Octob er 2020

Lecture Notes for 8.225 / STS.042, “Ph ysics in the 20th Cen tury”: Energy from Nuclear Fission

Da vid Kaiser

Center for The or etic al Physics, MIT

In tro duction

These note s discuss the earliest estimates for the energy released eac h time a hea vy , unsta- ble n ucleus undergo es n uclear fi ssion. The goal is to follo w argumen ts lik e those first pi eced together b y ph ysicists Lise Meitner and Otto Rob ert F risc h in Decem b er 1938, using ap- pro ximate order-of-magnitude estimates. Those ro ugh estimates w ere su ffi cien t to con vince scien tists all around the w orld — and, b efore long, p oliticians and m ilitary officials — that the energies in v olv ed in n uclear reactions lik e fi ssion w ere enormously gr eater than those asso ciated with familiar c hemical reactions. In particular, rough-and-ready estimates lik e these help ed to con vince leaders in m u ltiple coun tries to pursue n uclear w eap ons programs, righ t on the ev e of the o utbreak of fi gh ting in the Second W orld W ar.

The first section giv es a brief review of the kinds of n uclear transm utations that had already b ecome familiar to n ucle ar ph ysicists and c hemists b y the late 1930s. In the second section, w e turn to estimates of the energy released from n ucl ear fi ssion.

Reading these notes is optional ; the notes are mean t to fi ll in some of the gaps in v arious deriv ations that w e will n ot co v er during our class session.

Nuclear T ransm utations

Beginning in the mid-1 890s, led b y pioneering efforts b y Marie and Pierr e Curie, ph ysicists and c hemists studied and classi fi ed sev eral t yp es of ra dioactiv e transformations among c hem- ical elemen ts. F ollo wing Ernest Rutherford’s w ork in the early 1910s, scien tists’ fo cus on radioactivit y narro w ed to prop erties of the nuclei within atoms. During the 1910s and 1920s, man y naturally o ccurring radioactiv e pro cesses w ere classi fi ed in terms of the t yp e of ra dia- tion that w as emitted during deca y: α , β , or γ radiation. In time, α particles w ere iden ti fi ed

as the n uclei of Helium atoms, β particles as electrons, and γ ra ys as high-energy photons. 1 Nuclear scien tists could then piece together t ypical de c ay chains among radioact iv e n uclei, suc h as

U 234 − → Th 230 + α 4 ,

92 90 2

Th 230 − → Ra 226 + α 4 ,

(1)

90 88 2

Ra 226 − → Rn 222 + α 4 .

88 86 2

In this case, a n ucleus of a particular isotop e of uranium (U), with 92 proto ns and a total atomic mass of 234 units, could emit an α particle and transform in to a di ff eren t n ucleus. Because the α particle included t w o protons, the resulting n ucleus mo v ed two plac es lower on the p erio dic table; its n ucleus no w included o nly 90 protons instead of 92, corresp onding to the elemen t thorium (Th). Lik ewise, since the α particle included a total of 4 atomic mass units, the thorium n ucleus w ould ha v e a ma s s of 230 units, com pared to the 234 units of the original (radioactiv e) uranium n ucleus. The thorium n ucleus, in turn, w as itself radioactiv e. Up on emitting an α particle i t w ould pro duc e a n ucleus of radium (Ra) with 88 protons and atomic mass of 226 units; the r adium w ould deca y via α emission to y ield a n ucleus of radon (Rn), and so on. In short, eac h t ime a n ucleus deca y ed via emission of an α particle, the resulting n ucleus mo v ed t w o sp ots down the p erio dic table.

Other n uclei underw en t radioactiv e transfor mations in v olving the em ission of β particles rather than α particles. F or example, deca y c hains lik e these w ere also observ ed:

Th 234 − → P a 234 + β ,

90 91

P a 234 − → U 234 + β .

