Arthur Compton

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Arthur Compton
Arthur Compton in 1927
Born	Arthur Holly Compton September 10, 1892 Wooster, Ohio, United States
Died	March 15, 1962 (aged 69) Berkeley, California, United States
Nationality	American
Alma mater	College of Wooster Princeton University
Known for	Compton scattering Compton wavelength
Spouse(s)	Betty Charity McCloskey (d. 1980)
Children	Arthur Alan John Joseph
Awards	Nobel Prize for Physics (1927) Matteucci Medal (1930) Franklin Medal (1940) Hughes Medal (1940) Medal for Merit (1946)
Scientific career
Fields	Physics
Institutions	Washington University in St. Louis University of Chicago University of Minnesota
Doctoral advisor	Hereward L. Cooke
Doctoral students	Luis Walter Alvarez Winston H. Bostick Robert S. Shankland Wu Youxun
Signature
Notes
Brother of Wilson Compton and Karl Compton

Arthur Holly Compton (September 10, 1892 – March 15, 1962) was an American physicist who won the Nobel Prize in Physics in 1927 for his 1923 discovery of the Compton effect, which demonstrated the particle nature of electromagnetic radiation. It was a sensational discovery at the time: the wave nature of light had been well-demonstrated, but the idea that light had both wave and particle properties was not easily accepted. He is also known for his leadership of the Manhattan Project's Metallurgical Laboratory, and served as Chancellor of Washington University in St. Louis from 1945 to 1953.

In 1919, Compton was awarded one of the first two National Research Council Fellowships that allowed students to study abroad. He chose to go to Cambridge University's Cavendish Laboratory in England, where he studied the scattering and absorption of gamma rays. Further research along these lines led to the discovery of the Compton effect. He used X-rays to investigate ferromagnetism, concluding that it was a result of the alignment of electron spins, and studied cosmic rays, discovering that they were made up principally of positively charged particles.

During World War II, Compton was a key figure in the Manhattan Project that developed the first nuclear weapons. His reports were important in launching the project. In 1942, he became head of the Metallurgical Laboratory, with responsibility for producing nuclear reactors to convert uranium into plutonium, finding ways to separate the plutonium from the uranium and to design an atomic bomb. Compton oversaw Enrico Fermi's creation of Chicago Pile-1, the first nuclear reactor, which went critical on December 2, 1942. The Metallurgical Laboratory was also responsible for the design and operation of the X-10 Graphite Reactor at Oak Ridge, Tennessee. Plutonium began being produced in the Hanford Site reactors in 1945.

After the war, Compton became Chancellor of Washington University in St. Louis. During his tenure, the university formally desegregated its undergraduate divisions, named its first female full professor, and enrolled a record number of students after wartime veterans returned to the United States.

Early life

Arthur Compton and Werner Heisenberg in 1929 in Chicago

Arthur Compton was born on September 10, 1892, in Wooster, Ohio, the son of Elias and Otelia Catherine (née Augspurger) Compton, who was named American Mother of the Year in 1939. They were an academic family. Elias was dean of the University of Wooster (later The College of Wooster), which Arthur also attended. Arthur's eldest brother, Karl, who also attended Wooster, earned a PhD in physics from Princeton University in 1912, and was president of MIT from 1930 to 1948. His second brother Wilson likewise attended Wooster, earned his PhD in economics from Princeton in 1916 and was president of the State College of Washington, later Washington State University from 1944 to 1951. All three brothers were members of the Alpha Tau Omega fraternity.

Compton was initially interested in astronomy, and took a photograph of Halley's Comet in 1910. Around 1913, he described an experiment where an examination of the motion of water in a circular tube demonstrated the rotation of the earth. That year, he graduated from Wooster with a Bachelor of Science degree and entered Princeton, where he received his Master of Arts degree in 1914. Compton then studied for his PhD in physics under the supervision of Hereward L. Cooke, writing his dissertation on "The intensity of X-ray reflection, and the distribution of the electrons in atoms".

When Arthur Compton earned his PhD in 1916, he, Karl and Wilson became the first group of three brothers to earn PhDs from Princeton. Later, they would become the first such trio to simultaneously head American colleges. Their sister Mary married a missionary, C. Herbert Rice, who became the principal of Forman Christian College in Lahore. In June 1916, Compton married Betty Charity McCloskey, a Wooster classmate and fellow graduate. They had two sons, Arthur Alan and John Joseph Compton.

Compton spent a year as a physics instructor at the University of Minnesota in 1916–17, then two years as a research engineer with the Westinghouse Lamp Company in Pittsburgh, where he worked on the development of the sodium-vapor lamp. During World War I he developed aircraft instrumentation for the Signal Corps.

