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I want transmutation of elements (for example, changing lead into gold) to be possible in my world, using magic rather than nuclear physics. I want the overall energy input or output to be consistent with the real-world elements, but without the additional energy required for initiating the nuclear reaction that would normally be needed. I'd like to assume that a way has been found to change from one element to another without needing the extreme conditions of an exploding star.

Practical applications in the story would include knowing how much lead could be converted to gold using the energy available to a human from eating a meal (assuming they only use the energy that a human digestive system can extract). Alternatively if the change is exothermic, how much heat would the human have to absorb to change 1 gram of lead into gold? The energy calculations could be asked on Physics.SE and Biology.SE, but first I need to make the magical process self consistent and believable (hard magic, analagous to hard sci fi).

I'd be interested in anything else I'd need to take into account to make this believable. For example, would converting lead to gold involve releasing helium and hydrogen to account for the difference in atomic mass? Would this cause a risk of fire or asphyxiation if performed in a poorly ventilated space? Or would the extra mass become lithium, creating impure gold that then needs to be refined?

I'm trying to settle on a mechanism behind this magical process. I've been considering the following as possibilities:

  • Accelerating time in a small region containing the lead, so that it naturally decays into gold. Two problems with this:

    1. The decay products of lead may not include gold (it may decay directly into smaller elements).

    2. The huge amount of accelerated time required to see most of the lead decay may also see most of the gold produced decay too.

  • Magically inducing a rain of high energy neutrons to bombard the lead nuclei and split them into gold and H/He/Li. The two problems mentioned still apply here.

  • Some kind of "nuclear enzyme" to catalyse the nuclear reaction so that extreme conditions are not required. This seems the most arbitrary, but being specific to that one reaction it does not suffer from the two problems described.

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  • $\begingroup$ possible duplicate of Wondering what would happen if magic was constrained by Conservation of Energy $\endgroup$ – Telastyn Mar 2 '15 at 14:08
  • $\begingroup$ Since I'm looking for a consistent framework, and not just energy considerations, I don't see this as a duplicate. Both sides can be discussed on this meta question. $\endgroup$ – trichoplax Mar 2 '15 at 14:30
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    $\begingroup$ Before voting to close as duplicate, please see the Worldbuilding Meta question Magical transmutation - duplicate or not?. $\endgroup$ – a CVn Mar 2 '15 at 14:32
  • $\begingroup$ The 3 most common Lead (Pu) isotopes are stable, or only very slightly radioactive, as far as we know. Gold (Au) has one stable isotope, Au-197. Especially Pu-208, which is double magic. Unstable isotopes tend to decay into Hg, Tl or Bi, not Au. You're more likely to eventually get Platinum than Gold. $\endgroup$ – Serban Tanasa Mar 2 '15 at 15:56
  • $\begingroup$ @trichoplax This question still has no accepted answer... $\endgroup$ – 2012rcampion Apr 23 '15 at 21:57
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Edit: I got a bit carried away with the details and had to cut myself off at the end but the important bit is in the suggested procedure for your catalyst idea. In particular, the analogy with transparency and the wavelength of the particle initiating the transmutation.

One way that I imagine this transmutation magic working is through your fourth option - the use of a catalyst or activator for the nuclear reaction of lead into gold. In particular, I imagine the catalyst as being an exotic particle that a magic-user can generate from his body's energy. In particular, these particles would not interact with matter except through gravity and a short range (say ~1 nm range) exotic nuclear force (compare with how neutrinos pass harmlessly through kilometers of matter except for the rare weak nuclear interaction).

A stream of these particles passing through a block of lead would have some probability per meter of interacting with a lead nucleus, where the interaction could either be catalytic - conserving the exotic particle - or participatory - consuming the exotic particle as a reactant.

To apply these particles, the magic-user might need to cast them in a field between his hands. Imagine a person's hands out in from of him as if carrying a box but instead he is standing over a piece of lead on a table or on a stand while barely visible streams leap back and forth between his hands. Only lead between the hands of the magic-user is affected by the transmutation.

