Hans Geiger

How to use this course

Hans Geiger’s name is on an instrument known far beyond physics. Behind the clicking counter lies thirty-five years of work on a single problem: how to detect and count individual particles reliably. At Manchester, with Ernest Rutherford, Geiger counted alpha particles one by one, measured their charge, showed their emission is random, and with Ernest Marsden made the scattering measurements that revealed the atomic nucleus. With John Mitchell Nuttall he found the law linking alpha energy to half-life. In Germany he built the point counter, invented with Walther Bothe the coincidence method that tested the light-quantum hypothesis, and with Walther Müller made the Geiger–Müller tube.

His life also passed through two world wars and the Nazi dictatorship. Lesson 11 treats that period with the documented facts and their sources, including the points on which accounts differ.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 05 and 09, on counting statistics and on the counter itself, are the centre of the course.

Timeline

Neustadt and Erlangen, 1882–1906

Johannes Wilhelm Geiger, always called Hans, was born on 30 September 1882 in Neustadt an der Haardt, in the Palatinate, then part of the Kingdom of Bavaria. He was the eldest of five children of Wilhelm Geiger, a scholar of the languages and literature of India and Iran, and Marie Plochmann. In 1891 his father became professor at the University of Erlangen, and the family moved there. A younger brother, Rudolf Geiger, became a well-known climatologist.

Hans finished at the Erlangen Gymnasium in 1901 and did his year of military service. He studied physics and mathematics at Erlangen from 1902, with a semester at Munich in 1904, and passed the state examinations for teaching. His doctoral adviser was Eilhard Wiedemann, an experimental physicist who worked on electric discharges and luminescence. Geiger’s thesis, completed in 1906, measured radiation, temperature and potential in discharge tubes carrying strong currents. Work on gas discharges was the right training. Every counter he later built depended on controlling a discharge in a gas.

Manchester, 1906–1912

In 1906 Geiger went to the University of Manchester as an assistant to Arthur Schuster, the professor of physics, who had studied in Germany and kept close ties there. Schuster retired in 1907 and Ernest Rutherford came from McGill to succeed him. Rutherford inherited Geiger with the laboratory and quickly made him his main collaborator. Geiger also took on much of the training of research students in radioactive techniques.

The work of these six years is the core of Part II. In 1908 Rutherford and Geiger built an electrical counter for single alpha particles, counted the particles from radium and found their charge (Lesson 04). In 1910 they showed that the counts obey the laws of chance (Lesson 05). From 1908 Geiger studied how metal foils scatter alpha particles, and in 1909 he and Ernest Marsden, then an undergraduate, found that a few alpha particles bounce back from a metal plate (Lesson 06). Geiger measured the range and ionization of alpha particles (Lesson 07), and in 1911 he and Nuttall found the law that links range to half-life (Lesson 08).

Much of the counting was done by eye, watching scintillations on a zinc sulphide screen through a microscope in a dark room. Observers needed about half an hour for their eyes to adapt and could count accurately for only about a minute at a stretch. Rutherford wrote that Geiger was a demon at the work. In 1912 Geiger and Walter Makower published Practical Measurements in Radio-activity, a laboratory manual. By then Geiger was among the most experienced radioactivity experimenters in Europe.

The Reichsanstalt and the war, 1912–1925

In 1912 Geiger returned to Germany to head a new laboratory for radioactivity at the Physikalisch-Technische Reichsanstalt (PTR), the national standards laboratory in Berlin-Charlottenburg. He completed the scattering tests with Marsden by correspondence and visits; their final paper appeared in April 1913. In the same year he described a new detector, the point counter, in which a sharp needle replaced the central wire (Lesson 09).

Two young physicists joined him in 1913. James Chadwick came from Manchester on an 1851 Exhibition scholarship and used Geiger’s counters to show that the beta-ray spectrum is continuous. Walther Bothe became Geiger’s assistant and later his most important collaborator. When war came in August 1914 Chadwick was interned as an enemy alien, and Geiger, a reserve officer, was called up.

Geiger served through the war as an artillery officer. For a period he also served in the gas-warfare troops organised by Fritz Haber, as did several German physicists. He fought on the opposite side from several of his Manchester colleagues. The conditions left him with chronic rheumatism, which grew worse with age and shaped the last years of his life.