In 1919, Compton was awarded one of the first two National Research Council Fellowships that allowed students to study abroad. He chose to go to Cambridge University's Cavendish Laboratory in England. Working with George Paget Thomson, the son of J. J. Thomson, Compton studied the scattering and absorption of gamma rays. He observed that the scattered rays were more easily absorbed than the original source. Compton was greatly impressed by the Cavendish scientists, especially Ernest Rutherford, Charles Galton Darwin and Arthur Eddington, and he ultimately named his second son after J. J. Thomson.

For a time Compton was a deacon at a Baptist church. "Science can have no quarrel", he said, "with a religion which postulates a God to whom men are as His children."

Career

Compton on the cover of Time Magazine on January 13, 1936, holding his cosmic ray detector

Compton effect

Returning to the United States, Compton was appointed Wayman Crow Professor of Physics, and Head of the Department of Physics at Washington University in St. Louis in 1920. In 1922, he found that X-ray quanta scattered by free electrons had longer wavelengths and, in accordance with Planck's relation, less energy than the incoming X-rays, the surplus energy having been transferred to the electrons. This discovery, known as the "Compton effect" or "Compton scattering", demonstrated the particle concept of electromagnetic radiation.

In 1923, Compton published a paper in the Physical Review that explained the X-ray shift by attributing particle-like momentum to photons, something Einstein had invoked for his 1905 Nobel Prize–winning explanation of the photo-electric effect. First postulated by Max Planck in 1900, these were conceptualized as elements of light "quantized" by containing a specific amount of energy depending only on the frequency of the light. In his paper, Compton derived the mathematical relationship between the shift in wavelength and the scattering angle of the X-rays by assuming that each scattered X-ray photon interacted with only one electron. His paper concludes by reporting on experiments that verified his derived relation:

${\displaystyle \lambda '-\lambda ={\frac {h}{m_{e}c}}(1-\cos {\theta }),}$

where

${\displaystyle \lambda }$ is the initial wavelength,

${\displaystyle \lambda '}$ is the wavelength after scattering,

${\displaystyle h}$ is the Planck constant,

${\displaystyle m_{e}}$ is the electron rest mass,

${\displaystyle c}$ is the speed of light, and

${\displaystyle \theta }$ is the scattering angle.

The quantity ​^h⁄_mec is known as the Compton wavelength of the electron; it is equal to 2.43×10⁻¹² m. The wavelength shift λ′ − λ lies between zero (for θ = 0°) and twice the Compton wavelength of the electron (for θ = 180°). He found that some X-rays experienced no wavelength shift despite being scattered through large angles; in each of these cases the photon failed to eject an electron. Thus the magnitude of the shift is related not to the Compton wavelength of the electron, but to the Compton wavelength of the entire atom, which can be upwards of 10,000 times smaller.

"When I presented my results at a meeting of the American Physical Society in 1923," Compton later recalled, "it initiated the most hotly contested scientific controversy that I have ever known." The wave nature of light had been well demonstrated, and the idea that it could have a dual nature was not easily accepted. It was particularly telling that diffraction in a crystal lattice could only be explained with reference to its wave nature. It earned Compton the Nobel Prize in Physics in 1927. Compton and Alfred W. Simon developed the method for observing at the same instant individual scattered X-ray photons and the recoil electrons. In Germany, Walther Bothe and Hans Geiger independently developed a similar method.

X-rays

Compton at the University of Chicago in 1933 with graduate student Luis Alvarez next to his cosmic ray telescope.

In 1923, Compton moved to the University of Chicago as Professor of Physics, a position he would occupy for the next 22 years. In 1925, he demonstrated that the scattering of 130,000-volt X-rays from the first sixteen elements in the periodic table (hydrogen through sulfur) were polarized, a result predicted by J. J. Thomson. William Duane from Harvard University spearheaded an effort to prove that Compton's interpretation of the Compton effect was wrong. Duane carried out a series of experiments to disprove Compton, but instead found evidence that Compton was correct. In 1924, Duane conceded that this was the case.

Compton investigated the effect of X-rays on the sodium and chlorine nuclei in salt. He used X-rays to investigate ferromagnetism, concluding that it was a result of the alignment of electron spins. In 1926, he became a consultant for the Lamp Department at General Electric. In 1934, he returned to England as Eastman visiting professor at Oxford University. While there General Electric asked him to report on activities at General Electric Company plc's research laboratory at Wembley. Compton was intrigued by the possibilities of the research there into fluorescent lamps. His report prompted a research program in America that developed it.

