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Double screened Coulomb barrier accounts for neutrons production in cluster and other fusion experiments
PhysicsLettersA 164 (1992) 155—163 North-Holland
PHYSICS LETTERS A
Double screened Coulomb barrier accounts for neutrons production in cluster and other fusion experiments M. Rambaut 57H rue De La Hacquiniere, 91440 Bures-sur- Yvette, France Received 12 September 1991; revised manuscript received 22 january 1992; accepted for publication 23 January 1992 Communicated by J.P. Vigier
In a dense, fully ionized medium, containing fusible nuclei, a collision between two nuclei is accompanied by an electron concentration around them. By this, rate oftunneling is tremendously increased. The experimental results are in agreement with the calculations, the number of displaced electrons being typically in the range ofone to two thousand.
In a recent paper of the present author [1], which summarized a model developed with Vigier, it has been shown that a possible correlation exists between three kinds of observed fusion experiments, the so-called cold fusion experiments, the capillary fusion experiments and the cluster fusion experiments. This correlation was interpreted in terms of turbulence caused by longitudinal Ampere forces combined with Coulomb screening and quantum tunneling processes. The aim of this paper is to present a more detailed numerical computational justification of this model. In the majority of the three kinds of fusion experiments one is working with dense media. The partides are thus so close together that it is unrealistic to assume a Maxwellian velocity distribution whose extreme tails could give an explanation for the supposed existence of thermal fusion with deuterons, whose extreme energies would be in the so-called thermonuclear fusion range, i.e. typically superior to 10 keY. In fact it is more realistic to consider such dense media as non-ideal plasmas [2]. Many papers have recently been devoted to the screening problem between two neighbouring charged particles in dense media. Most ofthem do not propose numerical cornparisons with experiments, with the notable exception of a precise and complete calculation by Fedorovich, which investigates the screening of the electrical field of a positive charge located in a gas
ofa limited number offree electrons with low energy [3]. Despite the speculative appearance of this calculation, its results are in good agreement with experimental data, as shown in this Letter. Let us now assume that (1) the conducting medium is fully ionized, i.e. made up of two mixed gases, one an ion gas, the other an electron gas; (2) as a consequence of the great electron mobility in comparison with that of the ions, one can assume that there are two spatial distributions, one for the ions, the other for the electrons; (3) the ion spatial distribution is governed by a Poisson process [4]. This means practically that if in a specific volume one finds an average ion nurnber ,u, the probability F,, to find p ions in this volume is PP =(/p’)e~
(1)
(4) the specific dimension of the volume which contains one ion on average is typically ofthe order of 10~cm (4.64 x 10—8 cm if one assumes 1022 par~~ tides per cm3). Two deuterons will be a candidate for Coulomb barrier penetration if they are close enough together. In the space between positive charges, the mean value ofthe electric field potential is zero, given that positive and negative charges can be assumed to be equally distributed. In fact if two charges are suffi-
0375-9601 1921$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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ciently close together, this potential will grow locally, and if one takes into account the great electron mobility in comparison with the positive charge mobility, many electrons will be collected around the two positive charges and will form a cloud around them. It seems thus likely that all electron charges are collected, at a definite time, in the environment oftwo positive charges approaching each other. This situation changes rapidly after the two positive charges have interacted (nuclear reaction or simple diffusion). In fact all electrons are shared by those positive charge concentration points. The average number v of the electrons around a two-deuteron concentration (v being also the number of single positive charges in the environment) can be estimated in a simple way. The probability of a two-deuteron concentration at a close distance P2 is given by the Poisson law P2= ~p2e~,p being the mean deuteron number in the volume V. The rate of the two ions meeting in the volume V is thus equal to the ratio P2/P e~.If all electrons are involved in such a process, v is equal to the inverse of this ratio, so that v=2/u. With a volume whose typical dimension is 4.64 x 10—p cm, for example, one obtains ~U= 10—i and v = 2 x 1 Ø3~ The electron number in the cloud around the two deuterons can thus be rather high. The electron concentration around the two deuterons is ofcourse not instantaneous and their effective concentration in the cloud depends in fact on their velocities and on the number of collisions they have to perform before reaching the two deuterons. One can have a more precise idea of the necessary distance for the barrier penetration if one takes into account the results of Fedorovich [3] for an electron gas surrounding one positive charge. The resulting dimensionless electric field potential U(x) = rØ ( r) / e is given as a function of the dimensionless variable x= 2me2r/h2, r being the distance from the ion (fig. 1). This distance is measured with the unit /12/ 2me2 = 2.64 x 1 Q9 cm (i.e. half of what is conventionally called the Bohr radius), m being the electron mass. The parameter which characterizes the cloud is the electron density ô= v(/12/2me2)3. It represents the mean number of electrons which are inside 2
2
3
the elementary volume (/1 /2me ) v being the electron number around the two positive charges. For example the case ô= 0.025 describes approximately the presence of v = 26 electrons in the proximity of ,
156
U, W
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10
10.0
7.90
6.60
3.40
I /~“I 1.20
“~
Ij~~’ 5)’.
100
(1) 0.00
~
‘~ -
(~) 16.0
-
(3)
-
(4) 32.0
48.0
64.0
*
Fig. I. From ref. [3]. The distribution of the electrical dimensionless potential (curves (1), (2), (4) in solid lines) and the density of electrons (curves (5), (6), (8) in dotted lines) near the positive charge correspond, respectively, to three values of the dimensionless electron density (ct= 25, 1, 0.025). The conesponding total electron number around the positive charge is approximately 3 x 102, 90, 26 (in the same order).
one positive charge. This case thus describes two positive charges approaching each other, the great electron mobility having depopulated the nearby positive charges v by their associated electron. For a barrier penetration, the positive charge must be in the well part of the electric field potential pattern, i.e. approximately in a volume whose typical dimension is between 1 and 2.5 Bohr radius, according to Fedorovich’s results [31. Given the relative regularity ofthe positive-charge spacing, it is useful to use a naive picture which assumes that the two positive charges are at the center of an elementary cube surrounded by layers ofidentical cubes (fig. 2). One single positive charge is supposed to be in the middle of each of those cubes without any electron. The elementary cube number c for layer 1 is given by 2
c=81(21+l)+2(21—l) (2) This distribution of single positive charges implies a change of the electric field potential equal to the sum of the Coulomb potentials of all the single charges .
