X-ray Energy and Dose Calculation

In this example, we will calculate the energy of X-rays used in a chest X-ray and determine the dose received by the patient in terms of Gray (Gy) and Sievert (Sv).

Step 1: X-ray Energy Calculation

The energy of an X-ray photon can be calculated using the equation for photon energy:

\[ E = h f \]

Where:

\( E \) is the energy of the X-ray photon (in Joules, J).
\( h \) is Planck's constant, \( h = 6.626 \times 10^{-34} \, \text{J·s} \).
\( f \) is the frequency of the X-ray photon (in Hz).

For X-rays, the frequency is related to the wavelength \( \lambda \) by the equation:

\[ f = \frac{c}{\lambda} \]

Where:

\( c \) is the speed of light, \( c = 3.00 \times 10^8 \, \text{m/s} \).
\( \lambda \) is the wavelength of the X-ray (in meters, m).

Suppose the wavelength \( \lambda \) of the X-ray used in the chest X-ray is \( 1.0 \times 10^{-10} \, \text{m} \) (which is typical for diagnostic X-rays). Now, calculate the frequency \( f \):

\[ f = \frac{c}{\lambda} = \frac{3.00 \times 10^8 \, \text{m/s}}{1.0 \times 10^{-10} \, \text{m}} = 3.00 \times 10^{18} \, \text{Hz} \]

Now, calculate the energy of one X-ray photon:

\[ E = h f = (6.626 \times 10^{-34} \, \text{J·s}) \times (3.00 \times 10^{18} \, \text{Hz}) = 1.99 \times 10^{-15} \, \text{J} \]

So, the energy of one X-ray photon is approximately \( 1.99 \times 10^{-15} \, \text{J} \).

Step 2: X-ray Dose Calculation

The dose received by a patient during an X-ray procedure is usually measured in Gray (Gy). One Gray (Gy) is equal to one joule of energy absorbed per kilogram of tissue.

The dose \( D \) in Gray is given by:

\[ D = \frac{E_{\text{total}}}{m} \]

Where:

\( D \) is the dose (in Gray, Gy).
\( E_{\text{total}} \) is the total energy absorbed by the patient (in joules, J).
\( m \) is the mass of the tissue exposed to the X-ray (in kilograms, kg).

Suppose the total energy absorbed by the patient's body during a chest X-ray is 0.1 J, and the mass of the tissue exposed is \( m = 1.0 \, \text{kg} \). The dose in Gray is calculated as:

\[ D = \frac{0.1 \, \text{J}}{1.0 \, \text{kg}} = 0.1 \, \text{Gy} \]

So, the dose received by the patient is 0.1 Gray (Gy).

Step 3: Conversion to Sievert (Sv)

While Gray (Gy) measures the amount of energy absorbed, the Sievert (Sv) is used to measure the biological effect of the radiation on the body. The conversion from Gray to Sievert depends on the type of radiation (X-rays in this case) and its relative biological effectiveness (RBE).

For X-rays, the RBE is generally considered to be 1. This means that the dose in Gray is numerically equal to the dose in Sievert for X-ray radiation:

\[ \text{Dose in Sv} = \text{Dose in Gy} \times \text{RBE} \]

Since the RBE for X-rays is 1:

\[ \text{Dose in Sv} = 0.1 \, \text{Gy} \times 1 = 0.1 \, \text{Sv} \]

So, the equivalent dose received by the patient is 0.1 Sievert (Sv).

Step 4: Understanding the Result

A dose of 0.1 Sv (or 100 mSv) is a typical dose for a chest X-ray, and this is considered to be within safe limits for diagnostic procedures. However, doses at this level do carry some potential risk, which is why medical practitioners aim to minimize radiation exposure whenever possible.