Compton's first book, X-Rays and Electrons, was published in 1926. In it he showed how to calculate the densities of diffracting materials from their X-ray diffraction patterns. He revised his book with the help of Samuel K. Allison to produce X-Rays in Theory and Experiment (1935). This work remained a standard reference for the next three decades.

Cosmic rays

By the early 1930s, Compton had become interested in cosmic rays. At the time, their existence was known but their origin and nature remained speculative. Their presence could be detected using a spherical "bomb" containing compressed air or argon gas and measuring its electrical conductivity. Trips to Europe, India, Mexico, Peru and Australia gave Compton the opportunity to measure cosmic rays at different altitudes and latitudes. Along with other groups who made observations around the globe, they found that cosmic rays were 15 per cent more intense at the poles than at the equator. Compton attributed this to the effect of cosmic rays being made up principally of charged particles, rather than photons as Robert Millikan had suggested, with the latitude effect being due to Earth's magnetic field.

Manhattan Project

Arthur Compton's ID badge from the Hanford Site. For security reasons, he used a fake name.

In April 1941, Vannevar Bush, head of the wartime National Defense Research Committee (NDRC), created a special committee headed by Compton to report on the NDRC uranium program. Compton's report, which was submitted in May 1941, foresaw the prospects of developing radiological weapons, nuclear propulsion for ships, and nuclear weapons using uranium-235 or the recently discovered plutonium. In October he wrote another report on the practicality of an atomic bomb. For this report, he worked with Enrico Fermi on calculations of the critical mass of uranium-235, conservatively estimating it to be between 20 kilograms (44 lb) and 2 tonnes (2.0 long tons; 2.2 short tons). He also discussed the prospects for uranium enrichment with Harold Urey, spoke with Eugene Wigner about how plutonium might be produced in a nuclear reactor, and with Robert Serber about how the plutonium produced in a reactor might be separated from uranium. His report, submitted in November, stated that a bomb was feasible, although he was more conservative about its destructive power than Mark Oliphant and his British colleagues.

The final draft of Compton's November report made no mention of using plutonium, but after discussing the latest research with Ernest Lawrence, Compton became convinced that a plutonium bomb was also feasible. In December, Compton was placed in charge of the plutonium project. He hoped to achieve a controlled chain reaction by January 1943, and to have a bomb by January 1945. To tackle the problem, he had the different research groups working on plutonium and nuclear reactor design at Columbia University, Princeton University and the University of California, Berkeley, concentrated together as the Metallurgical Laboratory in Chicago. Its objectives were to produce reactors to convert uranium to plutonium, to find ways to chemically separate the plutonium from the uranium, and to design and build an atomic bomb.

In June 1942, the United States Army Corps of Engineers assumed control of the nuclear weapons program and Compton's Metallurgical Laboratory became part of the Manhattan Project. That month, Compton gave Robert Oppenheimer responsibility for bomb design. It fell to Compton to decide which of the different types of reactor designs that the Metallurgical Laboratory scientists had devised should be pursued, even though a successful reactor had not yet been built.

When labor disputes delayed construction of the Metallurgical Laboratory's new home in the Red Gate Woods, Compton decided to build Chicago Pile-1, the first nuclear reactor, under the stands at Stagg Field. Under Fermi's direction, it went critical on December 2, 1942. Compton arranged for Mallinckrodt to undertake the purification of uranium ore, and with DuPont to build the plutonium semi-works at Oak Ridge, Tennessee.

A major crisis for the plutonium program occurred in July 1943, when Emilio Segrè's group confirmed that plutonium created in the X-10 Graphite Reactor at Oak Ridge contained high levels of plutonium-240. Its spontaneous fission ruled out the use of plutonium in a gun-type nuclear weapon. Oppenheimer's Los Alamos Laboratory met the challenge by designing and building an implosion-type nuclear weapon.

Compton's house in Chicago, now a national landmark

Compton was at the Hanford site in September 1944 to watch the first reactor being brought online. The first batch of uranium slugs was fed into Reactor B at Hanford in November 1944, and shipments of plutonium to Los Alamos began in February 1945. Throughout the war, Compton would remain a prominent scientific adviser and administrator. In 1945, he served, along with Lawrence, Oppenheimer, and Fermi, on the Scientific Panel that recommended military use of the atomic bomb against Japan. He was awarded the Medal for Merit for his services to the Manhattan Project.

Return to Washington University

After the war ended, Compton resigned his chair as Charles H. Swift Distinguished Service Professor of Physics at the University of Chicago and returned to Washington University in St. Louis, where he was inaugurated as the university's ninth Chancellor in 1946. During Compton's time as Chancellor, the university formally desegregated its undergraduate divisions in 1952, named its first female full professor, and enrolled record numbers of students as wartime veterans returned to the United States. His reputation and connections in national scientific circles allowed him to recruit many nationally renowned scientific researchers to the university. Despite Compton's accomplishments, he was criticized then, and subsequently by historians, for moving too slowly toward full racial integration, making Washington University the last major institution of higher learning in St. Louis to open its doors to African Americans.