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According Vr=f(e/r-V~.,). to the description made byFedorovich (4) [3] it is straightforward to see how much this pedestal effect reduces the electric potential in comparison
I
- -
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-
with tential thepattern electron cloud approximately effect near thein nucleus anofinversely of the positive charge.varies The decrease of the zero the po-
Fig. 2. Sketch of the space supposed divided into elementary cubes. All those cubes contain one positive charge, except the central cube which is supposed to contain two positive charges. The side ofthe volume is thus 21+1,/being the row of the layer, and the number of cubes in the layer / being 8/(2/+ 1) + 21(21_1)2. The sketch corresponds to 1= 1.
contained in the layers surrounding the fusion point. To calculate the order of the range of this effective potential, one can consider an isolated unit constituted by the central cube containing the two meeting charges, and all the layers of positive charges which have contributed to make the electrons cloud around the two positive charges. The effect of other neighbouring units constituted by other layers of single positive charges surrounding another two meeting positive charges can be neglected, given the fact that one can consider their whole charge as being concentrated in their center and given the global neutrality of those neighbouring units. One deduces therefrom the expression for the electric field potential V~due to n single positive-charge layers, eln\/~ “C
V~=—~>~,
proportional way to log v both for the zero due to the pedestal and for the zero due to electron clustering around the two positive charges as is shown in fig. 3. The curve due to the pedestal is obtained from expression (3) and for the one due to electron clustering one has only three points from ref. [3] which are nevertheless sufficient to appreciate the relative efficiency of those two ways of screening. In fact the pedestal’s efficiency for reducing the electric barrier
7
6
5
(3)
s being the mean spacing between positive charges and the numerical factor expressing the mean value ofthe distance on a layer, which is bounded by si and s1.,,/~.The new level of reference is thus Vi,, instead of practically zero in an ideal layout where the positive and negative charges would be distributed regularly. The potential V~,will be conventionally called the pedestal. The electric potential of a specific positive charge in front of another positive charge is no longer Coulombic, but is the difference between the Coulomb potential and the pedestal V~.This difference multiplied by a factor f depending on the electron number v in the cloud gives the resulting potential
2
1
102
V
Fig. 3. Positions, relative to the deuteron center, of the zeros of the barrier, pedestal. (1) and (2) due to electron (3) potential the on potential given inclouding, units of and hthe 2/ 2,toand the abscissa is theLengths numberare v ofelectrons around 2me two meeting deuterons.
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is greater than the one of the electron clustering around the nucleus. For example, a pedestal V~,which would be due to approximately three layers of single positive charges would give a null resulting potential at a distance 8.69x 102 (in units of/12/2me2), i.e. approximately the twentieth part ofthe Bohr radius, The effect of clustering, as described by Fedorovich, yields a null value at approximately 0.3 Bohr radius. If one combines the two effects, one realizes that the most important effect is due to the pedestal, i.e. to the electron depopulation in the environment of the two deuterons. This result is similar to the proposal made by many authors, but in a phenomenological way. For example, Rabinowitz [5] proposes to describe the potential energy V between two deuterons by an effective expression as a function of the radius r outside the nuclear well, V=(e2/4x~ 0)(l/r—l/R), r1~r~R, (5) where e is the deuteron charge, r 1 is the nuclear-well radius. He also assumes that the potential energy is V= 0 for r> R. Here the pedestal effect is not taken into account despite a hint (in the same reference [5]) to the combined screening by ions and electrons which is claimed to be more efficient according to Takahashi’s calculations [61. The term 1 /R in that paper, like in other papers, is generally attributed to electron clouding near the nuclear well [~L Since its energy is very low, the behaviour of the incident deuteron is supposedly given by the timedependent Schrodinger equation. Using this equation in polar coordinates is compulsory if one wants to determine the partial amplitudes with non-zero orbital momentum L, M being the deuteron mass: 2 L2 / /12 ~j—~ d + + V(r))!P(r, 6,0, t) ~
=
~
EW( r, 9, 0, t) .
(6)
As emphasized by Fedorovich for the electron cloud [3], and following a general remark which also holds for the deuteron, those partial amplitudes are small if their wavelength is large compared with the action radius of the field. One can thus drop the L term. The Schrodingerequation becomes one-dimensional and can be written as 158
/1 2 d2
(_ ~
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+ V(r)) W(r, t) =EW(r, t),
(7)
E being the energy of the stationary state. This applies to states whose energy is definite, and the following changes of variables, W(r, t) = W(r) exp[ —i(E/h)t] V(r) = (h2/2M) U(r), E= (h2/2M)e, yields the simple second-order equation ~“+ [~—U(r)]!P=0.
(8)
For a constant value of U( r), it is well known that this equation implies the following solutions, if ~> U, P=A exp (ikr) + B exp ( ikr) —
(with k=(e—UY’2)) if(.czU, ~P=Aexp(hr)+Bexp(—hr), 1”2)). (with h=(U—e) The incident particle coming from positive r has the following wave function (k being a positive number), Y~exp{—i[kr+ (E/h)tJ} and the reflected wave is
,
(9)
P= exp{i [kr— (El/I )t] }. (10) In the interior of the barrier the forward and backward movement are described respectively by (h is a positive number) Wexp{— [hr+ (E/h)t]} and
(11)
W=exp[hr— (E/h)t] (12) The method used to calculate the values of h (or k) as a function of r is well known. It is a discrete approximation to the exact solution ofthe Schrodinger equation. It consists in splitting the potential barrier in a finite number of intervals s (in each of those intervals the potential is supposed constant and equal to the mean value of eVr) and expressing the continuity of Wand of its first derivative at each interface; this gives two equations for each interface. One thus obtains a linear system of equations with an number of unknowns equal to two times the interface number. The physical parameter of interest is F, .
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the transmission barrier factor. It is equal to the square of the ratio of the number of unknowns multiplied with the wave function penetrating the nuclear well to the wave function impinging upon the barrier. In fact the particle is represented by a wave packet at the entrance of the barrier, W(r, t)=
J
f(k’) exp{—i[k’r+(E’//I)t]} dk’
-~
,
(13)
and the duration of the whole barrier penetrationdepends on the phase difference between the phase of the entering wave and the phase of the wave penetrating the nuclear well. The phase 0 at the entrance of the nuclear well, (14) obeys the dependence condition dØ/dk= 0 on the wave vector whose modulus is k at the apex of the wave packet. It describes the movement of the wave packet’s center, r=(1/fz)(dE/dk)t+da/dk,
(15)
(1/h) (dE/dk) being the group velocity ofthe wave. The time lag 6 which is necessary for a positive charged particle to penetrate the screened barrier is 6=/I da/dE’.
(16)
Since the barrier penetration time is useful for the calculation of the nuclear reaction rate, as explained below, it is important to keep in mind all elements which are necessary for this task. According to the given elements the phenomenon depends on the macroscopic level on parameters which are different from the ones which are necessary for describing the thermonuclear concept. In the case of the thermonuclear concept the fusion reaction rate R per volume unit and per time unit is 2, (17) R=~n n being the number of particles which can be involved in a nuclear reaction per volume unit, a is the nuclear cross section and v is the deuteron mean velocity. One has to take into account a greater number of parameters than in the usual thermonuclear hypothesis. The extra parameters must express the fact that the barrier penetration is both a quantum and an
13 Api’il 1992
electrodynamic phenomenon. With the aim to derive a formula equivalent to (17), it is worthwhile to use a specific method which is an improvement of the known method of Yaschy [8]. This method, proposed in 1959 by Saint-Guilhem [9], has been deduced from group theory. It is based on a theorem stating that any physical relationship using a continuous set of system units, containing a finite number p of real variables x, can be written by means of m
variables V~,so that the bamer performances are only represented by F, i.e. the electrodynamic aspect is thus only contained in F. The inventory of the parameters which are necessary for describing the behaviour of the system is thus made of the rate R, the nuclear cross section a, the Planck constant /1, the whole barrier crossing duration 0, the barrier length L, the transmission barrier factor F, the number of particles n which can be involved in a nuclear reaction, the time t, and the deuteron energy E. The macroscopical process thus depends on p= 9 physical variables. Since those physical variables are of different species, the structural relationship depends on 9 physical species. One can list 6 independent equations describing the structure as a function of the non-dimensional multiplying factors describing a unit change. Those factors are represented by the same letters as the physical variables, but are here all upper case letters: E, L, R, T, N, H, E, 0. Those 6 independent equations are 1L2, HML2T’, R_L3T~ N=L3, E=MLT2, 0=HE’.
