Download R - University of St Andrews

Physics of Atoms
Donatella Cassettari
Email: dc43
Tel: 3109
Room 218 (office)
105 (lab)
Course Plan
•Atomic theory started around ~1900
with J.J. Thomson’s model of the atom
•Succession of atomic models:
Rutherford (1911), Bohr (1913), Bohr-Sommerfeld
“Old” quantum theory
or “semi-classical” theory
•Schroedinger equation for the Hydrogen atom
(1 electron, simplest case)
•Many-electron atoms (Alkali atoms, Helium atom)
•Spin-orbit interaction, spin-spin interaction
• Atoms in external fields (electric, magnetic, e.m.)
• Novel applications:
Laser cooling of atoms, Bose-Einstein condensation (BEC)
Why is atomic physics interesting?
Richard Feynman:
“The understanding of the structure of atoms represents the
greatest success of the theory of quantum mechanics”
•Modern physics is based on the understanding of atoms
•Applications: solid state, chemistry, biophysics, astrophysics,
lasers….
Material for this course:
• Textbook:
Haken and Wolf The Physics of Atoms and Quanta 7th ed (2005)
(only selected chapters)
• These notes!
• Tutorial problems
• Practice with past exams
Models of the atom – a short history
J J Thomson’s model ~ 1900
Known that an atom with atomic number Z contains Z electrons
“Plum-pudding model”: these electrons are embedded in a
continuous spherical distribution of positive charge
amounting in total to +Ze.
e
Diameter of atom
is ~0.1nm
e
e
e
e
e
e
+
This idea though was immediately proved invalid by
Rutherford’s expts in 1911 on the scattering of α-particles
(He nuclei) by atoms:
Measure no. of particles scattered between θ and θ +d θ.
In Thomson’s model, the spread out positive charge can’t
produce a large deflection, and the electrons are too light
to do so.
However, a significant fraction (1/10,000) of the α particles is found to be deflected thru angles θ greater
than 90 degrees, whereas the theory says this fraction
should be negligible.
“It was quite the most incredible event that has ever
happened to me in my life. It was almost as if you fired a
fifteen inch shell at a piece of tissue paper and it came
back and hit you.”
Rutherford Model 1911
Positive charge is concentrated in a very small nucleus. So αparticles can sometimes approach very close to the charge Ze
in the nucleus and the Coulomb force
1
( Ze)(2e)
F=
4πε o
r2
Can be large enough to cause large angle deflections.
Nuclear model of the atom
Detailed calculations now show that there is
a non-negligible probability of large angle
scattering
Stability problem
+
e
e
Because the electrons are in orbit, they are continuously
being accelerated and therefore, classically, should emit
radiation. They should lose energy and spiral into the
nucleus!
Bohr’s postulates
Bohr 1913: circumvented stability problem by
making two postulates:
1. An electron in an atom moves in a circular orbit for
which the angular momentum is an integral multiple of h
2. An electron in one of these orbits is stable. But if it
discontinuously changes its orbit from one where
energy is Ei to one where energy is Ef , energy is
emitted or absorbed in photons satisfying:
Ei − E f = hν
Photon frequency
Analysis based on Bohr postulates
for hydrogen-like atom (1 electron)
Force of
attraction
Ze 2 m0 v 2
F=
=
2
4πε o r
r
Kinetic
energy
2
1
1
1
Ze
KE = m0 v 2 =
2
2 4πε o r
Potential
energy
1
r
e
v
+Ze
(m 0= electron mass)
Ze 2
PE = −
4πε o r
1
Ze 2
Total E = −
4πε o 2r
1
E<0 because it’s
a bound state
1st postulate
eliminate
v from system
m0 vr = nh
1
Ze
m0 v =
4πε o r
2
2
2
2
2
nh
1 Ze
m0 2 2 =
4πε o r
m0 r
2
2
nh
r = 4πε o
2
m0 Ze
1
2 4
m0 Z e
E=−
2
2 2
(4πε o ) 2n h
Energy is quantised
For convenience we define the “Bohr Radius” as:
2
4πε o h
a0 =
= 0.0529nm
2
m0 e
And the energy unit 1 Rydberg (1Ry) as:
4
m0 e
= 13.6eV
2
2
(4πε o ) 2h
2
n
r = a0
Z
and
E= −
Z
2
n2
Ry
n=1
n=2
n=3
“excited states”
“ground state”
a 0 = radius of ground state (n=1) orbit for hydrogen (Z=1)
1 Ry=13.6eV is the ground state (n=1) binding energy for
hydrogen (agrees with experiment).
For larger Z-values, radius of orbit shrinks for given n. Binding
energies get much bigger.
Emission Spectra
Electron makes transition from initial quantum state ni to final
state nf . The frequency ν of the photon emitted satisfies:
2 4
1
m0 Z e
hν = Ei − E f =
2
2
(4πε o )
2h
 1

1
 − 
 n2 n2 
i 
 f
Express this in terms of wavelength
1
1
2 4
m0 Z e
=
2
3
λ (4πε o ) 4πh c
 1