Compton retired as Chancellor in 1954, but remained on the faculty as Distinguished Service Professor of Natural Philosophy until his retirement from the full-time faculty in 1961. In retirement he wrote Atomic Quest, a personal account of his role in the Manhattan Project, which was published in 1956.

Philosophy

Compton was one of a handful of scientists and philosophers to propose a two-stage model of free will. Others include William James, Henri Poincaré, Karl Popper, Henry Margenau, and Daniel Dennett. In 1931, Compton championed the idea of human freedom based on quantum indeterminacy, and invented the notion of amplification of microscopic quantum events to bring chance into the macroscopic world. In his somewhat bizarre mechanism, he imagined sticks of dynamite attached to his amplifier, anticipating the Schrödinger's cat paradox, which was published in 1935.

Reacting to criticisms that his ideas made chance the direct cause of people's actions, Compton clarified the two-stage nature of his idea in an Atlantic Monthly article in 1955. First there is a range of random possible events, then one adds a determining factor in the act of choice.

A set of known physical conditions is not adequate to specify precisely what a forthcoming event will be. These conditions, insofar as they can be known, define instead a range of possible events from among which some particular event will occur. When one exercises freedom, by his act of choice he is himself adding a factor not supplied by the physical conditions and is thus himself determining what will occur. That he does so is known only to the person himself. From the outside one can see in his act only the working of physical law. It is the inner knowledge that he is in fact doing what he intends to do that tells the actor himself that he is free.

Death and legacy

The Compton Gamma Ray Observatory released into Earth's orbit in 1991

Compton died in Berkeley, California, from a cerebral hemorrhage on March 15, 1962. He was survived by his wife and sons, and buried in the Wooster Cemetery in Wooster, Ohio. Before his death, he was Professor-at-Large at the University of California, Berkeley for Spring 1962.

Compton received many awards in his lifetime, including the Nobel Prize for Physics in 1927, the Matteucci Gold Medal in 1933, the Royal Society's Hughes Medal and the Franklin Institute's Benjamin Franklin Medal in 1940. He is commemorated in various ways. The Compton crater on the Moon is co-named for Compton and his brother Karl. The physics research building at Washington University in St Louis is named in his honor, as is the university's top fellowship for undergraduate students studying math, physics, or planetary science. Compton invented a more gentle, elongated, and ramped version of the speed bump called the "Holly hump," many of which are on the roads of the Washington University campus. The University of Chicago Residence Halls remembered Compton and his achievements by dedicating Arthur H. Compton House in Chicago in his honor. It is now listed as a National Historic Landmark. Compton also has a star on the St. Louis Walk of Fame. NASA's Compton Gamma Ray Observatory was named in honor of Compton. The Compton effect is central to the gamma ray detection instruments aboard the observatory.

Bibliography

Compton, Arthur (1926). X-Rays and Electrons: An Outline of Recent X-Ray Theory. New York: D. Van Nostrand Company, Inc. OCLC 1871779.

Compton, Arthur; with Allison, S. K. (1935). X-Rays in Theory and Experiment. New York: D. Van Nostrand Company, Inc. OCLC 853654.

Compton, Arthur (1935). The Freedom of Man. New Haven: Yale University Press. OCLC 5723621.

Compton, Arthur (1940). The Human Meaning of Science. Chapel Hill: University of North Carolina Press. OCLC 311688.

Compton, Arthur (1949). Man's Destiny in Eternity. Boston: Beacon Press. OCLC 4739240.

Compton, Arthur (1956). Atomic Quest. New York: Oxford University Press. OCLC 173307.

Compton, Arthur (1967). Johnston, Marjorie, ed. The Cosmos of Arthur Holly Compton. New York: Alfred A. Knopf. OCLC 953130.

Compton, Arthur (1973). Shankland, Robert S., ed. Scientific Papers of Arthur Holly Compton. Chicago: University of Chicago Press. ISBN 978-0-226-11430-9. OCLC 962635.

Nuclear shell model

From Wikipedia, the free encyclopedia

In nuclear physics and nuclear chemistry, the nuclear shell model is a model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels. The first shell model was proposed by Dmitry Ivanenko (together with E. Gapon) in 1932. The model was developed in 1949 following independent work by several physicists, most notably Eugene Paul Wigner, Maria Goeppert Mayer and J. Hans D. Jensen, who shared the 1963 Nobel Prize in Physics for their contributions.

The shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a filled shell results in greater stability. When adding nucleons (protons or neutrons) to a nucleus, there are certain points where the binding energy of the next nucleon is significantly less than the last one. This observation, that there are certain magic numbers of nucleons: 2, 8, 20, 28, 50, 82, 126 which are more tightly bound than the next higher number, is the origin of the shell model.

The shells for protons and for neutrons are independent of each other. Therefore, one can have "magic nuclei" where one nucleon type or the other is at a magic number, and "doubly magic nuclei", where both are. Due to some variations in orbital filling, the upper magic numbers are 126 and, speculatively, 184 for neutrons but only 114 for protons, playing a role in the search for the so-called island of stability. Some semimagic numbers have been found, notably Z=40 giving nuclear shell filling for the various elements; 16 may also be a magic number.

In order to get these numbers, the nuclear shell model starts from an average potential with a shape something between the square well and the harmonic oscillator. To this potential a spin orbit term is added. Even so, the total perturbation does not coincide with experiment, and an empirical spin orbit coupling must be added with at least two or three different values of its coupling constant, depending on the nuclei being studied.

Nevertheless, the magic numbers of nucleons, as well as other properties, can be arrived at by approximating the model with a three-dimensional harmonic oscillator plus a spin-orbit interaction. A more realistic but also complicated potential is known as Woods–Saxon potential.

Deformed harmonic oscillator approximated model

Consider a three-dimensional harmonic oscillator. This would give, for example, in the first two levels ("l" is angular momentum)

level n	l	ml	ms
0	0	±0	+​1⁄2
			−​1⁄2
1	1	+1	+​1⁄2
			−​1⁄2
		±0	+​1⁄2
			−​1⁄2
		−1	+​1⁄2
			−​1⁄2

We can imagine ourselves building a nucleus by adding protons and neutrons. These will always fill the lowest available level. Thus the first two protons fill level zero, the next six protons fill level one, and so on. As with electrons in the periodic table, protons in the outermost shell will be relatively loosely bound to the nucleus if there are only few protons in that shell, because they are farthest from the center of the nucleus. Therefore, nuclei which have a full outer proton shell will have a higher binding energy than other nuclei with a similar total number of protons. All this is true for neutrons as well.

This means that the magic numbers are expected to be those in which all occupied shells are full. We see that for the first two numbers we get 2 (level 0 full) and 8 (levels 0 and 1 full), in accord with experiment. However the full set of magic numbers does not turn out correctly. These can be computed as follows:

In a three-dimensional harmonic oscillator the total degeneracy at level n is . 

Due to the spin, the degeneracy is doubled and is ${\displaystyle (n+1)(n+2)}$.
 
Thus the magic numbers would be

for all integer k. This gives the following magic numbers: 2, 8, 20, 40, 70, 112, ..., which agree with experiment only in the first three entries. These numbers are twice the tetrahedral numbers (1, 4, 10, 20, 35, 56, ...) from the Pascal Triangle.

In particular, the first six shells are:

• level 0: 2 states (l = 0) = 2.
• level 1: 6 states (l = 1) = 6.
• level 2: 2 states (l = 0) + 10 states (l = 2) = 12.
• level 3: 6 states (l = 1) + 14 states (l = 3) = 20.
• level 4: 2 states (l = 0) + 10 states (l = 2) + 18 states (l = 4) = 30.
• level 5: 6 states (l = 1) + 14 states (l = 3) + 22 states (l = 5) = 42.

where for every l there are 2l+1 different values of m_l and 2 values of m_s, giving a total of 4l+2 states for every specific level.

These numbers are twice the values of triangular numbers from the Pascal Triangle: 1, 3, 6, 10, 15, 21, ....

Including a spin-orbit interaction

We next include a spin-orbit interaction. First we have to describe the system by the quantum numbers j, m_j and parity instead of l, m_l and m_s, as in the hydrogen-like atom. Since every even level includes only even values of l, it includes only states of even (positive) parity. Similarly, every odd level includes only states of odd (negative) parity. Thus we can ignore parity in counting states. The first six shells, described by the new quantum numbers, are