After the war he returned to the PTR. In 1920 he married Elisabeth Heffter, daughter of the Berlin pharmacologist Arthur Heffter; they had three sons. The same year he and Karl Scheel founded the Zeitschrift für Physik, which within a few years became the main journal of the new quantum mechanics. Geiger qualified as a university lecturer in Berlin in 1924, and with Bothe began the coincidence experiments of Lesson 10.

Counting single alpha particles

For the general reader. An alpha particle carries enormous energy for a single atom, but it is still only one atom. In 1908 Rutherford and Geiger found a way to make each one produce an electric pulse large enough to register on a meter. They counted how many alpha particles a known amount of radium emits per second and measured the total charge they carry. Dividing one by the other gave the charge of each particle: twice the charge of the electron.

Multiplication in a gas

An alpha particle of a few MeV crossing a centimetre of gas makes some tens of thousands of ion pairs. That charge, a few femtocoulombs, is too small for an electrometer of 1908. Rutherford and Geiger used a brass tube with a thin wire along its axis, filled with gas at a few centimetres of mercury and held at a voltage just below that at which a discharge starts. The alpha particles entered through a small window. Electrons freed by the particle drift towards the wire and gain enough energy between collisions to ionize further atoms. Each new electron does the same.

Because \(\alpha_T\) rises steeply with the electric field, the multiplication happens almost entirely in the strong field near the thin wire (Lesson 09). This was John Sealy Townsend’s mechanism of gas discharge, used for the first time to detect single particles.

The charge

Rutherford and Geiger counted the alpha particles from a thin layer of radium C, reached through a long tube so that only a few per minute entered the counter. They found that one gram of radium emits about \(3.4\times10^{10}\) alpha particles per second. In a separate experiment they measured the total charge carried by the alpha particles from a known source, and found about \(9.3\times10^{-10}\) electrostatic units per particle. That is close to twice the best estimates of the elementary charge, so they concluded that the alpha particle carries \(2e\). Taking the charge as exactly \(2e\) gave \(e\approx4.65\times10^{-10}\) esu, or \(1.55\times10^{-19}\) C, within 4% of the modern value.

They checked the electrical counts against scintillations. Every alpha particle produced a flash; the two methods agreed. This settled a doubt about the scintillation method, which was then used for the scattering work of Lesson 06.

The statistics of decay

For the general reader. If radioactive atoms decay at random, the number of alpha particles counted in a few seconds should fluctuate in a definite way, sometimes high and sometimes low. In 1910 Rutherford and Geiger counted alpha particles in 2608 short intervals. The mathematician Harry Bateman worked out what pure chance predicts. The match was close. This was the first precise demonstration that a law of physics governs single atoms only through probabilities.

Derivation. Let \(P_k(t)\) be the probability of \(k\) events by time \(t\). In \(dt\), either nothing happens (probability \(1-r\,dt\)) or one event happens (probability \(r\,dt\)). So \(P_k'=-rP_k+rP_{k-1}\), with \(P_0(0)=1\). Then \(P_0=e^{-rt}\), and induction gives \(P_k=(rt)^ke^{-rt}/k!\). The mean of \(k\) is \(\sum kP_k=m\), and \(\sum k(k-1)P_k=m^2\) gives variance \(m\). The waiting time exceeds \(\tau\) exactly when no event occurs in \(\tau\), with probability \(e^{-r\tau}\).

Radioactive decay fits the hypothesis exactly. A source of \(n\) atoms with decay constant \(\lambda\) has \(r=n\lambda\), and over the experiment \(n\) hardly changes. Each atom decays independently of the others.

The 1910 experiment

Rutherford and Geiger recorded the scintillations from a polonium source on a moving paper strip, then divided the record into intervals of one-eighth of a minute. Their table of how many intervals contained \(k\) particles, as reprinted in many textbooks, is:

The total is 2608 intervals and 10 097 particles, a mean of 3.87. The bottom row is \(2608\,P_k\) with \(m=3.87\). Bateman’s note, printed with the paper, gave the formula.

Scattering: single and multiple

For the general reader. When a narrow beam of alpha particles passes through thin gold leaf, it spreads out by a degree or so. In 1908 Geiger measured this spread carefully. A year later he and Marsden found something the spreading could not explain: a few particles came straight back. The small spread came from many gentle nudges by atoms; the rare reversals came from one violent encounter with a nucleus. Telling the two apart was the key to Rutherford’s atom.

