Trojan Horse as an indirect technique in nuclear astrophysics. Resonance reactions

Abstract

The Trojan Horse method is a powerful indirect technique that provides information to determine astrophysical factors for binary rearrangement processes $x + A \to b + B$ at astrophysically relevant energies by measuring the cross section for the Trojan Horse reaction $a + A \to y+ b + B$ in quasi-free kinematics. We present the theory of the Trojan Horse method for resonant binary subreactions based on the half-off-energy-shell R matrix approach which takes into account the off-energy-shell effects and initial and final state interactions.

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arXiv:0708.0658v1 [nucl-th] 5 Aug 2007
Trojan Horse as an indirect technique in nuclear astrophysics. Resonance reactions.
A. M. Mukhamedzhanov1, L. D. Blokhintsev2, B. F. Irgaziev3, A.
S. Kadyrov4, M. La Cognata5, C. Spitaleri5and R. E. Tribble1
1Cyclotron Institute, Texas A&M University, College Station, Texas, 77843, USA
2Institute of Nuclear Physics, Moscow State University, Moscow, Russia
3Faculty of Engineering Sciences, GIK Institute of Engineering Sciences and Technology, Topi-23640, N.W.F.P., Pakistan
4ARC Centre for Antimatter-Matter Studies, Curtin University of Technology,
GPO Box U1987, Perth, WA 6845, Australia and
5DMFCI, Universit` a di Catania, Catania, Italy and INFN - Laboratori Nazionali del Sud, Catania, Italy
The Trojan Horse method is a powerful indirect technique that provides information to determine
astrophysical factors for binary rearrangement processes x + A → b + B at astrophysically relevant
energies by measuring the cross section for the Trojan Horse reaction a+A → y+b+B in quasi-free
kinematics. We present the theory of the Trojan Horse method for resonant binary subreactions
based on the half-off-energy-shell R matrix approach which takes into account the off-energy-shell
effects and initial and final state interactions.
PACS numbers: 26.20.+f, 24.50.+g, 25.70.Ef, 25.70.Hi
I.INTRODUCTION
The presence of the Coulomb barrier for colliding charged nuclei makes nuclear reaction cross sections at astrophys-
ical energies so small that their direct measurement in the laboratory is very difficult, or even impossible.
Consequently indirect techniques often are used to determine these cross sections. The Trojan Horse (TH) method
is a powerful indirect technique which allows one to determine the astrophysical factor for rearrangement reactions.
The TH method, first suggested by Baur [1], involves obtaining the cross section of the binary x+A → b+B process
at astrophysical energies by measuring the two-body to three-body (2 → 3) TH process, a + A → y + b + B, in the
quasi-free (QF) kinematics regime, where the ”Trojan Horse” particle, a = (xy), is accelerated at energies above the
Coulomb barrier. After penetrating through the Coulomb barrier, nucleus a undergoes breakup leaving particle x to
interact with target A while projectile y flies away. From the measured a + A → y + b + B cross section, the energy
dependence of the binary subprocess, x + A → b + B, is determined.
The main advantage of the TH method is that the extracted cross section of the binary subprocess does not contain
the Coulomb barrier factor. Consequently the TH cross section can be used to determine the energy dependence of
the astrophysical factor, S(E), of the binary process, x + A → b + B, down to zero relative kinetic energy of the
particles x and A without distortion due to electron screening [2, 3]. The absolute value of S(E) must be found by
normalization to direct measurements at higher energies. At low energies where electron screening becomes important,
comparison of the astrophysical factor determined from the TH method to the direct result provides a determination
of the screening potential.
Even though the TH method has been applied successfully to many direct and resonant processes (see [4] and
references therein), there are still reservations about the reliability of the method due to two potential modifications
of the yield from off-shell effects and initial and final state interactions in the TH 2 → 3 reaction. Here we will address
the theory of the TH method for resonant binary reactions x + A → b + B.
II.TROJAN HORSE
The TH reaction is a many-body process (at least four-body) and its strict analysis requires many-body techniques.
However some important features of the TH method can be addressed in a simple model. Let us consider the TH
process assuming that nuclei y, x and B are constituent particles, i. e. we neglect their internal degrees of freedom.