• level 0 (n = 0): 2 states (j = ​¹⁄₂). Even parity.
• level 1 (n = 1): 2 states (j = ​¹⁄₂) + 4 states (j = ​³⁄₂) = 6. Odd parity.
• level 2 (n = 2): 2 states (j = ​¹⁄₂) + 4 states (j = ​³⁄₂) + 6 states (j = ​⁵⁄₂) = 12. Even parity.
• level 3 (n = 3): 2 states (j = ​¹⁄₂) + 4 states (j = ​³⁄₂) + 6 states (j = ​⁵⁄₂) + 8 states (j = ​⁷⁄₂) = 20. Odd parity.
• level 4 (n = 4): 2 states (j = ​¹⁄₂) + 4 states (j = ​³⁄₂) + 6 states (j = ​⁵⁄₂) + 8 states (j = ​⁷⁄₂) + 10 states (j = ​⁹⁄₂) = 30. Even parity.
• level 5 (n = 5): 2 states (j = ​¹⁄₂) + 4 states (j = ​³⁄₂) + 6 states (j = ​⁵⁄₂) + 8 states (j = ​⁷⁄₂) + 10 states (j = ​⁹⁄₂) + 12 states (j = ​¹¹⁄₂) = 42. Odd parity.

where for every j there are 2j+1 different states from different values of m_j.

Due to the spin-orbit interaction the energies of states of the same level but with different j will no longer be identical. This is because in the original quantum numbers, when ${\displaystyle \scriptstyle {\vec {s}}}$ is parallel to ${\displaystyle \scriptstyle {\vec {l}}}$, the interaction energy is positive; and in this case j = l + s = l + ​¹⁄₂. When ${\displaystyle \scriptstyle {\vec {s}}}$ is anti-parallel to ${\displaystyle \scriptstyle {\vec {l}}}$ (i.e. aligned oppositely), the interaction energy is negative, and in this case j=l−s=l−​¹⁄₂. Furthermore, the strength of the interaction is roughly proportional to l.

For example, consider the states at level 4:

• The 10 states with j = ​⁹⁄₂ come from l = 4 and s parallel to l. Thus they have a positive spin-orbit interaction energy.
• The 8 states with j = ​⁷⁄₂ came from l = 4 and s anti-parallel to l. Thus they have a negative spin-orbit interaction energy.
• The 6 states with j = ​⁵⁄₂ came from l = 2 and s parallel to l. Thus they have a positive spin-orbit interaction energy. However its magnitude is half compared to the states with j = ​⁹⁄₂.
• The 4 states with j = ​³⁄₂ came from l = 2 and s anti-parallel to l. Thus they have a negative spin-orbit interaction energy. However its magnitude is half compared to the states with j = ​⁷⁄₂.
• The 2 states with j = ​¹⁄₂ came from l = 0 and thus have zero spin-orbit interaction energy.

Deforming the potential

The harmonic oscillator potential ${\displaystyle V(r)=\mu \omega ^{2}r^{2}/2}$ grows infinitely as the distance from the center r goes to infinity. A more realistic potential, such as Woods–Saxon potential, would approach a constant at this limit. One main consequence is that the average radius of nucleons' orbits would be larger in a realistic potential; This leads to a reduced term ${\displaystyle \scriptstyle \hbar ^{2}l(l+1)/2mr^{2}}$ in the Laplace operator of the Hamiltonian. Another main difference is that orbits with high average radii, such as those with high n or high l, will have a lower energy than in a harmonic oscillator potential. Both effects lead to a reduction in the energy levels of high l orbits..

Predicted magic numbers

Low-lying energy levels in a single-particle shell model with an oscillator potential (with a small negative l² term) without spin-orbit (left) and with spin-orbit (right) interaction. The number to the right of a level indicates its degeneracy, (2j+1). The boxed integers indicate the magic numbers.

Together with the spin-orbit interaction, and for appropriate magnitudes of both effects, one is led to the following qualitative picture: At all levels, the highest j states have their energies shifted downwards, especially for high n (where the highest j is high). This is both due to the negative spin-orbit interaction energy and to the reduction in energy resulting from deforming the potential to a more realistic one. The second-to-highest j states, on the contrary, have their energy shifted up by the first effect and down by the second effect, leading to a small overall shift. The shifts in the energy of the highest j states can thus bring the energy of states of one level to be closer to the energy of states of a lower level. The "shells" of the shell model are then no longer identical to the levels denoted by n, and the magic numbers are changed.

