How to use this course
Lise Meitner is remembered today mainly for one paper: the two-page note of February 1939 in which she and her nephew Otto Robert Frisch explained how a uranium nucleus can split in two. That paper rested on three decades of experimental work on radioactivity, much of it done jointly with the chemist Otto Hahn, and on her own research on beta and gamma rays. This course presents the physics in the order she met it, alongside the story of a woman who entered science when universities barely admitted women and left Germany as a refugee at 59.
- First-year calculus and ordinary differential equations (for decay chains).
- Introductory modern physics: the Bohr atom, X-ray spectra, \(E=mc^2\).
- For Lessons 09 and 10, the idea of nuclear binding energy and the semi-empirical mass formula is helpful but is reviewed.
- No chemistry beyond the periodic table is assumed; Lesson 08 explains the carrier methods it uses.
Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Numbers use modern data unless a historical value is stated. The same events are told from the chemist’s side in the companion course on Otto Hahn; both courses give the same account of the discovery of fission and of the 1944 Nobel Prize.
Timeline
| 1878 | Born 7 November in the Leopoldstadt district of Vienna, third of eight children of the lawyer Philipp Meitner and Hedwig Meitner. |
| 1901 | Passes the external matura in July after private tuition; enters the University of Vienna in October and hears Ludwig Boltzmann lecture. |
| 1906 | Doctorate in physics, 1 February, on heat conduction in inhomogeneous bodies; the second woman to take a physics doctorate in Vienna. Begins radioactivity research with Stefan Meyer. |
| 1907 | Moves to Berlin to attend Max Planck’s lectures. Begins working with Otto Hahn in a converted woodworking shop at Emil Fischer’s Chemical Institute. |
| 1908 | Baptised a Lutheran. Nine joint papers with Hahn in 1908–1909, including the recoil method. |
| 1912 | Becomes Planck’s assistant. Moves with Hahn to the new Kaiser Wilhelm Institute for Chemistry in Dahlem, at first unpaid. |
| 1913 | Appointed a scientific member of the institute. |
| 1915–16 | X-ray nurse-technician with the Austrian army on the Eastern and Italian fronts. |
| 1917 | Head of her own physics section at the institute. With Hahn, identifies the long-lived parent of actinium. |
| 1918 | Paper announcing protactinium, element 91. |
| 1922 | Habilitation at Berlin, the first woman in physics in Prussia. Describes the radiationless emission now called the Auger–Meitner effect. |
| 1926 | Extraordinary professor at the University of Berlin, the first woman professor of physics in Germany. |
| 1930 | Confirms with Wilhelm Orthmann that beta-decay energy is not carried by the continuous spectrum’s endpoint. December: Wolfgang Pauli’s neutrino letter is addressed to her and Hans Geiger. |
| 1933 | 6 September: loses her university teaching right under Nazi law; keeps her institute post. |
| 1934 | Persuades Hahn to repeat Enrico Fermi’s neutron experiments on uranium; Fritz Strassmann joins the work. |
| 1938 | March: the Anschluss makes her subject to German law. 13 July: leaves Berlin with Dirk Coster for the Netherlands. August: arrives in Sweden; joins Manne Siegbahn’s institute in Stockholm. 13 November: meets Hahn in Copenhagen. 19 December: Hahn writes that the “radium” behaves like barium. Christmas at Kungälv with Frisch. |
| 1939 | 11 February: Meitner and Frisch, “Disintegration of uranium by neutrons”, Nature. |
| 1945 | November: the Royal Swedish Academy of Sciences awards the reserved 1944 chemistry prize to Hahn alone. |
| 1946 | Visiting professor at the Catholic University of America, Washington. |
| 1947 | Moves to the Royal Institute of Technology (KTH), Stockholm. |
| 1949 | Swedish citizenship. Max Planck Medal, shared with Hahn. |
| 1960 | Retires to Cambridge, England, near Frisch. |
| 1966 | Enrico Fermi Award, shared with Hahn and Strassmann. |
| 1968 | Dies 27 October in Cambridge, aged 89. Buried at Bramley, Hampshire. |
| 1997 | Element 109 named meitnerium. |
Vienna, 1878–1907
Elise Meitner was born on 7 November 1878 in Leopoldstadt, the Jewish quarter of Vienna. The community register gives 17 November; she always used the 7th. She was the third of eight children. Her father, Philipp Meitner, was among the first Jewish lawyers admitted to practise in Austria and a strong amateur chess player. The household was liberal and secular, and all the daughters were educated.
Austrian girls could not attend the academic gymnasium. Meitner finished school in 1892 and trained as a French teacher, the one profession then open to her. Austrian universities admitted women from 1897. She then prepared privately for the matura, catching up on Latin, Greek and mathematics in about two years, and passed it as an external candidate in July 1901 at the age of 22, together with Henriette Boltzmann, daughter of the physicist.
