Atomic Physics

 

28.1 Early Models of the Atom

 

 

 

The modern model of the atom is born. But:

 

1. Does not explain why there are spectral lines.

 

2. Classical electrodynamics rears its ugly head: orbiting electrons should continuously radiate light, lose energy, and fall into the nucleus. They don’t.

 

28.2 Atomic Spectra

 

 

Each element has its own characteristic “fingerprint”.

 

A heated gas emits emission lines.

 

The same gas, when placed between a bright continuous background source and the observer produces absorption lines at those same wavelengths.

 

 

Example: Problem #4

The “size” of the atom in Rutherford’s model is about 1.0 x 10¹⁰ m. (a) Determine the speed of an electron moving in a circle around the proton using the attractive electrical force between an electron and a proton separated by this distance. (b) Does this speed suggest that Einsteinian relativity must be considered when studying the atom? (c) Compute the de Broglie wavelength of the electron as it moves about the proton. (d) Does this wavelength suggest that wave effects, such as diffraction and interference, must be considered when studying the atom?

28.3 The Bohr Theory of Hydrogen

 

1913  Bohr postulates a “semi-classical” description of how the H atom works.

 

The total energy is  

We also have   

So  

 

Now, let’s look at what happens due to the condition of the angular momentum having fixed values.

 

The smallest allowed orbit occurs when n=1, and it is called the Bohr radius:

 

 

 

The radii of the rest of the orbits will simply be  

 

Substituting the equation for the radii into the energy equation gives:

 

 

According to Bohr, the frequency emitted by the atom when it goes from an initial energy state to a final energy state is then just

 

 

 

Example: Problem #14

A hydrogen atom initially in its ground state (n = 1) absorbs a photon and ends up in the state for which n = 3. (a) What is the energy of the absorbed photon? (b) If the atom eventually returns to the ground state, what photon energies could the atom emit?

28.4 Modification of the Bohr Theory

 

Successes of the Bohr Model:

 

• Explained all of the series of emission lines of H

 

• Predicted the correct value of R_H

 

·      Predicted the correct size of H

 

• Predicted the correct energy levels of H

 

It also worked for other “hydrogenic” atoms  those with a single electron

 

 

Example: Problem #27

(a) Find the energy of the electron in the ground state of doubly ionized lithium, which has an atomic number Z = 3. (b) Find the radius of its ground-state orbit.(Note: this is just Example 28.3 done for lithium instead of helium).

 

Did not work well with multi-electron atoms: modifications needed!

 

 

Maximum number of electrons per subshell is  

 

 

 

 

Fine structure splitting requires yet another one, the spin magnetic quantum number  

 

28.5 De Broglie Waves and the Hydrogen Atom

 

Okay, why did Bohr’s postulate for “quantized angular momentum” actually work for explaining the H atom? DeBroglie suggested that the electron’s wave pattern must be such that an integral number of waves would fit in the orbit, similar to a standing wave (but with its tail in its mouth!).

 

In this interpretation, the n=1 level contains 1 whole wave, the n=2 level has 2 whole waves, etc.

 

 

 

 

Example: Problem #31

Determine the wavelength of an electron in the third excited orbit of the hydrogen atom, with n = 4.

 

 

28.6   Quantum Mechanics and the Hydrogen Atom

 

Mathematically (without making any ad hoc assumptions), the full-blown solution to the wave equation automatically yields:

 

• the principle quantum number
  
   which determines the energies of the orbits

 

• the orbital quantum number
  
   

 

• the orbital magnetic quantum number  
  

 

• the rules that govern the values that these quantum numbers can have.

 

 

 

 

28.7   The Spin Magnetic Quantum Number

 

The splitting of the yellow sodium line into two separate lines of slightly different energies indicates that one more quantum number was required. This is the so-called spin quantum number .

 

 

Example: Problem #34

List the possible sets of quantum numbers for electrons in the 3d subshell.

 

 

28.8   Electron Clouds (read)

 

We no longer think of the electrons as particles orbiting in specific orbits. Rather, the wave function  (or more specifically its square) tells us the probability distribution of the electron.

 

 

28.9   The Exclusion Principle and the Periodic Table

 

Okay, atoms can have many energy levels. Why are there different elements? Why do they have different chemical properties? Why don’t all atoms have all of their electrons in the 1s orbital?

 

Pauli Exclusion Principle: No two electrons can ever be in the same quantum state; that is, no two electrons in the same atom can have exactly the same values for .

 

Each possible substate with 4 quantum numbers can only hold 1 electron! So in an atom with many electrons, the lower levels sequentially “fill up”, forcing the rest into higher

substates.

 

 

 

The Atomic Number which defines the element, is the number of protons in the nucleus. For a neutral atom, this must be equal to the number of electrons it has. The more it has, the higher will be the substates that contain these electrons. The chemical properties are governed by the number of electrons in the outermost substate only.

 

 

 

Example: Problem #36

(a)  Write out the electronic configuration of the ground state for oxygen (Z = 8). (b) Write out values for the set of quantum numbers n, , , and m_s for each of the electrons in oxygen.

 

28.11      Atomic Transitions (read)  The Truth about Transitions

 

Almost all electronic transitions that occur in atoms that involve photons fall into one of three categories:

 

• Stimulated absorption

 

• Spontaneous Emission

 

• Stimulated Emission