Abstract
A profusion of unbound, low-energy electrons creates a local electric field that reduces Coulomb
potential and increases quantum tunneling probability for pairs of nuclei. Neutral beam-target
experiments on deuterium-deuterium fusion reactions, observed with neutron detectors, show
percentage increases in fusion products are consistent with electron-screening predictions from
Schrödinger wave mechanics. Experiments performed confirm that observed fusion rate
enhancement with a negatively biased target is primarily due to changes to the fusion cross section,
rather than simply acceleration due to electrostatic forces.
- Introduction
The effect of electrons on fusion reaction rates has been investigated for many years to understand
its influence on stellar nucleosynthesis. The screening effects due to bound electrons have been
studied for various nuclear reactions, including D-D, D3He, 3He3He, p11B, and p7Li [1–6]. For
the reaction p11B, the screened cross section has been measured for energies between 17 and 134
keV [1]. In this range, the screening potential is only 0.25 to 2 percent of the total interaction
energy, resulting in enhancement of fusion cross sections of less than 10 percent [5]. Although the
bound electrons produce relatively high electric fields, the cumulative effect is largely cancelled
out by the positive charge of the nucleus, thus the net effect on the effective cross section is not
significant.
The screening effect due to thermal electrons, such as in two-component plasmas or neutral metals,
has also been studied extensively [5–14]. In an overall neutral system, the positively charged ions
are immersed in a sea of free electrons, which tend to cluster around regions near individual ions,
at the Debye-Hückel radius, having a charge neutralization effect similar to that of orbital electrons
around a nucleus. For metals, this screening distance is usually in the range of nanometers. The
effects of the thermal electrons on nuclear reactions have been measured with a deuterium target,
embedded in metals [9]. This magnitude of the thermal-electron screening is close to the effect by
bound electrons, around tens to hundreds of eV. The enhancement of cross sections is also limited
to only a few percent because the screening fields are not coherent [15].
The combined screening effects of bound and thermal electrons can be used to explain the minor
change of nuclear reaction rates in stars [16]. To produce fusion energy more effectively in a
laboratory setting, reaction rates must be raised by orders of magnitude compared to the processes
of stellar fusion. This can only be achieved with more aggressive screening [17].
Noting that both bound and thermal electrons produce measurable enhancement in fusion cross
sections, we wish to report on a process of using a profusion of low-energy, free electrons in target
materials to generate screening fields which may be able to reduce the Coulomb barrier
significantly, such that fusion cross sections are improved from vanishingly small to a level of
interest (~ 10–30 to 10–40 m2
) for significant reaction rates. This process involves ions as well as
high-density neutrals. The presence of neutrals in the system has the advantage of yielding high
beam densities and significant fusion events without causing plasma instabilities due to space
charge.
A presentation of a Schrödinger wave mechanics approach to predict the effects of screening fields
is given in Section 2.1 below. An experiment designed to use low energy neutral beams (centerof-mass energy of 25 keV) of deuterium to interact with a target of high-density deuterium (biased
up to –20 keV) demonstrates fusion through detection of energetic neutrons, 3He, Tritium and
protons. An increased fusion rate indicates the efficiency of shielding by free electrons. The
tunneling probability through the screened Coulomb barrier and the associated fusion cross
sections for D-D reactions is shown in Section 2.2. Alternative possible explanations of fusion
rate enhancement are considered in Section 5.
- Theory/Calculations
2.1 Schrödinger Equation and Screening Potentials
−
ℏ
2
2𝑚
𝛻
2𝜓 + [(
𝑍1𝑍2𝑒
2
4𝜋𝜖𝑟
− 𝑈𝑠) − 𝐸] 𝜓 = 0 (1)
The Schrödinger equation in Eq. 1 includes the screening potential Us to represent the electron
screening effect as a reduction of the Coulomb barrier between two nuclei of Z1 and Z2 within the
range where quantum wave properties of the particles are not negligible. At these interaction
distances the screening potential can be treated as a constant reduction of the internuclear Coulomb
potential. In such a case, Eq. 1 can be readily used to derive screened fusion cross sections for a
given beam energy.
We now consider the target as a cylinder with radius R1 located coaxially within a grounded
cylinder of radius R2, perpendicular to the beam path. When a target is negatively biased the field
and potential energy in the space between R1 and R2 can be found as:
𝐸(𝑅) =
𝑉
𝑅 𝑙𝑛 (
𝑅2
𝑅1
)
(2)
𝑈(𝑅) = −𝑒𝑍1
𝑙𝑛 (
𝑅2
𝑅
)
𝑙𝑛 (
𝑅2
𝑅1
)
𝑉 (3)
Here R is the initial radial position of the particle, V is the bias voltage, and eZ1 is the charge of
the projectile. On the surface of a 0.5 cm rod biased at –20 kV in a 1.5 cm tube, the electric field
E1 at the rod surface without any plasma in the medium is calculated to be approximately 4 MV/m.
