Effects Of Distortion On Trojan Horse Applications

Rosario Gianluca PizzoneRosario Gianluca Pizzone

INFN – Laboratori Nazionali del Sud CataniaINFN – Laboratori Nazionali del Sud Catania

The Trojan Horse Method

• Indirect Methods can improve Nuclear Astrophysics results. Among them the Trojan Horse Method (THM).

• It allows the study of reactions of astrophysical interest like x(A,C)c at energies as low as the astrophysical ones after selection of an appropriate a(A,Cs)c reaction, induced at energies greater than the Coulomb barrier in quasi free conditions. (C. Spitaleri’s talk)(C. Spitaleri’s talk)

For QF processes the binary cross For QF processes the binary cross section, as a function of the three body section, as a function of the three body one is:one is: a

quasi free break-up

s

A

x

virtual reaction in nuclear field

A + x c + C

c

C

HOES

d

dSpKF

Cdc

dc

dE

d

2)(

3

Measured 3-body cross section 2-body cross section

What is (ps)?

Outlook

• Spectator momentum distribution in TH nucleus is necessary for THM application

• TH nuclei (table) show a strong cluster configuration

• In recent years the impulse distribution of spectator inside TH nucleus has been extensively studied for 6Li as a function of the transferred momentum ( Pizzone et al. 2005);

Goal of the present work: to evaluate the momentum distribution distortion as a function of the transferred momentum also for the other nuclei used as TH nuclei

Knowing that:

ps): the momentum distribution

6Li

d6Li

• It represents the momentum distribution of the cluster s inside the Trojan Horse nucleus es. 6Li or 2H;

• A very well known case: 6Li=+d(Pizzone et al, PRC 058801 2005)

Step 1 to evaluate the distortion effects is the study of the -d momentum distribution for 6Li

Momentum distributionIf the reaction proceeds via a QF mechanism the process is a direct one and the impulse distribution of the spectator should be identical to the one it has inside TH nucleusSince :

Then if d/d ~ constant (small relative energy interval)

QF mechanism is present and can be separated from other contributions.

Hulthen function:Standard parametersa=0.2317 fm-1

b=1.202 fm-1

Only one variable in the fit, the normalization constant

Momentum distribution

Dots: experimental momentum distribution (n in deuteron case)

Red: DWBA calculation

Black line: PWIA

If -30<ps<30 MeV/c they almost agree (typical ranges for THM application, according to Shapiro prescriptions)

Momentum distribution for 6Li

By increasing the transferred momentum the momentum distribution widens up

Low transferred momentum higher transferred momentum

B

FWHM vs. transferred momentum variation: 6Li

• Step 2 investigate how momentum distribution shape changes with transferred momentum (reaction 6Li(6Li,)4He);

Correlation

FWHM – transferred momentum

6Li(6Li,)4He at different beam energies (2.1 – 44 MeV)

6Li(3He,p)4He at different beam energies (5-6 MeV)

Data from Barbarino et al. PRC 1980

f0=73 MeV/c; q0=324MeV/c

Is this behaviour independent from Trojan Horse Nucleus??

• Deuterium

• Helium 3

• Beryllium 9

• Tritium

FWHM vs. transferred momentum variation:

2H

With growing transferred momentum the distribution width get closer to its asymptotic

D=p+n

Additional study: momentum distribution for n in deuteron

Experimental points are compared with theoretical prediction (Lamia et al., PRC, 2012)D wave contributing up to 4%.

Remarks

• We see therefore that for decreasing transferred momenta qt distortions in the momentum distribution of n inside d shows up.

• This works also for other possible TH nuclei used in several experiments (Pizzone et al 2009)

FWHM vs. transferred momentum variation:

3He

3He=p+d

FWHM vs. transferred momentum variation:

9Be

9Be=a+5He

FWHM vs. transferred momentum variation: tritium

t=d+n

Parameter f0, q0 and k=(2

Eb)1/2 for examined nuclei

Correlation between k and q0

Conclusions

In all cases when transferred momentum is much higher than k the asymptotic value of the momentum distribution width is reached. Otherwise a narrowing shows up.That is a clear signature that distortion effects show up as soon as one get far from an “ideal” quasi free situation. One can take into account these effects by adopting in THM applications the effective momentum distribution instead of the asymptotic one.

qt>>k qt~k

Remarks: Distortions in THM• Distortions accounted for in THM by inserting in

the experimental momentum distribution width (in general different from the asymptotic one).

Two possible effects: - Impact of altered width in TH nucleus momentum

distributions- Impact of assuming only s-wave contributes to the d wave

function (e.g. for deuteron)

HOES

d

dSpKF

Cdc

dc

dE

d

2)(

3

• The S(E) factor for the 6Li(d,a)4He reaction has been extracted for different values of the momentum distribution FWHM (Pizzone et al. 2005).

FWHM = 70 MeV/c

FWHM = 60 MeV/c

FWHM = 50 MeV/c

No remarkable change (within experimental errors) differences around 5%

1

• The variation of the extracted S(E) factor for taking into account the d wave instead of the s wave alone in the n-p intercluster motion in d turned out to be negligible within experimental errors:

In both the examined cases the discrepancy is less than 0.5%

2

Bibliography: