Thermal Beam Equilibria in Periodic Focusing Fields*C. ChenMassachusetts Institute of Technology

Presented at Workshop on The Physics and Applications of High-Brightness Electron BeamsMaui, HawaiiNovember 16-19, 2009

Collaborators: T.R. Akylas, T.M. Bemis, R.J. Bhatt, K.R. Samokhvalova, J. Taylor, H. Wei and J. Zhou Thanks to the UMER group, especially S. Bernal.

*Research supported by DOE Grant No. DE-FG02-95ER40919, Grant No. DE-FG02-05ER54836 and MIT Undergraduate Research Opportunity (UROP) Program.

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*/33OutlineBackgroundImportance of thermal beamsHistorical perspectiveIssuesBeams in Periodic Solenoidal FocusingWarm-fluid and kinetic theoriesComparison between theory & experimentControl of chaotic particle motionBeams in Alternating-Gradient FocusingWarm-fluid theoryComparison between theory & experimentResearch Opportunities in Thermionic DC Beam Approach to High-Brightness, High-Average Power InjectorsConclusions Future Directions

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*/33Why is thermal beam equilibrium important?Beam losses and emittance growth are important issuesrelated to the dynamics of particle beams in non-equilibriumIt is important to find and study beam equilibrium states to maintain beam quality preserve beam emittanceprevent beam lossesprovide operational stabilitycontrol chaotic particle motion Control halo formationThermal equilibriummaximum entropyMaxwell-Boltzmann (thermal) distribution most likely state of a laboratory beamsmooth beam edgeQian, Davidson and Chen (1994)Pakter, Chen and Davidson (1999)Zhou, Chen, Qian (2003)Phase space for a KV beam

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*/33Applications of high-brightness charged-particle beamsInternational Linear Collider (ILC)Free Electron Lasers (FELs)Energy Recovery Linac (ERLs)Light Sources Large Hadron Collider (LHC)Spallation Neutron Source (SNS)High Energy Density Physics (HEDP) RF and Thermionic PhotoinjectorsThermionic DC InjectorsHigh Power Microwave Sources

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*/33University of Maryland Electron Ring (UMER)UMERCircumference = 11.52 mScaled low-energy e- beam Space-charge-dominated regimeLinear beam experimentsSolenoidal and quadrupole focusing experimentsDensity profile measurementsS. Bernal, B. Quinn, M. Reiser, and P.G. OShea, PRST-AB 5, 064202 (2002)S. Bernal, R. A. Kishek, M. Reiser, and I. Haber, Phys. Rev. Lett. 82, 4002 (1999)

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*/33Linear focusing channelWeak Focusing Strong Focusing

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*/33Rigid-rotor equilibrium in a uniform magnetic field*R. C. Davidson and N. A. Krall, Phys. Rev. Lett. 22, 833 (1969); A. J. Theiss, R. A. Mahaffey, and A. W. Trivelpiece, Phys. Rev. Lett. 35, 1436 (1975); L. Brillouin, Phys. Rev. 67, 260 (1945). dc Beam(non-neutral plasma column)BrillouinDensity

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*/33Thermal rigid-rotor equilibrium in a uniform magnetic fieldDavidson and Krall, 1971Trivelpiece, et al., 1975 Distribution function

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*/33Periodic Focusing Solenoid (weak focusing) Quadrupole (strong focusing) Single particle orbitsv=60o

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*/33Kapchinskij-Vladimirskij (KV) I. M. Kapchinskij, and V. V. Vladimirskij, in Proc. of the International Conf. on High Energy Accel. (CERN, Geneva, 1959), p. 274. Approximate (small v)R. C. Davidson, H. Qin, and P. J. Channell, Phys. Rev. Special Topics-Accel. Beams 2, 074401 (1999).

PeriodicQuadrupoleRigid-rotor kineticC. Chen, R. Pakter and R. C. Davidson, Phys. Rev. Lett. 79, 225 (1997).Cold-fluid beamR. C. Davidson, P. Stoltz, and C. Chen, Phys. Plasmas 4, 3710 (1997).Approximate (small v)R. C. Davidson, H. Qin, and P. J. Channell, Phys. Rev. Special Topics-Accel. Beams 2, 074401 (1999).

PeriodicSolenoidalCold-fluid beamR. C. Davidson, Physics of nonneutral plasmas (Addison-Wesley, Reading, MA, 1990).

Rigid-rotor kinetic R. C. Davidson, Physics of nonneutral plasmas (Addison-Wesley, Reading, MA, 1990).M. Reiser and N. Brown, Phys. Rev. Lett. 71, 2911 (1993). Warm-fluid beam S. M. Lund and R. C. Davidson, Phys. Plasmas 5, 3028 (1998).

Uniform

Other Beam Equilibria

Thermal Beam Equilibria Equilibria

FocusingPrevious equilibrium theories

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*/33Issues of previous theoriesThere was a lack of a fundamental understanding of beam equilbria beyond cold fluidKV-type equilbria are mathematical and cannot be realized or seen experimentally.Smooth-beam approximations were not accurate at high vacuum phase advance.

RMS envelope equations (Sacherer, 1971; Lapostolle; 1971)Assumption of a self-similar density distribution No self-consistent description of emittance evolutionNo self-consistent description of density evolutionSelf-similar density distribution0Constant-density contours are ellipses of the same aspect ratio

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*/33Warm-fluid equilibrium theory*(Solenoidal focusing)Continuity equationForce balance equationPoissons equationPressure tensorIdeal gas lawis ignored in paraxial treatment *K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007)

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*/33Warm-fluid equilibrium theory*(Solenodial focusing)Transverse beam velocity*K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007) Adiabatic equation of stateRMS beam radius

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*/33Warm-fluid equilibrium theoretical results*(Solenoidal focusing)perveancefocusing parameterrms beam radiusthermal rms emittancePoissons equationBeam rotationEnvelope equationBeam density*K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007)

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*/33Kinetic equilibrium theory*(Solenoidal focsuing)Vlasov equationSingle-particle HamiltonianParaxial approximation*J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008)Courant-Snyder transformation

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*/33Constants of motion and thermal distribution Angular momentum (exact):Scaled transverse Hamiltonian(approximate):Thermal distribution:J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008)

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*/33Beam envelope and densitycold beamwarm beam

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*/33UMER edge imaging experiment*5 keV electron beam focused by a short solenoid. Bell-shaped beam density profilesNot KV-like distributions*S. Bernal, B. Quinn, M. Reiser, and P.G. OShea, PRST-AB, 5, 064202 (2002)

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*/33Comparison between theory and experiment for 5 keV, 6.5 mA electron beam*Experimentaldataz=6.4cmz=11.2cmz=17.2cm*S. Bernal, B. Quinn, M. Reiser, and P.G. OShea, PRST-AB 5, 064202 (2002); K. R. Samokhvalova, J. Zhou, and C. Chen, Phys. Plasmas 14, 103102 (2007); J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 023102 (2008)

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*/33Chaotic phase space for a KV beamQian, Davidson and Chen (1994)Pakter, Chen and Davidson (1999)Zhou, Chen, Qian (2003)

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*/33Control of chaos in thermal beams (preliminary results) Thermal BeamNormalized RadiusNormalized MomentumKV BeamNormalized RadiusNormalized MomentumWei & Chen, paper presented at DPP09

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*/33Equation of state (adiabatic process)Transverse flow velocityBeam density profileWarm-fluid equilibrium theory (AG focusing)

Solenoidal LatticeQuadrupole LatticeForce-balance equation

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*/33Thermal beam equilibrium theoretical results (AG focusing)Beam densityPoissons equation4D thermal rms emittanceperveancefocusing parameterEnvelope equations

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*/33Beam equilibrium properties - Temperature effectsRms beam envelope increases with temperature.4D rms emittance is conserved.Transverse beam temperature is constant across the cross section of the beam.

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*/33Beam equilibrium properties - Density profileDensity profile on x-axisDensity profile on y-axis

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*/33Beam equilibrium properties - Equipotential anddensity contours

Equipotential contours are ellipses.

Constant density contours are also ellipses.

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*/33Elliptical symmetry but not self-similarThe density is not self-similarNumerical proof of self-field averages%1001-brmsbrmsyxba

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*/334 keV electron beam focused by 6 quadrupoles 2/3 of the beam is chopped by round apertureBeam density profiles are bell-shaped in the x-direction and hollow in the y-direction Cannot be explained by KV distributionUMER 6-quadrupole experiment**S. Bernal, R. A. Kishek, M. Reiser, and I. Haber, Phys. Rev. Lett. 82, 4002 (1999)10.48 13.43 17.13 26.83 35.28 42.43 49.88 57.98 66.08 73.98

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*/33Comparison between theory and experiment Z=13.43cmZ=17.13cmZ=26.83cmZ=35.28cm

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*/33Research opportunities in thermionic dc gun approach to high-average-power beamsCurrent state of the art1 A, 500 kV1.1 mm-mrad for 1.5 mm radius cathode (Spring-8 injector - Tagawa, et al., PRST-AB, 2007)

Is the intrinsic emittance achievable?0.25 mm-mrad per mm cathode radius

How can we control beam halo?

