[Enthalpy]

Replacing ⁹⁸Mo by a fissile material as a target for the tokamak's neutrons would seemingly ease the separation of the Mo produced then by fission. Just chemical separation of Mo would provide a big proportion of ⁹⁹Mo, and the other isotopes disappear quickly or are stable. Alas, this attempt is impractical.

The mini-tokamak itself is nonproliferating. It would produce too little Pu or ³H to make bombs. Keeping this advantage excludes targets of ²³⁵U and Pu. Natural U or ²³⁸U or ²³²Th can make nonproliferating targets, but they need fast neutrons to fission, like 14MeV, and then only very thick target blankets intercept the neutrons, and very little ⁹⁹Mo must be extracted from much fissile material.

A fission cross-section like 1b needs 4t/m² of ²³⁸U to intercept 14MeV neutrons with fair probability. It also needs more neutrons, as the yield to ⁹⁹Mo is like 5%.

Fission yields, cross-sections and more data is available there, thanks
  iaea.org - bnl.gov , also iaea.org - wikipedia

[Enthalpy]

All data sources indicate the mean energy of thermalized neutrons as kT.
I believe it should be 1.5×kT, for instance 39meV at 300K.

[Enthalpy]

[...] So 2.1×10¹⁵/s neutrons as well. [...]

Wrong! Every second D+D reaction produces T+p, but T has 1MeV recoil energy while the magnets confine only mean 20keV hydrogen. Tritium escapes the reaction zone.

2.1×10¹⁵n/s need a machine 8×8×8× smaller than Iter rather. Or better magnets, as these progress.

And the neutrons have 2.45MeV as emitted by the D(D, n)T reaction, not 14.1MeV as in T(D, n)⁴He.

[Enthalpy]

Natural uranium blankets and thermal neutrons could produce ⁹⁹Mo after all, but this takes a bigger tokamak, and the little molybdenum must be extracted from much uranium. At least, no isotopic separation is needed.

300K neutrons see 450b section for ²³⁵U fission, 67b for ²³⁵U absorption, 2.2b for ²³⁸U absorption. 0.72% and 99.28% abundances give natural uranium mean 5.9b with 55% chances to fission ²³⁵U. Natural C provides 4.9b for collisions (22mm mean free path) and 3.2mb for absorption. Accepting 1/3 neutrons lost in C, 923 moles C pass the neutrons many times through each mole U in a Brownian motion. Mean 510 collisions before absorption let a neutron spread by very roughly 0.5m rms. The uranium load is a bit over 1t.

Neutrons leaving a blanket inwards shall serve at opposite blankets. U absorbs too strongly between 1keV and 3eV so most neutrons must survive in well separated C. I didn't check deuterated polyolefins, light water nor the proliferating heavy water. Liquid oxygen could replace graphite, save a little bit uranium, and bring many drawbacks. Colder moderators bring little and liquid deuterium has big drawbacks. And the tokamak's coils must sit somewhere but will catch some neutrons.

Materials that absorb neutrons little: ethylene and propylene glycol carbonates and their eutectic, oxalates of alcohols or Be, including deuterated variants.

Some 30mg ⁹⁹Mo made in a day must be extracted from >1t U. As the fission destroys the U molecule, maybe the chosen environment could build a new Mo molecule easy to separate.

6.1% of the fissions yield ⁹⁹Mo. Maybe every second neutron misses the blankets or is absorbed elsewhere. 1/3 is lost in C. Among U events, 55% produce Mo. All this takes 100 neutrons per Mo atom, so the tokamak is only 1.7×1.7×1.7× smaller than ITER, alas. But producing no energy and needing no tritium, it's simpler. Adding U blankets at the top and if possible the bottom would reduce the size.

Marc Schaefer, aka Enthalpy

[Enthalpy]

I took initially an arbitrary size. So what tokamak size meets the demand for radioisotopes by fission of natural uranium by thermalized neutrons?

• The world uses 12kCi per week ⁹⁹Mo. 15 tokamaks around the world shall supply it for swift delivery and redundancy. That's 160Ci per tokamak and working day.
• The machines spend 1/3 of the time on ⁹⁹Mo, the rest on other radioisotopes. I take 4h irradiation spread on a day for ⁹⁹Mo, plus idle time to load, unload, maintain.
• 66h half-life need to produce 1.4×10¹⁴ atoms/s ⁹⁹Mo.
• 100 neutrons per ⁹⁹Mo atom need 1.4×10¹⁶ neutrons/s. 8³× smaller than Iter fed with D+D provides 2.1×10¹⁵ neutrons/s, so the machines would be 4.2³× smaller than Iter.