(19)
As the number of the physical species is larger than the number of equations, the dimensional analysis is applicable and one can, according to ref. [9], look 159
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for the p’ physical variables which can be present in the final expression. In fact it is easy to see that for 2EL if one = 5 one gets only one monomial RT/N takes into account only the dimensional parameters E, L, R, 0, N. The relationship which describes the macroscopic phenomenon thus has the following shape, using again the physical parameters,
production term” or, more shortly, the “production term” T,
g(F, R0/n2aL)=0.
entrance of the nuclear well. The best way is to rely on the results of high energy electron scattering on deuterons. Those experiments have been performed using 188 and 400 MeY electrons. It was shown that it was possible to fit the electron diffusion data with the extension range of the deuteron. It was estimated by McIntyre [10] that the best fit for the charge ex-
(20)
Dimensional analysis cannot tell more, but one can infer physically that the nuclear reaction rate R must be proportional to the barrier transmission factor F, and the relationship becomes (the usual ~ factor has been included) R=~n2aFL/0,
(21)
a formula which replaces in any fusion barrier process formula (17) which has been used largely in the past for thermonuclear processes. In fact the parameters which have disappeared (1, M, E, /1) are not useful directly for the result, but they are included implicitly in R for the time t, and in 0 for E and /1 and M. Nevertheless, according to the method of ref. [9] it was necessary to list them, even if in a natural way the process discards them from the result. But it is, however, important to emphasize the fact that the formulation is non-relativistic, and that the whole process would have to be reexamined in the case where the particle velocities would become relativistic, In other words, a more complete expression than (17) does exist, if one takes into account the relativistic behaviour of the particles in the therrnonuclear range ~‘. The application has been made here with a cornputer whose performances were limited to handling only a 10 x 10 complex matrix, which corresponds to a division ofthe barrier into four slices. Nevertheless it has been verified that the results obtained do not change drastically by using a greater number of matrix dimensions: the result of this relatively small number of matrix dimensions is only a lack of precision. In fact whatfor is R important to calculate is a(21), part of the expression according to formula this is a term which one can call “nuclear reaction ~ It seems that the expression for R (21) still applies in the relativistic case, but the transmission factor F will also depend at least on the velocity of light.
160
T= aFL/0. (22) To do this, one has to choose the proper slicing of the potential barrier and, before slicing the barrier, the abscissa of the first interface corresponding to the
tension of the deuteron corresponds with the assumption ofthe charge independence of the internal structure of nucleons. In other terms the effective radius of the deuteronwas estimated to be 1.70 X 10— i3 cm, with the assumption of a modification of the Coulomb law at small distances. So in our model the potential barrier, keeping a finite value, would extend outside the nuclear well from the effective radius of the deuteron nucleus to distances depending on the cumulative effect of the potential pedestal change and of electron clouding. To take into account those two effects it seems useful to combine them in the same semi-phenornenological formula. For this, the curve patterns of ref. [3] are represented by the product ofan exponential and a second order polynomial, the potential pedestal represented by V 0 is subtracted from the Coulomb potential, so the energy potential expression is U(r)=e(e/r— V0)(r—a)(r—b) exp(—Kr). (23) This choice for the nuclear-well radius implies also in fact the choice of a nuclear cross section. Inasmuch as the potential is crossed therebutis the no other uncertainty aboutbarrier the nuclear collision geometrical factor, depending only on the charge extension of the deuteron With isthis assumption one 2, #2which exactly the value gets a= 9 x 1026 cm 82
Except for a branching ratio equal to 0.5 of the two possible
reaction channels in the case of the D—D reaction. But there was no use to take into account the branching ratio in this paper, given the lack of precision due to the limited matrix dimensions.
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quoted by Lawson [11]. For the values of the electron number v in the cloud, it was necessary to extrapolate the results of Fedorovich [3] to higher values. Given that the effect of electron clouding is less stringent than the one due to the pedestal, an estimate of the electron clouding effect is possible in a rather large screening range without changing drastically the result, except for a higher number of electrons in the cloud. In fact the terms a, b, C, k were approximated by the following phenomenological formulas which are suited to the parameters a and b decreasing in an inversely proportional way with log v (fig. 3),
13 April 1992
3/s
o’FL m
e
.1 ~~30
~
10
h~ I
,‘
/
/ ,‘ ,‘
,‘
,//
a= 1.1 1/log v, b=3.48/log v, (24)
I
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/‘
//
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,///
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(4)
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Calculations have shown that for a specific value of the electron number v in the cloud (or equivalently for the same value of single positive charges in the environment of the two deuterons approaching each others) the values ofthe production term T, according to (22), are lying along a straight line in the diagram of log T as a function ofthe logarithm of the deuteron energy, logE (fig. 4). The experimental points (Z-pinch experiments, capillary fusion and cluster fusion) are the same as in ref. [1]. The Brookhaven point corresponds typically to a cloud of 3.3 x 1 0~electrons. The NRL 1 —CEA point seems a little apart and apparently corresponds to more than 5 x 1 0~electrons in the cloud. Of course the precision of the calculation would improve if one had used complex matrices with dimensions larger than 10 x 10. It has been shown however that between the use of one slice (the whole barrier) and the use of a 10 x 10 matrix, the change in the result is not more than some powers of ten for a specific case around T= l0~°rn3/s (fig. 5). The value grows from a 1 x 1 matrix to a 10 x 10 matrix and it seems possible to extrapolate a more precise T value around 1 0~above the values given by the 10 x 10 matrices. On the logarithmic diagram this is not a large change. In fact it will not change drastically the results of fig. 4. The cluster fusion point corresponds to a single ion number v in the environment which would be closer to .3 x 1 0~rather than above 3.3 x 1 0~,as was first proposed, using the raw calculation results. One has to remark that this value is not far from 2 x 1 0~ob-
/
,‘
I,
k=l.5/a2b.
I
,‘
KIEL BONPAS et Al.
e x
NRL1
+
BADUREK et AL.
