The One Man

by Andrew Gross

  “Good. Now to your question, boy, effusion is the rate of transference of a gas through a probe or, better yet, a membrane. Graham’s Law postulates that if the molecular weight of one gas is two times that of another, it will diffuse through a porous layer, or even an opening the size of a pinhole, at the rate of the other times the square of two. It is the key postulate in the separation of isotopes—which have the same molecular structure yet different atomic weights.”

  “Separating isotopes … Porous layers … Why do you need to teach me all this?” Leo shrugged, clearly already a bit bored.

  “For now, just let this all soak into your head, lad. Look—” Alfred took out a piece of chalk. He had a tin sheet, a scrap of metal left from the motor shed. “All that matters now is how it’s expressed as a formula.” Alfred scratched out:

  He asked, “Do you have it?”

  Leo stared at it, repeating it to himself. “I think so.”

  “Therefore the inverse of this equation is…” Alfred erased it with his sleeve and wrote out a new formula, “that the density of a gas is directly proportional to its molecular mass.”

  “Are you with me, son?” Alfred saw the boy’s eyes glaze.

  Leo nodded, a little fuzzily. “I guess.”

  “All right, then give it back to me, please. Just as I have written.”

  Leo shrugged. “The rate at which a gas diffuses is inversely proportional to the square root of its densities.”

  “Good. Now, here, write it out as a formula.” Alfred handed him the chalk and tin and covered up what he had done. “Exactly as I gave it to you.”

  Leo hesitated for a moment, blew out his cheeks, then wrote it,

  just as Alfred had conveyed.

  “Excellent. Now how about the inverse of that? For density?”

  Leo thought on it a second. “The density of a gas is inversely proportional to its molecular mass.”

  “No. Not inversely. The opposite of that. It’s directly proportional,” Alfred corrected him.

  “Excuse me, Professor…?” Leo scrunched his eyes.

  “The density of a gas is directly proportional to its mass. It’s the exact inverse of the first equation I gave you. You see—”

  “All right. Sorry. I think I’ve got it now.” Leo wrote out the formula,

  this time correctly. “Directly proportional. Then this would be the symbol—he drew it with a stylish flourish:

  ∝

  “Very good. Now, in gaseous diffusion, we’re dealing with the identical principle, except we are working with two radioactive isotopes. Uranium-235, which is a fissile property. Fissile meaning it can be split and is capable of creating what we refer to as a ‘chain reaction,’ if separated from its more plentiful, but not fissile, cousin, U-238.”

  “Two thirty-five? Two thirty-eight? Sorry, but my head is starting to feel like it’s fissile, Professor.”

  “Don’t try and understand it all now. You know of uranium, right?”

  “Yes. Its symbol is U. And I think it has the highest molecular weight of any element.”

  “Second highest. But no matter, plutonium is only newly isolated and probably wasn’t even on the element chart when you would have studied this. Uranium-235 occurs in a 0.139-to-one ratio in natural uranium ore. Meaning that only 7 percent of all uranium is U-235. The rest is 238. It’s quite rare. The trick, then, is to separate this rare, highly charged isotope, which has the same properties but a different molecular weight than its more common relative, U-238. To do that, or at least to do that in the kinds of quantities you would be seeking, you need not only a diffusing membrane but the only compound of uranium sufficiently volatile to engender this, uranium hexafluoride, UF-6, which is completely solid at room temperature but sublimes once it approaches—”

  “Sublimes?” Leo’s eyes started to glaze over again. “I’m afraid you’re beginning to lose me, Professor.”

  “Here…” Alfred took the tin again and scribbled a far more complex equation. “Just memorize this…”

  “Which is equal to, I think…” Alfred closed his eyes and went through a series of calculations in his head. “One point oh oh four two nine eight or something … Where—and this is important, Leo—RATE One is the rate of effusion of U-235 and RATE Two is the effusion of…?” He looked to Leo to finish.

  “U-238,” Leo replied, after a moment of thought.