Plasma size: H=1.6m D_i=1m D_o=3m. The volume is Iter /77.

[Enthalpy]

Tritium escapes the reaction zone.

No it doesn't. "Welcome to double checkers", as I said. For instance at 5T, 1MeV tritium and 3.5MeV helium turn in few cm. Which keeps in the plasma this part of the produced energy.

So, back to 0.5+0.5 neutron per D+D fusion, with energies 2.45MeV and 14.1MeV.

The tokamak meeting the radioisotope demand has a volume 150× smaller than ITER, dimensions /5.3³, plasma H=1.3m Di=0.8m Do=2.3m.

[Enthalpy]

A beam of protons can fission uranium into ⁹⁹Mo. But it isn't very convenient.

Protons above 20MeV fission ²³⁸U with 1.3b section. ²³⁵U is as sensitive, Pb and Bi less so, and I have no data for Th, so natural uranium is the target
  oecd-nea.org p784 (8MB)
I didn't check if deuterons or helions improve.

The section changes little up to 200MeV, a longer proton path in uranium improves fission chances. Braking from arbitrary 60MeV to 20MeV takes 3mm in metallic uranium
  physics.nist.gov
so mean 1.6b provide 2.4% fissions per proton. Starting from 30 or 40MeV needs much more beam power. 100 or 140MeV save some power but need a longer linear accelerator.

The cumulated yield of fission products resembles that of neutron fission
  escholarship.org - www-nds.iaea.org
so I take 6% ⁹⁹Mo per fission. Producing 1.4×10¹⁴ atoms/s as in the 05 Jul 2022 message needs 16mA beam intensity.

60MeV need then 1MW beam power, ouch. Niobium cavities lose little power, oven magnetrons are affordable (synchronization?), electricity costs 1M€/year of which 1/3 is for ⁹⁹Mo.

The target must evacuate 1MW heat. Sweeping a D=2mm spot at non-sine >100kHz isn't trivial and needs 10-100kVA; interrupting the beam can help. Maybe the beam can instead be defocussed to 1dm² and the 2mm thin target be strongly tilted, say as a deep cone backed with flowing water. Then some 3kW/cm² resemble the flux at the wall of a rocket engine. Multiple foils, wires, grains of uranium compound in water behind a thin wall would ease that.

Neutrons are produced by fission and spallation, possibly above 1MeV. Useless radionuclides too.

And now the good news: 0.2mg ⁹⁹Mo must be separated from 1kg uranium only, and chemical separation suffices.

Marc Schaefer, aka Enthalpy

[Enthalpy]

First alternative using a proton beam:
  ¹⁰⁰Mo(p,np)⁹⁹Mo (for chemists: ¹⁰⁰Mo + p ⁹⁹Mo + p + n)

¹⁰⁰Mo makes 9.6% of natural Mo, the target is highly enriched. The swept or defocussed proton beam passes a window (thin Be, carbon-carbon, thinner Ni...) and brakes from 75MeV (more would save electricity) to 20MeV in ¹⁰⁰Mo over 79kg/m² range (7.7mm if metallic) where the sought reaction has 0.14b section, according to
  oecd-nea.org p482 (8MB)
so 0.66% of the protons make a ⁹⁹Mo, and 1.4×10¹⁴ atoms/s need a 3.4mA 250kW beam. Bad, but less so than protons fissioning uranium.

The target can be immersed, for instance as biassed foils, wires, grains... in water, so cooling is reasonable. Maybe it's a concentrated solution.

I believe the Nb, Ru... byproducts must be eliminated before filling the "moly cow" with the mixture of ⁹⁹Mo and ¹⁰⁰Mo
  wikipedia
Whether 0.2mg/1kg suffice?  More compact cooling can reduce the target below 1dm². Or if the nuclear reaction breaks the molecule, separation can ease. After use, the customer returns the ¹⁰⁰Mo for reuse.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Other alternative with a proton beam: convert to neutrons for absorption by ⁹⁸Mo.

The table compares conversion nuclides. Data is from
  oecd-nea.org (8MB)
which lacks neutron data for W Re. I didn't check the more usual Pb Bi. I aim here 1.4*10¹⁴ neutrons/s as if the subsequent step were lossless. Deuterons may well outperform protons.

