1. Trojan Horse Method: Recent Results
OMEG07 : The 10th Int. Symp. On Origin of Matter and Evolution of Galaxies December 4-7 ,Hokkaido University
Trojan Horse Method: Recent Results
Claudio Spitaleri
Catania University-Italy
Laboratori Nazionali del Sud- Catania-Italy

2. Ideally
Direct measurement of cross sections at relevant energies is the best way to obtain the reaction rates
BUT…..
In the measurements between charged perticles two limits are present:
Coulomb barrier
Electron screening
To obtain the reaction-rates the measurements of nuclear cross sections are
NECESSARY
Nuclear reaction rates are basic input in many astrophysical models
(primordial nucleosynthesis,stellar evolution, novae, supernovae,….)
For various processes
(pp chains, CNO cycles, s,r,p,rp…)

3. DIRECT NUCLEAR REACTIONS BETWEEN CHARGED PARTICLES: LIMITS
Due to its presence
extremely small cross sections s(E) with strong energy dependence
astrophysical relevant energies EG (Gamov peak) usually not accessible
s(E) n b
3He(a,g)7Be
Ecm (Mev)
Gamow energy
The main limit in the charged particle cross section measurements at astrophysical energies is the presence of the Coulomb barrier between the interacting nuclei
in general,
direct evaluation of the cross sections is
-severely hindered
-and in some cases even beyond present technical possibilities.

4. A possible solution Extrapolation Ecm (Mev)
DIRECT NUCLEAR REACTIONS BETWEEN CHARGED PARTICLES: LIMITS
A possible solution
to evaluete these cross sections consist in using the
The main limit in the charged particle cross section measurements at astrophysical energies is the presence of the Coulomb barrier between the interacting nuclei
s(E) n b
3He(a,g)7Be
Extrapolation
extremely small cross sections s(E) with strong energy dependence
astrophysical relevant energies EG (Gamov peak) usually not accessible
Ecm (Mev)
through the Astrophysical S(E)-factor defined via the standard equation
Gamow energy
Cross-section
bare nucleus
Astrophysical factor
b(E) = exp(-2) Sb(E)
(uncertainties
in the extrapolation !!!!)
in general, their direct evaluation is
-severely hindered
-and in some cases even beyond present technical possibilities.

5. Ecm (Mev) (uncertainties in the extrapolation !!!!) s(E) n b
DIRECT NUCLEAR REACTIONS BETWEEN CHARGED PARTICLES: LIMITS
To avoid the problem of uncertainties
in the extrapolation procedure
Experimental technique were improved and some experiments were performed at Gamow energy.
BUT
NEW EFFECT WERE DISCOVERED
Enhancement in the cross sections due to
ELECTRON SCREENING
s(E) n b
3He(a,g)7Be
Ecm (Mev)
Gamow energy
(uncertainties
in the extrapolation !!!!)
(Assenbaum,Langanke,Rolfs:
Z.Phys.327(1987)461)

6. DIRECT NUCLEAR REACTIONS BETWEEN CHARGED PARTICLES: LIMITS
The second relevant source of uncertainty in extrapolating the S(E)-factor at astrophysical energies (down to zero energy) is the enhancement due to the electron screening effect !!!!
In the extrapolation of the cross section using the eq
it is assumed that the Coulomb potential of the target nucleus and projectile is that resulting from bare nuclei .
s(E)b= S(E)b(1/E) exp(-2ph)
Cross-section Astrophysical
factor
(bare nucleus)

7. DIRECT NUCLEAR REACTIONS BETWEEN CHARGED PARTICLES: LIMITS
However,.. in the laboratory,
the target and projectile nuclei are in the form of neutral atoms or molecules and ions, respectively.
The second relevant source of uncertainty in extrapolating the
S(E)-factor at astrophysical energies
(down to zero energy)
is the enhancement due to the electron screening effect !!!!
Due to this fact
a reduction of Coulomb barrier by electron cloud of target nucleus is seen
f lab(E) =
with
f lab(E) ~ exp(Ue/E)  1
and electron screening potential energy Ue
Ss(E)
Sb(E)
Enhancement flab(E) in the astrophysical Sb(E)-factor

