1 Introduction
Accurate measurements of Gamow–Teller strength distributions find important usage in three sub-fields of nuclear
physics: nuclear structure, neutrino physics and nucleosynthesis. To increase our understanding of nuclear structure, the
Gamow–Teller strength distribution is often used to obtain
information about the nuclear wave function. The Gamow–
Teller strength is usually characterised in terms of a B(GT)
value, which is defined as where Ji is the total angular momentum of the parent nucleus
and Ψi (Ψ f ) is the parent (daughter) nuclear wave function
[1]. τ ±, j is the isospin raising/lowering operator for the jth
nucleon [2] and σ j is the spin operator of the jth nucleon,
defined in terms of the x, y and z Pauli spin-matrices. The
± symbol defines whether the isospin z-component is raised
or lowered. Raising the isospin z-component corresponds in
a nucleus to a p → n transition and lowering corresponds
to an n → p transition (which is equivalent to performing
a (3 He, t) on that nucleus). B(GT) can be directly extracted
from measurable observables [3], and provides the Gamow–
Teller strength for the final state in the daughter nucleus.
In neutrino physics, nuclear matrix elements, and especially the Gamow–Teller ones (see Eq. (1)), frequently enter
the calculations of 2νββ-decay (two-neutrino double-beta
decay) [4–7]. This information can then be used, together
with other nuclear matrix elements [8], to understand the
0νββ-decay (neutrino-less double-beta decay), which is
especially interesting because even a single observation
(which has not been seen yet [4]) undeniably results in the
conclusion that the neutrino is a Majorana particle [4]. Moreover, such observations provide hints for Grand Unification
Theories (GUT), for SuperSymmetry (SUSY) [5,6,9], and
for understanding the matter-antimatter asymmetry in the
universe [10]. Calculations of the involved matrix elements,
and in particular the Gamow–Teller ones, usually contain
substantial uncertainties [11–13]. Hence, a direct measurement of B(GT ) values serves as a benchmark for theoretical calculations and thus helps to better understand possible observations of 0νββ-decay. Furthermore, such measurements also help in designing new detection techniques
of solar neutrinos [14].
Accurate measurements of B(GT) values also help to
understand nucleosynthesis for the elements heavier than
iron. The most common models to describe nucleosynthesis are the s-, r -, p- and rp-processes [12]. These processes
model the nucleosynthesis as long chains of neutron captures (s and r ), proton captures (rp), or photo-dissociations
( p), alternated by beta-decays. Hence, accurate knowledge
of relevant B(GT) values is important to understand these
processes.
Not only direct measurements of B(GT) values help to
increase our understanding of nuclear structure, neutrino
physics and nucleosynthesis, but also through comparison
to theoretical models, lead to more confidence in the application of the models in regimes not accessible to experiments. Having theoretical models with good predictive power
is important for being able to describe situations where direct
measurements are (almost) impossible to perform [12]. Especially comparisons between experimental data and theoretical models that extend up to high excitation energy are useful,
because this allows us to test the details of the theory. Such
a comparison is made in Sect. 7 for a QRPA+QPVC model
and for the data presented in this paper and indications for
improving the model were obtained. However, other models
(e.g., as described in Ref. [15]) may also benefit from such
comparisons.
In the present work, the cross sections of the 116,122 Sn3
( He, t)116,122 Sb charge-exchange reactions at 140 MeV/u
were measured at very forward angles, including zero
degrees. From these cross sections, the Gamow–Teller
strength distributions were extracted for 116 Sb and 122 Sb
between 0 and 28 MeV excitation energy. The experiment
was performed using the high-energy resolution Grand
Raiden spectrometer [16] at the Research Center for Nuclear
Physics (RCNP) [17] at Osaka University in Osaka, Japan.
There exist two reasons for specifically investigating the
116,122 Sn → 116,122 Sb Gamow–Teller transitions. The first
reason is that these isotopes provide important benchmarks
for theoretical studies that can have implications on nucleosynthesis processes. The second reason is that when measuring the mentioned Gamow–Teller transitions by a (3 He, t)
charge-exchange reaction, the spin-dipole (2− ) transitions
can also be measured at the same time. 122 Sb has a 2− g.s. and
116 Sb has a 2− excited state at 518 keV [18], both of which
are relatively strongly excited by a (3 He, t) charge-exchange
reaction at 140 MeV/u. For 0νββ-decay, the matrix elements
of the spin-dipole transitions are also involved. Hence, it is
important to find cases where such transitions occur (such as
these), in order to gauge the theoretical calculations [19,20].
These spin-dipole transitions will be the subject of another
paper, presenting the results of the data obtained in the same
experimental campaign.
The level densities of the chosen Sb isotopes are quite
high [18]. For this reason the Sn(3 He, t)Sb charge-exchange
reaction was used, since it allows for good energy resolution in the measurements. This is possible because both the
projectile and ejectile are charged particles and a magnetic
spectrometer has been used for detection of the ejectile [21].
Because of this, the energy resolution was good enough to
resolve the first few ΔL = 0, 1 transitions. Finally, a beam
energy of 140 MeV/u was chosen, because for this energy
a systematic study was done for the reliability of extracting B(GT ) values from differential cross sections (which
is about 5%) [21]. Earlier (3 He, t) measurements for Sntargets (especially 116 Sn and 122 Sn) were performed at lower
energies (e.g., at 67 MeV/u [22]). However, the extraction of
B(GT ) values from cross sections at these lower energies
is less reliable, since the spin–isospin term of the nucleon–
nucleon interaction is not dominant at these energies.
In this paper, the measurement and data-analysis procedures will be discussed briefly in Sects. 2 and 3. Further
details can be found in the PhD thesis of Ref. [23]. Subsequently, the measurement results (the B(GT) values) will be
presented in Sect. 4. These results are then discussed by comparing them to other experiments [22,24] in Sects. 5 and 6
and to theoretical calculations in Sect. 7. Finally, an outlook
is presented in Sect. 8 and our conclusions are summarised
in Sect. 9.
2 Measurement procedure
The measurement was done using the high-resolution Grand
Raiden spectrometer [16] at RCNP. A 3 He2+ beam of
140 MeV/u was produced with the coupled AVF and RING
cyclotrons at RCNP [17,25] and then transported through the
high-energy-resolution beam line called ‘WS course’ [26] to
Grand Raiden. Using the ‘WS course’, the lateral and angular dispersion-matching techniques [27] were employed to
obtain an energy resolution of σ ≈ 14 keV (σ means that
the resolution is expressed as a standard deviation). Contributions from differences in energy loss of tritons and 3 He
in the target are not included in this number (see Sect. 3 for
those contributions).
The used 116 Sn target had an areal density of 1.87 ± 0.01
mg/cm2 . For the 122 Sn target, it was 1.75 ± 0.01 mg/cm2 .