Boltzmann’s student
She entered the University of Vienna in October 1901. The lectures of Ludwig Boltzmann on theoretical physics, which she attended for several years, gave her both a training in statistical mechanics and a picture of physics as a search for truth. Her thesis, Wärmeleitung in inhomogenen Körpern, was supervised by Franz Exner and his assistant Hans Benndorf. It tested a formula of James Clerk Maxwell for conduction in mixtures. She was examined by Exner and Boltzmann on 19 December 1905 and received her doctorate on 1 February 1906, the second woman to earn a physics doctorate in Vienna after Olga Steindler.
Boltzmann died in September 1906. Meitner had begun to work on radioactivity with Stefan Meyer at the Vienna institute, and her experiments on the scattering of alpha particles in metal foils showed scattering growing with atomic mass. That autumn she decided to go to Berlin, with her father’s financial support, to hear Max Planck.
Berlin and the woodworking shop, 1907–1912
Prussian universities did not yet admit women. Meitner needed Planck’s personal permission to attend his lectures, and Planck, who on record opposed women’s higher education in general, made an exception for her. He became her mentor and later a close friend, and she was a regular guest at musical evenings in his house.
At the physics colloquium Heinrich Rubens introduced her to Otto Hahn, a chemist of her own age who had recently returned from work with William Ramsay in London and Ernest Rutherford in Montreal. Hahn wanted a physicist. Emil Fischer, director of the Chemical Institute, did not allow women in his laboratories, so the two worked in a former woodworking shop in the basement, which had its own street entrance. Until then she could not go upstairs. When Prussia opened its universities to women in 1908, Fischer relented and let her into the rest of the building.
The pairing worked because the skills were complementary. Hahn separated and identified radioactive substances chemically; Meitner measured their radiations and interpreted what they meant physically. In 1908–1909 they published nine joint papers. Among other results they developed the recoil method of separating decay products (Lesson 04) and found new members of the decay series. Meitner was paid nothing. She lived on a small allowance from her parents, and after her father died in 1910 her situation became difficult.
She was baptised a Lutheran in 1908. In 1912 Planck made her his assistant at the Institute for Theoretical Physics, grading students’ work. It was her first paid position and made her the first female scientific assistant in Prussia.
The Kaiser Wilhelm Institute, 1912–1933
The Kaiser Wilhelm Institute for Chemistry opened in Berlin-Dahlem on 23 October 1912, and Hahn moved there to head a small radioactivity section. Meitner came as an unpaid guest. In 1913 she was offered a post in Prague; to keep her in Berlin, the institute made her a scientific member with a salary. The joint group became known as the Hahn–Meitner laboratory.
War
When war came in 1914, Hahn was called up and later assigned to Fritz Haber’s chemical-warfare unit. Meitner trained as an X-ray technician and in July 1915 joined the Austrian army as an X-ray nurse, serving on the Eastern and Italian fronts until September 1916. Back in Berlin she carried on the search for the parent of actinium largely alone, writing to Hahn about each step. In January 1917 she was given her own physics section, and in 1918 the two announced protactinium (Lesson 05).
Recognition
In 1922 she habilitated, the first woman to do so in physics in Prussia. A newspaper is said to have reported the title of her inaugural lecture on the cosmic significance of radioactivity as “cosmetic”. In 1926 she became an extraordinary professor at the University of Berlin, the first woman professor of physics in Germany. Through the 1920s her section did leading work on beta and gamma rays (Lessons 06 and 07), and she was nominated for Nobel Prizes from 1924 onward. She received the Lieben Prize of the Austrian Academy in 1925. Her assistants in the early 1930s included Max Delbrück, later a founder of molecular biology.
In 1933 the new regime’s civil-service law reached her. As an Austrian citizen employed by a privately endowed institute she kept her research post, but on 6 September 1933 her right to teach at the university was withdrawn. She stayed. Leaving would have meant abandoning the laboratory she had built, and in 1934 a new problem arrived from Rome that drew her deeper into work (Lesson 08).
Decay chains and radioactive recoil
For the general reader. A radioactive atom changes into another, which may itself be radioactive, so that uranium or thorium sits at the head of a long chain of descendants. Untangling these chains was the main task of radioactivity research before 1914. Meitner and Hahn added a physical tool to the chemist’s kit: when an atom shoots out an alpha particle, the remaining atom recoils, and the recoil can carry it out of a surface onto a collecting plate.
- Solve the equations of a radioactive chain and understand secular equilibrium.
- Use momentum conservation to compute recoil energies.
- See how recoil separation identifies short-lived daughters.
The decay law
If \(N(t)\) nuclei of a species with decay constant \(\lambda\) are present, then \(dN/dt=-\lambda N\), so \(N=N_0e^{-\lambda t}\) and the half-life is \(T_{1/2}=\ln 2/\lambda\). The activity is \(A=\lambda N\).
For a chain \(1\to2\to\) with \(N_1(0)=N_0\), \(N_2(0)=0\) and \(\lambda_1\ne\lambda_2\),
\[ N_2(t)=\frac{\lambda_1N_0}{\lambda_2-\lambda_1}\bigl(e^{-\lambda_1t}-e^{-\lambda_2t}\bigr). \]If \(\lambda_1\ll\lambda_2\) (long-lived parent), then after a few daughter half-lives \(A_2=\lambda_2N_2\to\lambda_1N_1=A_1\): secular equilibrium. In a chain of \(n\) members in secular equilibrium all activities are equal.