(2)

91 92

Here an isotop e of thorium with 234 atomic mass units deca y ed in to a n ucleus o f proactinium (P a), whic h has 91 protons; and later the proactinium n u cleus emitted a β particle and transformed in to an isotop e of uranium with 92 proton s . Not long after James Chadw ic k iden ti fi ed the neutr on in 1932, Enrico F ermi suggested that these β -deca y c hains in v olv ed the transformation of a neutron within the radioactiv e n ucleus in to a proton plus t w o v ery ligh t particles that quic kly escap ed f rom the n ucleus: an electron (the β particle) and a new particle dubb ed the “neutrino”:

n 1 − → p 1 + e 0 + ν ¯ 0 . (3)

0 1 − 1 0

The electron w asn’t truly massless, but b y the 1930s it had already b ecome clear that its mass w as nearly 2000 times smalle r than that of the proton. Lik ewise, later studies clari fi ed

1 F or a brief and accessible review, see Emilio Segr ` e, F r om X-R ays to Quarks: Mo dern Physicists and Their Disc overies (San F rancisco: W. H. F reeman, 1980), c hap s . 2 and 3.

that the additional par ticle emitted with the electron w as an antineutrino ( ν ¯ ), rather than a neutrino ( ν ). Lik e the then-h yp othetical neutrino, the an tineutrino w as quic kly understo o d to b e electrically neutral and to ha v e a mass ev en smaller than that o f the electron. 2

Giv en F ermi’s new suggestio n ab out the β -deca y of a neutron, as in Eq. (3), the deca y pro cess for the thorium n ucleus in Eq. (2) could b e understo o d as

Th 234

n → p + e + ν ¯

234

90 − − − − − − → P a 91 + β + ν ¯ . (4)

Because the (an ti)neutrino w as electrically neutral and had a v er y small ma s s, it escap ed de- tection, whereas the c harged β ra y (an electron) could b e detected with common instru men ts lik e a Geiger coun ter. F or understanding these kinds of n uclear transformations, therefore, the masses of b oth the electron and the an tineutrino could b e neglected. The larger p oin t w as that n uclei t hat underw en t β deca y mo v ed up the p erio dic table b y one sp ot : a neutron within the n ucleus transformed in to a proton, thereb y increasing the total n um b er of protons within the n ucleus b y one unit.

By the early 1930s, sev eral researc h groups b egan to study “art ificial” or induced ra- dioactivit y: irradiating otherwise s t able elemen ts with some sort of radiation and inducing n uclear re actions. Marie and Pierre Curie ’s daugh ter Ir ` ene Joliot-Curie and her h usband F r ´ ed ´ eric Joliot-Curie b ecame esp ecially activ e in this area. So on after Chadwic k iden ti fi ed the neutron in 1932, ph ysicists lik e Enrico F ermi b egan systematically irradiating elemen ts with neutrons to induce radioactivit y . F ermi’s group marc hed all the w a y up the p erio dic table, finally coming to the hea viest-kno wn elemen t, uraniu m. They found that when they irradiated uranium with ne utrons, they could induce radioactivit y via β emission. Since the new reactions in v olv ed the emission of β ra ys, they assumed they had transformed the target n ucleus b y one step up the p erio dic table, m uc h as in the naturally o ccurring β deca ys of Eq. (2). That is, F ermi and his group assumed they w ere measuring

U 238 + n 1 − → U 239

n → p + e + ν ¯

X 239 + β + ν ¯ . (5)

92 0

92 − − − − − − → 93

In the first step, F ermi and his group reasoned, the hea vy uranium n ucleus a bsorb ed the incoming neutron, increasing its tota l mass b y one unit but lea ving its n um b er of protons (and hence its c h emical iden tit y) unc hanged. Next the neut ron w ou ld undergo β -deca y within the n ucleus, transforming in to a proton and emitting an electron and (an ti)neutrino as in Eq. (3). Within the rem aining n ucleus, there w ould no w b e 93 pro tons and a total mass of 239 units. F ermi w as con vinced that b y this pro cess, he and his group had created

