Bouncing Neutrons and Fundamental Physics

P. J. S. Watson

Jan 30^th, 2003

Outline

1. What is the Bouncing Neutron?
2. How can we see it?
3. How can we purify it?
4. What could we do with it?

Introduction: Ultra-Cold Neutrons

Neutrons with E ≤ 10^-5 eV are totally reflected from several metal surfaces, including Be. Note this is a quasi-classical phenomenon (coherent interaction of neutron with many atoms)

• Predicted (Zeldovitch 1959)
• first observed Dubna 1969
• ==> construction of Neutron Bottle
• Containment time > 100 s
• Note (surprisingly) very little energy transfer from walls: can have T ~ 1mK neutrons in a T ~ 10 K container
• f ~ 10^-9, since we are dealing with tail of M-B: however, since the flux is large, UCN's are plentiful.

Hence the "neutron bounce" state: a bound state of the neutron produced by the hard surface and gravity.

Theory:

Bounce eigenfunctions Z_n(z)

- 2 2m d2 dz2 Z( z )+( σz-E )Z( z ) = 0 , σ = mg

This can be converted to a dimensionless form via the substitution

y = βz-yn
β = (2mσ)1/3
     2

Totally reflecting "ground" ⇒Z(0) = 0.

d2Zn(y) +(y-yn)Zn( y ) = 0
dy2
Zn(-yn) = 0

Equation for the Airy function, y_n is n-th zero. Hence

energy En "bounce height" zn

En = 2β2 yn 2m zn = yn β

For neutron,

n	En	zn
1	1.41 peV	13.7 μ
2	2.46 peV	24.0 μ
3	3.32 peV	32.5 μ
4	4.08 peV	39.9 μ

Bound states transitions.

ν_2→1 = 254 Hz (∼ middle C!): implies

• Detector must be ∼1000 km in size
• Temp of detector ∼1 pK
• Lifetime of state ∼10⁸⁰ years

Observation: the Nesvizhevsky Experiment

(Nature 415,297 (2002).)

Proof of existence

Neutron absorber is lowered, extinguishing signal. Classical prediction is for signal to vanish smoothly, Q.M. is for sudden cutoff at height h

Compare classical prediction (UCN's of all bounce heights, but M-B distrib) to quantum prediction (no bounce height ≤ 13 μ)

Blow-up of lowest part of plot

So it exists: what do we do next?

The Neutron Centrifuge

The basic concept

UCN's are injected into the centre of the apparatus. High energy neutrons will bounce over and centrifuged out (B) Lowest energy bounce states will be trapped.(A)

Does this work in Q-Mechs?: 3-D Schrodinger equation which describes the neutron:

-2 ∇2Ψ(r) + (σz-E)Ψ(r) =0
  2m

subject to the boundary conditions Ψ(surface) = 0;

Solve:

initial probability P0(r,z) = | <Ψ0(r,z) |2

show the corresponding

probability P100(r,z) after 100 passes.

Purity of the ground state. The total flux is also shown

Can measure "purity" of ground state.

These are for l = 1: works better for large l

Note device is entirely passive: should separate lowest bounce state very effectively.

Finite Penetration effects:

( courtesy of Mike Pendlebury)

δV = δVR+iδVi where (for Be) δVR = 252neV,δVi = 1.26peV

Real shift:

δEn = PnEb∼0.01peV ∼ 1% 

V_i gives finite lifetime: arises from neutron absorption and inelastic interactions with H in the walls.

τ∼1.4×105s

Neutron halflife ∼885 s, so shouldn't be a problem

Applications

1. Tests of Quantum Mechanics
2. Magnetic Effects
3. EDM measurement
4. Short Range Behaviour of Gravity

Tests of Quantum Mechanics

e.g could apply physical oscillation , ,see Felber et al

Magnetic Effects

Static mag. field produces polarized neutron states

Spin ↑ and ↓ would have different bounce heights

Fmag = Fgrav dB = mg ∼ 1.7 Tm-1 dz dmag

Varying mag. field would allow excitation of state

B(z,r) = B0(z,r)sin(ωt)

Matrix elements vanish if the field is spatially uniform, but easy to arrange for varying field

Tn = <n|dmag.B|1>

The probability for resonant transition to the first excited state

P = Ω2 sin2( γt )
    γ2

where

γ2 = Ω2+δω2,
Ω2 = Tn2 
     2

δω2 = (ω1-ω2)2- ω2

The frequency γ must satisfy

γ << | ω1-ω2 | = 254Hz

which implies a maximum magnetic field of a few milligauss.

EDM measurement

"Natural" size of neutron EDM in absence of CP is 10^-13 e cm.

Expected values are model dependent: hence improving limit is important constraint (diagram taken from Ramsey)

Can use analog of mag. mom. argument

dmag.B  ⇒ del.E,  
E(z,r) = E0(z,r)*sin(ωt)

However process is dominated by neutron lifetime Limit obtainable ∼ 10-22 Current exptl limit ∼ 5 x10-25

Short Range Behaviour of Gravity

The gravitational potential is taken to be

λ in m^-1, so we would expect the stats to be sensitive to λ ∼ 10⁵. Current expts (variations of Cavendish expt.) give λ<10² m^-1for K∼G.

Putting slab of dense material below apparatus would shift energy levels:

Extra interaction is δV(z) = 2πKρmN λ2 ( λz+1 )e-λz (31) This is for K = 1: i.e. not a useful result.

However we could oscillate slab (amplitude z₀, frequency ω₀)to produce an extra pot. term

δV(z,t)=sin(ωt)2πKρmNe-λ(z+z0) ((λ(2z+z0)+4)I1(λz0)-2λz0I0(λz0)) 
            λ2

Transition time as a function of λ for K = 1. Could probably set limits for K ∼ 10-5 for λ ∼ 10 μ-1 better than anyone else, but not useful for K ∼ G

Conclusions

• Bouncing neutron is a novel quantum state
• Can be isolated in ground state
• Possible tests of Q mechs in classical regime
• Can be magnetically excited
• Does not provide a useful limit on EDM
• Could provide a useful limit on short dist. gravity-like effects (e.g. 5th force)
• Any of these experiments would be very hard, fluxes are small

A final thought:
What is magic about the neutron?
could we use a bouncing atom (e.g. He₃) or molecule (e.g. buckyball)?

Advantages

• Much easier to provide in large numbers (atomic fountain expts.)
• Stable (so we are not limited by half-life)

Disadvantages

• Van der Waals forces are always attractive, may bind atom to wall (e.g. He-He bound state exists with BE ∼ 10^-5 eV ∼ 10⁷ peV)
• Atomic X-sections are much larger, so lifetime may be limited by quality of vacuum, rather than interaction with walls

so maybe!