with angular
momentum S accounts for the deflection in SGE.
 Two equally spaced lined observed in SGE shows that
electron has two orientations with respect to magnetic
field.

The Stern-Gerlach Experiment

The Stern-Gerlach Experiment (SGE) is
performed in 1921, to see if electron has an
intrinsic magnetic moment.
A beam of hot (neutral) Silver (47Ag)
atoms was used.
 The beam is passed through an
inhomogeneous magnetic field along z
axis. This field would interact with the
magnetic dipole moment of the atom, if
any, and deflect it.
 Finally, the beam strikes a photographic
plate to measure,if any, deflection.


Why Neutral Silver atom

Why inhomogenous magnetic Field
◦ No Lorentz force (F = qv x B) acts on a neutral atom,
since the total charge (q) of the atom is zero.
◦ Only the magnetic moment of the atom interacts with
the external magnetic field.
◦ Electronic configuration:
1s2 2s2 2p6 3s2 3p6 3d10 4s1 4p6 4d10 5s1
So, a neutral Ag atom has zero total orbital
momentum.
◦ Therefore, if the electron at 5s orbital has a magnetic
moment, one can measure it.
◦ In a homogeneous field, each magnetic moment
experience only a torque and no deflecting force.
◦ An inhomogeneous field produces a deflecting force
on any magnetic moments that are present in the
beam.

In the experiment, they saw a deflection on the photographic plate.
Since atom has zero total magnetic moment, the magnetic
interaction producing the deflection should come from another
type of magnetic field. So electron’s (at 5s orbital) acted like a bar
magnet.

If the electrons were like ordinary magnets with random
orientations, they would show a continues distribution of pats. The
photographic plate in the SGE would have shown a continues
distribution of impact positions.

However, in the experiment, it was found that the beam pattern on
the photographic plate had split into two distinct parts.
Atoms were deflected either up or down by a constant
amount, in roughly equal numbers.

z component of the electron’s spin is quantized.
Electron Spin
Orbital motion of electrons, is specified by quantum number l.
Along the magnetic field, l can have 2l+1 discrete values.
L  l (l  1) 
l  0,1,2,,n 1
Lz  ml 
ml  l ,l  1, ,(l  1),l
Similar to orbital angular momentum L, the spin vector S is
quantized both in magnitude and direction, and can be
specified by spin quantum number s.
We have two orientations:
2 = 2s+1  s = 1/2
S
3
s(s  1)  1/2(1/2  1) 

2
The component Sz along z axis:
S z  ms 
ms  1 / 2
(spin up)
ms  1 / 2
(spin down)
It is found that intrinsic magnetic moment (μs) and angular
momentum (S) vectors are proportional to each other:
e
μs  gs
S
2m
where gs is called gyromagnetic ratio.
For the electron, gs =
2.0023
Spin Orbit Coupling

Interaction of the spin of
an electron with the
magnetic field generated
by its own orbital
motion
Larmor Precession
When a magnetic moment m is placed in a M.F.(B), it
experiences a torque.
 For a static magnetic moment or a classical current loop,
this torque tends to line up the magnetic moment with the
magnetic field B, so this represents its lowest energy
configuration.
 But if the magnetic moment arises from the motion of an
electron in orbit around a nucleus, the magnetic moment
is proportional to the angular momentum of the electron.
 The torque exerted then produces a change in angular
momentum which is perpendicular to that angular
momentum, causing the magnetic moment to precess
around the direction of the magnetic field rather than
settle down in the direction of the magnetic field. This is
called Larmor precession.


The precession
angular velocity
called Larmor
frequency is
ZEEMAN EFFECT
What is the Zeeman Effect




In absence of magnetic fields, electrons emit photons
of certain energy in atomic transitions
In presence of magnetic fields, atom forms a dipole,
wants to align itself with B-Field. So it has a potential
energy of orientation
This energy adds to the energy of atomic transitions,
changes the energy of the photon emitted.
Different energy = Different Wavelength. We can
observe and measure the shifts in wavelength
Two Different Types




Normal and
Anomalous
Zeeman effect
Magnetic dipole
is sum of orbital
and spin
If spins cancel,
normal Zeeman
effect
If not,
anomalous
Zeeman effect
Normal Zeeman effect

Observed in atoms with no spin.

Total spin of an N-electron atom is
N
S   si
i 1

Filled shells have no net spin, so only consider valence electrons. Since
electrons have spin 1/2, not possible to obtain S = 0 from atoms with odd
number of valence electrons.

