Gerlach's apparatus is shown in Fig. 10. A *Hohlraum* realization of a black body (Valentiner 1910), produces black-body radiation at 0◦ or 100 ◦C, defined, respectively, by the freezing and boiling points of water at atmospheric pressure. Upon passage through a diaphragm, the radiation is absorbed by detection stripes made of manganin (an alloy of copper, manganese, and nickel with a low thermal expansion coefficient) electroplated with platinum black (in order to suppress selective absorption). The detection stripes were held at a distance of half a millimeter from a thermopile, with an insulating layer of ambient air in between. The thermopile was of the type developed earlier by Paschen for his spectroscopic investigations (Gerlach 1912). The current produced by the thermopile was measured by a sensitive galvanometer. The measurement procedure was as follows: (a) the black body at 100 ◦C irradiates the detection stripes for as long as the galvanometer reading increases, reaching a steady-state value of, say, *i*0; (b) the black body at 100 ◦C is replaced with a black body at 0 ◦C and the detection stripes are electrically heated up until the galvanometer reading becomes equal to *i*0; (c) The measured Joule heat (electric power) equals the difference of the radiant power carried by the black-body radiation at 100 and 0 ◦C. In order to achieve good statistics, the black bodies were swapped every minute or two and the galvanometer read every 15 s. The value that came out of Gerlach's measurements was σ = (5.9 ± 0.057) × 10−12 W cm2 K−4 (after a correction for reflected radiation). Gerlach's detection scheme is sometimes referred to as Ångström-type pyrheliometer (Coblentz 1913).

Paschen lavished the highest praise on Gerlach's achievement (Paschen 1912b):

[Gerlach] was able to justify *ab ovo* every single aspect of the new method, which is one of the most difficult tasks of physics altogether.

However, when Gerlach's result, accompanied back-to-back by Paschen's endorsement, was published (Gerlach 1912), see also Fig. 11, the competitors, Ferdinand Kurlbaum and Siegfried Valentiner (1876–1971)—both from the *Physikalsch-Technische Reichsanstalt* (PTR) in Berlin—disagreed. A rather acrimonious public debate ensued that called for more work on Gerlach's and Paschen's part and led to two more investigations by Paschen and nine more by Gerlach, including Gerlach's *Habilitation* thesis.

Developing into a "war of attrition," the exchanges slowed down after the outbreak of World War One and ceased in 1916 (Gerlach 1916)—without resolving the issue. Throughout, Gerlach was troubled by the realization that a physics problem could not be brought to a closure, if possible in his favor. He would devise and implement new experimental schemes with a great persistence—but to no avail. In the end, the PTR made plans for resuming the measurements of σ—using Gerlach's method. Gerlach would demonstrate both his persistence and inventiveness in his later work

**Fig. 10** Schematic of the apparatus Gerlach built in Paschen's laboratory in Tübingen to perform precision measurements of the proportionality constant σ in the Stefan-Boltzmann law (Gerlach 1912). Gerlach's realization of the black body together with a diaphragm (D) and slits (b1 and b2) is shown on the left. The right-hand side shows the detector with the detection strips and thermopile (Th), the galvanometer (G), and apertures (B). The detector assembly is mounted on a dividing engine whose position can be accurately controlled

$$S = \sigma (t + 273)^{\bullet} \text{ .}$$

$$
\sigma = 6,30 \times 10^{-12} \text{ watt } \text{cm}^{-2} \text{ grad}^{--}.
$$

as well, most conspicuously in the Stern-Gerlach experiment. Apparently, when he got something into his head, it was difficult to dissuade him from it.

Interestingly, as part of the debates between Gerlach and Paschen on the one side and the PTR scientists on the other, Paschen pointed out, (Paschen 1912a), that the new value of σ would be of consequence for the values of the fundamental constants it was made out of according to Planck's law, cf. Eq. 3. Let us note that the currently accepted value of the Stefan-Boltzmann constant is (CODATA 2020)

$$
\sigma = 5.670374419 \times 10^{-12} \,\mathrm{W} \,\mathrm{cm}^{-2} \,\mathrm{K}^{-4} \tag{4}
$$

i.e., like Gerlach's value, between Kurlbaum's and Féry's values.

#### *3.2 Walther Gerlach and the Stern-Gerlach Experiment*

A detailed account of the purpose, outcome, and significance of the Stern-Gerlach experiment (SGE) can be found in Chap. 5 of this volume. Herein we emphasize Gerlach's contribution to the realization of the SGE and glean what the relationship between Stern and Gerlach was like from their mutual correspondence as well as from their correspondence with others.

As noted, in October 1920 Gerlach landed an assistantship at Wachsmuth's Institute for Experimental Physics at Frankfurt. The Frankfurt university recognized his *Habilitation* and, in addition, promoted him to the rank of *Extraordinarius* a month later. Max Born's adjacent Institute for Theoretical Physics was a more congenial environment for the curious and enterprising Gerlach than Wachsmuth's operation. All the more so that Born, with his assistants Otto Stern, Elisabeth Bormann, and Alfred Landé, was engaged in experiments as much as in theory and encouraged Gerlach to partake in their discussions as well as to give them a hand with their experiments. Born would even publish with Gerlach—on electron affinity (Gerlach and Born 1921a) and on light scattering (Gerlach and Born 1921b). However, Gerlach would also pursue his own agenda: it was at Frankfurt that he launched his investigations into the magnetic properties of materials that would bring him together with Stern and later take center stage in his research at Frankfurt and his subsequent stations. In particular, Gerlach was interested in the relationship between magnetization and structure (Bachmann and Rechenberg 1989), p. 10. In connection with his investigation of the magnetic properties of a bismuth alloy, the question arose as to whether atomic bismuth was para- or diamagnetic. Gerlach set out to answer this question in a molecular beam experiment, in which the deflection of a beam of bismuth atoms by an inhomogeneous magnetic field would be examined (Mehra and Rechenberg 1982), p. 436. Born tried to dissuade Gerlach from what seemed to be a hopelessly difficult undertaking. Whereupon Gerlach invoked a quip he heard from Edgar Meyer (1879–1960), his professor of theoretical physics at Tübingen: "No experiment is so dumb that it should not be tried" (Estermann 1975) and continued setting up his bismuth beam experiment and thus collecting experience in much of

**Fig. 12** Otto Stern (2nd from left), Edgar Meyer (5th from left), Walther Gerlach (6th from left) in Tübingen in about 1926. Courtesy of the Otto Stern Collection, Berkeley

what was needed for the SGE. Let us add that Edgar Meyer, Fig. 12, with whom Gerlach had worked on the photo-effect, contested the separation of physics into theoretical and experimental. Max Born was apparently of the same persuasion in this respect. In February 1921, he reported to Einstein (Born 1969), p. 82:

We have now Gerlach here with us, who is awesome: energetic, knowledgeable, skillful, ready to help.

In his 1977 talk, Gerlach told the story of his recruitment by Otto Stern for the SGE as follows (Gerlach 1977):

One day Stern would come to me and say: 'Do you know what space quantization is?' I would say: 'No, I have no idea.' 'But you should actually know that. Recently Debye and Sommerfeld published [papers] suggesting that the [anomalous] Zeeman effect can be explained by a quantum effect, by the so-called space quantization. That is, [the magnetic dipole of] a silver or sodium atom can only have two settings [orientations] in a magnetic field, it cannot adjust itself at will or precess, but can only have two very specific settings [orientations], or actually even three, namely perpendicular to the magnetic field or in ... the direction or against the direction [of the magnetic field] ...

Repeated discussions with Stern during our daily visits at *Café Rühl* finally led to a plan to make the experiment in such a way that there was hope of seeing space quantization.

Gerlach perhaps thought that he would just have to modify his current experiment on the magnetic properties of bismuth. Finally, he agreed: "Yes, I want to try it" (Gerlach 1977). But then

[Stern] would come back again: 'It isn't worth it, I've miscalculated, power of ten too little.' And then, it went back and forth a couple of times for a week or a fortnight and one day he would come back and say: 'Yes, now I've done [the calculations] properly and the thing only works if you get fields with an inhomogeneity of about ten or fifty thousand Oersted per centimeter—and that's not possible.' And then I said to him: 'Yes, I am almost there, I already have ten thousand [Oersted per centimeter], namely for my planned bismuth experiment'. 'So,' he said, 'let's try it.'

And they did. Stern first published the concept of what was to become the SGE, accompanied by feasibility calculations (Stern 1921), prompted to "patent" the idea by seeing the page proofs of a paper by Harmut Kallmann and Fritz Reiche on an electric analog of the SGE, see also Chaps. 5 and 20. The collaboration that ensued between Stern and Gerlach was in part so successful because of the complementarity of their skills and perhaps even working habits: while Stern had gained experience with molecular beams, Gerlach developed expertise in designing strong inhomogeneous magnetic fields. While Stern preferred to call it a day around 6 p.m. at that time, have dinner and go to the cinema, Gerlach liked to work at night, often doing with just three hours of sleep.

As described in Chap. 5, it took a tremendous effort to make the experiment work. Stern, who did not believe in the reality of space quantization to begin with, left on 1 October 1921 to assume a professorship at Rostock. Gerlach continued improving their apparatus and during the night of 4 November 1921 observed for the first time a broadening of a silver beam sent through an inhomogeneous magnetic field. This provided evidence that silver atoms carried a magnetic dipole moment—but did not suffice to demonstrate the existence of space quantization. During the Christmas recess, Gerlach and Stern reconfigured their apparatus again, but Gerlach's subsequent attempts to see space quantization had failed. At their meeting in Göttingen in early February 1922, Gerlach and Stern decided to try the experiment one more time. On the train back to Frankfurt, Gerlach remembered a modification he made earlier when examining crystals by X rays using the Debye-Scherrer method, namely to use a *slit* instead of a pinhole to boost both flux and spatial resolution. Gerlach had even reported on the improvement he thereby achieved at the German Physics Day in Jena in September 1921 (Huber 2015). Upon arrival in Frankfurt, Gerlach replaced the pinhole (of 0.05 mm diameter) defining the silver beam at the entrance into the inhomogeneous magnetic field by a rectangular 0.03 × 0.8 mm2 slit with its longer side perpendicular to the field direction (Gerlach and Stern 1922)—and during the night of 7 February 1922 achieved the ultimate success.

Wilhelm Schütz (1900–1972), who was in 1922 Gerlach's Ph.D. student, described the difficulties of the SGE as well as the final triumph as follows (Schütz 1969):

The old apparatus had only yielded a broadening of the silver beam [deposit on the glass plate] of the expected magnitude …due to the inhomogeneous magnetic field. A major improvement of the apparatus with the aim to further increase its resolution was [therefore] necessary. During this rebuilding period, Stern moved to Rostock to assume a Professorship for Theoretical Physics there. He would show up in Frankfurt every now and then (during Christmas 1921 and Easter 1922) for discussions and to measure the inhomogeneity of the magnetic field …Soon came the time when I was able to enter the holy premises of the laboratory and take a look at the pumps, when [the technician Mr.] Schmidt was not on duty and Prof. Gerlach had to sleep once in a while …Anyone who has not been through it cannot at all imagine how great were the difficulties with an oven to heat the silver up to about 1300◦K within an apparatus which could not be heated in its entirety [the seals would melt] and where a vacuum of 10−5 Torr had to be produced and maintained for several hours. The cooling was done with solid carbon dioxide and acetone or with liquid air. The pumping speed of the Gaede mercury backing pumps and the Volmer mercury diffusion pumps was ridiculously low compared with the performance of modern pumps. And then their fragility; the pumps were made of glass and quite often they broke, either from the thrust of boiling mercury …or from the dripping of condensed water vapor. In that case the effort of several days of pumping, required during the warming up and heating of the oven, was lost. Also, one could be by no means certain that the oven would not burn through during the fourto eight-hour exposure time. Then both the pumping and the heating of the oven had to be started from scratch. It was Sisyphus-like labor and the main load of responsibility lay on the broad shoulders of Prof. Gerlach. In particular, W. Gerlach would take over the night shifts. He would get in at about 9 p.m. equipped with a pile of reprints and books. During the night he then read the proofs and reviews, wrote papers, prepared lectures, drank plenty of cocoa or tea and smoked a lot. When I arrived the next day at the institute, heard the intimately familiar noise of the running pumps, and found Gerlach still in the lab, it was a good sign: nothing broke during the night.

Then I arrived at the institute one morning in February 1922; it was a wonderful morning: with cool air and fresh snow! W. Gerlach was once again at it, developing the deposit of an atomic beam that had been passing through an inhomogeneous magnetic field for eight hours. Full of expectation, we applied the development process, whereupon we experienced the success of several months of effort: The first splitting of a silver beam in an inhomogeneous magnetic field. After Master Schmidt and, if I remember correctly, E. Madelung had seen the splitting [the deposit was about 1.1 mm long and the splitting only about 0.06 to 0.1 mm], we went to Mr Nacken to the Mineralogical Institute to have the finding recorded on a microphotograph. Then I was tasked with sending a telegram to Professor Stern in Rostock, with the text: "Bohr is right after all!"

The consequences and impact of the stroke of luck for the emerging quantum physics that the collaboration between Otto Stern and Walther Gerlach at Frankfurt was are described in Chap. 5. We note that Albert Einstein and Paul Ehrenfest coined the term Stern-Gerlach experiment, in recognition of the fact that it was Stern who conceived it, although Gerlach largely carried it out (Einstein and Ehrenfest 1922). Moreover, Otto Stern was the pioneer of *quantitative* experiments with molecular beams.

