Role of rf electric and magnetic fields in heating of micro-protrusions in accelerating structures. Gregory S. Nusinovich and Thomas M. Antonsen, Jr.

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1 Role of rf electric and agnetic fields in heating of icro-protrusions in accelerating structures. Gregory S. Nusinovich and Thoas M. Antonsen, Jr. Abstract It is known that high-gradient operation in etallic accelerating structures causes significant deterioration of structure surfaces that, in turn, greatly increases the probability of icrowave breakdown. At the sae tie, the physical reason for this deterioration so far is not well understood. In the present paper, the role of two effects is analyzed, viz. (a) the icrowave heating caused by penetration of the rf agnetic field into icroprotrusion of a radius on the order of the skin depth and (b) the Joule heating caused by the field eitted current, i.e. the effect of the rf electric field agnified by a sharp protrusion. Corresponding expressions for the power densities of both effects are derived and the criterion for evaluating the doinance of one of these two is forulated. This criterion is analyzed and illustrated by the discussion of an exaple with paraeters typical for recent experients at the Stanford Linear Accelerator Center (SLAC) National Accelerator Laboratory. Key words: high-gradient accelerating structures, skin effect, Joule heating, dark current. 1

2 In the developent of linear accelerators of the next generation, one of the ain probles is providing reliable operation with low breakdown rate using the highest gradients possible. Theoretical and experiental studies aiing at iproving our understanding of possible causes of breakdown at high gradients has been an area of active research for a long tie. More than a decade ago it was widely accepted that the key factor liiting high-gradient operation is field eission fro sall protrusions, since this field eission causes the appearance of the dark current which, for a nuber reasons, can be considered a precursor of the breakdown [1]. This field eission described by the Fowler-Nordhei law is caused by the rf electric field which can be greatly agnified by a sharp protrusion. Later, significant attention was also paid to how the rf agnetic field causes deterioration of accelerating structures []. Experiental studies of both effects are in progress (see, e.g., recent papers [3, 4]). These experients have stiulated corresponding theoretical studies (see, e.g. [5, 6]). As a rule, the authors focus their attention priarily either on the effect of the rf electric field or on the effect of the rf agnetic field. (Ref. 6 is an exclusion of this rule.) At the sae tie, it sees reasonable to believe that the role of these fields depends on experiental conditions, viz. under one set of conditions the rf electric field plays a doinant role, while under another set of conditions the ost iportant is the rf agnetic field. Below, we analyze these contributions for the case of sall cylindrical icroprotrusion which ay exist on surfaces of accelerating structures after a certain period of high-gradient operation. Consider a thin cylindrical icroprotrusion on a structure surface. Then, the power (per unit length) of icrowave losses caused by the rf agnetic field penetrating into protrusion can be given as [7]: p S μ H skin = ωα. (1) In (1), ω is the wave frequency, α is the iaginary part of the agnetic polarisability of a cylinder, S is the cross-sectional area of the protrusion, μ is the pereability of free space (below we consider non-agnetic aterials) and H is the aplitude of the rf

3 agnetic field. Note that in (1) we took into account only the icrowave losses caused by the penetration of the rf agnetic field because estiates show [8] that under all reasonable conditions these losses are uch higher than contribution fro the iaginary part of the electric polarisability. Introducing the skin depth δ defined by a known forula [9] rewrite Eq. (1) as = / ωμ σ (where σ is the conductivity of the etal) allows us to δ p skin 1 S = σ δ α H. () Expressions for the agnetic polarisability of a cylindrical wire of a circular crosssection can be found elsewhere [7-9]. For other configurations of icro-protrusions they can be derived by using the ethod described elsewhere [7]. Let us now evaluate the power of Joule heating of this protrusion by an electron current. As a rule, it is assued that, when this current is eitted fro the protrusion apex, its density obeys the Fowler-Nordhei equation. In general, there are soe reasons [1] to take into account also theral contributions defined by the Richardson-Dushan equation to this current, but below we will discuss this current just bearing in ind the field eission defined by the Fowler-Nordhei law. In general, the power of Joule heating can be defined by the standard equation P = R I where angular brackets denote averaging over the rf period as well as possible other distributions over the angular position of an eitting site on apex, initial velocities etc (those distributions we will neglect below). Taking into account that the resistance per unit length is equal to R = ρ / S where ρ = 1/ σ, we can define the power (per unit length) of Joule heating caused by this electron current as 1 1 p J = I. (3) σ S Fro coparison of Eq. (3) with Eq. () it follows that the role of the rf agnetic field is ore iportant in protrusion heating than the role of the rf electric field causing eission of the dark current when the following condition holds: S I α >. (4) δ H 3