8. Evidences of electron screening
NUCLEAR REACTIONS BETWEEN CHARGED PARTICLES: ELECTRON SCREENING
3He(3He,2p)4He
S(E) (MeV b)
Ss(E)
Sb(E)
E (KeV)
Ecm (keV)
3He( 2H,p)4He
Evidences of
electron screening
Ss(E)
S(E) (MeV b)
Evidences of electron screening
Sb(E)
Ecm (keV)

9. DIRECT NUCLEAR REACTIONS BETWEEN CHARGED PARTICLES: LIMITS
Critical point:
due to electron screening,
EXTRAPOLATION
it is necessary as a “standard solution”
in order to extract cross sections at Gamow
energy

10. WHY INDIRECT METHODS ARE NEEDED? INDIRECT METHODS ARE NEEDED
Some INDIRECT METHODS
NEW METHODS ARE NECESSARY
to measure cross sections at never reached energies
a) - Coulomb dissociation
(talk Togano)
b) - Asymptotic Normalization
Coefficients (Anc)
Transfer reactions
d) b-delayed particle emission
e) The Trojan Horse Method
(THM)
f) Breakup of RNB
(talk Trache)
g) R-matrix
to retrieve information on electron screening effect when ultra-low energy measurements are available.
INDIRECT METHODS ARE NEEDED

11. - Trojan Horse
Basic idea:
It is possible to extract astrophysically relevant two-body cross section 
B + x  C + D
from quasi- free contribution
of an appropriate three-body reaction
A + B  C + D + S
Main application:
Charged particle bare nucleus cross section measurements at astrophysical energies
THM has roots in the field of direct processes and in particular in the quasi-free mechanism

12. QUASI-FREE REACTION MECHANISM: Generality
Three body reactions
A + B  C + D + S
Can be described by a Feynmam diagram
-The upper pole describes the
virtual break up of the target nucleus A into the cluster x
(participant) and S x B D C A S
-The S cluster acts as a spectator to
x+ B  C + D virtual reaction which takes place in the lower pole
-The A nucleus present a strong cluster structure:
A = x  S clusters

13. KF is a kinematical factor
In PWIA the cross section can be factorized into two terms corresponding to the two vertices
F(q)xs2 dΩ dσ
Half- Off energy shell KF
dEC dWC dWD
d3σ 
KF is a kinematical factor
|F(qxS)|2 describes the intercluster (x-S) momentum distribution
( ds/dW) is the (alf-)off-energy-shell differential cross-section for the two-body virtual reaction x + B C + D at the centre of mass energy Ecm x B D C A S

14. (under proper kinematical conditions)
ENERGY PRESCRIPTION
Ec.m. is given in postcollision prescription by
Ecm = EC-D - Q2B
Q2b is the two-body Q-value of the x + B C + D
reaction
EC-D is the relative energy between the outgoing
particles c and D x B D C A S
(under proper kinematical conditions)
Ecm~ 0

15. TROJAN HORSE Three body reactions A + B  C + D + S x B D C A S
A- in the quasi-free kinematical regime, the incoming “Trojan horse “ particle A is accelerated at energies EA above the Coulomb barrier energy (EAB)Coul. Bar
EA > (EAB)Coulomb Barrier
Three body reactions
A + B  C + D + S x B D C A S
-The upper pole describes the
virtual break up of the target nucleus A into the cluster x
(partecipant) and S
B- After penetrating through the Coulomb barrier, nucleus A undergoes breakup
-The S cluster acts as a spectator to
x+ B  C + D virtual reaction which takes place in the lower pole
Coulomb effects and electron screening are negligible
G.Baur: Phys. Lett.B178,(1986),135

16. INDIRECT TWO-BODY CROSS SECTION
F(q)xs2 dΩ dσ
half- Off energy shell KF
d3σ 
dEcdWcdWD
Below barrier correction for the Penetration factor is necessary
Ecm < Coulomb Barrier x-B
Calculated
Measured
Above barrier
Ecm > Coulomb Barrier x-B
d3σ
Calculated
half-off shell
Exp.
dEcdWcdWD
[Gl]
Penetrability
factor dσ = dΩ
KF |F(q xs)| 2
x + B C + D
Indirect 2-body
cross section R) (k F G 1 ) (q ax 2 l + =

17. INDIRECT TWO-BODY CROSS SECTION dΩ dσ
x + B C + D
Indirect 2-body
cross section
half-off shell
Direct 2-body
No absolute cross section is measurable
BUT
-If the excitation functions at energies below Coulomb barrier
is known from direct measurements
The absolute value of S(E) must be found by normalization to direct measurements at higher energies.

