The soul of the Manhattan Project, the most powerful problem solver of the 20th century [Part 1]

Bethe was a master in the field of physics and astrophysics. He made outstanding contributions to quantum mechanics, solid-state physics, nuclear physics, astrophysics, quantum electrodynamics and particle physics, and was awarded the 1967 Nobel Prize in Physics and many other awards. In the Manhattan Project, Bethe, as the director of the Theoretical Department, led a group of outstanding physicists to solve many key problems in the process of manufacturing the atomic bomb. Dyson called Bethe "the most powerful problem solver of the 20th century" because of his superb computing ability, numerous contributions and wide range of interests.

Written by | Wang Shanqin

When people talk about the Manhattan Project, the first person they think of is probably J. Robert Oppenheimer (1904-1967), known as the "Father of the Atomic Bomb." In fact, Oppenheimer did not participate in the research on the principles of atomic bomb explosions and the specific complex calculations. This task was undertaken by the Manhattan Project's Theoretical Department, which is the most important department in the Los Alamos Laboratory.

Hans Albrecht Bethe (1906-2005), director of the Theoretical Department, was at the top of this intellectual pyramid. Under his leadership, the Theoretical Department overcame various difficulties, solved many important theoretical problems in the development of the atomic bomb, and ensured the success of the project.

Bette. Image credit: Los Alamos National Laboratory

Bethe was also an outstanding master of physics and astrophysics. He was appointed director of the Manhattan Project's theoretical department because he was already a leading figure in the field of nuclear physics in the United States. His scientific research career lasted for at least 70 years, of which at least 50 years were at its peak. During this period, he moved to different fields and achieved important and even epoch-making results, showing extraordinary talent, diligence and creativity. Due to his systematic and in-depth study of the nuclear fusion process inside main sequence stars (including the sun), he won the 1967 Nobel Prize in Physics for "his contributions to the theory of nuclear reactions, especially his discoveries about the generation of energy in stars."

This article introduces Bethe's life and scientific contributions.

The pride of an academic family

On July 2, 1906, Bethe was born in Strasbourg, which was then part of Germany (now part of France).

Bethe's father, Albrecht Julius Theodor Bethe (1872-1954), was a physiologist who studied the nervous systems of invertebrates. Albrecht received his doctorate in philosophy from the University of Munich in 1895, worked at the Institute of Physiology in Strasbourg from 1896 to 1911, and received his doctorate in medicine from there in 1898.[1]

Bethe's maternal grandfather, Abraham Kuhn (1838-1900), was a professor at the University of Strasbourg. His daughter, Anna Kuhn (1876-1966), changed her name to Anna Bethe-Kuhn after marrying Albrecht. When Bethe was born, his maternal grandfather had passed away.

In 1911, Albrecht became a professor and director of the Institute of Physiology at the University of Kiel. In 1915, Albrecht became director of the Institute of Physiology at the University of Frankfurt [2]. These two appointments caused the Bethe family to move twice, and Bethe went to several schools to study.

Bette with her parents when she was 12 years old. Image source: Public copyright

In 1924, Bethe graduated from high school and entered the University of Frankfurt to study for a degree in chemistry. It turned out that Bethe was not suitable for this major because his experimental skills were poor and he made mistakes repeatedly. The worst case was that he spilled sulfuric acid on his lab coat. In this regard, he and his later friend Oppenheimer shared the same problem.

In April 1926, at the suggestion of his teacher, Bethe transferred to the University of Munich to study under the famous theoretical physicist Arnold Sommerfeld (1868-1951). Bethe, who was good at theoretical research, felt at home from then on. Sommerfeld suggested that Bethe use electron diffraction in crystals as a research topic, and he thus entered the field of solid-state physics.

In 1928, 22-year-old Bethe received his doctorate and joined the Technical University of Stuttgart the following year.

Young and promising

In 1929, Bethe published several papers on topics including the symmetry of hydrogen atom electron energy, the electron distribution of helium gas, and crystal separation, which involved quantum mechanics and solid-state physics. With Sommerfeld's recommendation, Bethe received a Travelling Scholarship from the Rockefeller Foundation, which paid $150 per month (equivalent to about $2,765 in 2023).

In 1930, Bethe published a 76-page paper entitled "The Theory of the Transmission of High-Speed ​​Particle Rays through Matter"[3]. Starting from the Schrödinger equation, this paper used Fourier transform to obtain the famous "Bethe formula". This formula describes the average energy loss of particles when they pass through a medium. Bethe later considered this the greatest paper he had ever written (no "one of the greatest"), when he was only 24 years old. This paper has been cited more than 6,000 times to date.

