Full text of DOI:10.1007/s100500070087

Eur. Phys. J. A 8, 355–360 (2000)
THE EUROPEAN
PHYSICAL JOURNAL A
c
ꢀ Societa` Italiana di Fisica
Springer-Verlag 2000
Nuclear temperature measurements with the double isotope
ratio technique: Influence of the experimental conditions
P.M. Milazzo1,a, G. Vannini2, M. Bruno2, N. Colonna3, M. D’Agostino2, F. Gramegna4, G.V. Margagliotti1,
P.F. Mastinu4, and A. Moroni5
1
Dipartimento di Fisica dell’Universit`a di Trieste and INFN, Sezione di Trieste, Italy
Dipartimento di Fisica dell’Universit`a di Bologna and INFN, Sezione di Bologna, Italy
INFN, Sezione di Bari, Italy
INFN, Laboratori Nazionali di Legnaro, Italy
INFN, Sezione di Milano, Italy
2
3
4
5
Received: 6 October 1999 / Revised version: 6 March 2000
Communicated by C. Signorini
Abstract. The dependence of the nuclear temperatures of highly excited systems, extracted by means of
the double ratios of the emitted isotopes, on the experimental conditions is investigated. Experimental
data obtained in the Xe+Cu 30 MeV/nucleon reaction are used to study the sensitivity of the method and
the effects of the energy thresholds on the obtained temperature values. We find that the temperatures
extracted using the He/Li ratios can be strongly influenced by the experimental energy thresholds which
are different for different elements. These distortions depend on the velocity of the emitting system and
on the detection angle and therefore particular care is needed in the choice of the detectors in those
experiments in which velocities are low and angles are large. The use of four isotopes of the same element
make negligible such effects.
PACS. 21.65.+f Nuclear matter – 25.70.Pq Multifragment emission and correlations
1 Introduction
directly affect temperature measurements when the dou-
ble isotope ratio technique is used. When the mass iden-
tification is carried out with the ∆E − E technique, it
has to be taken into account that the energy threshold is
different from one isotope to another, and therefore, the
temperature values extracted by means of the method of
double isotope-yield ratio, could be strongly affected by
the different energy range of each single isotope. The aim
of the present paper is to study how the experimental
conditions, such as the detectors thickness, the velocities
of the emitting systems, and detection angles, and their
interplay influence the temperature measurements. Such
an investigation has been carried out exploiting the ex-
perimental data on the isotope production and simulating
different energy thresholds.
In intermediate-energy heavy ion reactions (10 < E/A <
100 MeV/nucleon) several different decaying systems are
formed depending on the entrance channel and on the im-
pact parameter. These systems emit fragments (light, in-
termediate and heavy mass fragments), i.e. they behave
as fragment sources which differ in size, shape and exci-
tation energy. In particular for excitation energies above
2 MeV/nucleon multifragmentation appears as one of the
possible de-excitation process [1–12].
To gain more insight into the characteristics of the
intermediate mass fragment emitting sources, one has to
measure the temperatures of well-defined and experimen-
tally identified systems, with different excitation energy
and size, using reliable thermometers. Temperatures ex-
tracted from the isotopic abundances have stimulated a
large body of experimental and theoretical work, both
supportive and critical of the thermometric technique em-
ployed [13–21].
In sect. 2 a brief description of the temperature mea-
surement technique and of the experimental conditions
are given; in sect. 3 the results, regarding the induced dis-
tortions on temperature measurements are presented and
discussed. The conclusions are drawn in sect. 4.
Starting from the results of temperature measurements
of nuclear systems formed in Xe+Cu 30 MeV/nucleon col-
lisions we discuss here how the experimental constraints,
such as kinematical limits and detector inefficiencies, could
2 Experimental method
2.1 Nuclear temperature measurements
The reliability of the double ratios of isotope yields [22]
method is based on the following conditions:
a
e-mail: milazzo@trieste.infn.it

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The European Physical Journal A
i) free nucleons and composite fragments are contained ∆E − E technique [25]. A good mass resolution for light
within a certain volume V at a single temperature T and isotopes (up to carbon) was obtained (see for instance
are in thermal equilibrium;
ii) it is possible to use the Maxwell-Boltzmann statis-
tics;
fig. 1 of ref. [21]). Energy thresholds for mass identifica-
4
tion of 8.5, 10.5, 14 MeV/nucleon were achieved for He,
6Li and 12C nuclei, respectively.
