Atomic Structure

the nuclear atom

“Nothing exists except atoms and empty space; everything else is opinion.”—Democritus

Friday, February 4: Review, Q&A(no lecture notes for Friday, Feb. 4)

Monday, February 7: Exam 1

News Flash!We interrupt this lecture to bring you the following special message.

“In a paper to be published today in the Journal of Applied Physics, … three researchers from HP's Quantum Science Research Group describe how they created a tiny device that can perform one of the essential logic functions in computing devices just nanometers across.” –Computerworldhttp://www.computerworld.com/hardwaretopics/hardware/story/0,10801,99419,00.html?source=NLT_AM&nid=99419

Thank you for your attention and now back to our regularly-scheduled lecture.

Chapter 4Atomic Structure

4.1 The Nuclear Atom

Written records of the idea that matter consists of tiny, indivisible “building blocks” date back to the 5th century BC Greek philosophers Leucippus and Democritus.Of course, being philosophers, they specialized in “thought experiments.” They felt no need to verify that their thoughts were correct.

The modern idea of atoms was slowly (and sometimes painfully) developed over the century between the late 1700’s and late 1800’s.

“I would rather discover one scientific fact than become King of Persia.”—Democritus

As late as 1895, the great chemist Wilhelm Ostwald (1909 Nobel Prize for work on catalysts, equilibria, and reaction rates), publicly ridiculed the idea of atoms.*

*And contributed greatly to the suicide of Boltzmann, who we will learn of in Chapter 9.

The “discovery” of the electron in 1897 by J. J. Thomson led him to suggest in 1898 that atoms are positively charged matter with small, negatively charged electrons embedded. Plum pudding.

Beiser’s picture of the plum pudding model was not what Thomson was thinking of. That’s OK, “everybody” shows the wrong picture.

You may laugh at the plum pudding model today, but it was taken seriously in the early 1900’s.** The atomic model you probably believe right now is just as laughable.

**“Plum Pudding” Thomson won the 1906 Nobel Prize for his work on the conduction of electricity by gases.

Ahh, the power of plum pudding.*

*There are not now, nor have there ever been, plumbs in plum pudding. And no, that is not a typo. See http://www.whatscookingamerica.net/Cake/plumpuddingTips.htm.

Forgetting about plum pudding for now… if you have a mental picture (theory) of an atom, what are you going to do with it?See if it matches reality!

How?

Develop a theory. Experiment. Compare with theory.

Where do you start?

The simplest atom. Hydrogen.

The material on the next four slides is not in your text, but can we pursue it for a few minutes, because it foreshadows “correct” ideas?

The plum pudding model for hydrogen.

A lump of + charge (colored like pudding, just for your viewing enjoyment).

A small, negatively charged electron.

-+

Where does the electron go?

What if the electron gets displaced?

Restoring force!

What does the electron do?

Oscillates!

What does an oscillating (and therefore accelerating) electron do?

Emits radiation!

You can calculate the energy of the emitted photon, and the calculated value, 10 eV (HW problem 4.2, which is not assigned) agrees nicely with one of the major hydrogen spectral lines at 10.2 eV.

You think you’re on to something. What do you do next?More complex atoms!

Unfortunately, for anything beyond hydrogen, the modeling and calculations are too difficult! So what else do you try.

Study hydrogen some more?

When you pass a current at high voltage through hydrogen gas, the gas glows. You can measure the spectrum of light emitted by hydrogen. There are many spectral lines. Unfortunately, none of them except the 10 eV one are predicted by Thomson’s model.Dang!

What next?

If you want to find out what’s inside the pudding, do what Rutherford (one of the Plum Pudding believers) suggested…

Stick your finger in the pudding and see what’s there!

Ernest Rutherford, a physicist, won the 1908 Nobel Prize in Chemistry for his work on radioactive decay.*

*“I have dealt with many different transformations with various time-periods, but the quickest I have met was my own transformation from a physicist to a chemist.”—paraphrase of a quote by Rutherford

Rutherford was the world expert on alpha particles, which are produced in radioactive decay. An alpha particle is a helium nucleus—two neutrons, two protons, tightly bound together (as explained in chapter 12).Alpha particles have 8000 times the mass of an electron and Rutherford’s alpha particles were highly energetic. An alpha particle colliding with an electron would be like a human colliding with an ant.

In 1906, Rutherford discovered that alpha particles were only slightly deflected when they passed through matter. This was expected.

Analyzing this weak scattering could give insight into the distribution of charge and mass inside the atom. Rutherford pioneered the technique of studying matter by scattering particle (or wave) beams.

Rutherford’s co-worker was Hans Geiger (“Geiger counter”).

*“We know that nothing interesting is going to happen, but let’s have poor Marsden do it anyway. It’ll only waste a few months of his life. Ha ha ha...”

They had an undergraduate named Ernest Marsden working for them in 1909. They assigned Marsden the job of measuring the scattering at large angles* of alpha particles by a gold foil.

In the Thomson model, electric charge is smeared out over the atomic volume, and minimal interaction is expected between the charged alpha particles and the gold atoms.

