SLAC-PUB-9912 LBNL{49771, CBP Note{415 Probabilistic Model for the Simulation of Secondary Electron Emission M. A. Furman¤ and M. T. F. Piviy Center for Beam Physics Accelerator and Fusion Research Division Lawrence Berkeley National Laboratory, MS 71-259 Berkeley, CA 94720 (Dated: DRAFT: May 23, 2003) We provide a detailed description of a model and its computational algorithm for the secondary electron emission process. The model is based on a broad phenomenological ¯t to data for the secondary emission yield (SEY) and the emitted-energy spectrum. We provide two sets of values for the parameters by ¯tting our model to two particular data sets, one for copper and the other one for stainless steel. PACS numbers: 79.20.Hx, 34.80.Bm, 29.27.Bd, 02.70.Uu Keywords: Secondary electron emission, electron-cloud e®ect. I. INTRODUCTION. exceed the primary energy; and (4) the aggregate energy of the electrons emitted in any multi-electron event is also guaranteed not to exceed the primary energy. The The existence of the electron cloud e®ect (ECE) [1, 2], main challenge is the construction of the joint probability whose ¯rst and most prominent manifestation is beam- distributions for events in which two or more secondary induced multipacting [3, 4], has been ¯rmly established electrons are generated in such a way as to satisfy these experimentally at several storage rings [5{8]. Generally constraints. speaking, the ECE is a consequence of the strong cou- pling between a charged-particle beam and the vacuum The main purpose of this article is the description of chamber that contains it via a cloud of electrons in the the model and its computer implementation in the larger chamber. The ECE is detrimental to the performance of ECE simulation code. We also provide sample ¯ts to modern storage rings, which typically make use of intense existing data on the SEY and emitted-energy spectrum. beams, closely spaced bunches, and/or vacuum chambers With regards to this latter quantity, we provide here a of small transverse dimensions. correction to a previously used [9, Eq. 4.5] expression that was conceptually incorrect. Although the emitted- For the past several years we have been studying the energy spectrum is not computed nor used directly in ECE by means of multiparticle simulations with a code the ECE simulation code, it is used to extract param- that includes a detailed probabilistic model of the sec- eters from the data which are then fed as input to the ondary emission process [9, 10], which is one of the crit- simulation. Recent work has shown, in some cases, an ical contributors to the ECE. The input ingredients of unexpectedly strong sensitivity of the overall simulation the model are the secondary emission yield (SEY) ± and results on low-energy details of the SEY and the energy the emitted-energy spectrum of the secondary electrons spectrum [11, 12] that remains to be fully characterized d±=dE. The main result from our construction is the set and understood. Motivated by this, we have paid partic- of probabilities for the generation of electrons. This set ular attention to the above-mentioned low-energy details of probabilities is embodied in a Monte Carlo procedure in our model. Therefore, although the model involves a that generates simulated secondary emission events given fair number of adjustable parameters, and some of them the primary electron energy and angle. We represent ± cannot be uniquely pinned down by presently available and d±=dE by fairly general phenomenological ¯ts con- data, its mathematical consistency and its good over- structed to obtain good agreement with a broad range all agreement with secondary emission data ensure that of data. An additional virtue of the model is that it is the above-mentioned sensitivity cannot be attributed to mathematically self consistent; by this we mean that the mathematical artifacts of the model nor to inadequate event generator is constructed so that: (1) when averag- representation of the data. ing over an in¯nite number of secondary emission events, The Monte Carlo technique has been used before for the reconstructed ± and d±=dE are guaranteed to agree the description of the secondary emission process. In with the corresponding input quantities; (2) the energy a more traditional approach, the main ingredient is a integral of d±=dE is guaranteed to equal ±; (3) the en- microscopic model for the secondary-emission material, ergy of any given emitted electron is guaranteed not to typically speci¯ed by the electron and ion distributions, and by the elastic and inelastic cross sections for the col- Work supported in part by the Department of Energy Contract DE-AC03-76SF00515 lision of the primary and secondary electrons with the ¤Electronic