Light charged particles and fragments with charge up
to Z = 20 were detected at 23◦ < θlab <160◦ by the
phoswich detectors of the MSU Miniball hodoscope [26].
The charge identification thresholds were about 2, 3, 4
MeV/nucleon in the Miniball for Z = 3, 10, 18, respec-
tively. The geometric acceptance of the overall array was
greater than 87% of 4π.
iii) the system has reached the chemical equilibrium;
iv) the experimental yield of a fragment is proportional
to its density inside the volume V ;
v) all detected nuclei originate from a single source.
The double ratio R of the yields Y of four isotopes in
their ground states, prior to secondary decay, is then given
by
Y (A1, Z1)/Y (A1 + 1, Z1)
Y (A2, Z2)/Y (A2 + 1, Z2)
eB/T
R =
=
,
(1)
2.3 Data selection
a
Since the method of double ratios of isotope yields [22]
requires that the considered fragments be emitted from
the same source, we adopted a procedure of data analysis
which allowed to identify the emitting systems and as-
sured that all the selected fragments, used to extract tem-
perature values, could be safely assigned to well-defined
sources. Furthermore we selected events for a given range
of the impact parameter and verified that the recon-
structed sources emitted fragments isotropically as ex-
pected for thermal equilibrium.
where a is a constant related to spin and mass values, B =
BE(Z1, A1)−BE(Z1, A1+1)−BE(Z2, A2)+BE(Z2, A2+
1), and BE(Z, A) is the binding energy of a nucleus with
charge Z and mass A [22].
Moreover the emitting source is considered not to have
angular momentum and any dynamical behaviour (such as
compression and expansion of the system, pre-equilibrium
emission, etc.) is neglected.
As the fragments can be highly excited, secondary de-
cays from higher-lying states of the same and heavier nu-
clei cannot be neglected: they can lead to sizable correc-
tions to the measured ratios R [23]. To reduce the sensi-
tivity to such corrections, it is advisable to choose cases
The multiplicity of detected charged particles (Nc) was
used for the impact parameter b selection [27]:
ꢀ
ꢂ1/2
ꢁ
+∞
P(Nꢀc)dNꢀc
.
ˆ
for which B ꢀ T, since the uncertainties on T are pro-
b = b/bmax
=
T
Nc
portional to
.
Once the Bset of data coming from a single source is de-
fined, the isotopic resolution allows the extraction of tem-
peratures from a high number of isotope ratios. Since the
goal of this paper is to understand the effects induced by
Here P(Nc) is the charged particle probability distribu-
tion and π ·b2max is the measured reaction cross section for
Nc ≥ 3.
A careful check on the identification of the sources and
investigations on their characteristics was performed also
studying the angular and energy distributions [21,28].
In the following the discussion will be focused on the
temperature measurement of the quasi-projectile (QP)
emitting source, highly excited in peripheral collisions
instrument efficiencies, we limit here the discussion to two
6
of the most used thermometers, Li/7Li-3He/4He (THeLi
)
and 12C/13C-11C/12C (TC) [13,18–20].
∗
ˆ
2.2 Experimental set-up and the measurements
(b > 0.8, E =4 MeV/nucleon) [28]. In this impact pa-
rameter range the emitting systems (QP, quasi-target and
mid-rapidity source, if formed) have quite different rela-
tive velocities and it is possible to distinguish their own
decay products.
The data used in the present study refer to measurements
performed at the National Superconducting K1200 Cy-
clotron Laboratory of the Michigan State University. The
Xe+Cu at 30 MeV/nucleon [28] reaction was investigated
with the aim of observing the characteristics of the nu-
clear systems formed in central and peripheral collisions.
To this end, temperatures and excitation energies were
measured.
