Soft tissue has a linear attenuation coefficient in the range of
μ = 0.35 cm−1
at 30 keV and
μ = 0.16 cm−1
at 100 keV. For this problem we use
μ = 0.21 cm−1,
which applies at around 50 keV incident X-ray energy.
(a) What fraction of X-ray photons at 50 keV are passing through a person's body? Hint: the person's body thickness is about 19 cm. Use a soft tissue approximation, i.e., neglecting bones.
(b) We compare bone and soft tissue of 4 cm thickness each. Using
μbone = 0.57 cm−1
at 50 keV, what fraction of a 50 keV incident X-ray beam is stopped in bone and soft tissue respectively?
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To determine the fraction of X-ray photons at 50 keV passing through a person's body, we will use the linear attenuation coefficient of soft tissue, which is μ = 0.21 cm−1 at 50 keV.
(a) To calculate the fraction of X-ray photons passing through a person's body, we need to consider the thickness of the body. Given that the person's body thickness is about 19 cm, we can use the following formula:
Intensity_after = Intensity_before * e^(-μ * thickness)
Using this formula, we can find the fraction of X-ray photons passing through the body which is equal to Intensity_after / Intensity_before.
Plugging in the values:
- μ = 0.21 cm−1
- thickness = 19 cm
We can calculate the fraction of X-ray photons passing through the person's body as follows:
fraction_passing = e^(-μ * thickness)
fraction_passing = e^(-0.21 * 19)
Using a scientific calculator or software that supports exponential functions (like Python), we can calculate that the fraction of X-ray photons passing through a person's body is approximately 0.0002657, or about 0.02657%.
(b) Next, we need to compare the fraction of a 50 keV incident X-ray beam stopped in bone and soft tissue when they are each 4 cm thick.
For soft tissue:
- thickness = 4 cm
- μ soft tissue = 0.21 cm−1 at 50 keV
Using the same formula as above, we calculate the fraction of X-ray photons passing through the soft tissue as:
fraction_passing_soft_tissue = e^(-μ_soft_tissue * thickness)
fraction_passing_soft_tissue = e^(-0.21 * 4)
This gives us a fraction of approximately 0.4507, or about 45.07%.
For bone:
- thickness = 4 cm
- μ bone = 0.57 cm−1 at 50 keV
Using the same formula, we calculate the fraction of X-ray photons passing through the bone as:
fraction_passing_bone = e^(-μ_bone * thickness)
fraction_passing_bone = e^(-0.57 * 4)
This gives us a fraction of approximately 0.0487, or about 4.87%.
In summary:
(a) The fraction of X-ray photons at 50 keV passing through a person's body is approximately 0.0002657, or about 0.02657%.
(b) When comparing 4 cm thickness each of bone and soft tissue, the fraction of a 50 keV incident X-ray beam that passes through soft tissue is approximately 0.4507 (45.07%), while the fraction passing through bone is approximately 0.0487 (4.87%).