Atomic Spectra• Early experimental evidence for discrete energy

states in atoms: observation of the line spectra of hydrogen (Balmer, Rydberg, Ritz, 1885 – 1890)– Experimental setup:

• Light emitted at discrete wavelengths, separated by grating• Principle behind neon signs, sodium street lamps, etc.

(From Modern Physics for scientists and engineers, Thornton and Rex, 2nd ed.)

Line Spectra of Hydrogen• Balmer, Rydberg, and Ritz deduced an empirical

formula that predicted the observed wavelengths of lines in the hydrogen emission spectrum:

– RH = Rydberg constant = 1.0973732 107 m–1

– n, k are integers– k > n (always)– Understanding this equation theoretically was a hot topic in

the early 20th Century

22

111

knRH

Bohr Model of Hydrogen• To explain these results, Niels Bohr suggested

(1913) that the electron in hydrogen moved around nucleus (proton) in circular orbits– Similar to a miniature planetary system

– Coulomb attractive force keeps the electron in orbit (analogous to gravity)

• To maintain stable orbits, Bohr also suggested that the electron’s orbital angular momentum was quantized:

• In combination with Newton’s 2nd law, this leads to discrete radii for the electron orbits

– a0 = Bohr radius = 0.0529 nm

Bohr Model of Hydrogen

...3,2,12

nnnh

vrme

...3,2,1202

22

nnaekm

nr

een

Bohr Model for Hydrogen• From conservation of energy, the model predicts the

discrete total energies of the electron in hydrogen:

• Thus the energy levels form a series of states (like steps in a ladder), with transitions that can take place between the levels– Wavelengths of transitions are in

exact agreement with empirical result!

...3,2,1eV6.131

22 222

422

nnn

ekm

r

ekE eee

22

111

ifH nn

R

Interactive Example Problem: The Bohr Model

Animation and solution details given in class.

(ActivPhysics online problem 18.1, copyright Addison Wesley, 2006)

Bohr Model for Hydrogen• Although this model predicts the energies correctly,

it has several deficiencies:– Only useful for one-electron atoms (only considers e– –

nucleus force, not e– – e– forces)– Unable to account for doublets of spectral lines in

emission spectra (2 or more very closely spaced lines)– Cannot calculate transition intensities– Violates Uncertainty Principle (for circular orbit, we would

know r and pr exactly, which is not possible)

• The correct picture of the atom includes the wave attributes of the electron– Electron’s position is “smeared out,” and we are left with

electron “clouds” surrounding the nucleus

Examples of Electron Cloud Distributions

How Lasers Work• Laser = Light Amplification by Stimulated Emission

of Radiation• Stimulated

emission:

– Note difference from spontaneous emission• To produce more stimulated-emission transitions

than absorption transitions, need more atoms in excited metastable state (“population inversion”) then probability of absorbtion probability of stimulated emission

How Lasers Work• For helium – neon laser:

• Schematic of He – Ne laser:

1.20.6 eV

Helium (He)E1

E2

1. Absorption from electric current

2.

2. He atoms collide with Ne atoms

E3

4. 1.96 eV

4. Spontaneous emission of 1.96 eV photons ( = 632.8 nm, red light) produces stimulated emission of other 1.96 eV photons

Rear

Flat mirror, 100% reflective Concave mirror, reflects 99%, transmits 1%

FrontParallel laser beam

5.

5. Spontaneous emission transitions “recycle” process

E2

E1

E4 = 20.6 eV3.

3. E4 (metastable state) in Ne is populated (pop. inversion)

Neon (Ne)

Laser tube

The Atomic Nucleus• Nuclei consist of both protons and neutrons

(collectively called nucleons)

• For a given element, there are typically several nuclear isotopes having different A but same Z

Designate nucleus by: AZXN (example: )

Z = atomic number = # protons (names the element)

N = neutron number = # neutrons

A = mass number = Z + Nproton (p)

mass = 1.0072765 ucharge = +espin = ½

neutron (n)

mass = 1.0086649 ucharge = 0spin = ½

(1 u = 1.66054 10–27 kg = 931.49 MeV / c2)

6126 C

Image courtesy of ANL

Nuclear Sizes• Nuclear sizes can be inferred from scattering

experiments, where the “probe” must have a deBroglie wavelength on the order of the size being investigated (~ 1 fm)

• From these experiments, we learn:– Nuclear density is roughly a constant (2.3 1017 kg/m3)– Nuclear force radius Mass radius Charge radius– Can approximate nuclear radius assuming a spherical

charge distribution as R = r0 A1/3 where r0 = 1.2 fm

– 1 fm = 1 10–15 m (“femtometer” or “fermi”)

Beam of particles (e– , p, , n, etc.)

