Radioactivity inverse square law, absorption, and rates

1
Radioactivity inverse square law, absorption,
and rates

• presentation for Apr. 30, 2008 by
• Dr. Brian Davies, WIU Physics Dept.

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Inverse square law for radiation

• Particles and photons emitted by radioactive
  nuclei continue moving until they are absorbed by
  some material.
• The particles (or photons) that leave the sample
  will continue to move out in a radial direction.
  
• The number that pass through an imaginary sphere
  of radius r in some interval of time is the same
  for different radii, but the area of the sphere
  depends on the radius. A 4pr2
• Therefore, the number per unit area per unit time
  depends on the radius.

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Inverse square law

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Relation between source activity and intensity at
various radii

• Suppose that S is the number of particles emitted
  by the source per unit time.
• This S might be due to an activity which is
  written as S decays per second. (in Bq)
• At a distance r, the particles pass through a
  sphere of radius r and area A 4pr2
• Intensity is I S/A, the number of particles,
  per unit time, per unit area.
• Then I S/4pr2
• I decreases as the inverse square of the radius.

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Intensity for r, 2r, 3r, etc.

• If the intensity is I1 at a radius r1, then if we
  double the radius to 2r1, the intensity will be ¼
  as much, because the area is now 2x2 4 times as
  much.
• More generally, if we compare the intensity at
  radii r1 and r2, then we get a ratio of
  intensities that depends on the square of the
  radii
• I2 / I1 (r1/r2)2
• For example, if we compare radii of r and 3r, the
  intensities have a ratio of 1/9 (1/3)2

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Linear plot for inverse square law.

• We can plot I vs. r on a linear graph, but this
  is not always useful if the range is too large.

I 1/r2
I(1) 1
o
I(2) 1/4
I(3) 1/9
o
o
r
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Linear plot for 10000.r -2 over range 1 to 100.
not very useful !
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Log-log plot for inverse square law.

• If the intensity is a function of radius that is
  a power law rm (for example, inverse square is a
  power law, since I a.rm where m -2),
• then, we can plot I vs. r on a log-log graph.
• Applying the logarithm to both sides of the
  equation and using log(ab) log(a) log(b)
• log (I) log(a.rm) log(a) m . log(r)
• and if y log(I), x log(r), and log(a) b,
  we have the eq. of a straight line y m . x
  b

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log-log plot of I 10000 r -2
y log10(I)
I
4 3 2 1 0
r
0 1 2
log10(r) x
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Analysis of I 10000 r -2
How does this equation produce a straight line
on the log-log plot? log(I)
log(10000.r-2) log(10000) log(r-2)
log(104) (-2).log(r) Now define y
log(I) and x log(r) and then y m.x
b with b log(104) 4 (the
intercept) m -2 is the slope
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We study this in laboratory 10.
We use laboratory equipment to study the
distance dependence of the radiation from a
small source. This is examined experimentally
using log-log graph paper. We also examine
the use of shielding materials, which requires
semi-log paper to plot the absorption of gamma
rays.
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Absorption of X-rays and gamma rays

• X-rays and gamma rays can be very penetrating.
• Scattering of photons is not very important. It
  is more probable for the photon to be absorbed by
  an atom in the photoelectric effect.
• The photon is absorbed with some probability as
  it passes through a layer of material. This
  results in an exponential decrease in the
  intensity of the radiation (in addition to the
  inverse square law for distance dependence).

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Exponential absorption of X-rays

• I Io at detector, with no absorber

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Exponential absorption of X-rays

• With absorber in place, I Io exp(- m x)

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Exponential absorption of X-rays

• The exponential decrease in the intensity of the
  radiation due to an absorber of thickness x has
  this form
• I Io exp(- m x) Io e - m x
• where Io is the intensity without the absorber,
• I is the intensity with the absorber, and
• m is the linear absorption coefficient.
• m depends on material density and X-ray energy.

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Graph of the exponential exp(x)

• exp(x)

exp(0) 1

x
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Graph of the exponential exp(-x)

exp(0) 1
exp(-0.693) 0.5 ½

• exp(-x)

exp(-1) 1/e 0.37
x
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Half-thickness for absorption of X-rays

• For a particular thickness x ½ the intensity
  is decreased to ½ of its original magnitude. So
  if
• I(x½) Io exp(- m x ½) ½ Io
• we solve to find the half-thickness x ½.
• exp(- m x ½) ½ and m x ½ 0.693
• so x ½ 0.693 / m

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Calculation of half-thickness

• To calculate x ½ (of lead, Pb) we need to know
  m.
• As an example, for X-rays of energy 50 keV,
• m 88 cm-1 and x ½ 0.693/m
  so
• x ½ 0.693 / (88 cm-1) 0.0079 cm
• But for hard X-rays with energy 433 keV,
• m 2.2 cm-1 so
• x ½ 0.693 / (2.2 cm-1) 0.31 cm

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Graphs of linear attenuation coefficient m

• The linear attenuation coefficient m can be
  obtained from tables, or from automated databases
  such as the NIST database
• http//physics.nist.gov/PhysRefData/FFast/Text/c
  over.html

which produced this graph for lead (Pb)
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Tables of linear attenuation coefficient m

Data for Z 82, E 2 - 433 keV E (keV ) µ
Total (cm-1) 2.0004844E00 1.3412E04
2.0104868E00 1.3272E04 2.0205393E00
1.3133E04 2.0306420E00 1.2996E04
4.479101E01 1.1677E02 4.788159E01
9.8248E01 5.118542E01 8.2776E01 5.471721E01
6.9836E01 5.849270E01 5.8933E01
3.544049E02 3.1633E00 3.788588E02
2.7826E00 4.050001E02 2.4588E00 4.329451E02
2.1827E00
The NIST database produces this table of m for
lead (Pb)
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Half-thickness data from ORTEC-online. (link)
X
Gamma rays from Co-60
X
X
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Shielding of X-rays and gamma rays

• To reduce the intensity of radiation from a
  source, we can use an absorber in the path of the
  radiation. This is called shielding.
• To minimize I Io exp(- m x) we want to
  increase m or x. Then the exponential will be
  smaller, and I will be smaller for constant Io .
• To increase the absorption coefficient m we need
  to increase the density of the shielding.
• To increase the value of x we must use thicker
  shielding.

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Shielding of charged particles (alpha and beta
particles)

• The absorption of charged particles is quite
  different from the absorption of X-rays (or g).
• Charged particles lose kinetic energy
  continuously, instead of being absorbed in one
  single event like photons, and they also can
  scatter (change direction).
• The result is a range, a distance that only a
  small number of particles reach.
• Beyond the range, there is zero intensity.

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Range of alpha and beta particles

• The range of alpha particles is a few centimeters
  in air and much less in solids.
• Alphas may be completely absorbed by a single
  sheet of paper or by your skin
• Beta particles can travel a few meters in air or
  a few millimeters in organic materials, depending
  on their kinetic energy. One cm of polymer will
  usually stop beta particles. However, they can
  easily pass through skin or gloves.