Anti-Hydrogen FormationF. Robicheaux

Auburn University

In 2002, two experimental groups reported the formation of anti-hydrogen.

Both experiments were based on the nested Penning trap.

Anti-protons pass through a cold positron plasma in a strong magnetic field.

Anti-H forms through three body recombination giving Rydberg anti-atoms.

Cold, Magnetized PlasmasTwo groups (ATHENA & ATRAP) attempting to make ground state anti-hydrogen at CERN

G. Raithel group at U. of Michigan setting up experiments to investigate cold, magnetized plasmas in matter.

Previous theoretical investigation of atomic processes have poor approximations

Current, strong interest in atomic processes in cold, magnetized plasmas with almost no existing calculations

Atomic processes are inherently interesting

Cold, Magnetized PlasmasAnti-hydrogen formed from cold plasmas of positrons and anti-protons (B > 3 T)

Goal: precision spectroscopy of anti-hydrogen

Anti-hydrogen must be in ground state

What are properties of anti-hydrogen?

Where are they in the device?Physics TodayNovember 2002

Anti-Hydrogen Results (ATHENA)See Physics Today, Nov & Dec 2002, for general info

Anti-protons pass through cold positron plasma. Positron capture gives anti-hydrogen. Signal is annihilation on wall of trap.

A strong magnetic field, B, along the trap keeps the positrons and anti-protons from reaching the wall.

Atoms cross B-field and reach the wall.

Roughly 108 positrons, 104 anti-protons

Roughly 17% anti-p convert to anti-H

Rate decreases relatively slowly with T of positron plasma

Temperature &Time Dependence

The recombination rate decreases slowly with heating! T-9/2????

Should be possible to explain the ~1/2 s decay time.

Anti-hydrogen Results (ATRAP)

Schematic

SignalSignal from strip anti-H and capture the anti-p

Anti-H must travel ~5 cm to region where detected

Roughly 105 anti-protons and 106 positrons

MeasurementsNumber of anti-H vs number of positronsNumber of anti-H vs distance to detection region

MeasurementsNumber of anti-H vs field required to strip off positron

Gives information about the distribution of n-levels of the atoms.

Basic Ideas (B in z-direction)The strong magnetic field and low temperatures strongly modify atomic processes.

A charged particle in a magnetic field Bmoves in a circle with radiusr = m v/q B.The period of motion isτ = 2 π r /v = 2 π m/q B.

The angular momentumL = r m v = m2 v2/q B = m kB T/q B.At 4 K, an e- has v = (kB T/m)1/2 = 7.8 km/s. For B = 5 Teslar = 8.9 nm = 168 a0, τ = 7.1 X 10-12 s = 7.1 ps,h/τ = 6.8 K, L = 6.3 X 10-35 J s = 0.60 h

r

v

ConclusionsThe transverse motion of the positron might need to be quantized. Modification of scattering? (more likely to add energy into cyclotron motion)

In a classical calculation, it is hopeless to follow the full cyclotron motion of the positron (perhaps symplectic propagator would help).

Positron collision processes (TBR and positron-Rydberg) will be strongly modified.

The anti-H will be strongly modified down to low n; radiative decay completely changed at high n.

Guiding Center Approximation (matter)

Charged particles spiral along magnetic field lines. If there is a uniform and constant E-field perpendicular to the B-field, charged particles drift perpendicular to both fields and an average speed of E/B.

Take the charged particles position to be fixed in xy.

m az = Fz(x,y,z)

Glinsky & O’Neil used this approximation to compute the TBR rate. Found that the rate decreased by factor of 11 from field free rate.

However, the proton cyclotron radius is roughly 7000 a.u. Might need a better approximation.

Guiding Center ApproximationThe next level of approximation can allow the proton its full motion but keep the guiding center approximation for the electron. vy = -Ex/B and vx = Ey/B

30

2

zz

30

30

30

2

zz

x30

2

yy

y30

2

xx

R 4 Z)- (z e vm v z

R B 4Y) -(y e x

R B 4X) -(x e y

R 4z) - (Z e VM V Z

V B e R 4y) - (Y e VM V Y

V B e R 4 x)- (X e VM V X

επ

επεπ

επ

επ

επ

−==

=−=

−==

−−==

+−==

&

&&

&&

&&

&&

Guiding Center ApproximationThere are 4 constants of motionR2 = (x – X)2 + (y – Y)2 + (z – Z)2 + c2

zzz

yy

xx

0

22z

vm VM P

B e X) (x VM KB e Y) (y VM K

R 4e vm

21 V V M

21 E

+=

−−=−+=

−+⋅=επ

rr

When the electron is near the proton it circles with a frequency ω = e/(4 π ε0 B R3)

Three Body RecombinationWe computed the TBR rate by firing electrons randomly at a proton. The electrons have a thermal distribution of speeds. A recombination is determined to have occurred when an electron is bound by > 8 kB Te.

