Lecture 24-1 The Hydrogen Atom

According to the Uncertainty Principle, we cannot

know both the position and momentum of any particle

precisely at the same time.

The electron in a hydrogen atom cannot orbit the nucleus

in a circular orbit – or any other kind of orbit; otherwise,

both the position and momentum would be exactly known!

Instead, the probability to find an electron is given

by a 3D standing wave.

Standing waves of different shapes for different

states (and different energy levels).

Ground state wave function

Lecture 24-2 Quantum Numbers

The Bohr model quantum number which specifies the energy

level turns out to be only one of several such quantum numbers

that specifies the quantum state of the hydrogen atom:

2

13.61,2,3,...n n

eVE

nprincipal quantum number

There are other quantum numbers:

l for L orbital angular momentum

ml for Lz the “z-component” of L

s for s spin angular momentum

ms for sz the “z-component” of s

Lecture 24-3 Nuclear Structure

A nucleus is actually NOT a point charge. It has a

size that is O(1) fm (1 femtometer = 10-15m).

A nucleus is at least O(103) times more massive than an

electron and is positively charged.

A nucleus is composed of protons

and electrically neutral neutrons

(i.e., nucleons).

The number of protons, Z, is

called the atomic number. The

atomic number determines what

type of element the atom is.

A Z N

Atomic mass number

(or nucleon number)

Number of

neutrons

Lecture 24-4 Nuclear Structure

Each element has a fixed number Z of protons, but the

number of neutrons, N, can vary. These are called isotopes.

A Z N

8

18OA

Z Element Symbol

Shorthand notation for isotopes: e.g., Oxygen 18 has 8

protons (because it is Oxygen), the atomic mass number

18, and the neutron number N=10 (because A=Z+N).

Other examples: 16 2 3 12 14

8 1 1 6 6, , , ,O H H C C

Some isotopes are stable, others are unstable and radioactive.

Lecture 24-5

Physics 219 – Question 1 – April 11, 2012.

Isotopes of an element have the same number of ______

but different number of ______ . Fill the blanks with the

correct particle names.

A. electrons, protons

B. neutrons, electrons

C. protons, electrons

D. neutrons, protons

E. protons, neutrons

Lecture 24-6 The Strong Force

How are the protons (positive charge) and neutrons

(neutral) held together in the nucleus?

The answer is: by the strong force!

The strong force is one of nature’s 4 fundamental forces:

Force Relative Strength* Range (m)

Strong 1 10-15

Electromagnetic 10-2

Weak 10-6 10-17

Gravitational 10-43

The strong force holds a nucleus of multiple nucleons

together as well as the individual nucleons by themselves. It

competes with the electromagnetic repulsion among the protons.

(*for two u quarks separated by 0.03 fm)

Lecture 24-7 How large is a nucleus?

Mass of Nuclei 1 atomic mass unit (u) = 1/12 of a neutral 12C atom

= 1.660539 x 10-27 kg

Mass of a nucleon is approximately 1 u. That of an

electron is approximately 0.00055 u.

1 mole of nucleons ≈ 6.02 x 1023 u ≈ 10-3 kg = 1 g

Size of Nuclei

A Mass M volume V.

So the density ρ is roughly

independent of A.

34

3M r A

1/3

0r r A15

0 1.2 10 1.2r m fmwhere

fermi

Lecture 24-8 Binding Energy

The mass of a nucleus is less than the sum of the masses

of its parts!

m m(Z protons N neutrons) m(nucleus)

The mass defect, m, is the difference between the sum of the

masses of the protons and neutrons, and the mass of the nucleus.

2

BE m c The binding energy of the nucleus

represents the energy required to separate the nucleus into

individual nucleons.

Generally, a binding energy is the energy required to

separate a composite object into its constituent parts.

Lecture 24-9 How to find the binding energy

Mass of neutral atoms can be found in a table (e.g., NIST

table posted on the course home page under Lectures).

(Relative Atomic Weight in that table gives the atomic mass in u.)

To find the mass of the nucleus, you must subtract the

mass of the electrons contained within the neutral atom.

(But what about the binding due to electromagnetic forces?)

Example: 14N nuclear binding energy?

2 2( ) ( )m c m Z protons N neu m nuclt erons cus

Lecture 24-10

Neutral 14N atom = 14.003074 u

Mass of 7 electrons = 7 x me = 7 x 0.0005486 u = 0.003840 u

So 14N nuclear mass = 13.999234 u

Mass of 7 individual protons and 7 neutrons

= 7 x mp + 7 x mn = 7 x 1.0072765 u + 7 x 1.0086649 u

= 14.111589 u

So the mass defect

m = (14.111589 u) – (13.999234 u) = 0.112355 u

2 0.112355 931.494 /

104.659

BE m c u MeV u

MeV c2

Example: 14N nuclear binding energy?

Lecture 24-11 Nuclear Energy Levels

The nucleus has energy levels just like the electrons in an atom.

Protons and neutrons have separate energy levels.

They obey the exclusion principle and two of them can occupy

each level (one with spin up, one with spin down), like the electron.

The energy is lowest

with 6 protons and

6 neutrons, if

A=Z+N=12.

Lecture 24-12

Physics 219 – Question 2 – April 11, 2012.

Which description of the isotope is correct?

A. O (oxygen) with 8 protons and 6 neutrons

B. C (carbon) with 6 protons and 8 neutrons

C. Si (silicon) with 14 protons and 6 neutrons

D. Ca (calcium) with 6 protons and 20 neutrons

E. C (carbon) with 6 protons and 14 neutrons

14

6 ?

Atomic numbers of the above elements are:

carbon 6, oxygen 8, silicon 14, calcium 20.

Lecture 24-13 Composition of Nuclei

For smaller nuclides,

N=Z is most stable.

For bigger nuclides, the

Coulomb repulsion of

protons favor more

neutrons than protons to be

in the nucleus.

Some nuclides are

unusually stable: e.g.,

4 16 40 48 208

2 8 20 20 82, , , ,He O Ca Ca Pb

Lecture 24-14 Binding Energy Per Nucleon Curve

For smaller nuclides, binding gets tighter as

the mass number increases (as the nucleons

gain more neighbors to bind with). For

larger nuclides, the Coulomb repulsion

among the protons begins to make them less

tightly bound. The maximum binding occurs

around A=60.

tigh

ter

Lecture 24-15 Radioactive Decay

There are stable nuclides and unstable ones. An unstable

nuclide decays by emitting particles and/or radiation.

Most (~80%) of nuclides are radioactive,

including all those with Z > 83.

radioactive decay

There are 3 types of decays: alpha, beta, and gamma decays

Radioactive decays occur with a probability which depends

on the isotope and the type of decay.

Decays are random events, i.e., they don’t occur at predicted times.

Lecture 24-16 Conservation Laws in Radioactive Decay

1. The number of nucleons must remain the same

(though the types may change).

2. The total electric charge must remain the same.

3. The total energy must remain the same.

Energy here includes both the rest mass energy

and the kinetic energy. 2

0E mc

The sum of the masses of the decay products must

be less than the mass of the original nucleus in order

for a spontaneous decay from the nucleus at rest to

be possible. Disintegration energy is the name for that part of rest mass

energy of the original nucleus that is converted into other

forms of energy (such as kinetic energy or EM radiation).