Neutron-rich matter in heaven and on Earth

Characterizing the neutron-rich skin in ²⁰⁸Pb may help constrain nuclear models that aim to describe the nuclear dynamics of both atomic nuclei and neutron stars in a single unified framework. The link between the very small and the very large is particularly compelling given that a strong connection has been established between the thickness of the neutron skin of ²⁰⁸Pb and the radius of a neutron star.⁵ The dynamics behind such a correlation can be revealed by returning to the liquid-drop model, in which the nuclear binding energy is encoded in a handful of empirical parameters that represent volume, surface, Coulomb, and symmetry contributions:

The volume term a_v scales with the total number of nucleons A = Z + N, and that fact underscores both the short-range nature and saturation properties of the underlying nuclear force. A hallmark of nuclear dynamics is the existence of a saturation density of about ρ₀ ≈ 0.15 fm⁻³ , which is close but not equal to the nearly constant central density observed in atomic nuclei. The next two terms represent corrections to the energy that result from the development of a finite nuclear surface a_s and the Coulomb repulsion among protons a_c. A quantum correction is applied for asymmetric nuclei because of the Pauli exclusion principle. The last term—the symmetry energy a_a and especially its density dependence—is crucial in connecting the neutron-skin thickness of atomic nuclei to the radius of a neutron star.

Although the liquid-drop model successfully describes the smooth variation of the nuclear binding energy with Z and N, the atomic nucleus is not an incompressible liquid drop. So although highly insightful, the semiempirical mass formula fails to capture the response of the liquid drop to changes in density. That information is embodied in the equation of state, which dictates how the energy depends on the overall density and neutron–proton asymmetry of the system.

In the thermodynamic limit and ignoring the long-range Coulomb interaction, the energy per nucleon at the equilibrium density is given entirely by the terms of volume a_v and symmetry energy a_a. The volume term a_v accounts for the dynamics of a symmetric system having equal numbers of protons and neutrons, whereas a_a penalizes the system for breaking the symmetry.

So what happens as the system departs from its equilibrium position? Changes to the energy per nucleon with density are imprinted in the pressure. However, the contribution to the pressure from the symmetric term vanishes at the equilibrium density. Thus the entire contribution to the pressure at saturation density comes from the symmetry pressure. Often denoted in the literature by L, the quantity is closely related to the pressure at saturation density of a system made entirely of neutrons; that is, P₀ ≈ Lρ₀/3. The symmetry pressure, therefore, controls both the neutron-skin thickness of atomic nuclei and the radius of a neutron star.⁶ 

Where do the 44 excess neutrons in ²⁰⁸Pb go? Although the liquid-drop model favors the formation of a spherical drop of uniform density, it is unclear what fraction of the excess neutrons should reside at the surface or in the core. Surface tension favors placing them in the core, which tends to minimize the surface area. But the symmetry energy, which is larger at the core than at the surface, disfavors that arrangement. Conversely, moving them to the surface increases the surface tension but reduces the symmetry energy. Thus the thickness of the neutron skin is determined by a tug-of-war between the surface tension and the difference between the symmetry energy at saturation density and at the lower surface density. That difference is nothing more than the symmetry pressure L. If the pressure is large, then energy considerations favor the excess neutrons to move to the surface where the low symmetry energy results in a thick neutron skin.⁶ 

Where the neutrons go is nicely illustrated in figure 2, which displays neutron and proton densities for ²⁰⁸Pb as predicted by various models that successfully reproduce properties of finite nuclei and neutron stars.⁷ Given that the proton (or rather the charge) distribution of ²⁰⁸Pb has been measured with remarkable precision, no significant spread is observed in the model predictions. Instead, challenging parity-violating experiments are required for a clean measurement of neutron densities. And although PREX has provided an important first step, the precision attained was insufficient to distinguish between the various competing models. The result means that a large model spread remains for the neutron densities and consequently for the neutron-skin thickness, whose values are indicated in the figure 2 legend on the left and schematically depicted by the region between the two arrows. The running sum, which naturally terminates at 44, represents the total number of excess neutrons accumulated up to a distance r. Models with a large symmetry pressure L push the excess neutrons farther out to the surface.