  “Correct! And M1 is the molecular mass of U-235 and M2 the molecular mass of U-238. The slight difference in weights explains the 0.4 percent difference in the average velocities of their neutrons.” He looked at his student. “So how does that sit?”

  “To be honest, it sits a bit fuzzily, Professor.”

  “Keep at it. I know this might as well all be Greek to you right now…”

  “No, Greek I actually studied in school…” Leo rolled his eyes.

  “Look, you don’t have to understand it all now. But what’s important is that you have a basic grasp and commit the equations…” he drew a double line beneath the equation, “into that stellar brain of yours. So, look at it again. Let it sink in.”

  Leo ran his eyes over the equation again, then closed them.

  “Do you have it?”

  “RATE small 1 over RATE small 2 is equal to the square root of M small 1 over M small 2 … Must I read you back all the numbers, Professor? I’m pretty sure I can … One point, zero, zero, four…”

  “That’s not necessary. Anyone with a third-level math degree can calculate that out.”

  “Then, where Rsmall1 is the rate of effusion of … UF6-235, and Rsmall2 is the effusion for UF6-238, and M1…” Leo pressed a finger to his forehead, “the molecular mass of U-235, and Msmall2, the mass for 238. And so on, and so on, and so on…”

  “Bravo!” Alfred said, leaning forward and squeezing the boy’s knee. He coughed up a bit of phlegm.

  “One question, Professor, if you don’t mind…?”

  “Of course. Go ahead.”

  “I still don’t understand why you need to separate this U-235 from 238? And you said before ‘in sufficient quantities…’ Sufficient quantities for what?”

  “Let’s not get ahead of ourselves.” Alfred cast him a patient smile. “All that is to come, my boy. To come.”

  “So that’s it? That’s what you needed me to memorize? The physics that will save humanity?”

  “That’s Lesson One,” Alfred replied. “It’s enough for today.”

  “Lesson One…” Leo cocked his head a little warily. “One of…?”

  “Hundreds, my boy.” Alfred slapped him on the shoulder. “Hundreds. However, I must warn you, tomorrow it actually starts to get a little complicated.”

  TWENTY-THREE

  Weeks went by. They met whenever they were able, for a few minutes at a time after roll calls in the mornings, before meals, most every day.

  Other than Tuesdays, when Leo was usually summoned to play chess with the Lagerkommandant Ackermann’s wife.

  Which bristled Alfred. “Why do you choose to play a game when we have serious work to get done?”

  “Because that game, as you call it, may one day be the difference in saving my life. And yours too, I should remind you. I tell her all the time that I share the treats she gives me with my uncle, the Professor. She promises to watch out for us.”

  “Watch out for us … I think you just like to go because you are sixteen and you can stare at a woman’s tits. I may be old, but not so old that I can’t remember the pleasure in that.”

  “That as well, I suppose.” Leo blushed, with just a little shame. “Still, she is nice to me. And, I believe, she is not her husband when it comes to what goes on in here. I think she is genuinely reviled by what she sees him do. That’s why she helps with the sick at the infirmary.”

  “You think that’s so, huh? She shares all this with you?”

  “She does. While we play.”

  “I think that playing this game with you is how she puts her guilty conscience to rest
,” Alfred said. “In a way, you are her absolution.”

  “Ah, I see you are Dr. Freud now, too, as well as Dr. Mendl.” Leo sniffed with a roll of his eyes.

  “In this case, boy, studying the atom is as good as studying the mind. In the end, she is the Lagerkommandant’s wife. And you are just a Jew.” He turned over Leo’s arm. “With that number on your wrist. She’ll watch out for you until your time is up. Then she won’t give you a thought.”

  “We’ll see.” Leo shrugged. “In the meantime, the cakes and chocolates she gives me are nice.”

  “Yes, well, you’re right, we’ll see…” Alfred coughed, bringing up a little phlegm, and wiped his mouth with his hand. “Anyway, let’s get back to work.” The cough had worsened, growing a bit more hacking with each day, and his bones and ribs were starting to show through even more. “Sorry that all you get to look at here is me.”

