Quadrupole Magnetic Field Effects

Erik Gilson 
Joel Fajans 
Department of Physics, U.C. Berkeley 

This work is supported by the O.N.R. and by L.A.N.L.


Resonant particle transport has long been suspected as the primary cause of plasma loss in Malmberg-Penning traps , but there is no conclusive experimental evidence to support this claim. 

We have found experimental evidence for resonant particle transport when we apply a quadrupole magnetic field to our system. We have also measured the equilibrium shape of plasmas when a magnetic quadrupole perturbation is present. 

The results of this research apply directly to anti-hydrogen creation experiments proposed by the ATHENA and ATRAP collaborations. Malmberg-Penning traps will be used to confine positrons and anti-protons before creating anti-hydrogen, and quadrupole traps will be used to confine the neutral anti-hydrogen. 

The Malmberg-Penning Trap

Our plasma is comprised of electrons thermionically emitted from a tungsten filament. The plasma is confined in a cylindrical region as shown in Figure 1. The plasma is confined radially by an axial magnetic field and axially by potentials on the ends of the trap. 



Figure 1. The left ring is grounded to load electrons into the trap. The right ring is grounded to image plasma. When the right ring is grounded, the electrons stream along the magnetic field lines and strike the phosphor screen. 

Typical Parameters

L ~ 30 cm Rwall = 1.905 cm Bo = 20 - 2000 G
n ~ 107 cm-3 kT ~ 1 eV Rplasma ~ 1 cm


Quadrupole Magnetic Fields

We add an axially invariant transverse magnetic quadrupole field using the coils shown in Figure 2. The two sets of coils are rotated 45° from one another so that by varying the relative current in the coils, quadrupole patterns with arbitrary angles about the z-axis can be created. 



Figure 2. This photograph shows the various coils used to produce magnetic fields. 

The total field is the axial field, Bo, plus the transverse field. If only set#1 is used, then 

$$\vec{B}=B_o \hat{z}+/Beta_q(x\hat{x}-y\hat{y})$$




Figure 3. The magnetic field lines for a constant, axial field Bo with a small, transverse, quadrupole perturbation from set#1. 

The self electric fields of the plasma cause it to $\vec{E}  \times \vec{B} $ drift around the trap axis. When this rotation is slow, the electrons bounce back and forth along the magnetic field lines shown in Figure 3. The overall plasma shape looks like the shape shown in Figure 4. The plasma has a circular cross section in the middle and has elliptical cross sections at both ends. The ellipses are rotated 90° from one another. 



Figure 4. When the rotation is slow, electrons follow magnetic field lines, and the plasma has this shape. 

If the plasma rotation is fast compared to the bounce time, the plasma smears out into a cylinder. 



Figure 5. When the rotation is fast the plasma is cylindrical. 

Shape Experiments

When we image quickly and slowly rotating plasmas, we see the expected circular and elliptical shapes. 



Figure 6. bq/Bo=0.004 cm-1. (a) e=1.09, q=53.5°, Bo=32.43G. The plasma is rotating quickly. (b) e=1.26, q=-37.5°, Bo=500G. The plasma is rotating slowly. These images are from the data in Figure 7. 

We measure the ellipticity, e, and the orientation of the imaged plasma. Theoretically, e-1 should scale with bq/Bo, and the data shown in Figure 7 show that this is so. We do not understand the step in the data at Bo~400G. The variation in angle is reminiscent of the drive/response phase shift of a damped driven simple harmonic oscillator as it passes through resonance.



Figure 7. The scaled ellipticity and angle of the plasma as functions of Bo

We measure the quadrupole moment of a trapped plasma by measuring the voltage signal (V1+V3)-(V2+V4) that is induced on the wall of the trap. 



Figure 8. A larger image charge is induced on the top and bottom sectors than on the right and left sectors. 

When the plasma is rotating slowly, the shape of the plasma is determined by the geometry of the magnetic field lines, as in Figure 4. The quadrupole moment is zero in the center of the plasma and has equal and opposite values at the ends of the plasma. The quadrupole moment is proportional to bq






Figure 9. Measurements of quadrupole moment along the plasma’s length show the axial dependence and bq proportionality that we expect. 

As shown below, the quadrupole moment at the plasma end is proportional to the length of the plasma. 



Figure 10. The quadrupole moment as a function of L. The ratio bq/Bo is constant. 

Resonant Particles

If the rotation rate is such that an electron makes a quarter revolution each time it travels the length of the plasma, the electron can move ever outwards or inwards. The resonance condition can be written, 

$$B_0 = \frac{neL}{\pi \epislon_0 v_z}$$




Figure 11. The trajectory (red lines) of a resonant electron as it moves outwards. 

Resonant and near-resonant electrons traveling outwards can leave the plasma very quickly. Diffusion due to this mechanism can be very large. 

There are higher order resonances in which the electron makes N/4 (N odd) revolutions as it travels across the plasma, but these are less important. 