2 See Gino Segr ` e and Bettina Ho erlin, The Pop e of Physics: Enric o F ermi and the Birth of the A tomic A ge (New Y ork: Henry Holt, 2016), c h aps. 14-16, and F rancesco Guerra and Nadia Rob otti, The L ost Noteb o ok of Enric o F ermi: The T rue S tor y of the Disc overy of Neutr on-Induc e d R adio activity (New Y ork: Springer, 2018), c hap . 6.

the first tr ansur anic elemen ts: e lemen ts with mor e than 92 pr otons in eac h n ucleus, b ey ond the then-kno wn edge of the p erio dic table. 3 (Hence the placeholder lab el “X 93 ” in Eq. (5): there w as no kno wn elemen t b ey ond U 92 . An elemen t with 93 pro tons w a s later named “Neptunium”: just as the planet Neptune is the next furthest planet from t he Sun after Uran us, so the el emen t neptunium w ould app ear on the p erio dic table on place further than uranium.) This w as striking news; within just four y ear s , F ermi w as a w arded the Nob el Prize in Ph ysics for this w ork. 4

A t least one researc her at the time — n uclear c hemist Ida No ddac k — raised questions ab out whether F ermi’s group had really pro duced transuranic n uclei in these exp erim en ts. Dra wing on her training as a c hemist, she cautioned against iden tifying new elemen ts of the p erio dic table without thoroughly testing their c hemical prop erties, and c hec king for consis- tency with the c hemical b eha vior of kno wn elemen ts within the same columns of the p eri o dic table. 5 But No ddac k ’s cautions w ere almost univ ersally o v erlo ok ed at th e time. After all, F ermi’s in terpretation of his group’s exp erimen ts w as v ery m uc h in k eeping with decades of exp erience a mong n uclear scien tists b y that time: n uclear transformations t ypically mo v ed an elemen t up the p erio dic table b y one sp ot ( β deca y) or do wn the p erio dic table b y t w o sp ots ( α deca y).

Nuclear Fission

Other groups b egan rep eating F ermi’s induced-radioactivit y exp erimen ts, irradiating man y di ff eren t elem en ts with neutrons. A group in Berlin, including n uclear c hemists Otto Hahn and F ritz Strassmann a nd n uclear ph ysicist Lise Meitner, b ecame esp ecially activ e in this area. In the midst of those exp erimen ts — duri ng July 1938 — Me itner w as forced to flee German y . S he w as an Austrian citizen of Jewish bac kgr ound, so she only b ecame sub ject to the Nazis’ an ti-Semitic emplo ymen t la ws after the A nschluss of Marc h 1938, whic h formally incorp orated Austria within the German Reic h. 6

Unlik e F ermi, Hahn and Strassma nn w ere trained as c hemists, and they w ere esp ecially adept at p erform ing c h emical analyses of the reaction pro ducts. In one of the last exp eri-

3 E. F ermi, “P ossible pro duction of e l e men ts of atomic n um b er higher than 92,” Natur e 133 (1934): 898- 899; E. Amaldi, O. D’Agostino, E. F ermi, B. P on tecorv o, F. Rasetti, and E. Segr ´ e, “Artificial radioactivit y pro duced b y neutron b om bardmen t, I I,” Pr o c e e dings of the R oyal So ciety L ondon A149 (1935): 522-558.

4 Segr ` e and Ho erlin, Pop e of Physics , c haps. 17-18; Guerra and Rob otti, L ost Noteb o ok of Enric o F ermi , c haps. 7-10.

5 I. No ddac k, “ U ¨ b er das Elemen t 93,” Zeitschrift f u ¨ r A ngewandte Chemi e 47 (1934): 653-655; I. No ddac k, “Das P erio disc he S ys tem der Elemen te und Seine L u ¨ c k en,” Zeitschrift f u ¨ r A ngewandte Chemie 47 (1934): 301-305. See also Gildo Magalh ˜ aes San tos, “A tale of oblivion : Ida No ddac k and the ‘univ ersal abundance’ of matter,” Notes and R e c or ds of the R oyal So ciety 68 (2014): 373-389.