In order to identify the fragments coming from a com-
mon source, strong constraints on the data were necessary
[21,28]. In particular, only fragments emitted in the for-
ward direction with respect to the QP frame (θ <60◦) were
accepted for the analysis (the QP moves with a velocity of
The angular range 3◦ < θlab <23◦ was covered by the 6.3 cm/ns in the laboratory frame). Such constraints allow
MULTICS array [24]. It consisted of 48 telescopes, each to bypass problems due to the mixing of fragments coming
of which was composed of an Ionization Chamber (IC), from different sources (the quasi-target and mid-rapidity
a silicon position-sensitive detector (Si, 500 µm thick) sources are present inside the same event) as well as in-
and a CsI(Tl) crystal. Typical energy resolutions were efficiencies existing at backward emission angles (in the
2%, 1% and 5% for IC, Si and CsI, respectively. Energy laboratory frame fragments emitted backwards are less en-
calibrations were obtained by irradiating each telescope ergetic and below the mass identification threshold). With
with low intensity direct beams of He, 12C and 16O at these prescriptions it is possible to select a well-defined set
4
40 MeV/nucleon. The energy resolution of the silicon de- of data, while expecting negligible distortions due to other
tectors allows for mass discrimination by means of the experimental problems.

P.M. Milazzo et al.: Nuclear temperature measurements...
357
Fig. 1. Energies necessary to punch through increasing silicon
3
4
thicknesses (Z = 2: He open squares, He full squares; Z = 3:
6Li open circles, 7Li full circles; Z = 6: 11C full squares, 12C
open circles, 13C full circles.
3 Results
In order to test the influence of the energy threshold for
mass identification when applying eq. (1), the incident en-
ergy necessary to punch through increasing thicknesses of
the ∆E silicon detector was first evaluated. This thick-
ness has to include all the other layers preceding the stop
E detector, used for mass identification by means of the
well-known ∆E − E technique. In fig. 1 the minimum in-
cident energy necessary to punch through a silicon equiv-
alent thickness is plotted for different atomic species as a
function of the thickness itself.
Fig. 2. Energy distributions in the QP reference frame (left
panels) and in the laboratory frame (right panels): (a-b) 3He
dashed and 4He full lines; (c-d) 6Li full and 7Li dashed lines;
(e-f) 11C dotted, 12C full and 13C dashed lines.
Energy loss calculations were performed referring to
Anderson and Ziegler studies [25]. As expected fig. 1 shows
a strong dependence on the atomic number Z and a weaker
one on the mass A of the incident ion.
In fig. 2 the experimental energy distributions of the
same fragments in the QP frame and in the laboratory
frame are shown, on left and right side, respectively.
The energy distributions in the QP reference frame
show similar shapes and slopes (by the way, this can be
seen as a probe of a careful source identification [21,28]);
on the contrary the energy distributions in the labora-
tory frame are shifted towards higher energy values, the
12C, respectively) no significant cut is present in the en-
ergy distributions spectra and experimental data can be
safely used for temperature determination.
It is clear that for the experiment which we are deal-
ing with, the energy thresholds do not affect the obtained
temperature values, however, it is evident that higher en-
ergy thresholds or different experimental conditions, such
as lower source velocities or large detection angles, can
produce differences in the total yield used in eq. (1) and
influence the temperature value.
To have more insight into the role played by the en-
shift being larger for heavier isotopes. In fact in the lab- ergy threshold, i.e. by the thickness of the ∆E detector,
oratory frame one has to consider the contribution of the we analyzed the experimental data performing different
QP emitting source motion to the kinetic energy. This energy cuts, i.e. simulating different experimental energy
additional kinetic energy depends on the velocity of the thresholds as due to a different thickness of the ∆E de-
QP (vQP = 6.3 cm/ns in the present experimental situa- tector. The results of this procedure are shown in fig. 3,
tion [28]), on the mass of the considered isotope and on where the experimental yield ratios R between isotopes of
the detection angle. Limiting the discussion to forward the same atomic number are shown as a function of the
angles, as in the studied experimental data, this means silicon equivalent thickness preceding the stop detector
4
7
that, for instance, He, Li, 12C own at least ꢁ80, ꢁ150, (i.e. for increasing incident energy thresholds). Only frag-
ꢁ 250 MeV, respectively. This shift implies that up to a ments with energy higher than that necessary to punch
4
7
∆E thickness ꢁ2500, ꢁ2200, ꢁ900 µm (for He, Li and through the ∆E detector were considered to evaluate R.