Nuclide  Page   MeV in   MeV out  kg/m2   Thick     Barn   Conv    mA      kW
==============================================================================
  ²H       4      25       10      2.4     27m      0.15   1.1%     2.0    51
  ⁷Li      6       6        2      0.52   0.94mm    0.3    0.13%   17     102
  ⁹Be     11      18        3      4.1    0.22mm    0.1    0.27%    8.2   147
¹⁸¹Ta    673+     70       13     82      4.9mm     +++    6.0%     0.37   26
==============================================================================

I couldn't think of a good deuterium compound, so the target would be gaseous, under pressure for shortness, which needs a strong thin window: thin nickel, cold-drawn stainless, beryllium? For instance 1mm Be uses 3.4MeV above 25MeV. Deuterium seems the cleanest neutron source.

Lithium needs some vessel too, as I didn't find a good compound. It can have the natural composition. Beryllium instead could be a naked thin layer over the coolant, easier. It's monoisotopic naturally. The next light elements appear to need more and more power.

Heavier elements like lead expel more neutrons. Here refractory tantalum eases the design, like a naked thin layer over the coolant, or biassed foils in water. It's monoisotopic naturally. I cumulated the sections for 5, 4, 3, 2 neutrons over energy ranges.

The irradiated target produces many nuclides and particles, more so with heavy elements and high beam energy. But a neutron converter can at least be sealed.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Finally I've read a bit about how other Sapiens produce neutrons without uranium fission.

========== Electrostatic accelerators

They provide a big beam current like 1A but a low energy, like 0.2MeV
  Verbeke
so only D+D and D+T react. The beam smashes D and optionally T ions on a metal target that adsorbs them before new ions collide with them. The metal wastes much ion energy, and fusion reactions are rare at 0.2MeV. 1.4×10¹⁴ neutrons/s (if each one makes a ⁹⁹Mo!) then need 0.5MW beam power. As the D+D reaction is 20× slower than D+T at this energy, this neutron flux demands tritium, proliferating and obtained from a uranium reactor. Not my goal.

The targets are of Ti or Nb as they adsorb up to 2 hydrogen atoms per metal atom and are lighter than Pd hence brake protons less.

• Tri LiD and LiT? Li brakes protons less than Ti does. But LiH  isn't refractory, Li neither.
• Some alloys are abnormally light: NiTi, also Invar, bell bronze. Do they store hydrogen well?
• Thin Ti and Nb are deposited on Cu or cooling. Does warm Cu divert some hydrogen? A layer of impermeable material between Ti and Cu void avoid it. W, Ta, Mo, WC, BN, I don't quite know which is good.

========== Linear accelerators

Resonant cavities at 2K achieve a higher ion energy. 3MeV is reasonably short, 20MeV takes already a long building. This limits targets to Li, maybe Be and D.

========== Synchrotrons

High ion energy, optionally high beam current, focus, but look expensive. Small units are under development, interesting.

========== Cyclotrons

30MeV fits in a normal room, 90MeV is a bigger machine. Most cyclotrons work at room temperature, some use superconducting magnets.
  wikipedia
3mA is a record
  arxiv - arxiv - accelconf.web.cern.ch

Cyclotrons produce already varied medical radioisotopes by proton impact. Some hospitals are equipped (with low beam intensity). Cyclotrons also make neutrons by spallation, usually of Pb or Bi at higher proton energy. This is a path to ⁹⁹Mo.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Reactions cross-sections exist for protons, deuterons and alphas, ranges too, many thanks
  oecd-nea.org - oecd-nea.org - oecd-nea.org (8MB 3MB 5MB)
  physics.nist.gov - physics.nist.gov
and cyclotrons accelerate them all, so which is better?

========== Alphas are bad

They're known to extract a neutron from Be, but at meaningful 18MeV, their 58µm range obtains only 0.014 neutron for 100 alphas, while protons obtain 0.27 neutron.

Entering Ta with reoptimized 120MeV and 2.8mm range, the cumulated sections for emitting 8 to 2 neutrons provide 0.015 neutron per alpha. A 70MeV proton achieves 0.06 neutron.

I checked 70MeV alphas, accessible to the same cyclotron as 70MeV protons, in favourable ⁹⁷Mo. The beam power is 30× worse than for p hitting Ta.

========== Deuterons

To estimate deuterons range, I double the range of protons that have half the energy.

I suppose some D+D reactions make more neutrons than here fusion, but I have no data.

I limit the deuteron energy to 20MeV, more would improve much. The same cyclotron accelerates protons to 40MeV.

Nuclide  Page   MeV in   MeV end  kg/m²   Thick     Barn   Conv    mA      kW
==============================================================================
  ²H    Absent    20        1      1.1   7m 1bar    0.1    0.33%    6.7   134  Only fusion!
  ⁷Li      6      20        2      3.0    5.6mm     0.3    0.77%    2.9    58
 ¹¹B      19      20       12      3.2    1.4mm   2×0.045  0.16%   14     285
 ⁵¹V      77      20       10      2.6    0.43mm  2×0.6    0.41%    5.5   109  Not so dirty?
 ⁹⁸Mo    195      13        6      2.0    0.19mm    0.2    0.024%  81    1200  ⁹⁹Mo directly
==============================================================================

========== Protons update

Limiting the cyclotron size to 40MeV protons.

