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Physics with many positrons: positron sources and positron beams

The problem :

Many-positron experiments in matter need bunches of 10³ or more e+ within ca. 0.2 ns on spots of ca. 100 nm

NEPOMUC, the most intense current e+ source, delivers

ca. 1 e+/ ns ( 100 pA ) into a spot of 7 mm FWHM

( at E = 1keV , D E = 50 eV , B = 10 mT )

Compression required : in space by 5 , in time by 4 orders of magnitude

A possible solution : further upgrade of the NEPOMUC / Scanning Positron Microscope ( SPM ) facility at Munich

SPM at FRM-II:

• up to 106 events / second

• Reduced measuring time (factor 50, ca. 1 h / picture)

• improved lateral resolution (≤ 1 μm) intended: 50 - 100 nm lateral resolution

G. Kögel : Remarks on particle optics

( Phase space, paraxial optics, aberration correction, bunching, remoderation )

Case study of the Munich Positron Microscope

( Various beam lines, remoderation stages, application of optical principles in design and construction, outlook to many positron pulses )

Ch. Hugenschmidt : Positron sources, positron beams

W. Egger : Applications of pulsed positron beams

Remarks on particle optics Ray tracing : determine by finite element methods E(r) and B(r),

Then solve d²r/ dt² = q ( E(r) + v x B (r) ) ; v = dr/dt

Crossing of trajectories in real space is confusing

Representation in Phase Space x i , p i = d L ( x k , v k ) / d v i , i,k = 1,2,3 . . Then Liouville´s Theorem holds : D x Dp x = W x = const ; DE Dt = W E = const

A beam is defined by W = const ; Trajectories never cross in phase space

A beam is completely described by the energy E, intensity I, W x , W y , W z.

Brightness B = I E / ( W x W y ) = positrons / ( source area x solid angle )

Exercise : Which energy spread do you expect, if the cross section of the NEPOMUC beam was shrinked to the necessities of multi positron experimentation ?

Solution : W x W y =d² ( Dp )² = d² DE trans / 2m+ = 1 cm² x 50 eV /2m+

At d = 100 nm we get from Liouville`s Theorem

D E (at experiment ) = 500 GeV

Spatial focussing : The force should increase linearly with distance to axis

Perfect Solution : Quadrupole

E x = c x ; E y = - c y or B x = b x ; B y = - b y

⇒Strong focussing in y- direction strong defocussing in x–direction

Quadrupole optics ideally suited for a line focus

For a spot, one needs at least two quadrupoles in series expensive So far not applied for keV positron beams

Round lenses

Coaxial sets of tubes, cones, …where E and B have rotational symmetry

E r = - r/2 dE z / dz + r³/16 d³E z / dz³ + … ( also for B )

In the paraxial approximation to the equ. of motion, only linear terms contribute:

Same equ. for y(z). If there is a B-field, the x- and y- axis rotate around the z- axis by an angle Q(z) defined by

dQ /dz = qBz /2mvz ( rotating system ).

Only in the rotating system the x- and y- motions are independent ! In the lab system, positrons entering the B-field parallel to the z-axis from a field-free region gyrate around this axis with azimuthal angle Q (z) !

In the par. approx., any solution is a linear combination of two principal rays ( linear differential equ. of 2.order ). This is the foundation of linear optics.

0 )(4

)( 2)( 0

2

0 2

2

0 =−

+ +⎟

⎠ ⎞

⎜ ⎝ ⎛ − x

zUU

zB m q

dz Ud

dz dxzUU

dz d z

A typical electric lens with the two principal rays

The focal lengths f 1 , f 2 and the positions of the foci depend strongly on the potentials. They define the optical properties.

Magnetic single pole lens with magnetic field lines

Standard magnetic lens

Note : The radial fields at entrance and exit counteract. => least aberrations for a focus close to the maximum of E z

Liouville´s theorem and paraxial optics

Source paraxial system focal spot

2 a s 2 a f

D s , Dp s , E s D f , D p f , E f

D f = D s D p s / D p f => D² f = D² s ( a² s E s ) / ( a² f E f )

Note : The minimal spot size is always close to a focus. For an image, the beam must expand again, until each resolved pixel has ~ the size of the focal spot. This simplifies our task. But a f is limited by aberrations because of the third and higher orders in the expansion of radial focussing fields

Spherical aberrations

Disk of least confusion slightly before the paraxial focus

Focus location : field free region close to field maximum magn.lenses of SPM Aberrat. Const C s > 10 f C s ~f C s = 0.4 f

⇒Even under the most optimistic assumptions, the present NEPOMUC beam can not compressed to a spotsize below 2 mm

Scherzer´s Theorem (1937) : With only round lenses and no space charge between trajectories and the axis , the spherical aberration is always positive.