In the same year, Bethe used a scholarship to visit the Cavendish Laboratory at Cambridge University and worked as a postdoctoral fellow of Ralph Fowler (1889-1944). Patrick Blackett (1897-1974) hoped that he could generalize the "Bethe formula" to the relativistic case to describe extremely high-speed particles. Bethe met Blackett's wish and wrote the generalized formula into the paper "Relativistic Electron Deceleration Formula" [4], which was published in 1932.

While at Cambridge University, Bethe collaborated with young people in the same laboratory to fabricate a prank "paper" for the editorial department [5]. This "paper" calculated the fine structure constant at absolute zero in degrees Celsius to mock some physicists at the time for making up physics constants. The master of astrophysics Arthur Eddington (1882-1944) once used some numbers to make up the value of the fine structure constant. (Editor's note: See "He is a master of astrophysics, but also a stumbling block to the development of the discipline?") Bethe and others later apologized. [6]

As planned, Bethe used the remaining half of his scholarship to visit Enrico Fermi (1901-1954)'s physics laboratory at the University of Rome. Fermi's extraordinary intelligence impressed Bethe, and he felt that he had met him too late. On the other hand, Bethe was also considered the most outstanding person to visit Fermilab. Bethe inherited the strict style from Sommerfeld and the concise style from Fermi.

"Bate's hypothesis"

In March 1931, Bethe published his early representative work, “On the Theory of Metals. I. Eigenvalues ​​and Eigenfunctions of Linear Atomic Chains” [7]. This paper proposed the famous “Bethe ansatz”, which is used to accurately calculate the one-dimensional quantum many-body model problem and find the exact eigenvalues ​​and eigenfunctions of the wave function of certain quantum many-body models. To date, this paper has been cited more than 4,700 times. One of the topics to be learned written on the blackboard by the great physicist Richard Feynman (1918-1988) before his death was “Bethe ansatz problems”. When Bethe published this paper, he was not yet 25 years old.

During his visit to Rome, Bethe also collaborated with Fermi on the study of quantum electrodynamics (QED). QED is a branch of physics that describes the interaction between electrons/positrons (matter) and photons (radiation). Bethe and Fermi collaborated on a paper in the field of QED, entitled "The Interaction of Two Electrons"[8], which was published in 1932.

Bethe also wrote two reviews in 1932. The first was about the quantum mechanics of hydrogen and helium, and the second was about electrons in metals. In 1959, Robert Bacher (1905-2004) and Victor Weisskopf (1908-2002) carefully read Bethe's review on quantum mechanics in order to republish it, and found that it was profound and only needed minimal updating for reprinting.

Bethe-Heitler formula

After finishing his study tour, Bethe returned to Germany and became an assistant professor at the University of Tübingen in 1932. However, Nazi Germany soon began to discriminate against Jews. Because Bethe's mother was half Jewish, he was also implicated and fired from the university. With the help of British physicist William Lawrence Bragg (1890-1971), Bethe obtained a one-year lectureship at the University of Manchester in 1933 and quickly went to the UK.

While in the UK, Bethe became friends with Rudolf Peierls (1907-1995), who was also a German and had fled Germany because of his Jewish ancestry. Under his influence, Bethe began to study nuclear physics. Later, Peierls became the head of the British atomic bomb project (the "Alloy Tube Project") and met Bethe again in the late World War II to collaborate on the manufacture of atomic bombs.

Due to his outstanding academic ability, Bate was soon hired by the University of Bristol and Cornell University. Cornell University allowed Bate to join the university after completing his contract at the University of Bristol.

In 1934, Bethe and Walter Heinrich Heitler (1904-1981) jointly published a paper entitled "On the Stopping of Fast Particles and the Production of Positrons"[9], which studied the scattering of photons by atoms and molecules and the process of photon annihilation into electron-positron pairs. This paper proposed the famous "Bethe-Heitler formula". This classic article has been cited more than 2,500 times.

"Bet Bible"

In February 1935, Bethe joined Cornell University, where he did a flourishing research and became friends with Edward Teller (1908-2003) and others.

Between 1936 and 1937, Bethe published three major papers in the field of nuclear physics. The first paper was co-authored with Bacher (the second author) and discussed the stability of the atomic nucleus[10]; the second paper was written by Bethe alone and discussed the theory of nuclear dynamics[11]; the third paper was co-authored with Livingston (Milton Livingston, 1905-1986, the first author) and discussed the experiment of nuclear dynamics[12].

These three papers have a high status in the field of nuclear physics and were called "Bethe's Bible" by some scholars at the time.

In a letter to his mother, an exuberant Bethe wrote: "I am one of the leading theorists in the United States. This does not mean that I am the best. Wigner [Eugene Wigner, 1902-1995] is certainly better, and Oppenheimer and Taylor are probably as good as him. But I have done more and said more, and that is also important."[6]

In 1937, Bethe met Rose Ewald (1917-2019) while lecturing at Duke University. She also fled to the United States because of persecution in Nazi Germany. Rose's father, Paul Ewald (1888-1985), was a famous crystallographer and physicist, and a pioneer of X-ray diffraction; his doctoral supervisor was also Sommerfeld, so he was Bethe's senior. Because of this relationship, Rose met Bethe in Germany when she was a teenager. After meeting at Duke University, the two became lovers and got married in September 1939.

