# Test of weak and strong factorization in nucleus-nucleuscollisions atseveral hundred MeV/nucleon Page: 3 of 13

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3

Exit window Target Detectors

Trigger 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 [12]; 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

FACTORIZATION PROPERTY

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-

eters.

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

fragment;

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 verifyweak 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)

aF -(PTF)/ojPT~6fF>T(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(8)

for the weak factorization;

A T F [S P)TF)]2

P T F(9)

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.(7)

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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 September 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.