Many-Electron Atoms

symmetric and antisymmetricwave functions

periodic table

atomic structure

x-ray spectra

“Tomorrow is going to be wonderful because tonight I do not understand anything.”—Neils Bohr

Half integral spin particles (s=1/2, 3/2, etc.) have antisymmetric wave functions and are called fermions.

Electrons in a system are described by antisymmetric wave functions which change sign upon exchange of pairs of them. Other examples are neutrons (neutrons??--you should ask how they can have a spin if they have no charge) and protons. They are also fermions.

Only one fermion in a system can have a given set of quantum numbers!

7.3 Symmetric and Antisymmetric Wave Functionssummary from last lecture...

Integral spin particles (s=0,1,2, etc) have symmetric wave functions, and are called bosons.

Photons in a cavity are described by symmetric wave functions which do not change sign upon exchange of pairs of them. Other examples are alpha particles and nuclei with integral spins.

There is no restriction on how many bosons in the same system can have the same set of quantum numbers.

What about particles having symmetric wave functions?

We need to have a quiz next week.

Monday?

Wednesday?

Chapter 6?

Review: hydrogen atom radial probability densities. I can think (at the moment) of four things I might ask you to do with them. (Remember: P(r) dr = r2R2dr.*)

Calculate the value of P at some r, or compare P for two different electrons at some r (e.g. HW 6.19).

*Why the dr on both sides? –mathematically correct – remind us of uncertainty principle

Calculate the expectation value <r>. I could also call it the average value of r (e.g. HW 6.20).

Calculate the probability that an electron in hydrogen can be found between r1 and r2 (e.g. HW 6.21).

2

0r = R(r)* r R(r) r dr

2 2

1 1

r r 21 2 r r

P(r r r ) = P(r) dr= R(r)*R(r) r dr

1

2

3

Calculate the most probable value of r for an electron in hydrogen (e.g. HW 6.15).

What does “most” mean to you?What does “most” mean to you? The biggest? Maximum?

Maximum? Aha—calculus! Derivatives!

dP(r)0 =

dr

Solve for the root(s) of P(r)!

4

7.4 Periodic Table

I assume you have studied the periodic table a number of times in your academic careers, so I will just hit a few high points.

“Understanding matter is necessary to understanding the universe and ourselves, and therefore Mendeleev's Periodic Table is poetry.”—Primo Levi (chemist, writer, concentration camp survivor), paraphrased

Terms you should be familiar with:Terms you should be familiar with: period, Terms you should be familiar with: period, group,Terms you should be familiar with: period, group, metals,Terms you should be familiar with: period, group, metals, nonmetals,Terms you should be familiar with: period, group, metals, nonmetals, transition elements,Terms you should be familiar with: period, group, metals, nonmetals, transition elements, rare earths,Terms you should be familiar with: period, group, metals, nonmetals, transition elements, rare earths, actinides,

Terms you should be familiar with: period, group, metals, nonmetals, transition elements, rare earths, actinides, most active metal (why),

Terms you should be familiar with: period, group, metals, nonmetals, transition elements, rare earths, actinides, most active metal (why), most active nonmetal (why),

Terms you should be familiar with: period, group, metals, nonmetals, transition elements, rare earths, actinides, most active metal (why), most active nonmetal (why), inert gases.The lanthanides (rare earths) and actinides are also considered transition elements.

A search on Google for “interactive periodic table” produced about 159000 hits in fall 2002, 265000 hits in fall 2003, and 468000 hits in spring 2005.

There is no longer any excuse for not studying the periodic table!

Here are several good links:http://www.chemicalelements.com/http://www.webelements.com/http://pearl1.lanl.gov/periodic/default.htmhttp://site.ifrance.com/okapi/periodic3.htm

(all sites were active, 3-14-05)

7.5 Atomic Structures

Electronic structures of the elements are based on the following principles:

• Total energy is minimized. • Electrons obey the Pauli exclusion principle.

Electrons having the same n are roughly the same average distance from the nucleus, and have roughly the same energies.

We talk about atomic shells which correspond to different n's, and label them as follows:

n= 1 2 3 4 5K L M N O

The 2p6 above, for example, means that for n=2, ℓ=1, there are six electrons.

Electrons within a shell (i.e., having the same n) generally have lower total energies (bigger negative energies, and therefore larger binding energies) when they have lower ℓ. Electrons having the same ℓ are said to occupy the same subshell.

We write electronic configurations showing shells and subshells. For example, magnesium has a 1s22s22p63s2 configuration.

How many electrons can fit in a particular subshell?

2p means n=2, ℓ=1. The possible values of mℓ are 0,

1. That's three. But corresponding to each mℓ are ms= ½. Thus, a total of 6 electrons can fit in the 2p subshell.

Let's do the 2p subshell as an example.

n=2 ℓ=1

mℓ=+1

mℓ=0

mℓ=-1

ms=+½

ms=-½

ms=+½

ms=-½

ms=+½

ms=-½

n=2 ℓ=1

mℓ=-1ms=+½

Let's do the 4f subshell as another example.