The strong correlation between the neutron-skin thickness of ²⁰⁸Pb and the symmetry pressure L is evident in figure 3a, which shows predictions from a large number of models that utilize density functional theory in the spirit of the models⁷ displayed in figure 2. With a Pearson correlation coefficient of nearly 1, the correlation is strong indeed. Such a result indicates how a fundamental parameter of the equation of state of neutron star matter can be measured in a terrestrial laboratory. The error bars in figure 3a indicate the precision anticipated for upcoming campaigns: PREX-II at Jefferson Lab and the Mainz Radius Experiment at the future Mainz Energy-Recovering Superconductor Accelerator at Johannes Gutenberg University.

Remarkably, it is the same symmetry pressure L that determines the radius of a neutron star, as shown in figure 3b. In that case, however, the symmetry pressure pushes against the immense gravitational attraction encountered in the stellar interior. Yet regardless of whether the pressure pushes against surface tension or against gravity, both the neutron-skin thickness of ²⁰⁸Pb and the radius of a neutron star are sensitive to the symmetry pressure in the vicinity of saturation density. Despite a difference in size of 18 orders of magnitude, a powerful data-to-data relation emerges: The thicker the neutron-skin thickness of ²⁰⁸Pb, the larger the radius of a neutron star. The correlation is particularly strong for low-mass neutron stars in which the interior density is only slightly larger than saturation density. As shown in figure 3b, the correlation coefficient weakens from r = 0.99 to r = 0.95 in going from a neutron star with solar mass of 0.8 to 1.4.

Neutron stars are fascinating systems whose understanding requires a convergence of disciplines. Although the most common perception of a neutron star is that of a uniform assembly of neutrons packed to enormous densities, the reality is far different and much more interesting. First theorized in 1933 by Walter Baade and Fritz Zwicky, neutron stars—or more precisely the radio pulses they emit—were detected in 1968 by a talented Cambridge graduate student named Jocelyn Bell Burnell. The achievement famously won her doctoral adviser, but not her, a share of the 1974 Nobel Prize in Physics.⁸ Bell Burnell’s contributions were honored in 2018 with the Special Breakthrough Prize in Fundamental Physics, and she has announced that she will donate the full $3 million award to programs that support diversity in the field.

Nuclear physics is important for elucidating the structure and composition of neutron stars (see box 1). Unlike white dwarf stars, which are entirely supported against gravitational collapse by the pressure from their degenerate electrons, neutron stars get critical pressure support from nuclear interactions. Indeed, in a 1939 paper, J. Robert Oppenheimer and George Volkoff demonstrated that a neutron star supported exclusively by neutron degeneracy pressure will collapse into a black hole once its mass exceeds 0.7 solar masses (M_☉). Today, however, physicists know of at least two neutron stars with masses^9,10 as large as 2 M_☉.

The surface of a neutron star, though largely insensitive to nuclear dynamics, is of observational importance because it significantly influences estimates of the stellar radius. Assuming that the thermal emission from the surface follows a blackbody spectrum at a uniform temperature, then the stellar radius may be determined from the Stefan–Boltzmann law, which relates the luminosity to the temperature and radius of the star. Unfortunately, the determination of stellar radii by photometric means has been plagued by large systematic uncertainties arising from unreliable distance measurements and from distortions to the blackbody spectrum from a thin stellar atmosphere. In the past, those uncertainties revealed discrepancies in the extraction of stellar radii as large as 5–6 km. (Average neutron star radii are 10–15 km.) Fortunately, the situation has improved significantly through a better understanding of systematic uncertainties, important theoretical developments, and the implementation of robust statistical methods.¹¹ And while the uncertainty has now been reduced to about a couple of kilometers, a powerful new player has entered the game: gravitational-wave astronomy.

The first direct detection of gravitational waves, from a binary neutron star merger known as GW170817, by the collaboration of the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo has begun a new era of multimessenger astronomy.¹² Besides gravitational waves, electromagnetic counterparts associated with both a short gamma-ray burst and a long-term kilonova powered by the radioactive decay of r-process elements were also detected (see the article by Anna Frebel and Timothy C. Beers, Physics Today, January 2018, page 30). GW170817 has also provided fundamental new insights into the nature of dense matter.

Critical properties of the equation of state are encoded in the tidal polarizability, a property that describes the neutron star’s tendency to deform in response to the tidal field induced by a companion star. The tidal polarizability is highly sensitive to the stellar structure and scales as the fifth power of the compactness. That quantity is defined as the ratio of the stellar radius to the Schwarzschild radius—that is, the radius at which the star would become a black hole. The Schwarzschild radius is directly proportional to the stellar mass; for our sun it is approximately 3 km. So, as two neutron stars approach each other, the phase of the gravitational wave deviates from its point-mass nature characteristic of black holes, and those deviations are imprinted in the tidal polarizability.

