Measuring the Beta-Neutrino Angular Correlation in the Decay of Magneto-Optically Trapped Sodium-21 Atoms

Some Theory

The basic idea in the experiment is to examine the structure of the Weak Interaction between leptons (electrons and neutrinos) and quarks (which live inside the protons and neutrons in the nucleus).  The theory of the Weak Interaction is included in the larger theory called the Standard Model, which declares that the leptons and quarks interact Weakly by exchanging gauge bosons -- heavy, force-carrying particles whose properties are determined by the gauge group or symmetry group of the theory.  We have lots of reasons to believe that the Standard Model is not complete.  There might be other bosons of new gauge groups which have much higher energies or much weaker couplings.  If that is the case, we may be able to detect them in precise studies of nuclear beta decay by examining the kinematics of the decay products.  

You can calculate that the phase space for nuclear beta decay (loosely speaking, the decay probability) has several terms which depend on the momentum of the beta particle (e) and the neutrino (nu):

This very general result comes from a paper which is renowned in this field: J. D. Jackson, S. B. Treiman, and H. W. Wyld, Phys. Rev. 106, 517 (1957).  The "p" terms are momenta, the E terms are the energy, and the I terms are the nuclear spin.  The colored letters are correlation coefficients which tell you how much these various terms contribute to the decay rate.  The different terms contribute to the decay rate depending on how the vector variables in the decay are arranged.  In the case of the beta-neutrino correlation -- a_(beta-nu) -- the decay rate is larger if the momenta p_e and p_nu are parallel.  In sodium-21, a_(beta-nu) is positive and about equal to 0.55.  The correlation coefficients can be calculated if you know the half-life of the nucleus*, the total decay energy available*, the branching ratio (if there are other states that the parent nucleus can decay to)*, and the spins of the parent and daugther states*.  When you calculate these decay correlations, you assume that the Standard Model of the interactions between the leptons (beta and neutrino) and quarks (in the nucleus) is the correct description.  At low energy (the nulcear decay energy of 2.5 MeV) compared to the Weak scale (the Weak boson masses of 90 GeV), the Standard Model says that this semi-leptonic Electroweak interaction is pretty much just a contact interaction between the lepton currents and the quark currents, and that the only terms of this interaction are the ones that know about the vector and axial vector parts of these currents.  But what if there were other bosons besides the W, Z, and photon (gamma) which mediated interactions between leptons and quarks?  In that case, the most general hamiltonian for the interaction between leptons and quarks at these energies could be written:

Here the Psi's are the wavefunctions of the parent and daugther nuclei (psi_1 and psi_2) and the electron and neutrino wavefunctions.  The C's are coupling constants, which you just assume from knowing what the Standard Model gauge groups are.  The Standard Model says that the only terms which are non-zero are C_A and C_V.  Everything else is zero, according to the model.  

We measure the beta-neutrino correlation by inferring the momentum of the neutrinos from the momentum of the recoil of daughter ions from their time-of-flight. A precise measurement of the correlation coefficient can limit the existence of scalar or tensor currents from higher mass weak bosons present in some extensions to the Standard Model.  If we know these coupling constants, we can easily calculate the correlation coefficients in the beta decay phase space.  

If we measure something about the beta decay, we can extract the correlation coefficients, which then tell us whether we had the coupling constants right, or whether we should have to modify the Standard Model to include scalar and tensor coupling terms (C_S and C_T).  

Optically Trapped atoms as a Source

The atoms are produced here using the 88" cyclotron by bombarding a MgO pressed-powder target with 25 MeV protons. The sodium-21 created in a nuclear reaction with the protons and magnesium nuclei.  The sodium is boiled off from the target by heating an oven surrounding the target to roughly 1000 C. The hot atoms are collimated into a beam emerging from the target oven.  This beam of atoms is slowed down by a Zeeman slower.  The slow atoms are captured by a magneto-optic trap formed by lasers and a quadrupole magnetic field.  Up to one million radioactive atoms are confined at the center of a vacuum chamber in a roughly spherical blob about 0.3 mm across. The atoms undergo nuclear beta decay, releasing a beta particle (positron) and a neutrino, with about 2.5 MeV of total energy.  The daughter nucleus (neon-21) gets a small recoil energy from the decay of about 100 electron-volts.  The atom trap is placed between two microchannel plate detectors and between electrodes which create a smooth, constant electric field of about 1 kV per centimeter.
We can measure the beta-neutrino correlation (a_(beta-nu) ) by measuring the time-of-flight of the daughter nuclei.  The daughter (neon) emerges from the beta decay as a neutron or charged atom (ion), and the amount of time that it takes to strike the microchannel plate depends on its initial velocity.  One microchannel plate detects the low-energy atomic electrons that are shaken off of the neon atom after the beta decay process, while the other one detects a positively charged neon ion.  