If the region has a length L along the magnetic field, then the probability for launching an electron during intervalP = δt (N/L) (2 kB Te/π m)1/2

m v2 = –2 kB Te ln(y) where y random 0 < y < 1

The TBR rate is the inverse of the average time to recombination.

Three Body RecombinationDefining b = e2/(4 π ε0 kB Te) and ve = (kB Te/m)1/2, the TBR rate can be written as

Γ= C ne2 ve b5

The field free rate has C = 0.76.Glinsky & O’Neil (all charges pinned to field lines) found C = 0.070 (we found 0.072 for same approximation).The guiding center approximation gave C = 0.11 for T = 4, 8, 16 K and B = 3 and 5.4 T. Roughly 50% larger.

Transverse speed of the atoms roughly Maxwell at same T.

Dipoles small fraction of the possible size.

Only a small fraction can be stimulated to low n by photon.

Three Body RecombinationIn the experiments, the anti-protons pass through the positron gas with substantial speed. V0 = (2 kB Te/M)1/2 [V0 ~ 11 km/s at 4 K], Vz is speed of anti-proton, E is KE of anti-proton

1.270.0086/60.880.0115/60.560.0184/60.320.0313/60.140.0512/60.040.0811/60.000.1000/6

E (eV)CVz/V0

Modeling Anti-Proton MotionThe anti-protons are launched at a potential several V above the potential of the positron cloud. We need to model the motion of the anti-protons through the trap.

Positron CloudThe positrons are in thermal equilibrium in a strong magnetic field plus the E-fields from the electrodes. The one particle Hamiltonian is

....y 2B q π vm

z)y, U(x, π x2B q π y

2B q π

m 21 H

xx

2z

2

y

2

x

+=

+

+

−+

+=

The quantity πx y – πy x is a constant of the motion. The distribution function can be written as

exp{–[H + ω (πx y – πy x)]/kB T}

The shape of the cloud is determined by ω and U.

Positron Density

022

B

22

0

)/εrn( e )r U(

T k)/2y (x ω) m B (q ω )rU(exp n )rn(

rr

rr

−=∇

++−−=

The positron density is found by self consistent solution of the equations

with the condition that U/e match the potentials on the electrodes of the trap.

In practice, these equations are solved by iteration.

At low T, n is nearly constant near the center of the cloud 2 ω (q B + m ω) = – e2 n(center)/ε0

Density is roughly ellipsoid

ATHENA Geometry

red-through centerorange- radius of trap/32

ATHENA Geometry

e.g.

Potential decreases proportional to ρ2 through plasmaAnti-proton period computed by integrating 1/velocity

ATHENA Geometry

Electric field shorted in positron plasma

40 V/cm can strip n ~ 55

Energy Loss By Anti-ProtonsThe energy of an anti-proton determines the speed in the positron plasma. Affects TBR and final states due to motion itself and due to duration of interaction.

Energy dumped into plasma waves.

Energy into plasma due to individual anti-proton–positron collisions.

Stripping of weakly bound anti-hydrogen.

Stopping Power in Magnetized Plasma Nersisyan, Walter and Zwicknagel PRE 61, 7022 (2000) analyzed the energy lost per unit length to plasma waves in a magnetized plasma. The expression is quite complicated but not difficult to calculate. ∆E = ∆x dE/dx

This is energy loss to collective positron modes.