18. INDIRECT TWO-BODY CROSS SECTION dΩ dσ
x + B C + D
Indirect 2-body
cross section
half-off shell
Direct 2-body
Mukamedzhanov et al. :NPA (2006)
7Li(p,a)4He
6Li(d,a)4He
Theory
Energy dependence of the half-off-shell (red dashed line) and on-shell (black solid line) astrophysical factors for
(a) the 7Li(p,a)4He reaction
(b) 6Li(d,a)4He reaction
are the same !

19. Recent results

20. Depletion lights nuclei: Li, B, Be 1- p (11B ,8Be) a
INDIRECT REACTIONS
DIRECT REACTIONS
Depletion lights nuclei: Li, B, Be
1- p (11B ,8Be) a
2- p (10B ,7Be ) a
3- p (9B ,6Li ) a
11B + d  8Be + a + nspett.
10B + p  7Be + a + nspett.
9Be + d  6Li+ a + nspett.
The Fluorine problem in the AGB :
4- p (15N ,12C ) a
5 -p (18O ,15N ) a
15N + d  a + 12C + nspett.
18O + d  a + 15N + nspett.
Novae:
6 - p (17O ,14N ) a (poster section )
17O + d  a + 14N + nspett.
(p,a) reactions - inverse cinematic

21. I- Example of application: 11B + d  ao + 8Be + n
Study of the reaction
11B + p  ao + 8Be via the
11B + d  ao + 8Be + n 2H p
11B n
8Be α I II
d = p  n
p= participant
n= spectator
S(E) (MeV b)
Ecm (keV)
11B(p,a)8Be
Ss(E)
Sb(E)
E 11B = 27 MeV
TANDEM – LNS Catania
(2002)
E 11B = 27 MeV
TANDEM – LNS Catania
(2007)

22. dσ(E) dΩ F(q)2 F(q xs) 2= Necessary condition for existence
Check if the quasi free reaction mechanism is present and can be discriminated
F(q)2
Selected Momentum range
(-40 – 40) Mev/c
PS(MeV/c)
EαBe(MeV)
E~ const
ψ(r) =
ab(a+b)
2π(a-b)2
e-αr- e-βr r
Hulthe’n wave function
α= fm-1 ; β=1.202 fm-1
FWHM
60 MeV/c
dσ(E) dΩ
d3σ
Measured
d3σ dE d Ω d Ω
F(q xs) 2= C C D  dE d Ω d Ω C C D KF
dσ(E) KF dΩ
x + B C + D
Necessary condition for existence
quasi-free mechanism
Calculated

23. Verify that all direct data are reproduced -angular distributions - excitation functions
θCM(deg)
THM Data
E=.5 ± .1 MeV
E=.7 ± .1 MeV
E=.9 ± .1 MeV
THM
direct data
(Becker et al:
ZPhys.327,341,1987)
Direct data (Becker, 1987)
THM data (2003)
direct data
THM
Excitation function
Angular distributions

24. I- Example of application:11B + d  ao + 8Be + n ASTROPHYSICAL FACTOR Extracted via TH and direct data
S(0) ± DS(0)
MeVb
Present work
THM
1.98 ± 0.20
New RESULTS
Becker
2.1
Direct
data
Barker
2.4
R-matrix
RUN 2007 at LNS

25. Ii- Example of application: 15N + d  ao + 12C + n
Study of the reaction
15N + p  ao + 12C
via the
15N + d  ao + 12C + n 2H p
15N n
12C α I II
E 15N = 60 MeV
Cyclotron- TAMU
– College Station
La Cognata et al.
PRC in press

26. Necessary condition for existence
Check if the quasi free reaction mechanism is present and can be
discriminated from others.
Study of angular correlations energy spectra:
coincidence spectra projected for a fixed 1 and different 2
Events corresponding to a quasi-free mechanism
Coincidence yield attains a max. for pn approaching zero and decreases while moving far from this condition
15N(d, a 12C)n 
Quasi free angles
Necessary condition for existence
quasi-free mechanism