358
The European Physical Journal A
Fig. 4. Double ratio of isotope yields and related temperatures
for the two considered thermometers as a function of the silicon
thickness preceding the stop detector.
Fig. 3. Yield ratios R between isotopes of the same atomic
species as a function of the silicon thickness preceding the stop
detector.
the velocity of the QP system formed in mid-peripheral
The decreasing behaviour of the ratios as a function of
the ∆E thickness, is due to the fact that the lighter ions
(which contribute to the numerator of R) have in the labo-
ratory frame energy distributions relatively less expanded
towards the higher values than the heavier ones (which
contribute to the denominator of R).
From these experimental single isotope ratios, the dou-
ble ratios of isotope yields and the corresponding temper-
atures have been extracted and their values are shown in
fig. 4 as a function of the silicon thickness. The horizon-
tal lines correspond to the temperature values extracted
from non-distorted distributions [28]. One can clearly see
that no particular distortion appears in the carbon ther-
mometer: threshold effects do not play a significant role
when dealing on double isotope ratios at fixed Z value.
On the contrary, the temperature measurements involv-
ing He and Li isotopes can be influenced by the energy
ˆ
collisions at b ≈ 0.4 with an expected excitation energy
≈6 MeV/nucleon) the kinetic energies of Z = 2, 3 and
6 fragments are at least ꢁ65, ꢁ115, ꢁ 200 MeV, respec-
tively. The corresponding thicknesses of punching through
are 1700, 1300 and 500 µm. If for the same considered sys-
tem we have a beam energy of 20 MeV/nucleon, then, the
QP emitting sources have expected velocities of 5 cm/ns
(thinking at the 80% of the projectile velocity [13]). In
this case the kinetic energies of the emitted fragments are
at least ꢁ50 (Z = 2, thickness of 1000 µm), ꢁ90 (Z = 3,
850 µm), ꢁ155 MeV (Z = 6, 350 µm).
In the cases where the source velocity is not high
enough to give its decay products a strong burst, suffi-
cient to safely overcome the energy thresholds, an addi-
tional study on the phase space is needed.
In order to analyze how the different experimental con-
cuts. A strong dependence appears around 4 mm; for this ditions affect the isotope yields and, therefore, the temper-
thickness energies higher than ꢁ100 and ꢁ200 MeV are ature values, we built up a Montecarlo simulation in which
necessary to punch through, for Z = 2 and Z = 3, re- a source emits fragments isotropically in its c.m. frame and
spectively. These energy values correspond to the maxima with energy distributions given by the experimental ones
of the energy distributions and, therefore, the distorsions (see fig. 2a), c), e)). Therefore, we have studied the lab-
begin to play a significant role. Anyhow distorsions ap- oratory energy spectra of each considered (Z, A) product
pear in fig. 4 only for really thick ∆E detectors, as for as a function of the velocity of the source and of the detec-
instance lithium-drifted silicons (the typical thickness is tion angle. We report here the results obtained for three
3-5 mm) in multi-element telescopes. This holds for the values of the source velocity and for the case in which
data discussed here, i.e. for isotopes forward emitted by the decay products are emitted by the QP source with
a source travelling at vQP = 6.3 cm/ns. However, the en- the Maxwellian slopes given in ref. [28]. In fig. 5 the cal-
4
7
ergy thresholds are largerly influenced by the contribution culated energy distributions for He (left panels) and Li
of the emitting source motion to the total energy and the (right panels) with different angle selections are presented
∆E thickness for which distorsions arise becomes smaller for vQP = 6.3 cm/ns that is the experimental situation
for slower emitting sources (lower beam energies or higher presented here. In the first row the whole calculated en-
dissipation in the reaction). As an example, for the reac- ergy distribution is shown. In the middle row the full lines
tion here considered at a QP velocity of 5.6 cm/ns (i.e. refer to the energy distributions as seen by detectors cov-

P.M. Milazzo et al.: Nuclear temperature measurements...