Multiple scattering

Geiger’s 1908 apparatus was a glass tube about two metres long with a radioactive source at one end, a narrow slit in the middle and a zinc sulphide screen at the far end. With the tube evacuated, the image of the slit was sharp. A gold foil in the beam spread it out. In 1910 he measured the most probable angle of deflection for foils of gold, tin, silver, copper and aluminium: it grew with foil thickness and with atomic weight, and fell as the alpha particles were made faster.

Derivation. The sum of many independent random vectors tends to a Gaussian by the central limit theorem, with variance the sum of the variances. A two-dimensional isotropic Gaussian with \(\langle\Theta^2\rangle=\sigma^2\) has density \((1/\pi\sigma^2)e^{-\Theta^2/\sigma^2}\) per unit area in the angle plane. Integrating over \(\Theta>\theta\) with area element \(2\pi\Theta\,d\Theta\) gives the formula.

Single scattering

J. J. Thomson’s atom produces only small deflections in each encounter, so it can produce large angles only through this Gaussian tail. Geiger and Marsden’s 1909 observation, about one particle in 8000 reflected from a thick platinum plate, was impossible on that basis (Exercise 6.1). Rutherford’s single-scattering law instead gives a tail falling only as \(\sin^{-4}(\theta/2)\).

In 1913 Geiger and Marsden tested every prediction of that law. They mounted the source and foil in a sealed metal box that could be evacuated, with a microscope and screen that rotated about the foil. Over angles up to 150°, with silver and gold foils, the counts followed \(1/\sin^4(\theta/2)\) while varying by several orders of magnitude. They checked proportionality to foil thickness, the dependence on atomic weight, and the \(1/v^4\) dependence on velocity by slowing the alpha particles with mica sheets. They concluded that the nuclear charge is about half the atomic weight in units of \(e\).

The range of alpha particles

For the general reader. Alpha particles from a given substance travel a definite distance in air, a few centimetres, then stop abruptly. Geiger found how this range depends on speed: it grows as the cube of the speed. The rule became a working tool. Measuring the range of the particles from an unknown source told experimenters their energy, and through the next lesson, the lifetime of the source.

Range and stopping power

William Henry Bragg showed in 1904 that alpha particles from a single substance have a definite range, and that the ionization along the track rises towards the end before dropping sharply, the Bragg peak. Geiger measured the speed of alpha particles after they had passed through various thicknesses of absorber and in 1910 stated the relation that bears his name.

Derivation. The range is \(R=\int_0^E dE'/S(E')\), where \(S=-dE/dx\) is the stopping power. The Bethe formula, from 1930, gives \(S\propto\frac{z^2}{v^2}\ln\frac{2m_ev^2}{I}\), with \(I\) a mean excitation energy. Over a limited range of speeds the logarithm varies slowly and can be modelled as a power, so that \(S\propto v^{-1}\) roughly. With \(E=\tfrac12Mv^2\) and \(dE=Mv\,dv\), \(R\propto\int v\cdot v\,dv\propto v^3\). Geiger found the rule empirically, two decades before Bethe’s theory.

The Geiger–Nuttall law

For the general reader. Some alpha emitters last billions of years, others a fraction of a second. In 1911 Geiger and John Mitchell Nuttall found that the short-lived ones shoot their alpha particles further. Plotted on logarithmic scales, decay constant against range gave nearly straight lines. Classical physics could not say why. In 1928 George Gamow, and independently Ronald Gurney and Edward Condon, explained it with quantum tunnelling: a slightly more energetic alpha particle escapes the nucleus enormously more easily.

Since \(R\propto E^{3/2}\), a linear relation between \(\ln\lambda\) and \(\ln R\) is a relation between \(\ln\lambda\) and \(\ln E\). Over the narrow energy range of natural emitters, 4 to 9 MeV, it is hard to tell \(\ln E\) from \(1/\sqrt E\), which is what the theory gives.