6 See esp. Ruth Lewin Sime, Lise Meitner: A Life in Physics (Berk eley: Univ ersit y of California Press, 1996), c hap s . 6-10.

men ts that th e Berlin group conduct ed b efore Meitner w as forced to fl ee, they had found that the reaction pro ducts follo wing b om bardm en t of uranium with neutrons b eha v ed chemic al ly in a similar w a y to barium. Barium, w ith atomic n um b er 56, w as no where near uranium (atomic n um b er 92) on the p erio dic table, but it w as exactly one ro w ab o v e radium (ato mic n um b er 88), and hence barium and radium could b e exp ected to b eha v e simi larly in c hemical reactions. So the Berli n group had concluded, in their Ma y 1938 exp erimen t, that the irradi- ated uranium n uclei (p erhaps b y w a y of some unstable in termed iary) ev en tually underw en t a short α -deca y c hain, similar to that of Eq. (1), to pro duce radium among the reaction pro ducts. 7

Meitner co n vinced Hahn to re-do the exp erimen ts with uranium and to conduct mor e thorough c hemical tests of the reaction pro ducts. Hahn and Strassmann w ere able to com- plete the new exp erimen ts after Meitner’s forced departure. 8 They b ecame con vinced that they w e re not pro ducing a transuranic elemen t, on e step b ey ond uranium on the p erio d ic table, whic h migh t ha v e then undergone t ypical α deca ys to arriv e at radium. Rather, they found clear evidence of barium itself, not just som ething c hemically similar to barium. Y et barium w as almost half the size of u ranium — indicating a h uge transition from the n uclear reaction, do wn dozens of sp ots on the p erio dic table. Suc h a h uge leap had nev er b een iden ti fi ed b efore; all kno wn n uclear transformations had in v olv ed small steps up or do wn the p erio dic table. Hahn and Strassmann hastily wr ote up their results in Decem b er 1938, suggesting that ex p erimen ts lik e theirs and F ermi’s had actually corresp onded to

U 92 + n 0 − → Ba 56 + Kr 36 . (6)

They ac kno wledge d just ho w sho c king suc h a reaction w ould b e. In t heir closing paragraph, they wrote, “As c hemists, w e m ust actually sa y t he new particles do not b eha v e lik e radium but, in fact, lik e barium; as n uclear ph ysicists, w e cannot mak e this conclusion, whic h is in conflict with a ll exp erience in n uclear ph ysics.” 9

Meitner had narro wly escap ed Nazi German y during the summer of 1938; she w as giv en a te mp orary p osition at a ph ysics institute asso ciated with the Sw edish Acade m y of Sciences in Sto c kholm. In Decem b er 1938 she met up with her nephew, the y oung ph ysicist O tto

7 L. Meitner, F . S trass mann, and O. Hahn, “K u ¨ nstlic h e Um w andlungsprozesse b ei Bestrahlung des Tho- riums mit Neutronen; Auftreten isomer Reihen durc h Abspaltung v on α -Strahlen,” Zeitschrift f u ¨ r Physik 109 (1938): 538-552; see al s o Elisab eth Cra wford, Ruth Lewin Sime, and Mark W alk er, “A Nob el tale of p os t w ar i njustice,” Physics T o day 50 (Septem b er 1997): 26-32.

8 Cra wford, Le win Sime, and W alk er, “Nob el tale,” 26.

9 O. Hahn and F. Strassm an n, “ U ¨ b er den Nac h w eis und das V erhalten der b ei der Bestrahlung des Urans mittels Neutronen en tstehenden E rdalk alimetalle,” Naturwissenschaften 27 (1939): 11-15, on 15: “Als Chemik er m u ¨ ssten wir aus den kurz dargelegten V ersuc hen das ob en debrac h te Sc heme eigen tlic h um b enen- nen und statt Ra, Ac , Th die Sym b ole Ba, La, Ce einsetzen. Als der Ph ysik in ge wisser W eise nahestehende ‘Kernc hemik er’ k ¨ onnen wir uns zu disem, allen bisherigen Erfahrungen der Kernph ysik widersprec henden, Sprung no c h nigh t en tsc hliessen.”