)]/k v2 t cos( [1 1 )a(k t)( X(t)

X(t)] 2 t s exp[idt 2 s i 1 F(s) i G(s)

plasma of /width 1 k )r1/(scat k)cos()sin( 1 )cos( )cos(

v/)cos( V s

(s)F G(s)] [kF(s) )cos(ddkdk

4 e 2

dxdE

thc22

c2

0

minminmax

2

th

2

0222

D2

1

0

k

k

3

03

2D

2 max

min

ωµµ

ϕθµθµ

λϕµ

επλ π

−−+=

−+=+

==

−−=Θ

Θ=

++Θ

−=

∫

∫∫∫

∞

Stopping Power in Magnetized Plasma This treatment has two problems

There is only the dissipation part of the interaction. The fluctuation part of the interaction with plasma waves is not included. Set dE/dx to 0 when the anti-proton energy ~kB T

The collision with individual positrons not included. This comes from the kmax. This is included by direct solution of Newton’s equations (guiding center approx) for random positrons fired at anti-proton.

Positron – Anti-Proton Scattering The positron – anti-proton scattering can give a slowing along the field and thermalization of the cyclotron motion of the anti-proton.

The slowing along the field is a smaller effect since the kick along the field averages to ~ 0.

The thermalization time of the cyclotron motion is relatively rapid. For the conditions of the anti-hydrogen experiments, the transverse temperature is roughly that of the positrons

Positron Stripping An anti-proton captures a positron in a region of ~ 0 E-field. When exit the plasma, experiences an E-field. If strip the positron, the anti-proton will lose energy

∆E = -e ∆V

E lost

Strip

Strip

Positron Stripping Effect Change in energy and x,y position:

Calculate the E-field that will strip off the positron.

Follow the anti-H in its motion outside of the plasma.

When the E-field reaches this value, find the value of the potential using interpolation from a coarse grid of computed values.

Save the new x,y position and energy of the anti-proton.

Plasma Heating Anti-protons start with a few eV (1 eV ~ 1.2 X 104 K).

Slow in the plasma the positron plasma must heat.

The plasma cools by radiation.

ATHENA: the temperature hardly changes since the number of positrons/number of anti-protons ~ 104

ATRAP: ratio ~ 1-10. Cooling time/radiation time ~ 100. Rise in T ~ (5 X 1.2 X 104 K/3) ~ 20-100 K in version 1 & 5-20 K in version 2

TBR Revised The three body recombination calculations were steady state calculations.

The experiments have the anti-protons in the positron plasma for a short time.

At 3 km/s ~ 1/20 eV, theanti-proton spends0.3 mm/3 km/s = 0.1 µsin the ATRAP geometry &30 mm/3 km/s = 10 µsin the ATHENA geometry.

TBR Three Body Capture The time in the plasma is shorter than the time for a recombination.

The binding energy will be less than might be expected.

The ATRAP geometry particularly affected.

Preliminary Results of Simulation3 slowing mechanisms (excitation of plasma waves, positron/anti-p collisions, capture then strip). Solve Newton equation for anti-p and positrons.

ATHENA

30% recombine at 15 K, 12% recombine at 30 K: roughly correct amount of anti-H & T dependence

Motion of anti-H is highly directional

Lower T positron plasma gives more deeply bound anti-H

Velocity of Anti-H || B

Thermal~500 m/s

For deeply bound anti-H ½ > 4X thermal speed

How to stop a 2 km/s anti-H?

All anti-H, even those too weak to reach the wall

Energy Distribution

BE = 120 K corresponds to n = 36

Preliminary Results of SimulationATRAP

Thin positron plasma dominates processes---capture with ~no subsequent collisions.

Roughly 1/4000 – 1/10000 recombinations

Highly directional motion

Very weakly bound positrons

Binding Energy

Most atoms too weakly bound to survive trap E-fields.

Relatively little difference w/ T!

Vertical line marks states that survive 25 V/cm E-field

BE = 30 K corresponds to n = 72

Comparison

ATHENA: n = 2.5 X 108 cm-3, width = 32 mmATRAP: n = 4 X 107 cm-3, width 0.4 mm & 1.6 mmBE of 40 K needed to survive a 25 V/cm field

Comparison

Large fraction of the atoms have high velocity along the B-field.

FutureFull positron-Rydberg collision, can cyclotron motion of positron couple to motion of Rydberg positron?

Non-guiding center approximation?

Quantum mechanics of positron/anti-H collisions?

Radiation of anti-H to ground state

Suggestions for improvements (sacrifice width for thickness!) to configuration

Evolution of ATRAP positron plasma

Can Rydbergs be driven to deeper binding?

Other anti-H formation mechanisms?