For simplicity, we disregard the spins of the particles. The TH reaction amplitude is given in the post form by
˜
M(P,kaA) =< χ(−)
kyFΦ(−)
F|∆VyF|Ψ(+)
i
> . (1)
Here, Ψ(+)
χ(−)
kyF(rij) is the distorted wave of the system y +F, ϕiis the bound state wave function of nucleus i, rijand kijare
the relative coordinate and relative momentum of nuclei i and j, P = {kyF,kbB} is the six-dimesional momentum
describing the three-body system y, b and B in the final system, ∆VyF= VyF− UyF, VyF = Vyb+ VyB= Vyx+ VyA
i
is the exact a + A scattering wave function, Φ(−)
F
is the wave function of the system F = b + B = x + A,

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is the interaction potential of y and the system F and UyF is their optical potential. The surface approximation
suggested in [? ] was the first serious attempt to address the theory of the TH method. The surface approximation
assumes that the TH reaction amplitude has contributions from the external region where the interaction between the
fragments b and B (x and A) can be neglected and the wave function Φ(−)
form
F
can be replaced by its leading asymptotic
Φ(+)
F
≈ ϕb[eikbB·rbB+ FbBu(+)
kbB(rbB)] +
?vbB
vxAMbB→xA
1
2ikbB
u(+)
kxA(rxA), (2)
where Φ(+)
scattering amplitude, MbB→xAis the b+B → x+A reaction amplitude inverse to the binary reaction x+A → b+B and
vijis the relative velocity of nuclei i and j. The expression for the TH reaction amplitude in the surface approximation
is given by
F
≡ Φ(+)
kbB(F)and Φ(−)
kbB(F)= Φ(+)∗
−kbB(F), u(+)
kij(rij) is the outgoing spherical wave, FbB is the b + B elastic
˜
M(P,kaA) ∼ MbB→xA < χ(−)
kyFϕAu(−)
kxA(rxA)|∆VyF|ϕaϕAχ(+)
kaA(raA) >,(3)
where the exact initial scattering wave function Ψ(+)
describing the scattering of the nuclei a and A in the initial state of the TH reaction. For simplicity we don’t take
into account here the Coulomb interactions. However, in the case of the resonant binary reaction x + A → b + B the
dominant contribution comes from the nuclear interior where both channels x + A and b + B are coupled and where
the asymptotic approximation for Φ(+)
F
cannot be applied[12].
In this work we will address the theory of the TH method for the resonant binary subprocesses x + A → b + B
which explicitly takes into account the off-shell character of x. Eq. (1) can be used as a starting point to derive the
expression for the TH reaction amplitude. We assume that the resonant reaction x+A → b+B proceeds through the
formation of the intermediate compound state Φi, i. e. we neglect the direct coupling between the initial x + A and
final b + B channels, which contributes dominantly to direct reactions but gives negligible contribution to resonant
ones. An important step in deriving the resonant contribution to the TH reaction matrix element is the spectral
decomposition for the wave function Φ(−)
F
given by Eq. (3.8.1) [7]. It leads to the shell-model based resonant R
matrix representation for Φ(−)
F
which is similar to the level decomposition for the wave function in the internal region
in the R matrix approach:
i
is replaced by ϕaϕAχ(+)
kaA(raA) and χ(+)
aAis the distorted wave
Φ(−)
F
≈
N
?
ν,τ=1
˜VbB
ν
(EbB)[D−1]ντΦτ. (4)
Here N is the number of the levels included, EbB is the relative kinetic energy of nuclei b and B, Φτ is the bound
state wave function describing the compound system F excited to the level τ. Dντ is similar to the level matrix in
the R matrix theory and is given by Eq. (4.2.20b) [7]. Finally,
˜VbB
ν
(EbB) =< χ(−)
bBϕb|∆VbB|Φν> (5)
is the resonant form factor for the decay of the resonance Fν described by the compound state Φν into the channel
b + B. The partial resonance width is given by
˜Γν(EbB) = 2π|˜VbB
ν
(EbB)|2.(6)
Then the TH reaction amplitude is
˜
M(R)(P,kaA) ≈
N
?