Rob ert F ri s c h, for a ski holida y in Sw eden. A t the time F risc h w as a p ostdo ctoral researc her at Niels Bohr’s Institute for Th eoretical Ph ysics in Co p enhagen. Just b efore Meitner’s and F risc h’s trip, Meitner receiv ed a le tter from Hahn with an up date ab out the latest Berlin tests a nd their detection of barium among the reaction pro ducts. As F risc h later reca lled, “When I c ame out of m y hotel ro om after m y first nigh t in [the ski village] Kung ¨ alv I found Lise Meitner s tu dying a letter from Hahn and ob viously w orried ab out it. I w an ted to tell her of a new exp e rimen t I w as planning, but she w ouldn’t listen; I had to read that letter. Its con ten t w as indeed so startlin g that I w as at first inclined to b e sce ptical.” They con tin ued to puzzle through the implications of Hahn’s letter throughout the da y . A t one p oin t, F risc h recalled, “w e b oth sat do w n on a tree trunk (all that discussion had tak en place while w e w alk ed through the w o o d in the sno w, I with m y skis on, Lise Meitner making go o d her claim that she could w alk just as fast without), and started to calculate on scraps of pa p er. [...] The uranium n ucleus migh t i ndeed resem ble a v ery w obbly , unstable drop, ready to divide itself at the sligh test pro v o cation, suc h as the impact of a single neutron.” 10 While trudging through the sno w that da y — and o ccasionally pausing on a b enc h or tree trunk

— Meitner and F risc h w ork ed out the first-ev er ph ysical explanation of n uclear fi ssion.

They b egan b y reasoning that large n uclei, lik e uranium, m ust b e barely stable. 11 P erhaps the attractiv e n uclear force among protons and neutrons that k ept atomic n uclei from fal ling apart w as just barely able t o comp ensate for the electrosta tic (Coulom b) repulsion of nearly 100 p ositiv ely c harged prot ons closely pac k ed within a n ucleus, eac h rep elling the others. In that case, if the n ucleu s w ere p erturb e d b y an incoming neutron — esp ecially if that neutron had b een slowe d down to a lo w energy , thereb y stretc hing out its quan tum-mec hanical de Broglie w a v elength, λ = h/ ( mv ) — then the en tire, unsteady n ucleus could b egin to shak e or w obble lik e a liquid drop. The drop could then divid e in to t w o drops, eac h of roug hly equal size and in close pro ximit y to eac h other. F ollo win g division (or “ fi ssion”), there w ould no w b e t w o globs of n uclear matter, eac h fi lled with dozens of protons and exerting a strong electrostatic repulsion up on the other. Hence the t w o fi ssion pro ducts should eac h quic kly gain a large amoun t of kinetic ene rgy , as they raced apart from eac h other.

Meitner and F risc h first estimated the energy scales in v olv ed. Prior to fi ssion, a large n ucleus lik e uranium w ould ha v e a n uclear energy E n uc appro ximately balancing the electro- static energy arising from eac h of those protons in the n ucleus rep elling the others. They

10 Otto Rob ert F r is c h, What Little I R ememb er (New Y ork: Cam bridge Univ ersit y Press, 1979), on pp. 115- 116.

11 The follo wing discussion fills in the in termediate steps whic h Meitner and F risc h left tacit i n their first, brief rep ort ab out their new w ork: L. Meitner and O. R. F risc h, “Disin tegration of uranium b y neutrons: A new t yp e of n uclear reac ti on,” Natur e 143 (1939): 239-240.

