Electron screening in d(d,p)t for deuterated metals and the periodic table*
F.Raiola1, P.Migliardi1, L.Gang1,11, C.Bonomo1, G.Gyürky2, R.Bonetti3, C.Broggini4,
N.E.Christensen12, P.Corvisiero6, J.Cruz7, A.D`Onofrio8, Z.Fülöp2, G.Gervino9, L.Gialanella5,
A.P.Jesus7, M.Junker10, K.Langanke12, P.Prati6, V.Roca5, C.Rolfs1, M.Romano5, E.Somorjai2,
F.Strieder1, A.Svane12, F.Terrasi8, J.Winter13
1
Institut für Physik mit Ionenstrahlen, Ruhr-Universität Bochum, Bochum, Germany
Atomki, Debrecen, Hungary
3
Dipartimento di Fisica, Universita di Milano and INFN, Milano, Italy
4
INFN, Padova, Italy
5
Dipartimento di Scienze Fisiche, Universita Federico II and INFN, Napoli, Italy
6
Dipartimento di Fisica, Universita di Genova and INFN, Genova, Italy
7
Centro de Fisica Nuclear, Universidade de Lisboa, Lisboa, Portugal
8
Dipartimento di Scienze Ambientali, Seconda Universita di Napoli, Caserta and Napoli, Italy
9
Dipartimento di Fisica Sperimentale, Universita di Torino and INFN, Torino, Italy
10
Laboratori Nazionali del Gran Sasso dell`INFN, Assergi, Italy
11
Department of Nuclear Physics, China Institute of Atomic Energy, Beijing, P.R.China
12
Institute of Physics and Astronomy, University of Aarhus, Denmark
13
Institut für Plasmaphysik, Ruhr-Universität Bochum, Bochum, Germany
2
*Supported in part by BMBF (06BO812 and 05CL1PC1/1), GSI (Bo-Rol), DFG
(436UNG113-146), OTKA (T034259), and Dynamitron-Tandem-Laboratorium Bochum
Abstract: The electron screening effect in the d(d,p)t reaction has been studied for 29
deuterated metals and 5 deuterated insulators/semiconductors. As compared to measurements
performed with a gaseous D2 target, a large effect has been observed in the metals V, Nb, and
Ta, which belong to group 5 of the periodic table, as well as in Cr, Mo, and W (group 6), Mn
and Re (group 7), Fe and Ru (group 8), Co, Rh, and Ir (group 9), Ni, Pd, and Pt (group 10),
Zn and Cd (group 12), and Sn and Pb (group 14). In contrast, a comparatively small effect is
found in group 4 (Ti, Zr, Hf), group 11 (Cu, Ag, Au), group 13 (B, Al), for the insulator BeO,
and for the semiconductors C, Si, and Ge. An explanation of this apparently novel feature of
the periodic table is provided possibly by the classical Debye screening in metals.
1. Introduction
In the extrapolation of the cross section (E) of a charged-particle-induced nuclear reaction
to astrophysical energies one uses the equation 1
(E) = S(E) E-1 exp(-2(E)),
(1)
where (E) is the Sommerfeld parameter and S(E) the astrophysical S-factor. The equation
assumes that the Coulomb potential of the target nucleus and projectile is that resulting from
bare nuclei. However, for nuclear reactions studied in the laboratory, the target nuclei and the
projectiles are usually in the form of neutral atoms or molecules and ions, respectively. The
electron clouds surrounding the interacting nuclides act as a screening potential: the projectile
effectively sees a reduced Coulomb barrier, both in height and radial extension. This, in turn,
leads to a higher cross section for the screened nuclei, s(E), than would be the case for bare
nuclei, b(E). There is an enhancement factor 2,
flab(E) = s(E)/b(E) = b(E+Ue)/b(E)
= E (E+Ue)-1 exp(-2(E+Ue)+2(E)),
= E (E+Ue)-1 exp((E)Ue/E),
for S(E+Ue)  S(E)
for Ue/E  0.1
(2)
(3)
where Ue is an electron screening potential energy. In the adiabatic limit, Ue can be calculated
from the difference in atomic binding energies between the compound atom and the projectile
plus target atoms of the entrance channel 3. For the d(d,p)t reaction the adiabatic limit is Uad
= 39 eV for neutral atoms and 52 eV, if the projectile is a positively charged ion at the
moment of interaction.