We may then suppose that the highest j states for n = 3 have an intermediate energy between the average energies of n = 2 and n = 3, and suppose that the highest j states for larger n (at least up to n = 7) have an energy closer to the average energy of n−1. Then we get the following shells (see the figure)

• 1st shell:  2 states (n = 0, j = ​¹⁄₂).
• 2nd shell:  6 states (n = 1, j = ​¹⁄₂ or ​³⁄₂).
• 3rd shell: 12 states (n = 2, j = ​¹⁄₂, ​³⁄₂ or ​⁵⁄₂).
• 4th shell:  8 states (n = 3, j = ​⁷⁄₂).
• 5th shell: 22 states (n = 3, j = ​¹⁄₂, ​³⁄₂ or ​⁵⁄₂; n = 4, j = ​⁹⁄₂).
• 6th shell: 32 states (n = 4, j = ​¹⁄₂, ​³⁄₂, ​⁵⁄₂ or ​⁷⁄₂; n = 5, j = ​¹¹⁄₂).
• 7th shell: 44 states (n = 5, j = ​¹⁄₂, ​³⁄₂, ​⁵⁄₂, ​⁷⁄₂ or ​⁹⁄₂; n = 6, j = ​¹³⁄₂).
• 8th shell: 58 states (n = 6, j = ​¹⁄₂, ​³⁄₂, ​⁵⁄₂, ​⁷⁄₂, ​⁹⁄₂ or ​¹¹⁄₂; n = 7, j = ​¹⁵⁄₂).

and so on.

Note that the numbers of states after the 4th shell are doubled triangular numbers PLUS TWO. Spin-orbit coupling causes so-called 'intruder levels' to drop down from the next higher shell into the structure of the previous shell. The sizes of the intruders are such that the resulting shell sizes are themselves increased to the very next higher doubled triangular numbers from those of the harmonic oscillator. For example, 1f2p has 20 nucleons, and spin-orbit coupling adds 1g9/2 (10 nucleons) leading to a new shell with 30 nucleons. 1g2d3s has 30 nucleons, and addition of intruder 1h11/2 (12 nucleons) yields a new shell size of 42, and so on.

The magic numbers are then

•   2
•   8=2+6
•  20=2+6+12
•  28=2+6+12+8
•  50=2+6+12+8+22
•  82=2+6+12+8+22+32
• 126=2+6+12+8+22+32+44
• 184=2+6+12+8+22+32+44+58

and so on. This gives all the observed magic numbers, and also predicts a new one (the so-called island of stability) at the value of 184 (for protons, the magic number 126 has not been observed yet, and more complicated theoretical considerations predict the magic number to be 114 instead).

Another way to predict magic (and semi-magic) numbers is by laying out the idealized filling order (with spin-orbit splitting but energy levels not overlapping). For consistency s is split into j = 1⁄2 and j = -1⁄2 components with 2 and 0 members respectively. Taking leftmost and rightmost total counts within sequences marked bounded by / here gives the magic and semi-magic numbers.

• s(2,0)/p(4,2) > 2,2/6,8, so (semi)magic numbers 2,2/6,8
• d(6,4):s(2,0)/f(8,6):p(4,2) > 14,18:20,20/28,34:38,40, so 14,20/28,40
• g(10,8):d(6,4):s(2,0)/h(12,10):f(8,6):p(4,2) > 50,58,64,68,70,70/82,92,100,106,110,112, so 50,70/82,112
• i(14,12):g(10,8):d(6,4):s(2,0)/j(16,14):h(12,10):f(8,6):p(4,2) > 126,138,148,156,162,166,168,168/184,198,210,220,228,234,238,240, so 126,168/184,240

The rightmost predicted magic numbers of each pair within the quartets bisected by / are double tetrahedral numbers from the Pascal Triangle: 2, 8, 20, 40, 70, 112, 168, 240 are 2x 1, 4, 10, 20, 35, 56, 84, 120, ..., and the leftmost members of the pairs differ from the rightmost by double triangular numbers: 2 − 2 = 0, 8 − 6 = 2, 20 − 14 = 6, 40 − 28 = 12, 70 − 50 = 20, 112 − 82 = 30, 168 − 126 = 42, 240 − 184 = 56, where 0, 2, 6, 12, 20, 30, 42, 56, ... are 2 × 0, 1, 3, 6, 10, 15, 21, 28, ... .

Other properties of nuclei

This model also predicts or explains with some success other properties of nuclei, in particular spin and parity of nuclei ground states, and to some extent their excited states as well. Take ¹⁷
₈O (oxygen-17) as an example: Its nucleus has eight protons filling the three first proton "shells", eight neutrons filling the three first neutron "shells", and one extra neutron. All protons in a complete proton shell have total angular momentum zero, since their angular momenta cancel each other. The same is true for neutrons. All protons in the same level (n) have the same parity (either +1 or −1), and since the parity of a pair of particles is the product of their parities, an even number of protons from the same level (n) will have +1 parity. Thus the total angular momentum of the eight protons and the first eight neutrons is zero, and their total parity is +1. This means that the spin (i.e. angular momentum) of the nucleus, as well as its parity, are fully determined by that of the ninth neutron. This one is in the first (i.e. lowest energy) state of the 4th shell, which is a d-shell (l = 2), and since ${\displaystyle p=(-1)^{l}}$, this gives the nucleus an overall parity of +1. This 4th d-shell has a j = ​⁵⁄₂, thus the nucleus of ¹⁷
₈O is expected to have positive parity and total angular momentum ​⁵⁄₂, which indeed it has.

