This is one of those questions that been bugging me since I first read it, each time I go back, I find more details. It's also given me an excuse to work out how to use the ENSDF data sets with nudel and the NUBASE data set. (And I have a million tabs open and have probably got myself onto some government watch list, but hey-ho.)
It's a long time since I did nuclear physics formally, so I'll need other people to correct any errors.
A few notes:
- I'll be doing a lot of rounding to order of magnitudes because the numbers don't need to be that accurate.
- Calculating absorbed doses is hard so I'll be even vaguer there. If someone can do real dose calculations, that'd be nice.
- I'll be using the uranium to plutonium shielded by lead example from the question for several examples. That's not because this is an unusually bad choice (it's not), I just want to limit the examples I swap between.
Anyway, here are the things I've found.
Nuclear physics is out to kill you
I know it's considered bad form to anthropomorphise fundamental forces, but when it comes to radiation, assume it's in the corner, cackling at you, trying to decide between the quick thrill of instant vaporisation or the slow drawn out agony of radiation poisoning.
It may not seem professional, but you'll live longer. I hope by the end of this discussion you'll agree that I'm not over-reacting.
You're going into unknown territory
Perhaps its no surprise that most of the good quality data comes from isotopes with moderate or long half-lives. It's difficult to get good quality data when your experimental material disappears as you're trying to measure it.
That's not so say there's no data, just that it's lower quality.
The physics of nuclei is very complicated. Even though right at the basics, the underlying quantum mechanics is similar to atomic physics, most approximations fail. For example, nuclei may adopt a non-spherical shape to reduce their energy (see the Nilsson model). Theory often is not a good guide.
You're going to be pushing materials into short-lived isotopes. There may be surprises there.
At least you said a "modern" wizard. When preparing this answer I could look up lots of values (want to know the heat capacity of neptunium? no problem).
How are you balancing charge?
You correctly talk about elemental transformation, rather than nuclear transformation. Transforming just the nucleus would leave you with an imbalance of charge, each mole (at best, a couple of hundred grams) of material transformed would acquire 100 kC of charge, about a thousand times the charge in a lightning bolt, which is going to disintegrate the material.
So, the question is how are you balancing the charge. The natural answer to this is just to do it how beta decay does it which is to create or destroy the appropriate number of electrons.
This suggests a physical mechanism for the magic. If the magic alters the nuclear forces in a region, so that different isotopes were stable then beta decay would do the work of transforming the nuclei. This is the mechanism used in Isaac Asimov's The Gods Themselves, in that case by moving the material to a different universe with different physical laws. However, looking in more detail shows this to be a bad idea.
Balancing with positrons is a bad idea
Nuclei where it would be energetically favourable to decay to lower Z have two choices: electron capture or positron emission. If the difference in binding energy is low, only electron capture is possible. You really want electron capture. If using magic to transform to lower Z emitted positrons, then these would immediately annihilate with electrons in the material causing a burst of gamma rays. Each annihilation will produce two photons for a a total of about 10²⁴ gamma rays per mole of material. Each photon carries about half an MeV, for a total of about 100 kJ per mole of material which I think will be fatal for anyone standing close by.
Balancing with high energy electrons is a bad idea
Electrons emitted by beta decay normally have far too much energy to remain bound to the atom from which they came. This is going to leave ionised material. If every atom in a material ionised simultaneously then what you have now is a plasma. Containment is going to be a problem.
It's not an accident that the electrons aren't bound. Differences in mass excesses between nuclei is usually of the order of MeV and electron binding energies are of the order of eV.
You might think that if you were achieving the the transmutation by altering nuclear forces, you could arrange them so the left over energy was just enough to leave the electron bound. This is unlikely to be practical. If I recall correctly, the beta decay rate goes as the fifth power of the energy difference. So, reducing a energy difference from 1 MeV to eV, will reduce the decay rate by thirty orders of magnitude. That's going to be a long wait.
Molecules will be broken
So, let's assume the magic can take one neutral atom and convert it into another neutral atom with the electron in its lowest energy state. You'll have altered the number of valence electrons available for chemical bonds. If the atom were part of a molecule then the chemical bonds will be broken and the material is likely to disintegrate.
As the chemical bonds reform, you may end up with a sudden chemical reaction happening at every part of the material simultaneously. This is rate not normally seen in nature and could lead to much excitement.