Both targets were isotopically enriched above 95%. The
ejected tritons were guided to the spectrometer focal plane
in over-focus mode [28] to ensure a good sensitivity to the
vertical scattering angle (this resolution was σvert ≈ 0.16◦ ).
Thanks to the angular dispersion mode, sensitivity to the horizontal scattering angle was also guaranteed (σhor ≈ 0.1◦ ).
Data were taken for two positions of Grand Raiden, i.e. for
0◦ and for 2.5◦ . In over-focus mode, the horizontal acceptance was roughly ± 1◦ around the designated position and
the vertical acceptance was roughly ± 3◦ around zero.
Grand Raiden consists of two large dipole magnets for
momentum separation and two quadrupole and higher-order
magnets to focus the tritons and remove aberrations [16]. In
the 0◦ position, the unreacted 3 He2+ beam was stopped in a
Faraday cup inside the first dipole magnet. Due to the lateral
dispersive mode, this Faraday cup did not have a perfect
efficiency for stopping the beam [29]. In the 2.5◦ position,
the 3 He2+ beam was stopped in a Faraday cup behind the
first quadrupole magnet (with full efficiency). This setup is
illustrated in Fig. 1.
The focal-plane detection system consists of two MultiWire Drift Chamber (MWDC) detectors and two plastic scintillators (10 mm thick) behind them, with a 10 mm thick aluminium plate between those scintillators to prevent that secondary electrons from one scintillator can produce a signal
in the other. Both scintillators were equipped with a photomultiplier at each end. The trigger signal is a coincidence
between the two scintillators (four photomultipliers). This
signal provides a common start signal for the data taking.
Each MWDC detector consists of two layers of alternating
potential and sense wires. The first layer contains vertically
oriented wires and the second one contains diagonally ori-
ented wires. Hence, each MWDC detector is capable of providing two position coordinates of a passing triton track. With
4 such position coordinates in total, it is possible to reconstruct the horizontal and vertical positions and angle of incidence of the triton track at the focal plane.
The data of the MWDC detectors and of the scintillators
were first transported to a computer server and then saved on
hard-disk memory without performing software operations
of any kind. Subsequently, the data were converted offline to
ROOT [30] format (version 5.34) using Gey’s analyser code
[31], which is a modified version of Tamii’s analyser code
[25] in the sense that it now generates output in the ROOT
data format. All subsequent data analysis (see next section)
was done using ROOT.
To calibrate the excitation energy and the horizontal and
vertical scattering angles at the target, a sieve–slit measurement was performed. In such a measurement, a target with
known, sharply defined, states (95% isotopically enriched
13 C for the present experiment) is used while a sieve slit (a
multi-hole aperture) is placed between the target and the first
magnet of the spectrometer (see Fig. 2). The sieve slit was
given a regular rectangular pattern of small holes. The diameter of the holes was 2 mm, the horizontal spacing was 4 mm
and the vertical spacing was 5 mm. The sieve slit was placed
585 mm downstream of the target. Since such a sieve slit cuts
the scattered particle stream in a series of small pencil beams
for which both the horizontal and vertical scattering angles at
the target are known, the data of such a measurement allows
us to determine the correspondence between the position and
angle of incidence at the focal plane, and the horizontal and
vertical scattering angles at the target. The computation of
the correspondence is explained in the next section. When
the states of the recoil nucleus are also well-known (for 13 C,
Fig. 2 Illustration of the sieve slit measurement used to calibrate the
excitation energy and the horizontal and vertical scattering angles at the
target.
3 Data analysis
Using the raw data from the MWDC detectors, the triton
tracks passing through the focal plane were reconstructed,
according to the procedure of Ref. [33]. The resulting lines
were parameterised by (xfp , yfp , αfp , βfp ). x and y refer to the
horizontal and vertical positions of the track intersecting the
focal plane and α and β refer to the horizontal and vertical
angles of incidence at that intersection. The subscript ‘fp’
refers to the focal plane.
Subsequently, the parameters (xfp , yfp , αfp , βfp ) were
traced back to the target position event-by-event. The triton
tracks at the target were parameterised by (E t , αt , βt ), where
αt and βt are the horizontal and vertical scattering angles at
the target and E t is the kinetic energy of the ejected triton.
The back-tracing procedure was applied by a mapping function f between the two sets of parameters. This mapping
function was Taylor-expanded into (inverted) optical coefficients. Subsequently, these coefficients were fitted to the 13 C
data using the sieve–slit calibration procedure described in
the previous section (see Ref. [34] for more details). All coefficients up to third order were considered, along with the following higher-order terms: yfp 4 , yfp 4 xfp , yfp 4 xfp 2 , yfp 5 , yfp 6 ,
yfp 2 xfp 2 and xfp 3 θfp . Inspection of the 13 C data revealed that
these specific higher-order coefficients had to be included to
allow for an accurate fitting. However, because the sieve slit
only had five holes in the horizontal direction, the data did
not contain sufficient information to take other higher-order
coefficients along in the fitting.
After the (inverted) optical coefficients were fitted to the
13 C data, the full scattering angle θ of the triton was reconstructed as θ 2 = αt 2 + βt 2 , after shifting over the beam position, i.e. the 0◦ , obtained from the 3 He+ peak and by using the
approximation for small angles that tan(θ ) ≈ θ . The excita-
where n refers to a specific experimental run, θ refers
to the polar scattering angle, tn refers to the areal density
of the target in number of particles per area, Q n refers to
the total number of beam particles in the experimental run
(measured with the Faraday cups), An refers to the angular
acceptance in sr, n refers to the overall detection efficiency
(wire chamber efficiency, dead time correction and analysis
cuts) as a dimensionless number, and P(θ ) refers to the total
number of counts measured for the level we are interested in
(summed over all runs). This formula was derived from the
definition of the differential cross section [35].
The data were divided into bins of 0.3◦ for the scattering angle θ . The angular acceptance was extracted from the
data by plotting αt versus βt and defining a boundary where
the number of counts drops. The detection efficiency was
defined as the product of the efficiencies of the four MWDC
layers, multiplied with 1−τ where τ is the fraction of dead
time of the DAQ. τ was measured individually for each experimental run and was of the order of 2%. The efficiency of an
MWDC layer was defined as N4 /(N4 + N3 ), where N4 is
the number of events for which all four MWDC layers successfully detected a particle, and N3 is the number of events
where three MWDC layers successfully detected a particle
[33]. By reconstructing triton tracks through the focal plane
when only three MWDC layers produced a signal, and tracing
these tracks back to the target using the optical coefficients,
the numbers N4 and N3 could be determined specifically for
each bin in θ and for each individual run. By artificially setting βfp = 0, a track reconstruction through the focal plane
became possible with only three MWDC layers. Due to the
use of the over-focus mode, the information of βfp is not crucial in subsequent analysis anyway. The detection efficiency
per MWDC layer was found to be about 95% in the 0◦ mode
and about 89% in the 2.5◦ mode.
