Question

The maximum Compton shift in wavelength occurs when a photon isscattered through 180^\circ .

What scattering angle will produce a wavelength shift of one-fourththe maximum? Express the answer as a whole number indegrees.

Answer 1

Answer:
`90\°`

Step-by-step explanation:

The Compton Shift
`\Delta \lambda` in wavelength when the photons are scattered is given by the following equation:

`\Delta \lambda=\lambda_(c)(1-cos\theta)` (1)

Where:

`\lambda_(c)=2.43(10)^(-12) m` is a constant whose value is given by
`(h)/(m_(e)c)`, being
`h` the Planck constant,
`m_(e)` the mass of the electron and
`c` the speed of light in vacuum.

`\theta)` the angle between incident phhoton and the scatered photon.

We are told the maximum Compton shift in wavelength occurs when a photon isscattered through
`180\°`:

`\Delta \lambda_(max)=\lambda_(c)(1-cos(180\°))` (2)

`\Delta \lambda_(max)=\lambda_(c)(1-(-1))`

`\Delta \lambda_(max)=2\lambda_(c)` (3)

Now, let's find the angle that will produce a fourth of this maximum value found in (3):

`(1)/(4)\Delta \lambda_(max)=(1)/(4)2\lambda_(c)(1-cos\theta)` (4)

`(1)/(4)\Delta \lambda_(max)=(1)/(2)\lambda_(c)(1-cos\theta)` (5)

If we want
`(1)/(4)\Delta \lambda_(max)=(1)/(2)\lambda_(c)`,
`1-cos\theta` must be equal to 1:

`1-cos\theta=1` (6)

Finding
`\theta`:

`1-1=cos\theta`

`0=cos\theta`

`\theta=cos^(-1) (0)`

Finally:

`\theta=90\°` This is the scattering angle that will produce
`(1)/(4)\Delta \lambda_(max)`