Bulk materials aren't always safe
Even if there are no bonds between different elements, bonds between identical atoms can be problematic. You don't want to end up with a pile of atoms that Suddenly Want To Turn Back Into Elemental Nitrogen.
Maybe transforming one bulk metal into another bulk metal might be the least dangerous.
Or, extend the magic to place the electrons in a suitable position for the new material.
The physical change in the material is a problem
Even a bulk material may have a different preferred configuration from its starting material.
A dramatic case is transforming from an atom that wants to be a solid at room temperature to one that wants to be a gas (say, potassium to argon). This is going to cause an explosion for any non-trivial quantity.
However, even solid to solid transformations can be a problem. In the original question there's a suggestion of putting a shield of lead around the material on the assumption that when it transforms to bismuth it will still be a good shield. Even if that's true, bismuth is about 10% less dense than lead so the material will suddenly be under a lot of stress as the atoms would prefer to be further apart. That's assuming it doesn't want to change its crystal structure. There would be a serious question as to whether the material would retain enough structural integrity to act as a good quality shield.
Chemical impurities are going to kill you
As written, the magic transforms all atoms in the affected region. Getting high quality chemical purity is going to be a problem.
Let's suppose you left one microgram of carbon-12 somewhere in the material being transformed: perhaps a fingerprint on the outside, or some cleaning solvent trapped in the surface irregularities. Transforming carbon-12 up or down to boron-12 or nitrogen-12 gives a substance with a half-life of tens of milliseconds.
This is going to release about 10¹⁷ beta particles all at once (by the standard of human reaction times). This is equivalent to standing next to a 100 TBq source for about ten minutes.
In the case of transforming up to nitrogen-12, the beta particles are positrons so you get a similar number of annihilation gamma rays as well.
Transforming hydrogen in the material down will give a pulse of (presumably thermal) neutrons at a similar sort of level.
Isotopic impurities are going to kill you
Even if you can chemically purify your material, you still have to worry about isotopic impurities.
Taking the lead to bismuth shielding transformation from the question. The intention is to use lead-208 to give bismuth-208 with a half-life of 3×10⁵ years. However, without isotopically purifying the lead, only half of it will be lead-208. A quarter will be lead-206 (I'll ignore the remaining quarter of the lead). That transforms to bismuth-206 which has a half-life of six days.
Suppose the shielding used only 1 g lead, so the result is 0.5 g of bismuth-208 and 0.25 g of a gram of bismuth-206. The bismuth-208 has a radioactivity of about 100 MBq which isn't too bad. However, the bismuth-206 is about 1 PBq which is a problem.
If you are transforming chemically pure hydrogen then transforming it up is going to turn the deuterium into helium-2 which is unstable and will immediately decay into two protons.
Although it's possible to enrich material, it's an expensive process. Getting it isotopically pure is difficult. If we could cheaply purify bulk materials then we'd use it to reduce soft-errors in computer chips and we wouldn't need to recover lead from shipwrecks.
Even a thousand-fold reduction of the bismuth, in example above, merely reduces the radioactivity to 1 TBq.
There are a small number of elements where there's only one naturally occurring isotope. For these, you could use chemical techniques to get good isotopic purity. These are: ⁹Be, ¹⁹F, ²³Na, ²⁷Al, ³¹P, ⁴⁵Sc, ⁵⁵Mn, ⁵⁹Co, ⁷⁵As, ⁸⁹Y, ⁹³Nb, ¹⁰³Rh, ¹²⁷I, ¹³³Cs, ¹⁴¹Pr, ¹⁵⁹Tb, ¹⁶⁵Ho, ¹⁶⁹Tm, ¹⁹⁷Au, and ²⁰⁹Bi.
Strictly, Protactinium has only one naturally occurring isotope (²³¹Pa) but seeing that has a short half-life (30,000 years) it's present only because of decay of other particles and any sample is likely to become contaminated with its decay products so we can drop that.
Similarly, one source I used would put ²³²Th on this list as being the only naturally occurring isotope, but another source reports a 0.02% mixture of ²³⁰Th, so I've omitted it.
