2006-7 Module 113 - Lecture 2

Module 113- Quantum theory and atomic spectroscopy LECTURE 2 – Discrete energy levels In the late part of the 19th century, the nature of the so-called cathode rays and the reason for atomic absorption lines were unknown. The two problems were in fact both related to the internal structure of atoms. This lecture deals with how the so-called old quantum theory helped our understanding of the problem… but still failed to anticipate many new experimental details that complicated the basic simple model.

3. The one-electron atom Balmer and the study of hydrogen

Inspired by the desire to rationalise atomic spectra, the search began to try and bring some sense the hundreds of atomic lines seen in the visible region of the E.M. spectrum. In 1884, Johann Jakob BALMER, a Swiss teacher, wisely turned his attention to studying the visible absorption spectrum of hydrogen, known to be the lightest element discovered to that point and therefore probably the simplest atom to understand. By analysing the transition frequencies he discovered a simple formula that could predict the positions of all the lines in the visible hydrogen atom spectrum. The modern version of it (his was a bit different, but the essential features were there) is

⎛ 1 1 ⎞ − 2 ⎟⎟ 2 ⎝ n1 n2 ⎠

ν = RH ⎜⎜

(2.1)

ν

is the FREQUENCY of the light absorbed by the atom (the frequency of the spectral line)

RH

is a constant known as the Rydberg constant and n1 and n 2 are integers

With this simple formula, he could fit all the lines in the visible region IF n 2 = 2. The discrete lines of the atomic hydrogen spectrum could be explained in terms of a single constant and a set of integers. However, we must remember that this was a completely empirical result: Balmer, and for that matter no one else, knew what these integers actually meant!

Planck Max PLANCK was already a highly esteemed scientist when at the very end of the nineteenth century he decided to try and solve a few of the remaining puzzles in thermodynamics, a subject that seemed reasonably well understood. One problem in particular attracted him and that was black-body radiation. This was partly because probably the most important invention of the 19th century was… the light bulb. Not only had it solved the real hazard of providing light without (too much) unwanted heat, it had also spurred on the electrical supply industry to bring electricity to businesses and homes all over Europe and the US. This had led to the introduction of electrical gadgets in the home (in order so the power companies could make a profit) and had thus created what we call modern life. There was a lot of money to be made out of fabricating a better light bulb and the problem seems a simple one: what is the maximum energy I can extract from heating a filament? The ideal emitter/absorber of radiation is the so called black-body, and experiments on near-ideal black-bodies had produced a wealth of data. Trouble was…classical physics could not explain the operation of the perfect blackbody! For example, at a fixed temperature the light intensity increases as the wavelength falls till it reaches a peak and then the intensity declines to zero: classical models predicted that at any temperature the intensity would increase to lower wavelength, which would mean that the black-body could hold an infinite amount of energy!

The details I am afraid are rather dull so we won’t waste our time talking about it, but if the problem was boring the solution was awesome. Planck proposed in 1900 that the energy released from a hot body was related to the frequency of the emission lines and that since these, as in atoms, were quantised, the energy of the electrons must also be quantised. In his famous formula

E = hν

(2.2)

where h is Planck’s constant and E is of course energy. This idea was later refined to the picture we have today which is of a ladder of energy states in each atom. An electron in the atom will only absorb light that corresponds to the energy required to move from one energy level to another (and of course, emission is just the reverse of absorption). The integers in the Balmer formula, n, are now called the principal quantum numbers and of course refer to these different electronic energy levels (so-called because they involve the electrons). In hydrogen, which we know has just one electron, the lowest possible energy of that atom would consist of the single electron in n = 1. This we call the ground state. Supply the electron with energy by irradiating the atom, and the electron may change energy levels to leave it in an excited state. But supply too much, by using E.M. waves of very high energy, and the electron will escape from the nucleus: this is called ionisation. This is what x-rays can do as we saw with the experiments of Charles Barkla.

So the integers in Balmer’s formula reflects the fact that the energy levels are quantised. Fine. But why were the energy levels quantised? Niels BOHR proposed his infamous “planetary model” of the atom (which is completely useless and you must try and forget it ever existed) which contained one grain of truth: this was that certain restrictions on the nature of the energy levels, known as boundary conditions, limited the number of levels that could exist in an atom (much more about this later). Today we know what these boundary conditions are, but in 1915 they didn’t. A small digression: from optics, we know that the frequency and wavelength of light are related by a formula

c = νλ

(2.3)

where c is the speed of light, λ is the wavelength and ν is the frequency. Since c is a constant, the wavelength of length is inversely related to the frequency. So we can talk about the interval between energy levels in terms of either energy, frequency (thanks to Planck) and wavelength (see above). We often express atomic energies not in joules (J) but in electron volts (eV). This is because joules are great if you have a mole of atoms but are rather too big for a single atom. The conversion factor between the two is simply 1 eV = 1.60218 x 10-19 J The transfer of an electron from one energy level to another is called a transition. What would the set of transitions- the atomic spectra- of other elements be like?