Before I get to energy cost and the reaction itself, there are a few limitations (or rather features) that naturally come to mind for a transmutation process that works through this mechanism:

  1. Some of the exotic particles generated by the wizard will not interact with any lead. There should be some inefficiency defined for this process that perhaps more skilled magic-users can improve.
  2. As a catalyst or reactant for the nuclear reaction, the particles would only yield some finite reaction rate - an amount of lead to gold per minute that could be fine-tuned to the worldbuilder's whim. This finite rate could make the process slow enough to not easily ruin the economy.
  3. A magic-user transmuting lead into gold has to actually get the lead in front of him and concentrate his full attention (and both hands!) on the process. This procedure feels less soft than "point-and-transmute" or "midas-touching", since it superficially resembles other crafting procedures.
  4. The exotic particles would have a particular wavelength and would only interact with nuclei that had a particular spacing in its energy levels, in an analogous manner to the interaction of photons with electron orbitals based on discrete energy level spacing and photon wavelength. In this way, the magic-user could choose which transmutation reaction he would perform by consciously changing the wavelength of the exotic particles he is casting (limited of course by the availability of the particular isotope/element for that wavelength). By choosing a wavelength particular to lead -> gold, the magic-user would cast particles for which any material other than lead is transparent.

As for the reaction itself, for an exotic particle ε, a skeleton reaction is:

 Pb-208 + ε ==> Au-197 + Be-9 + other

Using known atomic masses, Pb-208 to Au-197 is a change in mass from $207.976 \space\text{u}$ to $196.966\space\text{u}$, where 3 protons and 8 neutrons are left out. Turning some of this waste into Be-9, which has an atomic mass of $9.012\text{ u}$, there remain 2 neutrons (the conversion of one neutron into a proton also results in an electron and an antineutrino). A total of $0.02\text{ u}$ of mass has seemingly been created here but this can be accounted for through other parameters of the transmutation and its exotic particle.

At this point, you could go several directions. The added mass can come from the exotic particle itself (i.e. it has a mass of $0.02\text{ u}$ or $18.6\space\text{MeV/c}^2$). For $1\text{ kg}$ of Pb-208, the energy supplied by the magic-user would need to be $$(2.895e21\text{ particles}) \times (18.6 \text{MeV}) \times (1.6e-13 \text{MeV/J}) = 8.6 \text{GJ}$$ or the equivalent energy from burning a barrel of oil. This calculation assumes $100\%$ efficiency which also doesn't make sense given the earlier description of the transmutation procedure.

Since a human cannot generate this much energy, you would have to imagine a much much smaller mass for the particle, one that cannot account for the added energy. To get around this issue, you would have to imagine that one of the 2 neutrons is converted into energy. The massive excess energy of $920\text{ MeV}$ per reaction (transmuting 1 kg of lead would release the energy of $10\text{ MOABs}$!) could be disposed of in the kinetic energy of neutrino-antineutrino pairs.

I could go on about how to make this better fit energy-momentum conservation, how to properly avoid violating lepton conservation, and how the range of the exotic nuclear force affects efficiency but I'm gonna cut this short. I hope this gave you some ideas! :)

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  • $\begingroup$ Very interesting. I don't see the energy requirement as a problem - I like the limitation on how much lead can be converted in a short time by a single person. Makes it more a demonstration of power rather than a way of creating arbitrary amounts of wealth to upset the value of gold. $\endgroup$ – trichoplax Mar 3 '15 at 20:27
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Your main problem is that lead and gold do not have the same ratio of protons to neutrons, so you won't be able to just rearrange your lead into gold. You will end up with some leftover neutrons however you do it.

An atom of gold has 79 protons, 118 neutrons, and -31.1 MeV of excess energy. An atom of lead has (on average) 82 protons, 125.24 neutrons, and -22.4 MeV of excess energy. A lone neutron has 8.1 MeV of excess energy.

If we want to achieve a maximum lead-to-gold ratio, we can simply emit all of the excess neutrons as neutron radiation. On average, each atom of lead will produce 1.04 atoms of gold, 2.76 neutrons, and -12.4 MeV of energy. That minus sign is a Bad Thing: it means that we have an energy deficit, i.e. we will need to put in that much energy for the reaction to take place. How much? About 50 Megawatt-hours per ounce of gold. You would also absorb on the order of tens of Sieverts of neutron radiation per ounce of gold produced. This is probably not good for your wizard.

