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Chapter 28:Atomic Physics
Homework : Read and understand the lecture note.
Atomic Spectra
 Emission and absorption spectrum
• Atoms emit and absorb light of specific wavelengths. Atoms from
different elements have different wavelengths of emitted and absorbed
light. These specific wavelengths are the same for
both emitted and absorbed
light.
• In 1885, Balmer found a
formula that described
these wavelengths for
hydrogen atom.
 1 1 
 RH  2  2  (n  3,4,5,...)

2 n 
RH  1.0973732 107 m -1 Rydberg constant
1
(n=3,=656.3 nm) etc.
Bohr Theory of Hydrogen (1913)
 Basic assumptions of Bohr theory
1. The electron moves in circular orbits
centripetal force
about the proton under the influence
of the Coulomb force of attraction.
2. Only certain electron orbits are stable.
These are orbits in which the hydrogen
atom does not emit energy in form of
EM radiation. Hence the total energy
of the atom is constant and classical
mechanics can be used to describe
the electron’s motion.
3. Radiation is emitted by the hydrogen atom when electron jumps
from a more energetic initial state to a less energetic state.
The frequency f of the radiation emitted in the jump is related to
the change in the atom’s energy and is independent of the
frequency of the electron’s orbital motion: Ei  E f  hf
4. The size of the allowed electron orbits is determined by a condition
imposed on the electron’s orbital angular momentum:
mevr  n (n  1,2,3,...)
Bohr Theory of Hydrogen (1913)
 Consequences of Bohr theory
• The electrical potential energy of the atom:
q1q2
(e)e
e2
PE  ke
 ke
 ke
r
r
r
• The total energy of the atom assuming that the nucleus is at rest:
1
e2
2
E  KE  PE  me v  ke
2
r
• From Newton’s 2nd law applied to the electron:
e2
v2
ke 2  me
r
r
ke e 2
1
2
me v 
2
2r
• From the 4th assumption and this :
n 2  ke e 2
2
v  2 2 
me r
me r
Bohr’s radius
2
a0 
 0.0529 mm
mke e 2
• Energy levels of electron :
E
ke e
2r
2
The negative sign
indicates the electron is bound to
the proton
n 2 2
rn 
(n  1,2,3,...)
2
me ke e
radii of allowed
electron orbits
rn  n 2 a0  n 2 (0.0529 nm) (n  1,2,3,...)
me ke2e 4  1 
13.6
En  


eV (n  1,2,3,...)


2
2
2
2  n 
n
Bohr Theory of Hydrogen (1913)
 Consequences of Bohr theory (cont’d)
• Ionization energy :
The upper most level corresponds to
E=0 and n=
and the energy needed
to remove the electron completely from
the atom (ionization energy = E1).
• Ground state :
The lowest energy state (n=1) is called
the ground state.
• Emitted/absorbed photon :
From the 3rd postulate, in the transition of
the electron from an orbit with principal
quantum number ni to another with nf, it
emits a photon of frequency f given by:
f 
Ei  E f
h
me ke2 e 4  1
1 

 2
3  2
4  n f ni 
Bohr Theory of Hydrogen (1913)
 Consequences of Bohr theory (cont’d)
• Wavelengths of emitted/absorbed photons
Since f=c,
 1

f me ke2 e 4  1
1 
1
 
 2  RH  2  2 
3  2
n

 c 4c  n f ni 
 f ni 
1
me ke2 e 4
RH 
4c 3
• Named transitions :
- The Lyman series nf=1, ni=2,3,4,…
- The Balmer series nf=2, ni=3,4,5,…
- The Paschen series nf=3, ni=4,5,6,…
 Bohr’s correspondence principle
• Quantum mechanics is in agreement with classical physics when
the energy differences between quantized levels are very small.
Modification of Bohr Theory
 Success of Bohr theory
• It explains the Balmer series and other series.
• It predicts correctly a value for the Rydberg constant.
• It gives an expression for the radius of the atom.
• It predicts the energy levels of hydrogen.
This theory gives a model of what the atom looks like and how it
behaves. With some refinements and modifications, it can be used
as a model for other atoms than hydrogen atom.
 Extension of Bohr theory to hydrogen-like atoms
• A hydrogen-like atom contains only one electron : He-, Li2-, Be3- etc.
• To extend the theory to hydrogen-like atoms, replace e2 with Ze2
where Z is the atomic number of the element.
me ke2 Z 2e 4  1 
En  
 