Need gun and beam matching theory including thermal effects Current research at MIT (Taylor, Akylas & Chen)

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*/33Experimental opportunitiesPeriodic solenoidal focusing channelNew design based on a patented high-brightness circular electron beam system (C. Chen, T. Bemis, R.J. Bhatt and J. Zhou, US Patent Pending, 2009).Minimize beam mismatch.Demonstrate adiabatic thermal beams in a long channel.

AG focusing channelNew design a patented high-brightness elliptic electron gun (R.J. Bhatt, C. Chen and J, Zhou, US patent No. 7,318,967, 2008) Minimize beam mismatch.Demonstrate adiabatic thermal beams in a long channel.R. Bhatt, T.M. Bemis & C. Chen, IEEE Trans PS (2006)T.M. Bemis, R. Bhatt, C Chen & J..Zhou, APL (2007)

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*/33ConclusionsAdiabatic thermal beam equilibria shown to exist inPeriodic solenoidal focusingAG FocusingAdiabatic equation of states assures the conservation of normalized rms emittance with space charge2D normalized rms emittance in periodic solenoidal focusing4D normalized rms emittance in AG focusingGaussian density distribution for emittance-dominated beamsFlat density in the center with a characteristic Debye fall off at the edge for space-charge-dominated beamsPredictions for AG focusing Conservation of 4D normalized rms emittance Elliptical constant density and potential contoursNon-self-similar density distribution

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*/33Future directionsPerform high-precision experiments to further test the adiabatic thermal beam equilibrium in periodic solenoidal focusing.

Perform high-precision experiments to test the adiabatic thermal beam equilibrium in AG focusing.

Develop a better understanding of thermal effects in thermionic electron guns and beam matching.

Apply the concept of adiabatic thermal beams in the research, development and commercialization of high-brightness, high-average-power electron sources and beams.

*The purpose of this thesis was to develop thermal equilibrium theory for periodically focused charged particles beam and study the properties of the beam in such equilibria*This thesis is devoted to studying the beam equilibrium. Why is the beam equilibrium important?Beam losses and emittance growth are important issuesrelated to the dynamics of particle beams in non-equilibrium statesIt is important to find and study beam equilibrium state to maintain beam quality preserve beam emittanceprevent beam lossesprovide operational stabilitystudy chaotic particle motion and halo formationIn this thesis we focus on finding thermal equilibrium because most likely state of a laboratory beamMaxwell-Boltzmann (thermal) distribution smooth beam edge

*This thesis is devoted to studying the beam equilibrium. Why is the beam equilibrium important?Beam losses and emittance growth are important issuesrelated to the dynamics of particle beams in non-equilibrium statesIt is important to find and study beam equilibrium state to maintain beam quality preserve beam emittanceprevent beam lossesprovide operational stabilitystudy chaotic particle motion and halo formationIn this thesis we focus on finding thermal equilibrium because most likely state of a laboratory beamMaxwell-Boltzmann (thermal) distribution smooth beam edge

*High-brightness charged-particle beams are used in a variety of applications, such as big accelerators and colliders, for example:

In Large Hudron Collider which has a circumference of 26km proton beams will be colliding at 14TeV. Spallation Neutron Source accelerates a H- beam to 1 GeV. High-intense proton beams are used to hit a target and generate neutronsin ILC high-intensity electron and positron beams are going to collide. 2 linacs, each 20 kilometers long, hurl beams of electrons and positrons toward each other atnearly the speed of light.In high energy density experiments and Heavy Ion Fusion experiments, multiple laser or particle beams are guided to converge on a small fusion fuel pellet or filament. The Fermilab NICADD Photoinjector Laboratory (FNPL) is an 18 MeV electron linac. It produces a high-intensity, low emittance beam. The principal components of the facility are a pulsed UV laser, a normal conducting RF gun with a csium telluride photo-cathode, a 9-cell superconducting cavity, a magnetic chicane, and a beam transport line. The accelerator is used for studies in accelerator science, plasma wake-field acceleration, and channelling radiation. A photoinjector is a high gradient electron accelerator used to produce a short pulse, high peak current, high quality relativistic electron beam. The resonant cavity builds up an accelerating field over several hundred nanoseconds until the peak gradient on axis is of the order 100 MV/m. Electrons are produced through photoemission at the peak of this accelerating field. Solenoidal focusing provides a means of minimizing transverse emittance in the electron beam by utilizing a known correlation between beam divergence and longitudinal position within the electron bunch [1].Another application of the high-brightness charged-particle beams high-power microwave source, such as a klystron, pictured here, which is designed at SLAC*Not all of the accelerators, that utilize the high-brightness charged-particle beams as large as the Machines youve seen on the previous slide.University of Maryland Ring (or UMER), for example has a circumference thats only 11.5 m.It uses a scaled low-energy electron beam and itAllows to access a space-charge-dominated regime at a much lower cost then bigger and more energetic machines.Is an ideal testbed for experimenting in pushing up the brightness of existing and future acceleratorsThere have been several linear beam experiments performed on UMER, as part of the injector experiment. Solenoidal and quadrupole fields were tested and the density profiles were measured. Later in this talk I will show the comparison between our theory and the experimental data from the University of Maryland straight focusing experiments.**In many charged-particle beams application, like the UMER experiment I just discussed, periodic focusing was used, The two magnetic field configurations widely used for focusing are Solenoidal and Quadrupole latticesIn Solenoidal focusing channel the beam has a circular cross-section, which provides a higher order symmetry,However, solenoidal focusing in weaker than quadrupole focusing, which is more often usedIn quadrupole focusing channel the beam has, in general, elliptic cross section, because every half-period of the latticeThe magnetic field is focusing the beam in one direction and defocusing it in the other direction, so the resulting beams cross section is elliptical. The trajectory of the particle in the periodic focusing channel can be described by the Hills equations, where kappaq is the focusing parameter, periodic with the period SThe solution to the Hills equation can be written in this form, where w(s) is the envelope function and psi(x) is the betatron function. If we plot the particles trajectory we can see that the particles oscillating trajectory has two characteristic periods. The fast period is equal to the period of the magnetic field S, the slower oscillation is called the betatron oscillation and can be characterized by the so-called vacuum phase advancevacuum phase advance is related to the envelope function and the magnetic focusing In this particular example, the vacuum phase advance was 60 deg, which means that every period of the envelope the betatron oscillation advanced 60 degrees. *There have been work done in this area and I wanted to discuss some of the equilibria that were found. In this table I classified the equilibria by type (thermal and all the other ones) and Focusing uniform, periodic solenoidal and periodic quadrupoleThere is a well established theory of the thermal beam equilibria in the uniform focusing field, in particular, there was a kinetic equilibrium found and there was also warm-fluid theory developedFor the periodic focusing fields, prior to this work, there was only one approximate equilibrium found that is valid for small vacuum phase advances, whereas most of the accelerators operate in the regime with moderate or high phase advances. Other beam equilibria were found for uniform focusing the cold-fluid beam equilibria that doesnt take into account the temperature effects.For the periodic solenoidal field there was kinetic rigid-rotor equilibrium and cold-fluid equilibriumFor the quadrupole focusing field there was a Kapchinkij-Vladimirskij or KV equilibrium found. KV equilibrium is characterized by the uniform density distribution and delta-function phase-space distribution. In my thesis, thermal beam equilibrium theory in developed for the beam in periodic solenoidal and quadrupole focusing field.*Ill start with the warm-fluid theory I developed.The starting equations will be the continuity equation and the force balance equation, where, as you can see, I take into account the self-magnetic field and the temperature effects. The self-field potential is governed by the Poissons equation and we assume that the pressure tensor is diagonalWe use ideal gas law for the transverse component of the pressure tensor and the parallel component of the pressure tensor can be ignored because our theory is paraxialTo bring closure to the fluid equations we need to choose an equation of state.