⁶⁹Ga makes 60% of the natural abundance, and the 40% ⁷¹Ga are decent too.

Nuclide  Page   MeV in   MeV end  kg/m²   Thick     Barn   Conv    mA      kW
==============================================================================
 ⁶⁹Ga    281...   28       15       9     1.5mm   2×0.4... 1.3%     1.7    47
 ⁷⁵As    324...   40       20      19    ~3.3mm   3×0.3... 1.2%     1.9    78
¹⁸¹Ta    673...   40       13      28     1.7mm   3×0.6... 2.1%     1.1    42
²⁰⁹Bi    738...   40       13      29     3.0mm   4×0.5... 2.2%     1.0    41  Looks dirty
==============================================================================

So both protons and deuterons are interesting, alphas are not, and 40MeV can suffice though suboptimum. I didn't check the paramount unwanted radioisotopes.

All these computations would need software.

The cyclotron draws power for its magnetic field, so less beam MeV buy many more kW. This also favours fewer production sites.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Thoughts about cyclotrons, here the coil for DC field. I haven't read about beam focussing etc, which constraint much the design, so exert due mistrust.

========== Wind the coil

Many DC electromagnet coils seem to use thick rectangular conductors, difficult to wind, and flow the coolant parallel to the current on a long narrow path.

Thin band fills well the volume and is easily wound around the magnet's axis as a spiral. Or two couterrotating spirals in an other, fed where they meet, let return the current from the extreme turns to ease the insulation. Uninsulated OFHC copper band (or pure aluminium) is easily available. A thin insulating film can separate the turns.

The example sketch takes generous 1.4T in 0.22m gap so 40MeV protons orbit with R=634mm. This needs 2×123kA×turn in the coils. 428 turns per coil take 287A, 5.65m×428turns of (not displayed to scale) 0.6mm×0.5m lukewarm copper resist 0.16Ω per coil for safe 46Vdc. Consumption is 13.3kW per coil. 1.0T in 0.12m gap would take only 2.0kW per coil.

Commercial 0.1mm thin insulating film can zigzag between the metal turns: Petp, Pi... Metal fills 85% of the volume. Some mm film width spaced by few unsupported cm leave room for the coolant. Some films are adhesive, even at heat (polyimide). Heat-resistant epoxy seems possible, glassfibres too.

Cores of DC magnets are often of pure Fe, from Armco or other. Fe saturates at 2.1T, expensive FeCo at 2.3T. For instance four 45° pillars close the flux loop at 2.0T. I believe the faces near to the RF acceleration voltage must be plated with Au, or at least Cu or Al.

========== Cool the coil

I compute with liquid commercial diphenyl oxide/biphenyl (Dowtherm A, Terminol...), but farnesane, phytane... would outperform them and fatty esters (exist as transformer oil) be biodegradable.

From 40°C to 50°C, the fluid absorbs 16kJ/kg so 0.81dm³/s per coil suffice. 428 turns spaced by 0.1mm on mean R=0.9m offer almost 2×0.12m² to the flow. The laminar speed parabola peaks at 10mm/s, its curvature over 50µm is 8.0Mm/s/m², so 2.3mPa×s viscosity drops 18kPa/m or 0.18bar over 0.5+0.5m.

The temperature drop across the copper and fluid thicknesses is tiny, the drop in copper across a spacer half-width too.

A complete coil can be encased in glassfibre composite for instance. Epoxy at the spacer film would resist reasonable fluid pressure, sewing across the coil thickness resists more.

========== More uses

Eddy currents in the wide bands prevent quick changes of the induction. Split width improves that. A DC electromagnet doesn't care and benefits from superior cooling, other uses too.

Take a copper coil, R=0.2 to 0.4m, H=0.3m. An axial flow of 80dm³/s of deionized water can extract 10MW from 40°C to 70°C. Water in 10% of the section needs only mean 2m/s while 90% copper resists 0.7µΩ/turn², so 10MW electricity provide 3.8MA×turn that create 30T in air. Counterrotating spirals can leave the duct at ground potential. The copper foil might be insulated. Forces must be addressed.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Some metals conduct much better at cold without superconducting. This can save power despite the chilling effort, as I already suggested there
  chemicalforums

Copper resists only 20pΩ×m at 20K instead of 20nΩ×m when lukewarm
  nist [5MB] page 40

========== Cyclotron

Imagine the previous cyclotron electromagnet at 2.0T instead of 1.4T, with the current ×1.43. Neglecting iron saturation, which I shouldn't, the losses climb to 2×27kW if lukewarm, but at 20K it's only 2×27W.

