Run the Stress Test

źlab Research · Field research outside energy · October 2026

The figure of merit no roadmap put a number on

Why Is There No ASML of Rare Earths?

Lithography turned one equation into public roadmaps and, after three decades, into EUV scanners built by ASML. Rare earth separation has its own equation, and for the hardest neighbours the number in it, in industrial use, has moved little since the 1960s. We compared the two industries on physics, roadmaps and the technical decisions that went wrong, and asked what would have to exist for separation to follow the same path.

Pr³⁺ 112.6 − Nd³⁺ 110.9 =1.7 pm· industrial β ≈ 1.5

Most of the difference chemistry has to grip between two neighbours: about 1.5% of an ion's size. This report follows it.

David Lagarejo · źlab

Executive summary

What we investigated. Why semiconductor lithography produced a chain of public roadmaps running since 1992 and, eventually, the EUV scanners built by ASML, while rare earth separation, the step that turns a mixed rare earth concentrate, from ore or scrap, into individual elements, did not. We read the physics, the roadmaps on both sides, and the public record of technical decisions that went wrong, and we had the cited sources reopened by an adversarial reviewer before publication.

The physics. Two neighbouring rare earths, praseodymium and neodymium, differ in ionic radius by about 1.7 picometres, roughly 1.5% of their size S1. That sliver is most of what chemistry has to grip. Industrial solvent extraction separates them with a separation factor near 1.5 S3, which is why plants run cascades of mixer-settlers that can number in the hundreds S5. A protein ligand reported in 2026, which its authors place among the best for any ligand, averages 2.1 across neighbouring pairs from Nd to Lu S6, against a series average of about 2.5 for the phosphoric acid extractants industry has used for decades S4.

The strongest finding. Lithography had public metrics that competitors shared and third parties measured, and a roadmap that coloured in red the targets for which Manufacturable solutions are NOT known S11. Rare earth separation has a figure of merit too, and the US Department of Energy wrote in 2011 that small improvements in separation factors would significantly decrease the cost S23. Yet the public plans we found, in China and the United States, set targets for recovery, purity, tonnage and emissions S24S25. None puts a number on the separation factor, and none marks which targets have no known solution.

What is not proven. We do not claim that a roadmap would have produced an ASML of rare earths. Measured like for like, the market is about a hundred times smaller than the one that paid for lithography. And in every rare earth project we examined, prices, finance or funding weighed at least as heavily as the technical variable.

Key terms in plain words

The article uses a few technical terms from chemistry, lithography and decision analysis. Each is defined once here.

Rare earth separation
Splitting a mixed rare earth concentrate, from ore or scrap, into individual elements such as neodymium or dysprosium. It comes after mining and before metal and magnet making.
Lanthanide contraction
The steady shrinking of the ions from lanthanum to lutetium. Each added electron goes into a buried 4f orbital that shields the growing nuclear charge poorly.
Separation factor (β)
The ratio of the two elements' distribution ratios between the two liquid phases in one contact, written so that it is at least 1. β = 1 means no separation; larger is easier.
Mixer-settler and cascade
A mixer-settler stirs two liquids together and lets them separate again. Many in series, with flows running in opposite directions, form a countercurrent cascade.
Fenske equation
A textbook relation, first written for distillation, giving the minimum number of ideal stages at total reflux. For a split where each product is p pure: stages = 2 ln(p / (1 − p)) / ln(β). At 99% and β = 1.5 that is about 23. It shows why small β means many stages.
Extractant
The organic chemical that pulls rare earths out of water into an oil phase. D2EHPA and PC-88A, both organophosphorus acids, are common industrial choices.
Rayleigh criterion
Lithography's resolution formula, R = k1 × λ / NA: wavelength λ, lens numerical aperture NA, and a process factor k1 whose physical floor, for a single exposure, is 0.25.
EUV
Extreme ultraviolet lithography, using 13.5 nm light made by hitting tin droplets with a laser. In volume production since 2019.
Technology roadmap and red cell
A public table of targets years ahead, agreed across an industry. In the semiconductor roadmap a red cell meant no manufacturable solution was known.
Second-law efficiency
The ideal minimum work of a separation divided by the work actually spent. It measures how far a process is from the physical floor.
Defensible, conditional, not supported
The three classes this research uses for a claim: supported by evidence today; true only if a named measurement comes out a certain way; or not supported by the evidence available.
Cheapest decision-changing measurement
The least expensive test whose result would change what a decision maker does next. It is the first thing to buy when a key claim is conditional.

