# 有關固體的問題

圖片參考：http://imgcld.yimg.com/8/n/HA00414085/o/7010120400...

The force-separation graph (dotted line) shows the interatomic force between two atoms. For r~r0, the curve is a straight line and the atoms vibrating about their equilibrum position are in simple harmonic motion (SHM). Is that true for the case that of many atoms? Briefly explain.

What is the minimum separation between two atoms for which they have already changed from solid into liquid?

The graph shows the interatmoic potential energy between two atoms represented by another curve. What does the value "ε" meant if the line XY up to the level in which they touched the point of inflexion?

The force-distance relationship is represented by the dotted line shown in the figure.

The minimun value of the force occurs at the seperation r=Z, is known as the breaking force.

What is this breaking force meant?

To change the solid into liquid?

Or it is the force that the material passes the elastic limit?

### 3 個解答

- 1 十年前最愛解答
(1) In a solid with many atoms, one can model this by considering connecting atoms by springs. This is different from thinking each atom independently sitting at a potential well. The atom-spring-atom description is a more realistic one because this has sound wave (or phonons) as a low energy excitation of the system.

(2) The point Z describes the length scale for solid-liquid transition. (The exact density depends on crystal details.) To explain this, we need to understand why crystalline order will be destroyed when inter-atomic spacing goes beyond Z. Consider for simplicity, a 1D chain of atoms (see footnote 1) like so: (a=atom)

......a......a......a......

Say due to temperature, there is a fluctuation of the middle atom

......a.........a...a......

If the atomic spacing is less than Z, the attraction of the left atom overwhelms the right one, and the middle atom wants to go back to the original position, i.e. to restore crystalline order.

If however the atomic spacing is longer than Z, the middle atom is more attracted to the right atom (see the dotted curve). And hence, it will not go back to the original configuration, and crystalline order is destroyed.

(3) The energy E is the excitation energy from the bottom of the potential profile (solid graph) to the inflexion point. The potential minimum is the energy a classical atom (see footnote 2) in a crystal would have in zero temperature, i.e. no oscillation. Since the inflexion point characterizes solid-liquid transition, E gives a rough energy scale for the critical temperature T_c, where E~k T_c and k is the Boltzmann constant. The phase transition temperature (melting point) should be close to T_c (meaning has the same order of magnitude at least in a mean field approximation).

Footnote:

1). In 1D, there is no long range order, so a solid liquid transition may be ill defined. But as an illustration it is fine, and one can easily generalize the same idea to higher dimensions.

2). Assume no quantum effects.