During our experiments, a plasma was formed in the vicinity of the target electrode. The ion
density in the sheath region creates a positive potential that modifies the electric field at the surface
of the target electrode. The net screening energy Us experienced by the impacting beam particles
and the target must be computed from the imposed external fields and the screening from plasmas.
The Debye length has been computed to be 10–3
cm.
2.2 Tunneling Probability and Fusion Cross Sections
A nucleus approaching another nucleus on the target surface experiences a generally repulsive
force as expected. It also experiences the collective, attractive force of the electrons grouped on
the biased target. The wave nature of the incoming beam nucleus allows it to propagate through
the barrier as an evanescent wave. The probability of its penetration, or tunneling, depends on its
velocity and the amount by which the Coulomb barrier is lowered. Fusion cross section is related
to this tunneling probability.
The probability of the particle tunneling through a 1-D barrier can be obtained using a simple wave
approach and the result is (derived in Appendix A):
3
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𝑃 = 𝑒𝑥𝑝 − 2√2𝑚 ℏ ∫ 𝑟2 𝑟1 √𝑈(𝑟) − 𝐸𝑑𝑟
Here r2 is the nuclear radius and r1 is the zero-velocity radius. For the modified Coulomb barrier,
the potential U and the energy E are given as
𝑈 =
𝑍1𝑍2𝑒
2
4𝜋𝜖𝑟
− 𝑈𝑠
; 𝐸 =
𝑍1𝑍2𝑒
2
4𝜋𝜖𝑟1
(5)
Here 𝑈𝑠
is the screening potential energy is assumed to be equal to the applied bias voltage (Vb)
multiplied by a shielding factor η where Us = ηeVb. This factor η depends on the characteristics
of plasmas produced around this target. The free-space potential profile is modified by the density
and temperature of plasma in front of the target. Our present experiment is designed to verify only
the functional dependence of the shielding as shown in Eq. 7 below. The absolute value η is to be
determined experimentally.
Using the WKB approximation,
𝑃 ≈ 𝑒𝑥𝑝 (−√
𝐸𝐺
𝐸
) (6)
where EG is the Gamow energy 𝐸𝐺 = (𝜋𝛼𝑍1𝑍2
)
22𝑚𝑟𝑐
2 = 986 𝑘𝑒𝑉 , mr is the reduced mass, and
the α is the fine-structure constant 𝛼 =
𝑒
2
ℏ𝑐
1
137.04
.
The fusion cross-section for a 3-D Coulomb potential is (derived in Appendix B):
𝜎(𝐸,𝑈𝑠
)~
𝑃
𝐸
𝑆(𝐸+𝑈𝑠
)
𝐸+𝑈𝑠
𝑒𝑥𝑝 (−√
𝐸𝐺
𝐸+𝑈𝑠
) (7)
We have included in σ(E), the astrophysical factor S(E), which represents the probability of nuclear
reaction once the projectile tunnels through the barrier [1–5].
The rate of fusion per unit volume can be calculated as
𝑑𝑛
𝑑𝑡
= 𝑛𝑏𝑛𝑡𝜎𝑣 (8)
where n is the number of fusion events, nb is the density of D in the beam, nt is the density of D
nuclei in the target, and v is the velocity of the beam.
An external electric potential as a DC bias in our experiments can affect the rate of fusion, dn/dt
from Eq. 7, in three ways—a change in nt, σ, or v. In our experiment we have carefully account for
the possibilities of changes to nt and v causing the percentage increase of the rate of fusion that we
observe, therefore, we present a model of enhancement of fusion rate to be compared with theory
relating to modification of the fusion cross section, σ.
4
The D-D fusion reaction has two branches of equal probability:
Neutronic reaction: 𝐷 + 𝐷 → 3𝐻𝑒 (0.82 𝑀𝑒𝑉) + 𝑛 (2.45 𝑀𝑒𝑉) (9)
Aneutronic reaction: 𝐷 + 𝐷 → 𝑇 (1.01 𝑀𝑒𝑉) + 𝑝 (3.02 𝑀𝑒𝑉) (10)
A count of any of the four emitted products (3He, n, T, p) from the system is, therefore, a reliable
measure of the overall rate of fusion. The neutron measurement is preferable to charged particle
detection for routine measurements since much more care is required to establish controls for
interfering signals when measuring the charged particles with the silicon detector than for the
neutron measurement, and the neutron detector can be located outside of the pressurized chamber.
