Test of weak and strong factorization in nucleus-nucleuscollisions atseveral hundred MeV/nucleon Page: 3 of 13
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Exit window Target Detectors
(Not to scale)
Figure 1: Scheme of a typical experimental setup chosen for the 2000 argon experiment (all distances are reported in cm).
related to the fact that the cross sections have been ob-
tained using several target thicknesses. The first type
of uncertainty is calculated using the error propagation
method, while the second is determined by examination
of the spread in measured cross sections obtained with
different target depths. Since the statistical error is typ-
ically dominated by the methodological error, the latter
can be taken as a reasonable estimation of the uncertain-
ties on the average cross sections.
B. Cross sections
In Table II the total and partial charge-changing cross
sections are listed for the argon projectile in all targets.
Hydrogen cross sections have been calculated using the
formula o(H) 0.5[(CH2) - u(C)].
Silicon cross sections are reported in ; the cross
sections for all the other projectiles are still preliminary
data and therefore are not listed here but will be soon
published in a separate paper.
IV. METHODS FOR TESTING THE
Two methods have been generally used for testing the
factorization property: a graphical approach which gives
a general idea of the data behavior and an analytical
method which allows calculating the factorization param-
With the graphical method we check that:
" for any given projectile P, and fragments F and F',
the ratio a(P,T,F) is independent on the target T;
or, equivalently, that
" for any given projectile P, fragment F and targets
T and T', the ratio TFis independent on the
These two statements test the degree to which weak
factorization holds. Moreover, if the second ratio is also
independent on the projectile, strong factorization holds.
Data for each projectile have been analyzed to verify
weak factorization while groups of two or three projec-
tiles have been used to test strong factorization.
In the first case a reference target was chosen for each
projectile and all the partial charge-changing cross sec-
tions relative to the other targets were divided by the
reference value; for each target, then, these values were
averaged over the fragments and the mean has been used
as a normalization factor:
o(P, T, F)/o(P, Te6f, F)
The quantity (o(P, T, F)/o(P, Tre f, F))T represents the
mean value of the ratio o(P, T, F)/o(P, Tref, F) for a
fixed target T. If weak factorization held all the ratios
should be around 1, independently on the fragment.
The graphical method for testing the strong factoriza-
tion is the same than the one described above for the
weak factorization, with the exception of the normaliza-
tion factor which in this case is averaged over all the pro-
jectiles and fragments involved. If strong factorization
held all the ratios should be around 1, independently on
the fragment and the projectile.
The analytical method is based on the minimization of
the x2 functions:
2 T [(P, T, F)- 0FPT]2
AID ~ ~[oj(P, T, F)]2
for the weak factorization;
A T F [S P)TF)]2
P T F
for the strong factorization.
So (P, T, F) represent the uncertainties on the cross sec-
tion values o(P, T, F). Since the parameters are defined
up to a multiplicative constant, we need to fix one of
them in order to uniquely determine all the others; the
value of one of the parameters was thus chosen to be 1.
The minimization has been performed using a program
developed by our group that gave also the corresponding
x2 value in the minimum. The algorithm chosen for the
minimization process (Levenberg-Marquardt) provided
an estimation of the coviariances matrix from which it
was possible to calculate the parameter uncertainties.
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La Tessa, Chiara; Sihver, Lembit; Zeitlin, Cary; Miller, Jack; Guetersloh, Stephen; Heilbronn, Lawrence et al. Test of weak and strong factorization in nucleus-nucleuscollisions atseveral hundred MeV/nucleon, article, June 21, 2006; United States. (digital.library.unt.edu/ark:/67531/metadc897399/m1/3/: accessed November 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.