  “Yes, the view is decidedly less appealing. Here, let me put a blanket over you, old man. Sorry, excuse me,” he grinned. “I meant, Professor, of course.” It was one of the thin, grimy pieces of burlap that did nothing to protect you from the cold.

  These past weeks, Alfred had begun to grow fond of the boy. And he thought Leo felt the same about him. You learned on your first day in this place it wasn’t wise to have feelings for another prisoner or to even invest in a person’s history. You never knew how the next day might unfold.

  “It’s nothing,” Alfred said. But he wrapped the cloth around himself nonetheless and, for a moment, it stopped the chill. “Thank you, boy.”

  * * *

  As Alfred warned, the work grew harder and more complex each day. Now he was taking Leo through something called Bessel functions—complex, mind-stretching equations that were almost like going through an entire chess game in his head. A dozen games, Leo felt, each requiring the concentration of playing against a master, though Alfred rattled off the detailed numbers and values without a moment’s hesitation, as familiar to him as was his own birth date or house address.

  “Remember, we are dealing with highly charged materials here,” he explained, “that are in flux from state to state and, in the case of diffusion, through a confined space, in this case, cylinders. So we must introduce the general neutron diffusion equation for such a state.” He scribbled on the back of a torn-down health notice:

  “Okay…” Leo stared at it, a little numb. “I see.”

  “You have to know it, Leo. Know it cold. That is a must.”

  “I’m trying, Alfred.”

  “Then try harder. You must focus more. The goal here”—Alfred coughed into his rag—“is to apply the neutron population within a cylinder. The spatial part of the neutron density, characterized as N, will be a function of the cylindrical coordinates (p, o, z), and is assumed to be separable and expressed as…” He leafed through the scraps of paper he kept from yesterday’s session:

  Npϕz (ρ,ϕ,z) = Nρ(ρ)Nϕ(ϕ)Nz(z)

  Leo looked at him blankly.

  “Are you with me, son?”

  The boy puffed out his cheeks and blew out a long blast of air. “You’re going too fast, Alfred. I’m not sure.”

  “Not sure? I thought we went over this yesterday.”

  “I know, but it’s not like chess. I don’t fully understand why it’s important.”

  “Right now, it’s important because I say it’s important. So let’s do it over again. What is the coordinate o in the equation, if you don’t mind?” Alfred asked him.

  “O…?”

  “Yes, small o. Where is your head, boy? We’ve been through this several times. It’s the angle between the cylinder’s width and radius. And p?”

  “P…? P must be its height then?” Leo answered tentatively.

  “Yes. Height. Dimension. I thought you were smart, Leo. I thought you could grasp this. You must concentrate, this is the easy part. Otherwise there is too much to learn.”

  “Can we take a break, Professor? My head’s about to explode. And what is the purpose of all this, anyway? Did you invent it or something? This precious, gaseous diffusion process? We just keep going over and over the same boring things!”

  “Because you must learn it, boy. Like you know your own name! Do you hear me?”

  “Yes, I hear you!” Leo leaped up from the cot. “I hear you. I hear you…” His head was bursting with all these numbers. A feeling of total frustration and pointlessness swept over him. “Maybe we should just call it a day.”

  Alfred looked at him, knowing he had pushed too far. He let the boy calm a moment. Then, “No, I didn’t invent it,” he said. He put down the sheet. “In fact, the Brits are developing it as well at the same time. And I’ve heard researchers at Columbia University in New York are on the same track as well.”

  “Then let them learn it,” Leo said testily.

  “That would be easy, wouldn’t it?” Alfred nodded, sitting back. “All I’ve done,” he said, “is simply to carry the data to a further state. Here…” He took the back of a poster on the spread of typhus that had been put in the block and drew out a rough, hand-sketched drawing. A kind of an interconnected system of tubes with long cylinders feeding into smaller tubes, through a network of coils and pumps. “If the uranium gas, hexachloride 6, which is extremely caustic, is pumped against a porous barrier of some kind, the lighter molecules of the gas, containing the enriched U-235, would pass through the cylinders more rapidly than the heavier U-238. Right?”

