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# 3.2 Conditions for degeneracy

Degeneracy can be described in two equivalent ways:

1. The separation between particles is less than their de Broglie wavelength,

Equation label:(3)

2. The number of particles per unit volume, n, is greater than the number of available quantum states nQ known as the quantum concentration,

Equation label:(4)

Before going further, complete the following activity to confirm to yourself that these two conditions are equivalent.

## Activity 2

Beginning with the second statement of the condition for degeneracy, , use the definition of the quantum concentration to derive an expression for the mean separation of the particles, in terms of the de Broglie wavelength. Show that this leads to the first statement for the condition of degeneracy.

Hint: if the mean separation of particles is , then the number of particles per unit volume is .

### Discussion

The second condition is , i.e. .

Because , substituting gives .

Taking the (1/3)-power and rearranging the result gives

That is, from the second degeneracy condition, , we obtain the first degeneracy condition, .

You have now shown that the two conditions for degeneracy are actually equivalent; they are merely alternative expressions of the same physics. In the following activity, you will see how a third such expression is derived, which is also equivalent.

## Activity 3

1. Rewrite the second description of degeneracy, , to find a limit for the thermal energy . State a third, equivalent condition for degeneracy as a limit on the temperature.

2. Calculate values of the quantity in the core of the Sun for (i) protons and (ii) electrons. Assume that the mass fractions of hydrogen and helium are and that the core density of the Sun is . Note that in the solar core, all atoms are fully ionised and that the number density is for each type of particle.

   The mass of an electron is kg, the mass of a hydrogen nucleus is kg and the mass of a helium nucleus is where u is the atomic mass unit given by kg. As usual Planck’s constant is and Boltzmann’s constant is .

3. Consider the results to part (b), and state whether (i) protons and (ii) electrons are degenerate in the Sun’s core. Assume that the core temperature of the Sun is .

### Answer

1. Since , so the degeneracy condition implies that .

   Taking the (2/3)-power and multiplying both sides by gives , i.e. .

   The third, equivalent condition for degeneracy, in relation to temperature, is: the gas is degenerate if its temperature .

2. (i) For protons, (i.e. hydrogen nuclei per m3).

   Therefore, for protons, the temperature-related degeneracy condition is

   (ii) In the solar core, all atoms are ionised. The electrons are provided by the hydrogen (1 electron per atom) and helium (2 electrons per atom), which each account for 0.5 of the composition by mass. So

   Therefore, for electrons, the temperature-related degeneracy condition is

3. The temperature condition for degeneracy is . Since , so the temperature in the core of the Sun is (i) much too high for proton degeneracy to have set in, and (ii) marginally too high for electron degeneracy to have set in.

You have seen in the previous activity that a third equivalent expression for the condition of degeneracy can be written as follows:

Equation label:(5)

Sometimes, this third condition is used to describe a degenerate gas as cold, because its temperature falls below some limit. However, this can be misleading because electrons may become degenerate at temperatures of millions of kelvins, which is obviously not cold in the common usage of the word.

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