Deixis Online

Density functional theory (DFT) can be used to determine densities of protons and neutrons making up a nucleus.

“If we can determine those densities precisely,” says Witold Nazarewicz, professor of physics at the University of Tennessee, “we can determine the binding energy – the energy stored in the nucleus.”

The energy density functional (EDF) in DFT is an integral of a function of those particle densities. The corresponding energy density is composed of proton and neutron densities, spins, momentum and more. Such an EDF includes variables, or coupling constants, that must be adjusted. The goal of the Universal Nuclear Energy Density Functional (UNEDF) collaboration is to find a universal EDF that works across the entire nuclear landscape, or chart of nuclides, which is a two-dimensional table that collects all nuclei represented by their proton and neutron numbers. Building that requires simulating the properties of thousands of nuclei and doing so repeatedly.

The first step in using DFT to model a nucleus is finding the densities that minimize the energy in a nucleus. This creates an optimization problem that can be tackled by solving the so-called Hartree–Fock–Bogoliubov (HFB) equations of the nuclear DFT.

For the coupling constants, says Nazarewicz, “some of those – but very few – are basically given by theory. Most of them have to be found by comparing DFT calculations with experiments.”

The results of the optimization experiments and starting values for coupling constants can be used to generate nuclear features, such as binding energies, radii, shape deformations and others.

The UNEDF team started with about a dozen coupling constants and used them in HFB equations to calculate more than 100 features of nuclei.

“We compare those observables with experiment and design a chi-square to see if the results are good or not,” Nazarewicz says. “Then we try to adjust the parameters so the chi-square becomes minimum. So this constitutes a huge optimization-minimization problem.”

Fortunately, some of the parameters cannot be varied broadly, because physical bounds limit some of them. That provides some small simplifications to this process.

To make this technique work, Nazarewicz and his colleagues came up with the form of the functional and selected the experimental fit-observables to use.

“We provided our codes to the Argonne group,” Nazarewicz says, “and they designed an optimized procedure that minimized the chi-square. That is POUNDERS (for “practical optimization using no derivatives for sums of squares”), which is a tremendous algorithm because it saved orders of magnitude of time.”

Without POUNDERS, the chi-square converged on a minimum so slowly that it would have taken hundreds of iterations. That takes too long when the minimization involves what Nazarewicz describes as “a problem that is highly nonlinear for more than 100 observables and a dozen or so coupling constants.”

With POUNDERS, DFT can be applied to more nuclei and other applications of this technique will surely keep emerging.