~
NRL 2
BROOKHAVEN
(1) i~-~° E (eV)
Fig. 4. Fusion production factor (in rn3/s) as a function of the energy of the deuteron (eV) m a logarithmic scale. The solid lines are the raw results of the calculation and the dotted lines are deduced from them by a correction due to the insufficient number ofmatrix dimensions (10 x 10). The experimental points have the same meaning as in ref. [I]. The number of electrons per cloud is (1) 342, (2) 1350, (3) 3374, (4) 5 x104. The curve D— D corresponds to the thermonuclear hypothesis.
tamed from pure stochastic considerations. Taking into account this rectification, due to the lack ofmatrix dimensions we obtain the dashed straight lines which are above the solid straight lines in fig. 4. This way of reasoning shows that the calculated values of the production term T are in the range of the ones which one infers from experimental results. It gives an estimate ofthe z value, so that our model gives a possible explanation for the so-called anomalous cross sections observed by Beuhler et al. [12]. One can thus assert that in the various kinds of experiments the cause of the observed fusion reaction is essentially the same: i.e. the simultaneous screening effects of the electron clouding around two approaching deuterons and the corresponding change 161
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PHYSICS LETFERS A
4) are just above thethat Brookhaven point (cluster fusion), which means the number of electrons in
(m /s)
i0~0
—-—
,..—- - —
-
-~
I
1
2
3
4
Number of sLices
Fig. 5. Precision test of the method. It shows that for four slices in the barrier which it is possible to handle with a lOx 10 matrix the fusion reaction production term T is still not close to the asymptotic value. The gap can be estimated to a factor 1 O~,which represents the distance, in loganthmic scale, between the dashed straight lines and the solid straight lines in fig. 4. On the X-axis is the number of slices and on the Y-axis is the fusion reaction production term.
of the electric potential pedestal in the environment of those two deuterons. But one has to take into account the presence of the non-fusible positive charges in the medium. Since the ionization energies of simple bodies like oxygen or nitrogen are lower than the one of deuterium, they could provide an extra number of electrons in the cloud and also contribute to the pedestal change in an efficient way. Their effect could be particularly marked in cluster fusion experiments [12]. In fact one can assume that in those experiments, the heavy water clusters impinging at very high velocity onto the target are fully ionized. The corresponding electron cloud phenomenon thus takes place in a limited medium. Deuteron meetings occur in a limited space, according to the statistical Poisson law, but a minimum number of D2O molecules is necessary to have a sufficiently efficient joint effect of electron cloud and change of pedestal. Beuhler et al. show in their paper [12] that the fusion reactions depends on the number of D2O in the impinging cluster, the maximum value being close to some hundreds. Our rectified computed values (dashed straight line 2 in fig. 162
13 April 1992
the cloud around the approaching deuterons was of the order of l0~,which is not contradictory with fig. 3 of ref. [12] reproduced in fig. 6 of this paper. If one assumes a full ionization of the cluster, and for less than approximately 500—600 molecules (producing typically l0~to l.5x l0~electrons by postimpact ionization), this means that one sole deuteron meeting point would appear statistically in this cluster. For a cluster containing more molecules, two deuteron meeting centers would appear, but with an insufficient number of available electrons and ions for creating a sufficient screening effect in the two deuteron meeting centers. Presumably one can recover fusion reaction production with clusters ofbigger sizes. To conclude this Letter one can make the following remarks: (1) This study is a result of the effects of higher electron densities than the ones given by Fedorovich [3]. It would thus be important in a new study to start calculating those high electron density screening effects and thus also compulsory to handle higher dimension matrices. (2) The slope of the straight lines in a logarithmic 10
8
8 ~ 6
~
-~
I
f 2
~
__________________________________ 20 60 Number of ~o te~°° 2000
10
Fig. 6. From ref. [121. It depicts the dependence of proton yield (which is proportional to the fusion production rate) on cluster size at a 300 kV accelerating potential. The ordinate scale is proton counts per 1000 s per nA ofcluster current. The pattern shows how the number of D20 per cluster becomes insufficient to sustam the fusion reactions by a crossing barrier process for two deuteron meeting points, beyond approximately one thousand D20.
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diagram, with a constant number of clouding electrons, is between 1.5 and 2, according to the root value. Those values are not far from the slopes which are conjectured in ref. [1], from experimental resuits, the production term being approximately proportional to a current law between j6 and V6 ~ It is interesting to underline that the model discussed here is in rather good agreement with the Z-pinch experiments, (3) The fusion time in the Z-pinch and capillary fusion is reached only when the current has largely begun to decrease, like in the Lochte-Holtgreven experiments [15], orjust at the apex ofthe current [4]. According to the model, the ionized deuterons must have a translational motion. This motion is given by the electrodynamic forces. They are lying in the same direction during the growing phase of the current; but as soon as the current begins to decrease, this decreasing evidently initiates a chaotic motion. It seems possible to calculate the behaviour of the system by assimilating the ions to harmonic oscillators submitted to a variable strength as a function of time according to the modern conceptsof chaos [16]. This chaos is thus the origin of collisions. It is easy to develop the classical macroscopic analogy of a train whose locomotive is slowing down, also producing a chaos-like behaviour. The less the maximum of the strength, the less the chaotic effects and the more they are delayed. (4) Fusion is possible by the process proposed here, not only for very light nuclei like deuterons and tritons, but also for many heavier nuclei, like the ones considered in astrophysics (for example 12C, ‘‘B, 7Li). (5) One can infer from those results that we now have both a theoncal and an experimental confirmation that an electric current passing through a dense non-ideal plasma [2] is not a simple process. The distribution of the electrons and ions is evidently submitted to a variable chaotic motion as has been noticed in a recent paper [17]. In fact on a ~ Any experimental result which would give an exponent less than 4 seems to be not reliable in any hypothesis, and due to an experimental artefact.
13 April 1992
macroscopic level, one observes only an average effect, even if locally there are electron clouds and potential pedestal changing effects, which involve thousands of particles and more. (6) This study has been limited to the non-relativisti~case, but one can infer from the proposed model that it is no longer possible to ignore that the conventionally called thermonuclear plasmas are in fact and above all relativistic plasmas. In such a plasma relativistic electrons play a role which one has to understand better to predict its evolution. This necessity is particularly important for the knowledge of star evolution. I am very grateful to J.P. Vigier for many discussions and exchanges during the elaboration of the model and also to Professors H. Rauch and P. Graneau for providing information on very recent progress in the field of Coulomb screening and fusion processes.
References [1] M. Rambaut, Phys. Lett. A 163 (1992) 335. [2] V.E. Fortov and I.T. Iabukov, Physics of nonideal plasma (Hemisphere, Washington, DC 1990).
[31G.V. Fedorovich, Phys. Lett. A 164
(1992)149. [4] J.L. Doob, Stochastic processes (Wiley, New York, 1953) p. 404. [5] M. Rabinowitz, Mod. Phys. Lett. B 4 (1990) 665. [6] H. Takahashi, Phys. Rev. 172 (1968) 747. [7] A.K. Bahtia et al., Phys. Rev. A 38 (1988) 340O~ L. Chatteijee, Phys. Lett. A 137 (1989) 4; Nature 342 (1989) 232. [8] Vaschy, Ann. Télégraphiques (1892) 25, 189. [9] R. Saint-Guilhem, Rev. Gen. Electr. (1959) 533. [lO]J.A. McIntyre, Phys. Rev. 103 (1956) 1464. [11] J.D. Lawson, Proc. Phys. Soc. London B 70 (1957) 6.
[121R.J. Beuhler,
G. Friedlander and L. Friedman, Phys. Rev. Lett. 63 (1989) 1292. [13] J.D. Sethian et al., Phys. Rev. Lett. 59 (1987) 892. [14] J.D. Sethian, Evolution ofdeuterium fiber Z-pinch driven by aLochte-Holtgreven,Atomkernenergie long current pulse, private communication. [15] W. 28 (1976) 150. [161 M.C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, Berlin, 1990). [17] M. Rambaut, Phys. Lett. A 154 (1991) 210.