It turns out none of these are interesting. Most don't have long-lived isotopes within two protons of them. The full list is (with half-lives, "a" is years):
| A |
Z−2 |
Z−1 |
Natural |
Z+1 |
Z+2 |
| 9 |
⁹He (very short) |
⁹Li (178.3 ms) |
⁹Be (stable) |
⁹B (0.54 keV) |
⁹C (126.5 ms) |
| 19 |
¹⁹N (271 ms) |
¹⁹O (26.88 s) |
¹⁹F (stable) |
¹⁹Ne (17.22 s) |
¹⁹Na (< 40 keV) |
| 23 |
²³F (2.23 s) |
²³Ne (37.25 s) |
²³Na (stable) |
²³Mg (11.3046 s) |
²³Al (446 ms) |
| 27 |
²⁷Na (301 ms) |
²⁷Mg (9.458 min) |
²⁷Al (stable) |
²⁷Si (4.15 s) |
²⁷P (260 ms) |
| 31 |
³¹Al (644 ms) |
³¹Si (157.36 min) |
³¹P (stable) |
³¹S (2.5534 s) |
³¹Cl (190 ms) |
| 45 |
⁴⁵K (17.81 min) |
⁴⁵Ca (162.61 d) |
⁴⁵Sc (stable) |
⁴⁵Ti (184.8 min) |
⁴⁵V (547 ms) |
| 55 |
⁵⁵V (6.54 s) |
⁵⁵Cr (3.497 min) |
⁵⁵Mn (stable) |
⁵⁵Fe (2.744 a) |
⁵⁵Co (17.53 h) |
| 59 |
⁵⁹Mn (4.59 s) |
⁵⁹Fe (44.490 d) |
⁵⁹Co (stable) |
⁵⁹Ni (7.6×10⁴ a) |
⁵⁹Cu (81.5 s) |
| 75 |
⁷⁵Ga (126 s) |
⁷⁵Ge (82.78 min) |
⁷⁵As (stable) |
⁷⁵Se (119.78 d) |
⁷⁵Br (96.7 min) |
| 89 |
⁸⁹Rb (15.32 min) |
⁸⁹Sr (50.563 d) |
⁸⁹Y (stable) |
⁸⁹Zr (78.41 h) |
⁸⁹Nb (2.03 h) |
| 93 |
⁹³Y (10.18 h) |
⁹³Zr (1.61×10⁶ a) |
⁹³Nb (stable) |
⁹³Mo (4.0×10³ a) |
⁹³Tc (2.75 h) |
| 103 |
¹⁰³Tc (54.2 s) |
¹⁰³Ru (39.247 d) |
¹⁰³Rh (stable) |
¹⁰³Pd (16.991 d) |
¹⁰³Ag (65.7 min) |
| 127 |
¹²⁷Sb (3.85 d) |
¹²⁷Te (9.35 h) |
¹²⁷I (stable) |
¹²⁷Xe (36.346 d) |
¹²⁷Cs (6.25 h) |
| 133 |
¹³³I (20.83 h) |
¹³³Xe (5.2475 d) |
¹³³Cs (stable) |
¹³³Ba (10.551 a) |
¹³³La (3.912 h) |
| 141 |
¹⁴¹La (3.92 h) |
¹⁴¹Ce (32.511 d) |
¹⁴¹Pr (stable) |
¹⁴¹Nd (2.49 h) |
¹⁴¹Pm (20.90 min) |
| 159 |
¹⁵⁹Eu (18.1 min) |
¹⁵⁹Gd (18.479 h) |
¹⁵⁹Tb (stable) |
¹⁵⁹Dy (144.4 d) |
¹⁵⁹Ho (33.05 min) |
| 165 |
¹⁶⁵Tb (2.11 min) |
¹⁶⁵Dy (2.332 h) |
¹⁶⁵Ho (stable) |
¹⁶⁵Er (10.36 h) |
¹⁶⁵Tm (30.06 h) |
| 169 |
¹⁶⁹Ho (4.72 min) |
¹⁶⁹Er (9.392 d) |
¹⁶⁹Tm (stable) |
¹⁶⁹Yb (32.018 d) |
¹⁶⁹Lu (34.06 h) |
| 197 |
¹⁹⁷Ir (5.8 min) |
¹⁹⁷Pt (19.8915 h) |
¹⁹⁷Au (stable) |
¹⁹⁷Hg (64.14 h) |
¹⁹⁷Tl (2.84 h) |
| 209 |
²⁰⁹Tl (2.162 min) |
²⁰⁹Pb (3.234 h) |
²⁰⁹Bi (2.01×10¹⁹ a) |
²⁰⁹Po (124 a) |
²⁰⁹At (5.42 h) |
For half-lives of over a million years (see below), there's Niobium-93 to Zirconium-93. For half-lives over a thousand years there's also Cobalt-59 to Nickel-59 and Niobium-93 to Molybdenum-93 and
Unshielded nuclei are either stable or going to kill you
People have been a bit blasé about letting materials beta decay after the transformation assuming that radiation isn't a problem. However, people are normally used to dealing with tiny quantities of radioactive materials with short half-lives where, in this context, "short" can mean years. As a rule of thumb, it's best to avoid interaction with gram (or kilogram or tonne) quantities of materials with half-lives below a million years. For milli- or microgram quantities you might be safe with shorter half-lives (thousands of years and years respectively).