In 1924, Einstein nominated, alongside with others, both Stern and Gerlach for the Nobel Prize in Physics (Schmidt-Böcking et al. 2019). By 1944, Gerlach and Stern had been nominated together thirty-one times for the Nobel Prize, cf. Fig. 1. Stern received fifty-two additional nominations for his other experiments with the molecular beam method and was awarded the Nobel Prize in Physics in 1944 for the year 1943. Gerlach ended up empty-handed, although Manne Siegbahn (1886– 1978), then chairman of the Nobel Committee for Physics, proposed Stern, together with Gerlach, in 1944 as the sole candidates. And Eric Hulthén (1891–1972) in his broadcast on Swedish Radio on 10 December 1944 honoring the award of the Nobel prize to Stern extolled almost exclusively the SGE. In the documents and reports of the Nobel Archives there is no indication as to why Gerlach was left out. The reason may have been Gerlach's high-level involvement in the Nazi research establishment, especially in the management of the nuclear program, see Sects. 1 and 4.

The personalities of Stern and Gerlach were quite different: while Gerlach enjoyed being in the driver's seat, Stern preferred the back seat. Only a few letters exchanged between them have been preserved. The following one, from 16 January 1924, concerns the last (Gerlach and Stern 1924) of their four joint publications, all of which dealt with the SGE (Schmidt-Böcking et al. 2019), p. 125:

Dear Gerlach, many thanks for your messages. I thought our paper had arrived at the *Annalen* [der Physik] a long time ago. In any case, I totally vote for the *Annalen*, and you do too, for such long claptrap is nothing for the [Zeitschrift für Physik]. I couldn't come in during the week, not to [Frankfurt], because I had to go to Breslau, and [going to] both [places] was a little too much for me. For [molecular beams] I invent ever more ingenious apparatus that only keeps working worse, z.[um] K.[otzen]! In contrast, the [electric molecular beams] are quite endurable. But it all goes so terribly slowly!

I hear that Schaefer got a call from Freiburg. He has to go there! Cordial greetings to all friends, your family, and yourself. Yours Otto Stern

When Gerlach succeeded Paschen at Tübingen, Stern sent him, on 16 November 1925, the following telegram (Schmidt-Böcking et al. 2019), p. 125:

= Cordial congratulations to the *Grossbonzen* [big shot] from Stern +

Whereupon Gerlach replied (Schmidt-Böcking et al. 2019), p. 126:

Dear Stern, it is Sunday, 22 November, and I just got your telegram. As I started writing the above, the furniture trucks have arrived ... So I begin my rant in the hope that my wife will leave me alone for a moment.

... Mr. S. made statements about the evaluation of our magneton experiments which—as we noticed from his multiple inquiries—give rise to the impression that our calculation could be 100% wrong; and furthermore that the evaluation did not take into account possible sources of error and uncertainties, and that, in particular, we missed out on taking the width of the slit into account. Although Mr S.'s reasoning is correct, his note is indeed likely to lead to misunderstandings.

Mr. S. namely always speaks about the distance between the locus of maximum intensity on the deflected strip and the locus of the ... narrow undeflected strip, for which case the formula we use would indeed give an almost 100% error. However, our measurements always refer to the center of the deflected strip, which Mr. S. only discusses at the end of his note; for this case, Mr. S. himself calculates a deviation of at most 20%.

Furthermore, Mr. S. seems to assume that we were not aware of the influence of the distance of the slit. [In our paper] we refer to the work of Stern where this influence was discussed and the corresponding formula ... that takes into account the Coriolis force was derived. Mr S. could have easily figured that out from the literature. At the time we just remarked as much ... and stated a possible error on the order of magnitude of ±10%. We insist that Mr. S.'s note doesn't bring forth any new thoughts and that its content pretty much coincides with our presentation. We only object to the manner of his attack.

Dear Stern, how are you health-wise? It's a pity that you weren't in Göttingen. Here [in Tübingen], there's a terrible mess [due to Gerlach's move]. Hopefully, it will sort itself out soon. I will then write to you about the atomic beam experiments. Please do publish something with [Immanuel] Estermann again! Cordial greetings, also from my wife, Yours W. Gerlach

Next in the chronology of the preserved letters that bear upon the relationship between Stern and Gerlach is a note written by Stern from Zurich to Lise Meitner (1878–1968) in 1957:

Dear Lise Meitner, ... So let's meet in Munich. However, I can only come for 1-2 days, for two reasons: (1) [I cannot be away from Zurich for more than 1–2 days, because I expect a visitor]; (2) I don't care about seeing the Munich physicist Mr. Gerlach. Therefore, I leave it entirely up to you when you and I will meet. Please just let me know as soon as possible. It was very nice to see [Otto Robert] Frisch again and to get to know his wife; they seem to fit very well together.

The two of us, the old ones, will have a lot to chat about and I'm hugely looking forward to seeing you again. Most cordially, Yours Otto Stern

Then there is a postcard to Stern penned jointly by Walther Gerlach, Otto Robert Frisch (1904–1979), Immanuel Estermann (1900–1973),William Nierenberg (1919– 2020), Hans Kopfermann (1895–1963), and Peter Toschek (1933–2020) from the Brookhaven Molecular Beam Conference that was organized by Hans Kopfermann and held at Heidelberg in 1959 (Schmidt-Böcking et al. 2019), p. 245:

Lichtstrahlen sind zum Brechen, Atomstrahlen z. K.! [zum Kotzen]. [This is a kind of affectionate "secret code" between Stern and Gerlach from their Frankfurt time—a pun expressing their occasional disgust with their difficult atomic/molecular beam experiments. "Brechen" means refraction as well as vomiting; "Kotzen" is a vulgar word for vomiting. A free translation, without the pun, would be: Light beams refract, atomic beams disgust.] Too bad that you aren't here, but we think of you warmly! Yours Walther Gerlach

Remarkably, I got to know Mr. Gerlach only here. But molecular beams have become awfully complicated! With cordial greetings, yours OR Frisch

Cordial greetings, Estermann

Best regards will see you soon! Nierenberg

We were very sorry not to have you here. Yours Hans Kopfermann

Cordial Greetings from yours P. Toschek

It can be gleaned from many letters held at Otto Stern's Estate (Schmidt-Böcking et al. 2019) that he had quite a friendly relationship with all his correspondents. The above-quoted letter to Lise Meitner from 22 April 1957 suggests that Stern's feelings towards Gerlach were/became less than cordial, at least at the time. Conversely, Walther Gerlach wrote and spoke about Stern with the highest respect and much affection. This transpires in particular in the obituary of Stern that Gerlach wrote for the *Physikalische Blätter* (Gerlach 1969):

Those who knew him appreciated his open-mindedness—he was a grand seigneur!—his unconditional reliability, the fruitful and—due to his fast thinking—difficult discussions, and—for those ho had a sense for it—his often nearly sarcastic but well-conceived assessments of things and people; bossing people or poor manners were anathema to him.

Although a theoretician by nature, Stern was full of experimental ideas, never at a loss for a new proposal if the implementation of the previous one failed. At our farewell from Frankfurt, I gave him, in memory of the months of hopeless striving to see space quantization, an ashtray with the inscription [Stern's and Gerlach's "secret code" in our translation] "Light beams refract, atomic beams disgust;" this ashtray endured all those years till Berkeley—but our experimental apparatus, lab books, and the originals of our results had burned during the Second World War.

A special tribute to the *"Stern-Stunden"* in Frankfurt and their importance for the development of quantum physics was given by Walther Gerlach in his lecture on 2 March 1960—still during Stern's life—at the *Physikalischer Verein Frankfurt* (Gerlach 1960):

Around 1910, the French physicist Dunoyer developed the method of the so-called atomic or molecular beams. These are atoms that fly along straight lines from an oven through a small orifice into a highly evacuated chamber. Here at this institute, Max Born, Elisabeth Bormann, and, foremost, Otto Stern took up this idea in 1920 and experimentally developed the atomic beam method. That was a risky undertaking as at the time the means to produce high vacuum were still extremely limited ... Stern succeeded in measuring the mean velocity of the atoms, Born and Bormann measured their mean free path, and in later years Stern also succeeded in measuring the velocity distribution in an atomic beam. In the meantime, this method has been so refined by [Immanuel] Estermann, who is now at Chicago, that it affords the best temperature measurement of gases or vapors at 2000 degrees or more. Finally, Stern was able to demonstrate that free-flying atoms follow a free-fall parabola like a projectile. Moreover, at this institute, the reality of space quantization was successfully demonstrated in an experiment that provided direct access to an atomic state predicted by quantum theory.

Upon finishing the SGE, Gerlach would return to what he called his "hobby," namely his research on radiation pressure that he had started already in 1913 in Tübingen (Huber 2015). The pursuit of this "hobby" was deemed to be about as difficult as the SGE (Rollwagen 1980). Gerlach's interest was likely triggered by the inherent connection between radiation pressure and the Stefan-Boltzmann law.

#### *3.3 Radiation Pressure*

Ludwig Boltzmann (1844–1906) succeeded in 1884 to derive the law, *I*(*T* ) ∝ *T* 4, cf. Eq. 2, that his teacher, Josef Stefan (1835–1893), found in 1879 empirically (Boltzmann 1884). In his derivation, Boltzmann invoked Maxwell's theory of electromagnetism and the second law of thermodynamics, prompted by an earlier attempt by Adolfo Bartoli (1851–1896) to arrive at Stefan's law by the same route. Boltzmann was able to show that substitution of the pressure *p* = *I*(*T* )/(3*c*) exerted by blackbody radiation of energy density *I*(*T* )/*c* into the second law of thermodynamics in the form *Tdp* − *pdT* = [*I*(*T* )/*c*]*dT* yields

$$\frac{dI(T)}{4I(T)} = \frac{dT}{T} \tag{5}$$

which upon integration indeed gives Stefan's law—since then also known as the Stefan-Boltzmann law.

During his detention at Farm Hall (see below), Gerlach reminisced (Gerlach 1945) about his early attempts to come to terms with the effects he observed with a Crookes radiometer (light mill), a contraption invented by William Crookes (1832–1919) in 1873:

In Tübingen in 1913/14, I tried to enhance the sensitivity of the radiometer [consisting of vanes mounted on a spindle in a partially evacuated bulb] by implementing alternative shapes of the vanes. This is when I observed a "negative" rotation of the vanes, i.e., in the direction opposite to that of the incident light.

Gerlach's original idea that he could measure radiation pressure with a Crookes radiometer turned out to be overly optimistic, as the processes involved in the radiometer physics are all but simple. It would take Gerlach and his coworkers two decades (1913–1932) to clarify the "positive" and "negative" radiometer effects and to carry out an absolute measurement of radiation pressure. Was it worth the effort? For sure it was, as those who were (and, in some quarters, still are) credited with first measurements of radiation pressure—Pyotr Lebedev (1866–1912), Ernst Nichols (1869–1924), and Gordon Hull (1870–1956)—did not and could not have measured anything else than spurious radiometer effects. As Gerlach and coworkers would show in their work, these only disappear at a vacuum better than 10−6 torr, which was not attainable during the period 1901–1903 when Lebedev, Nichols, and Hull published their radiation pressure studies.

Gerlach reentered the fray in 1919 when he published, jointly with Wilhelm Westphal, a theory of the radiometer (Gerlach and Westphal 1919) that, however, had to be quickly retracted (Westphal 1919):

More detailed considerations have shown ... that the theory is untenable, despite a very good agreement with experiment. In particular, Mr [Albert] Einstein gave me a friendly hint that [our theory] contradicts momentum conservation.

At the 1920 meeting of the German Physical Society in Berlin, Westphal noted (Westphal 1920):

The goal of the investigations [of the radiometer effects] is to collect a complete set of experimental data needed for a theory of the radiometer.

Gerlach answered the challenge implied by Westphal's talk with a series of four papers entitled*Untersuchungen an Radiometern I–IV* [Investigations of the Radiometer I–IV] published between 1923 and 1932. The first paper of the series opens with the bold statement (Gerlach and Albach 1923):

As is well known, there is no complete theory of the radiometer available.

The paper then describes a compensation radiometer consisting of a single vane with thermally insulated sides enclosed in a bulb filled with gas of variable pressure (in the range of 10−1–10−4 torr). One side of the vane is a receptor of radiation, the other is an electrically heatable bolometer. Like in his pyrheliometer, see Sect. 3.1, the carefully controlled electric heating of the bolometer side made it possible to compensate for the heating of the other side by the incident radiation. The compensation was carried

**Fig. 13** The torsional radiometer of Gerlach and Golsen in side-view (top) and top-view (bottom) (Golsen 1924). The vane (not shown) used in the first quantitative measurement of radiation pressure was made of platinum foil (1.45 × 1.05 cm2 and 7 µm thick). Its weight was balanced out by a platinum wire. The radiometer was housed in a glass ball (*Gl*) equipped with arms (*A*1–*A*5) for pumping and access and to allow to bring the radiation in and to take it out. It was evacuated by a Volmer diffusion pump combined with a cryo- and sorption pump (a *Volmeraggregat*) separated by a valve (*H*). The pressure was measured using a McLeod gauge and below 10−5 torr inferred from the damping of the torsional oscillations of the radiometer suspended on a 11 cm long quartz filament. A mirror (*S*) was attached to the filament to facilitate the read-out of the amplitude of the torsional oscillations. The radiation source was a tungsten arc lamp (*W*) whose output was focused on the vane by a camera lens (*Ob*). The power of the lamp was calibrated using a Hefner lamp and monitored during the measurements by a thermopile (*Th*) connected to a galvanometer (*G*). Except for the windows, the glass ball was shielded by a cotton-wool wrapping

out as a function of pressure for various absorption and thermal isolation materials. The instrument proved to be capable of sensitively measuring small changes of intense radiation.