address: [email protected] ions and with the other electrons in the material [13]. yElectronic address: [email protected]; address after Jan. 1st, 2003: In this approach, one may infer microscopic properties SLAC, P.O. Box 20450, Stanford CA 94309. of the material by comparing measured data for ± and Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 2 d±=dE with the corresponding quantities computed from II. MODEL OF SECONDARY ELECTRON the model. On the other hand, in the approach we EMISSION. present in this article, as mentioned above, the main in- gredient is the measured data for ± and d±=dE, and the A. Basic phenomenology. main result is the set of joint probability functions for the emission of secondary electrons. Thus our model is The two main quantities used in the experimental essentially phenomenological, and does not a®ord a di- study of the secondary emission process are the SEY ± rect insight into the properties of the material or the and the emitted-energy spectrum d±=dE [17, 18]. To de- theory of secondary emission. In particular, some of our ¯ne these, we consider a steady mono-energetic electron ¯tting formulas are of di®erent form from those based on beam impinging on a surface. The SEY is de¯ned by the theory of metals. However, although our formula for the true-secondary yield (Sec. III D 1) is di®erent from I ± = s (1) the conventional one [14], it does incorporate the well- I established range-energy relation [15]. 0 where I0 is the incident electron beam current and Is is the secondary current, i.e., the electron current emitted from the surface. The yield is a function of the kinetic en- ergy E0 of the incident electron beam, its incident angle θ0, and the type of surface material and its state of condi- In Sec. II we describe the secondary emission process 1 by ¯rst briefly recapitulating the basic phenomenology tioning. For applications to the ECE, we are primarily and then providing the probabilistic description of the interested in incident energies E0 below a few keV's, al- emission process in terms of the \most di®erential prob- though the framework presented here is formally valid for all energies. abilities" Pn, which constitute the basic building blocks for our model. This probabilistic description is quite gen- By applying a retarding voltage V in front of the sec- eral, and we believe its validity to be rooted in general ondary current detector one can select those electrons principles of the quantum theory for the secondary emis- that are emitted with individual energies Ek ¸ E = eV . sion process. In Sec. II C we de¯ne a speci¯c phenomeno- The cumulative emitted-energy spectrum S(E0;E) is then de¯ned to be logical model for the Pn's by following the principle of maximum simplicity consistent with the data. In partic- Is(E) ular, we strictly enforce the condition that the energy of S(E ;E) = (2) 0 I any secondary electron may not exceed that of the inci- 0 dent (primary) electron, a fact that is clearly exhibited where Is(E) is the secondary current that overcomes the by secondary energy spectrum data. In addition, we also retarding voltage (for notational conciseness we suppress impose the same restriction on the aggregate secondary a dependence of S on θ0). The emitted-energy spectrum energy. Although we are not aware of experimental data d±=dE is de¯ned to be supporting this latter restriction, we believe it to be true on account of general physical principles. In Sec. III we d± @S(E0;E) continue the de¯nition of our model by providing detailed = ¡ (3) dE @E parametrizations for each of the three components of the SEY based on various reviews of the theory and phe- where the ¡ sign ensures that d±=dE > 0 (the emitted- nomenology of the subject [14, 16{18]. In Sec. IV we energy spectrum d±=dE can also be measured directly by carry out the analytic calculation of the energy spectrum means of a magnetostatic or electrostatic energy analyzer within our model. In Sec. V we provide the algorithmic [19, 20]). Note that Is(0) in Eq. (2) is what is simply description of the probabilisitic model just constructed, called Is in Eq. (1), so that S(E0; 0) = ±(E0), hence as implemented in our ECE simulation code. In Sec. VI we use the energy spectrum, along with the three compo- Z1 d± nents of the SEY, to ¯t the data and extract the various dE = ±(E0): (4) parameters of the model. In Sec. VII we summarize our dE 0 conclusions. The various Appendices provide A: mathe- matical details of the analytic calculation of the energy For more detailed descriptions of the secondary emis- spectrum. B: a simpli¯ed alternative model for the Pn's sion process one may require additional variables or mea- that does not respect the above-mentioned constraints sured quantities.