We can do what Abulafia suggests in his answer and convert the excess neutrons into some light element. The best candidate is probably tritium, since it will take up two excess neutrons per lost proton. We will end up 7.9% tritium by mass. Unfortunately we still need to put in energy, around 73.8 MeV per atom of lead, since this is effectively doing fusion in reverse.

Another good candidate for a 'neutron sponge' is plutonium-244, with around 1.6 times as many neutrons as protons. We end up with around 34% plutonium by mass. This time our energy output is less, but still negative, with 18.0 MeV needed per lead atom. (This time, we're doing fission in reverse.)

I wrote a linear programming routine to find the highest-yield zero-energy transmutation from one given element to another, using all isotopes with half-lives over a given threshold. Here are some examples of transmutations into gold (percentages are mass fractions):

  • $\text{Pb}\to 8.5\%~^{126}\text{Sn} + 33.1\%~^{228}\text{Ra} + 58.4\%~\text{Au}$ (14 carat)
  • $\text{Pb}\to 13.7\%~^{126}\text{Sn} + 30.2\%~^{244}\text{Pu} + 56.0\%~\text{Au}$ (disallowing radium)
  • $\text{Pb}\to 15.0\%~^{136}\text{Xe} + 30.1\%~^{244}\text{Pu} + 54.9\%~\text{Au}$ (disallowing radioactive tin)
  • $\text{Pb}\to 14.3\%~^{136}\text{Xe} + 33.0\%~^{238}\text{U} + 52.7\%~\text{Au}$ (disallowing plutonium)
  • $\text{Pb}\to 0.4\%~\text{D} + 64.6\%~^{204}\text{Hg} + 35.0\%~\text{Au}$ (disallowing all radioactive elements)
  • $\text{Hg}\to 6.9\%~^{126}\text{Sn} + 11.7\%~^{250}\text{Cm} + 81.5\%~\text{Au}$ (just below 20 carat)
  • $\text{Hg}\to 6.6\%~^{126}\text{Sn} + 12.6\%~^{244}\text{Pu} + 80.7\%~\text{Au}$ (disallowing Curium)
  • $\text{Hg}\to 7.2\%~^{136}\text{Xe} + 12.6\%~^{244}\text{Pu} + 80.2\%~\text{Au}$ (disallowing radioactive tin)
  • $\text{Hg}\to 6.9\%~^{136}\text{Xe} + 13.8\%~^{236}\text{U} + 79.3\%~\text{Au}$ (disallowing plutonium)
  • $\text{Hg}\to 0.1\%~\text{D} + 26.7\%~^{204}\text{Hg} + 73.2\%~\text{Au}$ (around 18 carat) (disallowing all radioactive elements)
  • $\text{Ag}\to 70.5\%~^{58}\text{Fe} + 2.0\%~^{62}\text{Ni} + 27.5\%~\text{Au}$
  • $\text{Cu}\to 51.1\%~^{56}\text{Fe} + 44.5\%~^{62}\text{Ni} + 4.4\%~\text{Au}$
  • $\text{Fe}\to 4.8\%~^{54}\text{Fe} + 95.2\%~^{56}\text{Fe} + 0.1\%~\text{Au}$
  • $\text{Al}\to 0.9\%~\text{H} + 76.1\%~^{40}\text{Ca} + 23.1\%~\text{Au}$
  • $\text{H}_2\text{O}\to 15.6\%~\text{H} + 62.4\%~^{40}\text{Ca} + 22.1\%~\text{Au}$
  • $\text{Air}\to 8.3\%~\text{H} + 49.6\%~^{40}\text{Ca} + 42.1\%~\text{Au}$ (10 carat)

Zero-energy means that each of these methods take zero energy input or output. However, you will still need a way to purify the gold afterwards (perhaps magical separation?).

Note that elements closer to iron have worse yields, since they have much higher binding energies. Due to the zero-energy criterion, most of the mass must be turned into nearby high-energy isotopes to offset the creation of relatively low-energy gold.