ν,τ=1
˜VbB
ν (EbB)[D−1]ντ˜
Mτ(kyF,kaA), (7)
where˜
state Fτ of the system F = x + A = b + B:
Mτ(kyF,kaA) is the exact amplitude for the direct transfer reaction a+A → y+Fτpopulating the compound
˜
Mτ(kyF,kaA) =< χ(−)
yFΦτ|∆VyF|Ψ(+)
i
> .(8)
The direct transfer reaction is very well described by the DWBA amplitude, i. e. for the practical analysis we can
approximate Ψ(+)
i
≈ ϕaϕAχ(+)
Mτ(kyF,kaA) can be replaced by
aA. Correspondingly,˜
˜
MDW
τ
(kyF,kaA) =< χ(−)
yFΦτ|∆VyF|ϕaϕAχ(+)
i
> .(9)

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Correspondingly for the TH reaction amplitude we get from Eq. (7)
˜
M(R)(P,kaA) ≈
N
?
ν,τ=1
˜VbB
ν (EbB)[D−1]ντ˜
MDW
τ
(kyF,kaA). (10)
The DWBA amplitude takes into account the rescattering of nuclei a and A in the initial state of the TH reaction
and enters as a form factor into the TH resonant reaction amplitude reflecting the off-energy shell character of the
transferred particle x. Since in the TH method the astrophysical factor determined from the TH method is normalized
to the on-energy-shell (OES) S factor, the replacement of the exact transfer amplitude by the DWBA one, as we will
see, practically does not affect the final result.
A.Single resonance
The triple differential cross section for the TH process a+A → y+b+B proceeding through an isolated resonance
Fτ is given by
d3σ
dEbBdΩkbBdΩkyF
= λ3
ΓbB(τ)(EbB)|MDW
(ExA− ERτ)2+Γ2
τ
(kyF,kaA)|2
τ(ExA)
4
.(11)
Here, λ3 is the kinematical factor, ΓbB(τ)(EbB) is the observable resonance partial width in the channel b + B,
Γτ(ExA) is the total observable width of the resonance Fτ. Note that all functions T(E) are related to˜T(E) as
T(E) =˜T(E)/(1 − (d ∆ττ
dE)E=ERτ), where ∆ττ is the τ level shift. Also ERτis the resonance energy of the resonance
Fτ in the channel x + A. Thus the TH triple differential cross section, in contrast to the OES single-level resonance
cross section, contains the generalized form factor |MDW
τ
(kyF,kaA)|2rather then the entry channel partial resonance
width ΓxA(τ)(ExA) of the binary process x+ A → b +B. A simple renormalization of the TH triple differential cross
section allows us to single out the OES astrophysical factor for the resonant binary subprocess x + A → b + B:
S(ExA) = NF(ExA)
d3σ
dEbBdΩkbBdΩkyF
=
π
2µxAe2π ηxAΓbB(τ)(EbB)ΓxA(τ)(ExA)
(ExA− ERτ)2+Γ2
τ(ExA)
4
,(12)
where the normalization factor NF(ExA) is given by
NF(ExA) =
π
k2
xA
1
λ3ExAe2π ηxA
ΓxA(τ)(ExA)
|MDW
τ
(kyF,kaA)|2. (13)
Note that the DWBA amplitude MDW(kyF,kaA) remains practically constant on the interval of a few hundreds keV.
Eq. (12) explaines and justifies the phenomenological procedure used before successfully in the TH analysis (see [4]
and references therein). The renormalization factor can be rewritten as
NF(ExA) = e
2π?
ηxA−ηxA
??
ExA=ER1
?
ΓxA(τ)(ExA)
ΓxA(τ)(ER1)NF(ER1), (14)
where ΓxA(τ)(ExA)/ΓxA(τ)(ER1) = P(ExA) is the barrier penetration factor appearing in the R matrix theory. The
factor NF(ER1) can be found phenomenologically by comparing the experimental TH triple differential cross section
with the available OES experimental astrophysical factor at resonance energy. This phenomenological normalization
leads to the intermediate astrophysical factor
S′(ExA) =
π
2µxAe2π ηxA|ExA=ER1ΓbB(τ)(EbB)ΓxA(τ)(ER1)
(ExA− ERτ)2+Γ2
τ(ExA)
4
.(15)
The final astrophysical factor can be derived by multiplying S′(ExA) by the energy-dependent factor in Eq. (14)
2π?