Using an ion-implanted Silicon detector connected to a multi-channel analyzer to output the
detected particle energies (MeV) and number of particles (as pulse height), we observed particle
energies correlating to helium-3 (~0.5 MeV), tritium (~0.8 MeV), and proton (~3 MeV) as shown
in Fig. 1. The departure from tabulated energies is due to collisions between fusion products and
the plasma medium.
Figure 1: Fusion products (proton, tritium, and helium-3) observed with a silicon detector and multi-channel analyzer
from Ortec. The detector was placed within the target chamber with about 5 mTorr deuterium gas, during a D-D fusion
experiment a few inches from the target and perpendicular to the beam path. Energies were calibrated with respect to
an Am-241 source (calibration data not shown) and are consistent with those listed in Eqs. 9 and 10.
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- Material and Methods: Reduction in Coulomb Barrier by Free Electrons
An experiment was designed to quantify the lowering of the Coulomb barrier by electrons using a
beam-target interaction configuration. A reproducible, low-power (1.25 W) deuteron beam was
chosen to interact with an equally reproducible plasma to allow for digital sampling and signal
averaging of many repeated beam and target interactions per second.
The deuteron beam was produced from an ion beam source with a microwave-generated deuterium
plasma of density up to 1012 cm–3 with ion species up to 90% [18]. The ion source was floated to
high positive potential using an innovative DC blocking method [Unpublished results] which
allowed microwave power to be injected into the plasma chamber while holding off the high
voltage from ground. As a result, the ion beam produced was able to be sustained at a high energy
value of 25 to 100 keV with a grounded experimental chamber.
As the ion beam exits the ion source into the pressurized chamber, the ions traverse through a path
of sufficiently high neutral density that the deuterium beam becomes neutralized. The ion source
and the experiment chamber were both operated at the same pressure, verified by both convection
enhanced Pirani and hot cathode gauges. Based on previous work performed at ORNL [19], the
fraction of neutral beam along a drift tube has been analyzed based on published cross-sections of
the hydrogen species evolution. These involve all the combinations of H+
, H0
, H–
and their
corresponding molecular species. Based on the calculated value of the line density,
𝑥(𝐿) = ∫
𝐿
0
𝑛(𝑙)𝑑𝑙 (11)
where L is the total length traversed by the beam and n(l) is the neutral density as a function of
distance, n(l) being a constant for a given chamber pressure, the ion fractions of the beam can be
calculated. The line density was approximately 1016 cm–2 based on the measured values in this
system. Referring to the predicted ion fractionation [19] of the hydrogen species, the neutral
species fraction in a 40 keV deuterium at the calculated line density is predicted to be at least 80%.
The differences between the charge exchange interactions in hydrogen and those of the isotope
deuterium are assumed to be negligible.
Deuterons in the beam exit the ion source with a narrow distribution of kinetic energies about Ei
and can gain or lose energy through interactions with other deuterons or with electromagnetic
fields as they traverse the distance L through the chamber. The beam energy is significantly higher
than the average velocity of the deuterium gas at thermal equilibrium. The resultant net momentum
of the beam, therefore, remains roughly unchanged in direction by the charge exchange
interactions. Deuterium ions will be accelerated toward a negatively biased target, but this
interaction is weakest at the beam source, when the ion population in the beam is greatest, and for
a given deuteron is limited by the collision frequency. These combined effects result in a deuteron
beam that strikes the target consisting of mostly neutral atoms travelling at close to the initial ion
beam velocity.
Further neutralization of the beam was accomplished using a parallel set of biased plates along the
beam path to set up a perpendicular E-field in order to deflect the residual ion species to minimize
6
the number of ions that strike the target (see Fig. 2). A neutral beam cannot gain energy from a
biased target during its transit from source through the chamber. The ion fractionation model used
to calculate the collisional neutralization of the initial ion beam also predicts a certain degree of
ionization from the continual collisions after the removal of ions using the steering voltage. We
expect that the remaining beam that proceeds toward the target is predominantly neutral atoms.
The generally neutral character of the beam under these conditions was confirmed by pointing the
beam at a 30-degree angle to an axial magnetic field with no observable bending of the beam due
to Lorentz forces.
Ions approaching the target will be accelerated toward the negative bias. The net acceleration is
likely to be small for the entire beam, but there will likely be a greater spread in the distribution of
kinetic energy. Because of this ionization, some deuterons that would have just passed the target
when no bias is applied may get pulled in when the bias potential is present. This change in number
of beam particles striking the target in such cases must, therefore, be included in the calculation as
a change in nb.