163
PHYSICS LETTERS A
Double screened Coulomb barrier accounts for neutrons production in cluster and other fusion experiments M. Rambaut 57H rue De La Hacquiniere, 91440 Bures-sur- Yvette, France Received 12 September 1991; revised manuscript received 22 january 1992; accepted for publication 23 January 1992 Communicated by J.P. Vigier
In a dense, fully ionized medium, containing fusible nuclei, a collision between two nuclei is accompanied by an electron concentration around them. By this, rate oftunneling is tremendously increased. The experimental results are in agreement with the calculations, the number of displaced electrons being typically in the range ofone to two thousand.
In a recent paper of the present author [1], which summarized a model developed with Vigier, it has been shown that a possible correlation exists between three kinds of observed fusion experiments, the so-called cold fusion experiments, the capillary fusion experiments and the cluster fusion experiments. This correlation was interpreted in terms of turbulence caused by longitudinal Ampere forces combined with Coulomb screening and quantum tunneling processes. The aim of this paper is to present a more detailed numerical computational justification of this model. In the majority of the three kinds of fusion experiments one is working with dense media. The partides are thus so close together that it is unrealistic to assume a Maxwellian velocity distribution whose extreme tails could give an explanation for the supposed existence of thermal fusion with deuterons, whose extreme energies would be in the so-called thermonuclear fusion range, i.e. typically superior to 10 keY. In fact it is more realistic to consider such dense media as non-ideal plasmas [2]. Many papers have recently been devoted to the screening problem between two neighbouring charged particles in dense media. Most ofthem do not propose numerical cornparisons with experiments, with the notable exception of a precise and complete calculation by Fedorovich, which investigates the screening of the electrical field of a positive charge located in a gas
ofa limited number offree electrons with low energy [3]. Despite the speculative appearance of this calculation, its results are in good agreement with experimental data, as shown in this Letter. Let us now assume that (1) the conducting medium is fully ionized, i.e. made up of two mixed gases, one an ion gas, the other an electron gas; (2) as a consequence of the great electron mobility in comparison with that of the ions, one can assume that there are two spatial distributions, one for the ions, the other for the electrons; (3) the ion spatial distribution is governed by a Poisson process [4]. This means practically that if in a specific volume one finds an average ion nurnber ,u, the probability F,, to find p ions in this volume is PP =(/p’)e~
(1)
(4) the specific dimension of the volume which contains one ion on average is typically ofthe order of 10~cm (4.64 x 10—8 cm if one assumes 1022 par~~ tides per cm3). Two deuterons will be a candidate for Coulomb barrier penetration if they are close enough together. In the space between positive charges, the mean value ofthe electric field potential is zero, given that positive and negative charges can be assumed to be equally distributed. In fact if two charges are suffi-
0375-9601 1921$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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ciently close together, this potential will grow locally, and if one takes into account the great electron mobility in comparison with the positive charge mobility, many electrons will be collected around the two positive charges and will form a cloud around them. It seems thus likely that all electron charges are collected, at a definite time, in the environment oftwo positive charges approaching each other. This situation changes rapidly after the two positive charges have interacted (nuclear reaction or simple diffusion). In fact all electrons are shared by those positive charge concentration points. The average number v of the electrons around a two-deuteron concentration (v being also the number of single positive charges in the environment) can be estimated in a simple way. The probability of a two-deuteron concentration at a close distance P2 is given by the Poisson law P2= ~p2e~,p being the mean deuteron number in the volume V. The rate of the two ions meeting in the volume V is thus equal to the ratio P2/P e~.If all electrons are involved in such a process, v is equal to the inverse of this ratio, so that v=2/u. With a volume whose typical dimension is 4.64 x 10—p cm, for example, one obtains ~U= 10—i and v = 2 x 1 Ø3~ The electron number in the cloud around the two deuterons can thus be rather high. The electron concentration around the two deuterons is ofcourse not instantaneous and their effective concentration in the cloud depends in fact on their velocities and on the number of collisions they have to perform before reaching the two deuterons. One can have a more precise idea of the necessary distance for the barrier penetration if one takes into account the results of Fedorovich [3] for an electron gas surrounding one positive charge. The resulting dimensionless electric field potential U(x) = rØ ( r) / e is given as a function of the dimensionless variable x= 2me2r/h2, r being the distance from the ion (fig. 1). This distance is measured with the unit /12/ 2me2 = 2.64 x 1 Q9 cm (i.e. half of what is conventionally called the Bohr radius), m being the electron mass. The parameter which characterizes the cloud is the electron density ô= v(/12/2me2)3. It represents the mean number of electrons which are inside 2
2
3
the elementary volume (/1 /2me ) v being the electron number around the two positive charges. For example the case ô= 0.025 describes approximately the presence of v = 26 electrons in the proximity of ,
156
U, W
13 April 1992 *
10
10.0
7.90
6.60
3.40
I /~“I 1.20
“~
Ij~~’ 5)’.
100
(1) 0.00
~
‘~ -
(~) 16.0
-
(3)
-
(4) 32.0
48.0
64.0
*
Fig. I. From ref. [3]. The distribution of the electrical dimensionless potential (curves (1), (2), (4) in solid lines) and the density of electrons (curves (5), (6), (8) in dotted lines) near the positive charge correspond, respectively, to three values of the dimensionless electron density (ct= 25, 1, 0.025). The conesponding total electron number around the positive charge is approximately 3 x 102, 90, 26 (in the same order).
one positive charge. This case thus describes two positive charges approaching each other, the great electron mobility having depopulated the nearby positive charges v by their associated electron. For a barrier penetration, the positive charge must be in the well part of the electric field potential pattern, i.e. approximately in a volume whose typical dimension is between 1 and 2.5 Bohr radius, according to Fedorovich’s results [31. Given the relative regularity ofthe positive-charge spacing, it is useful to use a naive picture which assumes that the two positive charges are at the center of an elementary cube surrounded by layers ofidentical cubes (fig. 2). One single positive charge is supposed to be in the middle of each of those cubes without any electron. The elementary cube number c for layer 1 is given by 2
c=81(21+l)+2(21—l) (2) This distribution of single positive charges implies a change of the electric field potential equal to the sum of the Coulomb potentials of all the single charges .