1The 0.7 picometre problem

The rare earths are hard to separate because they are so alike. As you move along the lanthanide row the ions shrink steadily, an effect called the lanthanide contraction, because each added electron enters a 4f orbital that screens the growing nuclear charge poorly, pulling the outer shells inward. Those 4f electrons sit buried under filled outer shells, so they barely touch the outside world. Almost every lanthanide sits in water as a 3+ ion, and the main thing left to tell neighbours apart is size S2.

Figure 1ObservedFifteen ions, one gentle slope
98102106110114ionic radius, pmLaCePrNdPmSmEuGdTbDyHoErTmYbLuPr to Nd: 1.7 pm, about 1.5%La to Lu: 18.3 pm in 14 steps,about 1.3 pm per step (calculated)

Shannon effective ionic radii, trivalent ions, eight-fold coordination S1. The six-fold table gives a 0.7 pm Pr to Nd step, but its Pr value has only two decimals, so we plot the smoother eight-fold series. The heaviest neighbours are the closest, under 1 pm apart. For scale: if Nd3+ were a person 1.75 m tall, Pr3+ would be about 2.7 cm taller (calculated). Pm is radioactive and has no stable isotope; it is shown to keep the series continuous.

Size differences are tiny. In the standard Shannon tables, in eight-fold coordination, Pr3+ measures 112.6 picometres and Nd3+ 110.9, a gap of 1.7. Across the whole series the step between neighbours averages about 1.3 picometres, and the heaviest neighbours are under 1 apart S1. Neighbouring rare earths differ by less than 1.5 kcal/mol in how strongly they bind a common ligand such as EDTA S2.

Engineers capture this in one number, the separation factor, written β: how much more one element prefers the extracting phase than its neighbour. With β = 1, a stage does nothing. For the Nd/Pr pair in an industrial extractant, β sits near 1.5 S3. So a plant does not separate in one step. It repeats a small enrichment many times in a countercurrent cascade. The Department of Energy describes extraction trains of potentially hundreds of mixer/settlers, and says separating Nd from Pr alone can take 30 S5.

A textbook relation, the Fenske equation, gives the minimum number of ideal stages for a clean split. It shows how hard the number is to move:

Separation factor βMinimum ideal stages, 99% / 99% splitWhere this β comes from
1.5about 23Nd/Pr in industrial extraction (S3)
2.1about 12protein ligand, 2026, series average Nd to Lu, not Nd/Pr (S6)
2.5about 10phosphoric acid extractants, series average, not Nd/Pr (S4)
about 58about 2.3, ten times fewer than at β = 1.5calculated: what a tenfold smaller cascade would need

All stage counts in the table are calculated, at total reflux, and real plants need more. The point is the shape. Stage count falls only with the logarithm of β, so a tenfold smaller cascade for the hardest pair needs a separation factor near 58. That is the size of the gap between incremental chemistry and a different kind of plant.

Figure 2CalculatedMove β and watch the cascade shrink
Presets
Purity
23minimum ideal stages for a 99% / 99% split

Each block is one ideal mixer-settler stage at total reflux, from the Fenske relation. Real plants need more stages than this minimum. Raising purity from 99% to 99.99% doubles the count at any β; raising β cuts it only with its logarithm, which is why a full separation train can run to hundreds of real units S5. Preset values: S3S6S4.