However, Gerlach's goal was to directly measure light pressure rather than to investigate radiometer effects. To that end, he teamed up with Alice Golsen (Gerlach 1945):

With Ms. Alice Golsen from Wiesbaden—who, as it turned out, was my classmate in 1896—I did the first measurement of radiation [pressure] as a precision measurement—with absolutely measured radiation energy. It was arduous but beautiful, clean work, a recuperation **Fig. 14** The dependence of the vane amplitude (ordinate) on the logarithm of gas pressure, log *p* (abscissa). The negative and positive amplitudes of the platinum vane refer, respectively, to deflections against and along the direction of the incident light beam. The various series of data points (•, ×, and ) correspond to different irradiances and are all found to follow the same curve (Golsen 1924)

of sorts from the perpetual failures of the space-quantization experiments. In Ms. Golsen I found a wonderful collaborator, both scientifically and as a person.

Their collaboration resulted in the second paper (Gerlach and Golsen 1923) of the series as well as a detailed summary written by Alice Golsen (Golsen 1924). The aim of the experiment was to provide an unequivocal measurement of radiation pressure, free of radiometer effects and any disturbances. That meant that the radiometer measurements had to be done as a function of gas pressure all the way down to 10−6 or even 10−7 torr where a pressure dependence would vanish. A new apparatus was built, Fig. 13, that amounted to a torsion balance with a platinum vane attached to a quartz filament suspended in a glass ball. Its "rest-amplitude" observed at pressures below 10−6 torr was then attributed to radiation pressure. The measurements proceeded as follows: after several days of pumping, the dependence of the amplitude of the vane would be measured as a function of gas pressure at constant irradiation by a tungsten arc lamp, see caption to Fig. 13. The power of the lamp was monitored [normalized] by a thermopile. Achieving a steady-state amplitude lasted often for hours and was perturbed by outgassing as well as by the vibrations of the institute building. A typical dependence of the amplitude on gas pressure is shown in Fig. 14; it would take on the order of 100 h to acquire the data points shown. As one can see, at gas pressures between 1 torr and 10−4 torr, the amplitude is "negative," meaning that, upon irradiation, the vane moves against the incoming light beam. Only at pressures below 10−3 torr would the amplitude become "positive" (i.e., along the light beam direction), inching towards the pressure-independent "rest-amplitude" at pressures below 10−6 torr. In order to access the requisite pressure range, sorption pumping with charcoal and cryo-pumping with liquid oxygen (!) had to be applied—for days ... As stated by Gerlach and Golsen, cryo-pumping with dry ice had not sufficed to reach the "rest-amplitude" regime. The radiation pressure was then evaluated from the observed "rest-amplitude" and the measured properties of the torsion balance, such as its force constant. The measured light pressure (light force per illuminated surface area of the vane), *p*, and the calibrated irradiance, *I* ∗, were then compared and found to obey the relationship

$$p = \frac{I^\*}{c} \tag{6}$$

with an accuracy of about 2%. This was the first-ever quantitative measurement of radiation pressure.

Gerlach and Golsen summarized their results thus (Gerlach and Golsen 1923):

1. In a vacuum from about 10−6 to 10−7 torr a constant amplitude ["rest-amplitude"] of the radiometer was found that is interpreted as purely due to radiation pressure.

2. This amplitude is proportional to the incident energy [power] and independent of the wavelength of the radiation.

3. The radiation pressure calculated from the constant amplitude agrees with the theoretical value.

In the third paper of the radiometer series (Gerlach and Madelung 1923), Gerlach and Erwin Madelung (1881–1972) debunk the radiometer theory published in 1922 by Edith Einstein. Finally, in 1932 Gerlach and Wilhelm Schütz publish the final, fourth sequel of the series (Gerlach and Schütz 1932) that deals with the radiometer effects at "high pressures" and corroborates the recent model put forward by Paul Epstein (Epstein 1929).

In 1975, Gerlach wrote a rebuttal (Gerlach 1975) to an article published in *Physik in unserer Zeit* whose author repeated the claim that radiation pressure was measured for the first time in the experiments of Lebedev, Nichols, and Hull. We note that Gerlach provided an impetus in 1970 for the founding of *Physik in unserer Zeit*.

It is mind-boggling that Gerlach's work on radiation pressure is still not widely known and that most textbooks keep attributing the first measurements of radiation pressure to experiments in which it could have not been observed.

After completion of the radiation pressure work at Frankfurt, Gerlach moved to his second academic station—his alma mater—as *Ordinarius*. His appointment at Tübingen received a strong push from Albert Einstein (Rechenberg 1979). Figure 15 shows Gerlach during the Tübingen period. Figure 16 shows his extended family during that time.

In addition to his time-consuming research projects at Frankfurt, Gerlach wrote two books: *Experimentelle Grundlagen der Quantentheorie* (Gerlach 1921) and the acclaimed *Materie, Elektrizität, Energie* (Gerlach 1923), a survey of the development of atomism over the previous decade.

We note that among Gerlach's students at Frankfurt was Hans Bethe (1906– 2005), who began his physics studies in 1924. In his reminiscence (Bernstein 1979), Bethe acknowledged that Gerlach's stimulating lectures on atomic physics became a decisive influence on his further work in physics.

**Fig. 15** Walther Gerlach as director of the Physics Institute in Tübingen. Courtesy of Werner Kittel, Hamburg

#### **4 Gerlach's Involvement in the** *Uranprojekt*

The German *Uranprojekt* was no precision physics. Launched in reaction to the discovery of nuclear fission by Otto Hahn, Lise Meitner, Fritz Strassmann, and Otto Robert Frisch and in the wake of subsequent theoretical work by Niels Bohr and John Wheeler, the project started taking shape already several months before the outbreak of World War Two. Paul Harteck, the successor at Hamburg of the exiled Otto Stern, had written in April 1939 to the *Reichswehrministerium* [Ministry of Defence] about the promise of both a nuclear reactor and a nuclear weapon, amply described in the publications by the above. Harteck's letter ended up at the *Heereswaffenamt* [Army Ordnance Bureau]. In September 1939, the Bureau's Kurt Diebner (1905–1964) and former Heisenberg student Erich Bagge (1912–1996) enlisted leading German physicists—Walther Bothe (1891–1957), Hans Geiger (1882–1945), Heisenberg, Hahn, Harteck, and Carl Friedrich von Weizsäcker (1912–2007)—in a wide-ranging war-time nuclear program. This received additional support through an initiative by Göttingen's Wilhelm Hanle (1901–1993) and Georg Joos (1894–1959) from the Ministry of Education. The members of the group, also known as the *Uranverein*, got promptly down to work. Heisenberg produced a secret report in which he described a uranium nuclear reactor (*Uranmaschine*) and urged the Bureau's leadership to support isotope separation not only as the surest path to a functional reactor but also

**Fig. 16** The Gerlach family in Weimar in about 1927. From left: Walther Gerlach, Wolfgang Gerlach (brother of Walther Gerlach), Ruth Gerlach, neé Probst (2nd wife of Walther Gerlach), Valentin Gerlach (Walther Gerlach's father), Ingeborg Gerlach (elder daughter of Werner Gerlach and his wife Henriette "Henny" Syffert, who in 1943 married Wolfgang Kittel; they had two sons: Werner Kittel, born in 1945, and Gerd Kittel, born in 1948), Marie Gerlach, neé Niederhaeuser (mother of Walther Gerlach), Henny Gerlach, neé Syffert (wife of Werner Gerlach), and Werner Gerlach (brother of Walther Gerlach). Courtesy of Werner Kittel, Hamburg

to a nuclear bomb, without specifying the critical mass of U-235 needed (Cassidy 2017), p. 49. Based on the flawed research by Bothe on neutron capture by carbon, the *Heereswaffenamt* introduced the fatal mistake into the German nuclear program by branding graphite as an unsuitable moderator and relying on heavy water instead (Walker 1995), p. 225. Enrico Fermi's reactor at Chicago went critical in December 1942 using highly-purified graphite as a moderator. The loss of the heavy-water plant Norsk Hydro in Nazi-occupied Norway in early 1943 would then in effect upend the German nuclear program that relied on heavy water as a moderator. The *Uranprojekt* would continue, however, until the seizure of the German nuclear equipment by the American-led *Alsos Mission* in April-May 1945.

In 1941, several centers of German nuclear research emerged, all at first coordinated by Diebner and Bagge and concerned with aspects of the nuclear reactor as outlined by Heisenberg in his report. The most significant among them were Heisenberg's own institute at Leipzig and the Kaiser Wilhelm Institute (KWI) for Physics in Berlin, which fell under military command with Diebner installed as its acting director. Further reorganization saw Heisenberg appointed director of the KWI and Diebner relegated to an army research station in Gottow near Berlin. In August, Fritz Houtermans (1903–1966) and, independently, von Weizsäcker, demonstrated theoretically that Pu (plutonium) 239, produced in a uranium reactor from U-238 by neutron capture and subsequent β-decay, was at least as fissionable as U-235. As a result, an atom bomb suddenly appeared feasible. A controversial trip of Heisenberg and von Weizsäcker to see Bohr in Copenhagen followed. With the *Wehrmacht* defeated at Moscow and stuck at Leningrad, and the consequent mobilization of the German economy, the Army Ordnance Bureau approved funding, in February 1942, essentially only for Diebner's operation in Gottow (Cassidy 2017), p. 54. Heisenberg's KWI, however, had a sponsor in Abraham Esau of the Reich Research Council of the Ministry of Education and eventually of the *Reichsminister* Bernhard Rust himself. After a tantalizing conference, in February 1942, chaired by Rust on the prospects of a nuclear reactor, including its ability to breed fissionable plutonium, Esau was appointed, in December 1942, *Reichsbevollmächtigter* [Reich Plenipotentiary] for nuclear physics. But then the new Minister of Armaments, Albert Speer, induced Hitler to appoint Hermann Göring as head of the Reich Research Council whereby Esau became Göring's representative for nuclear issues. Already in July 1942, Heisenberg received a dual appointment in Berlin—as director of the KWI for Physics and professor of physics at the Berlin University. Heisenberg would use his expanded influence to push for Esau's replacement by a kindred spirit—Walther Gerlach. And indeed, as of 1 January 1944, Gerlach would become *Reichsmarschall's* Plenipotentiary for nuclear physics and remain in this position for sixteen months until his capture by the *Alsos Mission*.

Gerlach moved to his third academic station, *Ludwig-Maximilians-Universität* in Munich, on 1 October 1929 as the successor of Willy Wien. In 1935, a battle with the proponents of the so-called *Deutsche Physik*—Johannes Stark, Philipp Lenard, and their followers (Walker 1995)—flared up for the succession of the recently retired Arnold Sommerfeld (1868–1951), who held Munich's chair in theoretical physics. Gerlach headed the university's hiring committee, which chose Sommerfeld's former pupil, Werner Heisenberg—then already a Nobel laureate—to fill the vacant chair. The battle, which went through several stages and included public Nazi denunciations of the "White Jew" Heisenberg as well as an intervention by Heinrich Himmler (1900–1945) on Heisenberg's behalf, raged until September 1939 when Heisenberg was finally exonerated after an extensive SS investigation. However, in the meantime, the Munich chair went to a Nazi, Wilhelm Müller (1880–1968), an applied physicist. Whereupon Gerlach declared physics "dead" in Munich ... Heisenberg stayed put in Leipzig, until he received the call from Berlin.

Heisenberg and his *Uranverein* would hold additional presentations for both Speer and Göring and their staffs, carefully tailored to secure an autonomy of the physicists

**Fig. 17** From left: Otto Hahn, Walther Gerlach, and Carl Friedrich von Weizsäcker in Göttingen, late 1950s. All three were members of Göttinger Achtzehn. Creative Commons

in setting the goals for the nuclear program and avoiding being "ordered to build the bomb; since failure to do so at the height of war would surely have meant execution" (Cassidy 2017), p. 55. We note that Heisenberg's understanding of the functioning of the bomb was inadequate all the way down to Farm Hall, as his recorded lecture to and conversations with his detained colleagues attest. As a result, his estimate of the critical mass of U-235 was orders of magnitude too high and so was the time needed to accumulate it by isotope separation (Bernstein 2001), pp. 129–131. Figure 17 shows Gerlach later on with two of his Farm Hall fellow detainees and interlocutors, Hahn and von Weizsäcker.