*We consider the beam equilibrium to be adiabatic.It can be shown that the adiabatic equation of state can be written as transverse temperature times the characteristic transverse beam size is a constantRbrms is the rms beam radiusWe look for an equilibrium with transverse velocity that is linear in r and corresponds to the beam rotating with the frequency Omegab and radially pulatingUsing the transverse beam velocity, equation of state and the warm-fluid equations we can calculate*The beam density. Here the self-field potential is governed by Poissons equation The frequency of the beams rotation can be also calculated. Here this part is the frequency of beams rotation relative to the Larmor frame which rotates with half of the cyclotron frequency.The rms beam radius is governed by the envelope equation. Where you can see how the focusing force is being balanced by the space-charge force and the thermal pressure force. Also, the rotation relative to the Larmor frame is contributing to the thermal emittanceAlso, because of the adiabatic equation of state, the thermal rms emittance is conservedIll discuss the properties of this equilibrium later, I want to present the kinetic theory next

*The kinetic theory starts with Vlasovs equation and well have to find the distribution function. For the time-independent Vlasovs equation the distribution function which depends only on the constants of motion would satisfy the Vlasovs equation.So well need to find the constants of motion.We start with the single-particle Hamiltonian and consider paraxial approximation, We can then split the Hamiltonian into parallel and perpendicular parts.We then do two transformation of coordinates.First, we transform into Larmor Frame, which is rotating with half of the cyclotron frequency. Second, we do a Courant-Snyder transformation, which does a scaling of the coordinates by function w, which Ill define in the next slide.*The constants of motion are The angular momentum which is conserved because there is no theta dependence The second constant of motion is the scaled transverse Hamiltonian which is an approximate constant of motion.

Where is the transverse Hamiltonian and w function satisfies the following differential equation

We can then construct the transverse thermal distribution function using these constants of motion. Beta, C bar and omegab are constants. This distribution function is of Maxwell-Boltzmann type

To compute the statistical properties of the equilibrium, such as beam density, flow velocity, temperature we take moments of this distribution function

**5 keV electron beam focused by a short solenoid magnetic in one of the experiments of University of Maryland Electron Ring (UMER) The electron beam was generated by a gridded gun and exited the gun through an anode aperture at z=0 . Bell-shaped beam density profiles were imaged by a fluorescent screen while detailed velocity space distribution was not accessible.The bell-shape beam density profile and the change of the beam density shape as the beam propagates has not been well understood theoretically using previous equilibrium theories, such as the KV beam equilibrium.

Gridded electron guns are widely used today as electron sources for various devices such as microwave tubes and certain accelerators. Even though the beam is usually spacecharge dominated in the gun region, emittance growth that occurs there can be of significance either in its evolution of transverse beam modes at low energy, and also the overall envelope dynamics when the beam is accelerated to higher energy. At the University of Maryland, a gridded gun is being used in experiments to study the physics of spacecharge-dominated electron beams. These experiments include the University of Maryland Electron Ring (UMER) [1] and resistive-wall instability experiment [2], etc.In a gridded electron gun, a grid (referred as grid or cathode grid in the following text) located in front of the cathode is used to control the beam current and pulse shape. While this configuration provides the flexibility to control the beam, it has the potential disadvantage of reducing the beam quality through emittance growth in the grid mesh.*We take the profiles from the paper on compare the data with the theoretical curves. Here the dots are experimental points, the solid lines are the theory and the dashed line is the equivalent KV distribution. Equivalent KV distribution has the same rms beam radius and current as the actual distribution, and as can be seen it does not agree with the experimentally observed data, whereas our theory agrees with the experiment pretty well I would like now to move on to considering a thermal equilibrium theory for a beam in periodic quadrupole focusing field

*As in the solenoidal case we consider adiabatic process. We generalize the equation of state we had in the solenoidal field case to the quarupole field case. The effective transverse beam case in the case of the quadrupole field is ellipse. Xbrms and ybrms are the rms beam envelopes in x and y directions.The transverse flow velocity is linear in transverse coordinates Using the flow velocity and the equation of state we arrive at the following beam density profile Important part is that in this density profile we have the unknown envelope quantities that we have to self-consistently calculate. *The beam density depends on the self-field potential which is determined self-consistently using Poissons equation. The beam density depends on xrms beam sizes which are described by envelope equations. The 4D rms emittance of the beam is conserved because the beam is adiabatic. Before discussing the properties of the beam equilibrium I would like to show you that the self-field averages are in fact satisfied. *First of all, the beam temperature is constant across the beams cross-section and inversely proportional to the effective transverse area of the beamThe 4D thermal rms emittance is conserved which is the consequence of the fact that the beam is adiabaticThe rms beam envelope increases with temperature. On this figure the envelopes for a cold beam (dashed line) and warm-beam (solid line) are plotted. You can see that the rms envelope is increases for the case with temperature, although, not by a lot, because this is the case of a space-charge-dominated beam, with normalized perveance Khat =4

*Finally, here I plot the beam density along x and y directions. The beam density profile has a particular property that for a space-charge-dominated beam the beam density is flat near the center of the beam but then falls off within a few Debye length which is defined here: One very interesting property of this equilibrium is that the drop-off rate of the beam density is transversely isotropic.

