CH 101/Exercises 11

From WolfWikis

Energy levels

Kinetic energy

Particles can move at a lot different speeds, just like a car can. A car can go at 12.45 miles per hour but also at 12.46 or 75.21 mph and anything inbetween, although the driver might be ticketed in the latter case. Depending on the mass m and the speed v the particle (or car) has a certain kinetic energy: E_k=½ mv².

Standing and traveling waves

Waves are a bit different. They can either move or be caught in a standing wave pattern.

1. Traveling waves have a kinetic energy. This energy is continuous: it can have any value in a certain range.

   For a photon that is simply E=hν

   For an electron it is ½ mv². (And yes, an electron is a wave too, because of the duality of matter.)

2. Standing waves have a potential energy. The energy of a standing wave is discrete, i.e. it can only have very specific values and nothing inbetween.

   For electrons that energy depends on how tightly the electron is tethered to the nucleus it is held captive by. Each states is characterized by one or more quantum numbers. These are integers.

Schematic diagram of energy states

Bound and free electrons. The photoelectric effect

If an electron is bound, i.e. tethered to an atom (or a molecule) it is in a standing wave pattern with a certain discrete energy. If sufficient energy is added (e.g. by hitting it with a photon of the right wavelength) the tether can be broken. The smallest energy to do that is called the ionization energy (see the blue arrows in the diagram). It brings the electron to the bottom of the continuous vacuum states. These are traveling wave states so that the electron can travel away from the the atom, leaving behind a positively charged ion. If the photon has more than the required energy to ionize the atom (or molecule) the surplus energy will be given to the electron as kinetic energy (see the white arrows in the diagram). (In other words it will just fly away a bit harder.)

We can also look at the ionization energy upside down: We can also call it the binding energy of the electron (as it was before we hit it), although there are some subtle differences between these two terms.

It is convenient to choose the bottom of the vacuum continuum as the origin E=0 of the energy scale. The energies of the bound states are then all negative.

Photoelectron spectroscopy: measuring binding energies

The energies of the bound states can be measured, albeit a bit indirectly. You hit the material with photons of a particular well-known energy (read: wavelength). Usually this is a very high energy, like an X-ray photon. This means that more than enough energy is available to promote any bound electron into a vacuum state and then some. The surplus kinetic energy can now be measured by looking at how fast the electron is flying away. Because we know how much energy we provided (E_photon) and we know the surplus (E_kinetic of the ejected photoelectron.) we can figure out how strongly it was tethered before we hit it with the photon. Because all energies must add up to zero we get:

E_photon = E_binding + E_kinetic

(Actually, if you do this on a solid, it turns out that there is a small extra term, known as the work function Φ of the spectrometer.)

Calculating the binding energies

There are a host of ways to calculate binding energies and people make their living making such calculations. They often involve big computers and a lot of fancy math. The simplest model and the only one you'll do anything numerical with, is known as the Bohr model.

Interestingly this model is older than the theory of particle-wave duality, but it does allow us to calculate the possible binding energies of the single electron of a hydrogen atom. (Actually it also works well for any other entity that has only one electron, but those are rather exotic entities like He⁺, Li²⁺ or U⁹¹⁺.)

Each level is characterized by one single quantum number n and can be calculated form:

E_n= - hR(Z/n)²

h= Planck's constant

R= Rydberg constant

hR= 2.18x10^-18 Joule

Z= the charge of the nucleus = +1 for hydrogen.

(Notice the negative sign in the formula!)

Exercises

Consider a single hydrogen atom in vacuum.

1. Calculate the ionization (binding) energy of a H atom, when its electron is in the state n=2 and n=17.
2. The bottom of the vacuum continuum corresponds with n= infinity. Why?
3. What is the wavelength of the photon that can provide just enough energy to bring an electron with n=4 to the bottom of the continuum?
4. A UV light with a wavelength of λ=250nm is aimed at a gas of hydrogen atoms. Their electrons are either in the states n=1, 2 or 3. The kinetic energy of the liberated electrons are measured. Which value(s) do you expect to find for E_kinetic?
5. What is the energy required to promote an electron from n=1 to n=2?
6. What is the wavelength of a photon emitted when an electron goes from n=3 to n=2?