P(θ ) was determined from a Gaussian fit on top of a piecewise linear background for the peak corresponding to the
level of interest. A separate Gaussian fit was made for each
of the different bins in θ , but the counts from different runs
were added to each other before the fit was made. Subsequently, the total number of counts in the fitted peak P(θ )
was determined from the area under the Gaussian. Differential cross sections were also determined per excitation-energy
bin of 200 keV by using the total number of measured counts
in the bin as P(θ ). A special procedure was applied to the
so-called Isobaric Analogue State (IAS). The IAS is a Fermi
resonance with an intrinsic width that is significantly smaller
than the energy resolution for the present experiment [36].
It is, therefore, observed as a single peak in the excitationenergy spectrum. The IAS carries the full strength of the NonEnergy-Weighted Sum Rule (NEWSR) [1,37]. Just like the
Gamow–Teller strength is specified in terms of B(GT) values, Fermi strength is specified in B(F) values. The definition
of B(F) is identical to Eq. (1), except that the spin operator
σ j is not present in the equation. Since for (3 He, t) Fermi
β + transitions are blocked in nuclei with N > Z , the Fermi
NEWSR implies that for the IAS B(F) = |N − Z |, where N
and Z are the numbers of neutrons and protons in the parent nucleus, respectively [1]. Since the IAS typically gives a
very strong peak, the Gaussian fits were given exponentially
decaying tails, so that good fits could be obtained despite the
inaccuracies introduced by truncating our Taylor expansion
in the sieve–slit calibration. For the other states, the number
of counts was low enough so that normal Gaussian fits were
sufficient for the present data analyses.
The IAS fits were also used to determine the energy resolution of the measurements, which was then applied to the fits
of other states. These resolutions were phenomenologically
determined to be σ = 30.4 + 1.10θ 2 keV for the 116 Sn target (θ isin degrees). For the 122 Sn target, an energy resolution
of σ = 32.6 + 1.25θ 2 keV was obtained. These numbers
are larger than the σ ≈ 14 keV that was mentioned in the
previous section. The difference is attributed to differences
in energy loss of tritons and 3 He in the target, fluctuations in
the magnetic field, and, the natural width of the IAS being
the sum of its escape width and a spreading width due to
coupling between the IAS and isovector giant monopole resonance (see Ref. [37] for more details).
In principle, the energy resolution of the measurements
could also have been determined from a well-resolved lowlying state (below the proton and neutron emission thresholds). This is preferable, because unlike the IAS, those states
below the particle emission threshold do not have a natural
width due to particle emission. However, it turned out that
for the data presented in this paper (see Figs. 3 and 4 and
Ref. [18]) these low-lying states could not be individually
resolved, due to the limited experimental energy resolution.
Hence, for higher accuracy of the fits, it was decided to determine the experimental energy resolution from the IAS, which
had much higher statistics. The energy resolution determined
in this way turned out to be usable for accurate Gaussian fits
of the low-lying states.
The ΔL = 0 components were extracted from the differential cross sections with a Multipole-Decomposition Analysis (MDA). The details of this procedure can be found in
Ref. [24]. We performed an MDA based on contributions
from different values of ΔL. Contributions up to ΔL = 4
were considered. For each ΔL contribution, only a single
ΔJ contribution was considered in the MDA since contributions with the same ΔL but different ΔJ are very similar, so that any attempt to disentangle them would result in
large systematic errors. For ΔL = 0, we considered ΔJ = 1
(because we are interested in Gamow–Teller transitions). For
ΔL = 1, we considered ΔJ = 2 and for ΔL ≥ 2 we considered ΔJ = ΔL since those are usually the most dominant contributions. For the low-energy 1+ states it is known
beforehand that only the Gamow–Teller transition and the
ΔL = 2, ΔS = 1, ΔJ = 1 quadrupole transition can contribute. Hence, for such low-lying states only these two contributions were considered in the MDA.
The multipolarity contributions were computed with the
code FOLD. The code FOLD was developed by Cook and
Carr [38], based on the work of Petrovich and Stanley [39]
and then modified as described in Refs. [40] and [41]. The
One-Body Transition Densities (OBTDs) were computed in
the Normal Modes Formalism [37,42], using the code NORMOD [43,44]. For an overview of the theoretical formalism that is at the basis of the code FOLD, the interested
reader is referred to Ref. [7]. For the optical potentials in the
Distorted-Wave Born Approximation (DWBA), we assumed
a Coulomb term, real and imaginary volume terms of the
Woods–Saxon type, and an imaginary surface-Woods–Saxon
term (with the same radius and diffuseness as the imaginary volume term), according to Ref. [45]. This procedure
has already proven successful in Ref. [46] albeit without the
surface term. The parameters were obtained from a linear
interpolation of the parameters of the nuclei 12 C, 28 Si, 58 Ni,
90 Zr, and 208 Pb, obtained from Refs. [47,48]. The parameters of the outgoing channel were assumed equal to those of
the incoming channel, except that the depths VR , VI and Ws
were scaled by 0.85 [49]. The optical-potential parameters
used for the present nuclei are listed in Table 1.
After the DWBA results of FOLD were obtained, they
were smeared with σ = 0.2◦ (σ = 0.3◦ ) for the 116 Sn (122 Sn)
target to take the vertical and horizontal angular resolutions
into account as well as the binning in θ . The larger smearing
for the 122 Sn target was due to a slight deterioration of the
angular resolution of the beam. The smeared DWBA results
of FOLD that form the basis of our MDA are illustrated in
Figs. 5 and 6 for the 116 Sn target and excitation energies of
zero and 20 MeV.
After the ΔL = 0 component was extracted from the
MDA fit of the differential cross sections, this component was
extrapolated to zero momentum transfer q = 0 according to
the procedure described in Ref. [3], which means that extrapolation is done by scaling the experimental θ = 0◦ result by
the ratio of FOLD-computed results at θ = 0◦ and at q = 0,
which properly takes the Q-value of the ground state and
the excitation-energy dependence into consideration. Subsequently, B(GT) values were extracted using the following
equations [3,21]:
where K is a kinematic factor, N DGT is the distortion factor given by the ratio of DWBA and PWBA (Plane-Wave
Born Approximation) computed Gamow–Teller cross sections at q = 0 and |Jσ τ | is the volume integral of the central σ τ -component of the effective nucleon–nucleon interaction between the projectile and target nucleons [1,3]. The
product of these three quantities is the so-called unit cross
section. A similar equation also applies for Fermi cross sections, however, |Jσ τ | should then be replaced by |Jτ |, the
volume integral of the central τ -component, and N DGT by
N DF , the distortion factor for Fermi transitions. The Fermi
and Gamow–Teller unit cross sections at 140 MeV/u can,
with an uncertainty of 5%, be phenomenologically described
by [21]
found that the cross sections differed by 16%. Based on
Ref. [29], in which a similar discrepancy was encountered
and diagnosed as being due to incomplete charge integration
for the 0◦ setting, the cross sections for the 0◦ setting were
reduced by 16%.