In June 1942, a heavy non-nuclear accident damaged the nuclear research laboratory at Leipzig. Afterwards, significant reactor research continued at two locations only—Heisenberg's KWI in Berlin and Diebner's facility in Gottow. Based on his calculations, Heisenberg concluded that about three tons of cast uranium and one and half tons of heavy water were needed in order to achieve a chain reaction in a cylindrical arrangement with rolled uranium plates interspersed with heavy water, a reactor design Heisenberg started building in a bunker at his KWI. Diebner, on the other hand, bet on using cast uranium in the form of cubes suspended on chains and immersed in frozen heavy water. When the ordered amounts of uranium finally arrived from the *Auergesellschaft*, Diebner's design produced a much higher neutron multiplication than Heisenberg's. Once Gerlach took over as Plenipotentiary for nuclear research, he diverted resources toward Diebner's facility, but enabled Heisenberg's operation to run in parallel, thereby thinning key resources, especially the wanting heavy water. By then, the Allied aerial bombing raids on Berlin became heavy enough for the city to start evacuating. On Speer's order, a large part of the personnel of Heisenberg's KWI was moved to Hechingen, a rural place in Württemberg, not far from Tübingen. When Otto Hahn's KWI, a stone's throw from Heisenberg's, was destroyed in a targeted air raid, its personnel was moved to Tailfingen, not far from Hechingen. However, Heisenberg, his close associate Karl Wirtz (1910–1994) and their coworkers would stay on at the KWI for Physics in Dahlem and continue their attempts to get their reactor going. But at the end of 1944, with the Soviet Army reaching the left bank of the Oder river, Gerlach ordered both Heisenberg's and Diebner's groups to load their research equipment on trucks and move along with it to Hechingen. Once the convoy reached the experimental station of the *Reichsforschungsrat* in Stadtilm, about halfway, Gerlach pressed Diebner to stay there and make a final attempt to achieve chain reaction. Heisenberg's group, upon reaching Hechingen, set up a reactor in a cave—in fact a wine cellar—in a nearby village called Haigerloch. Their attempts, joined by von Weizsäcker, ended when the Haigerloch reactor was seized by the *Alsos Mission*. Gerlach's decision to enable Diebner his last-ditch effort is somewhat reminiscent of Gerlach's stubbornness in his own research that had so often paid off ...

Apparently, Heisenberg and Gerlach—and most others involved—struggled until the last moment not only out of scientific interest but also to salvage their scientific reputation. As David Cassidy put it (Cassidy 2017), p. 58:

For Heisenberg, success would have demonstrated the survival of decent German physics, and, perhaps equally [importantly], would have made German physicists influential figures in the postwar reconstruction of Germany.

In his conversation with Otto Hahn at Farm Hall secretly recorded after the atomic bombing of Hiroshima, Gerlach made a similar point but added yet another dimension to it (Hoffmann 1993), p. 157:

When I took [the *Uranprojekt*] over, I talked it over with Heisenberg and Hahn, and I said to my wife: "The war is lost and the result will be that as soon as the enemy enters the country I will be arrested and taken away." I only did it because, I said to myself, that [fission] is a German affair and we must see [to it] that German physics be preserved. I never thought for a moment of a bomb but I said to myself: "If Hahn has made this discovery, let us at least be the first to make use of it." When we get back to Germany we will have a hard time. We will be looked upon as the ones who have sabotaged everything. We will not remain alive [for] long there. You can be certain that there [will be] many people in Germany who [will] say that it is our fault. Now please leave me alone.

Gerlach withdrew from Haigerloch toMunich, "where he quietly resumed his pre-war work in his university laboratory" and was captured there on 20 April 1945 (Cassidy 2017), p. 75. He was first interned with a group of high-ranking Nazis and only on 15 June reunited with a group of detained German nuclear physicists. From 3 July 1945 until 2 January 1946, he was "detained as guest of His Majesty" (Gerlach 1978b) at Farm Hall in Cambridgeshire (Operation Epsilon), together with Erich Bagge, Kurt Diebner, Otto Hahn, Paul Harteck, Werner Heisenberg, Horst Korsching, Max von Laue, Carl Friedrich von Weizsäcker, and Karl Wirtz. The daily life at Farm Hall was described by Gerlach as follows (Gerlach 1978b):

Five prisoners of war were taking care of cooking, house cleaning, and service. There were no interrogations or tasks so that we could use most of our time for work, for which the necessary literature was provided; radio, a good library, and a large park were all at our disposal; there were occasional trips to London or Middle-England. Hahn was the "doyen," who would smooth out occasional disagreements with the American and British officers. The rapport with the two British attending officers, who would also partake in common lunches and dinners, was amicable to the point of being personal. The good atmosphere would be only seldom disturbed by a visit by a high inspector of the secret service.

In the Farm Hall Protocols, Gerlach was characterized as "cheerful" and "cooperative" but, "based on the recorded conversations," under suspicion of "having had connections to the Gestapo" (Hoffmann 1993), p. 64. We have not found evidence in support of this suspicion, but Gerlach's involvement with the Nazi regime still remains an open question. However, as for instance Paul Rosbaud's testimonial suggests, see Sect. 1, Gerlach harboured a strong anti-Nazi sentiment. And he apparently never joined the NSDAP. But his brother Werner Gerlach (1891–1963), a professor of pathology, was an early NSDAP member and held a high honorary rank in the SS (Simon 2002). Werner would have a falling out over his NSDAP membership with his principled father, Valentin Gerlach. We hope that ongoing research will provide more clarity.

Ironically, the Farm Hall Protocols recorded the following conversation (Hoffmann 1993), p. 100:

Diebner: I wonder whether there are microphones installed here?

Heisenberg: Microphones installed? (laughing) Oh no, they are not that cunning. I don't think they know the real Gestapo methods; they're a little old fashioned in this respect.

Upon his release from Farm Hall, Gerlach, along with his fellow detainees, was confined to the British Zone of Occupation. Nevertheless, within the British Zone, he was free to accept a professorship at the University of Bonn. In April 1948 he would be free to return to Munich, in the American Zone of Occupation. In postwar Munich, Gerlach dedicated much of his time and effort to the restoration of the German academia in general and the *Ludwig-Maximilians-Univerität* in particular, including the resurrection of its Institute of Physics. Figure 18 shows Gerlach at the General Assembly of the Max-Planck-Gesellschaft, on whose Senate he served since 1951. His success in helping to raise the country from the ashes would earn him the highest honours in the Federal Republic, such as the Order *Pour le Mérite für Wissenschaften und Künste* awarded to him in 1970 by the President of Germany. In the context of this volume we note that, in 1988, the Stern-Gerlach Prize (since 1993 the Stern-Gerlach Medal) was established as the most prestigious German award for work in experimental physics, cf. Chap. 5. As a further example of Gerlach's stature we show a recently recovered silver plate, Fig. 19, that Gerlach received on his 70th birthday from the Senate of the Max-Planck-Gesellschaft in recognition of the services he provided as a member of the body over several decades.

**Fig. 18** Walther Gerlach at the general assembly of the Max-Planck-Gesellschaft in Stuttgart in 1956. Courtesy of the Archiv der Max-Planck-Gesellschaft

**Fig. 19** Silver Plate presented to Walther Gerlach on the occasion of his 70th birthday by the Senate of the Max-Planck-Gesellschaft. Long after Gerlach's death it was passed on by his second wife Ruth, see Fig. 16, to her nephew, Werner Kittel. From him it was acquired in 2020 by Horst Schmidt-Böcking for the *Physikalischer Verein Frankfurt*. Photo H. Schmidt-Böcking, 2020

#### **5 Gerlach's Work in the History of Science**

From early on, Walther Gerlach cultivated a sense for the history of physics, perhaps in keeping with Goethe's maxim that "the history of a science is that science itself." Gerlach's first piece in the history *of* physics (Gerlach 1924) appeared at a time when he himself was making history *in* physics. As Gerlach's bibliography compiled by Margret Nida-Rümelin reveals (Nida-Rümelin 1982), this would be followed by about 500 additional publications on the history of physics/science, including about 60 scientific biographies, as well as outreach articles. During his distinguished career, Gerlach gave numerous talks on issues ranging from scientific funding to epistemological considerations, some of which would later be published. These are also included in the above number of 500.

Gerlach's sense for the history of science would also come to the fore in his capacity as educator. Like his academic mentor, Friedrich Paschen, Gerlach indulged his students in the spectacle of well-prepared experiments, some of which recapitulated chapters from the history of physics. The demonstration of Otto von Gericke's hemispheres, refuting the *horror vacui* theory, evacuated by Gerlach, a pioneer of high-vacuum technology, must surely have been a treat! Gerlach would also ask his students history questions during exams (Bachmann and Rechenberg 1989, p. 145). As Bachmann and Rechenberg report (Bachmann and Rechenberg 1989, p. 146):

When [Gerlach] realized *how* Newton brought out certain optical phenomena or Goethe observed phenomena that seemingly disproved them, he would be perhaps more pleased than if he discovered an altogether new physical effect.

Gerlach's writings on the history of science are based on his detailed knowledge of the subject—and its literature. He would have likely concurred with Steven Weinberg when he remarked (Weinberg 1998): "By assuming that scientists of the past thought about things the way we do, we make mistakes; what is worse, we lose appreciation for the difficulties, for the intellectual challenges, that they faced."

One of Gerlach's personal heroes was Johannes Kepler (1571–1630), whom he extolled not only as the first physicist in history worthy of the name, but also as a forerunner of humanism—"a priest of the book of Nature" (Gerlach 1972):

It was an unbearable thought ... for Kepler that, on the one hand, human reason enables insight into the wonders of Nature (and "only science reveals wonders"), into the *harmonic* order of the world, but, on the other, that human life generally passes in *disharmony*, driven by quarrel, conflict, hate, and war.

Gerlach also details Kepler's relationship with Galileo (1564–1642), who kept snubbing Kepler, whether about celestial mechanics or optics. But it was Kepler, Gerlach points out, who provided, through his third law (published in 1619) relating quantitative properties of the orbits of different planets, the most irrefutable evidence for the heliocentric system. Galileo would, however, never use it in his defense during the 1633 trial by the Inquisition. The lack of appreciation for Kepler in some quarters may have aroused special sympathy in Gerlach, as he too had not always received due recognition, see Sects. 1 and 3.3. However, there's no trace of complaint

**Fig. 20** Plaque at the entrance of the former *Physikalisches Institut* of the University of Frankfurt, Robert-Mayer Str. 2–4. Photo H. Schmidt-Böcking, 2002

about it in Gerlach's correspondence or any other source available to us. Secondly, Tycho de Brahe's measurements and their interpretation and analysis by Kepler of the eccentricity (0.0934) of Mars' orbit were revolutionary (in this case, also literally) precision measurements! And finally, Gerlach and Kepler were connected by the vicissitudes of their religious identity: they were both Protestants living in Catholic environments.

History of science was Gerlach's main preoccupation during the last twenty years of his life. His wide-ranging erudite historical writings deserve to be better known.

**Fig. 21** Double-portrait of Otto Stern and Walther Gerlach by Jürgen Jaumann. The schematic of the Stern-Gerlach experiment and its outcome was drawn by Theodor Hänsch. Photo H. Schmidt-Böcking, 2020

#### **6 In Conclusion**

Walther Gerlach lives on through his enduring legacy in physics, higher learning, and history of science. His estate, held at the Deutsches Museum in Munich, is comprised of sixteen thousand items. Walther Gerlach also lives on in a number of public depictions, among them the memorial plaque, Fig. 20, designating *Die Alte Physik* building in Frankfurt as the site where the Stern-Gerlach experiment was carried out. The Physics Department at Frankfurt also features a double-portrait of Stern and Gerlach, Fig. 21.

We close with Gerlach's credo (Gerlach 1978):

*Etwas Gutes kommt nie zu spät.* [It's never too late for something good to happen.]

#### **Appendix: Timeline of Walther Gerlach's Life and Career**

The timeline below has been translated and adapted from the catalogue of the 1989 centennial exhibition *Walther Gerlach—Physiker—Lehrer—Organisator* at the *Deutsches Museum* in Munich curated by Rudolf Heinrich und Hans-Reinhard Bachmann (Heinrich and Bachmann 1989).

#### • **August 1889–March 1908 Childhood, Youth**

#### **References**

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#### **Chapter 9 100 Years Molecular Beam Method Reproduction of Otto Stern's Atomic Beam Velocity Measurement**

**Axel Gruppe, Simon Cerny, Kurt Ernst Stiebing, Cedric George, Jakob Hoffmann, Maximilian Ilg, Nils Müller, Alienza Satar, Vincent Schobert, Leander Weimer, Markus Dworak, Stefan Engel, Gustav Rüschmann, Viorica Zimmer, Erich Zanger, and Horst Schmidt-Böcking**

**Abstract** The history of Otto Stern's pioneering measurement of the Maxwell-Boltzmann velocity distribution of a Silver atomic beam performed 1919 in Frankfurt is described. It is shown how Albert Einstein influenced Stern in his research. This experimental apparatus is not any more existing; therefore it was reconstructed in the workshops of the Physics faculty of the Goethe University in Frankfurt. The experimental verification of Stern's results was finally achieved by a team of Frankfurt high school students (Gymnasium Riedberg) under the supervision of their teachers Axel Gruppe and Simon Cerny. By fighting against a number of difficulties, they succeeded to get the reconstructed apparatus started and were able to reproduce the results from the early experiments of Stern.

#### **1 Otto Stern's Historic Atomic Beam Velocity Measurement**

Otto Stern was originally educated as a theoretical physical chemist. That he finally turned into one of the most genius experimenters in modern quantum physics is indeed astonishing. In 1912 he completed his dissertation with the title "Zur kinetischen Theorie des osmotischen Druckes konzentrierter Lösungen und über die Gültigkeit des Henryschen Gesetzes für konzentrierte Lösungen von Kohlendioxyd

A. Gruppe

K. E. Stiebing · M. Dworak · S. Engel · G. Rüschmann · V. Zimmer · E. Zanger · H. Schmidt-Böcking (B)

Institute for Nuclear Physics, University Frankfurt, 60438 Frankfurt, Germany e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Institute for Didactic of Physics, University Frankfurt, 60438 Frankfurt, Germany

A. Gruppe · S. Cerny · C. George · J. Hoffmann · M. Ilg · N. Müller · A. Satar · V. Schobert · L. Weimer

Gymnasium Riedberg, Friedrich-Dessauer-Str. 2, 60438 Frankfurt, Germany

**Fig. 1** Otto Stern and his brother Kurt as soldiers [3]

in organischen Lösungsmitteln bei tiefen Temperaturen" [1], which was partly experimental and partly theoretical. Thereafter he began his career in theoretical physics working with A. Einstein in Prague.

