Atomic Spectra If you “engergize” an atom in a variety of ways, it will give off light with characteristic wavelengths What wavelengths you get are a characteristic of the atom Diffraction Grating There were patterns in the wavelengths, but no one understood why Triplets were not uncommon Later, more lines were discovered Hydrogen had the simplest pattern

Sample Problem: For what values of m and n will you get red light, with wavelength between 625 and 740 nm? If m is 3 or bigger, you can’t get bigger than 1/9 = If m is 1, then n is two or bigger, and the difference is no smaller than 0.75 m = 2 You need to subtract at least 0.1  n is not too large n = 3

Rutherford Scattering Radioactive decay produces alpha particles Charge +2e Mass = Helium atom mass times heavier than electron Expected it would deflect – slightly – when it went through an atom 1909: Geiger and Marsden perform this experiment under the direction of Ernest Rutherford 

What they found: Alpha particles would deflect, occasionally, by large angles Even backwards Rutherford concluded that the positive charge was concentrated in a tiny region called the nucleus  +  b r q = 2e Q = Ze If you hit Gold (Au, Z = 79) with E kin = 3.0 MeV  -particles, 1.How close do you have to get to be deflected by 90  ? 2.What is the area of the target to be deflected by 90  ? Continued... b = impact parameter

Rutherford Scattering (cont.) fm deflects > 90  deflects < 90  Target area:  =  b 2 = 4512 fm 2 If you hit Gold (Au, Z = 79) with E kin = 3.0 MeV  -particles, 1.How close do you have to get to be deflected by 90  ? 2.What is the area of the target to be deflected by 90  ? The fact that this formula worked told Rutherford the nucleus was smaller than this, smaller than 38 fm in radius...

How small is the nucleus? How small a nucleus can one see by this method? You can’t see the nucleus until you get close enough Closest you can come happens when you are headed straight for the nucleus  + Conservation of energy gives you the closest approach Rutherford realized he needed higher energies and smaller Z He used Aluminim (Z = 13) with E kin = 7.7 MeV He saw that the scattering changed, and concluded that the nucleus was a few fm across

The atom and the Solar System The Solar system Sun’s mass is 1048 times Jupiter’s mass Radius of Neptune’s orbit is 6500 times radius of Sun Sun stays put and all planets orbit it Atoms are about a = 0.1 nm in radius Nucleus is about R = 10 fm in radius The hydrogen atom Nuclear mass is 1836 times electron’s mass Radius of atom is about 10,000 times radius of nucleus Could the atom be like a mini-solar system?

The Rutherford model of the atom 1911: Rutherford develops his model Nucleus at the center, nearly stationary Electrons orbit nucleus in ellipses, like planets + e-e- e-e- e-e- e-e- Problems with the model: Accelerating electrons should radiate, and drop into nucleus What about electron-electron interactions? No predictions of spectra of atoms

Equations you need Values of h, h-bar, e, k, k B, m e

The Bohr Model of the atom : Extension of Rutherford model, with three new assumptions 1.Electrons only go in circular orbits due to attraction of nucleus 2.When the electron changes its configuration, energy is emitted in the form of a single photon of energy 3.The electrons must have angular momentum that is an integer multiple of e-e- Electrostatic Attraction: Centripetal Force: Subs

The Bohr Model of the atom (part 2) + e-e- Define: Bohr Radius Diameter of H-atom for n = 1 is nm! Define: Fine Structure Constant Electrons are non-relativistic in hydrogen But wait, there’s more!

The Bohr Model of the atom (part 3) Energy has two components: Kinetic energy: Potential energy* * Note: the sign of this equation assumes the force F is directed inwards. + e-e-

The Bohr Model of the atom (part 4) What happens when the electron “jumps” from one level to another? 2.When the electron changes its configuration, energy is emitted in the form of a single photon of energy Define: Rydberg Constant Compare: + e-e- level n level m

Reduced Mass + e-e- Calculated treating nucleus as stationary Proton weighs 1836 times more than electron In fact, proton moves a very little bit as electron orbits it Can be corrected for by using “reduced mass” m e   The real formula for energy levels of Hydrogen: Spectroscopy can easily tell the difference Slight corrections for isotopes 0.027% for 1 H vs. 2 H This is why wavelengths were slightly off Huge correction for “unusual” atoms

Sample Problem Positronium consists of a bound state of an electron (m = MeV/c 2 ) and an anti-electron, or positron (same mass). What is the binding energy of positronium in the ground state? e-e- e+e+

Hydrogen-like atoms Atoms with one electron in them H, He +, Li +2, Be +3,..., U +91 Bohr model could be easily modified to make them fit e 2  Ze 2 Electrostatic Attraction: + Approx.Exact Note: This ignores relativity

Sample Problem What is the energy and wavelength of light emitted when a Ca +19 (Z = 20 ) electron jumps from level 7 down to level 6?

Multiple Electron Atoms: X-rays Could it be used to explain other atoms, with multiple electrons? Generally failed Succeeded approximately for innermost electrons Fast moving electron hits an inner electron and knocks it free Outer electron falls to lowest level and fills in the void Measure wavelength/energy of emitted photon + e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- Why it might work: Innermost electron is near nucleus with large charge Other electrons are outside of it ?

X-rays from inner electrons 1913 Henry Moseley measures X-ray spectra Finds this relationship almost works for inner electrons ? Why Z – 1? Innermost electron slightly screened by other electron Also found a relationship for next level out + e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- There are more electrons screening the nucleus

Testing one of Bohr’s Hypotheses 2.When the electron changes its configuration, energy is emitted in the form of a single photon of energy If the atom has only discrete energy levels, then there should be a minimum amount of energy that can be added to it Anything that collides with it with less energy must collide elastically When it does collide inelastically, the energy will be transmitted to the atom The atom then has to release a photon to lose a level + e-e- e-e- e-e- Predictions: When you hit the threshold of energy, scattering (energy loss) will suddenly increase The target atoms will start to glow at threshold energy

The Franck-Hertz Experiment +– +– e-e- Experiment performed, 1914 At moderate voltage, current is high Drops suddenly as you pass threshold Drops periodically at higher voltage Gas glows at predicted wavelength above first threshold heater thin gaswire screen V measure current

Assessing the Bohr Model The Good: Accurately predicted hydrogen line spectrum Predicted isotopic and other effects Did pretty well at predicting X-ray lines for heavier elements Clear evidence that atoms had “levels” of energy Evidence that quantum effects would help describe the atom The Bad: No idea how to proceed for heavier elements “Strength” of spectral lines was not predicted The Ugly: Arbitrary and unmotivated assumptions