4 For protrusions of a circular cross-section with radius I a, Eq. (4) reduces to 4 a a Φ π α > (5) The right-hand sides of Eq. (4) and Eq. (5) contains the electron current squared averaged over the rf period I. The distribution of I over the rf period has been analyzed elsewhere [6, 11]. It was shown [11] that for typical values of the work function 4-5 ev δ H and axiu surface gradients not exceeding 1 GV/, the ter I is less than.1 of ( ) ( ) its peak value I. The latter can be treated as its value in the DC electric field I. Below we will denote the ratio I / I by Ψ and present I as ( ) Ψ (I ). It should be noted that the DC coponent of the current I ( ) is equal to the current eitted fro the protrusion apex. The electron current density on the apex obeys the Fowler-Nordhei law that takes into account agnification of the rf electric field by a sharp end of the protrusion. Then, the total eitted current is the product of this density and the apex area. Typically, the area of this apex is uch saller than the cross sectional area of a central part of the protrusion. So the current density in this central part of protrusion is uch saller than on apex, but the total current reains the sae. Let us now discuss the iaginary part of the agnetic polarisability. The agnetic polarisability of a cylinder with a circular cross-section with respect to the agnetic field perpendicular to the cylinder axis can be defined as [7] where k ( 1+ i) / δ ( ka) ( ka) 1 J1 α = 1 (6) π ka J = is the coplex wave nuber. Corresponding dependence of the iaginary part of this polarisability on the a / δ -ratio is given in Ref. 8 where pulse heating by this rf field was studied with the account for the teperature dependence of conductivity and, hence, the skin depth. In the case when the protrusion radius a is uch saller than the skin depthδ, the iaginary part of this polarisability can be approxiated as [7] 1 a α =. (7) 8π 4

5 So the Poynting flux per unit length in such a case is equal to p skin 1 a = 8σ and the condition of doinance of the effect of rf agnetic field given above by (5) reduces to a Φ1 = π a li Φ 8 a / δ 6 4 H ( ) [ I ] > Ψ δ H This equation contains the sixth power of the radius-to-skin depth ratio. Partly, this originates fro the fact that in the case of sall radius of protrusion the losses caused by penetration of the rf agnetic field into it decrease with the conductor radius proportionally to the fourth power of a where a (8) (9) coes fro the conductor area (the rf field does not see a very thin conductor). Another ( a / δ ) ter coes fro the dependence of the iaginary part of the agnetic polarisability on the radius. At the sae tie, as the protrusion radius gets saller, its resistance gets larger, thus, joule heating of the conductor by the field eitted current increases. In the opposite liiting case, when the protrusion radius is uch larger than the skin depth, the iaginary part of the agnetic polarisability can be approxiated [7] as α = ( 1 / π )( δ a). Thus, instead Eq. (9) the following condition holds: / a Φ = π a li Φ a / δ 3 ( ) [ I ] > Ψ δ H Dependencies of all three functions, the general one ( ) a /δ Eq. (5), the function ( ) for large Φ a /δ 1 for sall / δ (1) Φ for arbitrary a / δ -ratios in a -ratios in Eq. (9) and the function Φ ( ) a /δ a / δ -ratios in Eq. (1) are shown in Fig. 1. As one can see, the sall a / δ -ratio approxiation works well up to approxiation gives reasonable agreeent starting fro a / δ values about one, while the large a / δ -ratio a / δ close to.5. 5

6 Figure 1. The function Φ and its approxiations as functions of the a / δ -ratio Let us illustrate these siple forulas with soe exaples. Consider the operation at the GHz frequency which for copper yields the skin depth of.63 icron. Let us liit our consideration by the case of thin protrusions (sall a / δ -ratio). Then, the range of protrusion radii we consider will be fro.1 to 1 icron. Assue that the ratio of the averaged to peak values of the squared current Ψ is close to.1. Now, we are left in (9) with only two paraeters: the total value of the Fowler-Nordhei DC current and the intensity of the rf agnetic field. As follows fro the analysis of the Fowler-Nordhei current in icroprotrusions given in Ref. 1 and supported by soe experiental data fro SLAC, the total dark current field eitted fro typical icroprotrusions can be as high as.1 A. Corresponding current densities fro sall apexes of these protrusions are on the order of a fraction of A/ μ. Regarding the possible level of intensity of the rf agnetic field let us note that recent experients at SLAC [1] were carried out at surface agnetic field values on the order of.5-.6 MA/. If we assue the total dark current is equal to.1 A and the rf agnetic field is equal to.5 MA/, we readily get fro Eq. (9) that penetration of the rf agnetic field is the doinant factor for heating of icroprotrusion with the radius exceeding.5δ, i.e. larger than 3 n. Note that to have intense heating not only should the absorbed power be large enough, but also the pulse duration should be short enough to avoid heat sink fro 6

