At least how many HVLs are needed to reduce the intensity of a monoenergetic photon beam to less than 10% of its original value?

To determine how many half-value layers (HVLs) are needed to reduce the intensity of a monoenergetic photon beam to less than 10% of its original value, we can use the concept that each HVL reduces the intensity by half.

Mathematically, after passing through n half-value layers, the remaining intensity can be expressed as:

[ I = I_0 \left( \frac{1}{2} \right)^n ]

where I is the final intensity, I_0 is the initial intensity, and n is the number of HVLs.

To find how many HVLs are necessary for the intensity to be less than 10% of the original intensity, we set up the inequality:

[ I < 0.1 I_0 ]

Substituting the equation for I gives:

[ I_0 \left( \frac{1}{2} \right)^n < 0.1 I_0 ]

This simplifies to:

[ \left( \frac{1}{2} \right)^n < 0.1 ]

Taking the logarithm of both sides, we can rewrite this as:

[ n \log(0.5) < \log(0