The rules for the ordering of the nucleus shells are similar to Hund's Rules of the atomic shells, however, unlike its use in atomic physics the completion of a shell is not signified by reaching the next n, as such the shell model cannot accurately predict the order of excited nuclei states, though it is very successful in predicting the ground states. The order of the first few terms are listed as follows: 1s, 1p​³⁄₂, 1p​¹⁄₂, 1d​⁵⁄₂, 2s, 1d​³⁄₂... For further clarification on the notation refer to the article on the Russell-Saunders term symbol.

For nuclei farther from the magic numbers one must add the assumption that due to the relation between the strong nuclear force and angular momentum, protons or neutrons with the same n tend to form pairs of opposite angular momenta. Therefore, a nucleus with an even number of protons and an even number of neutrons has 0 spin and positive parity. A nucleus with an even number of protons and an odd number of neutrons (or vice versa) has the parity of the last neutron (or proton), and the spin equal to the total angular momentum of this neutron (or proton). By "last" we mean the properties coming from the highest energy level.

In the case of a nucleus with an odd number of protons and an odd number of neutrons, one must consider the total angular momentum and parity of both the last neutron and the last proton. The nucleus parity will be a product of theirs, while the nucleus spin will be one of the possible results of the sum of their angular momenta (with other possible results being excited states of the nucleus).

The ordering of angular momentum levels within each shell is according to the principles described above - due to spin-orbit interaction, with high angular momentum states having their energies shifted downwards due to the deformation of the potential (i.e. moving from a harmonic oscillator potential to a more realistic one). For nucleon pairs, however, it is often energetically favorable to be at high angular momentum, even if its energy level for a single nucleon would be higher. This is due to the relation between angular momentum and the strong nuclear force.

Nuclear magnetic moment is partly predicted by this simple version of the shell model. The magnetic moment is calculated through j, l and s of the "last" nucleon, but nuclei are not in states of well defined l and s. Furthermore, for odd-odd nuclei, one has to consider the two "last" nucleons, as in deuterium. Therefore, one gets several possible answers for the nuclear magnetic moment, one for each possible combined l and s state, and the real state of the nucleus is a superposition of them. Thus the real (measured) nuclear magnetic moment is somewhere in between the possible answers.

The electric dipole of a nucleus is always zero, because its ground state has a definite parity, so its matter density (${\displaystyle \psi ^{2}}$, where ${\displaystyle \psi }$ is the wavefunction) is always invariant under parity. This is usually the situations with the atomic electric dipole as well.

Higher electric and magnetic multipole moments cannot be predicted by this simple version of the shell model, for the reasons similar to those in the case of deuterium.

Including residual interactions

Residual interactions among valance nucleons are included by diagonalising an effective Hamiltonian in a valance space outside an inert core. As indicated, only single-particle states lying in the valance space are active in the basis used.

For nuclei having two or more valence nucleons (i.e. nucleons outside a closed shell) a residual two-body interaction must be added. This residual term comes from the part of the inter-nucleon interaction not included in the approximative average potential. Through this inclusion different shell configurations are mixed and the energy degeneracy of states corresponding to the same configuration is broken.

These residual interactions are incorporated through shell model calculations in a truncated model space (or valance space). This space is spanned by a basis of many-particle states where only single-particle states in the model space are active. The Schrödinger equation is solved in this basis, using an effective Hamiltonian specifically suited for the model space. This Hamiltonian is different from the one of free nucleons as it among other things has to compensate for excluded configurations.

One can do away with the average potential approximation entirely by extending the model space to the previously inert core and treat all single-particle states up to the model space truncation as active. This forms the basis of the no-core shell model, which is an ab initio method. It is necessary to include a three-body interaction in such calculations to achieve agreement with experiments.

Related models

Igal Talmi developed a method to obtain the information from experimental data and use it to calculate and predict energies which have not been measured. This method has been successfully used by many nuclear physicists and has led to deeper understanding of nuclear structure. The theory which gives a good description of these properties was developed. This description turned out to furnish the shell model basis of the elegant and successful interacting boson model.

A model derived from the nuclear shell model is the alpha particle model developed by Henry Margenau, Edward Teller, J. K. Pering, T. H. Skyrme, also sometimes called the Skyrme model. Note, however, that the Skyrme model is usually taken to be a model of the nucleon itself, as a "cloud" of mesons (pions), rather than as a model of the nucleus as a "cloud" of alpha particles.