Lecture 3

The Atom : Bohr's atom, electron levels and spectra

The implications of Rutherford's results were quite astounding:

what appears as solid matter is in reality mostly empty space !!

The knowledge of the time had determined that the spatial extension of individual atoms was of the order of 10^-10 m. Proper numerical interpretation of Rutherford's results (i.e. of the probability for an $\alpha$particle to scatter at a given large angle) indicated that the extension of the nucleus was of the order of 10^-15 m : the electrons, carriers of the atom's negative charge, were at a distance from the central positive nucleus, where all of the atom's mass was concentrated, 5 orders of magnitude larger than the nucleus itself.

To use an often employed comparison:

imagine that the nucleus is the size of a basketball (r = 12 cm); to find the electrons, whizzing around the nucleus, you would have to move $12\times 10^5$ cm = 12 km (7 miles) away ! In between, it's all empty space !!


As surprising as it was, the picture was eventually accepted, and people started thinking of the atom structure as something analogous to the solar system, with the electrons (the planets) orbiting around the nucleus (the sun). Let us keep in mind anyway that this is only a picture with no real validity, even from the dimensional aspect: the average distance of the planets from the sun varies from 100 times (for Mercury, the planet nearest the sun) to 10,000 times (for Pluto, the farthest) the sun radius.

Further investigations of the nucleus showed that it contained positively charged particles, about 2000 times heavier but with exactly the same electric charge ($1.6\times 10^{-19}$ C) than the electrons : these particles were called protons. Many years later (in 1932) a neutral partner to the proton was discovered, the neutron. Neutrons have mass almost identical to the protons, but zero electric charge. More about this later (Chapter 14).

To recapitulate (Table 9.1):

The new picture of the atom as a microscopic solar system faced very soon fundamental difficulties, since it violated some very basic laws of the extremely successful ElectroMagnetic Theory.

Fundamental prediction of the EM Theory : a charged object, when undergoing some form of acceleration, will radiate energy in the form of ElectroMagnetic Waves.

The correctness of the theory was confirmed continuously by the ever growing technology of radio transmission: radio broadcast was carried by the electro- magnetic waves generated by accelerating electrons back and forth along the transmitter's antenna.

If electrons were in circular motion around the nucleus, this implied an acceleration. Accelerated electrons would gradually radiate their kinetic energy away, spiral down towards the nucleus where they would eventually be absorbed. Calculations were showing that this process would occur in some fraction of a second. This was in clear contradiction with the obvious stability of atomic matter.

A major impasse had been reached: either EM Theory was not valid in the atomic realm, or some new factor was at play.

The solution to the problem, produced in its first formulation by Niels Bohr in 1913, represents one of the major milestones in the history of science.

To reach the formulation of his revolutionary hypothesis, Bohr was assisted by some other established facts :

• at the end of the century, physicists had been trying to find the correct expression to reproduce the spectrum of radiation emitted by a body at a given temperature (the so called blackbody radiation). A successful formula was provided by Max Planck in 1900, but for it to work one had to make the hypothesis that energy is emitted not in a continuous way but in discrete packets that he called quanta
  
  lexical note : you are discreet if you mind your own business, but you are discrete if you come in individual units
  
  As part of his theory, Planck wrote down a fundamental expression, relating the energy of electromagnetic radiation (waves) to its frequency :
  $E = h\nu$
  h = Planck's constant = $6.626\times 10^{-34}$...
  
  
  Warning : this formula is not in your textbook, but I will expect you to know it
  
  
  Even though it represented a major deviation from Classical Physics (or maybe because of this), Planck's work did not generate too much attention at the time.

  QUESTION : given that $E = h\nu$,(E=Energy,
  $\nu$ = frequency, what are the units for h ?
  
  
  A Joules
  
  
  B Joules/sec
  
  
  C Watts
  
  
  D Joules sec
  
  
  E Not enough information is given to define h

• the next step came from the explanation of thephotoelectric effect, the process by which electrons can be extracted from a surface by shining light onto it. The measured charecteristics of such a process were in contrast with the expectations of Classical ElectroMagnetism. Einstein provided the solution in 1905, by extending Planck's hypothesis and assuming that light of a given frequency was really behaving like a discrete packet (or quantum) of energy, again given by $E = h\nu$. This time the quantization of energy had to be taken into serious consideration (and in fact Einstein got the Nobel prize for this, not for Relativity). Seen under this perspective, light, which by universal consensus had a wave-like behaviour, was also exhibiting some particle-like behaviour. Such particles of light were named photons.
• it was a well established fact that elements, when "excited" (i.e when receiving a certain amount of energy, either via electric currents or by heating ) were emitting light not in a continuous spectrum of colours but in well defined spectral lines. Each element would produce its own well defined and unique spectrum. Even though a lot of information was available on the spectra of numerous elements, there was no idea on what was the origin of such a behaviour.

Inspired by the above facts, Bohr put forward his revolutionary hypothesis:


Electron orbits around the nucleus (and, remember, there is a one to one correspondence between orbit radius and energy of the orbiting particle) are quantized. When circling the nucleus, electrons cannot have any arbitrary orbit radius (i.e. energy value), there are only some well defined values that are allowed.


This assumption solves the quandary associated with energy radiation in the Classical EM theory : in Bohr's view, electrons do not radiate in a continuous way, since this would cause them to move to a non allowed orbit (i.e energy state), and this has been declared to be an impossibility. In few words, electrons do not radiate because they can't.....

The only allowed emissions of energy are the ones corresponding to the difference in energy between two allowed orbits : when in a "higher" orbit, an electron can emit a quantum of energy corresponding exactly to the energy difference between the two orbits. This quantum of energy will appear as a photon of frequency corresponding to the energy, as given by $E = h\nu$.

Conversely, an atomic electron can only absorb energy in amounts corresponding to the energy difference between two allowed orbits. Such an absorption of energy will cause the electron to perform a quantum jump from a lower to a higher orbit. When in a higher (or excited) orbit, an electron is allowed to fall back to one of the lower orbits, and, in doing so, it will emit a photon of the correct energy/frequency. Even though it is difficult to visualize, when interpreting the quantum jumps we must assume that the transition takes place without going through any intermediate state. An electron can be either in state A or in state B, and it goes from A to B without going through all the intermediate positions between A and B....

One of the major consequences of Bohr's hypothesis (and obviously one of its motivations) is to provide the physical explanation for the atomic spectra: the series of emission frequencies associated with a given element correspond to the energy differences between the allowed energy levels for the given element.

QUESTION : if the atom of a given element has 4 allowed energy states, how many different spectral lines can it produce ?


A 1


B 3


C 4


D 6


E 8


F $\infty$