Foam would leak too much heat, but multilayer superinsulation can leak 10W per coil. It must be enclosed in bags for vacuum. This is available commercially.

If each coil needs 40W cooling, the common chiller consumes 4.2kW if 30% as efficient as Carnot's limit. This saves 50kW.

Cyclotrons operate in radiation bunkers, so gaseous helium shall chill the coils, not hydrogen. From 19K to 21K at 1atm it absorbs 10.5kJ/kg so each coil needs 3.8g/s. Density 2.4kg/m³ takes 1.6dm³/s per coil.

As previously, flow through 0.12m² means 19mm/s at the top of the laminar flow parabola. If the cooling channels are 25+25µm thin now, with curvature 62Mm/s/m², 3.5µPa×s viscosity take 220Pa/m, dropping only 2mbar in the coils.

More than 2.0T is possible without superconductors, but iron can't help then.

Chilled metal would also reduce the losses at the accelerating cavities or equivalent.

========== 30T electromagnet

Instead of 10MW warm as previously, copper would consume 10kW at 20K. Beware I didn't check the magnetoresistance, so this may be very wrong.

Heat leaks are small here. If 30% as efficient as Carnot's limit, the chiller consumes 520kW instead of 10MW for the warm coil.

In both examples, the conductors are much cheaper and easier to process than superconductors, the cold is easier to achieve, and the remaining consumption doesn't justify superconductors.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Transverse magnetoresistance of Cu and Al at cryogenic temperatures:
  lss.fnal.gov

• Strong field hampers much the high conductivity obtained by cold.
• Aluminium is abnormally insensitive to the field.
• The cyclotron coil remains valid, the 30T electromagnet is much affected.

In the more favourable transverse effect, the induction is perpendicular to the current. It happens at electromagnets like here.

========== Cyclotron coil

For 2t in the gap, each coil provides 176kA that create up to 0.7T or 7kOe in the 0.3m wide slit hosting the coil. From fig.1 in the pdf, the magnetoresistance would add 1.4× the zero-field resistivity of Al.

The 0.7T appear only near the open end of the slit, so the 1.4× is reduced by /3 as a mean, or rather by /2 as the current redistributes.

The pillars carrying the return flux cover only half of 360°. Where they're absent, the induction is around /3 and the 1.4× drops even more. Almost a factor /2 gained over 360°.

The pillars can stand farther away to reduce the induction through the coils. Improve easily /2. They can also be thicker and narrower to improve the gain over 360°.

So the magnetoresistance adds rather 0.35× at the sketched cyclotron coil, and with minimal optimization 0.1×. Fine.

========== 30T electromagnet

The field reaches 30T or 300kOe at the coil's inner face and almost as much at the outer face. Fig.2 shows Cu up to 100kOe, so if Kohler's law holds, the magnetoresistance adds up to 170× to the cold resistivity, ouch. Fig.1 shows Al measured up to 40kOe: if daring to extrapolate linearly the curve of 2100 residual resistivity ratio (RRR) to 300kOe, the magnetoresistance adds up to 6.2× to the cold resistivity, provided that the current distribution is stable.

The field increases linearly from zero at the section's centre point but decreases beyond a half-thickness, so we can count 2/3 of the 6.2×, that is 4.1×.

Purer Al can rescue the chiller consumption. Increasing the RRR from 2100 to 28200 worsens by 2.8 the magnetoresistance but it's relative to the zero-field cold resistivity which drops by 4 to 5, providing a net 1.6 gain.

Together, magnetoresistance makes the ohmic loss and chiller power 3.6× worse than I estimated in the previous message: 1.9MW instead of 10MW lukewarm. Still interesting. Reooptimize the temperature?

========== Laminar flow

I supposed a laminar helium flow, but η=35µP=3.5µPa×s doesn't help the intuition. ν=η/ρ=1.4mm²/s and 13mm/s in 100µm channel width define Reynolds Re=u×h/ν ~1 <<2000 so the flow is laminar.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Among the uses of strong magnets is magnetic resonance imaging MRI, and also NMR
  wikipedia - wikipedia
It goes without saying, but maybe better if I say it.

Most MRI machines have superconducting coils cooled by liquid helium to create 1.5T. Possibly aluminium coils around 20K can replace them for cheaper. I should throw a few figures at that some day.

Marc Schaefer, aka Enthalpy