Recently, the electron screening effect in d(d,p)t has been studied for the metals Al, Zr,
and Ta 4, where the deuterated metals were produced via implantation of low-energy
deuterons. The resulting S(E) data show an exponential enhancement according to equation 3.
However, the extracted Ue values (Ue = 19015, 2978, and 32215 eV for Al, Zr, and Ta,
respectively) are about one order of magnitude larger than the value found in a gas-target
experiment: Ue = 255 eV 5. An anomalous enhancement was reported earlier 6 for
deuterated Pd (Ue = 25015 eV) and a deuterated Au/Pd/PdO multilayer (Ue = 60123 eV),
while deuterated Ti and Au exhibited a normal (“gaseous”) enhancement: Ue = 3611 and
2311 eV, respectively. Our recent study of deuterated Ta has led to U e = 30912 eV 7
confirming the previous observation 4. In this letter we report on results for the other 5
metals studied previously as well as 23 additional metals and 5 insulators/semiconductors (for
details, see 8).
2. Experiment and results
The equipment, procedures, and data analysis have been described elsewhere 7. Briefly,
the 100 kV accelerator of the Dynamitron-Tandem-Laboratorium at the Ruhr-Universität
Bochum provided the deuteron beam, with a 54 A current on target. A liquid-nitrogen-
cooled Cu tube extended to within 5 cm of the target. Four Si detectors were installed at an
angle  = 130o around the beam axis at a 5 cm distance from the target and covered with a Ni
foil to stop the intense flux of elastically scattered particles. The target together with the
chamber and the detector holders (including the Ni foils) formed a Faraday cup for beam
integration. A negative voltage of 200 V was applied to the Cu tube for suppression of
secondary electrons.
Each deuterated target was produced in the following way. A fresh material “M” (provided
by Advent and Goodfellow, England, and Chempur, Germany, with a purity of better than
99%) was bombarded with 10 keV deuterons, whereby the proton yield of d(d,p)t was
recorded as a function of implantation charge: the yield reached usually a saturation level
after a charge of about 1 C, i.e. a stoichiometry MxD has been produced near the surface of
the target. The procedure was repeated at Ed = 30 keV. The deuteron distribution was
investigated subsequently via Elastic-Recoil-Detection-Analysis (ERDA) at the 4 MV tandem
accelerator in Bochum. For most of the materials the distribution was uniform within 10%
from the surface down to a depth consistent with the range of the implanted deuterons. An
exception to this procedure was the metal In (melting point = 159oC), where we could not
observe an increasing yield as a function of implantation charge and the observed yield was
dependent on beam current: we omitted thus In from further study, as well as other elements
with a low melting point.
The reaction yield of the infinitely thick target, Y(Ed,), was obtained in several runs at
Ed = 5 to 30 keV, with energy steps Ed of 0.5 keV at Ed = 5 to 10 keV and 1.0 keV at Ed = 10
to 30 keV. In order to arrive at a thin-target yield curve, Y(Ed,), the thick-target yield curve
was differentiated, i.e. the yield difference between two adjacent points Y(Ed,) and Y(EdEd,) was calculated and divided by Ed. The energy steps Ed correspond to about 50 and
100 atomic layers near the surface. The result is
Y(Ed,) =  eff(Ed)-1 (Eeff) ,
(4)
with the effective energy Eeff 1 and the constant , as measured using a radioactive source
7. The effective stopping power eff(Ed) for the MxD target is given by the expression
eff(Ed) = D(Ed) + x M(Ed).
(5)
From the compilation SRIM 9 one finds that the energy dependence of eff(Ed) at Ed = 5 to
30 keV is identical with M(Ed) to within 5% for a wide range of x values, mainly due to the
velocity-proportional stopping power; thus, the deduced energy dependence of (Eeff) is
nearly independent from the stoichiometric ratio x.
The resulting cross section (Eeff) (varying from a few nb to a few mb), i.e. the weighted
average of several runs, is illustrated in Fig. 1 in form of the astrophysical S(E) factor for the
examples Al, Pd, Au and Pb; the errors shown arise predominantly from the spread of the
(differentiated thick-target) thin-target yields from various runs. The absolute scale was
obtained by normalisation to previous work 5 at Ed = 30 keV including the effects of
electron screening where applicable; the normalisation led to a value for the stoichiometric
ratio x given in Table 1. We noted that the stoichiometry MxD was rather stable in time: e.g.
for Pt we observed the same reaction yield (within 10%) after an interruption of 2 months,
during which the deuterated Pt was stored in air at room temperature.
