Equivalent squares of circular electron fields

The results show that there is an exceptionally simple and remarkably accurate relationship for the equivalent square of circular electron fields, namely: X = 1.83 R, where R is the radius and X is the side length of the equivalent square. This is approximately midway between the two limiting values predicted by Fermi-Eyges theory ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:semantics> <mml:mrow><mml:msqrt><mml:mi>π</mml:mi></mml:msqrt> <mml:mo>≤</mml:mo> <mml:mi>X</mml:mi> <mml:mo>/</mml:mo> <mml:mi>R</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn></mml:mrow> <mml:annotation>$\sqrt \pi \le X/R \le 2$</mml:annotation></mml:semantics> </mml:math> ). For the 104 combinations of the parameters described above, the average ratio of the circular field output to the equivalent square output is 1.000, and the standard deviation is 0.003. In every case, the accuracy is better than 1% and, in most cases, better than 0.5%. Almost all the ratios fall within the ±0.4% accuracy expected based on the statistical uncertainty in the Monte Carlo calculations. It is shown that the equivalent square rule for circles is more accurate than the square root rule for a range of common widths, W, and L/W, where L is the length of the rectangle. For the clinical examples cited, the agreement between estimated and measured output is within a few percent.