The centrifugal force must equal the centripetal one, which is inversely proportional to r (the radius of the orbit):
Combining Equations 1.31 and 1.32, and solving for r, r = C3
In the above equations, C-, C2, and C3, are various constants. Thus, since the muon is 207 times more massive than the electron, the radius of the muonic deuteron must be 207 smaller than that of the normal deuteron; that is, it must be (approximately) 180 fm.
A positive heavy hydrogen ion (D-muon-tritium ion or D-^-T+) can be synthesized. The presence of the muon effectively shields the deuterium from the tritium, causing the distance between these atoms to be some 207 times smaller than in the D-electron-tritium case. This is close enough to allow tunneling, though the electrostatic barrier and the two atoms fuse, releasing the energy discussed when the D-T reaction (Equation 1.14) was discussed. Thus, the output of the system is 2.81 x 10--2 J (17.6 MeV) per muonic molecule. As we saw, 14 MeV are carried by the neutron, which can yield an additional 4.81 MeV when it is exothermally absorbed by lithium (Equation 1.19).
The muon survives the fusion and may live to catalyze more reactions. However, two mechanisms require the continuous replenishment of muons: its short lifetime and alpha-sticking, the fact that the muons tend to stick to the alphas from the fusion, being carried away. The energy balance pits the useful output against the cost of replenishing the muons. Muons can be produced by accelerating deuterons (800 MeV) and causing them to impact deuterium gas.
Thus, cold fusion certainly does occur. More than that, cold fusion occurs (almost certainly) even when not mediated by muons. S. E. Jones (1989) described an experiment that appears to prove just that. He used an electrolytic cell with a platinum anode and a palladium (sometimes, titanium) cathode, and D2O (heavy water) as electrolyte. Since water is a poor conductor, salts were added. Here is Jones's extraordinary recipe:
"The electrolyte is a mixture of about 160 g of deuterium oxide (D2O) plus various metal salts in about 0.2 g amounts each:
FeSO4, NiCl2, PdCl2, CaCOs, LiSO4, NaSO4, CaH4(PO4)2, TiOSO4, and a very small amount of AuCN."
A chemist might be horrified by the cocktail above—it would be hard to tell what is going on."
When a current was forced through the cell, a small flux of neutrons with a characteristic energy of 2.5 Mev was observed. Jones, a physicist, did a good job of neutron detection. Since 2.5 Mev is the energy of the neutrons in Reaction 1.29, this experiment tends to show that, indeed, fusion is going on.
Jones observed that some 8 hours after the start of operation, the neutron "signal" turned off by itself. This effect was attributed to the poisoning of the palladium electrode by deposition of metals from the solution. In fact, etching the electrode revived the cell.
The reaction rate observed by Jones was small, perhaps 10~20 fusions per deuterium pair per second. This could be explained if the deuterium molecules were somehow squeezed from 74,000 fm to half this distance by their residence in the palladium lattice."" Jones dubs this piezonuclear fusion.
Stanley Pons and Martin Fleishmann (1989) ran similar experiments but, being chemists not physicists, adopted a simpler electrolyte: an LiOH solution in D2O (heavy water). They also failed to make careful neutron measurements. What they reported is that, after prolonged precooking, some cells suddenly developed a great deal of heat, billions of times more than in the Jones experiment. Unfortunately, these results were never reproduced by other experimenters, and this casts severe doubts on their validity. Here is where I will don my devil's advocate mantle and, just for the fun of it, will defend the P&F results.
In a lecture delivered at the Utah University on March 31, 1989, Stanley Pons relates the most spectacular of his results. "A cube of palladium with a volume of 1 cm3 was used as cathode of an electrolyzer with lithium hydroxide dissolved in D2O as an electrolyte. A current of 250 mA/cm2 was applied for several weeks/months [sic] with nothing remarkable happening. A Geiger counter detected no radiation. The current was cut to 125 mA/cm2 late one day, and next morning the cube of palladium and the electrolysis cell were gone. A nearby Geiger counter was also ruined."""