In the analysis of the data (e.g. Fig. 1) we assumed a bare S(E) factor linearly increasing
with energy, Sb(E) = 43 + 0.54 E keV b (center-of-mass energy E in keV), as found
previously 5,7. Relative to this function, the data were fitted with the enhancement factor of
equation 2: the resulting Ue values are summarized in Table 1. The low (“gaseous”) Ue values
for Ti and Au and the high Ue value for Pd reported by 6 have been essentially confirmed,
while the high Ue values for Al and Zr reported by 4 could not be verified. Since the data for
all metals with large Ue values could be fitted well with equation 2 (e.g. Pd and Pb in Fig. 1),
the enhanced cross section is most likely due to electron effects of the environment of the
target deuterons. It should be pointed out that the unrealistic assumption of deuterons located
predominantly at the surface layer leads to Eeff  Ed and consequently to a reduction in
enhanced cross section and the corresponding Ue value by about a factor 2, still far away from
the gaseous value; for cases with a low Ue value, the assumption leads to a “negative”
enhancement compared to Sb(E). In turn, a deuteron concentration increasing with depth will
make the enhancements stronger and the deduced Ue values larger.
For one of the metals with a high Ue value, i.e. Pt, we tested the possible influence of the
deuteron beam current Id on the observed Ue = 44050 eV value: with Id = 5 A – rather than
Id = 54 A – we found Ue = 42050 eV. Similarly, the metal Ta was investigated at a target
temperature of T = -196oC resulting in Ue = 32030 eV, consistent with the value found at T
= -10oC (Table 1). For Ni and Pd we used also ultra-pure single crystals (provided by Mateck,
Germany: 16 mm diameter, 1 mm thickness) leading – for Id = 2.4 A and T = +20oC – to Ue
= 29070 and 82080 eV, respectively, in agreement with the foil-values (Table 1). In
another experiment, we used a deuterated Pt target and a 3He ion beam (I3He = 1 to 3 A) in
combination with the reaction d(3He,p)4He to study the associated electron screening effect.
The result is Ue = 73060 eV showing that high Ue values do not depend on the kind of ion
species but are a feature of the deuterated metals.
3. Discussion
A comparison of the Ue values with the periodic table indicates a common feature (Fig. 2):
where more than 1 element of a given group of the periodic table has been studied so far, the
corresponding Ue values are either low (“gaseous”) as for Ti, Zr, and Hf (group 4), Cu, Ag,
and Au (group 11), and B and Al (group 13), or high such as for V, Nb, and Ta (group 5), Cr,
Mo, and W (group 6), Mn and Re (group 7), Fe and Ru (group 8), Co, Rh, and Ir (group 9),
Ni, Pd, and Pt (group 10), and Zn and Cd (group 12). Group 14 is an apparent exception to
this feature: the metals Sn and Pb have a high Ue value, while the semiconductors C, Si, and
Ge have a low Ue value indicating that high Ue values are a feature of metals. The indication
is supported by the insulator BeO (metal compound) as well as by the deuterated metals Ti,
Zr, and Hf having a small x value (e.g. TiD2) and thus representing also the case of insulators.
It should be noted that the quoted Ue values depend on the energy dependence of the
stopping power values of deuterons in the materials at energies far below the Bragg peak,
where no energy loss data exist and the values derived from the compilation SRIM 9 are
based on extrapolations. Although it appears unlikely that the true stopping power values drop
– in comparison to SRIM - exactly with the inverse function of equations 2 or 3 for one
element (say, Pd) but that they are consistent with SRIM for a nearby element (say, Ag), one
cannot rule out rigorously this possibility. However, recent measurements of low-energy
stopping powers of protons in C, Al, Ni, and Au 10 have confirmed the SRIM
extrapolations, where Ni is one of the materials with a high Ue value. Additional
measurements of stopping power values for protons or deuterons at low energies in deuterated
materials are highly desirable, particularly for the lighter elements (e.g. Li, Be, B).
