One of the key questions that we have always asked ourselves is — what are we, and everything around us, made of? Our curiosity has led us to continually break bigger things into smaller things using the finest tools at our disposal. As a part of this process, through a series of experimental and theoretical advances that spanned more than a century, the atomic theory of matter came to be firmly established by the early 1900s. Subsequently it was discovered that, contrary to their name (‘atom’ comes from the Greek atomos, meaning ‘indivisible’), atoms are composed of yet smaller particles — negatively charged electrons that surround a dense, positively charged, central nucleus. The nucleus accounts for most of the mass of the atom, but occupies only a tiny fraction of the volume (the nuclear diameter is about 10⁻⁵ times the size of the atom). The nucleus is known to contain positively charged ‘protons’ and electrically neutral ‘neutrons’, collectively referred to as ‘nucleons’.
In an electrically neutral atom, the number of electrons that are electromagnetically bound to the nucleus has to be equal to the number of protons in the nucleus. The laws of quantum mechanics dictate that these electrons must organize themselves into a series of distinct orbitals, characterized by their energies and angular momenta. These arrangements of electrons into orbitals give rise to the unique chemical properties that are associated with the different elements.
In a similar vein, the structure of the nucleus can also be described in terms of energy levels, in what is known as the ‘nuclear shell model’ . This was developed in the mid-20th century and culminated in the 1963 Nobel Prize in physics (awarded jointly to Eugene Wigner, Maria Goeppert Mayer, and J. Hans D. Jensen) . Just as atoms with different numbers of electrons ‘behave’ differently, so do nuclei with different numbers of protons and neutrons. Just like a periodic table of the elements, one can likewise build a table of the nuclides¹ (see image below).
¹ ‘Nuclide’ is a term used for an atomic nucleus, as characterized by the number of protons and neutrons, just as the term ‘element’ is used for characterizing an atom based on the number of electrons (or equivalently, protons). For example, a ‘Deuteron’, an atomic nucleus consisting of one proton and one neutron, is a nuclide of Hydrogen.
That does, however, beg the question — how did we end up with a universe that contains such a wide variety of elements and nuclides? To find an answer, one must hark back to the origin of the universe itself. The most widely accepted theory for this today is the ‘Hot Big Bang’ theory. It predicts that the universe began in an extremely hot, dense state, and cooled as it expanded with time. The early universe also possessed a high degree of homogeneity and isotropy (i.e., uniformity across all locations and directions in space). But as it cooled, the tiny inhomogeneities that were present eventually grew and led to the formation of the structures we observe today — stars, galaxies, and gigantic galaxy clusters, interspersed between even bigger cosmic ‘voids’.
Part 1: Big Bang Nucleosynthesis
In a stable atom, the electrons and nucleons are bound together by the electromagnetic force. However, in environments with sufficiently high temperatures, the electrons can absorb enough energy to escape this electromagnetic pull and become free particles. This state of matter where electrons and nuclei can move about freely is referred to as a ‘plasma’. The early universe was in this state during the first 370,000 years of its history, when the temperature of its constituents was above 3000 Kelvins. The interiors of stars are also hot enough to be found in a plasma state.
In fact, during the first three or so minutes in the history of the universe, the temperature of the primordial plasma was so hot that even the atomic nuclei, which are held together by the ‘strong nuclear force’ (much stronger than electromagnetism), could not remain intact. Any nuclei formed in these conditions would immediately be dissociated by energetic photons back into their constituent nucleons. Therefore, the universe during this period consisted of a sea of free protons, neutrons, and electrons (and positrons), admixed with an abundant number of other particles such as photons and neutrinos². As the universe expanded and cooled from this state, the photons became less abundant and energetic, and the equilibrium between the forward and reverse processes, i.e., nucleonic fusion and photo-dissociation, continued to shift toward fusion, gradually reaching a stage where an appreciable number of nuclei could survive. Eventually, at a plasma temperature of around a billion Kelvin, the rates of the forward and reverse processes started to fall below the expansion rate of the universe, and as a result, these processes dropped out of equilibrium. At this juncture, the abundances of the nuclei thus formed became ‘frozen’ in time, not experiencing any further evolution as the universe expanded. This process is known as ‘NSE freeze-out’, where NSE stands for ‘Nuclear Statistical Equilibrium’.