could therefore estimate

E n uc

'

i,j = 1 , i / = j

n i n j e 2

. (7)

r ij

That is, they could sum up the energy asso ciated with the electro s ta tic repulsion of eac h proton (lab eled n i and p ossessing electric c harge e ) from ev ery other proton (lab eled n j and c harge e ), up to the tota l q protons con tained with the n ucleus; eac h pair n i n j w as separated b y some distance r ij . W e can think of the expression n i n j e 2 /r ij as a large q × q square matrix, e ac h elemen t of whic h represen ts the energy asso ciated with t he electr ostatic repulsion of a particula r pair of protons. A square q × q matrix in cludes q 2 elemen ts. As indicated in the sum b y the notation i / = j , they w ould not include the q en tries along the diagonal of the matrix, with n i = n j ; those w ou ld corresp ond to the rep ulsion of proton n i from itself. Hence the n um b er of elemen t s from the matrix to include is q 2 − q . F or q 1, this is appro ximately equal to q 2 .

T o simplify the problem — after all, Meitner and F risc h w ere in terested in a roug h order- of-magnitude estimate — they next assumed th at eac h of the pairs of protons ( n i , n j ) w as separated b y an a v era ge or t ypical dista nce r ij ∼ R n uc , roughly the size o f the uranium n ucleus. In that ca s e , the expression in Eq. (7) could b e simpli fi ed t o

( q e ) 2

E n uc ∼

R n uc

. (8)

F or a uranium n ucleus, they could estimate q ∼ 100, close to the actual n um b er of 92 proton s within the n ucleus. F or the t ypical size R n uc of a uranium n ucleus, they could reason as follo ws. Recall that the Bohr radius for the ground-state of an electron in a h ydrogen atom is a 0 ' 5 . 3 × 10 − 9 cm, and that Rutherford’s scattering exp erimen ts suggested that the radius of an atomic n ucleus w as t ypically ab out 10 5 smaller than the ra dius of the corresp onding atom. Meitner and F risc h estimated that the radius of a uranium atom should b e at least 100 times larger than the radius of a h ydrogen a tom, and henc e, in round n um b ers, t hey could estimate

R n uc ∼ 10 − 9 c m

r a dius of a h y drogen at o m

2

ratio ` of U to ˛ H ¸ atom i x c radii

− 5

ratio o ` f n ucle a ˛ r ¸ to ato m x ic radii

~ 10 − 12 cm .

(9)

Chemical reactions, o n the other hand, t ypically in v olv ed the transfer of a single (v alence) electron from one atom t o another across some distance R atom , and hence Meitner and F risc h could estimate

E c hem ∼ R

e 2

atom

. (10)

A toms larger than the ground-state of h ydr ogen w ould ha v e t ypical radii R atom ∼ 10 − 8 cm. Comparing Eqs. (8) and (10), using q ∼ 100, R n uc ∼ 10 − 12 cm, and R atom ∼ 10 − 8 cm yielded

a ratio

E n u c

( q e ) 2 R atom

2 10 − 8 cm 8

In other w ords, the t ypical energies in v olv ed in n uclear reactions amon g v ery large n uclei, suc h as uranium, should b e as m uc h as one hundr e d mil lion times lar ger than t ypical energies in v o lv ed in c hemical reactions!

By the late 1930s, n uclear ph ysicists t y pically measured the energies of v arious reactio ns in units o f electron-V olts, or eV. Recall that the ionization energy of a h ydrogen atom — that is, the energy required to remo v e its single electron — i s 13 . 6 eV; c hemical reactions t ypically

in v o lv e energies in the range E c hem ∼ O (1 − 10 eV ). Meitner and F risc h’s ba c k-of-the-en v elop e

calculation suggested that t ypical energies in v olv ed in n uclear reactions should instead b e E n uc ∼ O (10 8 − 10 9 eV ), that is, h undreds of millions of elec tron-V olts, O (10 2 MeV ), up to billions of electr on-V olts, O (1 GeV ). This w as an enormous shift in energy scale.