In the same year he followed Einstein to Zürich. In 1914 Einstein was appointed professor in Berlin. Stern accepted the offer by Max von Laue to become Laue's "Privatdozent" in Theoretical Physics at the newly founded University in Frankfurt. From 1914 until the end of 1918 Stern was soldier in World War One serving as weather observer (Fig. 1). In the second half of the year 1918 Stern was delegated to the Institute of Walter Nernst in Berlin, where, together with Max Volmer (Fig. 2), he performed several experimental investigations [2] in which Stern already demonstrated his ingenious skill of designing sophisticated physical experiments.

It therefore was not a surprise, that, after his return to Frankfurt in February 1919, Stern, the initially theoretically trained physical chemist continued performing experiments in physics. The Frankfurt Institute of Theoretical Physics (see faculty members in Fig. 3), directed by Max Born, owned a workshop with the young Adolf Schmidt as the only precision mechanic.

The first experiment that Otto Stern performed in 1919 in Frankfurt was the measurement of the Maxwell-Boltzmann-velocity distribution [5] of Ag atoms evaporated from a solid at the temperature of the melting point (Tm = 962 °C) [6]. He explained that he was interested in this experiment because, due to the influence of the "zero-point energy", he expected deviations from Maxwell's law at very low beam velocities. Together with Einstein he had published a theoretical paper on this issue in 1913 [7].

**Fig. 3** Members of the Frankfurt Physics faculty in 1920. From right: sitting Otto Stern, unknown, Max Born, Hedi Born and Richard Wachsmuth, standing: 3rd from right Alfred Landé, 4th Walther Gerlach, and next to Gerlach probably Elizabeth Bormann [3]

This pioneering experiment was the corner stone for Stern's famous molecular beam method MBM, which enabled the first ultra-high precision measurements of momenta of moving atoms or molecules in vacuum. With this experiment Stern established a method allowing the observation of inner atomic or even inner-nuclear ground state properties with unprecedented resolution, which, at least in 1919, was not achievable by energy spectroscopy (see Stern-Gerlach-Experiment SGE in 1922 [8]).

At a first glance, the measurement of the Maxwell-Boltzmann velocity distribution [6] looks simple. However, the authors of this paper, in their attempt to repeat this experiment, had to learn how difficult it really was, in particular, if one considers the very poor economic conditions in the year 1919 when the seminal experiment was performed.

The priming condition for the development of the MBM was the revolutionary progress in vacuum technology. Diffusion pumps became available creating vacuum in the low 10−5 torr regime. In such vacuum the free-path-length of moving atoms reaches several meters before they undergo a second collision. Stern benefited from his friendship with Max Volmer, who had developed a glass-made mercury diffusion pump, patented in 1918, Figs. 4 and 5. The Volmer mercury diffusion pump was fabricated in Berlin by Hanff and Buest. The rough vacuum was created by a rotating mercury pump invented by Wolfgang Max Paul Gaede (1878–1945) [9].

Otto Stern's experiment was inspired by the atomic beam experiment of L. Dunoyer in 1911 [10]. Dunoyer observed in his experiment that the beam particles move in vacuum like photons on straight lines. This is expected as long as the particles are not deflected by an external force or scattered by a gas molecules in the vacuum chamber. Vice versa one can use the transverse deflection in x and y direction (z is the direction of the velocity vector) by an external electro-magnetic force and thus determine electric or magnetic properties of the moving particle.

To perform deflection measurements and to obtain quantitative information on inner atomic properties one must measure the absolute value of the transverse momentum change. Therefore one has to know very precisely the direction of motion as well as the mass and the absolute velocity of the particle. In order to achieve this, one has to carefully prepare the atomic or molecular beam by a well aligned system of accurately manufactured slits. The principle of the transverse beam collimation is shown in Fig. 6.

The direction of motion is known from the geometry of the slit system. The velocity distribution of the atoms in the beam, generated by evaporating the atoms from a source at a defined temperature T was in 1919 only theoretically predicted but experimentally never measured. Thus, in order to later use the MBM for absolute momentum measurements, Stern had to verify Maxwell's theory [5] by measuring the Maxwell-Boltzmann velocity distribution of atoms, evaporated in a sufficiently good vacuum.

To perform these measurements Stern invented a kind of "streak camera" which is an ingeniously simple apparatus but which is very difficult to set into operation [6]. It is therefore astonishing and highly meritorious that the experiment had been accomplished, in particular when one considers the short period of one year from

**Fig. 4** The Volmer diffusion pump

beginning until getting a final result. One certainly has to anticipate that the help of the 26-year-old mechanic Adolf Schmidt was crucial for making the experiment a success.

In Fig. 7 Stern's "streak camera" is shown. The glass recipient had an inner diameter of 24 cm and was 30 cm high. With the help of the Gaede rotating mercury pump and the Volmer one-stage mercury diffusion pump the recipient could be evacuated to a pressure below 10−4 torr. The quality of the vacuum was measured by Geissler tubes. The pumping speed of both pumps was rather low (a few liters per second). The glass recipient was mounted vacuum-sealed on a 40 · 40 cm2 iron plate. The stationary frame (D) inside the recipient was fixed by screws on the iron plate. The streak camera (R) was adjusted inside D and could be rotated by a small motor (not seen on Fig. 7) with frequencies between 25 and 45 Hz. The axis of the motor was connected to the lower end of the main axis (A) of the apparatus by a short piece of vacuum hose. The other end of the motor axis was connected to a revolution counter by a flexible shaft. In the center of R a platinum wire (L) was mounted, the surface of

**Fig. 5** Historic pumps used in the original SGE. Left the Volmer mercury diffusion pump, right the Gaede diffusion rough pump [photo HSB]

**Fig. 6** Principle of the transverse beam collimation

which was covered by a thin layer (about 20 μm thick) of Ag. It could be heated by an electric current up to the melting temperature of Ag, emitting Ag vapor from the surface. It was very important that the wire remained stretched during heating. Two geometrically very thin beams were created by collimation by slits (S2), mounted symmetrically on both sides of the wire. These beams condensed on two polished

**Fig. 7** Stern's "streak camera" apparatus. G = Glass recipient, R = "rotating streak camera" (within the red dashed line), D = stationary frame, in which the "camera" is rotating, L = Platinum wire, from where the silver atoms evaporate, S1 and S2 = slits, P = detection plate. The pumps connected to the glass tubes and the motor are not shown [6]

brass plates (P) in 6 cm distance from the wire. The slits (S2) were halfway between L and P. A further set of two slits (S1) (8 mm distance from the wire) was mounted to ensure that the well-defined position of the Ag beam source did not change during the time of the measurement.

In order to measure the atom velocity, the streak camera (R) had to be rotated around the axis A (see Fig. 7, lower part, where the clockwise rotation is indicated by the arrows). In case of no rotation the slits and the wire were aligned on one straight line leading to a small streak in the middle of P. In case of rotation the streak on the detector plate is shifted in opposite direction of the rotation by about 0.4 mm (at a rotation frequency of 25 Hz). The reason is that, depending on their velocity, the atoms need some time to fly from the slit to the detector P while the detector has rotated forward. To obtain a better separation the system was rotated in both directions yielding about 0.8 mm separation.

Although the working scheme of the apparatus is rather simple, it required a number of skills in different experimental fields to make it run successfully: precision engineering, pumping and sealing to obtain a good vacuum, frequency and temperature measurement etc. One may anticipate that the help of the young mechanic Adolf Schmidt was essential for Otto Stern to make the apparatus run. In order to get a good velocity resolution the whole segment R had to rotate with a constant frequency. According to Stern, the required balance of the rotating part and the necessary vacuum sealing at the feedthrough of the rotating axis were the most difficult problems. For sealing the axis they used oil-soaked asbestos rope (see (St) in Fig. 7). Since this sealing was too leaky, they additionally had to evacuate the space M1/M2 where the axis A rotates in a tight-fitting but not touching brass tube (see Fig. 7). Because of frequent heating and cooling the Platinum wire got stretched and had to be adjusted frequently, in order to avoid bending when glowing.

On both detector plates (P) Stern observed two clearly separated lines one for rotating the system clockwise and the other for rotating counterclockwise (see Fig. 8). From the measured separation, the geometry of the streak camera and the rotation frequency he deduced a mean velocity of the beam of about 600 m/sec. Maxwell's theory, however, predicted only 534 m/sec for a temperature of the Ag melting point at 962 °C. Stern assigned this difference to a possible deviation between the measured and the real temperature at the wire.

However, Einstein in Berlin recognized that Stern had made a mistake in his analysis. He had overseen that the transmission flux of the beam through a slit depends on the third power of the atom velocity but not on the square. Walter Grotrian, who reported on Stern's experiment in a seminar in Berlin, where Einstein, Planck, Laue and Nernst [12] were in the audience, wrote in a letter to Stern (on July 30, 1920) and informed him about the discussion in this seminar in Berlin [12]:

#### Dear Stern!

… Your experiment appeared to all, who listened, also Franck and Reiche astounding and convincingly. After long discussions we were convinced, that also in case of sublimation of a solid, e.g. coal, the mean kinetic energy of the emitted atoms or molecules is 3/2 kT. Thus the issue was settled.

Then followed the discussion, which I will present in detail. It began with Nernst. His remark was related to experimental details. First he mentioned the rotating electric contact into the vacuum ("Öhse") and named it a master piece. After some insignificant remarks Laue asked, whether the evaporating molecules do really have the mean energy 3/2 kT. I tried first following your letter to turn the concerns down which were related to the evaporation

energy. Then Einstein stood up and went to the blackboard. He explained now, that one had to distinguish between velocity distribution per volume and the velocity distribution of these molecules which impact on or are emitted from a surface. The latter ones would be shifted to higher velocities.

From what he said and his later discussion with Planck it was not clear whether he was objecting your results or not.

We discussed yesterday again this issue in detail and came to the conclusion: … The question is: Is the mean square of the velocity distribution of a molecular beam penetrating in one direction per 1 cm2 through a slit equal to the mean square of the velocity distribution per 1 cm3 volume?…

We hope that you can inform us soon what is the answer to this question. It would be the best if you could visit us in Berlin.

Yours Walter Grotrian.

Einstein's concerns were proved to be true. On October 20 1920 Stern submitted an addendum to Z. Physik with a new analysis of his data based on Einstein's arguments [13]:

In the recent published communication [6] I have reported on experiments where the velocity of Ag atoms evaporated from a melting Ag surface into vacuum was measured with 600 m/sec. This value is within the error bars in agreement with the mean value calculated from the kinetic gas theory at the temperature of the melting point. This result seems to verify the assumption that the Ag atoms, which are emitted from the surface have the same velocity like the atoms of melting silver. But several people have now criticized this assumption, where the objection of Mr. Einstein is justified. The issue is the following: 1. We have a recipient filled with gas at a given temperature in equilibrium state. We now look at atoms that escape through a tiny hole into the vacuum. These atoms do not have the Maxwell velocity distribution of the equilibrium state inside the recipient in contrast to the analog case of black body radiation. The fast atoms have a higher probability to escape. Following known gas theoretical considerations the number dn'c of the molecules with velocity c escaping through the hole per time unit is equal to the number of atoms per volume unit dnc multiplied with the volume of a cylinder of length c. Therefore is dn'c not proportional to dnc but cdnc.

With this experiment Otto Stern for the first time confirmed that the Maxwell-Boltzmann theory on the velocity distribution of gases at temperature T agrees well with experimental results. By measuring the temperature T of the evaporating gas, Stern, with the help of Maxwell's theory, could deduce the velocity and thus the momentum of the moving atom or molecule. This experiment was the foundation of Stern's molecular beam method (MBM) enabling high-resolution momentum measurements. In the following decades even up to today numerous milestone experiments of quantum mechanics have been performed basing on this method.

In all his publications until his retirement in 1945, Stern never mentioned that his new method provided a high-resolution momentum spectrometer yielding a resolution never achieved before. He presented his data always as function of deflection angles. Therefor the extremely high momentum resolution was not obvious to the readers of his publications. It is important to note that the line width measured by Stern already in this first experiment corresponds to a transverse momentum width of sub-atomic size. This excellent momentum resolution can be estimated from Fig. 9 as follows: Let the velocity of the Ag atoms be 540 m/sec, corresponding to a momentum of about p = 50 au. The two lines in Fig. 9 are separated by 0.8 mm (on the detector plate) in 60 mm distance from the wire. Transformed into momentum space this distance corresponds to a momentum difference of p = (0.8:60) · 50 au = 0.67 au (see Fig. 9). The width of the left line is then less than 0.2 au, demonstrating the excellent momentum resolution achieved in this experiment. Reducing the slit width, the momentum resolution could even be improved.

In the conclusion of his paper [6] Stern revealed the motivation for measuring the Maxwell velocity distribution:

**Fig. 9** Line splitting as function of transverse momentum

A very precise examination of Maxwell's velocity distribution law would be of particular interest. According to the Quantum theory, small deviations occur in gases with a small molecular weight at high pressures and low temperatures, which are estimated to be about 1 percent for hydrogen at the boiling point under atmospheric pressure. Unfortunately, it is not possible to give more precise information about the type and amount of these expected deviations - except, for example, that assuming zero-point energy, the low velocities will occur more rarely than according to Maxwell - because the Quantum theory of translational motion encounters previously insurmountable difficulties. The experimental investigation of these deviations would be thus even more important, and it was precisely this problem that gave me the reason for the present investigation. Unfortunately, the conditions here are also very unfavorable for the experiment, but perhaps gravity will provide sufficient dispersion for the analysis of the low speeds.