4f means n=4, ℓ=3. The possible values of mℓ are 0, 1, 2, 3. For each mℓ the possible values of ms are 1/2. Thus 7x2=14 electrons can fit in the 4f subshell.

In general, a maximum of 2n2 electrons can fit in any shell.

4 3 +3 +½

4 3 +3 -½

4 3 -3 +½

4 3 -3 -½

Skip sections 7.6, 7.7, 7.8.

7.9 X-ray Spectra

We skipped an introductory section on x-rays back in chapter 2. Here's the idea of the section we skipped. Remember the photoelectric effect? Hit a metal with photons and get an electric current out.What if you zap a metal with high-energy electrons (in other words, pass a high-voltage current through it)? You get x-rays. Actually, you get highly-energetic photons, better known as x-rays.

You can call this the inverse photoelectric effect. You expect a spectrum like this:

Measured x-ray spectra have additional sharp lines.

The total x-ray spectrum looks like this.

We focus here on the sharp lines: where do they come from, and what good are they?

Let.s answer the “what good is this” first.

You can use a monochromator to pick out x-rays of a “single” wavelength.

http://hea-www.harvard.edu/HRC/facility/xmono.html

If you “shoot” these x-rays at a crystalline sample, you observe a scattered beam at angles for which Bragg’s law is satisfied.

Bragg’s Law:n=2d sin

a nearly monochromatic (single wavelength) beam of x-rays

observedcalculateddifference

The angles at which you observe reflected intensity tell you all the interplanar spacings in the crystal.

The intensities of the reflections tell you what atoms lie on the planes, and where they are within a plane.This information is enough to let you determine unknown crystal structures.

We used neutrons, not x-rays, for this experiment. Matter waves!

I’ve answered the “what good?”

Next: “where from?”

Here’s a schematic picture of an atom (we know it’s wrong, but let’s use it to help us think).

The outer electrons are easiest to remove, but if you hit the atom hard enough with a photon or another electron, you can knock out a K-shell electron.

K

L

What are we left with? An atom with a missing n=1 electron. Is this a tolerable situation?Is this a tolerable situation? No.

What happens?What happens? An electron from another shell drops into place.

+

Using a single + to represent all the protons.

What the outer electron “drops” into the K-shell, a photon is emitted.

The photon energy has a well-defined value (hence the sharp lines in the x-ray spectrum) and can be quite large.

X-rays originating from an electron dropping into the K (n=1, the innermost) level are called "K" x-rays.

Eventually, the hole left by the dropping electron must be filled. The atom will “find” an electron wherever it can, and “grab” it. In the case of a metallic laboratory x-ray source with a large current passing through it, there are plenty of “stray” electrons to “grab.”

X-rays originating from an electron dropping into the K shell from the L shell are called "K" x-rays.

X-rays originating from an electron dropping into the K shell from the M shell are called "K" x-rays.

X-rays originating from an electron dropping into the K shell from the N shell are called "K" x-rays.

K

L

MNO

K K K

K

L

MNO

K K K

L L L

X-rays originating from an electron dropping into the L shell from the M shell are called "L" x-rays.

X-rays originating from an electron dropping into the L shell from the N shell are called "L" x-rays.

X-rays originating from an electron dropping into the L shell from the O shell are called "L" x-rays.

Beiser calculates the frequency of a K x-ray photon (those last words are redundant, aren't they).

He uses our equations from chapter 4 for the energy of spectral lines. Those equations have to be modified to consider the different charges involved.

Beiser assumes the nuclear charge seen from the L shell is reduced by one by the shielding due to the remaining s electron. Beiser also assumes that the empty s state (into which the L shell electron falls) is also shielded by the remaining s electron.

K

L

+

The result for the frequency is

23cR Z-1

,4

f =

and for the energy of the photon in eV

210.2 eV Z-1 .E=

These equations come from replacing e4 by [ (Z-1) e2 ]2 in equations 4.15 and 4.18, and using ni=2 and nf=1 in equation 4.15.

“In a research which is destined to rank as one of the dozen most brilliant in conception, skillful in execution, and illuminating in results in the history of science, a young man twenty-six years old threw open the windows through which we can glimpse the sub-atomic world with a definiteness and certainty never dreamed of before. Had the European War had no other result than the snuffing out of this young life, that alone would make it one of the most hideous and most irreparable crimes in history.”—Robert Millikan, speaking of Moseley, who developed experimental uses for x-rays.

X-rays are used all the time in physics for understanding the structure and properties of collections of atoms.

Another different, but related, event can occur when we knock out an inner shell electron. This event does not involve the emission of a photon.

The Auger* effect is a two-electron process. One outer shell electron drops down to fill the inner shell vacancy. But no photon is emitted.

How can this happen?

Energy must somehow be carried away. In this case, an electron with the extra energy is ejected from the atom.

*“Auger” is pronounced “oh-zhay”, not “aw-ger.”

Auger effect.

K

L

+

The energy of the ejected electron is mostly the missing x-ray energy, but it is modified due to the chemical state of the atom.

We use Auger spectroscopy to learn what atoms are present in new materials. We can also learn the chemical state of the atoms; e.g., free carbon and carbon in an iron carbide compound give off different Auger electrons.