Meitner a nd F risc h w eren’t done. Their next step w as to consider ho w m uc h energy migh t t ypically b e r ele ase d eac h time a whole uranium n ucleus underw en t fi ssion. Prior to fi ssion, the n ucleus w ould ha v e energy

E whole ∼ E n uc ∼

( q e ) 2

R n uc

, (12)

with q ∼ 100. Afte r fi ssion, t w o smaller n uclei w ould remain, eac h with roughly equal n um b ers of c harges q / 2 pac k ed in to roughly eq ual volumes , V piece :

1

V piece = 2 V whole . (13)

The v olume of a sphere scales a s V ∼ R 3 , or R ∼ V 1 / 3 , so Meitner and F risc h could estimate

R piece =

R whole

V piece 1 / 3

V whole

1 1 / 3

' 0 . 8 , (14)

where R whole = R n uc , the radius of the whole (original) uranium n ucleus prior to fi ssion. That suggested that follo wing fi ssion, the energy of eac h piece, compared to the energy of the whole (or iginal) uranium n ucleus, should b e giv en roughly b y

E piece

q e 2

! R

n uc

1 2 1

' 0 . 3 ∼

1

. (15)

E whole

0 . 8 R n uc

( q e ) 2

2 0 . 8 3

In other w ords, follo wing fi ssion eac h piece w ould acq uire ab out one-thir d of the o riginal energy of the uranium n ucleus, E piece ∼ E whole / 3, with E whole ∼ E n uc ∼ O (10 2 − 10 3 MeV ). That result, in itself, w as remark able: a single incoming slo w neut ron, with kinetic ene rgy

~ O (1 eV ), could split a hea vy n ucleus and pro duc e t w o fi ssion fragmen ts, eac h with a kinetic

energy up to one h undred million times greater than the energy injected in to the system b y the neutron! Not only that, but the ener gy carried off b y the fi ssion fragmen ts w ou ld only accoun t for a fraction of the total energy released. An add itional amoun t of energy

1

∆ E = E whole − 2 E piece ∼ 3 E n uc (16)

w ould also b e released as r aw ener gy , eac h time a single uranium n ucleus underw en t fi ssion. Meitner and F risc h quic kly wrote a brief Letter to the Editor of Natur e describing their new ph ysical mo del of n uclear fi ssion; it w as receiv e d at the journal on Jan uary 16, 1939 and published in the F ebru ary 11 issue. In addition to describing the basic ph ysical mec ha nism, they describ ed their simple calculation, as sk etc hed in th ese notes, and predicted that th e fis- sion fragmen ts should eac h acquire t ypical kinetic energies of ab out 10 2 MeV. 12 As so on as he returned to Bohr’s Institute , F risc h w as also able to conduct new lab oratory measuremen ts, whic h quic kly confirme d exactly this energy scale. So on after that, researc hers in oth er lab o- ratories — b oth in Britain and in German y — indep enden tly measured comparable energies

in their o wn new exp erimen ts. 13

F risc h later recalled the swirl of ev en ts: he “rigged up” e quipmen t in the basemen t o f Bohr’s Institute o v er the course of a few da ys, “and then I w ork e d most of the nigh t to do the measure men ts b ecause the coun ting rates w ere v ery lo w. But b y thr ee in the morning I had the evidence of the big pulses [co rresp onding to the energetic fi ssion fragmen ts]. And I w en t to b ed at three in the morning, and then at sev en in the morning I w as kno c k ed out of b ed b y the p ostman who brough t a telegram to sa y that m y father had b een released f rom the concen tration camp.” 14

F risc h had giv en a cop y of the short pap er that he an d Meitner had just prepared for Natur e to h is sup ervisor, Niels Boh r, da ys b efore Bohr set off for a trip to the United Sta tes in Jan uary 1939. As so on as Bohr arriv ed for his sabbatic al visit at the I nsti tute for Adv a nced Study in Princeton, he b ega n w o rking with his colleague (and former p ostdo ctoral advisee) John Wheeler; Wheeler w as b y th en a professor of ph ysics at nearb y Princeton Univ ersit y . 15 They formalized Meitne r’s and F risc h’s ph ysical mo del of n uclear fi ssion, whic h had built

12 Meitner and F risc h , “Disin tegration of uranium b y neutrons,” on p. 239. W ithin a few w eeks, they published a brief follo w-up article as w ell: L. Meitner and O. R. F risc h, “Pro ducts of fission of the uranium n ucleus,” Natur e 143 : 471-472, rece iv ed at the journal on 6 Marc h 1939 and published in the 18 Marc h issue.