Gamow’s explanation

Model the alpha particle as preformed inside a nucleus of radius \(R_n\), striking the barrier with frequency \(f\). Outside \(R_n\) it feels the Coulomb potential \(V(r)=\kappa/r\), with \(\kappa=2Ze^2/(4\pi\varepsilon_0)\). It escapes with energy \(Q\), which equals \(V\) at the classical turning point \(r_c=\kappa/Q\). The WKB transmission probability is \(e^{-2G}\), with

Substitute \(r=r_c\cos^2u\) to evaluate the integral. For \(x\ll1\) the bracket is \(\tfrac\pi2-2\sqrt x\), so

The Geiger–Müller counter

For the general reader. Geiger’s 1908 counter worked only for alpha particles, and only with care. In 1913 he made a point counter that also caught beta rays. In 1928, at Kiel, he and his student Walther Müller found that a thin wire in a sealed tube, run at a higher voltage, gave a large pulse for any particle that ionized even a single gas molecule. The Geiger–Müller tube was cheap, robust and sensitive to every kind of ionizing radiation. It became the standard detector of nuclear and cosmic-ray physics, and of radiation protection.

Field near a thin wire

Derivation. By Gauss’s law, a line charge \(q\) per unit length gives \(E=q/(2\pi\varepsilon_0r)\). The potential difference is \(V=\int_a^bE\,dr=\tfrac{q}{2\pi\varepsilon_0}\ln(b/a)\). Eliminating \(q\) gives the formula. The field near the wire is \(b/a\) times stronger than at the wall. Electrons drift inwards gently, then multiply in an avalanche within a few wire radii of the anode.

Three regimes

At low voltage the tube is an ionization chamber: it collects the primary ions without gain. At higher voltage, multiplication sets in with a gain independent of the primary charge, and the pulse is proportional to the initial ionization. This proportional mode lets one distinguish an alpha particle from a beta particle. At still higher voltage, ultraviolet photons from the avalanche liberate electrons elsewhere along the wire, and the discharge spreads along the whole anode. The pulse is then the same size whatever started it. This is the Geiger regime. Its output is large, of order a volt, and needs no amplifier, but carries no information about the particle’s energy.

The discharge stops when the cloud of slow positive ions around the wire reduces the field below the threshold. The tube is then dead until the ions drift away. Early tubes relied on a large resistor in the external circuit to quench the discharge. Adding a little organic vapour such as alcohol to the gas made tubes self-quenching. Halogen-quenched tubes followed after the Second World War. Operating voltages for organic-quenched tubes are around 1000 V; the usable voltage range, the plateau, is where the count rate hardly changes with voltage.

Derivation. In unit time the counter is dead for a total time \(m\tau\). Particles arrive at rate \(n\) during the live time \(1-m\tau\), and all of those are recorded: \(m=n(1-m\tau)\).

Coincidences, Compton and cosmic rays

For the general reader. In 1924 Niels Bohr, Hendrik Kramers and John Slater proposed that energy might be conserved only on average in atomic processes. Bothe and Geiger devised a direct test. If an X-ray quantum scatters off an electron, do the scattered quantum and the recoil electron appear at the same instant? They put two counters on either side of a scatterer and recorded when each fired. The pulses came together far more often than chance allowed. Energy and momentum are conserved in each event. The coincidence method they invented became a basic tool of physics.

The Bothe–Geiger experiment

The BKS theory appeared in early 1924. Bothe and Geiger announced a test the same year. A beam of X-rays crossed hydrogen gas between two point counters. One counter, open to the gas, detected recoil electrons. The other, covered with thin platinum foil, detected scattered X-rays through the photoelectrons they released in the foil. The pulses moved the fibres of two string electrometers, which were photographed side by side on a moving film, so that simultaneous pulses could be identified by eye. The analysis took more than a year. In April 1925 they reported that coincidences occurred far more often than accident could explain, within a time window of about a millisecond. Arthur Compton and Alfred Simon reached the same conclusion with a cloud chamber later in 1925. Bohr accepted the result, and the BKS theory was abandoned. Bothe received the Nobel Prize in Physics for 1954 for the coincidence method, after Geiger’s death.

Derivation. Each pulse in counter 1 opens a window of width \(2\tau\), from \(\tau\) before to \(\tau\) after. The expected number of counter-2 pulses inside it is \(2\tau N_2\) for small \(\tau N_2\). Multiplying by the \(N_1\) windows per second gives the result.

