# Solution to Exercise 3.2-3

| How large will be the distance d between the (center of gravity) of the positive and negative charges for reasonable field strengths and atomic numbers, e.g. the combinations of 1 kV/cm 100 kV/cm 10 MV/cm , the last one being about the ultimate limit for the best dielectrics there are, z = 1 (H, Hydrogen) z = 50 (Sn (= tin), ...) z = 100 (?)
| From the backbone we have a relation for d as a function of z,m the radius R of the atom, and the field strength E:
dE = | 4 p e0 · R 3 · E z e
--- | ---
| We need to look up some number for the radius of the three atoms given (try this link), then the calculation is straight forward - let's make a table:
Atom | R | d(1 kV/cm) | d(100 kV/cm) | d(10 MV/cm)
z = 1 |  |  |  |
z = 50 |  |  |  |
z = 100 |  |  |  |
|  | Compared to the radius of the atoms, the separation distance is tiny. No wonder, electronic polarization is a small effect with spherical atoms!
| Calculate the "spring constant" and from that the resonance frequency of the "electron cloud" (assume the nucleus to be fixed in space).
| If you don't know off-hand the resonance frequency of a simple harmonic oscillator - that's fine. If you don't know exactly what that is, and where you can look it up - you are in deep trouble.
|  | Anyway, in this link you get all you need. In particular the resonance (circle) frequency w0 of a harmonic oscillator with the mass m and the spring constant kS is given by
w0 | = | æ ç è | kS m | ö ÷ ø | 1/2
|  | How large are the spring constants? That is question already answered in the backbone, so we import the equation
kS = | æ ç è | (z e) 2 4 p e0 · R 3 | ö ÷ ø
| Again, let's make a table for the answers:
Atom | Spring constant | w0
z = 1 |  |
z = 50 |  |
z = 100 |  |

With frame

Exercise 3.2-3

Harmonic Oscillator

© H. Föll (Advanced Materials B, part 1 - script)