2006-7 Module 113 - Lecture 3

Module 113- Quantum theory and atomic spectroscopy LECTURE 3 - Angular Momentum in Quantum Mechanics Why can six electrons sit in n = 2, l = 1 but only two can make a home in n = 2, l = 0? To find out, we shall take a look at the emission spectra of multi-electron atoms using the specific example of (i) helium, (ii) cadmium, (iii) sodium and (iv) silver.

6. The Zeeman Effect and what it tells us Non-hydrogenic atoms contain many electrons which have different n and l quantum numbers. How can we label all the different configurations of electrons in a multi-electron atom? If I am a spectroscopist, I would like to tell you about the research I am conducting: how can I describe this without resorting to a complete diagram of all the atomic levels? We use a system known as Term Symbols to label different configurations. Complete Term Symbols contain a lot of information but for now I will introduce just a small part, the orbital angular momentum part. Thus, we can describe an electronic configuration by the total orbital angular momentum, L, of all the electrons. We must remember though that angular momentum has a magnitude AND a direction (the axis) and so is a vector quantity. That means that the sum over all the individual angular momenta

∑

l

L

=

n

(3.1)

i

i

is a vector sum over n electrons: that means that when adding two values of l we don’t just have the solution l1 + l 2 but in fact a set of values L = l1 + l 2 , l1 + l 2 − 1,......., l1 − l 2 + 1, l1 − l 2 Where L is the total orbital angular momentum quantum number. You can see why from drawing two vectors and adding them together, remembering that the result must always be an integer value. Actually, the length of the vectors are really L = L(L + 1) (but you don’t need to know this much detail).

We will see more of this later, so for now we stick to something simple. So for our example of helium, n = 2. Both electrons are 1s electrons, so the total orbital angular momentum is 1s + 1s

L=0+0=0

When L equals 0, this is called an S state. Similarly, if L equals 1 it is a P state and so on. The lowest P state in helium is formed when one of the two electrons is no longer 1s but a 2p electron. Thus far we have used two quantum numbers to describe the electrons in an atom. Now we will meet a third quantum number, and this will help give some answers to why there can be up to six 2p electrons in an atom. Imagine having an electron in helium with quantum numbers n = 1, l = 0, and we give it the energy necessary so it has quantum numbers n = 2, l = 1. How can we describe this transition? In terms of the single electron, it is 2 p ← 1s In terms of TOTAL ORBITAL AM

2 P ← 1S

(again, the numbers reflect the principle quantum number of the highest energy electron)

Helium is present in the sun, so if we look at starlight in the ultraviolet (because of the very high energy separation between electronic levels in helium) we should see a dip according to the 2 P ← 1S transition as we disperse the light through a prism. Indeed we do. We can repeat this experiment in the laboratory, assuming we had a suitable light source and a sample of helium gas (there is not very much helium in the average laboratory!) Again, it would consist of a single line with a small width that is determined by how long the excited state can exist before it must decay (remember, all excited states are unstable and therefore must decay back to the ground state eventually). The dip corresponds to the 2 P ← 1S transition. Let’s now transport ourselves back some 100 years. There weren’t very suitable light sources then, but you can excite atoms using electrons and then look at the light they emit easily enough. Since emission is just the reverse of absorption, you would again see one line. Helium wasn’t a very good choice for lab work, so scientists tended to look at other elements like sodium. Again, you will see transition lines in the emission spectrum. The brightest seen corresponds to a P → S transition and you will meet this in the 205 laboratory work. After a while, your boss wants some new results, so what kind of experiments can you do on this emission line. What else was of interest to physicists at the time? Well, electric and magnetic fields were big news: can these fields influence the above spectrum? In 1896, Pieter ZEEMAN decided to repeat an earlier experiment by Michael FARADAY (1862) on the effect of magnetic fields on optical emission lines. Zeeman knew Faraday had failed, but magnets had improved a great deal and he was hopeful that with higher magnetic fields he would see an effect. Using his first magnets he noticed a broadening of the spectral lines in sodium. The following year with an even bigger magnet he tried again, but this time studying an emission line in atomic cadmium (the element is important here): he first noticed the line broadened and then split into three lines.

Cadmium spectrum

How? Well, the situation is analogous to what happens when helium atoms are placed in a magnetic field: there the P state has split into 3 components under the influence of the magnetic field. This is now known as the normal Zeeman effect. Another famous Dutch physicist, Hendrick Antoon LORENTZ tried to explain this in terms of a atom containing oscillating electrons (1897) blah blah and even came up with a formula that “predicted” the splittings of the absorption lines (in frequency):

ν0

and

ν0 ±

eBz 4πme c

Bz

= magnetic FIELD in z direction = JC-1m-1

(3.2)

which fitted the data quite well (not surprising really as he had Zeeman’s results in front of him). As a result, both Zeeman and Lorentz received the Nobel Prize for Physics in 1902. Only problem was Lorentz had got it wrong. This was probably the first time the prize was awarded for work that was later invalidated, but it would not be the last (Fermi got his for a load of old cobblers too) and it was only the 2nd prize for Physics given out so they were not off to a good start. This is the reason the Nobel Prize people wait a little while before handing out the prizes these days, to avoid any more red faces. Today we are pretty sure we know why the line splits in the way it does. In the case of helium, this is because this P state of an atom splits into three sublevels. Any angular momentum state > 0 has sublevels that in the absence of an external influence are degenerate. From angular momentum theory, we know that for any value L (or l )

there are

2L + 1 (or 2 l + 1) sublevels

These individual sublevels are represented by the third quantum number M L (or ml for a single electron) called the magnetic orbital quantum number. M L can take the values of

− (L ), −(L − 1),......, (L − 1), (L ) For example: An F state has L = 3, and therefore there are

2(3) + 1 = 7 sublevels.