13 O. R. F risc h, “Ph ysical evidence for the division of hea vy n ucle i under neutron b om bardmen t,” Natur e

143 (1939): 276. See also R. D. F o wler and R. W. Do dson, Natur e 143 (1939): 233; and W. Jen tsc hk e and

F. Prankl, “Un tersuc h ung der sc h w eren Kern bru s c hst u ¨ c k e b eim Zerfall v on neutr onen b es t rahltem Uran und Thorium,” Naturwissenschaften 27 (1939): 134-135.

14 A brief excerpt from a recording of an in terview with Otto Rob ert F risc h, from whic h this quotation is tak en, is a v ailable as p art of the online exhibit “The disco v ery of fission,” a v ailable at https://history. aip.org/history/exhibits/mod/fission/fission1/04.html (accessed 28 Septem b er 2020).

15 Niels Bohr and John A. Wheeler, “The mec h anism of n uclear fission,” Physic al R eview 56 (1939): 426- 450.

up on Bohr’s o wn (previous) liquid-drop mo del of large n uclei. Bohr and Whee ler also sho w ed that Meitner a nd F risc h’s initial estimate — that the energy released during the fi ssion pro cess should b e of order E n uc ∼ O (10 2 − 10 3 MeV ) — w as consisten t with what one could

estimate using Einstei n’s b y-then famous form ula, E = mc 2 . In particular, Bo hr and Wheeler

considered the nucle ar binding ener gy for the uraniu m n ucleu s prior to fiss i on, and for the fi ssion pro ducts follo wing the reaction.

The fi ssion reaction of Eq. (6) t ypically in v olv ed sp ecific isotop es of U, Ba, and Kr, as w ell as the release of a few excess neutrons:

U 235 + n 1 − → Ba 141 + Kr 92 + 3 n 1 . (17)

92 0 56 36 0

Here w e can see that b oth the total n um b er of proton s balances b efore and after the reaction (92), and the total n um b er of atomic mass units app ears to balance (236). Ho w ev er, the actual mass of a n ucleus is less than the sum of its constituen t parts; the difference b ecame kno wn as the “mass defect,” whic h could b e directly related to the n uclear binding energy b y means of Einste in’s relation b et w een energy and mass. Using mo dern v alues but follo wing the t yp e o f reasoning established in Bohr and Wheeler’s pap er, w e ma y consider the mass of a proton ( m p ) and a neutron ( m n ) in atomic mass units (am u) 16 :

m p = 1 . 0073 am u , m n = 1 . 0087 am u . (18) The mass of the n uclear constituen ts within the uranium n uc leus prior to fi ssion is therefore

m U , sum = 92 m p + (235 − 92) m n = 236 . 9157 am u . (19)

The actual, m easured mass of a U 235 n ucleus, on the other hand, is

m U , meas = 235 . 0439 am u . (20)

F or this isotop e of uranium, the mass defect is therefore giv en b y

∆ m U = m U , meas − m U , sum = − 1 . 8718 am u . (21) Using Einstein’s relation , this mass defe ct can b e expressed as a n uclear binding energy:

E U , bind = ∆ m U c 2

= ( − 1 . 8718 am u)

931 . 5 MeV /c 2

1 am u

c 2 = − 1743 . 58 MeV , (22)

up on con v erting from atomic mass units to the mass-scale MeV /c 2 . ( The binding energy in general is ne gative , since it is asso ciated with an attr active n uclear force; this is what

16 The n umerical v alues used here ma y b e found in the online edition of John R. Rum b le , ed., CR C Handb o ok of Physi cs and Chemistry , 101st edition (Bo ca Raton, FL: CR C Press / T a ylor & F rancis, 2020), a v ailable via the M I T Libraries w ebsite.