Nucleus

Scattered particles

Nuclear Binding Energy

The Nuclear Force• The nuclear force must be strong in magnitude and

attractive to prevent Coulomb repulsion from blowing the nucleus apart (known as strong force because it’s the strongest of the known forces)

• Nucleons themselves provide the forces• The most straightforward way to probe this force is

through scattering experiments• There are n–p, p–p, and n–n forces (and potentials)

n p

p

n

V(r)

rn–p

p p

p

p

V(r)

r

p–p

The Nuclear Force and Models• Protons and neutrons experience potential wells in

nucleus, and fill discrete energy states

• However, we do not fully understand the nuclear force or how nucleons interact inside the nucleus no unified descriptive theory exists for the nucleus!

• Variety of models are used to describe nuclear behavior– Nucleons move nearly independently in a common

nuclear potential (Independent Particle Models)– Nucleons are coupled strongly together and move

collectively (Strong Interaction Models)

rr

V(r)

~ – 43 MeV~ – 37 MeV

neutrons protons

Nuclear Models• Independent Particle Models

– Nucleons move nearly independently in a common nuclear potential

– Nucleons can be thought of as exerting forces on one another through the exchange of pions (an elementary particle with rest energy of 140 MeV)

– Spherical Shell Model– Deformed Shell Model

• Strong Interaction Models – Nucleons are strongly coupled together and move

collectively– Liquid Drop Model– Rotational Model– Vibrations

(image above from Krane, Modern Physics, 2nd ed.)

Basic Research in Nuclear Physics• Nuclear structure studies

– Individual particle motion: How occupation of single-particle orbits affect shape of nucleus

– Collective motion: How rotations and vibrations affect shape of nucleus

– Interplay between individual-particle and collective motion– Deformation studies from charge-distribution/mean lifetime

measurements– Pairing between protons and neutrons

• Nuclear reaction studies– Cross section (probability) measurements– Use polarized projectiles to learn how spin affects reactions

• Nuclear Astrophysics– Origin and synthesis of the elements

Applications of Nuclear Physics

• Medicine (gamma radiation used for imaging)

• Time dating (lead and carbon dating)

• Art (determines trace elements of paints)

• Agriculture (gamma radiation kills bacteria)

• Small power systems (used in Apollo crafts)

• Fission/fusion reactors

• Knowledge of the origin of the elements (production via nuclear reactions)

Image courtesy of ANL and NASA

How Nuclei are Made in the Lab

Radioactive Decay• A nucleus can release three forms of radiation

– Gamma () decay• Nucleus is in an excited energy state• A series of gamma rays (high-energy photons) is released until

the nucleus reaches its lowest-energy (ground) state• Nuclear isotope remains the same (no change in Z or A)

– Alpha decay• Heavy nucleus has too large a ratio of neutrons to protons relative

to stable nuclei• Nucleus emits particle (4He nucleus)• Nucleus loses two protons and two neutrons

– Beta decay• Nucleus has too many protons or neutrons relative to stable nuclei• If too many protons, p n and a positron (e+) is emitted to

conserve charge (+ decay)• If too many neutrons, n p and an electron (e–) is emitted to

conserve charge (– decay)

Radioactive Decay• Example of + decay

– 148O6 14

7N7 + + + (14O unstable, 14N stable)– + particle is a positron

• Example of – decay

– 146C8 14

7N7 + – + (14C unstable, 14N stable)

– – particle is an electron

• Example of decay

– 23092U138 226

90Th136 + (230U unstable, 226Th unstable and will also undergo decay)

– particle is a 4He nucleus

Radioactive Decay• All three forms of radioactive decay

follow the exponential decay law– = decay constant– N = number of radioactive nuclei present at time t

– N0 = number of radioactive nuclei present at time t = 0

– e = 2.718 … = Euler’s constant

• The decay rate (or activity R) follows the same exponential decay law– R = activity at time t

– R0 = activity at time t = 0

• R and N are related by:

teNN 0

teRR 0

NR

Radioactive Decay• Plot of N (or R) vs. time:

• The time it takes for half of a given number of radioactive nuclei to decay = half life = T1/2

693.02ln

2/1 T

Radioactive Decay Experiment

Example Problem #29.36

Solution (details given in class):

9.96 103 yr

A living specimen in equilibrium with the atmosphere contains one atom of 14C (half-life = 5730 yr) for every 7.70 1011 stable carbon atoms. An archeological sample of wood (cellulose, C12H22O11) contains 21.0 mg of carbon. When the sample is placed inside a shielded beta counter with 88.0% counting efficiency, 837 counts are accumulated in one week. Assuming that the cosmic-ray flux and the Earth’s atmosphere have not changed appreciably since the sample was formed, find the age of the sample.

Example Problem #29.51

Solution (details given in class):(a) 8.97 1011

(b) 0.100 J(c) 100 rad

A patient swallows a radiopharmaceutical tagged with phosphorus–32 (32P), a – emitter with a half-life of 14.3 days. The average kinetic energy of the emitted electrons is 700 keV. If the initial activity of the sample is 1.31 MBq, determine (a) the number of electrons emitted in a 10–day period, (b) the total energy deposited in the body during the 10 days, and (c) the absorbed dose if the electrons are completely absorbed in 100 g of tissue.