1 UNIVERSITA' DEGLI STUDI DI PADOVA Sede Amministrativa: Università degli Studi di Padova Dipartimento di Scienze MM.FF.NN. SCUOLA DI DOTTORATO DI RICERCA IN SCIENZA ED INGEGNERIA DEI MATERIALI XX CICLO 6 GHz CAVITIES: A METHOD TO TEST A15 INTERMETALLIC COMPOUNDS rf PROPERTIES Direttore della Scuola : Ch.mo Prof. Gaetano Granozzi Supervisore : Ch.mo Prof. Vincenzo Palmieri Dottorando : Silvia Maria Deambrosis DATA CONSEGNA TESI: gennaio 2008
2 CONTENTS i Contents Abstract Estratto Introduction ix 0.1 Cavity research ix A frontier particle accelerators ix Superconducting cavities x Cavity performance limitations xi 0.2 Alternatives to bulk niobium xii GHz cavities xiv 0.4 Organization of the dissertation xiv 1 Superconducting Resonant Cavities Cavity fundamentals and cavity elds Radio-frequency elds in cavities The accelerating eld Peak surface elds Power dissipation and the cavity quality Superconductivity essentials Superconductor surface resistance Critical magnetic eld Alternative materials to solid niobium A15 Superconducting Compounds and their Potential Application to rf Resonant Cavities Introduction A15 compounds crystal structure A15 superconducting critical temperature Experimental results Phase diagrams and T c Theory of T c in A15 compounds A15 critical elds The chosen A15 compounds v vii
3 ii CONTENTS 3 Methods to measure superconductors surface resistance Surface impedance measurements of superconducting lms by a ring microstrip resonator technique A method of surface resistance measurement with a niobium triaxial cavity working at 2 K rf surface resistance measurements on superconducting samples with vacuum insulated thermometers An instrument to measure the surface resistance of superconducting samples at 400 MHz GHz Cavities GHz Cavity geometry GHz cryogenic infrastructure Measuring bench Fundamental equations for rf test RF system Cavity measurements procedure Software RF system upgrading Nb 3 Sn Chemical and physical properties Variation in lattice properties Electron-Phonon interaction as a function of atomic Sn content T c as a function of atomic Sn content Nb 3 Sn production methods and applications Multilament wire fabrication techniques Superconducting cavities fabrication techniques V 3 Si Chemical and physical properties V 3 Si production methods and applications Multilament wire fabrication techniques Electronic devices fabrication techniques Superconducting cavities fabrication techniques Nb 3 Sn: Production and Results Introduction Nb 3 Sn: The chosen production method The Nb 3 Sn phase formation Nb 3 Sn: experimental apparatus Nb 3 Sn: 6 GHz cavities stand Nb 3 Sn samples Nb 3 Sn samples: preliminary surface treatments Nb 3 Sn samples: production Nb 3 Sn samples: dierent production methods Nb 3 Sn samples: lms obtained Nb 3 Sn samples: post-process treatments
4 CONTENTS iii 7.5 Nb 3 Sn samples: analysis SEM XRD SIMS EMPA PPMS Nb 3 Sn 6 GHz cavities Nb 3 Sn 6 GHz cavities: preliminary surface treatments Nb 3 Sn 6 GHz cavities: production Nb 3 Sn 6 GHz cavities: lms obtained Nb 3 Sn 6 GHz cavities: analysis and rf test Nb 3 Sn 6 GHz cavities: analysis Nb 3 Sn 6 GHz cavities: rf test V 3 Si: Production and Results The Chosen Production Method The V 3 Si phase formation V 3 Si: experimental apparatus V 3 Si: experimental apparatus modications V 3 Si samples V 3 Si samples: preliminary surface treatments V 3 Si samples: production V 3 Si samples: lms obtained V 3 Si samples: analysis Optical microscope Proler SEM XRD PPMS V 3 Si 6 GHz cavities V 3 Si 6 GHz cavities: preliminary surface treatments V 3 Si 6 GHz cavities: production V 3 Si 6 GHz cavities: lms obtained V 3 Si 6 GHz cavities: analysis and rf test V 3 Si 6 GHz cavities: analysis V 3 Si 6 GHz cavities: rf test Discussion Nb 3 Sn Nb 6 GHz cavities Nb 3 Sn 6 GHz cavities V 3 Si V 6 GHz cavities V 3 Si 6 GHz cavities GHz cavities as a novel tool for rf testing A15 materials
5 iv CONTENTS 10 Future Developments Nb 3 Sn: future developments Nb 3 Sn: double furnace system Nb 3 Sn: post-process surface treatments V 3 Si: future developments The "Plasma" process V 3 Si: post-process surface treatments Conclusions GHz cavities Nb 3 Sn by Tin thermal Diusion V 3 Si by thermal diusion A Visual Basic code 217 List of Tables 223 List of Figures 225 BIBLIOGRAPHY 235
