Get Your Website Here

Sunday, March 22, 2009

INTRODUCTION TO ATOMIC PHYSICS

1. About 2500 years ago, the ancient Greek philosopher Democritus asked the question:
what is the world made of? He conjectured that it is mostly empty, and that the
remainder i

s made of tiny "atoms". By definition these atoms are indivisible.
Then 300 years ago, it was noted by the French chemist Lavoisier that in all chemical
reactions the total mass of rea

ctants before and after a chemical reaction is the same.
He demonstrated that burning wood caused no change in mass. This is the Law of
Conservation of Matter.
• A major increase in understanding came with Dalton (1803) who showed that:
1) Atoms are building blocks of the elements.
2) All atoms of the same element have the same mass.
3) Atoms of different elements are different.
4) Two or more different atoms bond in simple ratios to form compounds.
2. Avogadro made the following hypothesis : "Equal volumes of all gases, under the same
conditions of temperature and pressure, contain equal numbers of molecules". Why?
Because we know that pressure is caused by molecules hitting the sides of the containing
vessel. If the temperature of two gases is the same, then their molecules move with
the same speeds, and so Avogadro's hypothesi
26
0
s follows for ideal gases. The famous number
N =6.023×10 per kilogram-mole is called Avogadro's Number.
3
3. Let's get an idea of the size of atoms. Amazingly, we do not need high-powered particle
accelerators to do so. Consider a cube of 1 1 1 . If the radius of an atom is , then
we have (1/ 2 )
m m m r
r
× ×
26
0
3
atoms in the cube. Now in 1kg.atom we have 6 10 atoms and
each atom occupies a volume ( / ) m , where atomic weight and density.
N
Aρ A ρ
= ×
= =
1/ 3
3
0
0
10
Hence = (1/ 2 ) / . This give 1 . Putting in some typical densities,
2
we find that 10 . This shows that atoms are mostly of the same size. This
is quite amaz
Ag Be
A N r A r
N
r r m
ρ
ρ

⎛ ⎞
× = ⎜ ⎟
⎝ ⎠
≈ ≈
ing because one expects a Be atom to be much smaller than an Ag atom.
4. Even if you know how big an atom is, this does not
mean that its internal structure is known. In 1895
J.J.Thomson proposed the "plum pudding" model of
an atom. Here the atom is considered as made of a
positively charged material with the negatively charged
electrons scattered through it.
5. But the plum-pudding model was wrong. In 1911, Rutherford carried out his famous
experiment that showed the existence of a small but very heavy core of the atom. He
arranged for a beam of α particles to strike gold atoms in a thin foil of gold.
If the positive and negative charges in the atom were randomly distributed, all ' would
go through without any deflection. But a lot of backscattering was seen, and some 's
were even def
α s
α
lected back in the direction of the incident beam. This was possible only if
they were colliding with a very heavy object inside the atom. Rutherford had discovered
the atomic nucleus.
6. The picture that emerged after Rutherford's discovery
was like that of the solar system - the atom was now
thought of as mostly empty space with a small, positive
nucleus that contained protons. Negative electrons moved
around the outside in orbits hat resembled those of planets,
attracted towards the centre by a coulomb force.
2 2
7. This sounds fine, but there is a serious problem: we know that a charge that accelerates
radiates energy. In fact the power radiated is , where is the charge and is the
acceleration
P∝ea e a
. Now, a particle moving in a circular orbit has an acceleration even if it is
moving at constant speed because it is changing its direction all the time. So this means
that the electron will be constantly radiating power and thus will slow down, collapsing
eventually into the nucleus.
8. This is not the only thing wrong with the solar system model. If you look at the light emitted
by any atom, you do not see a continuous distribution of colours (frequencies). Instead, a
spectroscope will easily show that light is emitted at only certain discrete frequencies.
In the above you see the emission spectrum of different atoms.
9. Similarly, if white light is passed through a gas of atoms of a certain type, only certain
colours are absorbed, and the others pass through without a hindrance.
The above shows the absorption spectrum of a certain atom. The wavelengths for both
emission and absorption lines are exactly equal. Classical physics and the Rutherford
model have no explanation for the spectrum.
10. Then came Niels Bohr. By this time it
was known that electrons had a dual
character as waves (De Broglie relation
and Davisson-Germer experiment). Bohr
said: suppose I bend a standing wave
into a circle. If the wavelength is not
exactly correct, wave interference will
make the wave disappear. So only integral
numbers of wavelengths can interfere
constructively.
11. Let us pursue this idea further. The electron has a wavelength and forms standing waves
in its orbit around the nucleus. An integral number of electron wavelengths must fit into
the circumference of the circular orbit. Hence 2 with 1,2,3 The momentum
is v . The angular momentum v is therefore
(2 / )
quantized in units of .
n r n
p m h h n L rm n
r n r
λ π
λ π
= = ⋅ ⋅ ⋅
= = = = 􀀽 = = 􀀽
􀀽
2 2
n
2
12. Now let us suppose that the electron moves in an
orbit of radius when it has . Equilibrium
demands that the centrifugal force be equal to the
coulomb attraction: v .
n r L n
m ke
r r
=
=
􀀽
2
2
2
From v
we find that the radius .
n
n
n
n
mr
r n
mke
=
⎛ ⎞
=⎜ ⎟
⎝ ⎠
􀀽
􀀽
8
0
2
0
For 1 the electron orbit which is closest to the nucleus, 0.53 10
(this is called the Bohr radius).
For higher , . The atom becomes huge for n 100, the so-called Rydberg
ato
n
n
n ra cm
n r a n
• = = ≡ × −
• = ≈
2
m. Such atoms are of experimental interest these days.
Note that the speed of the electron is smaller in orbits farther from the nucleus,
v . As becomes very large, the electron is very fa n
ke n
n

