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.” –Computerworld
http://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 4
Atomic 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.*
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.
*And contributed greatly to the suicide of Boltzmann, who we will learn of in Chapter 9.
Ahh, the power of
plum pudding.*
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.
*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.
**“Plum Pudding” Thomson won the 1906 Nobel Prize for his work on the
conduction of electricity by gases.
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).
Where does the electron go?
-
A small, negatively
charged electron.
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.*
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.
*“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
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”).
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.
*“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...”
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.
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
15 inch guns
USS Washington
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 randomlyplaced 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.
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.
*the neutron was not yet discovered
Skip the brief subsection Rutherford Scattering Formula, p. 122.
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.
Measuring the energies of alpha particles that don’t contact the
nucleus lets us place an upper bound on the nuclear diameter.
alpha
nucleus
Consider an alpha particle “infinitely” far from the nucleus.
The energy of the alpha particle is all kinetic.
alpha
nucleus
KE PE
As the alpha approaches the nucleus, KE becomes potential
energy.
KE PE
At closest approach, the energy is all potential energy.
KE PE
R
Afterwards, the alpha particle scatters away.
R
Solving for R (distance of closest approach) is a simple
conservation of energy problem:
Ef - Ei = [Wother ]if
K f + Uf - K i - Ui = [Wother ]if
1  2e  Ze 
= Ki
4 πε 0
R
We assume, as an
approximation, that
the target nucleus
remains at rest.
1 2Ze2
R=
.
4 πε 0 K i
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.
2  79  e2
1
R=
4 πε 0 7.7×106 ×e


The e in the denominator
converts ev to joules.

R = 9×10
9


2  79  1.6×10-19
7.7×10 

6
R = 2.96×10-14 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
λ=
λ=

h
2 × malpha × KEalpha

6.63×10-34
 

2 × 6.68×10-27 × 7.7×106 ×1.6×10-19
λ = 5.17×10-15 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
Just on the borderline of requiring a wave calculation.
22 slides just to get through 1 section of Chapter 4! Can we
make it to the end?