ΓxA(τ)(ExA)/ΓxA(τ)(ER1). Thus normalization of the triple TH differential cross section to the
experimental astrophysical factor at resonance energy achieved by multiplying Eq. (11) by the factor NF(ExA) plays
a very special role in the TH method.
e
ηxA−ηxA
??
ExA=ER1
?

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B. Two interfering resonances
For two interfering resonances we need to consider the two-level, two channel case. This requires the half-off-energy-
shell (HOES) R matrix formalism. Here we address this formalism for a simple case when the distances between two
resonances are significantly larger then their total widths. Then the OES reaction amplitude in the R matrix formalism
is given by the sum of the amplitude of each resonances (see Eq. (XII,5.15) [8]). The corresponding expression for the
HOES reaction amplitude can be obtained by the replacement of the resonance partial widths in the entry channel of
the binary reaction x+A → b+B by the corresponding generalized form factors MDW
triple TH cross section in the presence of two interfering resonances in the subsystem F = x + A = b + B is given by
τ
(kyF,ka), τ = 1,2. Thus the
d2σ
dEbBdΩkyFdΩkbB
= λ3
???
τ=1,2
Γ1/2
bB(τ)(EbB)MDW
ExA− ERτ+ iΓτ(ExA)
τ
(kyF,kaA)
2
??2.(16)
We assume that ER1< ER2. The goal of the THM is to determine the energy dependence of the astrophysical factor
at the astrophysically relevant energies. The ratio MDW
21
= MDW
in the interval of a few hundred keV, ExA≤ ER1. Normalizing the TH cross section to the OES S factor at E = ER1,
where the contribution from the second resonance can be neglected, gives the astrophysical factor determined from
the TH reaction
Γ1/2
bB(1)(EbB)
ExA− ER1+ iΓ1(ExA)
2
(kyF,kaA)/MDW
1
(kyF,kaA) is practically constant
STH(ExA) =π e2π ηxA
2µxA
ΓxA(1)(ExA)???
2
+
Γ1/2
bB(2)(EbB)MDW
ExA− ER2+ iΓτ(ExA)
21
2
???2. (17)
This astrophysical factor is to be compared with the OES astrophysical factor determined from direct measurements
S(ExA) =π e2π ηxA
2µxA
ΓxA(1)(ExA)???
xA(2)(ExA)/Γ1/2
(kyF,ka) is complex, but the ratio MDW
Γ1/2
bB(1)(EbB)
ExA− ER1+ iΓ1(ExA)
2
+
Γ1/2
bB(2)(EbB)γ(xA)21
ExA− ER2+ iΓτ(ExA)
2
???2.(18)
Here, γ(xA)21= γ(xA)2/γ(xA)1= Γ1/2
the channel x+A. Each amplitude MDW
The normalization of the TH S factor to the OES one at resonance energy plays a crucial role in the TH method.
After such a normalization, we need to know only the ratio of the DWBA amplitudes to calculate STH(ExA).
xA(1)(ExA) and γ(xA)τis the reduced width for the τ-th resonance in
2 21
may have a small imaginary part.
1. Plane wave approximation
Ratio MDW
proximation, because a simple plane wave approximation gives similar angular and energy dependence as the DWBA
but fails to reproduce the absolute value. It explains why a simple plane wave approximation works well in the TH
analysis [4]. Note that in the plane wave approximation MDW
21
can be approximated by the ratio of the corresponding amplitudes calculated in a plane wave ap-
τ
(kyF,kaA) is replaced by
M0
τ(kyF,kaA) =< eikyF·ryFϕyΦτ|VyA+ VxA|ϕaϕAeikaA·raA> . (19)
Note that the post and prior forms are equivalent but the post form is more convenient for our purpose. In the QF
kinematics for sufficiently high momentum of the the projectile A it will interact dominantly with the fragment x
while the contribution of the term with VyAis minimized. That is why in what follows we neglect the term containing
VyA. Then the transfer reaction amplitude in the plane wave approximation takes the form
M0
τ(kyF,kaA) ≈< eikyF·ryFIFτ
xA| < VxA>xA|Ia
yxeikaA·raA>,(20)
where IFτ
wave functions of A and x, Ia
a, x and y, and ϕx, < VxA>=< ϕAϕx|VxA|ϕxϕA>. The plane wave amplitude M0
factorized form
xA=< ϕAϕx|Φτ> is the overlap function of the wave function of the resonance state Fτand the bound state
yx=< ϕyϕx|ϕa> is the overlap function of the bound state wave functions of nuclei
τ(kyF,kaA) can be written in a
M0
τ(kyF,kaA) = [WFτ
xA(kA−mA
mF
kF)]∗Ia
yx(ky−my
maka).(21)
Here, Ia
yx(pyx) is the Fourier transform of the overlap function Ia
yx(ryx) and
WFτ
xA(kxA) =< eikxA·rxA| < VxA>xA(rxA)|IFτ
xA(rxA) > (22)
is the vertex form factor for x + A → Fτ. Then Eq. (16) for the TH triple differential cross section takes the form

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?
