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One of the most important elements of the analysis is a model
to fit the energy peak data. Recall the Doppler broadening phenomenon
from the "crash course". We know that high energy Ps
will have wide energy peaks, whereas low energy Ps will have narrow
energy peaks. Hence, the increase in peak width, due to the Doppler
effect, leads us to a value for the momentum carried by the Ps.
Our signal consists of several components, only one of which is
the desired, doppler-affected peak. In order to isolate this so-called
"narrow-peak," our model must recognize the background
and the wide-peak.
Background Signal:
The HPGe detector collects all radiation in the vicinity of the
experiment, including small amounts of natural, incoherent radiation
found on the surface of the earth. In addition to this flat background,
Ps may annihilate in three gamma rays, which produces radiation
between 0 and 511 keV with no definite peak. Thus, the left side
of the peak is raised, as shown.
Wide Peak:
Much to our dismay, positrons will form Ps and annihilate with
their respective electron only a fraction of the time. As such,
there are several other outcomes that regularly occur, including
slow positrons and the pickoff effect. The energy spectra from
these effects combine to form a wider peak, sharing a center with
the narrow peak at 511 keV.
Slow positrons are positrons that collide with the gas atoms
but do not form Ps. After a certain amount of kinetic energy is
lost, they are too slow to pull an electron out of its orbital
and create Ps, but instead annihilate with an electron. Because
the positron-electron pair gains kinetic energy from the electron’s
orbital speed, the resulting peak gains a significant amount of
width.
The pickoff effect refers to the case where Ps is formed, but
the positron annihilates with a different electron than its original
pair. Similar to the outcome of slow positrons, energy is gained
by the electron’s orbital velocity and the signal peak is
broadened.
Several analytical functions may be used to fit these components,
which are controlled by parameters such as intensity (total area
under the curve) and width (measured as the width of the peak
at half of its full amplitude, called the "full-width half-max").
Unfortunately, the gamma-ray detector does not provide a perfect
record of the incident radiation. Instead, the information is
slightly altered by the resolution of the detector, by an effect
called convolution.
In order for the fitting parameters to keep their physical significance,
we can accordingly convolve our fitting functions, once we know
the nature of the detector's resolution.
We can get a quantitative gauge for this convolution by collecting
the radiation from a Cs-137 source, which sits on top of the HPGe
detector for the duration of the experiment. Gamma rays from the
cesium source will travel directly into the detector and will
not undergo doppler-broadening. Atomic radiation
of this isotope releases exactly 661.7 keV of energy per decay,
so the physical distribution is a delta
function at precisely this energy. Hence, any increase
in peak width is due to detector resolution. (Note that
the energy peak from the cesium source is easily distinguished
by its decay energy. This means that we can simultaneously take
gas-scattering data and resolution data, to eliminate the effects
of a possible time-dependant resolution.) This information is
used in the peak fitting described above. Thus, we fit the respective
cesium peak to measure the magnitude of the convolution, before
performing the analysis on each 511 keV energy peak.
A single data collection may span the course of several days.
This collection is partitioned into hourly intervals, called acquisitions.
Each acquisition is divided into around 20 equal parts, or "time-slices,"
according to the lifetimes of the Ps. This means that a 72 hour
run could contain nearly 1500 distinct energy peaks, each of which
needs to be individually fit. To handle these large amounts of
data and automate the process, a computer program was written
in a language called “OriginC,” a superset of the
C programming language, made for use with the Origin software.
A schematic of the flow of information is shown here.
Additional analysis will be performed on the exported peak widths
(shown as the last step). This parameter is used to find the momentum
of the Positronium. Acquisitions and time-slices may be as numerous
as desired; the analysis program is designed to accommodate between
an hour and a week’s worth of data acquisitions.
You guys must have
some awesome results, right? --->
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