Derivation. \(dN_2/dt=\lambda_1N_1-\lambda_2N_2\). Multiply by \(e^{\lambda_2t}\): \(\frac{d}{dt}(N_2e^{\lambda_2t})=\lambda_1N_0e^{(\lambda_2-\lambda_1)t}\). Integrate from 0 with \(N_2(0)=0\) to obtain the formula. As \(t\to\infty\) with \(\lambda_1\ll\lambda_2\), \(e^{-\lambda_2t}\) dies away and \(\lambda_2N_2\approx\lambda_1N_0e^{-\lambda_1t}\).
Recoil
In alpha decay \(X\to Y+\alpha\) with the parent at rest, momentum conservation gives equal and opposite momenta \(p\). With non-relativistic kinetic energies \(T=p^2/2m\),
\[ T_Y=\frac{m_\alpha}{m_Y}\,T_\alpha,\qquad Q=T_\alpha+T_Y \ \Longrightarrow\ T_Y=\frac{m_\alpha}{m_\alpha+m_Y}\,Q. \]Recoil energies are tens of keV. That is tiny beside the alpha energy, but it is thousands of times a chemical bond energy, so a recoiling atom near a surface leaves it. Harriet Brooks and Rutherford had seen recoil effects earlier; in 1909 Hahn and Meitner turned the effect into a separation method. A plate coated with a parent substance, held near a negatively charged collector in a gas, transfers its recoiling daughters to the collector. Using this they isolated short-lived members of the thorium and actinium series, including the isotopes now called thallium-207 and bismuth-211, which chemistry alone could not separate quickly enough. Recoil also accompanies beta decay, with much smaller energy, and Hahn and Meitner used it there too.
Exercise 4.1Radium-226 decays to radon-222 with \(Q=4.871\) MeV. Find the recoil energy of the radon nucleus. Compare it with a chemical bond energy of a few eV.
Show solution
Exercise 4.2In the two-member chain, show that \(N_2\) reaches its maximum at \(t_m=\dfrac{\ln(\lambda_2/\lambda_1)}{\lambda_2-\lambda_1}\), and that at that moment \(A_1=A_2\).
Show solution
Exercise 4.3Uranium-238 (\(T_{1/2}=4.47\times10^9\) y) and radium-226 (\(1600\) y) are in secular equilibrium in an old ore. How many grams of radium accompany one tonne of uranium-238?
Show solution
Protactinium
For the general reader. Actinium, found around 1900, was clearly a product of uranium, yet it did not belong to the main uranium chain. Something had to stand between them. Meitner and Hahn found that missing parent in 1917: a new element, number 91, which Meitner named protactinium, the one “before actinium”.
- Place protactinium in the actinium (uranium-235) series using the displacement laws.
- Understand how a long-lived parent was found by the growth of its daughter.
- Estimate natural abundances from secular equilibrium.
The displacement laws
In alpha decay \(^{A}_{Z}X\to{}^{A-4}_{Z-2}Y\); in beta decay \(^{A}_{Z}X\to{}^{A}_{Z+1}Y\). Products with the same \(Z\) are chemically inseparable isotopes, whatever their mass or half-life.
Actinium (\(Z=89\)) could arise from an element of \(Z=91\) by alpha decay, or from \(Z=88\) by beta decay. In 1913 Kasimir Fajans and Oswald Göhring had found a short-lived isotope of element 91 (now protactinium-234m, half-life about 1.2 minutes) in the uranium-238 chain, which they called brevium. It could not be the actinium parent. The parent had to be long-lived, since actinium is always found in uranium ores in a fixed ratio.
The search
The strategy was chemical. Element 91 should resemble tantalum, its homologue in the periodic table of the day. Hahn and Meitner treated pitchblende residues with acids, carried the expected element down with tantalum compounds, and then waited. If the fraction contained the actinium parent, actinium would grow into it. Its growth would be revealed by the alpha activity of actinium’s short-lived descendants rising steadily over months. Working mostly alone in Berlin in 1917, with materials obtained with help from Stefan Meyer and Friedrich Giesel, Meitner saw exactly this. The paper of 1918 described the new element, with a half-life she estimated as very long (the modern value is 32 760 years). Frederick Soddy and John Cranston reported the same substance independently a few months later and accepted the Berlin priority. In 1949 the International Union of Pure and Applied Chemistry fixed the name protactinium and credited Hahn and Meitner as discoverers.
The series head, uranium-235, was not identified until the 1930s, when Arthur Dempster found it with a mass spectrograph in 1935. In 1917 the parent of protactinium was simply “actino-uranium”.
Exercise 5.1Natural uranium contains 0.72% uranium-235 by number. Assuming secular equilibrium, how much protactinium-231 is there in one tonne of natural uranium?
