phase space evolution - nc state...
TRANSCRIPT
Phase Space EvolutionPhase Space Evolution
Chen‐Yu LiuIndiana University
Nov. 9, 2012Neutron Lifetime Workshop (Santa Fe)
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UCN Trap Experiments
Measures the Storage Time
2Steyerl Walstrom
Material Bottle
A. Serebrov (2008)
D. Dubbers (2011)
How well to measure each non‐beta‐decay loss?
Each loss is a new degree of freedom, so this is a vector sum.
If the trap storage time is long, so that the correction to the beta‐decay lifetime is
12 days!
• need to measure τab at the 10% level of precision. 4
NIST Lifetime Experiment
• Discover marginally trapped UCN.
Fi ld l d t• Field lowered to 30% of the max value (1.1 T).
• 50% of UCN was flushed out
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flushed out.
Motion in Phase SpaceMotion in Phase Space
• Phase space trajectories don’t intersect, because of the p j ,deterministic mechanics.
• If the Hamiltonian is time‐independent (autonomous), the trajectories in phase space are invariant.
• Liouville’s Theorem: The Hamiltonian flow in phase space is incompressible. The density of particles is p p y pinvariant (if the particle number is conserved).
• The phase‐space volume of an statistical ensemble i i i t H ith h th “ i d”is invariant. However, with chaos, the “coarse‐grained” phase‐space volume is always increasing.
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Phase Space Evolution in UCN traps1. Fill : populate the phase space.
– Is the filling uniform? – How long does it take to establish static equilibrium?
Young
How long does it take to establish static equilibrium?
2. Spectrum Cleaning: define the trappable volume in the hphase‐space.
– How long does it take to clean sufficiently?Bowman
3. Storage: Phase space evolution– Additional loss (when the trappable volume morphs to intersect with loss boundaries)
Coakley
intersect with loss boundaries)
4. Detection: a fast sink in the phase‐space.d f f d ff– In‐situ detection: Uniformity of detection efficiency
– UCN drain: detection efficiency changes?7Golub, Morris
Trap Lifetime Experiment viewed in Phase‐Space
pField Potential
p
MaterialMaterialAbsorber
absorber(bottom Of the Trap)
vv
xp)
To detector
v During StorageAb ti /• Absorption/upscattering
• Spin flip• Slow
g
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Field Potential• Slow
evolution• Heating
UCN Dynamics in TrapsUCN Dynamics in Traps
• Integrable potentialg p– A Hamiltonian with N degrees of freedom is integrableif and only if there exist N independent isolating integrals (action variables: global invariants of theintegrals (action variables: global invariants of the motion).
– Produce regular orbits
• Near‐Integrable potential– D = 1,2: mixed dynamics, with regular trajectoriesD 1,2: mixed dynamics, with regular trajectories separating regions of stochasticity.
– D > 2: “web” of stochastic regions. Arnold diffusion.
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Asymmetric Trap Phase Space MixingL ( h i h fi ld i l ) i d i i• Low symmetry (together with field ripples) induces states mixing between circular orbits, through chaotic motion (or not).
• quick cleaning (~ seconds) of the quasi‐bound UCN with large tangential velocities.
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Halbach Array (NdFeB Permanent Magnets)
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UCN CleaningUCN Cleaning
6Emax = E0 + 6 neV
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Poincare’s Section of SurfaceTwo degrees of freedom
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Height vs. Transverse VelocityHeight vs. Transverse Velocity
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UCN motion becomes chaotic when the height reaches R/2.
Three degrees of Freedom=90
gDifferent azimuthal angels
=0
I i Increasing
15Increasing stochasticity with
Trap GeometrySymmetric vs Asymmetric
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Strong perturbation, KAM curves are destroyed.Thin stochastic regions bounded by KAM curves.
Arnold Diffusion =90
=03 degrees of freedom
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Spatial non‐uniformity of detection efficiency (acceptance)efficiency (acceptance)
Summary• Although quasi‐bound UCN can be cleaned from most traps, the evolution of the phase space motion and the p , p peffect of the experiment (fill, empty, detection, leakage) has not been even seriously considered in most previous experiments.
• Chaotic d namics ( ork in progress)• Chaotic dynamics (work in progress)• Small perturbations of an integrable potential could have narrow stochastic regions near resonances Thehave narrow stochastic regions near resonances. The evolution could have a long evolution time.
• Large perturbations (with discontinuous derivatives) increase stochasticity.