Finally, it should be noted that the above method allows for the first time to produce molecules of uniform speed, and e.g. to investigate whether condensation only takes place above or below a certain speed.

The first application of Stern's atomic beam method was performed by Max Born and Elisabeth Bormann [14]. They successfully used an Ag beam in 1920 in Frankfurt to determine the free path length λ of the Ag atoms in air. The Ag beam was collimated by a cascade of round copper screens, in each of which a centric hole was drilled for passing the beam through. When the air pressure in the recipient was gradually increased, more and more Ag atoms were scattered by the air molecule and were deposited on the copper diaphragms, which had been mounted in well-defined distances. The amounts of depositions were carefully measured and it was then possible to use a theoretical scattering model according to Jeans [15] to determine the "free path length" λ of the Ag atoms for the given pressure.

Stern himself used the atomic beam method for the first time in the famous Stern-Gerlach experiment SGE [8], which was carried out in Frankfurt from 1920 to 1922. The SGE demonstrated in an impressive manner what the MBM can achieve as a means for momentum measuring.

In 1928, when he worked as fellow in Stern's laboratory in Hamburg and later as professor at Columbia University in New York and at the MIT in Boston Isidor Rabi developed a new extremely powerful scheme for the application of MBM by using first a SGE approach to prepare atomic beams in well-defined quantum states. In a second interaction region the prepared states could be excited by resonant photon absorption into another quantum state with different magnetic quantum numbers. These states, excited by resonance absorption, moved into a third interaction region on a different trajectory and could be detected separately. He and his group very successfully applied this method to use the very narrow line width of photon absorption for high precision measurements of transition energies, like e.g. the Nobel Laureates [16] Willis Lamb and Polykarb Kusch for measuring the so-called Lamb-shift, Norman Ramsay to develop the Cs atomic clock (with 10−9 precision), Felix Bloch and Henry Purcell for developing the Nuclear Resonance technique etc.

Since 2002, when in Frankfurt the 80th anniversary of the SGE was commemorated, one of the authors (HSB) looked for remainders of the historical experimental set ups used by Stern at the various working places of Stern and Gerlach. The only parts he found were the microscope bought by Stern in 1919 from the company of Seibert in Wetzlar (found in Berkeley) and the Volmer diffusion pump (found in Frankfurt). Obviously the historic apparatus did no longer exist. Therefore, the idea was born to reconstruct both experimental apparatus of Stern: the set-up to measure the Maxwell-Boltzmann velocity distribution and the famous Stern-Gerlach experiment. While the Stern-Gerlach experiment is still waiting for its reconstruction, the first one was now reconstructed and was put into operation. In the following the reconstruction and the successful commissioning of the first apparatus is described.

#### **2 Reproduction of Otto Stern's Atomic Beam Velocity Measurement**

#### *2.1 Reconstruction of the Apparatus*

On the occasion of the 100th anniversary of Stern's appointment to Frankfurt, the initiative was taken to reconstruct the historic set up, which did no longer exist, and to reproduce Stern's famous measurement of the velocity of Ag atoms in an atomic beam. Based on the drawings and the detailed description in Stern's publication [6], and sponsored by Roentdek Handels GmbH, a number of identical copies were fabricated in the workshops of the Institute for Nuclear Physics in Frankfurt.

One of the copies was given to the Gymnasium Riedberg at Frankfurt with the requirement to repeat Otto Stern's measurements and, if possible, to verify his results. This task was adopted by a team of high school students from the 10th grade, who founded an "Otto-Stern-Arbeitsgemeinschaft" (OSAG) and, under the supervision of their physics teacher, Axel Gruppe, started to work on this project in summer of 2015. Their work was presented at the VDI Student Forum 2015 [17].

First, the students read Otto Stern's biography to get an impression of his scientific achievements. Then they began to examine the optically very appealing replica (Figs. 10 and 11) in view of its suitability in a real experiment, which Otto Stern described in great detail in his work from 1920 [6]. The following points were studied in detail: the trajectories of the silver atoms, the measurement of the rotation frequency, the mean free path of the silver atoms and the measurement of the temperature of the platinum wire.

#### *2.2 The Trajectories*

Anticipating the rotating frame to be at rest, it appears beyond question that the silver atoms, emitted from the Platinum wire, will fly through the slit diaphragms S1 and S2, which are aligned on a straight line with the emitting wire, and will impact on the collecting plate P exactly at the point, where the straight line hits P. But how does the trajectory change when the frame rotates?

**Fig. 10** Plan for the restoration of Stern's "streak camera" system

To understand this, one has to convert the path of the silver atoms, which in the laboratory system corresponds to a rectilinear, uniform motion, into the rotating coordinate system of the frame of the camera (R), with which the diaphragms and the collecting plate are firmly connected.

In [6] Stern describes the trajectory of the atoms in the co-rotating system as to represent a "horizontal throwing parabola" ("… by neglecting the centrifugal acceleration and the change in the Coriolis acceleration.").

In order to get an impression of the parameters of this parabola, the students developed an EXCEL worksheet for the coordinate transformation of the rectilinear,

**Fig. 11** Reconstructed apparatus to measure the Maxwell velocity distribution (without vacuum pumps and diagnostics)

uniform atomic motion into the rotating aperture system and thus studied the intricacies of Stern's experiment. In the upper part of Fig. 12 one can see the wire L in the coordinate origin of the rotating system X- /Y as well as the slit diaphragms S1, S2 and the collecting plate P.

Starting from L with a velocity of 500 m/s, the Silver atoms have to be emitted at an angle α = −0.55° relative to the X- - axis, to pass through slit diaphragm S2. It is evident that in this case the beam trajectory does not pass through S1 with the result that the silver atoms will not hit the detection plate P. To make the trajectory passing through S1 and S2, the location of the emission point must be shifted in the Ydirection (Fig. 12, lower part).

This is the reason why Otto Stern had rolled down the initially round wire (0.4 mm diameter) to a width of 0.6 mm in the Y direction [6]. By this, he wanted to ensure that the locations of emission of the Silver atoms enable trajectories that run through both S1 and S2. With these calculations, the students also realized how small the expected shift really is. At a rotation frequency of 25 Hz, the displacement of the point of impact for silver atoms with v = 500 m/s is only about 0.5 mm in the Y- direction!

**Fig. 12** Trajectories of the silver atoms in case of rotation (f = 25 Hz). Upper picture: the atoms start at L on the straight line passing through apertures S1 and S2. Since the trajectory has the shape of a parabola, no silver atom reaches the collecting plate P. Lower picture: The silver atoms must start from a point L shifted by dy in the Y direction from the center of rotation in order to reach the collecting plate P through the two apertures

#### *2.3 Measurement of the Rotation Frequency*

Since the slit diaphragms have a width of 0.2 mm, a 0.4 mm wide silver line is expected on the collecting plate. Under the most favorable conditions, the silver lines with the streak camera at rest and with the rotated camera would then be separated by just 0.15 mm. The students calculated that at a speed of 25 r/s a fluctuation of ±2 r/s would lead to a shift in the impact point of ±0.1 mm. Therefore the fluctuation of the rotation speed must not exceed this value if one wants to be able to separate the two silver lines and the rotation speed had to be controlled with sufficient accuracy. To achieve this a digital speed counter was designed and built by the students [18]. In this design, signals from chopping a light barrier by a slotted disc, which was attached to the axis of the built-in direct current motor, are read out and displayed as a frequency on a 4-digit 7-segment display by means of a tricky circuit.

#### *2.4 Mean Free-Path and Quality of the Vacuum*

Considering the dimensions of the original equipment, one may wonder, why it was not built larger, as the offset, and hence the effect would become larger, the longer the distance between source (L) and the screen (P) would be. At this point, the quality of the vacuum comes into play. The deposition lines of Silver will only sharply be imaged on the collecting plate (P) if the silver atoms do not undergo collisions with air molecules during their flight. Therefore the vacuum inside the apparatus must be good enough to avoid such collisions as far as possible. In order to estimate this influence, the students investigated the physics of the "mean free path" (MFP). MFP is the distance a molecule travels on average in a gas before it collides with another molecule. (According to the definition, the average number of molecules in an atomic beam, which have not yet collided with a residual gas atom, is only 1/3 after the beam has passed the length of MFP [19]). In addition to the temperature, MFP is essentially dependent on the gas pressure in the recipient.

For air at 20 °C the students found a value of MFP = 6.8 × 10−3 mbar cm/p [19]. Otto Stern mentions 1/10,000 mm (=1 × 10−4, torr = 1.33 × 10−4 mbar) as the required vacuum. This corresponds to an MFP of approximately 50 cm in Stern's apparatus. According to the nomenclature of vacuum technology, such a pressure is already termed "high vacuum". This means that in Stern's experiment only about 8% of the silver atoms collided with an air molecule on their 6 cm path to the screen.

#### *2.5 Measurement of the Temperature of the Filament*

Another difficulty was the measurement of the temperature of the silver-plated platinum wire. Stern writes in [6]: "Now the temperature of the evaporating silver was certainly higher than 962° because the molten silver contracts to form droplets and the parts of the platinum wire that have been freed from silver assume a higher temperature due to their higher resistance, which increases due to conduction to the Silver droplets. According to the brightness, the temperature should have been around 1200° …". The mention of the brightness of the wire by Stern suggests that he used a pyrometer to measure the temperature. Fortunately, a functional pyrometer from Hartmann and Braun ("Pyropto", manufactured in 1951 [21]) was found in the collection of the physical internship of the Institute of Applied Physics, and was kindly donated to OSAG.

#### *2.6 The Improved Experimental Setup and the Decisive Measurement*

After the essentially theoretical and preparatory work accomplished by this first OSAG at the Gymnasium Riedberg, the work was transferred back to the Goethe University. In discussions with physicists and mechanics of the Institute for Nuclear Physics Research (IKF), it became clear that the available replica of the Stern apparatus had to be changed in several details in order to successfully start with the next step, i.e. to reproduce the measurements from 1920:

In the school year of 2018/19, a new OSAG at the Gymnasium Riedberg started to perform measurements with the improved equipment. In the first test runs, it became clear that the necessary constancy of the rotation frequency could only be achieved with the help of a speed control of the DC motor. Using an Arduino® microcontroller, the students first built a simple linear control loop, the control oscillations of which, unfortunately, were still too large. However, reprogramming the Arduino to a more sophisticated proportional-integral-derivative (PID) controller led to the desired success (Fig. 13).

With the successful establishment of the PID control, all preparations for repeating the Stern experiment were completed in summer 2019. The final experimental setup is shown in Figs. 14, 15, 16, 17, 18 and 19.

The decisive measurement took place on August 30, 2019. Initially, the members of OSAG recorded a weak but clearly visible Ag line at a rotation speed of 45 r/s. During a second irradiation with the camera at rest, the lower part of the detector plate was covered with a plastic film to facilitate detection of the separation of the two lines. Figure 20 shows a scan of the brass plate which has been processed to enhance contrast. The evaluation shows a distance of the line centers of 0.62 mm with an estimated accuracy of 10% to 15%. This result must be compared with the value that Otto Stern presented in his publication [13]: "For 2700 tours (i.e. rotations/minute),

**Fig. 13** The proportional-integral-derivative (PID) control of the rotational speed. Above: the time course of the manipulated variable (pulse wave modulation). Below: the time course of the regulated rotation frequency. In this case the PID controller has settled after 20 s and keeps the rotation speed constant at 45 ± 0.5 r/s

the distance between the centers of the two lines created with right and left rotation was 1.26 mm". Otto Stern's value divided by 2 yields 0.63 mm, which is in excellent agreement with the value measured by OSAG (0.62 mm at 45 r/s = 2700 tours with all other experimental parameters kept identical as far as possible). After a few measurements, the platinum wire had to be readjusted regularly (Fig. 21).

The OSAG team was very much impressed by this excellent agreement 100 years after the original measurement by Otto Stern. The participants were very proud that they had managed to successfully complete this experiment, which was easy to understand but technically difficult to carry out.

From the teacher's point of view, this project was an ideal example of how to offer students deep insights into the interplay of theoretical and experimental physics with the help of an ambitious topic and authentic material.