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According Vr=f(e/r-V~.,). to the description made byFedorovich (4) [3] it is straightforward to see how much this pedestal effect reduces the electric potential in comparison
I
- -
13 April 1992
-
with tential thepattern electron cloud approximately effect near thein nucleus anofinversely of the positive charge.varies The decrease of the zero the po-
Fig. 2. Sketch of the space supposed divided into elementary cubes. All those cubes contain one positive charge, except the central cube which is supposed to contain two positive charges. The side ofthe volume is thus 21+1,/being the row of the layer, and the number of cubes in the layer / being 8/(2/+ 1) + 21(21_1)2. The sketch corresponds to 1= 1.
contained in the layers surrounding the fusion point. To calculate the order of the range of this effective potential, one can consider an isolated unit constituted by the central cube containing the two meeting charges, and all the layers of positive charges which have contributed to make the electrons cloud around the two positive charges. The effect of other neighbouring units constituted by other layers of single positive charges surrounding another two meeting positive charges can be neglected, given the fact that one can consider their whole charge as being concentrated in their center and given the global neutrality of those neighbouring units. One deduces therefrom the expression for the electric field potential V~due to n single positive-charge layers, eln\/~ “C
V~=—~>~,
proportional way to log v both for the zero due to the pedestal and for the zero due to electron clustering around the two positive charges as is shown in fig. 3. The curve due to the pedestal is obtained from expression (3) and for the one due to electron clustering one has only three points from ref. [3] which are nevertheless sufficient to appreciate the relative efficiency of those two ways of screening. In fact the pedestal’s efficiency for reducing the electric barrier
7
6
5
(3)
s being the mean spacing between positive charges and the numerical factor expressing the mean value ofthe distance on a layer, which is bounded by si and s1.,,/~.The new level of reference is thus Vi,, instead of practically zero in an ideal layout where the positive and negative charges would be distributed regularly. The potential V~,will be conventionally called the pedestal. The electric potential of a specific positive charge in front of another positive charge is no longer Coulombic, but is the difference between the Coulomb potential and the pedestal V~.This difference multiplied by a factor f depending on the electron number v in the cloud gives the resulting potential
2
1
102
V
Fig. 3. Positions, relative to the deuteron center, of the zeros of the barrier, pedestal. (1) and (2) due to electron (3) potential the on potential given inclouding, units of and hthe 2/ 2,toand the abscissa is theLengths numberare v ofelectrons around 2me two meeting deuterons.
157
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is greater than the one of the electron clustering around the nucleus. For example, a pedestal V~,which would be due to approximately three layers of single positive charges would give a null resulting potential at a distance 8.69x 102 (in units of/12/2me2), i.e. approximately the twentieth part ofthe Bohr radius, The effect of clustering, as described by Fedorovich, yields a null value at approximately 0.3 Bohr radius. If one combines the two effects, one realizes that the most important effect is due to the pedestal, i.e. to the electron depopulation in the environment of the two deuterons. This result is similar to the proposal made by many authors, but in a phenomenological way. For example, Rabinowitz [5] proposes to describe the potential energy V between two deuterons by an effective expression as a function of the radius r outside the nuclear well, V=(e2/4x~ 0)(l/r—l/R), r1~r~R, (5) where e is the deuteron charge, r 1 is the nuclear-well radius. He also assumes that the potential energy is V= 0 for r> R. Here the pedestal effect is not taken into account despite a hint (in the same reference [5]) to the combined screening by ions and electrons which is claimed to be more efficient according to Takahashi’s calculations [61. The term 1 /R in that paper, like in other papers, is generally attributed to electron clouding near the nuclear well [~L Since its energy is very low, the behaviour of the incident deuteron is supposedly given by the timedependent Schrodinger equation. Using this equation in polar coordinates is compulsory if one wants to determine the partial amplitudes with non-zero orbital momentum L, M being the deuteron mass: 2 L2 / /12 ~j—~ d + + V(r))!P(r, 6,0, t) ~
=
~
EW( r, 9, 0, t) .
(6)
As emphasized by Fedorovich for the electron cloud [3], and following a general remark which also holds for the deuteron, those partial amplitudes are small if their wavelength is large compared with the action radius of the field. One can thus drop the L term. The Schrodingerequation becomes one-dimensional and can be written as 158
/1 2 d2
(_ ~
13 April 1992
+ V(r)) W(r, t) =EW(r, t),
(7)
E being the energy of the stationary state. This applies to states whose energy is definite, and the following changes of variables, W(r, t) = W(r) exp[ —i(E/h)t] V(r) = (h2/2M) U(r), E= (h2/2M)e, yields the simple second-order equation ~“+ [~—U(r)]!P=0.
(8)
For a constant value of U( r), it is well known that this equation implies the following solutions, if ~> U, P=A exp (ikr) + B exp ( ikr) —
(with k=(e—UY’2)) if(.czU, ~P=Aexp(hr)+Bexp(—hr), 1”2)). (with h=(U—e) The incident particle coming from positive r has the following wave function (k being a positive number), Y~exp{—i[kr+ (E/h)tJ} and the reflected wave is
,
(9)
P= exp{i [kr— (El/I )t] }. (10) In the interior of the barrier the forward and backward movement are described respectively by (h is a positive number) Wexp{— [hr+ (E/h)t]} and
(11)
W=exp[hr— (E/h)t] (12) The method used to calculate the values of h (or k) as a function of r is well known. It is a discrete approximation to the exact solution ofthe Schrodinger equation. It consists in splitting the potential barrier in a finite number of intervals s (in each of those intervals the potential is supposed constant and equal to the mean value of eVr) and expressing the continuity of Wand of its first derivative at each interface; this gives two equations for each interface. One thus obtains a linear system of equations with an number of unknowns equal to two times the interface number. The physical parameter of interest is F, .
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the transmission barrier factor. It is equal to the square of the ratio of the number of unknowns multiplied with the wave function penetrating the nuclear well to the wave function impinging upon the barrier. In fact the particle is represented by a wave packet at the entrance of the barrier, W(r, t)=
J
f(k’) exp{—i[k’r+(E’//I)t]} dk’
-~
,
(13)
and the duration of the whole barrier penetrationdepends on the phase difference between the phase of the entering wave and the phase of the wave penetrating the nuclear well. The phase 0 at the entrance of the nuclear well, (14) obeys the dependence condition dØ/dk= 0 on the wave vector whose modulus is k at the apex of the wave packet. It describes the movement of the wave packet’s center, r=(1/fz)(dE/dk)t+da/dk,
(15)
(1/h) (dE/dk) being the group velocity ofthe wave. The time lag 6 which is necessary for a positive charged particle to penetrate the screened barrier is 6=/I da/dE’.
(16)
Since the barrier penetration time is useful for the calculation of the nuclear reaction rate, as explained below, it is important to keep in mind all elements which are necessary for this task. According to the given elements the phenomenon depends on the macroscopic level on parameters which are different from the ones which are necessary for describing the thermonuclear concept. In the case of the thermonuclear concept the fusion reaction rate R per volume unit and per time unit is 2
13 Api’il 1992
electrodynamic phenomenon. With the aim to derive a formula equivalent to (17), it is worthwhile to use a specific method which is an improvement of the known method of Yaschy [8]. This method, proposed in 1959 by Saint-Guilhem [9], has been deduced from group theory. It is based on a theorem stating that any physical relationship using a continuous set of system units, containing a finite number p of real variables x, can be written by means of m
variables V~,so that the bamer performances are only represented by F, i.e. the electrodynamic aspect is thus only contained in F. The inventory of the parameters which are necessary for describing the behaviour of the system is thus made of the rate R, the nuclear cross section a, the Planck constant /1, the whole barrier crossing duration 0, the barrier length L, the transmission barrier factor F, the number of particles n which can be involved in a nuclear reaction, the time t, and the deuteron energy E. The macroscopical process thus depends on p= 9 physical variables. Since those physical variables are of different species, the structural relationship depends on 9 physical species. One can list 6 independent equations describing the structure as a function of the non-dimensional multiplying factors describing a unit change. Those factors are represented by the same letters as the physical variables, but are here all upper case letters: E, L, R, T, N, H, E, 0. Those 6 independent equations are 1L2, HML2T’, R_L3T~ N=L3, E=MLT2, 0=HE’.