7
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The Debye sheath that forms around the target when a bias potential is applied decreases the
effective potential experienced by the beam ions as they approach the target. As the beam nuclei
pass through the Debye sheath, however, they will experience an increased local electric potential
gradient.
An aluminum rod target of 0.5 cm radius was placed 1.5 cm from the chamber ground and directly
in the path of the deuteron beam. The beam was either continuous or pulsed at a constant rate.
Loading of deuterons at the target surface, with no bias potential applied, until equilibrium was
achieved established a steady baseline fusion rate and maintained the reactant densities constant
for each bias and beam energy.
Neutron counts were measured with a proton-recoil fast neutron detector (FND) developed by
Alpha Ring (detector design is described in Appendix C). Additional monitoring of the average
neutron count rate by a Bonner sphere and a 3He thermal neutron detector with a cylindrical
polyethylene jacket to slow fast neutrons for detection so that the thermal neutron detector provides
information related to both fast and slow neutron counts. The fast neutron detector, however, can
also provide information on timing of neutron events in experiments where pulsed particle beams
are used.
After beam loading reached equilibrium, neutron counts were recorded for a selected time interval
with a bias potential applied to the target to accumulate negative charges. The neutron production
rate was measured at each bias value, thus with a different density of free electrons. An increase
in neutron production was observed as the target bias potential was varied from 0 to –20 kV. The
relative increase in fusion rate is reported here as a percent increase of counts during the
measurement period with respect to measured counts in the 0 kV case over the same duration.
- Results
The relative increase in fusion rate with increasing bias potentials of the target, as measured on
different days by neutron production, is shown in Fig. 3. The beam power used in these
experiments was constant at approximately 1.2 W throughout each data set indicating that the
density of ions initially accelerated did not change significantly.
Measurements shown in Fig. 3 were taken with the indicated neutron detectors for 20 minutes at
each bias value. Count rate data was collected on a per minute basis. The average 95% confidence
interval of the average neutron counts at each bias value, propagated through the calculation of
fusion rate enhancement, was calculated to be around ± 17% enhancement.
Because neutron data is obtained at only one location per detector, the comparison with theory is
limited to the demonstration of trend variation. Theoretical curves were derived from the
assumption that the enhancement in neutron yields is proportional to the percentage increase in
cross sections.
8
Figure 3: Percent increase in the neutron production rate with the variation of target bias potential. The ion beam at
25 keV and 0.05 mA is generated from a capacitively-coupled plasma by excitation at 2473 MHz, +9 dBm, and 100%
duty cycle with -100 V and 0.040 mA electron suppression. Beam neutralization, in a deuterium pressure of 5.3 mTorr
for set 1 and 5.4 mTorr for set 2, was further ensured using a -5 kV steering potential perpendicular to the beam path.
Each measurement period consists of repeated acquisition of neutron counts over 20 minutes at the specified target
bias. The theory curve (dashed line) represents an effective shielding energy calculated using the factor η = 0.22.
Experiment 1 and experiment 2 represent independent data sets from two different runs. Detector 1 represents data
from the FND, and detector 2 represents the detection of neutrons that have been slowed by the polyethylene housing
of the thermal neutron detectors for the specified data set. - Discussion
The increased fusion efficiency shown in Fig. 3 is consistent from one data set to another. The
relative increase calculated from fast neutron measurements only is also consistent with that
observed when neutrons of all energies are measured. The trend in observed enhancement in the
rate of fusion for each data set is within measurement uncertainties and distinct from that predicted
by an increased system velocity from the additional acceleration expected for ions approaching the
biased target.
The small population of ions in the beam at the time the beam encounters the target will be
accelerated toward the negative bias. Some of these deuterium ions in the fringes of the beam
profile that would otherwise pass the target when no bias is applied may add to the effective
9
reactant density when the bias potential is present. Since we have a mostly neutral beam, and this
density increase would come from regions of low deuteron density, it is unlikely that the observed
enhancement with increasing bias could be attributed to these additional collisions.
The most significant variable from Eq. 8 that can account for the higher fusion rates is, therefore,
an increase in the effective cross section, that is in turn related to Coulomb barrier reduction. The
combined effects of charges near the nuclei at each target bias potential is represented as an
effective shielding potential.