2How lithography turned physics into a roadmap

Lithography has its own one-line equation. The smallest feature a scanner can print is R = k1 × λ / NA, where λ is the wavelength of light and NA the numerical aperture of the lens S9. Every term is a lever a supplier can sell: shorter wavelength, larger aperture, smaller k1. Over four decades the industry moved from 436 nanometre light to the 13.5 nanometre extreme ultraviolet (EUV) light used today S9.

What turned the equation into an industry was coordination. In 1992 a committee of the US Semiconductor Industry Association, led by Gordon Moore, produced a national roadmap that set targets fifteen years ahead S10. Its successor, the International Technology Roadmap for Semiconductors, colour-coded every target, and red meant Manufacturable solutions are NOT known S11. The National Research Council could point at the 2001 edition and say a brick wall currently exists in 2007 S12. A roadmap that publishes its own ignorance tells every lab and supplier where the money is.

The path was long and expensive. EUV was first demonstrated in 1986, and the Department of Energy began funding it in 1993 S13. In 1997 Intel, AMD and Motorola committed $250 million to work with three national laboratories S14. Source power, the brightness of the light, was voted the top issue in 2002 S44 and stayed a central problem for about fifteen years. In 2017 ASML reported 250 watts at the source, against about 25 watts in 2012, in a source not yet shipped in tools S16. TSMC began volume production with EUV in the second quarter of 2019 S15. That is 33 years from first demonstration (calculated). Unnamed experts quoted by the trade press put the industry's research and development investment in ASML's EUV effort at about $7 billion by 2014 S17.

Figure 3CalculatedOne lever moved 32 times. The other is waiting for a number
Lithography: wavelength of light, nm436g-line365i-line248KrF193ArF13.5EUV436 to 13.5 nm: about 32 times shorter (calculated)Separation: β for Nd/Pr, log scale1251020501001.23D2EHPA1.4PC-88A1.5PC-88A*2.5DODGAA, lababout 58: ten times fewer stages than at β = 1.5 (calculated)industrial 1.23 to 1.5 · lab 2.5

Left: production wavelengths S9S48; EUV in volume production from 2019 S15. Right: Nd/Pr separation factors of 1.23 with D2EHPA, 1.4 with PC-88A and 2.5 with a new laboratory extractant, DODGAA, in one controlled comparison S47; *the same PC-88A extractant gave about 1.5 in another study under different conditions S3. Resolution scales with 1/λ, while stage count scales with 1/ln β, which is why the right axis is logarithmic. The dashed line is the β at which the minimum stage count is ten times lower than at β = 1.5, from the Fenske relation.

The roadmap also killed options, but not the way the legend says. Several rivals had shown the resolution required. They died on manufacturability. X-ray lithography absorbed far in excess of $1 billion and was abandoned in the mid-1990s S20, although its IBM lead still wrote in 1997 that it was not expected to be limited by fundamental physics S21. In January 2001 ASML and Applied Materials dissolved their electron projection venture, eLith, citing throughput, mask cost and key customers' preference for EUV S22.

The 157 nanometre generation is the clearest case of independent measurement at work. In 2001 the National Institute of Standards and Technology measured an intrinsic birefringence in calcium fluoride, the lens material, of (6.5 ± 0.4) × 10−7 at 157.10 nm, considerably larger than the industry's specification S18. The result was public, and it applied to every competitor at once. When Intel removed 157 nm from its roadmap in May 2003, the main reason it gave was the availability of calcium fluoride in commercial quantities, along with pellicles and resists S19. The same Intel engineer said water immersion at 193 nm was at best a one-generation tool S19. ASML shipped its first immersion scanner that same year S45. The machinery was good at correcting itself, not at forecasting.

3What separation improved, and what it did not

It would be wrong to say rare earth separation stood still. In the 1970s the Chinese chemist Xu Guangxian replaced trial and error with a design theory for countercurrent cascades. In 1975 a single 80-stage cascade at a Baotou plant produced 99.9% lanthanum oxide, 99.5% praseodymium oxide and 99.9% neodymium oxide at the same time, and in 1986 a plant was scaled up in one step, with no pilot trial S7. Rhône-Poulenc ran more than 1,000 separation units to reach purities above 99.999% S8. Process optimization cut ammonia use from about seven tonnes to one tonne or less per tonne of oxide S2.

