Lanzhou Summer School 2004 Frontiers in Atomic and Molecular Physics with Charged Particles Klaus Blaum European Organisation for Nuclear Research, CERN, Physics Department, 1211 Geneva 23, Switzerland and Gesellschaft für Schwerionenforschung, GSI, 64291 Darmstadt, Germany Email: [email protected]or [email protected]http://isoltrap.web.cern.ch/isoltrap
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Frontiers in Atomic and Molecular Physics with Charged ... · Frontiers in Atomic and Molecular Physics with Charged Particles Klaus Blaum European Organisation for Nuclear Research,
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Lanzhou Summer School 2004
Frontiers in Atomic and Molecular Physics with Charged Particles
Klaus BlaumEuropean Organisation for Nuclear Research, CERN,
Physics Department, 1211 Geneva 23, Switzerland
and
Gesellschaft für Schwerionenforschung, GSI, 64291 Darmstadt, Germany
Storage of charged particles in a Penning / Paul trap
BUT: sign of different !
no simultaneous trapping in 3 dimensions possible by purely electrostatic potentials
SOLUTION:
a: superposition of magneticfield in z-direction:
Penning trap
b: time varying voltage (RF) between ring electrode and endcaps:
Paul trap
r
Paul trap geometries
URF
3D confinement
The linear RFQ trap
end cap
ring electrode potential
Equation of motion in a Paul trap
( ) 00 =+∇ rmt,re &&ρφ
( ) 020
000 =+Ω+ ρρ
ρ&&
mtcos VUe
( )02
0
000 =+Ω+
− zz m
tcos VUe&&
ρ
mr = 0··∇
z = 0··
ρ = 0··
equation of motion:
220
00rz mr
Ue8a2aΩ
−=−=t2Ω
=τ ∼ U0substitution:
220
0042
Ω=−=
mrVe
qq rzu = x, y, z ∼ V0
( ) 0u 2cos q2ad
uduu2
2=τ−+
τMathieu differential equation
Ion motion in a Paul trap
Solution for small amplitudes:
( )ρω −
Ω−∝ ttqtu u
u cos cos 2
1)(
macro motion (slow)micro motion (fast, RF)
Ion trajectory
Stable motion only for special combinations of U0 and V0
STABILITY DIAGRAM
Stability diagram of a Paul trap
Principle of Penning traps
Cyclotron frequency: Bmqfc ⋅⋅=
π21
B
q/m
PENNING trapStrong homogeneous magnetic fieldWeak electric 3Dquadrupole field
z0
r0
ring electrode
end cap
Frans Michel Penning(Penning discharge 1936)
Hans G. Dehmelt(Nobel prize in physics 1989)
Confinement in a Penning trap
axial harmonicpotential
radial confinement with magnetic field
Penning trap configurations
mm
Uz (V)
0
50
100
-50
-100
0 40 80
Cylindrical Penning trap
Potentialdistribution
main electrodescorrection electrodes
0
50
100mm
0
5
10
Hyperbolical Penning trap
main electrodescorrection electrodes
Equation of motion in a Penning trap
F = −e0∇φ(r)+v×Bplus Lorentz force:
F = −e0(∇φ(r)+v×B) + mr = 0··equation of motion:
axial oscillation
02
20
00 =+⋅ zmzmd
Ue&&m z = 0··
20
002md
Uez =ω z or axial
frequency
radial oscillation
242
22zcc ωωω
ω −+=+
242
22zcc ωωωω −−=−
modified cyclotronfrequency
magnetronfrequency
iyxu +=
Bme
c0=ω
( ) tieutu ω−= 0
0uu2
ui2z
c =+ω
−ω &&&
substitution:
·u - ·u=
Ion motion in a Penning trap
Motion of an ion is the superposition of three characteristic harmonic motions:– axial motion (frequency fz)– magnetron motion (frequency f–)– modified cyclotron motion (frequency f+)
The frequencies of the radial motions obey the relation
c-+ fff =+
magnetron motion (f-)
modified cyclotronmotion (f+)
axial motion (fz)
zr
r-
r+
Typical frequenciesq = e, m = 100 u,B = 6 T⇒ f- ≈ 1 kHz
f+ ≈ 1 MHz
L.S. Brown, G. Gabrielse,Rev. Mod. Phys. 58, 233 (1986).
3
2
1
0n +
n+
0 1 2
0
n z
n_
Landau levels of an ion in a Penning trap
modifiedcyclotron frequency
axial frequency
magnetron frequency
Energy of harmonic oscillators:
E = ηω+(n++1/2) + ηωz(nz+1/2) - ηω-(n-+1/2)
amplitudes:
<ρ> ∼ n +12
⇒ magnetron motion is unstable !
cooling: quantum number n → 0
Excitation of radial ion motions
Dipolar azimuthal excitationEither of the ion's radial motions can be excited
by use of an electric dipole field in resonancewith the motion (RF excitation)⇒ amplitude of motion increases
without bounds
Quadrupolar azimuthal excitationIf the two radial motions are excited at their
sum frequency, they are coupled⇒ they are continuously converted
into each other
+Ud -Ud
r
r0
Ud
-Uq
-Uq
r0
r
+Uq+Uq
Uq
Magnetron excitation: ρ− Cyclotron excitation: ρ+
Conversion of radial motions
Penning and Paul traps at accelerators
• operating facilities • facilities under • planned facilitiesconstruction or test