² At even higher temperatures (above 2 trillion Kelvin), the nucleons themselves cannot exist as complete entities, but instead break down into their constituents, known as quarks and gluons. This was the case for the first 10⁻⁵ seconds or so, in the history of the universe.
In parallel with the aforementioned processes of fusion and dissociation, another set of nuclear reactions were also taking place. These consisted of inter-conversions between protons and neutrons via captures of electrons, positrons, neutrinos or antineutrinos, as well as decays of free neutrons into protons. Each of these processes are mediated by the so-called ‘weak nuclear force’ or ‘weak interaction’. Very early on, the universe contained almost an equal number of protons and neutrons, but as it cooled, the equilibrium shifted towards more protons, on account of the neutron being slightly heavier than the proton. As a result, by the time of the NSE freeze-out epoch, the protons in the universe out-numbered the neutrons by nearly 7 to 1.
On the whole, the process of combining free nucleons into nuclei via NSE freeze-out can be best described as a competition between nuclear binding and disorder. On one hand, combining free nucleons into nuclei results in a configuration with a lower total energy, which is favorable. But on the other hand, the early universe is characterized by a very high entropy-per-nucleon³. Practically, this means that the number of photons in the primordial plasma far exceeds the number of nucleons — by more than a billion to one. Under such circumstances, it is extremely difficult for nuclei with large numbers of nucleons to form. This is because packing many nucleons into a single large nucleus represents an ordered configuration, whereas a high entropy, such as in the early universe, tends to favor disorder. The net result of this tug-of-war is that, by the time of NSE freeze-out, nearly all the free neutrons pair up with protons to form nuclei of the Helium-4 isotope (also known as alpha particles — consisting of two protons and two neutrons). The reason behind Helium-4 being so strongly favored in the early universe is its appreciably high binding energy (making it energetically favorable), whilst also being not too large of a nucleus (and therefore also favorable from the disorder standpoint).
³ Entropy is a concept in thermodynamics and statistical physics that can be treated as a measure of the amount of disorder present in a system. The second law of thermodynamics states that the change in entropy of an isolated system is always non-negative, implying that any isolated system left to its own volition always tends to drift towards a maximally disordered state.
Thus, the universe after NSE freeze-out consists of about 25% of all its nucleons locked up into alpha particles, and the remaining 75% continuing to exist as free protons. In addition, trace amounts of other nuclides, such as Deuterium and Lithium-7, are also produced. In addition to the high entropy, another bottleneck that stymies the production of heavier elements during BBN is the lack of any stable nuclides with 5 or 8 nucleons.
This entire process of light nuclide formation in the early universe, involving an interplay of all four fundamental forces (gravitation, electromagnetism, and the strong and weak interactions), is referred to as ‘Big Bang Nucleosynthesis’, or BBN for short. Through astronomical observations in the present-day universe, it is possible to directly or indirectly measure the abundances of light nuclides that were produced during BBN, and for the most part, the observations so far have shown excellent agreement with theoretical predictions, lending credence to the Hot Big Bang model.
The original theory of Big Bang Nucleosynthesis was developed by George Gamow and his PhD student, Ralph Alpher, and first published in the famous Alpher-Bethe-Gamow paper  from 1948 (also known as the 'αβγ' paper⁴). However, that study reached the erroneous conclusion that heavier elements could also be produced during BBN. This likely resulted from the fact that not much was known at the time regarding the entropy of the early universe, which led Gamow and Alpher to mistakenly model the early universe as a dense neutron gas (which might have been more appropriate had the entropy been a lot lower). Subsequently, the correct physical description of BBN was worked out by Robert Wagoner, William A. Fowler, and Fred Hoyle in 1967 .