2 2  n 2 
(n  1,2,3,...)
me ke2 Z 2 e 4  1
1 



4c 3  n 2f ni2 
1
Modification of Bohr Theory
 Sommerfeld’s extention of Bohr theory
• Sommerfeld extended the Bohr theory to include elliptical orbits.
• Sommerfeld model introduced, in addition to the principal quantum
number n, a new quantum number called orbital quantum number l ,
where the value of l ranges from 0 to n-1 in integer step.
For a given n, l =0,1,…,n-1 : n=2->l =0,1….
• An electron in any one of the allowed energy states of a hydrogen
atom may move in any one of a number of orbits corresponding to
different l.
• All states with the same principal quantum
number n are said to form a shell, which is
identified by the letter K,L,M,…corresponding
to n = 1,2,3,…
• The letters s, p, d, f, g,… are used to
designate the states with l = 0,1,2,3,4,…
Modification of Bohr Theory
 Orbital magnetic quantum number ml
• Another quantum number was introduced when it was discovered
that the spectral lines of a gas are split into several closely spaced
lines when the gas is placed in a strong magnetic field (Zeeman effect).
Orbital magnetic quantum number ml :
-l =< ml =< l (2l+1 states)
 Spin magnetic quantum number ms
• Yet another quantum number was introduced
when it was discovered that the spectral lines
of a gas are actually split into two closely
spaced lines (fine structure) even without a strong magnetic field due to
spinning of electrons.
Spin magnetic quantum number ms : ms =-1/2,+1/2
 Number of allowed states with n and l
2(2l+1)
Example : p subshell has 2(2x1+1)=6 possible states
De Broglie Waves and Hydrogen Atom
 Quantization of angular momentum and de Broglie waves
• De Broglie found an interpretation of the Bohr’s angular momentum
quantization in terms of his wave theory.
An electron orbit would be stable (allowed) only if it contained an
integral number of electron wavelengths.
2r  n (n  1,2,3,...)
The de Broglie wavelength of an electron is:
  h /( me v)
2r  nh /( me v)
me vr  n
This argument strengthened the wave theory of matter.
Hydrogen Atom
 Quantum numbers
Spin Magnetic Quantum Number
Electron Clouds
 Wave function and electron clouds
• The solution of the wave equation yields a wave function Y that depends
on the quantum number n, l, and ml.
• The quantity |Y|2 DVP gives a probability of
finding the electron in a small volume DVP
around a point P.
• The maximum of the probability coincides
with the Bohr radius.
Quantum theory predicts that the electron
is not located at a fixed point.
Probability per unit length of finding
the electron at a distance r from the
nucleus for 1s state of hydrogen atom
Exclusion Principle and Periodic Table
 Pauli exclusion principle (1925)
No two electrons in an atom can ever have the same set of values
for the set of quantum numbers n, l, ml, and ms.
• The Pauli exclusion principle explains the electronic structure of
complex atoms as a succession of filled levels with different quantum
numbers increasing in energy, where the outermost electrons are
primarily responsible for the chemical properties of the element.
 General rule for the order that electrons fill a subshell
• Once one subshell is filled, the next electron goes into the vacant
subshell that is the lowest in energy.
• If the atom were not in the lowest energy state available, it would
radiate energy until it reached that state.
• A subshell is filled when it contains 2(2l+1) electrons.
For hydrogen atom at the ground state : (n,l, ml, ms) = (1,0,0,1/2)
or (1,0,0,-1/2)
1 electron 1s1
Exclusion Principle and Periodic Table
 General rule for the order that electrons fill a subshell (cont’d)
For helium atom at the ground state : (n,l, ml, ms) = (1,0,0,1/2)
2 electrons 1s2
and (1,0,0,-1/2)
For lithium atom at the ground state : two electrons in 1s subshell
one electron in 2s subshell
2 electrons 1s2
2
1
1s 2s
1 electron 2s1
( 2p subshell has higher energy)
Exclusion Principle and Periodic Table
 Periodic table (Mendeleev 1871)
Characteristic X-Rays
 Characteristic x-rays
• The discrete lines in an x-ray spectrum
are called characteristic x-rays.
• When an electron beam bombards
a metal to produce x-rays, knocked-off
electrons leave vacancies at the energy
levels at which they were located.
• Then these vacancies are filled in by
electrons that were at higher energy
states. In such a transition, a photon
is emitted with an energy corresponding
to the difference in energy between the
initial and the final energy of the electron
that has filled a vacated state.
Atomic Transitions
 Stimulated absorption process
• When an atom is hit by a photon
that carries an energy equal to
the energy difference between
two states, an electron at the
lower energy state can move up
to the higher energy state by
absorbing the incident photon.
• This process is called stimulated
absorption process and the higher
energy state is called an excited
state.
• When this happens, there is a
probability that the electron that
has moved up to the excited state
goes back to a lower energy state
by emitting a photon (spontaneous
emission).
Atomic Transitions
 Stimulated absorption process (cont’d)
• When an atom is in the excited
state E2 and a photon with energy
hf=E2-E1 is incident on it, the
incoming photon increases the
probability that the excited atom
will return to the ground state by
emitting a second photon with the
same energy (stimulated emission).
The emitted photon is exactly in
phase with the incident photon.
Laser
 Population inversion
• When an incident photon causes atomic transitions, stimulated
absorption and stimulated emission happen with the equal
probability.
• When light is incident on a system of atoms, there is usually a net
absorption. This is because when the system is in thermal equilibrium
there are many more atoms in the ground state than in excited states.
• If there are more atoms in excited states than in the ground state
(population inversion), a net emission can result.
• A mechanism exists to realize the population inversion: laser.
Laser
 Laser (light amplification by simulated emission of radiation)
• There are three conditions to achieve
laser action:
- The system must be in a state of
population inversion.
- The excited state of the system must
be metastable state (longer lifetime
than otherwise short liftime). Then
stimulated emission occurs before
spontaneous emission.
- The emitted photons must be confined
within the system long enough to allow
them to stimulate further emission from
other excited atoms.