6 v Abstract Since the International Committee for Future Accelerators recommended that the Linear Collider design has to be based on the superconducting technology, the scientic world interest is now focused on further developments of new resonant cavities fabrication techniques and cost reduction. It is important to pursue research on new materials: the goal will be the achievement of superconducting cavities working better than the Nb ones at 4.2 K. For superconducting alloys and compounds, at a given operating temperature, the best rf performances (low surface resistance and λ, high relevant critical elds) are obtained for high T c and low ρ n materials. Among the possible candidates, A15 compounds appear to be the most promising. We needed a fast, easy and performing way to characterize A15 superconducting materials for their potential application to accelerating resonators. The idea is to build microcavities completely equal in shape to the real scale model. The rf characterization of samples is an useful diagnostic tool to accurately investigate local properties of superconducting materials. However, a common limitation of systems used for this, often consists in the diculty of scaling the measured results to the real resonator. In this work we will proof that 6 GHz resonators can simply become our cavity shaped samples. Our attention was focused on two materials: V 3 Si that has a really high RRR value and Nb 3 Sn that is the only A15 material already used for a resonant accelerating structure . The process parameters optimization necessary to improve the A15 phase superconducting properties, crystal structure and morphology is going on through the small sample production: this is fundamental but still not enough. We are perfectly aware that having satisfactory results with A15 samples, doesn't mean obtaining good superconducting cavities with ease. Our solution is to work directly with cavities. Obviously using 1.5 GHz resonant structures would be time wasting and a cost limited approach. In the best situation, working very hard, one can produce and measure one resonator every two weeks. 6 GHz cavities are made from larger cavities fabrication remaining material, they don't
7 vi Abstract need welding (even for anges) and they can be directly measured inside a liquid helium dewar. Finally it is possible to perform more than one rf test per day!
8 vii Estratto Dal momento in cui la International Committee for Future Accelerators ha stabilito che il nuovo Linear Collider dovrà essere basato sulla tecnologia superconduttiva, l'interesse del mondo scientico si è spostato verso ulteriori sviluppi nell'ambito della fabbricazione di cavità e riduzione dei costi. E' estremamente importante promuovere la ricerca su nuovi materiali che possano fornire strutture risonanti con migliori prestazioni di quelle del niobio a 4.2 K. Nel caso di leghe e composti, le migliori performance in radio frequenza (bassa resistenza superciale e lunghezza di penetrazione del campo magnetico, alti campi critici), ad una data temperatura di lavoro, si ottengono nel caso in cui, non solo la temperatura critica sia elevata ma il comportamento sia metallico (bassa resistività in stato normale). Tra i possibili candidati, i composti A15 sembrano essere particolarmente promettenti. Avevamo bisogno di un modo facile e veloce di caratterizzare i materiali A15 vista la loro potenziale applicazione a strutture acceleranti superconducttive. Abbiamo deciso di costruire micro-cavità che siano esattamente identiche a quelle reali, solo più piccole. La caratterizzazione in radiofrequenza di piccoli campioni è un mezzo diagnostico accurato per investigare le proprietà locali dei materiali superconduttori. Una evidente limitazione di tali sistemi spesso consiste nelle dicoltà di risclare i risultati ottenuti ad una cavità reale. Mediante questo lavoro proveremo che le strutture risonanti a 6 GHz possono divenire i nostri "campioni a forma di cavità". Sono stati scelti due composti A15: il V 3 Si, che mostra elevati valori di RRR ed il Nb 3 Sn, ovvero il solo A15 già utilizzato per la costruzione di cavità acceleratrici. L'ottimizzazione dei parametri di processo, necessaria per migliorare le proprietà superconduttive della fase A15, la struttura cistallina e la morfologia viene condotta attraverso la produzione dei campioni: naturalmente questo punto è fondamentale ma non suciente. Siamo perfettamente consapevoli del fatto che ottenere risultati soddisfacenti con i campioni non signichi necessariamente ottenere facilmente buone cavità superconduttive. La nostra soluzione è quella di lavorare direttamente con le cavità. Per ovvi motivi non sarebbe possibile servirsi di strutture risonanti ad una frequenza pari a 1.5 GHz: sarebbe troppo dispendioso sia in termini di tempo che di costi. Anche nelle migliori condizioni e
9 viii Estratto lavorando molto è possibile produrre e misurare una cavità ogni due settimane. Le cavità 6 GHz vengono fatte tramite la tecnica dello spinning, con gli scarti di produzione delle cavità più grandi, non necessitano di saldature, nemmeno per le ange ed inoltre possono essere misurate una volta inserite in un semplice dewar di elio liquido. Vista la rapidità di rareddamento e lo scarso consumo di elio è plausibile eettuare più di un test rf al giorno!