The angular distributions of the IAS were used to finetune the optical-potential parameter Ws used in the DWBA
calculations to reduce the uncertainty in the MDA procedure. It was found that by including an imaginary surface
Woods–Saxon potential, the location of the first minimum in
the experimental angular distribution could be better reproduced by the calculation. Therefore, this surface Woods–
Saxon potential was included for all DWBA calculations.
Based on the IAS distributions, Ws was determined to be
12 MeV. Ws was the only parameter of Table 1 that was finetuned in the DWBA from the original value of zero [46] to
12 MeV.
The above data-analysis procedure resulted in B(F) = 16.0
± 0.9 for the 116 Sn target and in B(F)= 22.5 ± 1.2 for the
122 Sn target. Since these are in good agreement with the
expected B(F) = |N − Z |, and the red. χ 2 values are close to
unity, we conclude that our analysis procedure can be used
to reliably extract B(GT) values.
4 Results
The use of Eqs. (3) and (4) and Table 1 was first tested on
the IAS for both targets. The IAS presents an excellent benchmark for this, since for this state B(F) = |N − Z |. The results
are shown in Figs. 7 and 8. In these figures, the reduced χ 2
value (red. χ 2 ) and the probability of the fit are also shown.
The fit probability is the area under the reduced χ 2 distribution to the right of the red. χ 2 value of the fit.
For the runs taken at 0◦ and 2.5◦ , different Faraday cups
were used to integrate the charge deposited by the beam (see
Fig. 1: D1FC was used for 0◦ and Q1FC was used for 2.5◦ ).
By comparing the angular distributions in the range in which
the 0◦ and 2.5◦ settings overlap (1.5◦ ≤ θ ≤ 3.3◦ ), it was
The B(GT) values for the low-lying states were extracted
from the differential cross sections of observed discrete
states. Since both target nuclei have a 0+ ground state, any
0+ → 1+ transition in our excitation-energy region is only
possible through a Gamow–Teller or a quadrupole (ΔL = 2,
ΔS = 1, ΔJ = 1) excitation. Therefore, as discussed in the
previous section, these contributions were the only two contributions considered in the MDA. All low-lying states in
the excitation-energy spectrum that could be resolved, were
considered. The corresponding peaks are illustrated in Figs. 9
and 10. If the differential cross section versus scattering angle
of the state showed a 1+ character (illustrated in Figs. 11 and
12 for the states labelled 1), the B(GT) value of the state
was reported in Tables 2 and 3, together with the measured
mean excitation energy (Meas. E ∗ ), the Number of Degrees
of Freedom (NDF) and the reduced χ 2 value of the MDA fit
(red. χ 2 ). The NDF is different for different states, because
for weaker states it was not always possible to obtain accurate Gaussian fits for some of the angular bins. These bins
and the corresponding datapoints were omitted in the MDA
fit, causing the NDF to vary slightly. For some states, the
measured mean excitation energy could also be matched to
the NNDC database [18]. In the situations where this matching was possible, the 1+ character of the state was confirmed
and the corresponding excitation energy from ref. [18] was
also shown (Lit. E ∗ ). The measurement errors of the B(GT)
bin of 200 keV. In this situation, the MDA as described in
the previous section (with all contributions up to ΔL = 4)
was used to extract the ΔL = 0 contribution. The resulting
Gamow–Teller strength distributions are shown in Figs. 13
and 14. The systematic error from the MDA is shown as a grey
band in Figs. 13 and 14 and has been estimated by fitting the
differential cross sections without the ΔL = 4 contribution
in the MDA.
The Gamow–Teller strength distributions were obtained
for two different situations. In the first case, the distributions were extracted without subtracting the quasi-free background, meaning that the full ΔL = 0 contribution to the
differential cross section at 0◦ was used as input to Eq. (3)
after extrapolation to q = 0. This procedure is similar to that
of Ref. [24]. The different multipolarity contributions to the
differential cross section at 0◦ are illustrated for this situation
in Figs. 15 and 16.
In the second case, the quasi-free background [22,37] was
subtracted before the MDA was performed. To subtract the
quasi-free background, the quasi-free differential cross section was subtracted from the differential cross section per
bin of 200 keV (which was determined from the measured
number of counts in that bin). To compute the quasi-free differential cross section, the model from Ref. [22] was used:
values contain a statistical contribution, a 1% contribution
from the extrapolation to q = 0 and a 5% contribution from
the uncertainty of Eq. (4).
To study the Gamow–Teller transitions at higher excitation energies, the differential cross section was extracted per
where E t is the kinetic energy of the ejected triton, which
depends on θ and on the excitation energy E ∗ . The other
quantities are model parameters. Equation (5) applies only
in the situation where E ∗ > S p (the quasi-free background is
zero otherwise [22,37]). In our analysis, T and W were fixed
to the same values used in Refs. [22,50]: T = 100 MeV and
W = 22 MeV. W accounts for the Fermi momentum of the
neutron in the target nucleus that is transferred and T has the
characteristics of a temperature. In Ref. [50], searches for the
values of W and T were performed. The obtained values for
W are about the same for (3 He, t) for all studied target nuclei
as well as for pion-exchange, namely 22 MeV. Furthermore,
in Ref. [50], only a weak temperature dependence was found.
According to Ref. [22], the parameter E 0 was modelled as
E 0 = E t (E ∗ = 0) − Sp , where Sp is the proton-separation
energy of the recoil nucleus. This leaves the parameters N0
and E QF to be fitted to the data. By varying E QF , it was
confirmed that the shape produced by the model in Eq. (5)
is not very sensitive to the precise value of E QF . However,
the value given in Ref. [22] (E QF = 180 MeV) should be
adapted to a beam energy of 140 MeV/u, which is why we
did refit the parameter E QF .
On the other hand, the choice of N0 is very important,
because it determines the overall normalisation and, therefore, the total amount of quasi-free background. Unfortunately, it is difficult to accurately determine N0 due to a
lack of knowledge of the quasi-free background [37]. Therefore, we chose to check two extremes. The first one being
N0 = 0, which corresponds to no subtraction of quasi-free
background. The other one is where the quasi-free background is as large as possible, and this scenario will be
denoted as N0 being maximal. In this scenario, we assume
that all contributions to the total differential cross section
above the IV(S)GDR come from the quasi-free background.
This scenario corresponds to what was used in Ref. [22] (see
Fig. 17 for an illustration). However, as the IV(S)GDR is very
broad, this does not necessarily mean that at 28 MeV, the total
differential cross section consists of only quasi-free background. Moreover, the analysis procedure used in Ref. [22]
assumes that in addition to the IAS, there is only the Gamow–
Teller Resonance (GTR), pygme resonances, the IV(S)GDR
and the quasi-free background, which is a simplification of
reality. As a result, it is still possible that after subtracting
the quasi-free background, a proper MDA analysis reveals a
small ΔL = 0 contribution at high excitation energies, possibly corresponding to the low-energy tail of the IV(S)GMR
[36]. With the assumption of N0 being maximal, Eq. (5) can
be fitted to the data and this situation corresponds to the
largest possible quasi-free background contribution.