⁴ Gamow insisted on including fellow nuclear physicist Hans Bethe’s name as an author, just to create the αβγ wordplay, a decision that Alpher reportedly wasn’t too pleased about.
Part 2: Stellar Nucleosynthesis
Big Bang Nucleosynthesis, while fascinating in its own right, still falls short of explaining how any nuclides heavier than Lithium-7 are formed in the universe. To seek an explanation, one must turn to the stars. The first stars formed when the universe was a few hundred million years old. As the tiny inhomogeneities that were present in the early universe continued to grow, the associated gravitational potential wells got deeper. The effect of that deepening, in combination with the cooling of the gas clouds, enabled these clouds to collapse gravitationally into stars.
By the early-to-mid 20th century, the contributions of various scientists such as Francis Aston, Arthur Eddington, Subrahmanyan Chandrasekhar, George Gamow, Carl Friedrich von Weizsäcker, and Hans Bethe, among others, led to the understanding that stars (including the sun) produce energy through nuclear fusion processes. Since nuclear fusion involves combining lighter nuclei into heavier ones, stars seemed to be the natural candidates for producing nuclides heavier than the ones synthesized during BBN.
In 1957, a group of four scientists — Margaret Burbidge, Geoffrey Burbidge, William A. Fowler, and Fred Hoyle, published a seminal paper (known as the B²FH paper) that laid out the theory of stellar nucleosynthesis in detail . The paper was built upon earlier works by Hoyle in 1946 and 1954, and incorporated knowledge from the observed elemental abundances (from the Burbidges), as well as laboratory studies of nuclear reactions (from Fowler).
To understand how nucleosynthesis occurs in stars, one must first review the stellar life cycle. Ordinary stars such as the sun generate most of their energy via the fusion of hydrogen into helium in their cores. Once the core of the star begins to run out of hydrogen, it contracts and heats up, triggering a shell of hydrogen fusion around its periphery. Subsequently, if the star is sufficiently heavy, the helium core becomes hot enough to permit fusion of helium into carbon. The net outcome of this process is an inner carbon core surrounded by a helium shell.
In stars that are even more massive, this process can repeat so that increasingly heavier nuclides (oxygen, neon, magnesium, silicon, calcium, iron, in that order) can successively form in the inner regions of the core, surrounded by shells of lighter nuclides — resulting in an onion-like structure, as shown in the figure below. This spontaneous chain of nuclear fusion events in the stellar cores terminates once the iron group of nuclides (iron, cobalt, and nickel) is reached. This is because the iron group of nuclei have the highest binding energies per nucleon, and as a result any process that could produce heavier nuclides requires energy to be added to the system, rendering these processes non-spontaneous.
At this point, two questions still remain unanswered — how are the nuclides heavier than iron produced, and how do nuclides get distributed through the cosmos, instead of being forever locked up in the stellar cores?
Once again, the answer lies in stellar evolution, and for that one must examine what happens to massive stars once spontaneous nuclear fusion ends. In stars not much heavier than the sun, the stellar core continues to contract gravitationally until it is stopped by what is known as ‘electron degeneracy pressure’. This pressure arises because electrons belong to a class of particles called ‘fermions’, which obey the principle that no two fermions can occupy the same quantum state. This sets a limit on how tightly a gas of electrons can be compressed, and it is this quantum mechanical pressure that holds up the core against further gravitational contraction. At this stage, the star becomes what is known as a ‘White Dwarf’.
However, in more massive stars, the gravitational force that causes core collapse is far too strong for even the electron degeneracy pressure to counteract. In such a situation, gravitational contraction forces the tightly packed electrons to combine with protons to produce neutrons, and the core continues to collapse further. The process of combining protons and electrons into neutrons is also accompanied by the emission of neutrinos. Neutrinos are electrically neutral, nearly massless particles that travel at almost the speed of light, and interact with other particles only through the weak interaction, which is much weaker than the electromagnetic force at the relevant energies. As a result of these properties, neutrinos can efficiently transport energy away from the stellar interiors, and are in fact responsible for carrying a majority (~99%) of the gravitational binding energy released by the stellar core as it collapses.
