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Summary of Lecture 43 – INTRODUCTION TO ATOMIC PHYSICS
- 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:
- Atoms are building blocks of the elements.
- All atoms of the same element have the same mass.
- Atoms of different elements are different.
- Two or more different atoms bond in simple ratios to form compounds.
- 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 0 26
s follows for ideal gases. The famous number N = 6.023 × 10 per kilogram-mole is called Avogadro's Number.
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
× ×
0 26 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
A ρ A ρ
= ×
1/ 3 0 3 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
N r A r A N r r m
ρ ρ −
× = ⎛^ ⎞
ing because one expects a Be atom to be much smaller than an Ag atom.
- 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.
- 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.
- 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
- 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 ∝ e a 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
- 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.
- 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
= = = = =^ = = =
n^2 2
- 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.
r L n n
m (^) ke r r
(^22) 2
From v
we find that the radius.
n n
n
n mr
r n mke
= ⎜⎛^ ⎞
8 0
0 2
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 r a 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 (^) n ke^. As n becomes very large, the electron is very fa 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 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 (^12)
2 2 2 2 2 4 2 2 2
- We can compute the energies of the various orbits: v
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
= ⎛^ ⎞− = −
= −⎜⎛^ ⎞⎛^ ⎞
= − ⎛^ ⎞ = −
- 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, hv = E (^) f − E
(^8 7 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
= = − = − ⎛^ − ⎞
- 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 multi- electron 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 ( , ) of the electron. r t G
3p 6
Principal quantum number n = 3
Number of electrons in subshell = 6
Angular momentum quantum number l = 1 (p)
- 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 must remain unchanged. In other words, (1,2) 2 (2,1) 2 , where 1 and 2 denote the positions of the first and second particles. But something very interesting happens now because either one o
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
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
te. This is the famous Pauli Exclusion Principle.
- Let us apply the Pauli Exclusion Principle to the multi-electron atom where each electron has the quantum numbers { , , n l m m (^) l , (^) s }. Only one electron 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
lectron configuration is how the electrons are distributed among the various atomic orbitals in an atom. A common notation is of the type here:
- Building the shell structure of multi-electron atoms through n =4 using the Pauli Principle.