Each scattered photon direction corresponds to a definite electron direction. Bothe and Geiger’s coincidences showed the timing; Compton and Simon’s photographs showed this angular correlation event by event.

Cosmic-ray showers

The Geiger–Müller tube made coincidence counting easy, because its pulses are large and fast. Bothe and Werner Kolhörster used two tubes in coincidence in 1929 to show that cosmic rays at sea level include charged particles that penetrate thick gold and lead. At Tübingen and later in Berlin, Geiger and his students studied cosmic-ray showers, bursts of many particles that fire several counters at once. With Otto Zeiller he measured their spatial spread and the ratio of electrons to photons; with W. Stubbe, the frequency and size of the extensive air showers that Pierre Auger had found in 1938 to cover areas many metres across.

Kiel, Tübingen, Berlin, 1925–1945

Kiel and Tübingen

In 1925 Geiger became full professor of physics at the University of Kiel, his first university chair. There, with Walther Müller, he developed the counter of Lesson 09; their first paper appeared in the Physikalische Zeitschrift in 1928. In October 1929 he moved to the University of Tübingen as professor and director of its physics institute. He turned increasingly to cosmic rays, using arrays of counters in coincidence. In 1929 the Royal Society awarded him the Hughes Medal for his methods of counting alpha and beta particles. From 1926 to 1929 he edited the Handbuch der Physik with Scheel.

1933

The Nazi government’s Law for the Restoration of the Professional Civil Service of 7 April 1933 led to the dismissal of civil servants of Jewish descent and of political opponents, including university staff. At Tübingen, Hans Bethe had held a junior teaching post in theoretical physics since 1932, while Geiger was professor of experimental physics. Bethe, whose mother was of Jewish descent, was dismissed. In a letter of 11 April 1933 to Arnold Sommerfeld, Bethe described the very stiff reply he had received from Geiger when he asked about his position. Geiger declined to help him. Sommerfeld restored Bethe’s fellowship in Munich, and Bethe went to England later that year and to Cornell in 1935. Other accounts record that Geiger helped some students affected by the racial laws, for example by supporting a student classed as half-Jewish through to the diploma. Historians’ judgements of his conduct differ, and the sources do not allow a single verdict.

Berlin and the defence of physics

In 1936 Geiger moved to the Technische Hochschule Berlin-Charlottenburg as director of its physics institute. The chair had been held by Gustav Hertz, who had been forced out under Nazi racial policy. In the same year, after Scheel’s death, Geiger became editor of the Zeitschrift für Physik, and he edited it through the war years. He was elected to the Leopoldina in 1935 and to the Prussian Academy of Sciences in 1936.

From the mid-1930s Philipp Lenard and Johannes Stark led a campaign for a “German physics” that denounced relativity and quantum theory as Jewish. In 1936 Geiger joined Werner Heisenberg and Max Wien in drafting a petition to the Reich Education Ministry, signed by 75 physicists, that defended theoretical physics against these attacks and against political interference in appointments. The petition had little immediate effect. It is the clearest documented public act by Geiger in the politics of science under the regime. The sources consulted for this course record no Nazi party membership, and he made no public statements either for or against the regime.

The war

After the discovery of fission, the German Army Ordnance Office convened physicists to study military uses of uranium. Geiger was among those invited to the meeting of 16 September 1939 in Berlin that started the uranium project, often called the Uranverein, and he took part in its early discussions. According to German accounts, his advice at that meeting to pursue the research carried weight, and in 1942 he argued against continuing the work on a large scale. From about 1942 a severe rheumatic illness confined him to bed for long periods and he withdrew from most of his duties. His institute continued work on counters, cosmic rays and artificial radioactivity.

Death, honours and legacy

Geiger lived in Potsdam-Babelsberg at the end of the war. In June 1945, with the area set aside for the Potsdam Conference of the Allied leaders, Soviet forces requisitioned his house and the family had to leave. Already severely ill, he died in Potsdam on 24 September 1945, six days before his 63rd birthday. He was buried in the Neuer Friedhof in Potsdam.

His honours included the Hughes Medal of the Royal Society (1929), the Arrhenius Prize (1934) and the Duddell Medal of the Physical Society of London (1937). He was a member of the Prussian Academy of Sciences and the Leopoldina. A lunar crater and the asteroid (14413) Geiger are named after him.