If the mass of the core is roughly between 1.4 solar masses (known as the ‘Chandrasekhar limit’) and 3 solar masses, then the core collapse is eventually stopped by nuclear repulsion — through a combination of the repulsive strong nuclear force⁵ and neutron degeneracy pressure (neutrons are also fermions). When that point of equilibrium between gravitation and nuclear repulsion is reached, the inner core rebounds, sending a powerful shockwave through the outer layers of the star, and ripping it apart in a violent explosion known as a 'core-collapse' supernova or a Type II supernova⁶. The core gets left behind as an extremely dense remnant, known as a ‘neutron star’ on account of it being composed predominantly of neutrons. In stars with cores heavier than 3 solar masses, even nuclear repulsion isn’t enough to overcome gravitation, and the core collapses into a black hole.
⁵ The strong nuclear force is attractive at distances farther than about 0.7 femtometers (1 fm = 10⁻¹⁵ m), but it becomes repulsive at distances closer than that.
⁶ Numerical simulations have actually shown that the expanding shockwave loses energy and often stalls before it can traverse through the outer envelope of the star. However, since core-collapse supernovae are observed in nature, some physical process (or a combination of various processes) must be responsible for reviving the shock wave and giving it enough energy to blow up the star. The exact mechanism behind this shock revival is still a subject of ongoing research.
In the context of nucleosynthesis, core-collapse supernovae serve a two-fold purpose. First, they provide an energetic and potentially neutron-rich environment for nuclides heavier than iron to be synthesized. Secondly, the explosion triggered by the shockwave also scatters these nuclides throughout the galactic neighborhood of the star. Eventually, when the next generation of stellar systems are formed, they contain traces of this nuclear debris.
Nuclides heavier than iron can be produced via a number of different mechanisms. Each of these mechanisms involves a chain of nucleonic capture events, interspersed with radioactive decays such as beta, inverse beta, or alpha decays⁷. Nuclides that are neutron-rich (i.e, containing many more neutrons than protons) are produced via two different processes, namely the slow and rapid neutron capture processes (s-process and r-process, for short). Here the names ‘slow’ and ‘rapid’ reflect the rate of neutron capture relative to the rate of radioactive decays within the reaction chain — in the s-process, neutron capture can occur over thousands of years, whereas in the r-process, it occurs on a timescale of seconds. Aside from these, there are a couple of other processes, called p-process (proton capture), and 𝜈p process. These are responsible for producing some of the more proton-rich nuclei that cannot be produced via the s- or r-processes (‘proton-rich’ here does not mean having more protons than neutrons, it just refers to nuclides that contain fewer neutrons compared to the s- and r-process nuclides).
⁷ Beta decay refers to the conversion of a single neutron inside a nucleus into a proton, accompanied by the emission of an electron and an antineutrino. Inverse beta decay is the conversion of a proton into a neutron, while emitting a positron and a neutrino. Alpha decay is the emission of an alpha particle from a nucleus.
The s-process is known to occur mainly in stars known as ‘Asymptotic Giant Branch’ (AGB) stars, which are stars with a mass comparable to the sun (or a little heavier) that have exhausted the hydrogen in their core and are in the process of fusing helium into carbon. In these stars, iron nuclides left behind by previous supernovae act as seeds for the capture of free neutrons, which are produced through reactions such as alpha particle capture on a Carbon-13 nucleus. The s-process can account for abundances of some of the nuclides up to Pb-206, an isotope of lead. However, the extremely slow rate of neutron capture renders this process unsuitable for producing even heavier nuclides such as Uranium and Thorium, as well as for explaining the entirety of observed abundances of other neutron-rich nuclides such as silver and gold. These can only be accounted for by r-process nucleosynthesis.