It is not a common procedure to subtract the quasi-free
background before applying MDA. The reason for this is
that the use of equation (5) can bias the MDA procedure.
For example, an overestimation of N0 might result in an
underestimation of Gamow–Teller strength. This will be discussed in more detail in Sect. 6. However, not subtracting
any quasi-free background could lead to an overestimation
of the Gamow–Teller strength. Since the correct quasi-free
contribution is very difficult to determine [37], we decided
to apply both strategies (N0 = 0 and N0 being maximal as
discussed above), so that it is known for sure that the true
Gamow–Teller distribution is between the two distributions
presented in Figs. 13 and 14. Another reason for pursuing
both strategies is that we wanted to be able to compare our
results to those published in Ref. [22] (where putting N0 to
its maximal value was chosen) and to those published in Ref.
[24] (where putting N0 = 0 was chosen). These comparisons
are discussed in the next two sections.
We further note that, since the IAS is identified as a
ΔL = 0 contribution in the MDA, it shows up in the B(GT)
distributions of Figs. 13, 14, 15 and 16. However, its contribution is, of course, removed before further discussing
Gamow–Teller strength in the data.
As a final remark, it is noted that the extraction of Gamow–
Teller strength from a charge-exchange reaction always contains an additional systematic uncertainty due to interference
between the ΔL = 0 and ΔL = 2 amplitudes for ΔJ = 1
excitations. This interference is mediated by the tensor-τ
component of the nucleon–nucleon interaction and cannot
be removed through the MDA analysis. It has been shown
[51,52] that, for the (3 He, t) reaction, B(GT ) measurements
deviate from the true value. These deviations are relatively
stronger for weak Gamow–Teller transitions, and they are, on
average, zero. The magnitude for this systematic uncertainty
could be estimated on a state-by-state basis by the following
relationship (see Eq. (6) and Fig. 6 in Ref. [51] and Eq. (4)
and Fig. 4 in Ref. [52]): where ΔB(GT)tensor denotes the absolute systematic
uncertainty due to the interference described above. The systematic uncertainty from Eq. (6) was not shown in Tables 2
and 3 and also not in Figs. 13 and 14, because it only becomes
significant for (very) small B(GT) values due to the logarithmic behaviour. For this reason, it should be noted that the
Gamow–Teller (ΔL = 0) cross section gets very small in the
excitation-energy region of 16−28 MeV for the 116 Sn target.
It is much smaller than the ones for the major ΔL = 1 and
ΔL = 2 components. Thus, extraction of the small ΔL = 0
component is difficult in this region, and leads to a large
uncertainty in the ΔL = 0 cross section. The uncertainty in
the extraction of the Gamow–Teller strength is, therefore,
increased due to the tensor effect described above and other
effects.
To further investigate the reliability of the measurements
presented in this paper and to explore the uncertainty in N0
of the quasi-free background, we will compare the measurements presented in this paper to the results presented in Refs.
[22] (Pham et al.) and [24] (Guess et al.) in the next sections.
5 Comparison to the results presented in Ref. [22]
Pham et al. [22] studied the Gamow–Teller strength distributions in the Sn(3 He, t)Sb reaction for several isotopes (among
them are the two isotopes discussed in this work), but for a
beam energy of 67 MeV/u. We have compared the data presented in this paper to theirs for the 116 Sn and 122 Sn targets.
Pham et al. studied the Gamow–Teller distribution by fitting
the excitation-energy spectrum to a sum of Gaussians. Five
Gaussians (labelled GT1-GT5) were used to fit the Gamow–
Teller (GT) resonances: the GTR [36] and the pygmy resonances [22]. A sixth Gaussian was used to fit the IAS and a
seventh was used to fit the isovector (spin) giant dipole resonance (IV(S)GDR) [36]. The quasi-free background model
of Eq. (5) was also added. This procedure is illustrated in
Fig. 17 for the data presented in this paper. For this figure, it
should be noted that, technically, Pham et al. applied a condition of −0.3◦ ≤ αt ≤ 1.3◦ to their data, while we applied
0◦ ≤ θ ≤ 1.3◦ to the present data (see Fig. 17 and note
that θ 2 = αt 2 + βt 2 ). Since the vertical acceptance (βt ) of
the spectrometer used by Pham et al. was only ±0.5◦ , the
scattering angle could be approximated by θ ≈ |αt | and a
condition on αt alone was sufficient. However, the vertical
acceptance (βt ) of the present experiment is of the order of
±3◦ , which makes it necessary to use the full scattering angle
θ in the condition for the data presented in this paper.
Pham et al. reported the Gamow–Teller cross sections as
determined from the Gaussian fits illustrated in Fig. 17. The
data in the range of − 0.3◦ ≤ αt ≤ 1.3◦ were used for these
fits and the ΔL = 0 multipolarity contributions were not
subtracted. However, B(GT) values were not extracted from
these cross sections. Hence, we have taken the cross sections
reported by Pham et al., corrected them for the smearing
effects of − 0.3◦ ≤ αt ≤ 1.3◦ and extrapolated them to q = 0.
This was done using the code FOLD introduced in Sect. 3
and the optical potentials from Ref. [45]. These potentials
are different from Table 1 because of the difference in beam
energy.
Subsequently, the B(GT) values were extracted using Eq.
(3). To determine the Gamow–Teller unit cross section at
67 MeV/u, a two-point MDA was performed on the data
published in Ref. [22]. For 118 Sb, Ref. [22] provided the
differential cross section under the condition − 0.3◦ ≤ αt ≤
1.3◦ and under the condition 1.3◦ ≤ αt ≤ 2.9◦ (see Fig. 2
in that paper). These results provided two data points like
the ones in Figs. 11 and 12. With two data points only, an
MDA can at most distinguish two multipolarity contributions
(hence, a two-point MDA). We chose a ΔL = 0 contribution
(The Gamow–Teller one) and a ΔL = 2 contribution, which
are the same two contributions used to calculate the numbers
in Tables 2 and 3. With this two-point MDA, the Gamow–
Teller multipolarity contribution to the 118 Sb ground state
cross section was determined to be (87 ± 9)%.
Since for all other isotopes, Ref. [22] only provided differential cross sections under the condition −0.3◦ ≤ αt ≤
1.3◦ , 118 Sb was the only isotope for which the Gamow–
Teller contribution to the ground state could be established
at 67 MeV/u. For this reason, we chose to first determine the
unit cross section for the 118 Sn target, and then to extrapolate
this result to the isotopes of interest. For this extrapolation,
theoretical estimates of K , N D and |Jσ τ | (see Eq. (3)) were
used, so that the ratio of theoretical unit cross sections for the
116 Sn and 118 Sn targets could be computed. This ratio was
then multiplied with the 118 Sn unit cross section determined
from the two-point MDA. We found a Gamow–Teller unit
cross section of 2.53 mb/sr for the 116 Sn target at 67 MeV/u.