Even number of electrons can produce S = 0 state (e.g., for two valence
electrons, S = 0 or 1).

All ground states of Group II (divalent atoms) have ns2 configurations
=> always have S = 0 as two electrons align with their spins antiparallel.
B ˆ
ˆ  L


Magnetic moment of an atom with no spin will be due entirely to orbital
motion.


Interaction energy between magnetic moment
and a uniform magnetic field is
DE    B

Assume B is only in the z- direction. Hence,
interaction energy of the atom is
DE  z Bz  B Bz ml
where ml is the orbital
magnetic quantum
number.
This equation implies
that B splits the
degeneracy of the
ml states evenly.
Normal Zeeman effect transitions
Selections rules for ml:
Dml = 0, ±1.
 Consider transitions between l=0 and l=1 atomic
levels. Allowed transition frequencies are
therefore,

h  h 0   B B z
h  h 0
h  h 0   B B z

Dml  1
Dml  0
Dml  1
Emitted photons also have a
polarization, depending on
which transition they result from.

Anomalous Zeeman effect

Occurs in atoms with non-zero spin => atoms with odd
number of electrons.

In LS-coupling, the spin-orbit interaction couples the spin
and orbital angular momenta to give a total angular
momentum according to
J  LS

In an applied B-field, J precesses
about B at the Larmor frequency.

L and S precess more rapidly
about J to due to spin-orbit
interaction.
Spin-orbit effect therefore stronger.

Interaction energy of atom is equal to sum
of interactions of spin and orbital
magnetic moments with B-field
DE    z Bz
 (  zorbital   zspin ) Bz
B
 Lz  gs S z

Bz
where gs= 2, and the < … > is the
expectation value.
The normal Zeeman effect is obtained by
setting ˆ
Lˆ z  ml .
Sz  0

In the case of precessing atomic magnetic
system neither Sz nor Lz are constant. Only
Jˆz  m j
is well defined.

Must
 therefore project L and S onto J and
project onto z-axis
ˆ
ˆ 
J
J
B
ˆ  | Lˆ | cos1  2 | Sˆ | cos2
| Jˆ |
| Jˆ |

The angles 1 and 2 can be calculated from the
scalar products of the respective vectors
Lˆ  Jˆ | L || J | cos 1
Sˆ  Jˆ | S || J | cos  2
which implies that
Lˆ  Jˆ
Sˆ  Jˆ  B ˆ
ˆ  
2
J
2
2
ˆ
ˆ

|J|
|J|

ˆ
Sˆ  Jˆ  L
Sˆ  Sˆ  ( Jˆ  Lˆ )  ( Jˆ  Lˆ )  Jˆ  Jˆ  Lˆ  Lˆ  2 Lˆ  Jˆ
Now, using
Lˆ  Jˆ  (Jˆ  Jˆ  Lˆ  Lˆ  Sˆ  Sˆ ) /2

so that
2
ˆ
ˆ
j(
j

1)

l(l

1)

s(s

1)
/2
L J



2
j( j  1) 2
| Jˆ |
j( j  1)  l(l  1)  s(s  1)


2 j( j  1)


Similarly, Sˆ  Jˆ  ( Jˆ  Jˆ  Sˆ  Sˆ  Lˆ  Lˆ ) /2
and
j( j  1)  s(s  1)  l(l  1)
Sˆ  Jˆ


2
2 j( j  1)
| Jˆ |

We can therefore write
 j( j  1)  l(l  1)  s(s  1)
j( j  1)  s(s  1)  l(l  1)B ˆ

ˆ  

2
 J
2 j( j  1)
2 j( j  1)




This can be written in the form
B ˆ
ˆ  g

J
j
where gJ is the Lande g-factor given by
g j  1

j( j  1)  s(s  1)  l(l  1)
2 j( j  1)

This implies that
z  g j B m j

and interaction energy with the B-field is
DE  z Bz  g j B Bz m j
Mathematical Relation
DE   Bg m' j
b
Energy Shift
Landé g-factor
Bohr Magneton,

b

e
2m
Magnetic Field
g  1
j ' ( j '1)  s' ( s'1)  l ' (l '1)
2 j ' ( j '1)
Quantum number, due
to combination of
orbital and spin
magnetic dipoles

Anomalous Zeeman effect spectra

Selection rules for J and mj
Dj  0,1
Dm j  0,1
Na d-lines produced by 3p  3s
transition