Measuring the size of the proton

The size of an atom's nucleus is roughly five orders of magnitude smaller than the size of the atom. Consequently, the nuclear corrections to atomic energy levels are very tiny. However, these corrections have become important in recent high-precision measurements of transitions in the hydrogen atom, which are being performed to test quantum electrodynamics (QED) and to determine related fundamental constants. Today, the interpretation of this data is only limited by the uncertainties in the size of the nucleus, which in the case of hydrogen is a single proton.

The charge radius r_p of the proton is a physical parameter that characterizes important aspects of the effective size of the proton. Its value is used together with the Rydberg constant in the calculations of bound-state QED involving hydrogen atoms as well as muonic hydrogen atoms that have a muon orbiting the nucleus rather than the electron. (Muons are like electrons but 200 times heavier.) A recent spectroscopic study of muonic hydrogen resulted in a new measurement of the proton root mean square (rms) charge radius.¹ The reported value, r_p=0.84184(67)fm, differs by 5 standard deviations from (and is a factor of 10 more precise than) previous determinations. These earlier values come from three very different methods that are based mostly on electronic hydrogen data. Specifically, a compilation of physical constants (CODATA) gives r_p=0.8768(69)fm,² the Lamb shift of electronic hydrogen results in r_p=0.883(14)fm,^{3, 4} and electron scattering from hydrogen yields r_p=0.895(18)fm.⁵ It is noteworthy that QED considerations show⁶ that the electron scattering experiments and the atomic hydrogen spectroscopy determine the same radius. Discussions about the correctness of the theory, experiments, or the adequacy of the models are not leading to satisfactory explanations of this relevant discrepancy.

One way to address the disagreements over the proton charge radius is to measure a different but complementary aspect of the proton electromagnetic structure. The proton Zemach radius, R_p, is a fundamental parameter related to the spatial distribution of both the electric charge and the magnetic moment of the proton. The value of the Zemach radius (and its uncertainty) set limits on the value and uncertainty of the proton charge radius, and vice versa.^{7, 8} A measurement of the Zemach radius will, therefore, impose restrictions on the proton charge distribution that will exclude one of the currently competing values of the proton charge radius or part of the models used in the calculations. Alternatively, it will lead to some new conclusions.

Figure 1. Block diagram of the layout of the first approach to a dedicated nonlinear laser source.

There exist three possibilities to determine the numerical value of R_p. One approach based on the analysis of the world data on electron-proton scattering gives R_p=1.086±0.012fm.⁹ Another method uses the comparison of theoretical and experimental results for the hydrogen atom to obtain R_p=1.037(16)fm,⁷ or a similar value.⁸ We suggest a third method, based on the comparison of experimental data and theoretical results for muonic hydrogen. Similar to the recent proton charge radius experiment performed at the Paul Scherrer Institute,¹ our proposed experiment is a spectroscopic investigation of muonic hydrogen that targets a different transition. In particular, we aim to measure the hyperfine splitting of the ground state energy level of muonic-hydrogen by inducing a laser-stimulated singlet-to-triplet transition (³S₁−¹S₀), as suggested in previous works.^{10, 11}

Figure 2. The laser source and the team at Elettra Lab in Trieste.

The basic outline of the experimental design is as follows. To produce the muonic hydrogen, pulses from an intense, low energy muon beam strike a hydrogen target. The target shall have cylindrical symmetry, with a cross section on the order of a few square centimeters and a length of a few centimeters. The muon beam pulses should have a repetition rate of 50Hz and a pulse duration of 70ns. Measuring the energy difference (³S₁−¹S₀) in the muonic hydrogen requires a tunable laser source at a mid-IR wavelength of 6785nm and a line-width <0.07nm, tunable in the range 6785+xxx−3nm with a high power output. All of these parameters will be subject to further simulation and optimization to reach the final experimental set up.

At present, the main experimental challenge is the realization of the laser source, together with a suitably designed external multi-pass cavity to achieve sufficient transition probability and signal-to-noise ratio. We are pursuing two different approaches. In the first, we consider a set-up based on one or more properly engineered distributed-feedback quantum cascade lasers (QCLs). We acquired suitable QCLs and fully characterized them. Our results show that these sources are stable and simple to operate. They are also tunable over the specified wavelength range, with a satisfactory line-width. One limitation is the delivered energy density, so we are investigating how matrixes of QCLs might operate.

In the second approach, we are studying the realization of a tunable, narrow-bandwidth mid-IR source based on a nonlinear optical system (see Figures 1 and 2). Different nonlinear crystals and frequency down-conversion schemes are under evaluation and testing.

In summary, our present task is to demonstrate the feasibility of a laser source able to induce a sufficient transition probability for the (³S₁−¹S₀) hyperfine splitting of muonic hydrogen atoms produced in a suitable system of hydrogen target, optical-cavity, and muon beam. The successive step will be to plan and execute this high-precision spectroscopic experiment. It is worth noting that while the development of a source for a highly sensitive system in the mid-IR is important for our experiment, it also has potential impact on medical applications and gas detection in security and environmental monitoring. Such wide applicability should stimulate industrial implementation.

The authors are very grateful to the International Society for Optics and Photonics (SPIE) for generously supporting this work through its Anchor Research Program at ICTP (Quantum Cascade Laser project).