FEgen (v.1): Field Emission Distribution Generator Freeware Based on Fowler-Nordheim Equation
FFEgen (v.1): Field Emission Distribution Generator Freeware Based onFowler-Nordheim Equation
Emily Jevarjian,
1, 2, a) Mitchell Schneider,
1, 2, b) and Sergey V. Baryshev c) Department of Physics and Astronomy, Michigan State University, MI 48824,USA Department of Electrical and Computer Engineering, Michigan State University, MI 48824,USA
INTRODUCTION
As field emitters are poised to become the preferredelectron source for next-generation electron acceleratorsand other vacuum electronics microwave devices movingup in operating frequency for higher peak power ratingand compactness, a computational toolbox must bedeveloped to realistically model the particle dynamics.Unlike in photoemission, where an ultrashort high powerdensity laser pulse is synchronized (in other words, phasematched) with an rf/microwave drive signal, electrons aregenerated by and interact with the rf/microwave drive cy-cle in a much wider phase window, regardless of whethera field emission cathode is operated in an ungated orgated fashion (by means of a physical gate electrode, har-monics mixing, or multicell gun design). An extendedinteraction phase window is of paramount importance tocorrectly reveal the longitudinal phase space of the re-sulting beam, which may promote delayed emission andsecondary effects in the injector ultimately leading tobeam loading, multipacting, and cathode field screeningeffects. Compactness of a high frequency system (and thecorresponding small emitting area of an electron sourcerequired to emit high charge) poses challenges in regardto correctly tracking and accounting for vacuum spacecharge effect and beam expansion/explosion. Currently,all in-demand beam tracking software that is capable ofaccounting for space charge effects, such as ASTRA, IMPACT-T, or GPT, does not incorporate a particledistribution generator suitable for field emission analy-ses. There are costly PIC codes, VSim and Michelle for example, that account for the space charge effect yetcontain field emission models likely based on the conven-tional Fowler-Nordheim equation. However, the exactmechanism of how Michelle and VSim determine theirfield emission distributions is proprietary. This work isaimed to reengineer the generator function designed forASTRA using Python, implement the Fowler-Nordheimequation to allow for observing temporal/phase (and thuscomplex longitudinal) beam processes, and make it afreeware as the ASTRA generator is only available as anexecutable file and has a number of application issues.We compare our results to those obtained using the AS- a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]
TRA generator to verify our results for the coordinate-momentum space distributions. The generated distribu-tions have the same format and structure as those ob-tained from ASTRA and can be directly translated intoboth ASTRA and GPT, the latter of which was used forour detailed application example. In particular, this workprovides the ability to design and simulate transverselyinherently shaped beams using array field emission cath-odes providing new means for improving wakefield struc-ture or plasma accelerators. The paper is laid out asfollows: Section I shows the momentum distribution, Sec-tion II shows the spatio-temporal distribution, and Sec-tion III shows a GPT application example for an L-bandinjector design. Section IV briefly discusses further devel-opments for FEgen beyond the Fowler-Nordheim equa-tion for future releases.To download Python software package go to https://github.com/schne525/FEgen
I. MOMENTUM DISTRIBUTION
The momentum distribution is based off an isotropicdistribution which arises from the emittance on the cath-ode surface as the electrons tunnel through the barrier.This results in a momentum spread that is uniformly dis-tributed over a half sphere, where the base is the surfaceof the cathode. This is referred to as an isotropic distri-bution . To create the isotropic momentum distributions,first the maximum energy of each particle is calculated.Using Numpys random normal distribution function, athree-dimensional array of values with a normal distri-bution between -1 and 1 in each dimension is created toserve as unit vectors for each particles momentum in the x , y , and z direction. To compare to ASTRA’s generator,the x and y components of the unit vector array are mul-tiplied by the maximum energy for a particle, creating auniform kinetic energy distribution. Subsequently, thisproduct is multiplied by a scaling factor which accountsfor the uncertainty in the original x and y momentumdistributions created by ASTRA. We found this scalingfactor for uncertainty to be needed only in the x and y dimensions, and that the z component of the momen-tum distribution