Pion correlations as a function of atomic mass in heavy ion collisions Page: 24 of 130
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CHAPTER 3. MOTIVATION AND THEORY
is suppressed. Last, it is possible to justify Eq. 3.8 by saying that for Ax - Ap < h = 1
and AE - At < A (where Ax = 1ri - rnl, AP = Ipi - P2 and so on) then the uncertainty
principle allows the wave functions to overlap, and hence to interfere.
3.3 Correlation Function Used in the Fit
Generally it is not possible to invert the Fourier transform (particularly since what is
measured is the square of the absolute value of the Fourier transform), s. a source density
function with several free parameters is assumed. Its Fourier transform is then calculated,
and the free parameters are adjusted for best agreement with the data. The choice of the
parameterization used is made according to the physics tw be explored.
Fourier transforms have the general property that large scal features transform to
small scale features. For example, the overall normalization of the number of pion emitters,
a property of the integral of p($, t) over all space, transforms to the property that the
intercept of the correlation function at Q = 0 is C2(Q - 0) = 2 (this is best seen by
setting P1 = P2 in Eq. 3.2 and then substituting the result into Eq. 3.3). Because the
finite amount of data that can be taken in a realistic experiment restricts the resolution
of the correlation function, which is roughly the Fburier transform of the pion source
density, only the larger features of the pion source density are well determined. These are
the lifetime and the radius parameters in the two directions that the spectrometer can
determine, the radius parameter parallel to the beam and the radius perpendicular to the
beam (the spectrometer cannot determine the impact plane).
In this experiment the source density p(l, t) is assumed to be the Gaussian
p(S, t) cc a i): l)-*) (3.12)
Where the notations .1. (and 11) mean perpendicular (and parallel) to the axis through
the collision defined by the beam axis. The parameters R.l, All Paid r are the two radius
parameters and the lifetime parameter. The Gaussian source density is chosen because
the the transform is particularly simple to calculate. Note that other paramceterizatihns
will yield similar forms for the correlation function, but the parameters may differ by a
multiplicative factor. When comparing data it is essential to check the parameterizations
used.14
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Chacon, A.D. Pion correlations as a function of atomic mass in heavy ion collisions, report, November 26, 1989; Berkeley, California. (https://digital.library.unt.edu/ark:/67531/metadc1055804/m1/24/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.