1
 − 
 n2 n2 
i 
 f

 1
1
2

= R∞ Z
−
 n2 n2 
λ
i 
 f
1
1
4
m0 e
7
−1
R∞ =
= 1.0974 × 10 m
2
3
(4πε o ) 4πh c
is the Rydberg constant
There are infinite spectral lines….
Let’s organize them in families.
Family = all transitions with the same final state
Note that for each family there is a series limit for
For the Lyman series, this is at
1
λ
= R∞ = 91nm
ni → ∞
well within the
UV range
The Balmer series lies in the near UV and visible region,
and the others are all in the infrared.
At time of Bohr’s proposal, only Balmer-Paschen series
were known, and the remaining series were therefore
predicted in advance of the discovery (triumph for Bohr
theory).
Absorption spectrum
General formula above also applies to the case where an electron
gains the energy of a suitable photon having energy hν exactly
equal to the difference between initial and final states.
Normally, electron will start off in ground state so only the
Lyman series is observed in absorption spectrum.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Next, we are going to consider two extensions
of the Bohr model:
• Finite nuclear mass
• Elliptical orbits (Bohr-Sommerfeld model)
Finite Nuclear Mass
In the theory above, we assumed that the nucleus is so
massive that it is effectively at rest. But the mass of the
nucleus is finite and the classical model of the atom therefore
envisages that the electron and the nucleus both revolve
around their common centre of mass.
Therefore, any theory should be modified by replacing the
mass of the electron by the “reduced mass” of the system.
m0 M
m0
µ=
=
m0 + M 1 + m0
M
where m0 = electron mass
M = nuclear mass

 1
1
2

= R.Z
−
 n2 n2 
λ
i 
 f
1
4
µe
R =
2
3
(4πε o ) 4πh c
1
µ
R
M
1
=
=
=
m0
R∞ m0 m0 + M 1 + M
R∞
is the limit of R as M tends to infinity
Difference between R and R∞ is small (~ 1 part in 1800)
but shows up in precise measurements of spectral lines.
For hydrogen
7
−1
7
−1
R = 1.0968 × 10 m
Cf.
R∞ = 1.0974 ×10 m
Isotope shift of spectral lines:
It occurs because different isotopes of the same atomic species
have different reduced masses, hence different values of R.
Example:
Positron “atom” =system containing 1 positron (like an e- with a
positive charge) and 1 electron.
How does emission spectrum differ from hydrogen?
Set M=m0 , so
R∞
R∞
R=
=
m0
2
1+
M
1  1
R∞  1
=
−
=
×
λ
2  n 2f ni2  2


1
previous formula
All wavelengths are doubled
Effect on energy and radii of quantum states?
We replace m0 in the formulae above by
m0
µ=
2
Therefore, all energies are halved in magnitude, radii
doubled.
Elliptical orbits (Bohr-Sommerfeld)
When viewed at high resolution, transitions split.
Transitions between any two Bohr energy states involve
several spectral lines. This is known as fine structure.
Explanation: each energy level actually consists of several
distinct states with almost the same energy.
The first theory that justified this was done by Wilson and
Sommerfeld: they conjectured that electron orbits can be
elliptical, of which a circular orbit is a special case.
Each orbit is specified by 2 parameters instead of 1.
Geometrically by semi-major and semi-minor axes a,b, no just
radius r. Therefore we have now two quantum numbers.
e
nθ b
=
n a
a
+Ze
b
nθ = “azimuthal” quantum number
Thus, energy levels turn out to be dependent on two quantum
numbers, but only when one takes relativistic considerations
into account.
Without relativity, we get the same formula for E as before.
Relativistic correction: electrons in very eccentric orbits have
large velocities when they are near the nucleus, so v/c is NOT
negligible.
Each energy level n is split into several sub-levels
corresponding to orbits of different eccentricity (i.e. different nθ )
fine structure
Final remarks on the early atomic models
1) Bohr’s theory works! It generates the correct formula for the
energy levels in a hydrogen atom.
This is because the Bohr postulate: m0 vr = nh inadvertently
makes use of the de Broglie wave associated with an electron.
Supposing the classical orbit is circular, let’s look at that the
associated de Broglie wave following the motion:
in order for the wave to return to its initial value (i.e. we are
requiring that the wave be single valued), we must have
2πr = circumference = nλ
de Broglie wavelength
p=
h
h
λ= =
p m0 v
h
λ
2πrm0 v = nh
m0 vr = nh
The Bohr condition
2) Inadequacies of the Bohr Theory
•Does well to describe hydrogen, but can be extended only to
1-electron atoms, i.e. hydrogen-like, with higher Z values.
•Theory does not explain rate at which transitions occur
between states, i.e. the relative intensities of spectral lines.
•The theory is ad hoc and lacks a satisfying basis.
Superseded by Quantum Mechanics, initiated by de Broglie
(1924) and Schroedinger (1926).