=
􀀽
r out and very slow.
In the above 0 is strictly not allowed. As you can see, none of the formulae
make any sense for this case. The minimum angular momentum that the electron
can have is . (I
• n=
􀀽 n proper quantum mechanics the minimum is 0 and this is a big
difference with the Bohr model of the atom).
􀀽
1 2
2
2 2 2
1
2
2 2
2 2
2 4
2 2 2
13. We can compute the energies of the various orbits:
v
2
Hence, 1
2
1 13.6 eV
2
n
E K U m U
ke ke ke
r r r
E ke mke
n
mk e
n n
= + = +
⎛ ⎞
= ⎜ ⎟− =−
⎝ ⎠
⎛ ⎞⎛ ⎞
= −⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
⎛ ⎞
= −⎜ ⎟ = −
⎝ ⎠
􀀽
􀀽
14. In the Bohr model, electrons can jump between different orbits due to the absorption or
emission of photons. Dark lines in the absorption spectra are due to photons being
absorbed, and bright lines in the emission spectra are due to photons being emitted. The
energy of the emitted or absorbed photon is equal to the difference of the initial and
final energy levels, f hv= E −E
8 7 2 2
. The picture below shows the electron in the 7 and
8 levels. The photon emitted has 13.6 1 1 ev.
8 7
i n
n hv E E
=
= = − = − ⎛⎜ − ⎞⎟
⎝ ⎠
15. The Bohr model gave wonderful results when compared against the hydrogen spectrum.
It was the among the first indications that some "new physics" was needed at the atomic
level. But this model is not to be taken too seriously - it fails to explain many atomic
properties, and fails to explain why the H atom can exist even when the electron has
no orbital angular momentum (and hence no centrifugal force to balance against the
Coulomb attraction). It cannot predict all the lines observed for H, much less for multielectron
atoms such as Oxygen. The real value of this model was that it showed the way
forward towards developing quantum mechanics, which is the true physics of the world,
both microscopic and macroscopic. I have discussed some elements of QM in the last
lecture, in particular the wavefunction ψ (r􀁇,t) of the electron.
a0 = 0.529Ao r
2 P = ψ (r􀁇,t)
16. Quantum mechanics gives a picture that is
quite different from the solar-system model.
We solve the "Schrodinger Equation" to
find the wavefunction ( , ), whose square
gives
ψ r􀁇 t
the probability of finding the electron at
the point . In the lowest energy state, the
electron can be viewed as a spherical cloud
surrounding the nucleus. The densest regions
r􀁇
of the cloud represent the highest probability
for finding the electron.
2
The principal quantum number 1,2,3 determines the allowed energy levels. An
electron can only have energy 13.6 eV. Miraculously this is the same result
as in the Boh
n
n
E
n
• = ⋅ ⋅ ⋅
= −
r model.
The orbital angular momentum is determined by the number , and ( 1) .
Allowed values of are, 0,1,2,3 , 1.
The magnetic quantum number determines the p l
l L ll
l l n
m
• = +
= ⋅⋅⋅ −