**Fig. 14** The final experimental setup of the Otto Stern experiment in 2019: In the center: glass recipient with pressure gauge on an iron plate. Inside, the fixed frame with the rotating slit system is visible. On the right: parts of the vacuum system with backing pump (red, in front) and turbomolecular pump (rear). In the foreground in front of the glass bell stands the pyrometer ("Pyropto", Hartmann & Braun®) for temperature measurement. Right front: PID controller for speed control on breadboard with an ARDUINO® controller. Rear right: controllable DC power supplies for the motor and the heating wire. Below is the vacuum measuring device with the vacuum display. The Pirani and Penning sensors (Balzers®) are positioned under the base plate and are not visible. On the left: Laptop for setting and logging the rotation speed during the measurement

**Fig. 15** The view through the pyrometer. The bent pyrometer wire is adjusted to the same brightness as the vertical platinum wire (left: at room temperature, right: at 1050 °C)

**Fig. 16** The Members of the Otto-Stern-AG during a measurement (from left to right: Leander Weimer, Nils Müller, Simon Cerny, Jakob Hoffmann)

**Fig. 17** The glowing platinum wire in the center of the rotating slit system at a rotation speed of 45 r/s

**Fig. 18** The slit system with the detector plate after the experiment. On the right is the platinum wire; on the left, next to it, is the first slit aperture, which is completely covered with silver. To the left of it is the second slit aperture on which one can clearly see the "shadow" of the first slit aperture. On the far left, the detector plate is to be seen, on which a faint, brownish Ag line is just visible

**Fig. 19** According to legend, Otto Stern "developed" the faint traces of Ag on the brass plates by cigar smoke, as Leander Weimer and Nils Müller also tried after the experiment

**Fig. 20** Contrast-enhanced scan of the collecting plate. The longer thin line on the left was recorded at a speed of 45 r/s. The shorter line on the right was taken with the camera at rest. The distance between the line centers is 0.62 mm

**Fig. 21** The mechanic Stefan Engel balancing the equipment

**Acknowledgements** We would like to thank all those who have contributed to this project. In particular the commitment of the members of the IKF workshop and the accelerator group is gratefully acknowledged. We also express our gratitude to the Pfeiffer Vacuum Company [20] for providing us with a powerful HV pumping system consisting of a turbo-molecular pump in combination with a backing pump. The Pyrometer for measuring the temperature of the Platinum wire was donated by the Institute of Applied Physics. The experiments have been carried out within the framework and in the premises of the Goethe Lab-Schüler-Labor "Radioaktivität und Strahlung" at the Goethe University, Frankfurt. The project would not have been possible without the generous financial support by Roentdek Handels GmbH, Kelkheim, Germany.

#### **References**

**Open Access** This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

#### **Chapter 10 Wilhelm Heinrich Heraeus—Doctoral Student at the University Frankfurt**

**S. Jorda and H. Schmidt-Böcking**

The 702nd WE-Heraeus Seminar, conducted from September 1st–5th 2019 in the historic physics building in Frankfurt, commemorated the great discoveries in Quantum Physics made between 1919 and 1922 at the Frankfurt university in particular by Otto Stern and Walther Gerlach. These milestone discoveries were made in the theoretical institute of physics under the directorship of Max Born and Erwin Madelung by Otto Stern, Walther Gerlach, Max Born, Elisabeth Bormann and last not least by Alfred Landè. In this period in the early twenties Wilhelm Heinrich Heraeus was working on his doctoral thesis in Frankfurt and met Gerlach and probably Stern, Bormann, Landè and Born too (see Fig. 1). He was thus a contemporary witness to these great pioneering achievements, many decades before he and his wife Else would establish the Wilhelm and Else Heraeus Foundation.

Wilhelm Heinrich Heraeus was born on February 3rd 1900 in Hanau/Hessen, which is located about 25 km east of Frankfurt. He was the grandson of Wilhelm Carl Heraeus (\* March 6th 1827; † September 14th 1904), the founder of the Heraeus company in Hanau. He described the first 23 years of his life in a short curriculum vita when he enlisted in 1923 at the University of Frankfurt for the Ph.D. examination. According to this, he had to leave grammar school, having just turned 17, with the emergency certificate, before he worked for a year, until Easter 1918, in the "patriotic emergency service". His subsequent studies of physics and natural sciences in Bonn were interrupted after only a few months by another military service, and the same was true of his subsequent studies in Göttingen, where Heraeus took part in lectures and practical exercises by Debye, Hilbert and Courant. From the fall of 1921 he finally studied in Munich, where he worked for "Geheimrat" (Privy Councilor) Wien and attended lectures by Sommerfeld, among others, before moving

S. Jorda

Wilhelm und Else Heraeus-Stiftung, Kurt-Blaum-Platz 1, 63450 Hanau, Germany

H. Schmidt-Böcking (B)

Institut für Kernphysik, Goethe Universität, Max-von-Laue-Str.1, 60438 Frankfurt, Germany e-mail<EMAIL_ADDRESS>
to Frankfurt in late 1922. There he took part in lectures and practical exercises by Gerlach, Madelung, Lorentz, and carried out his doctoral thesis with "Geheimrat" Wachsmuth and Gerlach.

Professor Richard Wachsmuth, the director of the experimental institute, together with Professor Walther Gerlach (the co-author of the famous Stern-Gerlach experiment) supervised the dissertation of Wilhelm Heraeus. The experimental doctoral research studies were performed in the laboratory of the Heraeus company in Hanau. The results of Wilhelm Heraeus' work are described in the 26 pages of his dissertation with the title: "Die Abhängigkeit der thermoelektrischen Kraft des Eisens von seiner Struktur" (The dependence of the thermo-electrical force of iron on its structure). Wachsmuth had to apply to the dean Professor Fritz Drevermann, that the submitted scientific work of Wilhelm Heraeus fulfilled the requirements of a dissertation. Wachsmuth wrote in July 1923:

A paper appeared in the Annals of Physics about 1 year ago, in which the author Borelius, by measuring thermoelectric forces between two samples of pure iron, one of which was pretreated by heating to temperatures up to 500°C and subsequent quenching, the other was untreated, has shown that in this interval a large number of internal iron conversion processes have now been found. Mr. Heraeus, to whom the resources of the father's company were available, seemed to me to be the suitable man for a revision. When the experiment was

**Fig. 1** Wilhelm Heraeus' student card [Archiv, Goethe-Universit¨at Frankfurt, Frankfurt, Germany]

**Fig. 2** The new experimental apparatus of Wilhelm Heraeus [Archiv, Goethe-Universit¨at Frankfurt, Frankfurt, Germany]

extended up to 950 °C, the results were so interesting that this little extra work became a doctoral thesis.

Not only has the author of the present work discovered the errors in the Borelius's arrangement and the thermoelectric curves between "iron heated" and "iron unheated" up to the Borelius border without its discontinuities, but he has also been able to increase the temperature further. He found a decrease in the thermoelectric force between 500 and 790 °C and a new increase between 800 to 870 °C. He also associated this phenomenon with signs of recrystallization and checked it for clarification by means of appropriate tests and documented it with metallographic images. The work was carried out very carefully and conscientiously, but also completely independently. … I propose the work to the faculty for acceptance and apply for the rating "very good".

[Archiv, Goethe-Universit¨at Frankfurt, Frankfurt, Germany]

In the introduction of his thesis Wilhelm Heraeus asserts: In order to develop a theory of the electrical conductivity of metals, detailed thermoelectric studies are necessary. So far, there is little data on this topic existing, as pure metals have so far hardly been available. Low impurities can have a decisive influence on the thermodynamic behavior of the metals. For this reason, for the present investigation, melted electrolytic iron was chosen, which was treated in the cold state with special care in order to avoid contamination from the carbon-containing iron of the rolls and from the material of the drawing dies. The investigation of the thermoelectric behavior of annealed versus un-annealed iron is also of particular interest insofar as there are already studies by Borelius and Gunnesson [1]. The results of this work have led to conclusions on structural changes in the temperature range from 60 to 500 °C, which are in sharp contradiction with other methods of investigation.

The new measuring device built by Wilhelm Heraeus allowed to heat an electrolytic iron wire in an electric oven with temperature steps of 10 °C (see Fig. 2).

**Fig. 3** Typical data sets [Archiv, Goethe-Universit¨at Frankfurt, Frankfurt, Germany]

After heating, the wire was quenched to room temperature in a water bath below. The two ends of the quenched iron wire were then connected to a galvanometer and the thermal force was measured. The measurement series were reproducible. Even if the galvanometer was connected in the opposite direction, the same values were obtained for both directions.

The Borelius' investigations were repeated by Wilhelm Heraeus. The samples were brought to a certain temperature T by means of an electric furnace and then quenched in water. Then the voltage difference was measured relative to another untreated sample. An experimental setup was chosen which was identical to that of Borelius. Even the samples were identical, since Borelius also obtained the iron wires from the Heraeus company. Wilhelm Heraeus then states: The curves (temperature on the abscissa and the measured voltage on the ordinate) of Borelius are characterized by their inconsistent, complicated course (see Fig. 3). The order of magnitude is the same for Borelius and Heraeus. However, what is completely missing is the reproducibility of each individual measurement. If the same wire was taken out of the arrangement several times and put back in again, the galvanometer deflections were completely different. It even turned out that the galvanometer showed a rash, even though the copper block was not warmed. The rashes could not therefore have resulted from a thermoelectric force; the investigation showed that there were contact forces between iron and copper, i.e. the pressing forces of the clamps. Even the smallest changes in temperature affect the measurement results (1 °C results in 0.05 to 0.15 microvolts). The fluctuations observed by Borelius can therefore all be explained by these influences.

Wilhelm Heraeus also took photomicrographs of the samples to identify structural changes. He was able to clearly identify recrystallization in his investigations, which also has a significant influence on the properties of iron. The type of recrystallization could also be influenced by the duration of the heating.

Wilhelm Heraeus concluded his written dissertation as follows. The available studies therefore show:

Finally in the acknowledgment Wilhelm Heraeus mentioned as his supervisors Wachsmuth and Gerlach as well as Professor Fränkel.

After passing the dissertation exam he acquired first experience in the precious metals company of his uncle in Newark, N.J., USA. In 1925 he joined the W.C. Heraeus company, and in 1927 he became a third generation member of the executive board. As head of technical development, he expanded the group's product range and, after 1945, also managed the reconstruction of the destroyed plant. In 1965 he changed from the management to the supervisory board of the family company.

Together with his wife Else († 1987) (they had no children of their own) he founded in 1963 what is now the Wilhelm and Else Heraeus Foundation, which supports scientific research and education with an emphasis on physics. Organizing WE-Heraeus Seminars is the most important and oldest funding activity of the foundation. In 1983, two years before he passed away, Wilhelm Heraeus became an honorary member of the German Physical Society (DPG).

#### **Reference**

1. G. Borelius, F. Gunnason, Annalen der Physik, 67 227 and 238 (1922); Annalen der Physik, 68 67 (1922)

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## **Part II Foundations of Quantum Physics and Precision Measurements**

#### **Chapter 11 Quantum or Classical Perception of Atomic Motion**

**John S. Briggs**

**Abstract** An assessment is given as to the extent to which pure unitary evolution, as distinct from environmental decohering interaction, can provide the transition necessary for an observer to perceive quantum dynamics as classical. This has implications for the interpretation of quantum wavefunctions as a characteristic of ensembles or of single particles and the related question of wavefunction "collapse". A brief historical overview is presented as well as recent emphasis on the role of the semi-classical "imaging theorem" in describing quantum to classical unitary evolution.

#### **1 Introduction**

In describing the motion of the silver atom beam through their apparatus in 1922, Gerlach and Stern [1] naturally assumed classical mechanics; only the internal angular momentum was quantised. Later, following Schrödinger's 1926 introduction of wave mechanics, more attention was given to the quantum state of motion of the atoms. For example, Bohm in 1951 [2] described the beam translational motion by quantum mechanics and gave particular attention to the question of interference between the waves of the two beams leaving the magnet region. An extensive discussion of the classical and quantum aspects is given in Gottfried and Yan [3] in terms of particle wave packets.

Here, the general description of the motion of atomic particles over macroscopic distances is considered. It is shown how classical features appear autonomously and that the perception of classical or quantal behaviour depends upon the extent and accuracy of detection of such motion. This has consequences for the meaning assigned to a wave function and to its interpretation as to describing the motion of a single particle or the motions of an ensemble of identical particles.

J. S. Briggs (B)

Institute of Physics, University of Freiburg, Freiburg, Germany e-mail<EMAIL_ADDRESS>
Department of Physics, Royal University of Phnom Penh, Phnom Penh, Cambodia

#### *1.1 Particle or Wave or Particle Ensemble?*

In the scattering of electromagnetic waves around a sharp material object, the nature of the perceived outline depends upon the resolution and sensitivity of the instrument. For visible light, in the case of the human eye, usually a sharp outline of the object ascribable to a ray description of the light would be inferred. However, from measurement with an instrument able to resolve at sub-wavelength accuracy, a blurred outline corresponding to a diffraction pattern and ascribable to the wave nature of the light would be inferred. The instrument resolution is understood as the accuracy of position location. Important also is the sensitivity of the instrument, whose limit is taken here as the ability or not to register reception of a single quantum particle e.g. a photon, electron, atom or molecule.

If the detector sensitivity is sufficient one can monitor the arrival of individual particles. In the case of photons, their arrival at a screen appears in a seemingly arbitrary pattern until enough photons are counted. Then the statistical distribution gradually assumes the structured diffraction or interference form expected on the basis of the classical wave picture of electromagnetism. This is the wave-particle duality of light. With increasing resolution and sensitivity of the measurement, there are three levels of perception, classical ray trajectory, the wave picture and the ensemble of (quantum) particles picture.

A similar situation arises in the wave-particle duality of matter. For an ensemble of identical particles with a mass which is very large on an atomic scale, one assigns to their motion a unique classical trajectory so long as the resolution of, say position detection, is not itself on the atomic scale. When the mass of the particles *is* on the atomic scale it is necessary to calculate the average of their motion from the wave picture of quantum mechanics. Increasing the sensitivity to detect individual particles leads to a seemingly arbitrary pattern until the statistics are sufficient that a pattern predicted by the wave description emerges, for an experiment with electrons see Ref. [4]. Again there are three levels of perception of the ensemble; unique classical trajectory, many particles registered as a wave pattern or the statistical pattern from individual quantum particles. Which description is appropriate depends both upon the the resolution and the sensitivity of the measurement.