You can see from the example above, that bismuth-208, with a half-life just below the million year mark I gave, is just about handleable in at the gram level. Radioactivity rates are inversely proportional to half-life. So, a material with a half-life of one year has a million times the radiation flux of a material with a half-life of a million years.
I'm not kidding, even the heating will kill you
In the same example, the goal is to transform uranium-238 to plutonium-238 via neptunium-238 to make a nuclear battery. Let's assume everything is isotopically pure
A radioactive heater unit might contain 34 g of plutonium-238. I think that's going to produce about 20 W of heat which is containable in 34 g of bulk material (it's about 5 °C/s of heating). The neptunium-238 has a half-life about ten thousand times shorter. After correcting for the different decay energies, the 34 g of neptunium produces about 100 kW and heats up at 20,000 °C/s (assuming the radiation is absorbed within the bulk of the material) meaning it vaporises about 200 ms after it's formed. A boiling cloud of neptunium (emitting about 400 PBq of radiation) is going to shatter a thin shield, melt its way through a thick shield and ruin your day.
Two nucleons are better than one
With a half-life threshold set at a million years, there are so few single nucleon transformations that give a suitably long-lived product, that it's possible to list them all:
- ¹⁰B → ¹⁰Be (1.51×10⁶ a)
- ⁴⁰Ar → ⁴⁰K (1.248×10⁹ a)
- ⁴⁰K → ⁴⁰Ar (stable)
- ⁴⁰K → ⁴⁰Ca (stable)
- ⁵⁰Ti → ⁵⁰V (2.65×10¹⁷ a)
- ⁵⁰V → ⁵⁰Ti (stable)
- ⁵⁰V → ⁵⁰Cr (> 1.3×10¹⁸ a)
- ⁵⁰Cr → ⁵⁰V (2.65×10¹⁷ a)
- ⁵³Cr → ⁵³Mn (3.7×10⁶ a)
- ⁸⁷Rb → ⁸⁷Sr (stable)
- ⁸⁷Sr → ⁸⁷Rb (4.97×10¹⁰ a)
- ⁹²Zr → ⁹²Nb (3.47×10⁷ a)
- ⁹²Mo → ⁹²Nb (3.47×10⁷ a)
- ⁹³Nb → ⁹³Zr (1.61×10⁶ a)
- ⁹⁷Mo → ⁹⁷Tc (4.21×10⁶ a)
- ⁹⁸Mo → ⁹⁸Tc (4.2×10⁶ a)
- ⁹⁸Ru → ⁹⁸Tc (4.2×10⁶ a)
- ¹⁰⁷Ag → ¹⁰⁷Pd (6.5×10⁶ a)
- ¹¹³Cd → ¹¹³In (stable)
- ¹¹³In → ¹¹³Cd (8.04×10¹⁵ a)
- ¹¹⁵In → ¹¹⁵Sn (stable)
- ¹¹⁵Sn → ¹¹⁵In (4.41×10¹⁴ a)
- ¹²³Sb → ¹²³Te (> 9.2×10¹⁶ a)
- ¹²³Te → ¹²³Sb (stable)
- ¹²⁹Xe → ¹²⁹I (1.57×10⁷ a)
- ¹³⁵Ba → ¹³⁵Cs (2.3×10⁶ a)
- ¹³⁸Ba → ¹³⁸La (1.03×10¹¹ a)
- ¹³⁸La → ¹³⁸Ba (stable)
- ¹³⁸La → ¹³⁸Ce (> 4.4×10¹⁶ a)
- ¹³⁸Ce → ¹³⁸La (1.03×10¹¹ a)
- ¹⁷⁶Yb → ¹⁷⁶Lu (3.76×10¹⁰ a)
- ¹⁷⁶Lu → ¹⁷⁶Yb (stable)
- ¹⁷⁶Lu → ¹⁷⁶Hf (stable)
- ¹⁷⁶Hf → ¹⁷⁶Lu (3.76×10¹⁰ a)
- ¹⁸⁰Ta → ¹⁸⁰Hf (stable)
- ¹⁸⁰Ta → ¹⁸⁰W (1.8×10¹⁸ a)
- ¹⁸⁷Re → ¹⁸⁷Os (stable)
- ¹⁸⁷Os → ¹⁸⁷Re (4.33×10¹⁰ a)
- ²⁰⁵Tl → ²⁰⁵Pb (1.70×10⁷ a)
That's a total of 39 possibilities. If you're prepared to drop the half-life safety threshold to one thousand years, you can add another 15.
It's not an accident that there are no good single nucleon transformations.
You'll notice that, in all the cases I listed, one of the two nuclei is radioactive, albeit with usefully long half-lives. This is consequence of the Mattauch isobar rule.
Beta decay will exploit quite small differences in mass excesses (by the scale of the total binding energy of the nucleus). You can imagine that nominally, a plot of mass excess versus protons for constant mass, has a U shape and isotopes beta decay down the sides to the middle. If this were the case there'd be only one stable isotope per mass.