?
?
FIG. 1: Comparison of the calculated astrophysical factor STH(E) for
direct data [9, 10, 11].
15N(p,α)12C (solid line), where E ≡ ExA, with the
d2σ
dEbBdΩkyFdΩkbB
= λ3|Ia
yx(ky−my
maka)|2???
τ=1,2
Γ1/2
bB(τ)(EbB)[WFτ
ExA− ERτ+ iΓτ(ExA)
xA(kA−mA
mFkF)]∗
2
??2. (23)
Now we can get the HOES cross section for the binary subprocess x + A → b + B from the triple differential cross
section
?
dσ
dΩc.m.
?HOES
∝
?
d2σ
dEbBdΩkyFdΩkbB
?
1
λ3|Ia
yx(pyx)|2,(24)
where pyx= ky−my
TH cross sections. Note that in a strict approach the triple differential cross section is expressed in terms of the
overlap function Ia
maka. Eq. (24) explains and justifies the procedure used in IA [4] to connect the triple and binary
yxrather then the two-body bound state wave function ϕa. Note that Ia
yxand ϕaare related by
Ia
yx= S1/2
yxϕa, (25)
where S1/2
goal is the TH astrophysical factor which can be determined by normalization of the triple differential cross section to
the OES astrophysical factor in the first resonance peak and is given by Eq. (17). In the plane wave approximation
MDW
21
is replaced by
yx is the spectroscopic factor. The binary reaction HOES cross section is only intermediate result. The final
M0
21=
[WFτ
[WFτ
xA(kA−mA
xA(kA−mA
mFkF)]∗
mFkF)]∗. (26)
If M0
ER1. In Fig. 1 the astrophysical factor STH(ExA) for15N(p,α)12C calculated using Eq. (17) for the TH reaction
15N(d,nα)12C is compared with the experimental S(ExA) obtained from direct measurements. There are two 1−
interfering resonances at ER1= 312 keV and ER2= 962 keV. The best fit has been achieved for ΓxA(1)≡ Γp(1)= 1.1
keV, ΓbB(1)≡ Γα(1)= 93.4 keV, ΓxA(2)≡ Γp(2)= 95.31 keV and ΓbB(2)≡ Γα(2)= 45 keV. To find M0
(22) in which the overlap function IF(i)
potential calculated in the internal region by a procedure similar to that used in R-matrix method to calculate the
level eigenfunctions. We find that M0
shown in Fig. 1 is in an excellent agreement with the direct data.
We presented the expression for the resonant S factor determined from the TH reaction taking into account the off-
energy-shell effects within the HOES R matrix formalism and justified a simple plane wave approximation. Validating
this makes it clear why the TH method is such a powerful indirect technique for nuclear astrophysics.
21≈ γ(xA)21, the astrophysical factor STH(ExA) reproduces the OES S factor S(ExA) at energies ExA ≤
21we used Eq.
xAis approximated by a single-particle15N−p wave function in the Woods-Saxon
21≈ 1.13 while γ(xA)21= 1.1 ± 0.1. It explains why the calculated STH(ExA)

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This work was supported in part by the U.S. DOE under Grant No. DE-FG02-93ER40773.
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[12] Generally speaking one must be very careful in using the asymptotic approximation for the scattering wave function Φ(−)
because the matrix element with the exact wave function in the initial state and ingoing spherical wave u(−)
final state vanishes after transformation of the volume integral into a surface integral [6].
F
kxA(rxA) in the