The thermal motion of the target atoms has been considered 7 assuming a value of 1 eV
due to the lattice vibrations, which leads to a Doppler energy spread of 88 eV corresponding
to a beam energy increment of 1.2 eV and a cross section enhancement of 0.1%. The
consideration is consistent with the observed independence of the Ue values from the target
temperature (for Ta) and the deuteron beam current (for Pt). The latter feature excludes
secondary effects such as beam-induced lattice vibrations.
We also considered the channeling effect 7: the deuteron beam could be guided by the
lattice into planes or axes, whereby an interstitial atom such as deuterium could be hit with an
increased probability. The critical angle for channeling scales with the inverse square root of
the incident energy and the channeled flux scales thus with the inverse of the incident energy.
Thus, one may expect an enhancement factor proportional to 1/E, which is however not
observed. Similarly, the channeled flux scales with Z (Z = atomic number of the lattice atoms)
and one may thus expect an enhancement mainly among the heavy metals, which is again not
observed. Furthermore, the random orientation of the polycristalline foil-materials as well as
radiation damage of the crystalline structure by the intense deuteron beam lead to large
dechanneling effects. It was concluded therefore that the channeling phenomenon is not the
primary cause of the large observed cross section enhancements. The conclusion is consistent
with the observed Ue values for the single crystals Ni and Pd, which are identical to the foilvalues, within experimental error.
The diffusion property or the conductivity of the materials also provides no consistent
picture 7: for example, the diffusion coefficient for Zr is at least 3 orders of magnitude
smaller than that for Al and Ta, while the observed Ue values do not reflect such a trend.
Similarly, the conductivity of Zr is at least a factor 4 smaller than that for Al and Ta. A
comparison of other target features such as crystal structure, electron configuration, and Hall
coefficient do not provide an explanation of the observed “group” properties, nor does the
deduced stoichiometric ratio x (Table 1).
It may be of interest to note that a fusion of D with Ta would lead to a screening potential
energy of about Uad = 2 keV, in the adiabatic limit. If the electron clouds of the Ta atoms in
the lattice would be such that about 10% of them form a potential well around the interstitial
deuteron atom, one might arrive at a possible explanation. One must await however the results
of detailed calculations, although some theoretical work in this direction has been performed
already (7 and references therein).
In the “Fermi shuttle” acceleration mechanism 11-13, the d+d collision partners can
acquire additional energy (up to a factor 2) in a sequence of collisions with the heavy partners
of the target lattice. Since the elastic scattering of both the deuteron projectile and the
deuteron recoil (after a first direct scattering) on the lattice atoms leads to a Z4 dependence,
one may expect this mechanism to operate mainly among the heavy elements, which is
however not observed. Furthermore, it is unlikely that this mechanism leads to an
enhancement function nearly identical to equation 2.
Theoretical Ue values were derived from binding energy differences obtained by means of
ab initio density-functional [14,15] calculations along lines similar to those used in [16]. For
example, in the case of deuterated Pd, Ag, and Ta we find the same small value, Ue = 57 eV.
The “solid state contribution” to the screening energies as calculated in this way is only
around 5 eV. This is a small correction to the adiabatic atomic result, and it cannot account for
the large Ue values observed for several metals. In the case of the reaction d(3He,p)4He and
the deuterated Pt, calculations similar to those just described for the d(d,p)t case give Ue = 122
eV for Pt, again close to the adiabatic limit for neutral atoms, Uad = 119 eV, but much smaller
than the observed Ue value.
If one considers the metallic electrons as quasi-free, one may apply the classical plasma
theory of Debye leading to an electron sphere of radius 1,18
RD = (o kT / e2 neff a)1/2 = 69 (T / neff a)1/2 m
(6)
with the temperature of the free electrons T in K and the atomic density a in m-3; neff is the
effective number of free valence electrons of a given metallic atom. For T = 293 K, a =
6x1028 m-3, and neff = 1 one obtains a radius RD, which is about a factor 10 smaller than an
atomic radius. With the Coulomb energy at RD set equal to Ue, one obtains Ue = 300 eV, the
order of magnitude of the observed Ue values. A comparison of the calculated and observed
Ue values leads to neff given in Table 1: for most metals neff is of the order of one. Since the
Coulomb energy scales with the nuclear charge of the incident ion, the 3He beam case on
deuterated Pt should lead to a Ue value a factor 2 larger than that for deuterons on deuterated
Pt, consistent with observation (ratio = 1.70.4). Furthermore, the metals Be, Ti, Zr, and Hf
have a low Ue value, because they were in form of an insulator (e.g. BeO and TiD2), thus
having no free electrons; similarly, B is known to be an insulator. It may be surprising that the
classical Debye screening can explain most of the observed data. This aspect as well as the
exceptional case of group 11 (the “nobel” metals Cu, Ag, Au) with low Ue values (and
relatively high x values) must be explained in the future by improved theoretical work
including quantum and lattice effects.