Now Id like to present the comparison between this theory and the UMER experiment

*Next Id like the discuss the elliptical properties of the distribtuion.The equipotential contours, plotted here, are ellipces. On the right I show the fit of the contours to ellipses. The constant density contours are also ellipses, but the density is not self-similar, as I mentioned previously. The fit is shown on the right.*First of all, we discovered, that the equilibrium beam density is not self-similar, which is the simplest elliptical symmetry.The contours of constant density are ellipses, which Ill demonstrate to you shortly. But the contours of constant density are not ellipses with the same ratio of the semi-axes In this figure I plotted the difference in percent between the ratio of the semi-axes of the ellipses and the rms beam sizes. If the beam density was self-similar, the line would be straight. However, I found that the self-field averages relations are satisfied. On this figure I plotted the difference in percent between the right hand side of the slef-field average expression which was computed using the envelope equation and the left-hand side which was computed using the numerically calculated density profile. The difference is less than 1 %, so we can our theory is valid. Now Id like to discuss the properties of the beam equilibrium*We looked at the UMER 6-quadrupole experiment. In this experiment 4 keV electron beam focused by 6 quadrupoles 2/3 of the beam is chopped by round apertureThe beam was pictured with the phosphor screen and CCD cameraBeam density profiles are bell-shaped in the x-direction and hollow in the y-direction Cannot be explained by KV distributionOn this pictures you can see the beam at different axial positions. On the first 2 pictures the beam appears to be cropped because the screen was not big enough to picture the whole beamHere, I plotted the normalized focusing coefficient And the beam rms envelopes with the dots being the experimental data.For this experiment I collaborated with Dr. Santiago Bernal from University of Maryland and he gave me the original pictures that I then analyzed.*For several other axial positions. The top row shows the densities in the x directionThe bottom row shows the densities in the y directionAs on the previous slide you can observe god agreement in the drop-off region in the x-direction and not so good agreement in the y-direction*I want to finish with outliing a few directions of future research In this thesis We have gained a fundamental understanding of the thermal equilibrium of the intense charged-particle beams in periodic focusing channels. Our theory, however, is limited to two-dimensional continuous dc beams. Because many charged-particle beamsapplications require 3D beams or bunched beams, it is important to further explore the physics of space-charge-dominated 3D beams and study the thermal equilibrium properties of such beams. It is well-known that 2D KV beam distribution cannot be extended to bunched beams. It would be interesting to explore the possibility of generalizing our 2D thermal equilibrium theory to a 3D thermal equilibrium theory. It would also be interesting to propose an experiment dedicated to studying the beam equilibrium. Such an experiment would address issues that have prevented the beam frombeing in a true equilibrium (i.e., beam aperture, magnets misalignment, and magnetic field nonlinearities). Comparison with the data from such experiments will provide betterinsight into the applicability of the theoretical thermal beam equilibria presented in thesis. Another direction of the further research could be a theoretical and numerical study of chaotic particle motion and the possibility of chaos control in the beams in thermalequilibria. Results of such a study could be of significant practical interest to the design and operation of future beam systems. *I have also presentedThermal Beam Equilibrium Theory in Periodic Quadrupole Focusing FieldsWarm-fluid theory developed for periodically focused beam in an adiabatic processConservation of 4D beam rms emittance Uniform temperature across the beams cross-sectionDensity of the space-charge-dominated beam is flat near the center but falls off within a few Debye lengths Transversely isotropic density fall-off rate Elliptical density and self-field potential but not self-similarReasonably good agreement between theory and experimental data*I have also presentedThermal Beam Equilibrium Theory in Periodic Quadrupole Focusing FieldsWarm-fluid theory developed for periodically focused beam in an adiabatic processConservation of 4D beam rms emittance Uniform temperature across the beams cross-sectionDensity of the space-charge-dominated beam is flat near the center but falls off within a few Debye lengths Transversely isotropic density fall-off rate Elliptical density and self-field potential but not self-similarReasonably good agreement between theory and experimental data