For the 122 Sn target, a similar procedure was used and we
found a Gamow–Teller unit cross section of 2.40 mb/sr. The
inaccuracies of the two-point MDA and of the 118 Sb data
from ref. [22] cause the above cross sections to contain an
uncertainty of 20% (which is a correlated error between the
116 Sn and 122 Sn targets).
The comparisons between the B(GT) values obtained from
the Pham et al. data and the B(GT) values obtained from the
present experiment are shown in Tables 4 and 5.
The first columns in Tables 4 and 5 show the labelling of
the states according to Fig. 17. The second columns show
the B(GT ) values obtained from the cross sections reported
by Pham et al. (B(F) is obtained from the Fermi NEWSR).
The third columns show the B(GT) values obtained from the
present experiment, but analysed according to the procedures
of Pham et al. (Gaussian fits of the excitation-energy spectrum like in Fig. 17, i.e. no MDA and a single angular bin of
0◦ ≤ θ ≤ 1.3◦ ). The fourth columns are obtained from fitting
Gaussians like in Fig. 17 not to the excitation-energy spectrum, but to the Gamow–Teller distributions in Figs. 13 and
14 (the case where the quasi-free background is subtracted,
since that is what was done in Ref. [22]). The IAS values in
these columns were obtained from Figs. 7 and 8. This essentially means that the difference between the third and fourth
columns is the elimination of the ΔL = 0 multipolarity contributions. For the third and fourth columns, the unit cross
sections of Eq. (4) were used. For the second column, the
derived unit cross sections (discussed above) at 67 MeV/u
were used.
Obviously, a fair comparison to Pham et al., requires the
use of the same analysis procedures. Hence, a comparison
should only be made between the second and third columns
of Tables 4 and 5. In these two columns, there is only a significant deviation between the present data and those of Pham
et al. for the IAS and GT1 of the 122 Sn target. The deviation
in the IAS is easily explained, as Pham et al., simply assumed
B(F) =|N − Z | for the IAS. The number in the third column,
on the other hand, was obtained from a pure Gaussian fit
(see Fig. 17). However, in Sect. 3 it was already discussed
that a pure Gaussian was not suitable for fitting the IAS and
that exponential tails had to be included. This inclusion is
reflected in the numbers presented in the fourth columns.
Hence, the fourth columns should show the correct numbers.
The deviation in GT1 could be a result of systematic uncertainties in the fitting procedure of Fig. 17. The present data
show that the excitation-energy spectrum of the 122 Sn target
is almost flat between GT1 and GT2 (see Fig. 4), while that of
the 116 Sn target shows a clear dip (around 6 MeV; see Fig. 3).
Hence, fitting Gaussians to GT1 and GT2 may result in a significant systematic uncertainty for GT1 for the 122 Sn target.
These systematic uncertainties were not included in Tables
4 and 5 because due to the flatness of the excitation-energy
spectra, they could not be accurately determined. This could
explain the GT1 discrepancy. The data in the figures shown
in Pham et al. [22] do not disagree with this explanation.
Pham et al., also reported cross sections for the individual low-lying Gamow–Teller states. B(GT) values could be
extracted from these cross sections in a similar way as was
done for the states GT1–GT5 in Tables 4 and 5, but the B(GT)
values obtained from the Pham et al., cross sections are much
higher than those presented in Tables 2 and 3. The reason for
this, is that Pham et al., did not include the description of
a piecewise linear background in the determination of the
cross sections (like we did). This background is significant
for all states in Tables 2 and 3, except for the first ones. For
those states, the cross sections of Pham et al. provided us
with B(GT)= 0.39 ± 0.09 for the first Gamow–Teller state
of the 116 Sn target. For the first Gamow–Teller state of the
122 Sn target, this was B(GT)= 0.33 ± 0.07. The differences
between these values and those in Tables 2 and 3 are not too
big, but also not insignificant. To obtain these two B(GT)
values from fitting the data of Pham et al., we assumed that
the Gamow–Teller contribution to the cross section was 87%.
This assumption came from the two-point MDA we used to
obtain the Gamow–Teller unit cross section for the 118 Sn
target at 67 MeV/u (see discussion above). There, it was
established that the Gamow–Teller contribution to the total
cross section was (87 ± 9)%. However, this number contains
a large uncertainty and secondly, there is no a priori reason
to assume that this number is the same for the first Gamow–
Teller states of the 116 Sn and 122 Sn targets (Ref. [22] does
not contain the required data to obtain these numbers for the
116 Sn and 122 Sn targets). If the Gamow–Teller contribution
would be somewhat lower than 87%, the discrepancies would
disappear.
Finally, there is the issue that Pham et al., claim that GT1
contains (65 ± 3)% of the Ikeda sum rule for all isotopes
investigated. However, the second columns of Tables 4 and 5
claim that GT1 of the 116 Sn target, as it was determined from
the cross sections presented in Pham et al., only contains
(14 ± 3)% of the Ikeda sum rule. For the 122 Sn target, this is
(19 ± 5)%. The percentage of (65 ± 3)% from Pham et al.,
was computed with the following equation [22,36,53,54]:
where σGT is the measured Gamow–Teller cross section,
σIAS is the measured IAS cross section (which was reported
in Ref. [22]), kfIAS /kfGT is the ratio of the wave numbers
of the outgoing ejectile (usually around unity) and D is
the ratio of Fermi and Gamow–Teller unit cross sections.
Pham et al., applied Eq. (7) with D = (E/E 0 )2 = 1.48
(E 0 = 55.0 ± 0.4 MeV) [54]. However, D = (E/E 0 )2 =
1.48 only applies to ( p, n) reactions, while Pham et al.,
used the (3 He, t) reaction. With the IAS cross sections from
Pham et al. (which carry an uncertainty of 10% [50]) and
our derived Gamow–Teller unit cross sections at 67 MeV/u
(which carry an uncertainty of 20% as discussed above), we
find D = 5.2 ± 1.1 for (3 He, t) at 67 MeV/u. This explains
the large difference. Taking this into consideration, our
results are in reasonable agreement with the data presented
in Ref. [22] if we follow the same analysis procedure. This
confirms the reliability of our measurements.
6 Comparison to the results presented in Ref. [24]
We chose to include a comparison between the present
data and the Gamow–Teller strength distribution of the
150 Nd(3 He, t)150 Pm reaction studied by Guess et al. [24]
at 140 MeV/u to further validate our data analysis and to
explore the issue of whether the quasi-free background
should be subtracted or not (see Sect. 4). Even though a
completely different target nucleus was studied, the experiment of Guess et al., is particularly suited for this purpose,
because the experimental setup, the beam energy and the
data-analysis procedures followed are completely identical
to ours. The only difference is that Guess et al., did not subtract the quasi-free background, implying that their analysis
procedure corresponds to our N0 = 0 scenario introduced in
Sect. 4.