Show solution
Exercise 5.2A freshly purified protactinium sample contains no actinium. Ignoring actinium’s daughters, how long until the actinium-227 activity reaches half the protactinium activity? Why did this make the 1917 experiment slow?
Show solution
Exercise 5.3Use the displacement laws to explain why protactinium-231 must be an alpha emitter if it is to produce actinium-227, and identify the beta-emitting parent of protactinium-231.
Show solution
Beta spectra and the energy crisis
For the general reader. Electrons from radioactive nuclei come out with a smooth spread of energies, plus a few sharp lines. Meitner first believed that every nucleus should emit electrons of one definite energy, and spent a decade arguing about where the spread came from. When experiments, including her own, proved the spread was real, the loss of energy seemed to break the conservation law. Wolfgang Pauli’s rescue was a new invisible particle, proposed in a letter addressed to her.
- Distinguish conversion lines from the continuous beta spectrum.
- Understand the calorimetric test of Ellis and Wooster and its repetition by Meitner and Orthmann.
- See how a third particle restores energy conservation.
Lines and continuum
From 1910 Hahn, Meitner and Otto von Baeyer used magnetic deflection onto photographic plates to study beta rays, and found sharp lines. Meitner first took these as the primary nuclear electrons. In 1922 she gave the interpretation that holds today for the lines: a nucleus left excited after a decay emits a gamma ray, or transfers the energy directly to an atomic electron, which leaves with a definite energy. This is internal conversion. She argued that the gamma ray comes from the daughter nucleus, after the decay.
If a nuclear transition of energy \(E_\gamma\) ejects an electron from shell \(i\) with binding energy \(B_i\), the electron energy is
\[ T_i=E_\gamma-B_i\qquad(i=K,L_1,L_2,\dots). \]Line differences \(T_L-T_K=B_K-B_L\) match the X-ray levels of the daughter element, which identifies which atom emitted them.
The continuous spectrum was the disputed part. James Chadwick showed in 1914 that it was real. Charles Drummond Ellis held that the primary electrons were continuous; Meitner argued that secondary processes smeared out a sharp primary energy. The decisive test was calorimetric. If each decay released the endpoint energy, a thick absorber would heat at the rate of (activity) × (endpoint). If not, it would heat at the rate of the mean spectral energy.
For radium E (bismuth-210, endpoint about 1.16 MeV in modern data), the heat deposited per decay is about 0.34 MeV, close to the mean energy of the continuous spectrum. Each decay releases a different amount of energy into the electron.
Meitner and Wilhelm Orthmann repeated the measurement with an improved calorimeter, confirmed it, and accepted the result. Niels Bohr was prepared to give up energy conservation in individual nuclear events. Pauli refused. On 4 December 1930 he wrote an open letter to the “radioactive ladies and gentlemen” meeting at Tübingen, naming Meitner and Hans Geiger, proposing a light neutral particle emitted with the electron. Enrico Fermi named it the neutrino and built the theory of beta decay on it in 1933–34.
Kinematics of a three-body decay
For \(n\to p+e^-+\bar\nu\) inside a nucleus, the decay energy \(Q\) is shared between electron and antineutrino (nuclear recoil is negligible), so \(0\le T_e\le Q\). Fermi’s theory gives the allowed spectrum
\[ \frac{dN}{dT_e}\ \propto\ F(Z,T_e)\,p_e\,E_e\,(Q-T_e)^2, \]where \(E_e=T_e+m_ec^2\), \(p_e\) is the electron momentum and \(F\) is the Fermi function describing the Coulomb distortion. The factor \((Q-T_e)^2\) comes from the neutrino phase space; the spectrum ends at \(T_e=Q\) for a massless neutrino.
Exercise 6.1Lead-214 beta-decays to bismuth-214, and a 351.9 keV transition follows. Conversion electrons are observed at 261.4 keV and 335.5 keV. The K and L\(_1\) binding energies are 88.0 and 15.9 keV for lead, 90.5 and 16.4 keV for bismuth. Identify the shells and decide which atom emits the electrons.
Show solution
Exercise 6.2A radium E source has activity \(3.7\times10^{9}\) Bq (0.1 Ci). Compare the heating power expected in a thick calorimeter if each decay deposits (a) the endpoint energy 1.16 MeV, (b) the mean energy 0.34 MeV.
Show solution
Exercise 6.3Show that in a two-body decay \(X\to Y+e\) at rest the electron energy is fixed, so a continuous spectrum is impossible without a third particle.
Show solution
The Auger–Meitner effect
For the general reader. When an inner electron is knocked out of an atom, an outer electron drops into the hole. The energy released usually goes into an X-ray. Sometimes, instead, it throws out another electron. Meitner described this radiationless process in 1922 while sorting out her beta-ray lines; Pierre Auger discovered it independently in 1923 in cloud-chamber tracks and studied it in depth. It is now a basic tool of surface analysis.
- Compute the energy of a KLL electron from atomic levels.
- Understand the competition between X-ray fluorescence and electron emission.
- See why the effect identifies elements at surfaces.