The r-process requires a specific set of conditions, namely a high ratio of neutrons to seed nuclei. Such conditions can be present in very few environments. Identifying astrophysical sites that are suitable candidates for producing r-process nuclides remains an open question to this day. Whether or not supernovae can produce appreciable yields of r-process nuclides depends on the details of neutrino evolution and interactions in these environments, since neutrinos and antineutrinos are present in large numbers at these sites. Interactions of neutrinos are responsible for converting neutrons into protons and vice versa, and therefore, they are an important input in determining the ratio of neutrons to seed nuclei. Accurately modeling the physics of neutrinos in these environments continues to be an extremely challenging computational endeavor.
More recently, the first observation of a binary neutron star merger (i.e., the in-spiraling and subsequent collision between two neutron stars in orbit around one another) has helped shed an enormous amount of light on this problem . The event, dubbed ‘170817’ (indicating the date at which its signals arrived on the earth), was first detected via its ‘Gravitational Wave’ signal at the LIGO and VIRGO gravitational wave detectors⁸ . Subsequent follow-up optical observations using various wavelengths of light have since confirmed that r-process nucleosynthesis did indeed happen at the site of this cataclysmic astrophysical event [8, 9, 10]. However, these events are exceptionally rare, and whether they happen often enough and produce enough r-process output to explain the observed abundances of many heavy nuclides today, remains a mystery.
⁸ Gravitational waves are ripples in the fabric of space and time that propagate at the speed of light. These waves result in alternate stretching and squeezing of space along the directions transverse to their propagation. As a result, they can be detected by measuring the relative variation in the lengths of two perpendicular arms of a giant optical interferometer. The current state-of-the-art detectors built for this purpose are sensitive enough to detect a relative change in length of less than one part in 10²².
Each nucleosynthesis process has a characteristic chain of nuclides that it ends up producing. Aside from Big Bang nucleosynthesis, stellar nuclear fusion, and the aforementioned neutron and proton capture processes in stars and core-collapse supernovae, there are also a few other mechanisms such as cosmic ray fission, or thermonuclear supernovae (i.e., white dwarfs that explode after accreting enough matter from a companion star) that account for a small number of nuclides that are found in the universe. Nearly all of these processes still remain a subject of active research, in some form or another.
All in all, the task of explaining how the elements came to be and led to the universe being the way we see it today, continues to be an exciting and perplexing problem. The allure of it is how it necessitates drawing upon the expertise from many different areas of physics, such as elementary particle physics, nuclear physics, plasma physics, and gravitational physics, among others. Moreover, ongoing research that is connected to this subject involves the absolute cutting edge of observation, theory, experimentation, and computation. No doubt this is a problem that has led to, and shall continue to lead to, many a sleepless night for physicists all around the world!
Cited in text:
Nuclear shell model (Wikipedia)
Nobel lecture by Maria Goeppert Mayer
The Origin of Chemical Elements, by R. A. Alpher, H. Bethe, and G. Gamow, Phys. Rev. 73, 803 (1948)
On the Synthesis of Elements at Very High Temperatures, by Wagoner, Robert V., Fowler, William A., Hoyle, F., Astrophysical Journal, vol. 148, p.3 (1967)
Synthesis of the Elements in Stars, E. Margaret Burbidge, G. R. Burbidge, William A. Fowler, and F. Hoyle, Rev. Mod. Phys. 29, 547 (1957)
Multi-messenger Observations of a Binary Neutron Star Merger, B. P. Abbott et al., ApJL 848 L12 (2017)
GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, B. P. Abbott et al., Phys. Rev. Lett. 119, 161101 (2017)
Identification of strontium in the merger of two neutron stars, Darach Watson et al., Nature volume 574, pages 497–500 (2019)
GW170817 --the first observed neutron star merger and its kilonova: Implications for the astrophysical site of the r-process, Daniel M. Siegel, The European Physical Journal A volume 55, Article number: 203 (2019)
Not cited in text:
Populating the periodic table: Nucleosynthesis of the elements, by Jennifer A. Johnson
Origin of the Chemical Elements, by T. Rauscher and A. Patkós