359
Fig. 5. Energy distributions for different emission angle ranges
in the laboratory frame.
ering in the laboratory frame angles between 3 and 23◦ as
for the experimental data of ref. [28] (but without energy
threshold cuts); the dashed lines refer to the obtained ex-
perimental energy distribution [28]; in this case the tem-
peratures were extracted performing a careful study on
angular and energy distributions since the phase space
has to be uniformly measured for all the investigated ions
[21,28]. The last row presents the expected energy distri-
butions considering only angles higher than 30◦ and 45◦
in the laboratory frame for He and Li, respectively. In
this case the energy distributions mainly cover values for
which the effects of energy threshold in the mass iden-
tification take place heavily. Thus it is clear, already at
first sight, that temperature measurements result less re-
liable if obtained from experimental data collected with
detectors placed at large angles.
Fig. 6. Double ratio of isotope yields and related tempera-
tures for the THeLi thermometer as a function of the silicon
thickness preceding the stop detector: a) vQP = 6.3 cm/ns,
b) vQP = 5.6 cm/ns, c) vQP = 5.0 cm/ns. Only fragments emit-
ted at angles higher than 30◦ in the laboratory frame are con-
sidered.
4
7
of this kind could give rise to important difficulties in data
interpretation.
In conclusion, one has to stress that as in experiments
in which temperature measurements are performed with
detectors placed at large laboratory angles, a careful data
selection, that takes into account the efficiency of phase
space covered, is required. The importance of this problem
increases when the sources are relatively slow, for instance
for QP with low energy beam (<20 MeV/nucleon) or for
central collisions in which incomplete fusion takes place
(vQP = 5.2 cm/ns, for Xe+Cu 30 MeV/nucleon).
To see it more quantitatively, we considered different
energy thresholds due to different detector thicknesses and
we extracted the temperatures from the data collected at
angles higher than 30◦ as a function of the detector thick-
ness for the given source velocity (fig. 6a). In fig. 6b) and
c) we show the same results for different source velocities.
For fast QP emitting sources (vQP = 6.3 cm/ns, fig. 6a)
it is evident that little differences on extracted THeLi val-
ues appear even for small thicknesses (200 µm); however
in this case distortions remain of the order of 1-2% deal-
ing on small thicknesses (up to 500 µm). The situation is
different for slower emitting sources: systematic errors of
5% (vQP = 5.6 cm/ns, fig. 6b) and 10% (vQP = 5 cm/ns,
fig. 6c) take place, even for ∆E detectors relatively thin
(500–600 µm).
4 Conclusions
In summary isotope yields emitted from highly excited nu-
clei produced in Xe+Cu 30 MeV/nucleon collisions were
measured in experiments performed at the National Su-
perconducting K1200 Cyclotron Laboratory of the Michi-
gan State University.
If, for instance, the measurements are devoted to the
study of a possible liquid-gas phase transition, where tem-
peratures range between 4 and 6 MeV, systematic errors mentally defined emitting sources have been performed by
Temperature measurements of well-defined and experi-

360
The European Physical Journal A
means of the double ratio of isotope yields technique. Ef-
fects due to energy threshold in mass discrimination have
been investigated. It has been observed that instrumen-
tal effects can play a significant role in the calculation
of ratios involving isotopes with different Z values (He,
Li), because of the strong dependence on Z of the energy
losses in the material. The importance of this problem in-
creases when the emitting sources are relatively slow and
detectors are placed at large laboratory angles. Systematic
errors of the order of 10% can be easily introduced in the
THeLi temperatures, even for thin ∆E detectors (500-600
µm silicon equivalent).
The analysis here presented, together with those of ref.
[21,28] indicates how it is possible to limit the effects of
experimental identification thresholds by a careful study
of the phase space and by applying eq. (1) to isotopes
for which the phase space has been measured with similar
efficiencies. In this respect it appears that using isotopes
with the same Z value strongly reduces the effects induced
by the energy threshold for discrimination.
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