mimicks that of ASTRAs without anadjustment for uncertainty (see Fig. 1).In Fig. 1, the absolute value of the z dimension of theunit vector array is multiplied by the maximum energyfor a particle, given that momentum in the z direction a r X i v : . [ phy s i c s . acc - ph ] S e p evarjian, Schneider and Baryshev MSU MAM Report z direction; (b) the distribution for the magnitude of the momentum vector.must be positive. The magnitude of each particles mo-mentum vector is then calculated and used to create anarray of each particles energy which is then used in ScipysStatistical Kolmogorov-Smirnov test (kstest) to generatea p -value indicating the uniformity of the energy distri-bution by testing against Scipys Statistic Uniform dis-tribution function. The given p -value is then comparedto a significance level of 0.01. When the generated mo-mentum arrays meet this significance condition, the mo-mentum values are accepted and stored in the programto later be written into the output file along with thespatio-temporal components. II. SPATIO-TEMPORAL DISTRIBUTION
Fig. 2a illustrates how the radial distribution im-planted in ASTRA resembles a spiral pattern, as if itsrandom number generator uses the Fibonacci spiral withthe ratio of π − . The spatial radial distribution is uni-form over a given radius. For FEgen, Numpys randomdistribution was used to produce the radial distribution,as illustrated in Fig. 2a. This conceptual difference doesnot affect the resulting individual distributions of the x and y coordinates as clearly emphasized by Fig. 2c and2d.The FEgen has additional features such that, beyondhaving a single emitter, one can design a variety of emis-sion patterns to simulate custom emitter arrays. A usercan pick not only the radius of the emitter but also de-sign an emitter grid and a custom pattern of emissionpoints. An additional benefit is if the user knows thetotal charge of the beam or the total charge over the en-tire emission pattern region, FEgen can calculate thenthe charge for each emitter. This is useful in the case ofsimulating an emission grid of only a few emitters to rep-resent a uniform emission which may have thousands ofemitters on the cathode surface to maintain the ratio ofemission area to charge to accurately simulate the space charge forces on the beam downstream. The interface ofthe initial particle distribution FEgen, containing all ofthe aforementioned features, is shown in Fig. 3.The temporal distribution is determined upon whetherthe field emission source is operated in a dc pulsed power(dc) or rf (ac) environment. In the dc environment, thefield emission current is constant with time. Therefore,the temporal distribution follows a uniform distributionwhere the output charge is found by inputs for the pulselength and current.In the rf environment, the Fowler-Nordheim equationis time-dependent. When averaged over an rf cycle, theFowler-Nordheim equation is transformed into a formthat reads I F ( t ) = 1 . × − × . φ − . A e [ βE c ( t )] φ × exp[ − . × φ . βE c ( t ) ] , (1)here the external electric field is modeled as a time vari-ant sinusoidal oscillation which is a result of only con-sidering the longitudinal component. Eq. 1 is then fittedto a Gaussian distribution to determine the mean andstandard deviation over the emission phase as specifiedby the input parameters (exemplified in Fig. 4).Generally speaking, there is no intrinsic gating in fieldemission, and the current is allowed to emit over 360 ◦ of the rf cycle, and only electric field strength in Eq. 1dictates when the emitting charge quenches. On theother hand, Eq. 1 is highly non-linear and it is thereforehypothesized that the emission phase window is muchshorter than 360 ◦ , often assumed to be equal to 60 ◦ ( ± ◦ around the rf cycle electric field crest). This is ofcourse not a fundamentally defined threshold, and FE-gen interface offers to input a specific rf phase wherethe emission to occur thus allowing, e.g., for finding bestagreement between simulations and experimental data.page 2evarjian, Schneider and Baryshev MSU MAM Report x and y distributions obtained with ASTRA and FEgen.FIG. 3: FEgen interface for initial particle distribution generation. Functionalities include 1) rf and dc pulsedpower environments, ability to design 2) uniformly spaced grid of emitters and 3) custom grid of emitters. NotePulsed power function uses the pulse length of a dc system and the current to calculate the charge. As in any dcenvironment, the output current is constant with respect to time and does not follow the Gaussian-like distributionassociated with the Fowler-Nordheim equation containing time varying electric field. FEgen interface functionalityfor (a) rf and (b) dc pulsed power environments. page 3evarjian, Schneider and Baryshev MSU MAM Report β ).The FEgen code was