27. Firsth example of quasi-free nuclear reaction
How to discriminate the quasi-free contribution?
9Be(3He,a a)4He
Study of angular correlations energy spectra:
Firsth example
of quasi-free nuclear reaction
coincidence spectra projected for a fixed 1 and different 2
Events corresponding to a quasi-free mechanism
Coincidence yield attains a max. for pn approaching zero and decreases while moving far from this condition 
Quasi free angles
Necessary condition for existence
quasi-free mechanism
A.Kasagi . et al. :Nucl. Phys.A,239(1975),233

28. - excitation functions including resonances -angular distributions
Verify that all direct data are reproduced
- excitation functions angular distributions
Direct
data
Direct
data
THM
THM
- excitation functions including resonances
-angular distributions

29. II- Example of application; 15N + d  ao + 12C + n :
S(0) ± DS(0)
MeVb
Present work
Zyskind 79
Redder 82
THM
R-matrix
Direct data
62 ± 10
64 ± 10
78 ± 6
65 ± 4
La Cognata et al. (2005)
Comparison with direct data
Total cross section extracted after integrating d/d over 4
-To be compared indirect data should be normalized to direct ones
- Experimental error:
stat. + norm. + background
- Energy resolution: 20 keV

30. III- Example of application; 10B + d  ao + 7Be + n
Study of the reaction
10B + p  ao + 7Be via the
10B + d  ao + 7Be + n 2H p
10B n
7Be α I II
extrapolation
E 10B = 27 MeV
TANDEM – DFN ,San Paolo
L.Lamia et al.NPA 787,309c(2007)
Resonance at Ecm = 10 keV corresponding to the
11C* (8.70 MeV)
E 11B = 27 MeV
TANDEM – LNS Catania
GAMOW
ENERGY

31. IV- Example of application: 18O + d  ao + 15N + n
Study of the reaction
18O + p  a + 15N
via the
18O + d  a + 15N + n 2H p
18O n
15N α I II
PRELIMINARY RESULTS
d = p  n
p= participant
n= spectator
E 18O = 60 MeV
Cyclotron- TAMU, College Station
E 18O = 60 MeV
TANDEM – LNS Catania

32. V- Example of application: 17O + d  ao + 14N + n
Study of the reaction
17O + p  a + 14N
via the
17O + d  a + 14N + n 2H p
17O n
14N α I II
d = p  n
p= participant
n= spectator
E 17O = 41 MeV
TANDEM – LNS Catania
Poster session

33. Perspectives

34. New application at CNS : 18F + d  ao + 15O + n
Study of the reaction
18F + p  a + 15O
via the
18F + d  a + 15O + n 2H p
18F n
15O α
d = p  n
p= participant
n= spectator
E 18F = 50. MeV
CNS, Tokyo Univ.
(p,a) reaction

35. New application : 12C + 12O a + 20Ne + d
Study of the reaction
12C + 12C  a + 20Ne
via the
12C + 16O  a + 20Ne + a
16O a
20Ne
12C
12C α
Tandem , LNS , Catania

36. It is possible to study nuclear reactions induced by
IN PRINCIPLE:
It is possible to study nuclear reactions induced by
light nuclear particles (both stable and unstable).
Nucleus Trojan Horse
clusters
Inter
cluster momentum
l-relative
Bindind energy
(MeV) 1 d
p-n
2.225 2 t
d-n
6.257 3
3He
d-p
5.494 4
6Li
d-a
1.475 5
7Li
t-a
2.468 6
7Be
3He-a
1.587 7
9Be
5He-a
2.467

37. IN PRINCIPLE:
It is possible to study nuclear reactions induced by
light nuclear particles (both stable and unstable).
Indirect
Beam
“Trojan Horse nucleus” 1 n
d, 3H 2 p
d, 3He 3 d
3He, 3H, 6Li 4 t
7Li 5
3He
7Be 6 a
6Li, 7Li, 7Be, 9Be 7
5He
9Be

38. VII- Example of application: 6Li+ d  a + t + p2H n
6Li p t α I II
Study of the reaction
6Li+ n  a + t
via the
6Li+ d  a + t + p
d = p  n
p= participant
n= spectator
Deuteron-Beam as a virtual
Neutron-beam
Tandem –LNS, Catania (2004)
Tandem- LNS, Catania
(2006)