We have chosen to make the comparison by plotting the
integral of the Gamow–Teller distributions between 0 and a
certain excitation energy versus that excitation energy. This
way, implications for the general quenching phenomenon of
the Gamow–Teller resonance can be easily visualised. The
plots are shown in Fig. 18. The data from Guess et al., include
the systematic error contribution from the MDA. Hence, for
the present data we have chosen to include this contribution
as well. For completeness, the results with the quasi-free
background subtracted (with N0 being maximal) are shown
in Fig. 19 for the present data. Corresponding results (N0
being maximal) concerning the 150 Nd(3 He, t)150 Pm reaction
were not available in Ref. [24].
From Fig. 18, it is clear that all three isotopes overlap near 17 MeV. The differences below that energy could
be explained by the facts that different nuclei have their
Gamow–Teller resonance located at different excitation energies [36] and that deformation effects (150 Nd is a deformed
nucleus) could influence the distribution of the Gamow–
Teller strength. However, at higher excitation energies, the
results of the 116 Sn target starts to deviate again from the
other two. This deviation is still within the error bands.
Nevertheless, it could be an interesting feature due to
the general quenching phenomenon of the Gamow–Teller
strength. This phenomenon is well-documented in the literature [36,37,55,56] and results in that only about 50−60%
of the Ikeda sum rule is exhausted by the Gamow–Teller
strength in the Gamow–Teller resonance region. Without
subtracting the quasi-free background, Guess et al., found
(52 ± 5)% of the Ikeda sum rule below 28 MeV for the 150 Nd
target and we found (48 ± 6)% for the 122 Sn target, both in
agreement with this phenomenon. However, for the 116 Sn
target, we found only (38 ± 7)%.
The Gamow–Teller distribution where the quasi-free
background is not subtracted goes through zero near 20 MeV
for the 116 Sn target (see Fig. 13). Because of the general
shape of the quasi-free background (see Fig. 17 and Eq. (5)),
we conclude that the contribution from the quasi-free background to the ΔL = 0 multipolarity contribution is small
for the 116 Sn target (the only zero-point in Fig. 17 is at
S p = 4.08 MeV [18] for this target). Note that due to the
large systematic errors, we can only conclude that the contribution is small for the 116 Sn target, not that it is exactly zero.
On the other hand, the Gamow–Teller distribution for the
150 Nd target published in Ref. [24], as well as the Gamow–
Teller distribution shown in Fig. 14 (where no quasi-free
background was subtracted) do not become zero above the
Gamow–Teller resonance. Hence, in those distributions, the
contribution from the quasi-free background may be larger.
Unfortunately, it is very difficult to determine the correct
contribution from the quasi-free background (the value of
N0 in Eq. (5)) [37]. The reason for this is that the region
near 28 MeV may contain Gamow–Teller strength that is
moved to higher excitation energies due to 2p–2h couplings
[57,58], and it may contain tails from the isovector giant
monopole resonance (IVGMR) and the isovector spin giant
monopole resonance (IVSGMR) [37]. Since all these contributions are ΔL = 0 and the quasi-free background also has
a ΔL = 0 contribution, an MDA cannot distinguish between
these contributions (it would give a significant contributing
to the systematic uncertainties; see Sect. 3). Therefore, due
to the absense of an (almost) zero-point in the region above
the Gamow–Teller resonance for the 122 Sn and 150 Nd [24]
targets, there is no reliable method to estimate the quasi-free
background contributions for those targets. The contributions
may be as small as for the 116 Sn target, but they may also be
significantly larger. We conclude that the data do not contain
enough information to accurately determine the quasi-free
contribution. For this reason, Guess et al., chose not to subtract the quasi-free background at all (which is equivalent to
putting N0 = 0), so they would be ensured that no Gamow–
Teller, IVGMR, and IVSGMR contributions were neglected.
To summarise, we conclude that only (38 ± 7)% of the
Ikeda sum rule is found for 116 Sn, which seems to have little
quasi-free background in its spectra, and that the percentages
of the Ikeda sum rule measured for the 122 Sn and 150 Nd isotopes, which may, or may not have a significant quasi-free
background contribution in their spectra, agree with the general trend of the quenching phenomenon. However, it should
be noted that the percentage of the Ikeda sum rule for the
116 Sn target ((38 ± 7)%) is not necessarily in disagreement
with the observed general quenching phenomenon if one considers the large measurement error of 7% (which contains a
6% systematic contribution). This issue raises the need for
accurate determination of the quasi-free background contribution, in order to have a better understanding of the quenching phenomenon of the Gamow–Teller strength.
7 Comparison to QRPA+QPVC calculations
The measurements of the 116 Sn and 122 Sn targets were also
compared to Quasi-Particle Random-Phase Approximation
with Quasi-Particle Vibration Coupling (QRPA+QPVC) calculations performed with the Skyrme SkM* interaction [59].
For a more detailed discussion of the formalism of QRPA+QPVC calculations, as well as for an application to 120 Sn,
the interested reader is referred to Ref. [60]. The calculations
presented in this section were performed specifically for the
present paper.
In the previous section, it was discussed that the quasifree background contribution is small for the 116 Sn target.
Therefore, overestimating the subtraction of this quasi-free
background could result in losing Gamow–Teller strength at
higher excitation energies (2p–2h couplings). For this reason,
we chose to compare the calculations to our results where no
quasi-free background was subtracted.
Our self-consistent calculations should fulfill, by construction, the Ikeda sum rule. In practise, the QRPA+QPVC
calculations reproduce 97% of Ikeda sum rule when the
strength is integrated up to an excitation energy of 80 MeV
[60]. The inclusion of the QPVC effect could shift 10 − 15%
of the Gamow–Teller strength to excitation energies above
25 MeV, however, this is still not enough to fully explain
the general quenching phenomenon. Therefore, in order to
compare with experimental data, the theoretical calculations
were artificially normalised to (0.75)2 · 3|N − Z |, in agreement with Ref. [24]. The comparison is illustrated in Figs.
20 and 21.
From Figs. 20 and 21, we conclude that the QRPA+QPVC calculations were able to reasonably predict the IAS
and the Gamow–Teller resonance. If the systematic uncertainties are included, the theoretical result is always within
a 2σ distance of the experimental data. However, in the
region of 3−5 MeV excitation energy, the QRPA+QPVC
calculations show a large, broad peak, which has not been
observed experimentally. This peak is the result of the lowest main state of the QRPA+QPVC model, which corresponds to a single-particle excitation of back spin-flip type
( j = l − 1/2 → j = l + 1/2) [60]. Also, the inclusion of
attractive isoscalar pairing interactions in the QRPA+QPVC
model is partially responsible, as it increases the height of
this peak by about 20% (it also improves the agreement with
experimental data in the region of the Gamow–Teller resonance) [60]. The peak in the 3−5 MeV region is a known
discrepancy between the QRPA+QPVC model and experimental data. It also showed up for 120 Sn in Ref. [60]. Possible extensions of the QRPA+QPVC model of Ref. [60] may
be envisaged (e.g., by including further correlations). At the
same time, the Gamow–Teller peaks result from the interplay
of the single-particle spin–orbit splittings and of the residual interaction between particle–hole excitations. Therefore,
refinements on the effective interactions to increase their
accuracy in this respect should also be investigated, although
this may be quite demanding and show the limitations of the
present Skyrme ansatz.