There was a long delay (several days, at least) before heat evolved. Since the Jones cell poisons itself in 8 hours, this cell will never reach the primed state and no heat can be observed.
t Jones was trying to create a chemical environment somewhat like the one in the soil because he was trying to show that some of the internal heat of our planet is generated by deuterium fusion.
tt A possible cause of the squeezing would be the increase of the electron mass to a few times its free mass.
tttAs related by Patrick Nolan, 1989 (paraphrased).
Why such a delay? Hydride hydrogen storage systems (see Chapter 11) are well known and are commercially available. One popular system uses a TiFe alloy to absorb H2. Many other metals and alloys will do the same. Palladium, in particular, is a notorious H2 absorber. It is not used commercially owing to its high price.
When TiFe powder (after duly activated) is exposed to hydrogen, it will form a (reversible) hydride, TiFeH. If the amount of hydrogen is small, there will be a mixture of TiFe and TiFeH in the powder. This mixture, called ^-phase, has the empirical formula TiFeH^, where x becomes 1 when all the material has been hydrided.
After full hydridization, addition of more hydrogen will cause the formation of a di-hydride, TiFeH2 (7-phase). Clearly, the hydrogen is more densely packed in the (di-hydride) 7-phase than in the ,3-phase. It is, therefore, plausible that the fusion will proceed faster once the 7-phase is reached. How long does it take to reach this 7-phase?
In the described experiment, Pons used a current density of 250 mA/cm"2 —a total current of 1.5 A when added over the six sides of the cubic cathode. This corresponds to a production of 9.4 x 1018 deuterons/second. Each cubic centimeter of palladium contains 68 x 1021 atoms. Thus, it takes 7200 seconds, or 2 hours, for the palladium, in this particular experiment, to start becoming di-hydrided. This assumes that all the deuterons produced are absorbed by the palladium, and, thus, the time calculated is a rough lower limit.
Could the heat have resulted from a chemical reaction? The highest enthalpy of formation of any palladium salt seems to be 706 MJ/kmole, for palladium hydroxide. Atomic mass of palladium is 106 daltons, and density is 12 g cm"3. This means that one gets 80 kJ cm"3 chemically. Pons and Fleishmann have (they say) gotten 5 MJ cm"3, two orders of magnitude more than chemistry allows.
1. The heat produced cannot be due to classical fusion reaction (insufficient neutrons, tritium, and 7-rays).
2. The heat produced cannot be due to chemical reaction.
3. Then, simplistically, the heat was not produced.
There is at least one more possible reaction that occurs very rarely:
As written above, this reaction cannot take place because two particles are converted into a single particle, and it is impossible to conserve simultaneously energy and momentum under such conditions. For the reaction to proceed, it is necessary to shed energy, and, in classical physics, this is done by emitting a 16-MeV 7-ray. Pons did not report 7-rays. There is still an outside possibility that the energy can be shed by some other mechanism such as a phonon, although physicists tell me that this is nonsense. Observe that Reaction 1.34 produces one order of magnitude more energy per fusion than do Reactions 1.28 and 1.29.
So far, we have attempted to explain the hypothetical cold fusion as the result of deuteron-deuteron reaction. It has been difficult to account for the absence of the expected large fluxes of neutrons or gamma rays. It is even more difficult to imagine such a reaction proceeding when common water is used in place of heavy water. Nevertheless, some experimentalists make exactly such a claim.
There have been suggestions that cold fusion actually involves nuclear reactions other than those considered so far. Let us recapitulate what has been said about cold fusion.
1. The results, if any, are not easily reproduced.
2. No substantial neutron flux has been detected. This seems to eliminate the deuteron-deuteron reactions of Equations 1.28 and 1.29.
3. No substantial gamma ray flux has been detected. This eliminates the classical form of the deuteron-deuteron reaction of Equation 1.34.
4. Reactions are reported to be highly dependent on the exact nature of the palladium electrode.
5. Reactions have been reported with an H2O instead a D2O electrolyte.
The following cold fusion mechanism fitting the above observations has been proposed.