39. Comparison between direct-indirect excitation function
6Li(n,a)3H studied through the 6Li(d,a 3H)p reaction MeV
E6Li=14 MeV
Comparison between direct-indirect excitation function
7Li(p,a)4He studied through the 7Li(d,aa)n reaction
………………MeV
PWIA
Tandem- LNS
PWIA
Tandem- LNS
PWIA
Tandem- LNS
direct data
THM
Resonances reproduced
Tandem- LNS
PWIA
d = p  n
p= participant
n= spectator
6Li a n
Qf kinimatic conditions
Qf kinematic conditions
+ LENS EFFECT
3He p d
Tumino et al. EPJ

40. MAIN LIMITATIONS OF THE METHOD
A- Preliminary study of quasi-free mechanism.
Presence of different 3-body reaction mechanisms
(Sequential Decay)
B- Tests of validity are necessary.
C- Measurements with high angular and energy
resolutions are needed;
D-Theoretical analysis is needed:
- PWIA, MPWBA, DBWA, …..

41. SUMMARY
-The main advantages of the THM are that the extracted cross section of the binary subprocess does not contain the Coulomb barrier factor.
No Coulomb barrier effects
-TH cross section can be used to determine the energy dependence of the astrophysical factor, S(E), of the binary process x+ B c + C,down to zero relative kinetic energy of the particles x and B without distortion due to electron screening.
No extrapolation
No electron screening effects
- It is possible to measure excitation function in a “ relatively” short time because typical order of magnitude for a three- bod cross- section is of oder mb - Possibility of application to the radioactive beam measurements;
- No complex experimental apparatus.
-At low energies where electron screening becomes important, comparison
of the astrophysical factor determinated from the TM Method to the
direct result provides a determination of the screening potential.

42. The University of Tokyo, Tokyo, Japan N. IWASA
S. CHERUBINI, V.CRUCILLÀ, M.GULINO, M.LA COGNATA, M.LAMIA, R.G.PIZZONE, S.PUGLIA, G.RAPISARDA, S.ROMANO, L.SERGI, C.SPITALERI, S.TUDISCO, A.TUMINO
I N F N, Laboratori Nazionali del Sud, Catania, Italy and Università di Catania, Italy
S. KUBONO, S. HAYAKAWA, Y. WAKABAYASHI,H. YAMAGUCHI Center for Nuclear Study, Graduate School of Science,
The University of Tokyo, Tokyo,  Japan
N. IWASA
Department of Physics, Tohoku University, Sendai, Japan
S. KATO
Department of Physics, Yamagata University,Yamagata, Japan
S. NISHIMURA RIKEN (The Institute of Physical and Chemical Research), Wako, Japan
T. TERANISHI Department of Engineering, Kyushu University, Kyushu, Japan
A.MUKHAMEDZHANOV, B.TRIBBLE, L.TRACHE,V.GOLDBERG
Ciclotron Institute, Texas A&M University, Usa
A.COC,
CSNSM, Orsay,France
F.HAMMACHE, N. DE SERVEILLE
IPN, Orsay, France

43. Ciae, Bejin, Cina V.BURJAN, V.KROHA, J. MRAZEK
Nuclear Physics Institute, Academic of Science,Rez, Czech Rep
S. ZHOU, LI CENGBO
Ciae, Bejin, Cina
Z.ELEKES, Z.FULOP, G.GYURKY, G.KISS, E.SOMORJAI
Inst. of Nuclear Research ofAcademic of Science Debrecen,Ungaria
G.ROGACHEV
FSU, USA
N.CARLIN, M.GAMEIRO MUNHOZ, M.GIMENEZ DEL SANTO, R.LIGUORI NETO, M.DE MOURA, F.SOUZA, A.SUAIDE, E.SZANTO, A.SZANTO DE TOLEDO
Dipartimento de Fisica Nucleare, Universidade de Sao Paulo,Brasil

44. Thank you ETNA in Sicily
OMEG07 : The 10th Int. Symp. On Origin of Matter and Evolution of Galaxies December 4-7 ,Hokkaido University
Thank you
ETNA in Sicily