Diffusion Experiments

Above resonance, when the plasma is rotating slowly, there are many resonant electrons and the quadrupole field has an immediate effect as shown in Figure 12(a). Well below resonance, when the plasma is rotating quickly, there are few resonant electrons and there should not be a large effect due to the quadrupole field. Indeed, for Bo = 20 G (Figure 12(b)), we see that the diffusion is suppressed at first. However, because the plasma density decreases over time the electrons become resonant. At the point indicated by the arrow in Figure 12(b), the diffusion becomes greatly enhanced. 



Figure 12. By comparing the time evolution of the central density with the quadrupole field on and off, we can separate the effects of the quadrupole field from other plasma loss mechanisms. 

From a series of images taken at successive times, we measure the diffusion coefficient, D. The plasma images measure the z-averaged radial density profile n(r,t), from which we compute N(t), the integrated signal within some arbitrary radius, R. 



When we write the diffusion equation, dndt, in polar coordinates and integrate once with respect to r to yield 





All q variations have been neglected because the quadrupole fields used in the diffusion experiments are typically small. 

In Figure 13 we keep bq/Bo fixed, as would be the case if the quadrupole field were due to imperfections in the main magnet coils. When the quadrupole fields are on, D is the sum of the diffusion due to the quadrupole field and the diffusion due to background processes. Below resonance, the quadrupole field has little effect, but above resonance, it enhances diffusion. 


Figure 13. Below resonance, no electrons meet the resonance condition and no electrons can be lost via this resonance process. The graph on the right shows that, above Bo~200G, the diffusion due to the quadrupole field scales roughly like Bo2, while below Bo~200G, it is weaker. The green curve is D(n,kT,Bo) from our theory, assuming n = 8 106 cm-3 and kT = 6 eV. However, the temperature and density vary from data point to data point. Evaluating D(n,kT,Bo) at each value of Bo, assuming kT = 6 eV, but allowing the density to vary gives the black curve. 

If we hold bq fixed, the resonance is sharper. For large axial fields, the diffusion due to the quadrupole field becomes small and the background processes dominate the diffusion. 


Figure 14. Both below and above resonance the diffusion due to the quadrupole field is weak. Near resonance, diffusion is enhanced. The green curve is D(n,kT,Bo) from our theory, assuming n = 8 106 cm-3 and kT = 6 eV. However, the temperature and density vary from data point to data point. Evaluating D(n,kT,Bo) at each value of Bo, assuming kT = 6 eV, but allowing the density to vary gives the black curve. 

Note the anomalous bump in the background (D(bq=0)) data at Bo~400G. This bump needs to be understood before further study can be completed. 

At Bo=175G, the diffusion due to the quadrupole field scales like bq1.84, as shown in Figure 15. This is consistent with the prediction that diffusion from the quadrupole magnetic field scales like bq2

Figure 15. Diffusion due to the quadrupole field as a function of bq/Bo. 
Figure 15. Diffusion due to the quadrupole field as a function of bq/Bo. 

Measuring the relative lifetimes using three different plasma lengths, we see that the location of the resonance moves in agreement with the change in the resonant condition. To find the plasma’s lifetime, we measure the time it takes for the central density to drop to ~70% of its initial value. We do this both with the quadrupole field on and off, then compute the relative lifetime. 


Figure 16. Graphs of the relative lifetimes versus magnetic field show that when the resonance condition is met, particle loss is enhanced. The resonance location is length dependent. 


We construct a diffusion coefficient, D = l²nf, where l is the average step size of a resonant electron, n is the frequency of collisions that knock an electron out of resonance, and f is the fraction of electrons that satisfy the resonance condition. Because an electron stays on a resonant trajectory only until a collision occurs, l is inversely proportional to n. Though it is not obvious, f is linearly dependent on n. Thus, D is independent of n

Near resonance, many electrons participate in the diffusion process and D is large. Below resonance, the average electron's velocity is well below the resonant velocity. Few electrons participate in the diffusion process so D is small. Well above resonance, the step size becomes small and diffusion is also reduced. 



Figure 17. If Bo ~ Bresonant, then there are many particles available to participate in resonant diffusion (left picture). If o < Bresonant (right picture), there are no particles available to diffuse. 

We must sum over the higher order resonances to finally obtain an expression for D. The result is D = SN Odd DN, where 



Because DN~N-5, D~D1.



Figure 18. D1 as a function of Bo. The maximum of D1 occurs at Bm. The value of Bm is proportional to the value Bo that satisfies the resonance condition. 


Clear evidence for resonant particle transport as the mechanism for plasma loss in Malmberg-Penning traps has been lacking. When applying a magnetic quadrupole perturbation, we observe resonant behavior that could help to explain plasma loss in Malmberg-Penning traps. 

If operating in suitable parameter regime, experiments planned by the ATHENA and ATRAP collaboration may be able to use both Malmberg-Penning traps and quadrupole traps.