For a vacancy in shell \(W\) filled from shell \(X\), with ejection of an electron from shell \(Y\), the electron energy is approximately
\[ E_{WXY}\approx E_W-E_X-E_Y^{*}, \]where \(E_W,E_X\) are binding energies of the neutral atom and \(E_Y^{*}\) is the binding energy of \(Y\) in the ion already missing an electron in \(X\) (roughly \(E_Y\) of the element with \(Z+1\)). The energy depends only on atomic levels, not on whatever made the first vacancy.
Meitner’s paper of 1922, “Über die Entstehung der β-Strahl-Spektren radioaktiver Substanzen” in the Zeitschrift für Physik, described secondary electrons whose energies matched such level differences. Auger’s 1923 cloud-chamber photographs showed pairs of tracks from a single atom ionized by X-rays: the photoelectron and a second electron of fixed energy. The process carries Auger’s name in most textbooks; the name Auger–Meitner effect has gained ground since around 2019.
Fluorescence yield
A K vacancy decays either radiatively (probability \(\omega_K\), the fluorescence yield) or by electron emission (probability \(1-\omega_K\)). The radiative rate grows roughly as \(Z^4\) while the Auger–Meitner rate is nearly independent of \(Z\), giving the empirical rule
\[ \omega_K\approx\frac{Z^4}{a+Z^4},\qquad a\approx10^6. \]Light elements de-excite mainly by electron emission; heavy elements mainly by X-rays. Because Auger–Meitner electrons of a few hundred eV escape only from the top few nanometres of a solid, their spectra reveal the chemical composition of surfaces.
Exercise 7.1Use Henry Moseley’s law \(E_{K\alpha}\approx10.2\ {\rm eV}\times(Z-1)^2\) to estimate the K\(\alpha\) energy of copper (\(Z=29\)). Then, with \(E_K=8.98\) keV and \(E_{L_{2,3}}\approx0.93\) keV, estimate the KL\(_{2,3}\)L\(_{2,3}\) electron energy.
Show solution
Exercise 7.2With the rule \(\omega_K=Z^4/(10^6+Z^4)\), find the atomic number at which K vacancies are equally likely to decay radiatively and by electron emission. Estimate \(\omega_K\) for carbon and for lead.
Show solution
The transuranic puzzle, 1934–1938
For the general reader. In 1934 Enrico Fermi’s group in Rome bombarded uranium with neutrons and found new radioactivities that seemed to belong to elements beyond uranium. Meitner saw a physicist’s problem in it and asked Hahn to join her. For four years the Berlin team, with Fritz Strassmann, built an elaborate structure of “transuranic” elements. It was wrong, and the way it went wrong is instructive.
- Explain why neutron capture followed by beta decay was expected to give element 93.
- See how carrier chemistry assigns an activity to an element, and how it can mislead.
- Understand Meitner’s 1937 interpretation and its difficulties.
The expected reaction
A neutron, having no charge, is not repelled by the nucleus. Capture produces a heavier isotope with a neutron excess, which should beta-decay, raising \(Z\) by one:
\[ ^{238}_{\ 92}\mathrm U+n\ \to\ ^{239}_{\ 92}\mathrm U+\gamma,\qquad ^{239}_{\ 92}\mathrm U\ \xrightarrow{\ \beta^-\ }\ ^{239}_{\ 93}\mathrm X. \]Fermi found that slowing the neutrons in paraffin greatly increased many activities, since a slow neutron spends longer near a nucleus (capture cross sections often vary as \(1/v\)). Ida Noddack objected in 1934 that the nucleus might break into large fragments, but she offered no mechanism or experiment, and nobody followed it up. Aristid von Grosse suggested that one Rome activity was protactinium. Hahn and Meitner, the discoverers of protactinium, tested this and showed it was not, which committed them to the problem.
Carrier chemistry
Radioactive products were formed in amounts of perhaps \(10^{6}\)–\(10^{8}\) atoms, far too few to weigh. A chemist adds milligrams of a stable carrier element and performs a precipitation; whatever activity follows the carrier is assigned to an element of similar chemistry. The Berlin team assumed the new elements would resemble rhenium, osmium, iridium and platinum, the homologues of 93–96 in the periodic table of the day. Activities precipitated with platinum-group sulphides were therefore labelled elements 93 to 96. In fact the actinides form a separate series, and the activities were fission products, among them isotopes of technetium and other mid-weight elements.
By 1937 the group had about ten half-lives arranged in three parallel chains, each starting from uranium-239 and climbing to “eka-platinum”. Meitner concluded that capture by uranium-238 had to produce three isomeric states of uranium-239, and she wrote that this was very hard to reconcile with existing ideas of the nucleus. The same work established the 23-minute half-life of uranium-239, formed by resonance capture, which was correct.
The real daughter of uranium-239, neptunium-239 (half-life 2.36 days), was identified by Edwin McMillan and Philip Abelson in 1940. In 1938 Irène Joliot-Curie and Pavel Savitch reported a 3.5-hour activity that behaved like lanthanum, which Berlin doubted. Hahn and Strassmann then claimed radium isotopes formed by the emission of two alpha particles (told from the chemist’s side in the companion course on Hahn). Meitner, writing from Stockholm, pressed them to check this.