  Leo nodded. That much he had learned.

  “Which is exactly what this formula represents. You must have this down cold, Leo. No matter how dull or complicated it may seem. This is the heart of what you need to know.”

  “‘Need to know’ why…? Who gives a fart about this stupid diffusion process anyway? Or is it ef-fusion?” Leo snatched the drawing from the cot, crumpled it into a ball, and flung it into the corner. “Do you know what I saw today…? I watched as six men were pulled from my line at work and ordered to lie down. Then told to ‘Get up!’ And then, ‘Lie down’ again. Snap, snap, double time. And then, ‘Up again! Then down! Faster! Faster!’ And then to ‘Run in place!’ and then ‘Squat! Squat ten times!’ And then to ‘Get up again! Quick. And then lie down. On the double! Schnell! Schnell! Faster.’ Until, one by one, they all just stopped in complete exhaustion, totally out of breath, and were finished off with clubs while they attempted just to suck oxygen into their lungs. The last one, red in the face, barely able to lift his legs, the guards laughing at him as if he were a marionette on a vaudeville stage. Until they beat him dead as well. So tell me, what did Graham’s Law do for them?”

  Alfred just looked at him.

  “And did you hear? Two nights ago, everyone in Block Forty-Six was marched into the night and never came back. Did all these cylinders and diffusion equations save them? Soon it will be us. You’re a fool, Alfred, to think the Germans will ever let any of us leave. Any of us! You know that as well as me. We’re all going to die in here. You and I. So what does it matter, in the end, if it’s small p or large P … U-235 or 238? My head is bursting, Alfred. Every day we do this. Over and over. And why…? You force these things in my brain and you won’t even tell me why?”

  Alfred nodded. He sat back against the wall and let out an understanding breath. “It matters a lot, my boy. You’re right, I probably will die in here. But you … The war has turned, Leo. You hear it from the new people coming in. The German Army is in tatters in the East. The Allies are set to invade. You can see it in the guards’ eyes. They are growing concerned. One day you may well get out of here, and I will give you the names of people to ask for. Respected people. Because what I am showing you on these torn slivers of paper and on the backs of these filthy food labels is worth more than all the gold the Germans take out of our teeth. A thousand times more.”

  “I know. You keep saying that, Alfred. But why…?”

  The professor bent down and picked up the crumpled diagram Leo had thrown against the wall and smoothed it out on the cot
. “Right now, in laboratories in Britain and in the United States, even in Germany, the most accomplished scientists, ones who make me seem as dull as an oxen, are going over the very same things…”

  “So then what do they need you for?” Leo pressed. “And all these equations you’re jamming into my head?”

  “Ultimately, they don’t.” Alfred shrugged. “Except that I know this one thing, and know it very well. And that is how to assemble a sufficiently large mass of uranium to capture and use the secondary neutrons before they escape through the surface of the material. And though this may not seem like much to you, Leo, because there’s no chessboard to mull over or pieces to move, be assured that whoever understands this process, understands it first … it is they who will win the war. And all the guns and tanks and planes in the sky won’t be able to stop them.”

  “This effusion process…?” Leo squinted back at him. “Or diffusion, whichever?”

  “Diffusion.” Alfred nodded with a smile.

  “You keep using the words ‘sufficient quantities,’ Alfred. Quantities sufficient for what?”

  This time Alfred just looked at him, with an elder’s gravity that it was time to explain difficult things to a boy who would now become a man. “You asked me once, what is the purpose of separating U-235 from U-238…?”

  “I can see now, it’s clearly some form of harnessing energy,” Leo said. “Maybe some kind of powering device? An engine. For a tank, perhaps? Or a ship?”

  “Yes, but much, much larger than that, I’m afraid. And with a far more devastating effect.”

  “You’re talking a bomb?” Leo’s eyes grew wide.

  Alfred sank his back against the wall and smiled with a kind of resignation. “A small part of one, yes. But a larger and far more destructive bomb than the world has ever seen. More like a thousand bombs, Leo. In one.”