(19)
As the number of the physical species is larger than the number of equations, the dimensional analysis is applicable and one can, according to ref. [9], look 159
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PHYSICS LETTERS A
13 April 1992
for the p’ physical variables which can be present in the final expression. In fact it is easy to see that for 2EL if one = 5 one gets only one monomial RT/N takes into account only the dimensional parameters E, L, R, 0, N. The relationship which describes the macroscopic phenomenon thus has the following shape, using again the physical parameters,
production term” or, more shortly, the “production term” T,
g(F, R0/n2aL)=0.
entrance of the nuclear well. The best way is to rely on the results of high energy electron scattering on deuterons. Those experiments have been performed using 188 and 400 MeY electrons. It was shown that it was possible to fit the electron diffusion data with the extension range of the deuteron. It was estimated by McIntyre [10] that the best fit for the charge ex-
(20)
Dimensional analysis cannot tell more, but one can infer physically that the nuclear reaction rate R must be proportional to the barrier transmission factor F, and the relationship becomes (the usual ~ factor has been included) R=~n2aFL/0,
(21)
a formula which replaces in any fusion barrier process formula (17) which has been used largely in the past for thermonuclear processes. In fact the parameters which have disappeared (1, M, E, /1) are not useful directly for the result, but they are included implicitly in R for the time t, and in 0 for E and /1 and M. Nevertheless, according to the method of ref. [9] it was necessary to list them, even if in a natural way the process discards them from the result. But it is, however, important to emphasize the fact that the formulation is non-relativistic, and that the whole process would have to be reexamined in the case where the particle velocities would become relativistic, In other words, a more complete expression than (17) does exist, if one takes into account the relativistic behaviour of the particles in the therrnonuclear range ~‘. The application has been made here with a cornputer whose performances were limited to handling only a 10 x 10 complex matrix, which corresponds to a division ofthe barrier into four slices. Nevertheless it has been verified that the results obtained do not change drastically by using a greater number of matrix dimensions: the result of this relatively small number of matrix dimensions is only a lack of precision. In fact whatfor is R important to calculate is a(21), part of the expression according to formula this is a term which one can call “nuclear reaction ~ It seems that the expression for R (21) still applies in the relativistic case, but the transmission factor F will also depend at least on the velocity of light.
160
T= aFL/0. (22) To do this, one has to choose the proper slicing of the potential barrier and, before slicing the barrier, the abscissa of the first interface corresponding to the
tension of the deuteron corresponds with the assumption ofthe charge independence of the internal structure of nucleons. In other terms the effective radius of the deuteronwas estimated to be 1.70 X 10— i3 cm, with the assumption of a modification of the Coulomb law at small distances. So in our model the potential barrier, keeping a finite value, would extend outside the nuclear well from the effective radius of the deuteron nucleus to distances depending on the cumulative effect of the potential pedestal change and of electron clouding. To take into account those two effects it seems useful to combine them in the same semi-phenornenological formula. For this, the curve patterns of ref. [3] are represented by the product ofan exponential and a second order polynomial, the potential pedestal represented by V 0 is subtracted from the Coulomb potential, so the energy potential expression is U(r)=e(e/r— V0)(r—a)(r—b) exp(—Kr). (23) This choice for the nuclear-well radius implies also in fact the choice of a nuclear cross section. Inasmuch as the potential is crossed therebutis the no other uncertainty aboutbarrier the nuclear collision geometrical factor, depending only on the charge extension of the deuteron With isthis assumption one 2, #2which exactly the value gets a= 9 x 1026 cm 82
Except for a branching ratio equal to 0.5 of the two possible
reaction channels in the case of the D—D reaction. But there was no use to take into account the branching ratio in this paper, given the lack of precision due to the limited matrix dimensions.
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PHYSICS LETTERS A
quoted by Lawson [11]. For the values of the electron number v in the cloud, it was necessary to extrapolate the results of Fedorovich [3] to higher values. Given that the effect of electron clouding is less stringent than the one due to the pedestal, an estimate of the electron clouding effect is possible in a rather large screening range without changing drastically the result, except for a higher number of electrons in the cloud. In fact the terms a, b, C, k were approximated by the following phenomenological formulas which are suited to the parameters a and b decreasing in an inversely proportional way with log v (fig. 3),
13 April 1992
3/s
o’FL m
e
.1 ~~30
~
10
h~ I
,‘
/
/ ,‘ ,‘
,‘
,//
a= 1.1 1/log v, b=3.48/log v, (24)
I
+
/
~Ii
-50
,~~
10 ,‘
/‘
//
/
,///
~‘
(4)
(2)
(3)
/
%
/~
,/
/
,‘ I
,i /
//
/
/
~ 0
/
“/
// /‘/
/
/
/
I/
/
~/
/
,//
/
// ,‘
/
J’
/1 // /
Calculations have shown that for a specific value of the electron number v in the cloud (or equivalently for the same value of single positive charges in the environment of the two deuterons approaching each others) the values ofthe production term T, according to (22), are lying along a straight line in the diagram of log T as a function ofthe logarithm of the deuteron energy, logE (fig. 4). The experimental points (Z-pinch experiments, capillary fusion and cluster fusion) are the same as in ref. [1]. The Brookhaven point corresponds typically to a cloud of 3.3 x 1 0~electrons. The NRL 1 —CEA point seems a little apart and apparently corresponds to more than 5 x 1 0~electrons in the cloud. Of course the precision of the calculation would improve if one had used complex matrices with dimensions larger than 10 x 10. It has been shown however that between the use of one slice (the whole barrier) and the use of a 10 x 10 matrix, the change in the result is not more than some powers of ten for a specific case around T= l0~°rn3/s (fig. 5). The value grows from a 1 x 1 matrix to a 10 x 10 matrix and it seems possible to extrapolate a more precise T value around 1 0~above the values given by the 10 x 10 matrices. On the logarithmic diagram this is not a large change. In fact it will not change drastically the results of fig. 4. The cluster fusion point corresponds to a single ion number v in the environment which would be closer to .3 x 1 0~rather than above 3.3 x 1 0~,as was first proposed, using the raw calculation results. One has to remark that this value is not far from 2 x 1 0~ob-
/
,‘
I,
k=l.5/a2b.
I
,‘
KIEL BONPAS et Al.
e x
NRL1
+
BADUREK et AL.