Even though the QRPA+QPVC calculations were able
to reasonably predict the IAS and the Gamow–Teller resonance, no conclusions about the quasi-free background or the
general quenching phenomenon could be drawn from this,
which was our hope when making this comparison. Since
the calculations satisfy the sum rule when the Gamow–Teller
strength up to 80 MeV is integrated, an improvement of the
3−5 MeVregion would increase the strength either in the
Gamow–Teller resonance region, or in the excitation-energy
region above 25 MeV. This needs to be further checked by
a proper effective interaction that can better reproduce the
low-energy data. Hence, the current agreement in the region
of the Gamow–Teller resonance could be fortuitous. On the
other hand, we can conclude from this comparison that the
data presented in this paper do provide a useful reference for
improving the current theoretical calculations, especially by
giving a guideline for the improvement of the spin–isospin
terms of the effective interactions in the nuclear medium.
No QRPA+QPVC calculations were done above 22 MeV
for this work. We would like to note that in order to
extend the calculations to 28 MeV, contributions from the
IVGMR, IVSGMR, shifted Gamow–Teller strength and
the quasi-free background would all have to be included,
because MDA cannot reasonably distinguish between these
contributions. Including the quasi-free background in the
QRPA+QPVC calculations may be challenging, because the
QRPA+QPVC formalism was originally not meant to model
this phenomenon. However, from Sect. 6, it follows that the
quasi-free background could be significant for some of the
isotopes and some quasi-free contribution has to be included
in the theoretical calculations if the goal is to achieve agreement with experimental data.
8 Outlook
In Sect. 6, we concluded that more research is required to
study the quasi-free background. This is needed to accurately
estimate the quasi-free background contributions not only
in the data presented in this work, but also in other data.
These data may include what has been presented by Guess
et al. [24], so that the general quenching phenomenon of the
Gamow–Teller resonance can be better understood.
A possibility would be to repeat the measurements of this
work, the 116 Sn(3 He, t)116 Sb and 122 Sn(3 He, t)122 Sb reactions at 140 MeV/u, while measuring the proton emissions at
backward angles. The requirement of a coincidence between
emitted tritons at forward angles and emitted protons at backward angles has proven to be a powerful technique to suppress
the quasi-free background [37]. However, it should be noted
that this will also reduce the Gamow–Teller strength distributions, because states that cannot decay by proton emission
will be removed from the distribution.
Therefore, it would be a better strategy to measure the
proton emissions of the recoil nucleus over the full 4π solid
angle. It is known from Ref. [37,61] that the quasi-free background corresponds almost exclusively to protons emitted
in the forward hemisphere (with respect to the recoil axis).
On the other hand, in the LAB-frame, all multipole distributions have a symmetric distribution in the forward and
backward hemisphere with respect to the recoil axis. Hence,
if one would integrate the measured data over these two
hemispheres and subtract the two, all multipole contributions to proton decay should cancel out and only the contribution from the quasi-free background should remain. This
remaining quasi-free background contribution can be used to
directly determine N0 in Eq. (5). This is because the quasifree background in Figs. 13 and 14 is due to charge-exchange
knock-on proton emission. The only uncertainly may come
from the small quasi-free contribution that might still be
present in the backward hemisphere.
With such an experiment, it would be possible to confirm
that the quasi-free contribution to the ΔL = 0 component
of the 116 Sn(3 He, t)116 Sb reaction at 140 MeV/u is indeed
small. The experiment can also be used to see whether a
proper subtraction of the quasi-free background (which is
only possible when N0 is known) results in a better agreement
between the Ikeda sum rule percentages of the two Sn-targets.
However, it should be noted that the experiment described
above to measure N0 might be challenging. The reason for
this is that the proton emission is generally expected to be
very strong at forward angles, which leads to very high count
rates in the solid-state detectors. Moreover, at these forward
angles the detectors are also expected to be bombarded by
products of other nuclear reactions than the one of interest.
A QRPA+QPVC calculation with the inclusion of contribution from excitations of other multipolarities up to
28 MeV/u would also be helpful to better understand the general quenching phenomenon. For this purpose, the calculation would have to be extended to higher excitation energies,
and contributions from the IVGMR, IVSGMR and shifted
Gamow–Teller strength would have to be included to allow
for a good comparison with experimental data.
9 Concluding remarks
The Gamow–Teller strength distributions of 116,122 Sn →
116,122 Sb were measured with the (3 He, t) charge-exchange
reaction at a bombarding energy of 140 MeV/u. The results
were compared with the same distributions obtained in previous measurements at a lower incident energy of 67 MeV/u.
When the same analysis procedure was applied, we found
agreement, which confirms the reliability of the new measurements. However, the old analysis procedure showed several inaccuracies, and we improved the analysis method (see
Sect. 3 for the description of the new method).
The data (with the new analysis method) were also compared to 150 Nd → 150 Pm data and to QRPA+QPVC calculations. Without subtracting the quasi-free background, we
found (48 ± 6)% of the Ikeda sum rule in the region below
an excitation energy of 28 MeV for the 122 Sn target. This
agrees with the general trend of the quenching phenomenon of the Gamow–Teller strength. For the 116 Sn target, we
found (38 ± 7)% in the same excitation-energy region, which
appears to be somewhat lower (the errors include all systematic uncertainties). These summed Gamow–Teller strengths
were extracted as usual by using the unit cross section in the
region up to an excitation energy of 28 MeV, well above the
main Gamow–Teller resonance, after extrapolating the cross
sections to q = 0.
Our comparisons suggest that there could be a difference
in the amount of quasi-free background that is present for
both Sn-targets. The quasi-free background contribution is
believed to be small for the 116 Sn target, but may, or may not
be larger for the 122 Sn target. The present data for the 122 Sn
target are inconclusive about this. When the largest possible
quasi-free background contribution was subtracted [22], we
found (29 ± 7)% ((35 ± 5)%) of the Ikeda sum rule for the
116 Sn (116 Sn) target in the same excitation-energy region.
From these numbers, we conclude that more research is
needed to better understand the general quenching phenomenon. From the experimental side, measurement techniques
are needed to determine the size of the quasi-free background.
From the theoretical side (once the quasi-free background
contribution is established), more investigations are necessary to determine whether the lower percentage of the 116 Sn
target indeed is in disagreement with the general trend of the
quenching phenomenon.