Bette's wife Ewald (1967). Image source: Public domain

Prometheus stole fire: Unraveling the mystery of stellar energy

As early as 1920, Eddington pointed out in his paper that the energy of stars during most of their lifetimes does not come from the contraction of stars, but from the fusion of hydrogen nuclei (protons). However, Eddington did not give the specific process of hydrogen fusion into helium.

In 1937, George Gamow (1904-1968) and Carl von Weizsäcker (1912-2007) proposed that protons in the core of the sun fuse into helium through a "proton-proton chain" (pp chain) reaction, releasing energy. In addition, Weizsäcker also proposed the carbon-nitrogen-oxygen (CNO) cycle process in 1937 and 1938. However, these works have not yet given some important specific processes.

The basis of the pp chain is the reaction of proton fusion into deuterium (D), namely the pp reaction: two protons fuse into deuterium, releasing a positron and a neutrino at the same time. Weizsäcker suggested that Bethe study the pp reaction. Almost at the same time, Gamow also asked his student Charles Critchfield (1910-1994) to calculate the pp reaction. The latter completed this calculation in early 1938, and Gamow suggested that this paper be sent to Bethe for review, because Bethe had done a lot of calculations on dinuclear reactions [13]. Bethe confirmed that Critchfield's calculations were correct. The two therefore collaborated on the paper "Proton Combination to Form Deuterium". [14]

The process calculated by Bethe and Critchfield is as follows: two protons combine to form deuterium, deuterium combines with a proton to form helium-3, helium-3 and helium-4 combine to form beryllium-7, beryllium-7 decays to form lithium-7, and lithium-7 combines with a proton to form two helium-4s.

Later studies showed that there are four types of pp chains. Bethe and Critchfield calculated what is now called type II pp chain. The core temperature of the sun is 15.7 million K, and the main mode of hydrogen fusion in its core is type I pp chain, which contributes 81.6% of the sun's energy; type II pp chain contributes 16% of the sun's energy. Although they did not consider other types of pp chains, their calculations on type II pp chain are important and outstanding enough.

What troubled the two was that if the temperature of the sun's core was 40 million K as Eddington had previously estimated, after substituting this value into the calculation, the brightness obtained would far exceed the observed brightness of the sun.

On March 17, 1938, Bethe was invited to attend the Fourth Annual Conference on Theoretical Physics in Washington, D.C., hosted by Gamow and Taylor. The theme of the conference was "The Generation of Stellar Energy." Bethe did not want to accept the invitation because his interest was still in QED. However, under Taylor's persuasion, he still attended the conference. [13]

At this conference, Bengt Strömgren (1908-1987) announced that based on his analysis and calculation of the chemical composition of the sun, the temperature of the sun's core is about 15 million K, not 40 million K. After substituting 15 million K into Bethe and Critchfield's calculations, the resulting solar brightness was consistent with the observed brightness. This was an encouragement to Bethe and others.

After the meeting, Bethe thought about nuclear reactions in more massive stars. The more massive a star is, the hotter its core is, and the higher the internal energy rate is. Bethe knew that among the elements heavier than helium-4, lithium, beryllium, and boron were too rare, so he thought that carbon was a possible starting point for the reaction.[13]

After two weeks of thinking and calculations[13], Bethe rediscovered the CNO cycle reaction. The cycle process discovered by Bethe is: carbon-12→nitrogen-13→carbon-13→nitrogen-14→oxygen-15→nitrogen-15→carbon-12. In the whole process, carbon, nitrogen, and oxygen act as catalysts and are not consumed themselves.

Type I CNO cycle process. In the figure, H, He, C, N, O, ν and γ are hydrogen, helium, carbon, nitrogen, oxygen, neutrinos and gamma photons respectively. Image source: Borb

After that, experimental physicists bombarded carbon-12 targets with high-speed protons and soon found evidence of nitrogen-13 decay. This proved that Bethe's calculations were correct. Later studies showed that there are multiple channels for hydrogen in stars to fuse into helium through the CNO cycle process. Both Weizsäcker and Bethe discovered the Type I CNO cycle process, which is therefore called the "Bethe-Weizsäcker cycle".

Bethe wrote his research results in a paper titled “The Generation of Stellar Energy”[15]. In this paper, Bethe further carefully calculated the reaction rate of the pp chain and pointed out that the internal energy of smaller stars, such as the sun, mainly comes from the pp chain reaction; the internal energy of massive stars mainly comes from the CNO cycle. This conclusion is still correct today.

