Consider a collision between an x-ray photon of initial energy 50.0keVand an electron at rest, in which the photon is scattered backward and the electron is knocked forward.

(a) What is the energy of the backscattered photon?

(b) What is the kinetic energy of the electron?

Short Answer

Expert verified

(a) The energy of the backscattered photon is41.8 keV.

(b) The kinetic energy of the electron is 8.2 keV.

Step by step solution

01

Write the given data from the question.

Initial energy of the photon,

02

Determine the formulas to calculate the energy of the backscattered photon and kinetic energy of the electron. 

The expression to calculate the initial energy is given as follows.

…(i)

Here,h is the plank’s constant and c is the speed of light.

The expression to calculate the change in the wavelength is given as follows.

Δλ=hmc(1cosθ) …(ii)

Here,m is the mass of the electron.

The expression to calculate the kinetic energy of the electron is given as follows.

K=E-E' …(iii)

Here, E'is the energy of backscattered photon.

03

Calculate the energy of the backscattered photon.

The value of plank’s constant is, 6.62×1034 mkg/s.

The value of mass of electron is,9.11×1031 kg.

Since the photon is backscattered and electron is knocked forward, therefore the angle, θ=180°.

Substitute180°forθinto equation (ii).

Δλ=hmc(1cos180°)Δλ=hmc(1(1))Δλ=2hmc

The new wavelength is given by,

role="math" localid="1663071812908" λ'=Δλ+λλ'=λΔλλ+1

Calculate the energy of photon when it is backscattered.

E'=hcλ'

Substitute λΔλλ+1forλ'into above equation.

E'=hcλΔλλ+1E'=hcλΔλλ+1

Substitute E forhcλinto above equation.

E'=EΔλλ+1

Substitute2h/mcforΔλand hc/Eforλinto above equation.

E'=E2hmchcE+1E'=E2hEmhc2+1E'=E2Emc2+1

And,

mc2=9.11×1031 kg×(3×108 m/s)2=8.199×1014 J=8.199×1014×6.242×1015 keV=511.78 keVmc2511 keV

Substitute50 keVfor E and 511 keVfor mc2into above equation

E'=50 keV2×50 keV511 keV+1E'=50 keV100511+1E'=50 keV1.1956E'=41.8 keV

Hence the energy of the backscattered photon is .41.8 keV

04

Calculate the kinetic energy of the electron.

Calculate the kinetic energy of electron.

Substitute41.8 keVfor E'and 50 keVfor E into equation (iii).

K=50 keV41.8 keVK=8.2 keV

Hence the kinetic energy of the electron is8.2 keV.

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