Exercise 8.1The capture cross section of a nucleus for slow neutrons often varies as \(\sigma\propto1/v\). By what factor does it increase when a 1 MeV neutron is thermalized to 0.025 eV? Why did paraffin help Fermi?
Show solution
Exercise 8.2Show that beta decay of uranium-239 followed by beta decay of element 93 leads to an isotope of element 94 with mass number 239, and identify it. Why could the 1930s team not detect the second step easily?
Show solution
The liquid drop splits
For the general reader. Just before Christmas 1938, Hahn wrote to Meitner that the “radium” formed from uranium behaved like barium, an element about half as heavy. Meitner and Frisch, walking in the snow at Kungälv, asked how a nucleus could do this. Their answer was that a uranium nucleus is like a drop of liquid barely held together; a small kick makes it wobble, stretch and divide, and the two charged halves fly apart with an enormous energy.
- Set up the liquid-drop energy balance between surface tension and Coulomb repulsion.
- Derive the fissility parameter and the stability limit.
- Estimate the energy released, two ways, as Meitner and Frisch did.
The drop
George Gamow and then Bohr had described the nucleus as a drop of incompressible nuclear fluid of radius \(R=r_0A^{1/3}\), \(r_0\approx1.2\) fm. In the semi-empirical mass formula of Carl Friedrich von Weizsäcker (1935), the surface and Coulomb energies of a sphere are
\[ E_S^{(0)}=a_SA^{2/3},\qquad E_C^{(0)}=a_C\frac{Z^2}{A^{1/3}},\qquad a_S\approx17.8\ {\rm MeV},\ a_C\approx0.71\ {\rm MeV}. \]Deform the sphere into a spheroid with small quadrupole parameter \(\varepsilon\) at fixed volume. To second order (Bohr and Wheeler, 1939),
\[ E_S=E_S^{(0)}\Bigl(1+\tfrac25\varepsilon^2\Bigr),\qquad E_C=E_C^{(0)}\Bigl(1-\tfrac15\varepsilon^2\Bigr). \]The change in energy is \(\Delta E=\tfrac{\varepsilon^2}{5}\bigl(2E_S^{(0)}-E_C^{(0)}\bigr)\). A spherical drop is stable if and only if the fissility
\[ x=\frac{E_C^{(0)}}{2E_S^{(0)}}=\frac{a_C}{2a_S}\,\frac{Z^2}{A}<1, \qquad\text{i.e.}\qquad \frac{Z^2}{A}<\Bigl(\frac{Z^2}{A}\Bigr)_{\rm crit}=\frac{2a_S}{a_C}\approx50. \]Meitner and Frisch did not have the second-order formula; they argued that the effective surface tension of a charged drop falls as the charge grows and estimated that it would vanish near \(Z\approx100\). Uranium, with \(Z^2/A\approx36\) and \(x\approx0.7\), is stable at rest but close enough to the limit that the energy of a captured neutron, around 6 MeV, can drive it over a barrier of similar height.
The energy
Meitner and Frisch estimated the energy two ways. First, two fragments of charges \(Z_1,Z_2\) starting at contact repel with Coulomb energy
\[ E\approx\frac{e^2}{4\pi\varepsilon_0}\,\frac{Z_1Z_2}{R_1+R_2},\qquad\frac{e^2}{4\pi\varepsilon_0}=1.44\ {\rm MeV\,fm}, \]which is roughly 200 MeV. Second, Meitner remembered the packing fractions (mass defects) of the elements: uranium is heavier than the sum of two mid-weight nuclei by about one-fifth of a proton mass, which by \(E=mc^2\) is about 200 MeV. The agreement convinced them. The fragments, having the neutron-to-proton ratio of uranium, would be neutron-rich and undergo chains of beta decays, which is why Berlin saw so many half-lives.
Meitner and Frisch wrote their note by telephone between Stockholm and Copenhagen in early January 1939. Frisch borrowed the word fission from cell biology. On 13 January he detected the large ionization pulses of the fragments in an ionization chamber lined with uranium. The note, “Disintegration of uranium by neutrons: a new type of nuclear reaction”, was sent to Nature on 16 January and published on 11 February 1939; Frisch’s experimental note followed on 18 February.
Exercise 9.1Estimate the Coulomb energy of barium-141 and krypton-92 at contact, with \(r_0=1.2\) fm. Compare with the measured total kinetic energy of fragments, about 170 MeV. Why is the estimate high?
Show solution
Exercise 9.2The binding energy per nucleon is about 7.6 MeV for uranium and about 8.4 MeV for nuclei near \(A=120\). Estimate the energy released when uranium-236 splits, and express the mass loss as a fraction of a proton mass.
Show solution
Exercise 9.3Compute \(x\) for uranium-235 and for thorium-232 with \(a_S=17.8\) MeV and \(a_C=0.711\) MeV. Which is closer to instability? Estimate the \(Z\) at which \(x=1\) for nuclei with \(A\approx2.5Z\).