6 focus

2 S

focus||,

source||, 2 focus

2 source2

source 2 min 4

α α α C

E E

dd +=

Feasibility of aberration – corrected positron beam systems ?

In electron microscopes, corrections by sextupoles and multipoles of higher order have been achieved eventually, after 60 years of failures.

Since for multi-positron experiments the spot sizes are orders of magnitude larger than in electron microscopes, and since we need no image of the source, the correction by the induced charge in grids is feasible without intolerable perturbation by the meshes of the grid

Correction by space charge: The main deflection is due to a magnetic lens ( not shown ). The positrons experience only outward directed correction forces

Also correction by multipole optics could be considered

Compression in the time domain : bunching

Liouv. Theor. => D t (bunch) > D t D E / D E (bunch)

Because of aberrations, D E (bunch ) < E / 10 ⇒T drift > 20 D t

L drift > 40 D t [ ns ] ( E [ keV] ) ½ [ cm ]

The voltage at the buncher gap, U(t), must change rapidly : dU / dt = D E / D t (bunch ) = 10 …. 100 V/ns

Usually the buncher is part of a rf circuit operated in resonance

D E (bunch)

Buncher types

Coaxial resonator

( Q ~ 100 , L = 1.5 m at 50 Mhz )

Double gap buncher,

Q ~ 300 – 1000

Disadvantages of resonant (sine wave ) bunchers

Only ca. 10 % of a continuous beam are bunched, 90 % produce background

Available options : 1. Sawtooth prebuncher ( SPM )

U ~

U~

Up to 70 % in bunch

2. Storage trap / harmonic prebuncher

Slow e+ ( 0.2 mm/ns at 100 meV ) are collected for µs and released by suddenly switching to the harmonic potential

Beam chopper : Removal of the unbunched background

U(t)

open

t

a= 2 U(t) l / (Uc w)

Typically : pulse FWHM 3 ns , rise time of U (t) ~ 1 ns , amplitude 2V

The fringing fields at the entrance to and exit from the chopper plates modulate as well the longitudinal beam energy since the exit of the positrons is delayed relatively to the entrance by the transit time v/l of the positrons !

w

End of the tutorial introduction into fundamentals of particle optics and functional elements of the Munich positron beams

Conclusion

Without new tools from outside the range of traditional particle optics, there is no chance for true multi-positron experiments, at least at NEPOMUC .

Remoderation, The key concept of positron particle optics

W

Re-emitted positrons constitute a narrow , pulsed source with energy spreads in the few kT range ( details depending on material, surface properties and vacuum conditions ). The ( more intense) backwards emitted beam must be separated from the parent beam.The easily extracted beam in forward direction (transmission mode) depends on very thin ( ~ 100 nm ) remod. crystals.

crystal effic. ( reflexion ) cleaning technique further remarks W(100,110) ~ 20 % ~ 2500 K in UHV very reliable ( > 10 years) Ni(100) ~ 30% sputtering, H2 - n-SiC ~ 60% uncleaned, 10 y in air energy spreads unknown

Summary on particle optics in relation to positron experiments

Phase space considerations give us a reliable, simple insight into the feasibility of a project or the usefulness of existing parts.

Numerical simulations ( ray tracing )are only needed for the optimization of critical components, e.g. spot forming lenses.

The main handicap of positron beam physics is the low brightness of all positron sources (when compared to electron sources)

This is partially compensated by remoderation of positrons. Also correction of spherical aberration should be more efficient and less expensive as in electron optics.

To remove the enormous adiabatic heating of the beam due to the compression in an ( optical ) beam system , a sequence of remoderation stages is necessary for spots / pulses in the sub µm / sub ns region. This introduces dramatic losses in int