Boron is a common impurity in palladium. Natural boron exists in the form of two isotopes, with the relative abundance of 20% for 10B and 80% for 11B. Thus, under some special circumstances, the two triple-alpha reactions of Equations 1.25 and 1.27 might occur. They emit neither neutrons nor gamma rays and can occur with either normal water or heavy water.
The boron impurity may be interstitial, or it may collect in grain boundaries. The reaction may only occur if the boron is in one or the other of these distributions. It may also only occur when the amount of impurity falls within some narrow range. Thus, a palladium rod may become "exhausted" after some time of operation if the boron concentration falls below some given limiting concentration.
Perhaps the worst indictment of the P&F experiment is its irrepro-ducibility. No one has claimed to have seen the large heat production reported from Utah. Pons himself states that his experiment will only work occasionally; he claims that there is live palladium and dead palladium. This could be interesting. Hydrogen absorbed in metals is known to accumulate in imperfections in the crystal lattice. It is possible that such defects promote the high concentrations of deuterium necessary to trigger the reaction.
I still have an old issue of the CRC handbook that lists the thermoelectric power of silicon as both +170 mV/K and —230 mV/K. How can it be both positive and negative? Notice that the determination of the sign of the Seebeck effect is trivial; this cannot be the result of an experimental error. In both cases "chemically pure" silicon was used. So, how come? We have a good and classical example of irreproducibility. That was back in the 1930s. Now any EE junior knows that one sample must have been p-silicon, while the other, n-silicon. Both could be "chemically pure"—to change the Seebeck sign, all it takes is an impurity concentration of 1 part in 10 million. Is there an equally subtle property in the palladium that will allow fusion in some cases?
In April 1992, Akito Takahashi of Osaka University revealed that his cold fusion cell produced an average excess heat of 100 W over periods of months. The electric power fed to the cell was only 2.5 W. The main difference between the Takahashi cell and that of other experimenters is the use of palladium sheets (instead of rods) and of varying current to cause the cell to operate mostly under transient conditions. The excess heat measured is far too large to be attributed to errors in calorimetry. Disturbing to theoreticians is the absence of detectable neutrons. See D. H. Freedman's (1992) report.
In spite of being saddled with the stigma of "pseudo-science," cold fusion does not seem to go away. The September 2004 issue of IEEE Spectrum published a report titled "Cold Fusion Back from the Dead," in which recent work on cold fusion by reputable laboratories is mentioned. It quotes the U.S. Navy as revealing that the Space and Naval Systems Center (San Diego) was working on this subject.^ It also mentioned the Tenth International Conference on Cold Fusion that took place in Cambridge, Massachnets, in August 2003. It appears that by 2004, "a number of groups around the world have reproduced the original Pons-Fleishmann excess heat effect." Mike McKubre of SRI International maintains that the effect requires that the palladium electrode be 100% packed with deuterium (one deuterium-to-one-palladium atom). This coincides with our wild guess at the beginning of this subsection.
At the moment, cold fusion research has gone partially underground, at least as far as the media are concerned. Yet, the consensus is that it merits further study. This is also the opinion of independent scientists such as Paul Chu and Edward Teller who have been brought in as observers.
Osaka University, mentioned a couple of paragraphs above, seems to be keeping the cold fusion flame alive. In two papers (February and March 2008) published in the Journal of the High Temperature Society of Japan, physics professor Yoshiaki Arata claims the reproducible production of tit is reported that Stanislaw Szpak (of the SNSC) has taken infrared pictures of mini-explosions on the surface of the palladium, when cold fusion appears to be taking place.
excess heat and helium-4 from samples of zirconium oxide/palladium nano powder charged with deuterium gas. In May 2008, Professor Arata made a public demonstration of this phenomenon, which could be interpreted as a confirmation of cold fusion.
It may be that cold fusion will one day prove practical. That is almost too good to be true and, for the classical fusion researchers, almost too bad to be true.
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