Calculation of the positronium hyperfine pressure shift in buffer gases

Abstract

An approximate quantitative theory of the positronium hyperfine pressure shift (HPS) is discussed. Adopting the approach of Adrian, the fractional shift Δ (R) is calculated as a function of the positronium–buffer gas separation R; the measured HPS is obtained by a thermal average of Δ (R) over R. A quantum mechanical averaging technique is used, rather than classical methods, in which the Boltzmann factor exp[−V (R)/kT], where V (R) is the intermolecular pair potential, is replaced by the positronium center-of-mass wavefunctions. The fractional shift Δ (R) is divided into two parts; a negative long range van der Waals contribution and a short range positive exchange contribution. It is argued that in the positronium center-of-mass approximation the short range contribution to Δ (R) is the same for both hydrogen and positronium and therefore existing calculations on hydrogen are utilized. Perturbation theory is used to obtain the long range contribution of Δ (R) and two models are used to evaluate the matrix elements that occur. In either model the long range contribution is related to the van der Waals interaction and can be obtained from a power series in R−1. A hard core potential model of the problem is solved exactly and predicts the HPS to be nearly temperature independent, a result essentially due to the long deBroglie wavelength of thermal positronium. Finally, the wave equation is numerically integrated with realistic potentials to find the positronium center-of-mass wavefunctions and the pressure shifts. Results of the numerical calculation also indicate a relatively small temperature coefficient of the HPS. The calculations of the positronium hyperfine pressure shift have substantially better agreement with measured pressure shifts than do calculations on hydrogen.

The hyperfine frequency shift of atomic hydrogen in the presence of helium buffer gas

“…I. They resemble each other in the case of ~ositronium (9). In this Paper we Present a the sense that both are positive and steeply sloped for quantum mechanical treatment of the thermal average small R and both are negative and fall off as ~-6 for of the HFS for H atoms in the presence of He buffer gas, large R. The zero-crossing of the HFS occurs slightly making use of the best available data for the H-He outside that of the potential.…”

Precision Measurements in Positronium

“…Since the theory of the Ps hyperfine pressure shift is not sufficiently advanced, 114 ' the systematic shift is removed by extrapolating to zero gas density. Fig.…”