What changed was the design method, the reagent bill and the purity. What barely changed was the equipment class and the size of the gap between neighbours. Peppard introduced the extractant family still in use in 1957, and a 1972 study already reported separation factors of up to 2.5, with adjacent rare earths near one S32. Very different ligand families, from phosphoric acids to engineered proteins, land near the same average separation factor S4S6. The cost structure follows. The Department of Energy, citing the Oeko-Institut, called oxide separation perhaps the most capital-intensive step, at about 60% of production cost S23.

Meanwhile the United States let its own capability lapse. The Mountain Pass separation plant in California closed in 1998 following regulatory problems with the main wastewater pipeline S35. Ames Laboratory phased out its separation chemistry, and the Rare-earth Information Center closed in August 2002 S37. In 2010 Ames's senior rare earth scientist wrote, in responses to a House committee, that US research across extraction, separation, processing and recycling was virtually zero, while noting that some separation chemistry continued at national laboratories for nuclear waste S37.

4The roadmaps that exist measure something else

Separation does have public plans, and some of them are numeric. China's rare earth industry plan for 2016 to 2020 set goals to raise separation recovery from 90% in 2015 to 92% by 2020 for light rare earths, and from 94% to 96% for ionic clays. It also set a 20% cut in the intensity of pollutant emissions, and named ammonia-free separation and online detection as technology goals S24. In 2023 the US National Energy Technology Laboratory set metrics for a demonstration-scale facility: 1 to 3 tonnes a day of mixed oxide, separated oxides above 90% purity and, pending supply chain specifications, metals above 99.9% S25. The Department of Energy's Critical Materials Institute, created in 2013, reports extractants with separation factors two to three times those of existing extractants, for an unspecified pair S26.

Figure 4ObservedWhat the plans count, and what they leave blank
What a plan could targetStatusWhat we found
Separation recoverynumeric targetraise from 90% in 2015 to 92% by 2020 (light); from 94% to 96% (ionic clay) S24
Emissionsnumeric target20% cut in pollutant emission intensity S24
Throughputnumeric target1 to 3 t/day mixed oxide, demonstration scale S25
Puritynumeric targetindividual or binary separated oxides above 90 wt%; metals above 99.9 wt%, pending specifications S25
Energynumeric targetshare of firms meeting energy standards, 40% to 90% S24
Separation factorno numbernamed as a research aim (S23); CMI reports 2 to 3 times better extractants as a result, not a target (S26)
Distance to the thermodynamic floorno numberno published second-law efficiency found
Targets with no known solutionno numbernothing marked red

Public plans read as a roadmap would read them. Lithography's roadmap also had a status this table cannot show: a red cell, the last of its four colours, meaning Manufacturable solutions are NOT known S11.

Read against lithography, the difference is what is being targeted. These plans set goals for what a plant produces and emits. The Department of Energy has named higher separation factors as a research aim since 2011 S23, but no plan we found puts a number on the separation factor of the hardest pairs, publishes who measures it, or marks a target as red. We found no published figure for the thermodynamic efficiency of rare earth separation at all. The ideal work to unmix a 50/50 binary pair is RT ln 2, about 1.7 kilojoules per mole of mixture (calculated), but no one we found has set the real process against it. The most specific energy estimate for solvent extraction we found is described by its own authors as a rough estimate, built as a proxy from yttrium and uranium plants S8. Lithography's roadmap tracked its distance to physical limits, such as the floor of k1 = 0.25 S9. For separation we found no such distance written down.