􀀽
rojection (or component) of the
vector angular momentum on to any fixed axis, . Allowed values are,
, 2, 1,0,1,2, , .
17. Electrons can be thought of as little sp
z l
l
L L m
m l l
=
= − ⋅ ⋅ ⋅ − − ⋅ ⋅ ⋅
􀁇
􀀽
inning balls of charge. All electrons spin at the
same speed (more accurately, their spin angular momentum is the same and equals /2).
An electron can spin in only one of two possible direct
􀀽
ions.
When charges move around in a circle, that constitutes a
current. As you know, currents give rise to magnetic fields.
This is why electrons are also little magnets that interact
with other magnets. Now go back to the Stern-Gerlach
experiment described in the previous lecture, and you will
understand better why the electrons were deflected by the
applied magneti
1 1
2 2
c field. Now that we have learned that the
electron has spin, we can describe the two spin states by giving the "magnetic" quantum
number where , . These two states have the same en s s m m= − ergy except when there
is some magnetic field present.
18. States (or orbitals) having 0,1,2, are called s,p,d, . The s-states are spherical. As
increases, the s-orbitals get larger and the wavefunction is larger away from the nucleus.
l
n
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
3p6
Principal quantum
number n = 3
Number of electrons
in subshell = 6
Angular momentum
quantum number l = 1 (p)
19. In the world we are used to, we can always tell apart identical particles (same mass,
charge, spin,...) by simply watching them. But in QM, their idetities can get confused
and identical particles are indistinguishable. Suppose that A and B are particles that are
identical in every possible way, and we exchange them. Of course, the probability of
finding one or the other 2 2 must remain unchanged. In other words, (1,2) (2,1) ,
where 1 and 2 denote the positions of the first and second particles. But something
very interesting happens now because either one o
Ψ =Ψ
f two possibilities can be true:
(1,2) (2,1) or (1,2) (2,1). Particles obeying the first are called ,
while those obeying the second are called . What if we bring t
bosons
fermions
Ψ =+Ψ Ψ =−Ψ
wo fermions to
same point in space? Then: (1,1) (1,1). This means that (1,1) 0 ! In other
words, two identical fermions will never be at the same point or in the same quantum
sta
Ψ =−Ψ Ψ =
te. This is the famous Pauli Exclusion Principle.
20. Let us apply the Pauli Exclusion Principle to the multi-electron atom where each electron
has the quantum numbers { , , , }. Only one electr l s n l m m on at a time may have a particular
set of quantum numbers. Now for some definitions:
Shell - electrons with the same value of
Subshell - electrons with th
• n
• e same values of and
Orbital - electrons with the same values of , , and
Once a particular state is occupied, other electrons are excluded from that state. The
e
l
n l
• n l m
lectron configuration is how the electrons are distributed among the various atomic
orbitals in an atom. A common notation is of the type here:
21. Building the shell structure of multi-electron atoms through n = 4 using the Pauli Principle.

No comments:

Post a Comment