Indeed, most of Marsden’s alpha particles passed straight through the gold foil.

A few were scattered at large angles.

Some even bounced straight back.

“It was as if you fired a 15-inch shell at a sheet of tissue paper and it came back to hit you.”—Ernest Rutherford

USS Washington

15 inch guns

In the Thomson model, electric charge is smeared out over the atomic volume, and minimal interaction is expected between the charged alpha particles and the gold atoms. That’s because there’s no local electric field to deflect a charged particle.

OK, I lied on the previous slide (for dramatic effect). The battleship Washington had a main battery of nine 16-inch guns. I spent a long time looking for pictures of 15-inch guns and shells, with humans in them to illustrate the scale.

Lots of 16-shell pictures, like this one. No 15-inch. But if you see one of these coming at you, are you going to argue about that extra inch of diameter?

“It was as if you fired a 15-inch shell at a sheet of tissue paper and it came back to hit you.”—Ernest Rutherford

15-inch shell: weight—2700 pounds; muzzle velocity—1570 miles per hour; range—21 miles

Splat!

If the Thomson model were correct, an alpha particle scattered in exactly the same direction by every one of the gold atoms in the 400-atom thick foil would only deflect by a degree or so. The only way to explain the experiment is if the scattering centers are compact, massive, and highly charged. Like this... or this. (Fowler’s modern physics lectures are fascinating, if you like getting the historical background.)

-++++

-

-

-

Beiser’s picture of Rutherford’s atom, and many others, shows a nucleus at the center, and apparently randomly-placed static electrons. The actual thinking at the time (ca. 1910) was that electrons orbit the nucleus, but any orbit is allowed.

At the time of Rutherford’s model, our understanding that the atom is a collection of protons in the nucleus* and the same number of electrons outside, was finally taking shape. The relationship between periodic table and the number of protons and electrons in an atom was even beginning to make sense.

*the neutron was not yet discovered

To help him understand atomic structure, Rutherford developed a theory of alpha particle scattering by nuclei. The theory takes six pages of an appendix to chapter 4 to develop. Do you mind if I skip it?

I thought so. Besides, the theory is entirely classical and part of the reason it works is just luck, plus the fact that the alpha particles in the theory never get close enough to the nucleus for their wave nature to be apparent.

There is a subsection in Chapter 4 on Nuclear Dimensions that can be confusing.

You can estimate the size of a nucleus based on how close a highly energetic alpha particle can get to it.

Any alpha particle which follows the Rutherford scattering formula (which we skip), couldn’t have contacted the nucleus.

Skip the brief subsection Rutherford Scattering Formula, p. 122.

Measuring the energies of alpha particles that don’t contact the nucleus lets us place an upper bound on the nuclear diameter.

alphanucleus

Consider an alpha particle “infinitely” far from the nucleus.

The energy of the alpha particle is all kinetic.

alphanucleus

As the alpha approaches the nucleus, KE becomes potential energy.

KE

KE PE

At closest approach, the energy is all potential energy.

KE PE R

PE

Afterwards, the alpha particle scatters away.

R

Solving for R (distance of closest approach) is a simple conservation of energy problem:

fi other i fE - E = [W ]

ffi i other i fK + U - K - U = [W ]

0πε i

2e Ze1 = K

4 R

We assume, as an approximation, that the target nucleus remains at rest.

0

πε

2

i

1 2Ze R = .

4 K

In this equation, R is the distance of closest approach, +2e is the charge on the alpha, Ze is the nuclear charge, and Ki is the alpha particle initial kinetic energy.What would you do if I gave you a proton instead of an alpha particle?

Example: calculate the distance of closest approach of 7.7 MeV alpha particles on gold nuclei, Z=79.

0

πε

2

6

2 79 e1 R =

4 7.7×10 ×eThe e in the denominator converts ev to joules.

-19

9

6

2 79 1.6×10 R = 9×10

7.7×10

-14 R = 2.96×10 m

This is an upper bound on the gold nucleus: its radius must be smaller than this. In fact, the radius is about 5 times smaller.

Does this classical calculation have any validity? Remember, the alpha particle has a wave nature. What should we check?If the alpha particle wavelength is much much less than R, then the classical calculation is reasonable.

The alpha particle rest mass is roughly 4x1000 MeV/c2 so an 8 MeV alpha particle is nonrelativistic. Relativity is not going to be a problem here.

An alpha particle has a “rest energy” of roughly 4,000 MeV/c2, so the alpha particle kinetic energy of 7.7 MeV is small enough to allow us to use a nonrelativistic calculation of the wavelength.

h λ =

p

alpha alpha

h λ =

2 × m × KE

-34

-27 6 -19

6.63×10 λ =

2 × 6.68×10 × 7.7×10 ×1.6×10

-15 λ = 5.17×10 m

Interesting—the alpha particle wavelength is (coincidentally) about the same as the gold nuclear radius, and about 1/6 of the distance of closest approach.

6

22 slides just to get through 1 section of Chapter 4! Can we make it to the end?

Just on the borderline of requiring a wave calculation.