Legacy

• The nuclear atom: the Geiger–Marsden measurements of 1909 and 1913 supplied the evidence for Rutherford’s nucleus and tested every term of his scattering law.
• Counting statistics: the 1910 Poisson test made randomness at the level of single atoms a quantitative fact, and set the standard for error analysis in counting experiments.
• The Geiger–Nuttall law: a regularity found by measurement became, through Gamow’s explanation, the first triumph of quantum mechanics in nuclear physics.
• The coincidence method: from the Bothe–Geiger test of energy conservation to cosmic-ray showers, timing correlations between detectors became the basic logic of particle physics experiments.
• The counter: the Geiger–Müller tube is still used in radiation protection, geology, medicine and teaching laboratories.
• Students: his doctoral students included Walther Müller, Otto Haxel, Ernst Stuhlinger and Helmut Volz. Chadwick and Bothe learned their craft in his laboratory.

Sources and further reading

Biography and history

1. Hans Geiger, Wikipedia; Hans Geiger (Physiker), German Wikipedia (Nazi period, Bethe correspondence, Uranverein).
2. E. Fünfer, Geiger, Hans, Neue Deutsche Biographie 6 (1964), 141–142.
3. Hans Geiger, Encyclopedia.com (World of Physics; complete dictionary entries).
4. American Physical Society, This month in physics history: Hans Geiger, APS News, June 2012.
5. Hans Geiger, Encyclopaedia Britannica.
6. W. Bothe, Die Geigerschen Zählmethoden, Die Naturwissenschaften 30 (1942), 593–599.
7. M. von Laue, Hans Geiger, Jahrbuch der Deutschen Akademie der Wissenschaften zu Berlin 1946–1949 (1950), 150–158.
8. K. Hentschel (ed.), Physics and National Socialism: An Anthology of Primary Sources, Birkhäuser, 1996 (includes the 1936 Heisenberg–Wien–Geiger petition).
9. A. D. Beyerchen, Scientists under Hitler: Politics and the Physics Community in the Third Reich, Yale University Press, 1977.
10. Hans Bethe, Wikipedia (dismissal at Tübingen, 1933).
11. Uranverein, Wikipedia (participants of the 1939 meetings).

Physics

1. E. Rutherford, H. Geiger, An electrical method of counting the number of α-particles from radio-active substances, Proc. R. Soc. A 81 (1908), 141–161.
2. E. Rutherford, H. Geiger, The charge and nature of the α-particle, Proc. R. Soc. A 81 (1908), 162–173.
3. H. Geiger, E. Marsden, On a diffuse reflection of the α-particles, Proc. R. Soc. A 82 (1909), 495–500.
4. E. Rutherford, H. Geiger, with a note by H. Bateman, The probability variations in the distribution of α particles, Phil. Mag. 20 (1910), 698–707.
5. H. Geiger, The scattering of the α-particles by matter, Proc. R. Soc. A 83 (1910), 492–504.
6. H. Geiger, J. M. Nuttall, The ranges of the α particles from various radioactive substances and a relation between range and period of transformation, Phil. Mag. 22 (1911), 613–621.
7. H. Geiger, E. Marsden, The laws of deflexion of α particles through large angles, Phil. Mag. 25 (1913), 604–623.
8. H. Geiger, Über eine einfache Methode zur Zählung von α- und β-Strahlen, Verh. Dtsch. Phys. Ges. 15 (1913), 534–539.
9. W. Makower, H. Geiger, Practical Measurements in Radio-activity, Longmans, Green, 1912.
10. W. Bothe, H. Geiger, Über das Wesen des Comptoneffekts; ein experimenteller Beitrag zur Theorie der Strahlung, Z. Phys. 32 (1925), 639–663.
11. H. Geiger, W. Müller, Elektronenzählrohr zur Messung schwächster Aktivitäten, Phys. Z. 29 (1928), 839–841.
12. G. Gamow, Zur Quantentheorie des Atomkernes, Z. Phys. 51 (1928), 204–212.
13. G. F. Knoll, Radiation Detection and Measurement, 4th ed., Wiley, 2010 (gas multiplication, Geiger–Müller counters, dead time, counting statistics).
14. K. S. Krane, Introductory Nuclear Physics, Wiley, 1988 (range, alpha-decay theory).