4. The two-electron atom - helium The short answer to the above is: very, very complicated! The reason is because we cannot so much think of a simple series of energy levels, but on the different possible arrangements of electrons within those levels. To see what I mean, we can take helium as an example. Helium has just one extra electron. Let us simplify matters and instead of thinking of our infinite series of levels as before, now just consider an atom with two energy levels. This two-level model of atoms is very useful and is one of the favourite tools of scientists to solve problems in atomic (and molecular) physics. In our two level atom, one level has higher energy than the other; that is the levels are non-degenerate. This is not a moral judgement: it just means they don’t have the same energy. Degeneracy is an important consideration in quantum physics, as we will see in the next lecture.

If we have one electron, there can be just two different arrangements for the electron. It can either be in the lower, or in the upper level. But with two electrons, we have a set of four arrangements as shown below:

The total energy of each of these configurations depends on the energy of the individual electrons. The lowest energy configuration is the one on the left because the individual electrons have their lowest possible energy. This configuration is the ground state. The three other configurations are at higher energies because at least one electron has additional energy. These configurations are known as excited states. Excitation of the atom, say be smashing the atom hard into a target, can excite the electrons into any one of the three “excited states”. In classical physics, it turns out that the energy to excite (2) and (3) is exactly the same, because they both involve the raising of a single electron. But now we meet a purely quantum effect: although these two arrangements seem to have the same energy, quantum theory would distinguish between the two because they are not identical arrangements. What quantum theory predicts is that we have a mixture- a superposition- of c2 and c3 and results in two new levels that do not have the same energy. We will consider this more closely in the final lecture, where hopefully this will be a little clearer! To reach configuration (4) one needs to raise both electrons, so more energy is needed. So despite having a two level atom, we have four energy states that are manifest in the atomic spectrum. The important point I am trying to make is that spectroscopy consists of transitions from an initial energy state to a final energy state.

In electronic spectroscopy radiation is absorbed by a single electron. Transitions between the different arrangements can make atomic spectra look very complex indeed. Luckily, in absorption spectroscopy we are only usually interested in transitions that begin with atoms in the ground electronic state. But what determines the arrangement of electrons in an atom? Obviously it will be in such a manner as to minimise the total energy, so it boils down to how the energy levels are distributed in energy and how many electrons can occupy a particular energy level. Now we immediately see that electrons don’t just fall into the very lowest energy level because then we would expect a simple linear relationship between the ionisation energy of an element and the atomic number (the increased charge will pull the electrons closer to the nucleus and hence deeper them in energy). A quick look at the ionisation energy of the elements reveals that this cannot be the case.

We need a technique to determine the energy of the electron levels and the number of electrons in each level and luckily there is a simple way: photoelectron spectroscopy. This is an interesting form of spectroscopy because the final state of the atom is always the same: the removal of an electron completely from the atom.

5. Ionisation Energy Note how in the hydrogen spectrum the lines bunch together- they merge into a continuum. The lines tend towards an energy limit. If the electron gains more energy than this limit, it is lost from the atom- the atom is ionised and so this limit is known as the ionisation energy (I.E.) Short wavelength radiation like X-rays is excellent for this. Now, if monochromatic light is used (single frequency radiation) then by simple energy balancing with the kinetic energy

1 me ve2 of the expelled electron (it 2

cannot have any other energy) will reveal the ionisation energy of the electron via.

1 hν = I .E. + me ve2 2

(2.4)

This is the basis of photoelectron spectroscopy. For hydrogen, the photoelectron spectrum will consist of a single peak- the energy required to remove an electron in n = 1. We can find the energy necessary to do this from eqn. (1) by setting n 2 = ∞:

⎛ 1

1⎞

⎛ 1

⎞

R

H ⎟⎟ = R H ⎜⎜ 2 − 0 ⎟⎟ = ν IP = RH ⎜⎜ 2 − ∞ n12 ⎝ n1 ⎝ n1 ⎠ ⎠

(2.5)

Since in hydrogen , n1 = 1

ν IP = RH The helium ion, He+, also has one electron - we say it is a hydrogenic atom. However, the binding energy of the electron depends on the nuclear charge Z , which is clearly greater in helium. For such hydrogenic atoms:

⎛ 1 1 ⎞ − 2 ⎟⎟ 2 ⎝ n1 n2 ⎠

ν = Z 2 RH ⎜⎜

(2.6)

and so

ν IP = Z 2 R H

(2.7)

The helium ion photoelectron spectrum has one peak, as expected. But the helium atom also has a single peak in the photoelectron spectrum- that is because both electrons belong to the same atomic subshell. Different peaks therefore correspond to different subshells in a photoelectron spectrum (since the final electron energy is n 2 = ∞ the only thing that can change is the initial energy of the electron in the atom).