A fluffy or large-radius neutron star is much easier to polarize than a corresponding compact star with the same mass but a smaller radius. Given the sensitivity of the gravitational-wave signal to the neutron star structure, limits on the tidal polarizability inferred from GW170817 rule out overly large stellar radii and thereby provide a powerful complementary approach to the traditional photometric techniques.^13,14 Additional observational limits have been obtained on both the maximum stellar mass and the minimum radius of a 1.6 solar-mass neutron star.^15,16 As shown in figure 4a, the limiting values of stellar radii and maximum masses are now starting to paint a detailed picture of the mass-versus-radius relation.

How do all the new developments illuminate the connection between GW170817 and atomic-scale laboratory experiments? In particular, given their sensitivity to the symmetry pressure, how do the inferred limits on stellar radii reflect on the neutron-skin thickness of ²⁰⁸Pb? Considering that GW170817 disfavors overly large stellar radii, the inferred neutron-skin thickness is well below the central value measured by the PREX collaboration¹³ and is clearly illustrated in figure 4b. To reduce the experimental uncertainty by a factor of three, the follow-up PREX-II experiment is scheduled to run at Jefferson Lab in 2019. After it and its sister campaign on calcium-48 are completed, the lab will pass the baton to the Facility for Rare Isotope Beams (FRIB) at Michigan State University that will study exotic nuclei with thick neutron skins.

The third observing run by the LIGO–Virgo collaboration began in April 2019 with the promise of many more detections of binary neutron star mergers. A PREX-II confirmation that the neutron-skin thickness of ²⁰⁸Pb is large would imply that the symmetry pressure is also large or “stiff” at the typical densities found in atomic nuclei. If at the same time the LIGO–Virgo collaboration validates the relatively small stellar radii suggested by GW170817, then it will imply that the symmetry pressure is small or soft at about twice the saturation density. The evolution of the symmetry energy from stiff at typical nuclear densities to soft at slightly higher densities may indicate an exotic phase transition in the neutron star interior. In a recent reanalysis of GW170817 data, the LIGO–Virgo collaboration obtained limits on the tidal polarizability that are even more stringent than reported in the original discovery paper.

The determination of the symmetry pressure L—and more generally the density dependence of the symmetry energy—has far-reaching consequences in areas of physics as diverse as precision tests of the standard model using atomic-parity violation, the collision of heavy ions, and nuclear and neutron star structures. However, the search for new physics beyond the standard model is hindered by large uncertainties in the neutron radius, which, as previously discussed, is highly sensitive to L. Above saturation density, the symmetry pressure may be determined by means of experiments involving the collision of heavy ions, the only way to probe vast regions of the nuclear equation of state in terrestrial laboratories. Past experiments with energetic heavy ions enabled nuclear matter to be compressed to several times the nuclear saturation density and allowed researchers to extract the equation of state of symmetric nuclear matter. Current uncertainties in the density dependence of the symmetry energy are large, yet ongoing international efforts, such as the RIKEN Nishina Center for Accelerator-Based Science in Japan, FRIB, and the Facility for Antiproton and Ion Research at the GSI Helmholtz Center for Heavy Ion Research in Germany, are poised to probe neutron-rich matter at suprasaturation density and will offer a better understanding of its properties.

Although the multimessenger era is still in its infancy, the first observation of a binary neutron star merger is already providing a treasure trove of insights into the nature of dense matter. In the new era of multimessenger astronomy, the strong synergy between nuclear physics and astrophysics will grow even stronger. As illustrated in box 2, ultrasensitive gravitational-wave observatories, Earth- and space-based telescopes operating at various wavelengths, and new terrestrial facilities probing atomic nuclei at the limits of their existence are poised to answer 2 of the 11 science questions for the next century:¹⁷ What are the new states of matter at exceedingly high density and temperature? How were the elements from iron to uranium made? The future is very bright indeed!

We thank our many colleagues who have contributed to this research and the US Department of Energy Office of Nuclear Physics for its support, award number DE-FG02-92ER40750.

Jorge Piekarewicz is a professor in the department of physics at Florida State University in Tallahassee, and Farrukh Fattoyev is an assistant professor in the department of physics at Manhattan College in New York City.