Trapped atoms are great for this type of experiment because:

• source scattering effects are negligible(**)
• the sample is isotopically pure -- the laser trap only catches sodium-21 atoms
• the trap is localized in a small volume, so we can reliably calculate the time of flight
• the atoms decay at rest, so we don't have to worry about any other initial velocity
• they can provide a polarized sample if you use lasers to manipulate the nuclear spin hyperfine state

Beta-Coincidence Measurement

We completed a measurement of the beta-neutrino correlation coefficient in which the ion detection was triggered on positron detection, using a separate positron detector. The measured coefficient was a=.5243(0.0092), a deviation from the Standard Model at the 3.6-sigma level. The Standard Model predicted a=0.558(0.003). Shown below is the observed time-of-flight spectrum. The spectrum was fitted to a Monte-Carlo simulation, and the beta-neutrino coefficient, was extracted. The details of this work can be found in Nick Scielzo's thesis, and were published in Physical Review Letters, volume 93, p. 102501 (2004).  Had we just found an indication of Beyond Standard Model physics?  Unfortunately, not.  Our answer seemed to depend on the number of atoms we had in the trap, as if there were some density-dependent effect happening.  Like collisions... 

Why the Deviation from the Standard Model?

One of the key assumptions we made about how great atom traps were turned out to be wrong.  Our atoms do not necessarily have negligible source scattering.  Magneto-optic traps are a great location for atoms to form molecules by undergoing "cold" chemical reactions.  When an atom is trapped in a typical  magneto-optic trap, it is constantly cycling from its ground to excited state (so it glows brightly as it emits photons).  This is a nice feature because it allows us to see where the trap is.  But if two atoms collide while they are in their excited states, they can stick together to form a molecule and emit photons with a slightly different energy which is the sum of the molecular binding energy (quite small) and the atomic excitation energy (pretty big).  The molecules of sodium (Na_2) can be confined by the magnetic field of our magneto-optic trap.  If one of the atoms beta decays, the daughter nucleus now scatters off of its little bound partner, sharing its final momentum with its molecular mate.  This fouls up our measurement of the momentum/energy of the recoil nucleus, which contained the information we wanted.  We discovered this problem when we began using a new technique, described in Nuclear Physics A .  

Polarized Atom Traps

You can polarized the nucleus by manipulating the hyperfine state of the atom.  The hyperfine interaction allows you to change the nuclear spin by using the valence electron's spin as a crowbar.  The hyperfine state is labeled with the quantum number F, the vector sum of the nuclear spin I and the total electron spin J.  The ground state of the sodium atom has two possible hyperfine states F=1 and F=2, separated by about 2 gigahertz.  If you put the atom into the F=2 hyperfine state, then separate out the Zeeman magnetic sublevels of this state, you find that the state with F=2, m_f = 2 (or -2) has a nuclear spin with I = 3/2 and m_I = 3/2 (or -3/2), as large as it can be (stretched).  If you put the atoms into this state, then the nuclei must all have their spins fully aligned to the magnetic axis.  You can do this by optical pumping, but to do that, you have to turn off the magneto-optic trap, which scrambles the polarization state of the atoms (and hence the nuclei).  What you need is a dark trap -- something that holds on to the atoms without letting them interact with the laser light after you have optically pumped them to the state you want.  

What about a red-detuned Far-Off-Resonance-Trap (FORT)?

It's great!  A red-detuned FORT traps atoms at the highest intensity of light in a single, focused laser beam.  And if you tune the laser to a magic frequency between the D1 and D2 transitions of an alkali, you can selectively trap the atoms in the stretched hyperfine state!  Too bad it doesn't work for sodium, where the D1 and D2 transitions are rather close together, and the photon frequency (or energy) is just too dang large, so that the scattering rate of photons in a red-detuned trap is unacceptably high.  Rats.  You can do this with other alkalis, like cesium and rubidium, though.

Blue-Detuned Far-Off-Resonance-Traps

Well, in this case, the atoms are repelled from the high intensity region of a laser beam.  That means that you have to get a little tricky to make a trap.  The atoms will run away from the laser, so you have to build a "blue box" of light for the atoms to rattle around in.  One way is to scan the laser beam around in a little circle, so that you make a little ring of light, and have the scanning frequency of the laser be very fast compared to how quickly the atoms move around in the trap, so that they always see a wall.  
Here's a little scanned circle trap using a 2D acousto-optic modulator controller.