5The physical handles nobody tabled

Ionic size is not the only difference between rare earths. It is just the one solvent extraction uses. The public literature describes others, each with a different reach:

Physical handleHow large the difference isPublic status
Volatility of the metalYtterbium is volatile and lutetium is not; a patent claims Yb removed from Lu at over 1000:1 by vacuum sublimation (S31)Patented, laboratory example; aimed at medical isotope supply
Oxidation stateSelective reduction of samarium followed by distillation gave a Nd/Sm factor of about 570 (S2)Laboratory
Magnetic momentFree ion values of about 10.6 Bohr magnetons for Dy3+ against 3.6 for Nd3+ (calculated from Hund's rules)Laboratory physics
MassPr weighs 140.91 and Nd 144.24 in standard atomic weight (S33), but Nd and Sm share isotopes of mass 144, 148 and 150, so mass alone cannot split every pairModelled for rare earths only (S27)

Some of these give separation factors far beyond solvent extraction, but mostly for pairs that are not neighbours, and none has a public table saying which pair, at what factor, at what throughput, measured by whom.

Plasma mass separation is the most instructive history. In 2002 Archimedes Technology Group published quantitative targets for a plasma mass filter to process Hanford nuclear waste surrogates: 0.13 moles per second of feed at under 500 electron volts per ion, with first results in early 2003 S28. That is exactly what a roadmap asks for: numbers, in advance. A later Princeton Plasma Physics Laboratory review says the device was able to demonstrate separation, but that difficulty in creating the plasma and operating the end electrodes prevented adequate results S29. We found no public record of measured values against the 2002 targets. The company ceased operation in 2006 after failing to receive federal funding S30. For rare earths, a 2018 analysis concluded that full scale demonstration of the entire plasma separation process thus remains to be made S27. The targets were public; we found no published measurements against them.

6Where governing the evidence would have helped, and where it would not

A decision rests on claims. Some are defensible on the evidence at hand, some hold only if a specific thing is measured, and some are not supported. Sorting them before money moves, and naming the cheapest measurement that would change the decision, is a small discipline. Applied to the public record of the cases above, it reads like this. These are our classifications for illustration, made from public documents.

Figure 5IllustrationFrom a claim to the measurement that would change the decision
1 Claim19,050 t a year fromMountain Pass by 20122 ClassConditional on thereagent loop working3 Cheapest testrun the loop atcommercial scale4 Decisioncommit, stage it,or measure firstThree classes: defensible today · conditional on a named measurement · not supported by the evidenceThe prospectus said its chlor-alkali reagent and water technology was not yet proven at commercial scale (S34)

The method in four steps, with the Molycorp 2010 restart as the example. The classification is the author's illustration from public documents S34S36.

It is just as important to say where this would not have helped. Molycorp's Resources segment average selling price fell from $82.00 to $36.12 per kilogram between 2011 and 2012 S34. In March 2015 the company said its chlor-alkali plant was performing well S36, and the company still filed for Chapter 11 on 25 June 2015 S36. Lynas, in its first quarter and in a weak market, realised just over A$5 per kilogram on a low-value product mix while customers were still qualifying its output, against a cost of about A$15 S38. Archimedes ended for lack of funding. In each case the technical variable was real, but on the public record it was not the only thing, and probably not the main thing, that shaped the outcome. Governing the evidence narrows engineering claims. It does not set prices, raise capital or write policy.

7Comparing like with like

The obvious objection is scale, and it is right. Comparing the value of a material with the revenue of a machine maker mixes two levels of the chain. The fair comparison is between the markets that pay each tool industry:

Same levelChipsRare earths
Market that pays the toolmakers$796 billion, semiconductors, 2025 (S39)about $7 billion, rare earths sold in 2024 at 2024 to 2025 prices, consultancy estimate (S40)
The tool market itselfabout $29 billion, lithography equipment, market research estimate; ASML alone reported €32.7 billion of 2025 sales, including service (S41, S46)not published by anyone we found
Tool market as a share of its customersabout 3.5% to 5% (calculated)the same share would be roughly $250 to $350 million a year (calculated)
Figure 6CalculatedThe market that would have to pay for an ASML of rare earths
$0.1B$1B$10B$100B$1,000BSemiconductor market, 2025$796BASML sales, 2025, with serviceabout $36.9BLithography equipment, about$29BRare earths sold, 2024$7BSame tool share of rare earths$0.25B to $0.35B (calculated)

Each gridline to the right is ten times larger, so $796B is about 114 times $7B. Sources: WSTS S39, ASML, converted at the 2025 average of about 1.13 dollars per euro S46, market research S41, a consultancy estimate S40. The last row is our calculation: lithography tools take about 3.5% to 5% of what the chip market is worth (3.7% from the market estimate, 4.6% for ASML alone with service), applied to rare earths. Years and price bases differ; read it as orders of magnitude.

