Question

If the temperature of a good conductor decreases, how does the relaxation time of electrons in the conductor change?

Answer 1

The Short Answer

If the temperature of a good conductor decreases, the relaxation time of electrons increases.

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The Detailed Explanation

To understand why, let's break down the concepts involved.

1. What is Relaxation Time (τ)?

In a conductor like a metal, there is a "sea" of free electrons moving randomly. When a voltage is applied, these electrons are accelerated and drift in a specific direction, creating an electric current.

However, their path is not clear. They constantly collide with things inside the conductor. Relaxation time (τ) is the average time an electron travels freely between two successive collisions.

• Short Relaxation Time: Electrons collide very frequently.
• Long Relaxation Time: Electrons travel for a longer time before colliding.

2. What Do Electrons Collide With?

In an ideal, perfectly still crystal lattice, electrons could move almost without resistance. However, in a real conductor, electrons primarily collide with:

• Vibrating Lattice Ions (Phonons): The positive metal ions are not stationary; they vibrate around their fixed positions. The energy of these vibrations is directly related to the temperature of the conductor. These vibrations are the main source of scattering in a good conductor.
• Impurities and Crystal Defects: These are imperfections in the crystal structure. Their effect is largely independent of temperature.

3. The Role of Temperature

Temperature is a measure of the average kinetic energy of the particles in a substance.

• High Temperature: The metal ions have a lot of thermal energy, causing them to vibrate with a larger amplitude and higher frequency. This creates more "obstacles" and chaos for the drifting electrons, leading to more frequent collisions.
• Low Temperature: The metal ions have less thermal energy and vibrate much less. The lattice becomes more orderly and "calm." This provides a clearer path for the electrons, leading to less frequent collisions.

Putting It All Together

Let's trace the cause and effect of decreasing the temperature:

1. Temperature Decreases: The conductor cools down.
2. Lattice Vibrations Reduce: The metal ions have less thermal energy and vibrate with a smaller amplitude.
3. Fewer Collisions: With a more orderly lattice, the probability of an electron colliding with a vibrating ion decreases.
4. Relaxation Time Increases: Since collisions are less frequent, the average time an electron travels between collisions (the relaxation time, τ) gets longer.

Analogy: The Crowded Corridor

Imagine you are trying to run down a corridor filled with people (the lattice ions).

• High Temperature: The people are jumping around randomly and energetically. It's very difficult to run without bumping into someone. Your time between collisions is very short.
• Low Temperature: The people are standing almost perfectly still. It's much easier to run a long distance without hitting anyone. Your time between collisions is long.

Connection to Resistivity

This change in relaxation time directly explains why a conductor's resistance changes with temperature. The formula for electrical resistivity ($ρ$) is:

$ρ = m / (n e^2 τ)$

Where:
$m$ = mass of an electron (constant)
$n$ = number density of free electrons (constant for a given metal)
$e$ = charge of an electron (constant)
$τ$ = relaxation time

As you can see, resistivity ($ρ$) is inversely proportional to relaxation time ($τ$).

Therefore:
When temperature decreases, relaxation time ($τ$) increases.
As a result, resistivity ($ρ$) decreases.

This is why good conductors become even better conductors (less resistive) at lower temperatures.