Bethe's paper gives the relationship between the energy production rate of the two energy production modes and temperature (unit: 1 million K). The dotted line is the pp chain, the dashed line is the CNO cycle, and the solid line is the sum of the two. When the core temperature of the star is lower than 15 million K, the pp chain contributes most of the energy; conversely, the CNO cycle contributes most of the energy. Image source: Reference [15]

Bethe submitted "The Generation of Stellar Energy" to Physical Review. Soon after, Bethe's doctoral student Robert Marshak (1916-1992) noticed that the New York Academy of Sciences was offering a $500 reward (equivalent to $10,915 in 2023) for the best paper on solar and stellar energy, provided that the paper had not yet been published. [13]

Marshak immediately told Bethe the news. Bethe quickly withdrew his paper and sent it to the New York Academy of Sciences, winning the $500 prize. He gave Marshak $50 for the information. He then sent $250 to the German government to ensure that all of his mother's belongings would be properly taken care of when she was about to flee Germany. [13]

Finally, this epoch-making paper was resubmitted by Bethe to Physical Review and published in March 1939. The results of this paper apply not only to the sun, but also to all stars in the main sequence stage (stars in the state of core hydrogen fusion). Stars spend most of their lives in the main sequence stage.

The man behind the Manhattan Project

After the outbreak of World War II in Europe, many scholars began to devote themselves to the subjects related to weapon design. Bethe was no exception. He worked with Taylor to study the theory of shock waves when a warhead passes through a gas. He also studied the armor penetration theory, but the theory was immediately classified by the military, so Bethe, who had not yet become a US citizen, could not get involved further.

In March 1941, Bethe obtained American citizenship, which removed the biggest obstacle for him to engage in military research. In December 1941, Bethe finally obtained security clearance and joined the Radiation Laboratory of the Massachusetts Institute of Technology. There, he invented the "Bethe-hole directional coupler" that can be used in radar groups.

After the Manhattan Project was officially launched, Oppenheimer was appointed as the scientific director, coordinating all departments. Among these departments, the theoretical department was responsible for theoretical calculations and determining the feasibility of various plans, so it was the most critical department. Oppenheimer wanted to serve as the director of the theoretical department himself.

However, when Oppenheimer asked his good friend Isidor Rabi (1898-1988) for his opinion on the Manhattan Project, Rabi gave two suggestions: do not wear military uniforms; invite Bethe to be the director of the theoretical department. Although Oppenheimer was unruly, he was respectful to Rabi and obeyed his advice. Moreover, he knew that although Bethe was still very young (35 years old), he was already a leader in the field of nuclear physics. Therefore, Bethe was invited to be the director of the theoretical department.

A photo of Bethe during the Manhattan Project, with ID number K3. Image source: Los Alamos National Laboratory

After taking office, Bethe led the members of the theoretical department to calculate key issues such as the critical mass of uranium 235 (the minimum mass that allows a chain reaction to proceed), efficiency, fission proliferation, fluid dynamics of the explosion, neutron initiators, and radiation propagation of the explosion. He also developed a formula for calculating the explosive equivalent of an atomic bomb with Feynman, a member of the theoretical group. [16]

At the critical moment of intense research and development, Bethe played a role in stabilizing the morale of the troops: Taylor calculated that a nuclear explosion would cause nitrogen in the earth's atmosphere to fuse into magnesium and release helium ions, releasing huge energy to burn the atmosphere; Bethe immediately determined that this calculation was wrong. Then he proved his judgment through rigorous calculations and pointed out that Taylor's calculation was based on a wrong assumption. Bethe's calculations provided Oppenheimer with enough confidence. (This is also a scene in Nolan's movie "Oppenheimer".)

Due to the Manhattan Project, Bethe's research in the field of pure science was greatly reduced. In 1944, he seemed to have more time and published a paper on the diffraction of electromagnetic waves by circular holes[17], which made a new and in-depth study of the ancient diffraction problem. This article has been cited more than 3,700 times to date.

On July 16, 1945, participants in the Manhattan Project conducted the first nuclear test in human history, the Trinity test, and the world's first atomic bomb was successfully detonated. The theoretical team led by Bethe made a crucial contribution to its success. Various data measured after the explosion verified the accuracy of the theoretical department's calculation results.

During the implementation of the Manhattan Project, the Theoretical Department was the department with the least cost and the highest reputation. As the director of the Theoretical Department, Bethe's role was no less than that of Oppenheimer, who coordinated the overall situation. Facts also proved Rabi's vision: Bethe not only had outstanding physics talent, but also had excellent team leadership ability. It can be said that Bethe was the soul of the Manhattan Project.

This article is supported by the Science Popularization China Starry Sky Project

Produced by: China Association for Science and Technology Department of Science Popularization

Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd.