Statistics: Lecture 8

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Statistics Lecture 8

• Peak analysis
• Integration
• Centroid identification
• Width determination
• Peak fitting
• Gaussian
• Non-linear backgrounds
• Chi-squared minimisation
• Examples

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Peak area measurement

• A typical gamma-ray spectrum consists of a large
  number of channels (bins) in each of which are
  accumulated those counts which fall within a
  small energy range.
• We might have 4096 (4k) channels representing an
  energy range of 2048keV, each channel
  representing the number of counts within a 0.5keV
  energy window.
• In principle, the peak area measurement, requires
  no more than a simple summation of the number of
  counts in each of those channels we consider to
  be part of the peak and the subtraction of an
  allowance for the background beneath the peak.
• The background beneath the peak arises from many
  sources. Generally, however, the background
  presents the Compton continuum from other gamma
  interactions in the detector/shielding.

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Peak area measurement
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Simple peak integration

• A number of simple algorithms exist for simple
  peak area calculation. The Covell method
  involves
• Locate the highest channel (called the centroid).
• Mark the peak limits an equal number of channels
  away from the peak centroid on either side of the
  peak.
• The background level is estimated using the
  channel contents at the upper and lower edges of
  the peak region.

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Simple Peak Integration

• Take the first channel either side of the region,
  which we consider to be the peak, to represent
  the background.
• The gross (integral) area of the peak is
• Where Ci are the counts in the ith channel. The
  total background beneath the peak is estimated
  as
• Where n is the number of channels within the peak
  region.
• This background area is mathematically the area
  of the background trapezium beneath the peak.

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Simple Peak Integration
Centroid
L
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Background n11
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Simple Peak Integration

• It can be useful to think of the background as
  the mean background count per channel beneath the
  peak, multiplied by the number of channels within
  the peak region. The net peak area, A, is then
• We can calculate precisely the number of counts
  within the peak region (G).
• We can only estimate the number of background
  counts beneath the peal. It is impossible to know
  which counts within the peak are due to
  background and which are due to peak counts.
• In certain circumstances, in particular small
  peaks on large backgrounds, the uncertainty in
  the background estimate can dominate the total
  uncertainty in the peak area.

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Simple Peak Measurement

• The background estimate can be made more precise
  by using more channels to estimate the mean count
  per channel under the peak.
• Instead of a single channel, m channels beyond
  each side of the peak are used to estimate the
  background.
• For the measurement of peak area to have meaning
  we must estimate the statistical uncertainty. For
  AG-B then the variance of the net peak area is
  the sum of the variances we have (Poisson
  statistics)

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Peak Area Uncertainties

• Some important remarks
• This method assumes that the background is linear
  from the bottom to the top edge of the peak.
• This is not strictly true it is found
  empirically that well-defined peaks have a step
  function beneath the peak.
• However, this method gives reasonable results for
  well separated peaks.
• Do NOT use for overlapping peaks!

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Peak position and width

• Other parameters of interest are the position (P)
  and width (W), the latter specified by the full
  width at half-maximum (FWHM).
• A measure of the peak position is obtained from
  the weighted mean channel number (expectation
  value of i) for the counts in the peak. The
  centroid of the peak.
• Each channel is weighted by the number of counts
  in the peak at that channel
• Where the weighting factor wiCi-Bi, where Ci is
  the number of counts in the channel Bi is the
  background.

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Peak position and width

• The width of the peak may be known for a given
  instrument, eg. the energy resolution of a
  detector.
• It may also be estimated from the variance of the
  channel distribution of observed counts about the
  centroid.
• From which the standard deviation .
  Each channel is again weighted by the number of
  peak counts in that channel.
• If the peak is Gaussian in shape, then the FWHM
  W2.35s.
• The statistical uncertainty in P depends on the
  number of counts in the peak. If a peak
  consisting of n counts has a FWHM of W, the
  standard deviation for the error on P is

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Optimising Conditions

• Background channels
• It has been determined that the uncertainty in
  the background estimate must be dependent upon
  the number of background channels used.
• It appears that the more channels used in the
  background implies a better background estimate.
• However, there are decreasing returns, you must
  not forget the possibility of including
  neighbouring peaks, which cause a wide background
  region to be non-linear.
• Most Multi-Channel-Analyser (MCA) programs use
  3,4 or 5 channel backgrounds.

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Optimum Conditions

• Spectrum size
• The optimum spectrum size for a HPGe or Silcon
  detector is 4096 or 8192 channels. This provides
  best compromise for statistics/spread.
• Counting time
• If we want to analyse a batch of samples, we need
  to decide to what precision the final result is
  required.

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Peak fitting with composite curves

• Many peak fitting procedures involve determining
  the parameters of a peak or peaks that are
  superimposed upon a smoothly varying background.
• Consider a function which corresponds to a
  Gaussian shape on a second degree polynomial
  background. The peak is represented by the
  equation
• Where a0 is the peak height above the background.
  A least-squares fitting routine is used to find
  the optimum values of parameters a0, a1, a2, a3,
  C and s, which minimise c2.

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Area Measurement

• The area of the peak provides a measure of the
  intensity of a particular transition or the
  strength of a reaction channel.
• When peaks are not well separated, or when the
  contribution from the background is substantial,
  a least squares fitting procedure can provide a
  consistent method of extracting such information
  from the data.
• Remember the importance of consistency, other
  experiments may wish to check your results.
  Understand your chosen method.
• The method least squares is considered an
  unbiased estimator of the fitting parameters and
  all parameters are presumed to be estimated as
  well as possible.
• Remember use a valid fitting function.

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Area measurement

• If the shape of the peak is well fitted by our
  2nd degree polynomial function, then the area of
  the peak is given by the Gaussian distribution
• Alternatively, when the peak is not well fitted
  by a simple distribution, we fit a specified
  region of the background to the function
• By determining the optimum values for the
  parameters a1, a2 and a3. The area of the peak
  can then be determined from the number of events
  in the peak above the background summed over a
  region between C-d and Cd, where d is chosen to
  encompass the peak.

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Area measurement uncertainties

• We should estimate and correct for events in the
  tails of the peak distribution i.e. those events
  outside the arbitrarily selected limits Cd.
• However, this method ignores some of the
  improvements in the area estimate resulting from
  the fitting procedure.
• If an area is calculated from the previous
  equation, then the uncertainty should be
  estimated from the uncertainties in the
  parameters from the shape of the c2 minimum in
  the least-squares fitting routine.
• For example, the uncertainty in the peak position
  parameter C is obtained by calculating c2 as a
  function of the centroid C in the formula for the
  peak area.

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88Y Gamma-ray Spectrum
y0.4995x-0.9811
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Our Example Peak Fitting
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Our Example Peak fitting
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Peak fitting on high background
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Data analysis Tools
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Data Analysis Tools
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Data Analysis Tools
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Summary of lecture 8

• Peak analysis
• Integration
• Centroid identification
• Width determination
• Peak fitting
• Gaussian
• Non-linear backgrounds
• Chi-squared minimisation
• Examples