Production of X-rays (AQA A Level Physics): Revision Note

X-rays are a type of electromagnetic wave with wavelengths in the range 10⁻⁸ to 10⁻¹³ m

If the accelerating potential difference in an X-ray tube is 60 kV, determine if the photons emitted fall within this range.

[3]

Answer:

Step 1: Write out known quantities

• Charge on an electron, $e=1.6\times {10}^{-19}\mathrm{C}$

• Accelerating potential difference, $V=60000\mathrm{V}$

• Planck’s constant, $h=6.63\times {10}^{-34}\mathrm{J}\mathrm{s}$

• Speed of light, $c=3\times {10}^8\mathrm{m}{\mathrm{s}}^{-1}$

Step 2: Determine the maximum possible energy of a photon

• The maximum possible energy of a photon corresponds to the maximum energy an electron could have:

$E_{max}=eV$

Step 3: Determine an expression for minimum wavelength

Planck relation: $E=hf$

Wave equation: $c=f\lambda$

• When energy is a maximum:

$E_{max}=eV=h{f}_{max}$

• Maximum energy corresponds to a minimum wavelength:

$eV=\frac{hc}{{\lambda }_{min}}$

• Rearrange for minimum wavelength, ${\lambda }_{min}$:

${\lambda }_{min}=\frac{hc}{eV}$

Step 4: Calculate the minimum wavelength ${\lambda }_{min}$

${\lambda }_{min}=\frac{\left(6.63\times {10}^{-34}\right)\left(3\times {10}^8\right)}{\left(1.6\times {10}^{-19}\right)\left(60000\right)}$ [1 mark]

${\lambda }_{min}=2.1\times {10}^{-11}\mathrm{m}$ [1 mark]

Step 5: Comment on whether this is within the range for the wavelength of an X-ray 

• X-ray wavelengths are within 10⁻⁸ to 10⁻¹³ m

• The minimum wavelength for a 60 kV supply is 2.1 × 10⁻¹¹ m, which means the photons produced will be X-rays [1 mark]

A typical spectrum of the X-ray radiation produced by electron bombardment of a metal target is shown below.

Explain why:

(a) A continuous spectrum of wavelengths is produced. [2]

(b) The gradient is steeper at shorter wavelengths. [2]

(c) The spectrum has a sharp cut-off at short wavelengths. [2]

 Answer:

Part (a)

Step 1: Consider the path of the electrons from the cathode to the anode

• Photons are produced whenever a charged particle undergoes a large acceleration or deceleration

• X-ray tubes fire high-speed electrons at a metal target

• When an electron collides with the metal target, it loses energy in the form of an X-ray photon as it decelerates [1 mark]

Step 2: Consider the relationship between the energy of the electron and the wavelength of the photon

• The wavelength of a photon depends on the energy transferred by a decelerating electron

• The electrons don't all undergo the same deceleration when they strike the target

• This leads to a distribution of energies, hence, a range, or continuous spectrum, of wavelengths is observed [1 mark]

Part (b)

Step 1: Identify the significance of the intensity 

• The intensity of the graph signifies the proportion of photons produced with a specific energy, or wavelength

• The higher the intensity, the more photons of a particular wavelength are produced

• In other words, the total intensity is the sum of all the photons with a particular wavelength [1 mark]

Step 2: Explain the shape of the graph

• When a single electron collides with the metal target, a single photon is produced

• Most electrons only give up part of their energy, and hence there are more X-rays produced at wavelengths higher than the minimum (or energies lower than the maximum)

• At short wavelengths, there is a steeper gradient because only a few electrons transfer all, or most of, their energy [1 mark]

Part (c)

Step 1: Identify the relationship between minimum wavelength and maximum energy

• The minimum wavelength of an X-ray is equal to

${\lambda }_{min}=\frac{hc}{E_{max}}$

• The equation shows the maximum energy of the electron corresponds to the minimum wavelength, they are inversely proportional 

${\lambda }_{min}\propto \frac{1}{E_{max}}$

• Therefore, the higher the energy of the electron, the shorter the wavelength of the X-ray produced [1 mark]

Step 2: Explain the presence of the cut-off point

• The accelerating voltage determines the kinetic energy which the electrons have before striking the target

• The value of this accelerating voltage, therefore, determines the value of the maximum energy

• This corresponds to the minimum, or cut-off, wavelength [1 mark]