There are a number of effects that prevent the sides of that plot being smooth. The most important is that nucleons like to pair up. If you're familiar with filling electron shells in atoms you'll be aware that electrons preferentially fill each shell with one electron before going back and filling in its pair. For nuclei it's different. It's energetically favourable to pair the nucleons first. So, nuclei with even number of protons and neutrons are more tightly bound than those with odd numbers.
This superimposes an even-odd zig-zag on the U shape. So, at the bottom of some Us there can be two local minima separated by two protons. The Z between these two is typically unstable, preferring to beta decay to either, or both, of the two adjacent nuclei.
So, if you can move by two nucleons, you don't create unpaired nucleons and there are many more possibilities: 136 with a million year half-life cutoff, and only an additional two with a thousand year cutoff (see also Isobar stability).
A couple of interesting ones are:
- ¹⁹⁶Hg → ¹⁹⁶Pt (stable)
- ¹⁹⁸Hg → ¹⁹⁸Pt (stable)
Mercury-196 is only about 0.15% of natural mercury but mercury-198 is about 10%. Needless to say, these will need to be isotopically pure. The other isotopes of mercury, 199, 200, 201, 202 and 204 do not have stable platinum isobars. The most stable has a half-life of 12.5 hours, the least stable is unbound.
Two nucleons once is better than one nucleon twice
It's already been suggested that the wizard can perform the magic twice to move by more than one proton. The problem is that the element between two stable isobars often has a very short half-life. Taking the mercury to platinum transformations I just listed, in both cases, the intermediate gold nuclide has half life of a few days and by now you know where this is going. These will produce petabecquerels of radiation per gram, and about a kilowatt of power per gram.
You need to change the rules
If you don't change the rules of your wizard's magic there's no good ending. The wizard has only useless transformations and the transformation of trace contaminants will be fatal.
As already noted, changing the wizards magic to move two nucleons is a big help. The next is to change the selectivity. If the wizard can target specific nuclides then the contamination problem disappears.
Maybe you can work the selectivity into the story as the wizard's special ability. It's easy to assume that the problem with magic is power, but maybe it's control. Perhaps, by analogy, when casting fireball the problem is not getting the fire: just open a portal to the elemental plane of fire. Perhaps 99% of the fireball spell is getting the fire to its target instead of boiling the wizard's brain.
So, if your wizard can control their magic, tune it to the right nuclides, let the transformation go slowly to give the material time to relax and cool from atomic shifts, then maybe they can survive to do something useful.
