The spin-dependent nd scattering length—a proposed high-accuracy measurement
Introduction
In the past few years, a new strategy has been developed to describe nuclear forces at low energy. Chiral perturbation theory (χPT) is an effective field theory, which describes interactions of pions and between pions and nucleons (N). It leads to a systematic expansion of the scattering amplitude in powers of ratios of small momenta and low-energy input parameters like the pion mass over the breakdown scale of the theory. For the first time the accuracy of calculations can be estimated in a model-independent theory of nuclear interactions, providing reliable predictions of many important low-energy quantities. These are ground-state properties of bound systems and processes involving external and exchange currents, as e.g. cross-sections relevant for big-bang nucleosynthesis and stellar fusion [1], [2]. Also the determination of fundamental properties of the neutron from experiments on few-nucleon systems mandates a model-independent subtraction of nucleon binding and meson exchange effects.
Weinberg pointed out that chiral three-nucleon (3N) forces appear naturally in χPT [3]. The most relevant processes are: a two-pion exchange, a 2N contact interaction with pion exchange, and a 3N contact interaction [4]. The contact interactions parameterise the short-range physics. As in Fermi's theory of weak interaction, they are characterised by effective couplings, called low-energy constants (LECs). They have to be fixed by measured data of two independent low-energy 3N observables. Very recently, a first complete analysis of nd scattering at next-to-next-to leading order has been performed with impressive results [5]. All observables are expanded in powers of momenta and the pion mass over the χPT-breakdown scale of about .
An even simpler approach to the nuclear few-body problem is an effective field theory without pionic degrees of freedom [6], [7]. This theory starts out from point-like interactions between nucleons, which only have to respect the symmetries of QCD. Like χPT, it describes phenomena in a systematic way, but is applicable only at energies well below a breakdown scale set by the pion mass. Again, only two LECs characterising 3N forces are required to predict observables with an accuracy of less than 1% in processes involving three nucleons (Fig. 1).
However, this accuracy can only be achieved if the experimental inputs are known correspondingly well. The binding energy of the triton and the doublet nd scattering length a2 are particularly well suited to determine the LECs for χPT and for the pion-free theory. First, there are no Coulomb effects to be considered. Second, a2 is very sensitive to 3N forces: in the quartet channel, the incident neutron has its spins parallel to the one bound in the deuteron, so that the Pauli principle prohibits 3N forces to play any sizeable role at low momentum transfer. In contradiction, the s-wave in the doublet channel allows for a momentum-independent 3N interaction. This turns out even necessary to achieve results which are not sensitive to physics at high energy scales, or respectively, at short distances beyond the range of applicability of the theory. While the triton binding energy is known with an accuracy of 5×10−7, the experimental knowledge of the nd doublet scattering length is only 6%.
Section snippets
Present situation and accuracy goal
The scattering length of a neutron with spin s and a nucleus with spin I is given bywhere a+ and a− denote the scattering lengths in the state with total spin , respectively, . Since one cannot prepare the latter state, it is not possible to measure a− directly. Experimentally accessible are the spin-independent, coherent scattering length ac, and the factor ai which parameterises the spin-dependence (sometimes called “incoherent”
Method
The spin-dependent scattering length induces a spin-dependence of the neutron refractive index. As a result, bi can be determined directly with a polarised neutron beam passing through a polarised target, via detection of pseudomagnetic neutron precession around the axis of nuclear polarisation [10], [11]. The pseudomagnetic precession angle is given bywhere Nk is the number density, Ik the nuclear spin and Pk the nuclear polarisation, with the sum index k extending over
Some practical comments
The choice of the sample is governed by several factors. First, we consider the isotopic composition. Best sensitivity is attained for . DNP keeps the spin temperatures of protons and deuterons equal [18]. Using the Brillouin functions in the high-temperature limit,From Eq. (9),Thus, for Nd/Np≈4%. On the other hand, to keep the systematic uncertainties induced by non-linearities of the NMR resonance circuit small, requires , which happens to
Acknowledgements
This work has been funded by BMBF (contract number 06MT 197).
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