7 protrusion to the body of a etallic structure. The latter eans that the height of a icroprotrusion h should be larger than the heat propagation distance l h = Dτ (here D is the diffusion coefficient and τ is the icrowave pulse duration). For copper the diffusion coefficient is close to.1 μ / ns. For exaple, in the case of 1 ns pulses, this condition h > l h is valid for icroprotrusions with a height exceeding 4 icrons. Therefore we should expect significant heating caused by the skin effect in the case of icro-protrusions with a height-to-radius ratio on the order of 1. Note that just such thin and long icro-protrusions were found responsible [13] for the dark current in experients with the DC field. Foration of such icroprotrusions on the originally sooth, well polished surface of an accelerating structure is, in turn, an area of active research. In this regard, first, let us note that often the footnotes of breakdown events are found in the parts of accelerating structures where both rf electric and rf agnetic fields are present (see, e.g., Ref. 14). Then, Ref. 5 should be entioned where surface heating by the rf agnetic field up to its elting and subsequent foration of Taylor cones is analyzed. Clearly, when the shape of such conical protrusion sharpens, this akes the radius of protrusion (at least of its part close to the apex) saller. So, it ay happen that in the course of this evolution of the protrusion shape the distribution of roles changes: while in the initial stage the key role belongs to the rf agnetic field, then, in the case of increasing height-to-base ratios, the rf electric field becoes ore iportant. This issue of dynaic evolution of the protrusion shape, however, goes beyond the scope of the present paper. This work was supported by the Division of High Energy Physics of the U.S. Departent of Energy. The authors are indebted to S. Tantawi for insightful discussion and D. Kashyn for help in preparation of the figure. 7

8 References: 1. J. W. Wang and G. A. Loew, SLAC Report No. SLAC-PUB-7684, D. P. Pritzkau and R. H. Sieann, Phys. Rev. ST Accelerators and Beas, 5, 11 (). 3. L. Laurent, S. Tantawi, V. Dolgashev, C. Nantista, Y. Higashi, M. Aicheler, S. Heikkinen and W. Wuench, Phys. Rev. ST Accelerators and Beas, 14, 411 (11). 4. N. S. Ginzburg, I. I. Golubev, A. K. Kainsky, A. P. Kozlov, S. V. Kuzikov, E. A. Perelstein, N. Yu. Peskov, M. I. Petelin, S. N. Sedykh, A. P. Sergeev, A. S. Sergeev, A. A. Vikharev, and N. I. Zaitsev, Phys. Rev. ST Accelerators and Beas, 14, 41 (11). 5. P. B. Wilson, 1 th AAC Workshop, AIP Conf. Proc., Melville, New York, 877, 7 (6). 6. A. Grudiev, S. Calatroni, and W. Wuench, Phys. Rev. ST Accelerators and Beas, 1, 11 (9). 7. L. D. Landau and E. M. Lifshitz, Electrodynaics of Continuous Media (Pergaon Press, Oxford, 196). 8. G. S. Nusinovich, D. Kashyn, and T. M. Antonsen, Jr., Phys. Rev. ST Accelerators and Beas, 1, 111 (9). 9. S. Rao and J. R. Whinnery, Fields and Waves in Modern Radio, John Wiley & Sons, New York, London, 196, Section K. L. Jensen, Y. Y. Lau, D. W. Feldan and P. G. O Shea, Phys. Rev. ST Accelerators and Beas, 11, 811 (8). 11. G. S. Nusinovich, D. G. Kashyn, A. C. Keser, O. V. Sinitsyn, T. M. Antonsen, Jr., and K. Jensen, Effects Liiting High-Gradient Operation of Metallic Accelerating Structures, US High-Gradient Research Collaboration Workshop, February 9-1, 11, SLAC, Menlo Park, CA; 1. V. Dolgashev, S. Tantawi, Y. Higashi, and B. Spataro, Appl. Phys. Lett., 97, (1). 13. R. P. Little and W. T. Whittney, J. Appl. Phys., 34, 43 (1963). 14. V. Dolgashev, AIP Conf. Proc., 199, Melville, New York, 1, p