Future experimental work involves additional elements of the periodic table, including the
lanthanides. The studies will use also target compounds, such as metal oxides. Furthermore,
improvements in ERDA should help to measure more precisely the deuteron distribution near
the metallic surface and the use of Rutherford-Backscattering-Spectroscopy (RBS) and other
analytic methods should provide information on the cleanliness near the surface, all needed to
improve the precision and accuracy of the deduced Ue and x values. The observations may
require eventually an Ultra-High-Vacuum (UHV) setup.
We thank J.Roberts (Edinburgh) for assistance during part of the experiments, J.V.Pinto
and J.P.Ribeiro (Lissabon) for some ERDA analyses, J.U.Andersen (Aarhus) and G.Fiorentini
(Bologna) for fruitful discussions, and H.J.Maier (München) for provision of several metals.
One of us (L.G.) thanks the support from the Major State Basic Research Development
Program of China (G200077400) and the National Natural Science Foundation of China
(10025524). The help of the technical team of the Dynamitron-Tandem-Laboratorium is
highly appreciated.
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Fig. 1: Astrophysical S(E) factor of the reaction d(d,p)t as obtained for the deuterated metals
Al, Pd, Au, and Pb (E = effective center-of-mass energy). The dotted curve represents the bare
S(E) factor while the solid curve includes the enhancement due to electron screening
(equation 2) with the Ue value given.
Fig. 2: Periodic table showing elements studied so far, where those with low Ue values (Ue 
100 eV, small effect) are lightly shadowed and those with high U e values (Ue  100 eV, large
effect) are heavily shadowed.
Table 1: summary of results
-------------------------------------------------------------------------------------------materiala
Ue (eV)b
stoichiometry xc
neff
----------------------------------------------------------------------------metals
Mgd
44040
8.7
3.1
Al
30
3.7
0.010
Ti
30
0.75 f
0.011
V
35030
16
1.1
Cr
22020
7.8
0.40
Mn
35040
3.7
1.0
Fe
21050
18
0.35
Co
20020
3.1
0.3
i
Ni
45080
26
1.5
Cu
4320
6.6
0.015
Zn
14020
12
0.20
Y
32040
8.3
2.3
Zr
8320
0.42f
0.11
Nb
40040
8.7
1.9
Mo
22020
7.0
0.51
Rud
22030
5.6
0.44
Rhd
84070
11
6.5
Pd
80070j
35
5.9
Ag
2310
10
0.0061
Cdd
39060
6.2
2.2
Sn
20020
36
0.76
Hf
8720
0.55 f
0.11
Tah
34014e
9.1
1.4
W
22020
3.3
0.52
d
Re
70070
4.9
4.8
Ird
33030
4.4
1.0
Ptk
44050g
14
2.0
Au
6120
2.6
0.042
Pb
44050
18
4.0
insulators/semiconductors
BeOd
30
4.0
0.0050
Bd
30
2.6
0.0046
Cd
5220
3.1
0.010
Sid
4520
4.3
0.027
Ged
6020
1.8
0.055
--------------------------------------------------------------------------------a
For a target temperature T = -10oC and a current Id = 54 A
b
Using equation 2
c
Estimated uncertainty is of the order of 20%
d
For T = +20oC and Id = 2.4 A
e
From equation 3 one obtains Ue = 30912 eV 7
f
The minimum x value is expected 17 to be 0.50, e.g. Ti0.5D or TiD2
g
Ue = 42050 eV for Id = 5 A
h
Ue = 32030 eV for T = -196oC
i
Ue = 29070 eV for single crystal
j
Ue = 82080 eV for single crystal
k
Ue = 73060 eV for d(3He,p)4He