Show solution
Neutrons, uranium-235 and the chain reaction
For the general reader. Within weeks of the fission papers, physicists asked two questions. Which uranium isotope splits? Does fission release neutrons that can split further nuclei? Bohr answered the first, using in part Meitner’s 1937 measurements; groups in Paris and New York answered the second. The answers made a chain reaction, and a bomb, possible. Meitner refused to take part in building one.
- Understand why uranium-235 fissions with slow neutrons and uranium-238 does not.
- Define the multiplication factor and follow a chain reaction.
- Estimate the energy content of fissile material.
Odd and even
Bohr pointed out early in 1939 that slow-neutron fission must come from the rare isotope uranium-235. Capturing a neutron into uranium-235 forms an even–even nucleus, uranium-236, and the pairing energy makes the excitation energy larger (about 6.5 MeV) than the fission barrier (about 6 MeV). Capture into uranium-238 gives odd-neutron uranium-239 with about 4.8 MeV, below its barrier, so it fissions only with fast neutrons above about 1 MeV and otherwise forms the 23-minute uranium-239 that Meitner had measured. Bohr and John Wheeler set out the full theory in Physical Review on 1 September 1939. Separated uranium-235 confirmed the assignment in 1940.
Let each fission release on average \(\nu\approx2.4\) neutrons (for uranium-235 with thermal neutrons). If on average \(k\) of the neutrons from one fission cause another fission, then after \(n\) generations the number of fissions per generation is \(k^n\). The system is subcritical for \(k<1\), critical for \(k=1\), and supercritical for \(k>1\).
Hans von Halban, Frédéric Joliot-Curie and Lew Kowarski in Paris, and Fermi and Herbert Anderson in New York, showed in the spring of 1939 that fission emits more than one neutron. Frisch and Rudolf Peierls in Birmingham showed in 1940 that a few kilograms of pure uranium-235 would suffice for a weapon. Frisch went on to Los Alamos. Meitner declined an invitation to join the British mission there.
Exercise 10.1How many uranium-235 atoms are in 1 kg? If every one fissions with 200 MeV, find the energy in joules and in kilotons of TNT (1 kt = \(4.184\times10^{12}\) J). How many generations with \(k=2\) would be needed to fission them all starting from one neutron?
Show solution
Exercise 10.2Natural uranium contains 0.72% uranium-235. Explain why a lump of natural uranium in air cannot sustain a chain reaction, while natural uranium arranged in a lattice with graphite or heavy water can.
Show solution
Exile and the 1944 Nobel Prize, 1938–1946
On 12 March 1938 Germany annexed Austria. Meitner’s Austrian passport became invalid and German law now applied to her fully. Carl Bosch tried to obtain permission for her to leave and was refused; scientists were forbidden to emigrate. Her Dutch colleagues Dirk Coster and Adriaan Fokker, with Peter Debye in Berlin and Bohr in Copenhagen, arranged an escape. Coster arrived in Berlin on 11 July. On 13 July, after a last day at the institute, she left with him by train with a small suitcase and crossed into the Netherlands at Nieuweschans. Hahn and the publisher Paul Rosbaud helped. Hahn gave her a diamond ring that had belonged to his mother, for emergencies.
From the Netherlands she went to Copenhagen and then to Sweden, where she joined Manne Siegbahn’s new Nobel Institute for Physics in Stockholm, on a small salary supported by Nobel funds. She had no assistants or equipment of her own and felt unwelcome. She kept up a constant correspondence with Hahn about the uranium work. On 13 November 1938 she met Hahn secretly in Copenhagen and urged him to check his radium claims. On 19 December Hahn wrote that his “radium” behaved like barium; Hahn and Strassmann sent their paper to Die Naturwissenschaften on 22 December. She spent Christmas at Kungälv with her friend Eva von Bahr and with Frisch, and the interpretation of Lesson 09 followed.
The Nobel Prize
The 1944 Nobel Prize in Chemistry was decided in an unusual way. In 1944 the Royal Swedish Academy of Sciences, on the advice of its chemistry committee, reserved the prize for a year. In November 1945 it awarded the 1944 prize to Otto Hahn alone, “for his discovery of the fission of heavy nuclei”. Hahn was then interned with nine other German scientists at Farm Hall in England; he received the prize in Stockholm on 10 December 1946. Strassmann was not included, nor were Meitner and Frisch. Meitner was nominated 49 times in all, 19 times for chemistry (1924–1948) and 30 times for physics (1937–1967), by nominators including Planck, Bohr, Max Born and Arthur Compton, but never received a prize.
When the committee papers were opened, the historians Elisabeth Crawford, Ruth Lewin Sime and Mark Walker analysed them in 1997. The chemistry committee, advised by Theodor Svedberg and Arne Westgren, had judged the discovery a chemical one and treated the physical interpretation as secondary. They concluded that the exclusion of Meitner reflected disciplinary bias, political obtuseness, ignorance and haste. Meitner herself wrote that Hahn fully deserved the prize, while holding that she and Frisch had contributed something not insignificant to understanding fission. Her relations with Hahn survived, though she was hurt that he described the discovery after the war as the result of chemistry alone. In an unsent letter of June 1945 she reproached her German colleagues for having worked for Nazi Germany without even attempting passive resistance.