For n = 1, we find that we can have only a maximum of two electrons in this subshell. If we have a third electron, like in lithium, this new addition is added to the next energy level or shell, n = 2 (naturally). The difference in energy between the two levels is enormous, on the order of 1000 eV (which is, take my word for it, a lot of energy). How many electrons can this energy level hold? Well, we find lithium has two peaks, and so too does beryllium. But when we try boron, we find that another peak appears, close to the second.

Ionisation energy (x 1000 kJ/mol-1) The energy separation is far too small for it to be n = 3, so this must be a new subshell within n = 2. How many subshells and electrons can n = 2 hold? You can try all the elements up to neon and you get the same type of spectrum as boron (though the energy of the peaks is different), then for sodium a new subshell appears, with the enormous change in energy one expects for a change in n. Thus, n = 2 can hold 8 electrons: 2 in one subshell, and 6 in another. For n = 3, we find there are 3 subshells and 18 electrons. The number of subshells = the principal quantum number, and to describe the pattern of subshells they introduced a new quantum number l . For any shell,

l = 0, 1… (n-1) We label orbitals according to their principal quantum number n and their so-called orbital quantum number l , thus If

n = 1, n = 2, n = 2, n = 3, and so on

l l l l

=0 =0 =1 =0

we have a 1s electron we have a 2s electron we have a 2p electron we have a 3s electron

This pattern was very confusing for the early atomic physicists, and its explanation in terms of angular momentum is one of the major triumphs of quantum physics, as we will see (by about the end of lecture 5). Finally a note about the energies of these electrons: for hydrogenic (one-electron) atoms, there is no difference in energy for a 2s and 2p electron: in fact, all subshells with the same principal quantum number n will have identical energy. It is the presence of other electrons in helium and higher atoms that causes the energy shift in the p, d etc electrons, relative to their s electron cousins. We shall revisit this later. As the charge on the nucleus increases, we find that the electrons experience a greater Coulombic force and hence the energy of each subshell does become increasingly negative. Henry MOSELEY demonstrated that the frequency of the X-ray lines increases in proportion to the square-root of the atomic number i.e. the nuclear charge. This confirmed the prediction made by Neils Bohr on the energy levels of atoms, but it turns out that Bohr’s model wasn’t quite right though his predicted energies were correct. Again, this is an example of a coincidence between prediction and data: further tests, however revealed that the truth lay elsewhere. The ionisation energies of the elements depend on the energy and identity of the highest energy electron in an atom, and this can change as we move from one element to next.

Below is the energy level structure of hydrogen. Each level is a subshell and is labelled by its n and l quantum number. The number in square brackets is the maximum number of electron pairs that can be accommodated in that subshell. But why can a p sublevel hold 6 electrons? The angular momentum model of the atom allows us to understand why there are 6 electrons in n = 2, l = 1, as we shall see in the next lecture.

The previous diagram is the energy level structure of hydrogen: it is unique amongst the elements in that all the subshells with the same value of n have the same energy. Just as Balmer had found with his ground breaking analysis of the hydrogen spectrum, the study of the simplest atom was crucial to understanding all atoms…but that must wait for a later lecture!!!! So the next question we need answering is: •

Why do different subshells hold a different number of electrons?

We’ll see next time that this requires a brand new quantum number. Key points in lecture 2 • • • • • •

In 1900, Planck proposed that thermal radiation was not a continuous spectrum but produced by a finite number of microscopic oscillators with unique energies. The energy levels of atoms are quantised and each atom possesses a unique absorption spectrum. The spectrum is determined by the arrangement of electrons amongst the atomic energy levels. The energy levels can be arranged into shells and subshells. Each subshell can hold a maximum number of electrons determined by the l quantum number. The energy of the subshells can be found using photoelectron spectroscopy. Atomic energy levels are expressed by negative energies, because the ionisation limit is usually set as zero. As the atomic charge increases, the sub-shells become more negative in energy, as revealed by X-ray spectroscopy, because of the increased Coulombic potential. The ionisation energy depends on the highest energy electron in the atom, and this depends on which sub-shell it belongs too.