originally intended for the ArgonneCathode Teststand (ACT) where the default frequencyis the L-band operational frequency of 1.316 GHz. Theinitial energy distribution at the cathode surface is de-faulted to 0.1 eV, as for most materials the initial en-ergy distribution is a fraction of an eV. As this cur-rent model uses the Fowler-Nordheim equation, all par-ticles are assumed to emit at z =0, which is the locationof the cathode surface in the simulation given that thecathode surface is of a planar geometry. Even so, someuseful results can be obtained when simulating a fieldemission cathode which consists of a specifically designedpatterned array of emitting tips this is illustrated in thenext section. III. APPLICATION EXAMPLE
Currently, photoemission sources are used to producetransversely shaped electron beamlets by means of trans-versely shaped lasers or masks on the cathode surfaceto construct a desired pattern. A transversely shapedlaser pulse using additional optics complicates overallsystem design and may diminish the overall power of thelaser. However, a field emission source specifically engi-neered with a specific pattern/array of emitters enablesnew means for direct transverse multi-beamlet generationin a high efficiency fashion. Emittance exchange tech-niques further allow for conversing transversely shapedbeams into longitudinal bunch trains critical for the de-velopment of the next generation wakefield structure orplasma accelerators. One example of recent successfuldemonstration of transversely shaped beams is describedin Ref. : specifically, it was demonstrated that using a grid of pyramidal nanodiamond emitters a transverselyshaped beam could be transported nearly intact ∼ .The pyramids were simulated as radially uniform planaremitters with a radius equal to 15 nm spaced 450 µ maway from each other. The total charge is the charge pereach of 8 emitters (where the total beam charge reportedin Ref. was 60 pC), acquired over an rf pulse lengthof 6 µ s. When the option for charge per pulse length isselected in FEgen, it determines the charge in a single rfcycle by determining the number of rf cycles within the rfpulse length based upon the gun operating frequency en-tered (1.316 GHz in this case). The phase shift is given asa default value of 180 ◦ to ensure that the reference parti-cle is at the field crest at the time of emission. This valuecomes from the phase shift of the field by 90 ◦ due to co-sine field used in Superfish and sine field used by ASTRAand GPT. The field crest is found at an additional shift of90 ◦ , hence the total phase shift is set at a default of 180 ◦ .The local field represents the product of the applied fieldtimes the field enhancement factor β , which was foundto be 450, yielding a local field of 6.795 GV/m (givenan operating applied field of 15.1 MV/m. ) The cath-ode radius was determined using the effective emissionarea calculated from the Fowler-Nordheim equation witha total effective emission area of 5,490 nm for all eightemitters. This led to the radius of the emitter for eachof the pyramids being 14.78 nm which closely matchedwith the actual physical size of nano-diamond pyramidtips. (See Appendix A for more details on input param-eters.)FIG. 5: Parameters set for creating initial fieldemission distribution and the custom emission patternreplicating one described in Ref. . page 4evarjian, Schneider and Baryshev MSU MAM Report z =0 and t =0,while Fig. 6b shows the transverse electron distributionat a location along the beamline that would correspondto the position of the imaging screen YAG3 ( z =2.54 m).This results demonstrates and proves computationallythat the transversely shaped beam, once generated, canbe transported downstream for a long distance. Anotherconclusion from this simulation is that it was primarilypossible to achieve thanks to relatively low space chargeeffect. When coupled with beam tracking software, thisresult further proves that a useful means was created toeasily design custom emission patterns allowing for rapidR&D of intrinsically shaped beams, finding beamline set-tings to transport them, and comparing to experiment. IV. OUTLOOK FOR FUTURE RELEASES
A limitation from the application example is that thedesigned field emitter is a pyramid; however, at this cur-rent time only the very tip of the emitter ( z ≈