A d electron has l = 2, and therefore there are

2(2) + 1 = 5 sublevels.

We sometimes call these sublevels the atomic orbitals. In essence, an orbital is really just a set of labels for an electron consisting of the quantum numbers n, l , ml .

At the risk of making the Nobel Prize committee look really foolish, in 1897, four years before the prize, Thomas PRESTON had discovered a completely different pattern in sodium, which became known as the anomalous Zeeman effect. This was the first of many different such “anomalous” patterns, all due simply to the fact that the electronic states involved were different than the case studied in cadmium.

Indeed, in 1897 they had no idea why sodium was a “doublet” line without the magnetic field, let alone split into ten lines with a magnetic field, but more on that later in the course! Why was the “wrong” Lorentz model not picked up sooner? Two reasons 1) Lorentz was a big and famous name and therefore people laughed at his poor jokes and accepted his theories too readily. 2) The Lorentz equation earlier did in fact describe the normal Zeeman effect quite well. The second point is very clear if we look at the modern formula for the splitting of an orbital angular momentum state in a magnetic field, which is

∆E M L =

ehBz ML 4πme c

(3.3)

Since M L is either –1, 0 or + 1 for a P state, we can see how this result exactly matches the wrong Lorentz theory (using Planck’s relationship E = hν). (Note: in Atkins, he uses B = magnetic induction which is not the same a magnetic field. The unit for magnetic induction is T. This is strictly speaking more correct than the old style equation I am using here, but hey, you hate me at this point anyway! Magnetic field strength has units of JC-1m-1 or Tms-1. You can open your eyes again, I’ve finished)

7. Optical transitions in atoms We can now form a rudimentary picture of what happens in the optical absorption process. Light is absorbed by an electron occupying a single energy level if the atom and is promoted to a vacant site in another energy level. But if the electron is carrying angular momentum, the transition must surely obey conservation of angular momentum as well. We now know that light can induce an electric dipole transition in an atom and when it does we need a change in the electron angular momentum by a single unit of angular momentum 1h and therefore Initial angular momentum of electron = final angular momentum of electron ± 1 Thus an electron with l = 1 can undergo a transition into an electron with l = 0 or l = 2, but not l = 3, 4 etc. This is an example of a selection rule: these rules are very useful for unravelling atomic spectra because not only do they simplify the spectra they also allow a successful analysis.

There is a selection rule as well for the principal quantum number n for the hydrogen spectrum only ∆n ≠ 0 Therefore, we don’t see 2s ↔ 2p transitions, which is obvious really since in hydrogen these energy levels are degenerate.

8. Space quantization of angular momentum Why do the three sublevels with l = 1 possess different energies in a magnetic field? Well, Arnold SOMMERFELD used the concept of space quantisation to explain this: Take an angular momentum vector and assume that it’s projection along a particular axis may only take certain integer values.

In essence, the vector can only take certain orientations in space with respect to this axis. The projections of L onto the magnetic field direction in Zeeman’s experiment had to take these specific values, determined by M L .

Everyone believed this was a rather good little mathematical model to help solve quantum theory problems, but few in the early 1920s actually believed this was the physical reality. But a German experimental physicist Otto STERN took a different line- he believed that space quantisation was literally true. So at the University of Frankfurt with his colleague Walter GERLACH, they attempted to demonstrate space quantisation in an atomic beam of silver atoms. If space quantisation was real, then an inhomogeneous (i.e. changing with position) magnetic field should transform the different M L values of the atoms into a spatial separation of the atoms. The silver atoms passed through the magnetic field to a glass plate (today a photographic plate would be used) which was then removed from the vacuum chamber to observe the final position of the silver beam.

At first they couldn’t see the spots, but the legend has it when Stern was smoking his cigar near the plate the smoke reacted with the atoms and revealed there were a couple of spots instead of just one if the beam hadn’t split (hmm, not sure about this cigar part)! The result was a massive vindication of the space quantisation concept. But there was just one oddity…there were just two spots.

Why is this so strange? Think back to degeneracy of the orbital angular momentum states- they all had an odd number of sublevels. In fact, what Stern and Gerlach had discovered by chance was a completely new concept in angular momentum- intrinsic angular momentum or spin. However, in 1922 they knew nothing about spin so we will come back to this later. For the moment, the important message is that together the quantum numbers l and ml are the result of the angular momentum that the electron possesses. Angular momentum is thus of crucial importance to understanding electrons, atoms and molecules. For example, in a molecule the arrangement of atoms can be rotated about a particular axis. This will be described by an angular momentum quantum number (usually N) and therefore any rotational state or level will have 2N + 1 sublevels.

So we now know that the number of electrons in the l = 1 subshell is limited to six because there are three sublevels and they can hold a maximum of two electrons. We tend to call these sublevels by another name- the orbital. This leads us naturally to another question: •

Why can an orbital hold just two electrons?

We look at this next time! Key points in lecture 3 • • • •

The total orbital angular momentum for an atom is the vector sum of all the individual electron orbital angular momenta. The subshells of atoms are affected by a magnetic field. This can be seen as a splitting of spectral lines in a magnetic field (Zeeman effect). The splitting of the lines reveals that atomic energy levels can be degenerate: this degeneracy depends on the orbital and spin quantum numbers. Upon absorption or emission, individual electrons undergo a change in orbital angular momentum ∆ l = 0. The Stern - Gerlach effect demonstrates space-quantisation: that in a magnetic field the orbital angular momentum will only adopt a small set of orientations with respect to the magnetic field.