~
NRL 2
BROOKHAVEN
(1) i~-~° E (eV)
Fig. 4. Fusion production factor (in rn3/s) as a function of the energy of the deuteron (eV) m a logarithmic scale. The solid lines are the raw results of the calculation and the dotted lines are deduced from them by a correction due to the insufficient number ofmatrix dimensions (10 x 10). The experimental points have the same meaning as in ref. [I]. The number of electrons per cloud is (1) 342, (2) 1350, (3) 3374, (4) 5 x104. The curve D— D corresponds to the thermonuclear hypothesis.
tamed from pure stochastic considerations. Taking into account this rectification, due to the lack ofmatrix dimensions we obtain the dashed straight lines which are above the solid straight lines in fig. 4. This way of reasoning shows that the calculated values of the production term T are in the range of the ones which one infers from experimental results. It gives an estimate ofthe z value, so that our model gives a possible explanation for the so-called anomalous cross sections observed by Beuhler et al. [12]. One can thus assert that in the various kinds of experiments the cause of the observed fusion reaction is essentially the same: i.e. the simultaneous screening effects of the electron clouding around two approaching deuterons and the corresponding change 161
Volume 164, number 2 o~FL —i-—
PHYSICS LETFERS A
4) are just above thethat Brookhaven point (cluster fusion), which means the number of electrons in
(m /s)
i0~0
—-—
,..—- - —
-
-~
I
1
2
3
4
Number of sLices
Fig. 5. Precision test of the method. It shows that for four slices in the barrier which it is possible to handle with a lOx 10 matrix the fusion reaction production term T is still not close to the asymptotic value. The gap can be estimated to a factor 1 O~,which represents the distance, in loganthmic scale, between the dashed straight lines and the solid straight lines in fig. 4. On the X-axis is the number of slices and on the Y-axis is the fusion reaction production term.
of the electric potential pedestal in the environment of those two deuterons. But one has to take into account the presence of the non-fusible positive charges in the medium. Since the ionization energies of simple bodies like oxygen or nitrogen are lower than the one of deuterium, they could provide an extra number of electrons in the cloud and also contribute to the pedestal change in an efficient way. Their effect could be particularly marked in cluster fusion experiments [12]. In fact one can assume that in those experiments, the heavy water clusters impinging at very high velocity onto the target are fully ionized. The corresponding electron cloud phenomenon thus takes place in a limited medium. Deuteron meetings occur in a limited space, according to the statistical Poisson law, but a minimum number of D2O molecules is necessary to have a sufficiently efficient joint effect of electron cloud and change of pedestal. Beuhler et al. show in their paper [12] that the fusion reactions depends on the number of D2O in the impinging cluster, the maximum value being close to some hundreds. Our rectified computed values (dashed straight line 2 in fig. 162
13 April 1992
the cloud around the approaching deuterons was of the order of l0~,which is not contradictory with fig. 3 of ref. [12] reproduced in fig. 6 of this paper. If one assumes a full ionization of the cluster, and for less than approximately 500—600 molecules (producing typically l0~to l.5x l0~electrons by postimpact ionization), this means that one sole deuteron meeting point would appear statistically in this cluster. For a cluster containing more molecules, two deuteron meeting centers would appear, but with an insufficient number of available electrons and ions for creating a sufficient screening effect in the two deuteron meeting centers. Presumably one can recover fusion reaction production with clusters ofbigger sizes. To conclude this Letter one can make the following remarks: (1) This study is a result of the effects of higher electron densities than the ones given by Fedorovich [3]. It would thus be important in a new study to start calculating those high electron density screening effects and thus also compulsory to handle higher dimension matrices. (2) The slope of the straight lines in a logarithmic 10
8
8 ~ 6
~
-~
I
f 2
~
__________________________________ 20 60 Number of ~o te~°° 2000
10
Fig. 6. From ref. [121. It depicts the dependence of proton yield (which is proportional to the fusion production rate) on cluster size at a 300 kV accelerating potential. The ordinate scale is proton counts per 1000 s per nA ofcluster current. The pattern shows how the number of D20 per cluster becomes insufficient to sustam the fusion reactions by a crossing barrier process for two deuteron meeting points, beyond approximately one thousand D20.
Volume 164, number 2
PHYSICS LETTERS A
diagram, with a constant number of clouding electrons, is between 1.5 and 2, according to the root value. Those values are not far from the slopes which are conjectured in ref. [1], from experimental resuits, the production term being approximately proportional to a current law between j6 and V6 ~ It is interesting to underline that the model discussed here is in rather good agreement with the Z-pinch experiments, (3) The fusion time in the Z-pinch and capillary fusion is reached only when the current has largely begun to decrease, like in the Lochte-Holtgreven experiments [15], orjust at the apex ofthe current [4]. According to the model, the ionized deuterons must have a translational motion. This motion is given by the electrodynamic forces. They are lying in the same direction during the growing phase of the current; but as soon as the current begins to decrease, this decreasing evidently initiates a chaotic motion. It seems possible to calculate the behaviour of the system by assimilating the ions to harmonic oscillators submitted to a variable strength as a function of time according to the modern conceptsof chaos [16]. This chaos is thus the origin of collisions. It is easy to develop the classical macroscopic analogy of a train whose locomotive is slowing down, also producing a chaos-like behaviour. The less the maximum of the strength, the less the chaotic effects and the more they are delayed. (4) Fusion is possible by the process proposed here, not only for very light nuclei like deuterons and tritons, but also for many heavier nuclei, like the ones considered in astrophysics (for example 12C, ‘‘B, 7Li). (5) One can infer from those results that we now have both a theoncal and an experimental confirmation that an electric current passing through a dense non-ideal plasma [2] is not a simple process. The distribution of the electrons and ions is evidently submitted to a variable chaotic motion as has been noticed in a recent paper [17]. In fact on a ~ Any experimental result which would give an exponent less than 4 seems to be not reliable in any hypothesis, and due to an experimental artefact.
13 April 1992
macroscopic level, one observes only an average effect, even if locally there are electron clouds and potential pedestal changing effects, which involve thousands of particles and more. (6) This study has been limited to the non-relativisti~case, but one can infer from the proposed model that it is no longer possible to ignore that the conventionally called thermonuclear plasmas are in fact and above all relativistic plasmas. In such a plasma relativistic electrons play a role which one has to understand better to predict its evolution. This necessity is particularly important for the knowledge of star evolution. I am very grateful to J.P. Vigier for many discussions and exchanges during the elaboration of the model and also to Professors H. Rauch and P. Graneau for providing information on very recent progress in the field of Coulomb screening and fusion processes.
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(1992)149. [4] J.L. Doob, Stochastic processes (Wiley, New York, 1953) p. 404. [5] M. Rabinowitz, Mod. Phys. Lett. B 4 (1990) 665. [6] H. Takahashi, Phys. Rev. 172 (1968) 747. [7] A.K. Bahtia et al., Phys. Rev. A 38 (1988) 340O~ L. Chatteijee, Phys. Lett. A 137 (1989) 4; Nature 342 (1989) 232. [8] Vaschy, Ann. Télégraphiques (1892) 25, 189. [9] R. Saint-Guilhem, Rev. Gen. Electr. (1959) 533. [lO]J.A. McIntyre, Phys. Rev. 103 (1956) 1464. [11] J.D. Lawson, Proc. Phys. Soc. London B 70 (1957) 6.
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G. Friedlander and L. Friedman, Phys. Rev. Lett. 63 (1989) 1292. [13] J.D. Sethian et al., Phys. Rev. Lett. 59 (1987) 892. [14] J.D. Sethian, Evolution ofdeuterium fiber Z-pinch driven by aLochte-Holtgreven,Atomkernenergie long current pulse, private communication. [15] W. 28 (1976) 150. [161 M.C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, Berlin, 1990). [17] M. Rambaut, Phys. Lett. A 154 (1991) 210.
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