0) was con-sidered. It is known from experimental measurementsthat the emitter first emits from the tip and then, as theexternal field increases, more of the side walls begin toemit, as the turn-on field for the pyramid is a functionof z and the geometry of the tip emitter. Thus, the nextstep is to expand this work with Fowler-Nordheim equa-tions for non-planar geometries to model such a physicsaccurately. The potential distribution implementationover curved surfaces will strongly affect the momentumand the spatio-temporal distributions.The next generation of field emission sources, asshown in the application example, are most likely tobe semiconductor materials, such as those made of car-bon nanotubes and diamond materials. Recent re-sults have shown a divergence from the classical Fowler-Nordheim conditions, which is attributed to thecurrent model failing to account for an emitter being asemiconductor. Future work will implement the semi-conductor effects using the Stratton-Baskin-Lvov-Furseyformalism, expanding into the rf environment, temper-ature and patchy/varying work function effects. Such im-plementations will be completed for future releases. Thegoal in mind is to modify FEgen to account for theseeffects and to ultimately understand the new emissionphysics that can be seen through experimental studies. FIG. 6: (a) Initial emission pattern on the cathodesurface. Indicated by an arrow is the location of eachemitter with a initial radius of 14.78 nm. The circlerepresents the emitter that was denoted as being placedat the origin. This emitter was selected to be at theorigin because it was the brightest emitter in Ref. ,indicating that that emitter is generated on-axis; (b)Transverse electron beamlet distribution at 2.54 mdownstream from the cathode ( z =0). YAG screen hasradius of 0.0254 m, therefore the entire emission patternwould be projected onto the YAG screen without loss. ACKNOWLEDGMENTS
We would like to thank Dr. Zenghai Li of SLAC andJiahang Shao of ANL for illuminating discussions. Thiswork was supported by the US Department of Energy,Office of Science, High Energy Physics under Coopera-tive Agreement award No. DE-SC0018362 and Michi-gan State University, the College of Engineering, Michi-gan State University, under the Global Impact Initiative.This material is also based upon work supported by theU.S. Department of Energy, Office of Science, Office ofHigh Energy Physics under Award No. DE-SC0020429.page 5evarjian, Schneider and Baryshev MSU MAM Report
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For all entry of parameters using scientific notation, use e notation to prevent errors when inputs are processed (e.g.enter 1.2e3 for 1200, enter 1.2e-6 for 0.0000012). Details on entering parameters are provided in the table below. Parameter Details Unit
One Emitter Select for a single emitter –Emitter Grid Select to create a square grid of emitters. See parameters for Size of Grid and GridSpacing for details on specifying grid size –Custom EmissionPattern Select to create a custom pattern of emitters. See parameter for Locations of Emittersfor details on specifying emitter locations –Size of Grid The length of one side of a square grid of emitters (e.g. Size of Grid = 7 will createa 7 × x , y ) pairs given. To enter locations, separaterespective x and y with a comma and place a semicolon between ( x , y ) pairs. Do notplace a semicolon after the last ( x , y ) pair. Straying from this format will result inan error. See Fig. 5 as an example mCurrent For the pulsed power option, current and pulse length are used to calculate charge q = I × τ dc where τ dc is the dc pulse length ATotal Charge Charge per emitter (e.g. for 1 nC charge over 10 emitters, one would enter 0.1 nC) nCCharge per rf Cycle Select for charge in single rf cycle nCCharge per PulseLength Select for charge over multiple rf cycles. Pulse length and frequency are used tocalculate charge per cycle q = q total f × τ rf where f is the operating frequency and τ rf isthe rf pulse length nCCharge per EmitterRadius Select to scale the total charge to the emitter radius nCCharge per CathodeRadius Select to scale the total charge to a smaller cathode radius q = q total ( r emitter r cathode ) nCInitial Energy Initial energy distribution at cathode surface (default is 0.1 eV) eVLocal Field Equal to β times applied field where β is the field enhancement factor GV/mPhase shift Accounts for phase shift of the ac field. Default value is 180 ◦ due to a 90 ◦ phase shiftbetween the cosine and sine field (used in Superfish and ASTRA/GPT, respectively)and an additional 90 ◦ shift such that the reference particle is at the peak field at thetime of emission degreerf Phase Phase window of the rf cycle. Default is 360 ◦ , phase window can be specified uponentry degreeFrequency Default option provided is for the Argonne Cathode Teststand (ACT) for which FE-gen was originally intended where the default frequency is the L-band operationalfrequency of the ACT. Alternate frequencies can be entered by user Hz, phase window can be specified uponentry degreeFrequency Default option provided is for the Argonne Cathode Teststand (ACT) for which FE-gen was originally intended where the default frequency is the L-band operationalfrequency of the ACT. Alternate frequencies can be entered by user Hz