Indeed the analogy between the classical wave equations of electromagnetism (Helmholtz equations and paraxial approximation) and the wave equations of quantum mechanics (time-independent and time-dependent Schrödinger equations) is very close mathematically. This leads to the similarity of perception alluded to above. The semi-classical limit of quantum mechanics, used extensively below, corresponds to the eikonal approximation for electric wave propagation. Thus the quantum to classical limit for material particles corresponds to the wave to beam limit of electric field propagation. In optics, the large separation between source and observer is used to derive the Fraunhofer diffraction formula which is in close analogy to the asymptotic "Imaging Theorem" of quantum mechanics derived in section III.

One must be clear on this point. The quantum to classical transition in particle dynamics is the transition from wave to particle perception. Paradoxically, the quantum to classical transition in photon dynamics is the opposite, from particle to wave perception. It is the *wave to beam* transition of classical optical dynamics which is the analogue of the *quantum to classical* transition in particle dynamics.

A key element of quantum mechanics, not present for classical light, is the interpretation of the modulus squared of the wavefunction as a statistical probability. Here, two points of view have emerged. The first, to be called the ensemble picture, is that the probability describes the percentage of members of an ensemble of identical, and identically-prepared, particles having a particular value of a dynamical variable. The second, to be called the single-particle (SP) picture, considers that it is the probability with which an individual particle exhibits a given value out of the totality of possibilities. That is, on measurement the wavefunction "collapses" into one eigenstate of the observable with one definite value of the variable observed. The difference is that in the ensemble interpretation, only measurements on the whole ensemble are meaningful. In the SP picture meaning is assigned to a measurement on a single particle.

Here it is argued that only the ensemble interpretation of the wavefunction is tenable. However, it should be made clear from the outset that "ensemble" refers to an ensemble of *N measurements*, not necessarily *N* particles. The initial conditions have to be identical and the wavefunction gives statistical information on the outcomes. The measurements can be simultaneous or sequential. In the case of *N* particles the particles must be indistinguishable. This specification of ensembles of measurements is necessary since, unthinkable to the founders of quantum mechanics, experiments today can be made on trapped single electrons, atoms or molecules. Then the wavefunction gives only statistical information on a sequence of measurements in which the same particle initially is brought into the same state e.g. experiments on quantum jumps.

Furthermore, if the wavefunction refers to an entangled state of several particles, then that group of particles is to be regarded as a single member (single "particle") of the ensemble. Correspondingly, a feature that is very important but has been often neglected in the past, is the occurrence or otherwise of many-particle *good quantum numbers* describing eigenstates of some many-particle mechanical variable. This is because, for this special case, the measurement of the corresponding many-particle mechanical variable gives the same sharp value for all members of the ensemble. For the simple case of single-particle ensembles in an eigenstate, there is no difference between the SP and ensemble pictures, although a repeated sequence of measurements is still required to confirm the eigenstate unless defined uniquely by the preparation.

The object of this work is to re-appraise, in the light of the SP and ensemble pictures, the transition from quantum to classical mechanics by emphasising the role of the "Imaging Theorem" (IT) [5–10] in determining what an observer perceives as a consequence of the experimental resolution, sensitivity and information extraction. The IT was proved as long ago as 1937 by Kemble [5] whose aim was to show how a particle linear momentum vector could be measured and assigned in a collision experiment. Although largely forgotten until recently, here a more fundamental consequence of this theorem with regard to the quantum to classical transition is suggested, following Ref. [6].

The IT shows that any system of particles emanating from microscopic separations describable by quantum dynamics will acquire characteristics of classical trajectories simply through unitary propagation to the macroscopic separations at which measurements are made. Specifically, the IT equates the final position wavefunction -(*rf* , *tf* ) at a detector at *macroscopic* position and time, to the initial momentum wavefunction -( ˜ *pi*, *ti*) at *microscopic* position and time but, importantly, where the variables *r*, *p* and *t* are related by a *classical trajectory*. This justifies the standard approach of experimentalists who use classical trajectories to trace the motion of particles from reaction zone to detector, even though the particle correlations indicate existence of a quantum wavefunction. In fact the relevance of great advances in multi-particle coincident detection [11] to the criteria for quantum or classical perception given here cannot be underestimated.

The IT connects the *momentum* wavefunction in the microscopic collision zone with the *position* wavefunction at macroscopic distance. The initial position in the collision zone cannot be defined precisely but, for free motion, the initial momentum is conserved out to the detector. This feature of the IT is completely in correspondence with the supposition of Schmidt-Böcking et.al. in Chap. 12 of this volume [12], who demonstrate that the momentum of particles emanating from a microscopic collision can be determined with sub-atomic precision but the initial position can never be measured with comparable precision.

As emphasised in the chapter by Schmidt-Böcking et al. the ensemble picture is essential for the correct interpretation of the uncertainty relation (UR) for position and momentum. According to the Robertson formulation of the UR [13] the statedependent spread in the measurement of position and momentum refer to ensemble averages. Therefore, by definition, the individual measurements must have an *accuracy* much less than the spread. The fact that, in an individual single measurement, the product of accuracies of simultaneous momentum and position measurements can be less than is demonstrated convincingly in Chap. 12.

The IT defines the asymptotic wavefunction of any quantum complex after interacting with particle or photon beams in a microscopic collision region. Such collisions *always* result in entangled many-particle wavefunctions. Indeed, although the recognition of entanglement is usually attributed to Schrödinger, initially he considered only effective one-particle problems, the hydrogen atom [14], the harmonic oscillator and the rotor [15].

It was Max Born [16] who first treated the two-particle problem of an electron colliding with an atom (to explain the experiment of Franck and Hertz) and wrote down an entangled collision wavefunction. In the same paper Born gave the statistical interpretation of the wave function which is of course the whole basis of the ensemble picture. The theory of entanglement in collisions, involving both the continuous variables of position and momentum and the discrete variables of binding energy, spin and angular momentum has been developed with great sophistication and in countless works in atomic, molecular and nuclear physics, since Born's pioneering paper, see for example [17–20].

The subject of many-particle entanglement has enjoyed enormously renewed interest in the last few years in the fields of quantum information and quantum computing, for example see Ref. [21]. However, this development, often with the limited purview of entangled photon states [22], has been made largely without reference to the study of entanglement in collision physics.

On the basis of the IT, it emerges that whether one ascribes (a) classical dynamics (a single trajectory analogous to a light ray), (b) a quantum wave description of the ensemble as a whole or (c) single particles registered separately whose statistical distribution corresponds to a wave, to the movement of material particles depends upon the precision and extent to which the dynamical variables of position and momentum are determined by the measurement. This is equally true for ensembles of many-particle systems involving entangled wavefunctions as it is for ensembles described by single particle wavefunctions.

The elimination of the overtly quantum effects of entanglement and coherence as a prerequisite for the transition to classical mechanics has been ascribed to interaction with the environment [23–26]. It goes under the broad name of "decoherence theory" (DT). This theory is part of the wider study of open quantum systems and these approaches usually involve propagation of the quantum density matrix in time.

The principal feature of such models is that the interaction with an environment leads to a suppression of the off-diagonal elements of the density matrix, which is considered a key element of the transition to classical behaviour.

Without doubt DT can explain many features of the quantum to classical transition but, according to DT, unitary evolution *in the system Hamiltonian alone* does not contribute to this transition. One main aim of the present work is to show that this is not the case.

Generically, from the result of the IT, a quantum system wavefunction or corresponding density matrix propagating in time without environmental interaction will develop such that the position and momentum coordinates change according to classical mechanics. In particular, the off-diagonal density matrix elements acquire oscillatory phase factors such that, except under a high-resolution measurement, they average to zero. In this sense, the IT does not negate any predictions of DT, rather it is complementary to it. However the unitary propagation transition occurs over time and position increments which are still of atomic dimensions and thus largely obviate any additional changes to the density matrix ascribable to the environment.

An exhaustive discussion of DT with an appraisal of its notable successes but also its limitations is given in the reviews of Schlosshauer [25, 26]. It is clear that this theory is anchored firmly in the SP interpretation of the wavefunction since wavefunction collapse, or apparent collapse, plays a prominent role.

In the present work only continuum quantum states are considered of relevance in the transition to classical mechanics. Bound states and quantised internal degrees of freedom (e.g. intrinsic spin) are viewed as wholly quantum features. They are not affected by the unitary propagation according to the IT of particles as a whole. However, the quantised angular momentum may be used to affect the classical trajectories [27] exactly as in the original Stern-Gerlach experiment.

Here, there is no discussion of the measurement process itself. In the particle detectors considered in this work the quantum particle is intercepted by a macroscopic detector involving an enormous number of atomic degrees of freedom, giving a completely irreversible transformation. The particle energy is absorbed through ionisation or photon emission in the detector and amplified to give a recorded signal. Such measurements are often called "strong" measurements. Hence, here we do not consider so-called "weak" measurements, usually performed on light [28], which involve additional manipulation of the wavefunction e.g. change of polarisation state, during transit.

#### **2 Interpretation of the Wavefunction**

Here a simple but sufficient interpretation of the wavefunction is applied. This involves the minimum of supposition required to explain modern multi-hit coincident detection of particles emanating from complexes of atomic dimension. The following rules are adopted in connection with the detection of moving particles.

Consequent on this ensemble view, the popular expression that a particle can also behave as a wave is redundant.What is detected is always a particle. The wavefunction simply assigns a probability amplitude that a particle from an ensemble of identical particles will be detected to have particular values of the dynamical variables.

As will be shown in the following, the IT provides many of the features of wavefunction propagation ascribed to decoherence due to environmental interaction. However, since the propagation is unitary, classical features emerge autonomously.

Wavefunction "collapse" is a widely-accepted aspect of quantum mechanics. This concept is peculiar to the SP picture. In the ensemble picture the need to invoke collapse of the wavefunction does not arise.

#### **3 The Imaging Theorem**

The result known as the imaging theorem can be expressed in a few equations. Details of the original proof for free asymptotic motion can be found in the book of Kemble [5] and its generalisation for arbitrary motion, e.g. in external electromagnetic fields, in Ref. [6].

The propagation in time of a localised quantum state defined at time *t* can be written

$$\mid \Psi(t) \rangle = U(t, t') \mid \Psi(t') \rangle,\tag{1}$$

where*U*(*t*, *t* - )is the time-development operator. Projecting this equation into position space gives

$$
\Psi(\mathbf{r},t) = \int K(\mathbf{r},t;\mathbf{r}',t')\,\Psi(\mathbf{r}',t')\,d\mathbf{r}',\tag{2}
$$

where the function *K*(*r*, *t*;*r*- , *t* - ) ≡ *r*| *U*(*t*, *t* - )|*r*- is called the space-time propagator. The IT rests on the asymptotic large *r*, large *t* limit when the action becomes much greater than and the propagator can be approximated by its semi-classical form [29]

$$K(\mathbf{r}, t; \mathbf{r}', 0) = \frac{1}{(2\pi i\hbar)^{3/2}} \left| \det \frac{\partial^2 S}{\partial \mathbf{r} \partial \mathbf{r}'} \right|^{1/2} e^{i\mathbf{S}(\mathbf{r}, t; \mathbf{r}', 0)/\hbar},\tag{3}$$

where *S*(*r*, *t*;*r*- , 0) is the *classical* action function in coordinate space and the initial time *t* is taken as the zero of time.

Now it is recognised that the *r* integral is confined to a small volume, of atomic dimensions, around *r*-≈ 0, so that the action can be expanded around this point as

$$S(\mathbf{r}, t; \mathbf{r}', 0) \approx S(\mathbf{r}, t; 0, 0) + \frac{\partial S}{\partial \mathbf{r}'}\Big|\_{0} \cdot \mathbf{r}'.\tag{4}$$

Then, using the classical relationship ∂*S*/∂*r*- |0 ≡ −*p*, substitution in the integral Eq. (2) gives a Fourier transform and the result

$$
\Psi(\mathbf{r},t) \approx \left(i\right)^{-3/2} \left(\frac{d\mathbf{p}}{d\mathbf{r}}\right)^{1/2} e^{iS(\mathbf{r},t;0,0)/\hbar} \tilde{\Psi}(\mathbf{p},0),\tag{5}
$$

which is the IT of Kemble, here generalised to arbitrary classical motion.

One notes that the IT rests upon two approximations. The first is the semi-classical approximation of *K* in Eq. (2). However, in the integral over *r*- , all possible values of *r* contribute to the asymptotic wavefunction at *r*, *t*. It is the recognition that the quantum wavefunction at time zero is limited to a microscopic extent, Eq. (4), that associates a fixed classical momentum *p* to each final coordinate *r*(*t*). That is, each initial [(*r*- ≈ 0), *p*] value is connected to a fixed final *r*, *t* by a classical trajectory. For free motion the connection is simply *r* = *p t*/*m*, where *m* is the particle mass.

The essence of the IT result is that the position and momentum coordinates evolve classically but within the shroud of the quantum wave functions. Indeed, one can show [8] that the asymptotic position wave function of Eq. (5) can be viewed as an eigenfunction of both the position and momentum operators. Hence, to the accuracy of the IT, these operators commute and, as emphasised in Ref. [12], there is no obstacle in measuring both momentum and position with arbitrary accuracy.

Then the probability density for detection of a particle of the ensemble is given by

$$\left|\Psi(\mathbf{r}(t))\right|^2 \approx \frac{d\mathbf{p}}{d\mathbf{r}} \left|\tilde{\Psi}(\mathbf{p}, 0)\right|^2. \tag{6}$$