In January 1946 she went to the United States as a visiting professor at the Catholic University of America, lectured at Princeton, Harvard and Columbia, and was named Woman of the Year by the National Press Club.
DiscussionThe chemistry committee treated fission as a discovery made in a test tube. Using Lessons 08 and 09, list what the Berlin team knew on 22 December 1938 and what was added by 11 February 1939. Which of these were chemical and which physical, and does the distinction hold?
Later years and legacy, 1947–1968
In 1947 Meitner moved to the Royal Institute of Technology (KTH) in Stockholm, where Gudmund Borelius had set up a nuclear research group, and she contributed to the work that led to Sweden’s first research reactor, R1. She became a Swedish citizen in 1949, keeping her Austrian nationality. The same year she and Hahn shared the Max Planck Medal of the German Physical Society. She received the Otto Hahn Prize in 1955, was elected a Foreign Member of the Royal Society in 1955, and joined the order Pour le Mérite for Sciences and Arts in 1957.
She retired in 1960 and moved to Cambridge, England, near Frisch. In 1966 the United States Atomic Energy Commission gave the Enrico Fermi Award jointly to Hahn, Meitner and Strassmann: Frisch received it on her behalf. She died in Cambridge on 27 October 1968, aged 89, three months after Hahn. She was buried at St James’ Church, Bramley, Hampshire, near her brother Walter. The inscription Frisch wrote for her headstone calls her a physicist who never lost her humanity.
Legacy
- Nuclear physics: the first physical interpretation of fission and its energy; the 23-minute uranium-239 and early neutron cross sections that Bohr used to assign fission to uranium-235.
- Radioactivity: protactinium, the recoil method, and the identification of many members of the decay series.
- Beta and gamma rays: the origin of conversion lines, the place of gamma emission after decay, the calorimetric confirmation that set up the neutrino, and the radiationless process now named for her and Auger.
- Honours after death: element 109, meitnerium (1997); the Hahn-Meitner-Institut in Berlin (1959); the Lise Meitner Prize of the European Physical Society; numerous schools and streets in Austria and Germany.
- History: the case of the 1944 prize is now a standard study of how credit is assigned across disciplines and how exile erases contributions.
Final projectChoose one result from Part II. Write a two-page account for a strong undergraduate that states it, works one numerical example in full, and explains what physical reasoning, as opposed to chemical identification, it required.
Sources and further reading
Biography and history
- Lise Meitner, Wikipedia (with extensive references).
- R. L. Sime, Lise Meitner: A Life in Physics, University of California Press, 1996.
- E. Crawford, R. L. Sime, M. Walker, A Nobel tale of postwar injustice, Physics Today 50(9) (1997) 26.
- Nobel Prize nomination archive, Lise Meitner (49 nominations).
- The Nobel Prize, The Nobel Prize in Chemistry 1944.
- Discovery of nuclear fission, Wikipedia.
- O. R. Frisch, What Little I Remember, Cambridge University Press, 1979.
- O. R. Frisch, Lise Meitner 1878–1968, Biographical Memoirs of Fellows of the Royal Society 16 (1970).
Physics
- O. Hahn, L. Meitner, Die Muttersubstanz des Actiniums, ein neues radioaktives Element von langer Lebensdauer, Physikalische Zeitschrift 19 (1918).
- L. Meitner, Über die Entstehung der β-Strahl-Spektren radioaktiver Substanzen, Zeitschrift für Physik 9 (1922) 131–144.
- C. D. Ellis, W. A. Wooster, The average energy of disintegration of radium E, Proc. R. Soc. A 117 (1927).
- L. Meitner, W. Orthmann, Über eine absolute Bestimmung der Energie der primären β-Strahlen von Radium E, Zeitschrift für Physik 60 (1930).
- L. Meitner, O. Hahn, F. Strassmann, Über die Umwandlungsreihen des Urans, die durch Neutronenbestrahlung erzeugt werden, Zeitschrift für Physik 106 (1937).
- O. Hahn, F. Strassmann, Über den Nachweis und das Verhalten der bei der Bestrahlung des Urans mittels Neutronen entstehenden Erdalkalimetalle, Naturwissenschaften 27 (1939) 11–15.
- L. Meitner, O. R. Frisch, Disintegration of uranium by neutrons: a new type of nuclear reaction, Nature 143 (1939) 239–240.
- O. R. Frisch, Physical evidence for the division of heavy nuclei under neutron bombardment, Nature 143 (1939) 276.
- N. Bohr, J. A. Wheeler, The mechanism of nuclear fission, Physical Review 56 (1939) 426–450.
- Auger effect, Wikipedia (history of the Meitner and Auger papers).
- K. S. Krane, Introductory Nuclear Physics, Wiley, 1988 (decay chains, beta spectra, fission).