Transmutations between nearby elements work better, like mercury and gold. Another example, turning air (nitrogen and oxygen) into diamond (carbon) and ruby or sapphire (corundum/aluminum oxide):

  • $\text{Air}\to 1.5\%~\text{D} + 0.03\%~^{10}\text{Be} + 98.5\%~\text{C}$ (The beryllium changes to plutonium-244, uranium-238, and mercury-204 in roughly the same proportion as we increase the minimum half-life)
  • $\text{Air}\to 1.7\%~\text{H} + 6.4\%~\text{D} + 91.9\%~\text{Al}_2\text{O}_3$

Note that since aluminum has one more neutron than an even split (like nitrogen and oxygen), in this case we have to dump excess protons instead of excess neutrons.

A possible explanation for the magical transmutation could be a sort of probability manipulation, forcing the quantum-mechanical waveform to collapse into the desired state. How you justify your magic is ultimately up to you though.

Update

While playing around with my code, I stumbled across a much more efficient transmutation: turning mercury into platinum:

  • $\text{Hg}\to 0.32\%~\text{n} + 0.04\%~\text{H} + 99.6\%~\text{Pt}$

This produces three nines fine platinum (999.6‰ purity after neutrons escape, same as bullion) with a small amount (0.038%) of hydrogen impurity. You don't have to worry about hydrogen embrittlement with platinum, so you can use it as-is, or smelt the transmuted metal into bars (unfortunately you can't cast bars by pouring the mercury into molds, since it will shrink by 37% during the transmutation process). This method produces a burst of thermal neutrons, but you should be able to block those with water shielding.

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  • $\begingroup$ 125.24 neutrons? What's that, 2/3 of an up quark? Also, I'm not sure you're taking into account which Au isotope you're generating. Only one is stable, if I remember by extended isotopic mendeleevian chart well. $\endgroup$ – Serban Tanasa Mar 2 '15 at 20:04
  • $\begingroup$ I said "on average," meaning that those number are all averaged, weighted by typical isotope fraction. And yes, I'm generating only Au-197. (For all the data I used Mathematica, e.g. ElementData["Lead", "IsotopeAbundances"] to get the composition, and then IsotopeData to get the masses and proton/neutron numbers.) $\endgroup$ – 2012rcampion Mar 2 '15 at 20:10
  • $\begingroup$ That's all in Mathematica? I'm dumping R tomorrow. Ps: the quark thing was a joke, of course. $\endgroup$ – Serban Tanasa Mar 2 '15 at 20:11
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    $\begingroup$ Note that seawater contains around $10^{-11}~\text{kg}/\text{L}$ of gold. Using a 100% efficient (i.e. magical) method, it would take only around 11 kJ, or 3 Calories of energy to extract an ounce of gold. $\endgroup$ – 2012rcampion Mar 3 '15 at 0:16
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    $\begingroup$ Haha, it's easier to make gold out of seawater than out of lead! That's a great piece of trivia! $\endgroup$ – Serban Tanasa Mar 3 '15 at 2:28
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Turning lead into gold releases a wizard-like amount of power.

If we assume that magic can re-arrange the nucleus at will, we can take one end-result and calculate what amount of energy would have to go in or out to achieve that result.

Lets say you start with the primordial lead-isotope Lead-204 with proton number 82 and relative atomic mass 203,97. You turn this into the stable gold isotope Gold-197 with proton number 79 and relative atomic mass 196,97. That leaves 3 protons and 7 neutrons which we will conveniently assume were turned into Lithium-7, which has a relative atomic mass of 6,94.

This leaves no free particles, but a change in relative atomic mass of 0.06. In percentages of original weight this is $0.06/203.97 = 0,02941\%$. So in short, about $0.03\%$ of the mass of the lead is turned into energy. About 0.3 grams that is.

How much energy is this? We can use $\text{E=mc}^2$ to calcualte this for $1\text{kg}$ of lead:

$\text{E = 1kg}\times 0.03/100 \times (299792458 \text{m/s})^2 = 7.49 \text{GWh}$. (About $2.7 \times 10^{13} \text{Joules}$)

The nice webservice Wolframalpha not only allow us to calculate the above without messing up the units, but also tells us that $7.49$ gigawatt hours is equivalent to half the energy output of the "Little Boy" nuclear bomb [1]. Other comparisions include 3 times the fuel capacity of an airbus A380 and twice the kinetic energy of the international space station, should it crash into something at full speed. [2]

Bottom line is, if your wizard can transmute elements, he won't be doing it to peddle nuggets of gold, he'll be demolishing cities of the enemy.