5.1 Target Loading
The first of two possible alternative explanations considered here was the phenomenon of target
loading [20], wherein the target may produce an uneven emission of neutral D atoms due to
progressive heating of the target by the beam impact. Because fusion rate depends on target
density as well as beam density, the observed enhancement in the fusion rate could be explained
by this variation of the target density. - To address this possibility of a progressive beam loading effect, we used a stochastic sampling of
the arrangement of target voltage, Vb values: 0, –4, –8, –12, –16, and –20 kV. Data was collected
for each Vb in this randomly structured queue. The queue was re-populated with another random
shuffle of this set of values after every 6th iteration. The use of a constant set of values ensured
that every Vb element was equally represented in the pooled data set for statistical comparison, and
the random order in which the Vb values were set averaged out the effect, if any, of target loading.
Each Vb was set for a period of 5 seconds (reducing the effect of long-time constant temperature
effects) and neutron counts were measured. Neutron counts were then averaged for each Vb value
and compared with the case without bias (0 keV). The observed results were consistent with the
original experimental results and showed a clear correlation between an increase in neutron yield
and increasing negative biases on the target independent of the beam-loading history. We,
therefore, assert that the results of this stochastic sampling experiment eliminate target loading as
a significant influencing factor for the observed increase in fusion rate. - 5.2 Beam Re-Ionization from Electron Emission
A second possible alternate explanation for the fusion rate enhancement is that the neutralized
beam may have become re-ionized by electrons emitted from the biased target. Since fusion
depends on the probability of the two atomic nuclei penetrating the Coulomb barrier, an increase
in final relative velocity may enhance the likelihood of nuclei to fuse. Though the neutral beam
would be effectively unaffected by the target bias during most of the transit through the chamber,
the acceleration of the newly formed ions toward the negatively biased target would be the
equivalent of a grounded target and a higher-energy beam.
The neutral particles in the beam could become ionized in our system by energetic electrons
emitted from the negatively biased target through mechanisms that include thermal electron
emission, secondary electrons from impact of other beam particles on the target surface, and field
emission of electrons. In each case the interaction distance where reionization can occur will be
10
very close to the target surface. And the extent of ionization depends on the electron flux, the
kinetic energy of the ejected electrons, and the collision probability between the ejected electron
and a neutral beam particle.
To investigate this alternate explanation, the re-ionization cross section for the incoming beam and
the electron flux emitted from the target must be accurately characterized to distinguish the effect
of re-ionization phenomena the fusion rate enhancement to from that of Coulomb barrier reduction.
Assuming a significant degree of ionization is possible, however, we can create an upper limit case
for effective increase in fusion rate due only to beam acceleration.
Considering the interaction distance of the ejected electrons to ionize the incoming beam particles
to be on the order of the mean free path at the deuterium pressure in the chamber, we can predict
the approximate magnitude of the effect of reionization on the rate of fusion events. For a chamber
pressure of 5.3 mTorr at 300 K, the mean free path is 27 mm. For a target bias of –20 kV an ionized
deuteron, initially at the beam velocity, gains only an additional 0.5 keV of kinetic energy over
this distance using the velocity calculation 𝑣
2 = 𝑣0
2 + 2𝑎𝜆, where a is the acceleration 𝑎 =
𝐸𝑞
𝑚
,
and λ is the mean free path of a deuterium atom using the Van der Waals radius of the deuterium
molecule.
The velocity of the accelerated ions at the time of collision with a target deuteron results in a fusion
cross section calculated from the new system energy instead of the initial beam energy. The
expected enhancement of fusion probability for a neutral beam particle with a speed v0 that is
ionized within one mean free path of an electron from the target, based on Eq. 8, and assuming the
beam and target deuteron densities remain constant, is given by
(𝑅−𝑅0
)
𝑅0
(𝑣𝜎−𝑣0𝜎0)
𝑣0𝜎0
(12)
For an initial beam energy of 25 keV, therefore, we would expect an enhancement of fusion rate
of no more than 14 percent with a target bias of –20 kV if the beam reionization is complete. This
is significantly less than the >60 percent enhancement illustrated in Fig. 2. We can, therefore,
conclude that the enhancement of fusion that we observe with the applied bias is due primarily to
a modification in the effective fusion cross section.
- Conclusion
The beam-target experiments described in this report demonstrated an enhancement of neutron
yields when the target is negatively biased. Results are consistent with Alpha Ring’s theory of
Coulomb barrier reduction caused by electron shielding. The use of a neutral beam ensures that
beam particles do not gain substantial energy from the target bias, and residual ion acceleration
alone is unable to create sufficient acceleration to account for observed effects under these
conditions. Experimental conditions eliminate reactant densities and target loading irregularities
as a possible alternate explanation for fusion rate enhancement.
