# Planck's Quanta

/Physics students often ask how Max Planck's concept of energy quanta explains the shape of the blackbody radiation curve at all frequencies. How is it if we make the assumption * E=hf* we are able to avoid the prediction of classical wave theory that an infinite amount of energy is released at high frequencies (the ultraviolet catastrophe) ?

### What is Planck's quantum hypothesis?

In 1901 Max Planck proposed that the vibrating atoms in the walls of hot objects can only have a discrete **set of energy values**. The energy of the oscillator is said to be **quantised** as it can only have certain **quantities** of energy given by the equation * E=nhf* where n=1,2,3.... The energy emitted by the atoms is in bundles of value

*where*

**hf****is the frequency of oscillation and**

*f**is Planck's constant.*

**h**### How does quantisation explain the shape of the blackbody radiation curve?

The first point to note is that **not all of the atoms in the hot object are vibrating with the same energy**. The number of atoms vibrating at energy * E* is proportional to

*raised to the power of*

**e***where k is Boltzmann's constant and T is the kelvin temperature.*

**-E/kT***The exponential factor is a statistical factor that describes the spread of vibration energies throughout the object in much the same way as there is a range of heights of people in the population. The second point to note is that when the average vibration energy of an atom is calculated*

**the energy quantisation rule**(

*) causes a geometric series to be formed that has a*

**E = nhf****limiting sum and so the ultraviolet catastrophe is avoided.**The result is that

**a very large number of atoms**in the hot object

**vibrate at low frequencies, a large number of atoms vibrate at intermediate frequencies**and

**a**

**relatively small number vibrate at high frequencies**. The energy-frequency graph therefore has a

**peak in the midrange**(where most of the energy is released due to the large number of vibrating atoms in this range) and is

**small in height at the extremities**(since a very large number of atoms vibrating at a low frequency gives a low energy output and a small number of atoms vibrating at a high frequency also gives a low energy output).