Story-wise, you could imagine that an adept wizard re-route the energy output into other transmutations, that absorb rather than emit energy.

[1] http://www.wolframalpha.com/input/?i=7.49+gigawatt+hours&lk=1 [2] http://en.wikipedia.org/wiki/Orders_of_magnitude_(energy)

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  • $\begingroup$ What happens if you suppose that among the free particles, as many as possible are turned into gold too? $\endgroup$ – Maxime Lucas Mar 2 '15 at 16:51
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    $\begingroup$ Well, do you know how much of that is neutrinos and how much photons? It would probably make a bit of a difference if it's 98% neutrinos... $\endgroup$ – Leprechaun Mar 2 '15 at 16:59
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    $\begingroup$ Pb-204 makes up only around 1.4% of all lead, so you'd have to magically 'enrich' the lead first. $\endgroup$ – 2012rcampion Mar 2 '15 at 17:45
  • $\begingroup$ @skysurf3000: That would also produce energy. When individual particles or atoms lighter than iron are fused together, energy is released. Say hydrogen into helium like in our sun. For elements heavier than iron, energy is released when they are split which is what happens with uranium in a nuclear reactor. $\endgroup$ – Abulafia Mar 2 '15 at 21:11
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    $\begingroup$ @MikeNichols: No, there are no integer masses, except for Carbon-12 which has mass 12u, because that's used as the definition for the "u" unit. The thing with atoms is that two free particles have more mass than when they are together. You can verify this for the deuterium nucleus, which is one proton and one neutron. A proton has mass 1.007276466812u, a neutron has mass 1.00866491600u, which sums to 2.0159..u But the nucleus has a mass of only 2.0135.. u The missing mass has been released as energy. Gamma radiation in the case of deuterium I believe, but not sure. $\endgroup$ – Abulafia Mar 3 '15 at 10:33
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The only conceivable energy sink large enough to allow for such transmutations in noticeable amounts without triggering (or absorbing) the equivalent of a small nuclear bomb would be Zero-Point-Energy.

Moreover, 97% of lead isotopes are stable or stable for tens of billions of years, so destabilizing them would be difficult (nucleic neutronic bombardment?).

Even worse, most of the decay products are other elements like mercury, thallium or bismuth. Very few decay paths from lead lead to a stable gold isotope.

In fact, gold is so hard to create, it's not generated inside normal stars, only in the most extreme supernovas and perhaps in neutron-star collisions.

If you want to transmute something into gold, I would suggest using the Mercury isotope Hg-196, (0.15% prevalence in natural Mercury). It can be converted to gold by neutron capture, and after electron capture-decay it will turn into Au-197 with slow neutron emission.

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    $\begingroup$ +1 I like the Hg isotope idea a lot! I was actually going to answer in a very similar way $\endgroup$ – Serban Tanasa Mar 2 '15 at 20:00
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Your ideas imply doing things with magic, that we do with machines. The awesome thing about magic is: You can do things with it that would not be possible (or only very difficult to do).

My personal approach would be not to follow some established scientific methods, but to go a bit deeper. Instead of trying to get a reaction, micromanage it: What I mean is, split whatever you want to turn into some other material into its smallest parts (electrons, neutrons, protons) and put them back together afterwards.

So instead of changing lead indirectly, rearrange the basic building blocks to get gold

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    $\begingroup$ I'm looking for a self consistent framework that will provide not only capabilities but also restrictions, so a realistic story can be built. Having complete control and no restrictions means any arbitrary story can be told, without a sense of why it is happening. $\endgroup$ – trichoplax Mar 2 '15 at 14:10
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I think the unappreciated problem here is economic. If your average magician can easily turn lead into gold, then gold is pretty quickly going to be worth about as much as lead, which renders the conversion pointless.

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  • $\begingroup$ This is why I am looking for a believable framework that has associated costs and risks, rather than simply allowing arbitrary amounts to be converted. $\endgroup$ – trichoplax Mar 5 '15 at 13:32

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