The years, sources and price bases differ, and rare earth prices swing hard, so treat these as orders of magnitude. The Department of Energy put the rare earth market at about $3 billion in 2010 S23. Even so, the gap is roughly a hundredfold. An ASML of rare earths cannot be paid for the way the real one partly was: by Moore's Law and by customers rich enough to fund their supplier directly, as Intel, TSMC and Samsung did in 2012 when they committed €1.38 billion of research funding over five years S42.

If such a thing ever exists, it will need a different payer. One is beginning to appear. From 1 January 2027, a US defense contract clause restricts, with exceptions, neodymium-iron-boron magnets made with material from China, Russia, North Korea or Iran at any step from mining to the finished magnet S43. A buyer that pays for security of supply is not Moore's Law. But it is a buyer that needs to know, in advance and in numbers, which separation claims are defensible.

8The questions nobody asks

None of these requires new science. They are the questions a roadmap forces, and the public record of rare earth separation does not answer them.

  1. What separation factor, for which pair, would change the plant? For Nd/Pr, a tenfold smaller cascade needs β near 58 (calculated). Is anyone aiming at a number, or only at improvements?
  2. Which targets are red? Which goals have no known manufacturable solution today, written down where every lab and supplier can see them?
  3. Who measures, and is it someone other than the claimant? Lithography's clearest turning point, at 157 nm, came from a third-party number published to everyone at once.
  4. Which physical handle is being used, and for which pair? Size, volatility, oxidation state, magnetic moment and mass reach different pairs at different factors. A claim that does not name the pair cannot be checked.
  5. How far is the process from the thermodynamic floor? We could find no published second-law efficiency for rare earth separation. Lithography's roadmap tracked its distance to physical limits.
  6. Which unit has run continuously on the real feed? Not a peak rate, not a pilot on clean material. The Molycorp and Lynas records turn on exactly this.
  7. What is the cheapest measurement that would move a red cell? In 2001 it was one polarimetric measurement of a crystal. It is rarely the most expensive study.

Conclusion

Lithography did not win because its physics was easier. It spent years and billions against a power target it kept missing. Part of why it won is that the physics was written into a public table: shared metrics, some measured by third parties, with the unknowns coloured red. Money, talent and suppliers could aim at that table.

Rare earth separation has its own equation, and its key number for the hardest neighbours, in industrial use, has moved little since the extractants of the 1950s and 1960s. The plans that exist count tonnes and purity. None puts the separation factor, the distance to the physical floor, or the open problems in one place. The market is too small to buy an ASML. A public table of what has been measured, by whom, and what is still unknown costs far less, and we found none.

Methods and sources

We reviewed the physics of lanthanide separation, the history of lithography roadmaps and the public record of technical decisions in both industries, using sources retrieved between 15 and 20 October 2026. Four research passes gathered sources. Three adversarial passes, on the research, on this text and on the figures, then reopened the cited sources, checked quotes, numbers, dates and scope, and attacked the argument. Claims it could not confirm were removed or reworded. Values marked calculated are our own arithmetic from cited inputs and are not published figures.

Sources are graded: A peer-reviewed study, government document or regulation; B company filing or release, patent, thesis, book chapter or interview; C news, trade press or consultancy estimate. The classifications in section 6 are the author's illustration from public documents, not the output of any analysis run on these companies. This article is not legal, engineering or investment advice, and nothing here alleges that any named organization acted improperly.

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