Slides for Statistics, Precision, and Solid Angle

22.01 – Intro to Radiation October 14, 2015

22.01 – Intro to Ionizing Radiation Precision etc., Slide 1 Solid Angles, Dose vs. Distance

• Dose decreases with the inverse square of distance from the source:

1 퐷표푠푒 ∝ 푟2

• This is due to the decrease in solid angle subtended by the detector, shielding, person, etc. absorbing the radiation

22.01 – Intro to Ionizing Radiation Precision etc., Slide 2 Solid Angles, Dose vs. Distance

• The solid angle is defined in steradians, and given the symbol Ω. • For a rectangle with width w and length l, at a distance r from a point source: 푤푙 Ω = 4푎푟푐푡푎푛 2푟 4푟2 + w2 + 푙2 • A full sphere has 4π steradians (Sr)

22.01 – Intro to Ionizing Radiation Precision etc., Slide 3 Solid Angles, Dose vs. Distance

http://www.powerfromthesun.net/Book/chapter02/chapter02.html • Total luminance (activity) of a source is constant, but the flux through a surface decreases with distance

Courtesy of William B. Stine. Used with permission.

22.01 – Intro to Ionizing Radiation Precision etc., Slide 4 Exponential Gamma Attenuation

• Gamma sources are attenuated exponentially according to this formula:

Initial intensity Mass attenuation coefficient 흁 Distance through − 흆 흆풙 푰 = 푰ퟎ풆 material Transmitted intensity Material density • Attenuation means removal from a narrowly collimated beam by any means

22.01 – Intro to Ionizing Radiation Precision etc., Slide 5 Exponential Gamma Attenuation

Look up values in NIST x-ray attenuation tables: http://www.nist.gov/pml/data/xraycoef/

Initial intensity 흁 Distance through − 흆 흆풙 푰=푰ퟎ풆 material Transmitted intensity Material density

22.01 – Intro to Ionizing Radiation Precision etc., Slide 6 Exponential Gamma Attenuation

Look up values in NIST x-ray attenuation tables: http://www.nist.gov/pml/data/xraycoef/

Initial intensity 흁 Distance through − 흆 흆풙 푰=푰ퟎ풆 material Transmitted intensity Material density For compound materials, just add their total attenuation coefficients: 휇 휇 휇 = 𝜌 + 𝜌 +⋯ 푡표푡푎푙 𝜌 1 𝜌 2 1 2 22.01 – Intro to Ionizing Radiation Precision etc., Slide 7 Mass Attenuation Coefficients

http://physics.nist.gov/PhysRefData/XrayMassCoef/ComTab/blood.html

Public domain, from U.S. NIST.

22.01 – Intro to Ionizing Radiation Precision etc., Slide 8 Statistics, Counting, Uncertainty

Confidence increases with counting time and counting rate:

푐표푢푛푡 푟푎푡푒 𝜎 = 푐표푢푛푡푖푛푔 푡푖푚푒 Count rates are expressed in counts per time plus or minus standard deviations:

푐표푢푛푡푠 푝푒푟 푚푖푛푢푡푒 = 퐶푃푀 ± 𝜎

22.01 – Intro to Ionizing Radiation Precision etc., Slide 9 Statistics, Counting, Uncertainty

Remember to measure a background count rate with its own uncertainty:

퐶푃푀푏 𝜎푏 = 푡푏 Express total uncertainties in quadrature:

퐶푃푀푡표푡푎푙 퐶푃푀푏 퐶푃푀푛푒푡 = 퐶푃푀푡표푡푎푙 − 퐶푃푀푏 𝜎푛푒푡 = + 푡푡표푡푎푙 푡푏

22.01 – Intro to Ionizing Radiation Precision etc., Slide 10 MIT OpenCourseWare http://ocw.mit.edu

22.01 Introduction to Nuclear Engineering and Ionizing Radiation Fall 2015

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