10 ix Introduction 0.1 Cavity research A frontier particle accelerators In the past century, physicists have explored smaller and smaller scales, cataloguing and understanding the fundamental components of the universe, trying to explain the origin of mass and probing the theory of extra dimensions. And in recent years, experiments and observations have pointed to evidence that we can only account for a surprising ve percent of the universe. Scientists believe that the remaining 95 percent is a mysterious dark matter and dark energy, revealing a universe far stranger and more wonderful than they ever suspected. The global particle physics community agrees that a machine like the International Linear Collider will answer these questions about what the universe is made of and provide exciting new insights into how it works. Using unprecedented technology, discoveries are within reach. This could stretch our imagination with new forms of matter, new forces of nature, new dimensions of space and time and bring into focus Albert Einstein's vision of an ultimate unied theory. The International Linear Collider will give physicists a new cosmic doorway to explore energy regimes beyond the reach of today's accelerators. The proposed electron-positron collider (ILC) will complement the Large Hadron Collider, a proton-proton collider at the European Center for Nuclear Research (CERN) in Geneva, Switzerland, together unlocking some of the deepest mysteries in the universe. Consisting of two linear accelerators that face each other, the ILC will hurl some 10 billion electrons and their anti-particles, positrons, toward each other at nearly the speed of light. Accelerator cavities give the particles more and more energy until they smash in a blazing crossre at the center of the machine. Stretching approximately 35 kilometers in length, the beams collide 14,000 times every second at extremely high energies (500 billion-electron-volts(gev)). Each spectacular collision creates an array of new particles that could answer some of the most fundamental questions of all time. The current baseline design allows for an upgrade to a 50-kilometers, 1 trillion-electron-volt (TeV) machine during the second stage of the project. Planning, designing, funding and building the proposed International Linear Collider
11 x Introduction Figure 1: A photograph of a bulk niobium TESLA-type 9-cells cavity. will require global participation and global organization. An international team of more than 60 scientists and engineers leads the Global Design Eort (GDE) for the ILC. The GDE team sets the design and priorities for the work of scientists and engineers around the world. From the senior physicist to the undergraduate student, about 2000 people from more than 100 universities and laboratories in over two dozen countries are collaborating to build the ILC, the next-generation particle accelerator (http://www.linearcollider.org/cms/) Superconducting cavities Cavities are the devices used to provide energy to the particles. Radio frequency (rf) accelerating structures are the mostly employed and an example of them is shown in Figure 1. In the past, copper cavities were utilized for acceleration (e.g., at SLAC): however, over the last 20 years, superconducting bulk niobium technology has proven itself as the alternative. Although not completely loss free above T = 0 K, as in the dc case, superconducting cavities dissipate orders of magnitude less power than normal conducting accelerating structures. The dramatically reduced resistivity translates into a number of very important advantages. They include : 1. Operating cost savings: Even when taking into account the cost of refrigerating superconducting cavities, their power demand in cw applications is more than two orders of magnitude less than that of equivalent copper cavities. 2. Capital cost savings: The reduced power requirements translate into capital cost
12 0.1 Cavity research xi savings, since fewer (and sometimes simpler) klystrons are needed. 3. High gradient: The relatively low power consumption also enables superconducting cavities to operate at high cw gradients. 4. Reduced impedance: The aperture of superconducting cavities is large, thereby minimizing disruptive interactions of the cavity with the beam, higher currents can therefore be accelerated. Superconducting bulk niobium resonant structures has been successfully used in many machines: among them HERA and TESLA (DESY, Hamburg, Germany), CEBAF (Thomas Jeerson Laboratories, Newport News, Virginia, USA), the KEK B-factory (KEK, Tzukuba, Japan), the LHC (CERN, Geneva, Switzerland). Superconductors have played a pioneering role at both the energy frontier and the high current one. Extensive research has therefore been performed to understand the performance limitations of superconducting cavities and to improve upon the achieved accelerating gradients. Going by these remarks the International Committee for Future Accelerators recommended that the Linear Collider design has to be based on the superconducting technology: the scientic world interest is now focused on further developments of new resonant cavities fabrication techniques and cost reduction Cavity performance limitations A limit on the maximum accelerating gradient of superconducting cavities is imposed by the superheating magnetic eld. At no point of the cavity surface may the magnetic eld exceed the superheating eld, otherwise the superconductor goes normal conducting ("quenches"). Niobium cavities (TESLA shape, se Figure 1), for example, are therefore limited to an accelerating gradient of about 50 MV/m. Though, for a number of reasons, such high accelerating gradients are dicult to achieve in practical cavities because of some limiting mechanisms as eld emission and thermal breakdown. In the presence of high surface electric eld, rf power is lost to electrons that tunnel out of the cavity wall at very localized points. The emitted electrons are accelerated by the electromagnetic elds and, upon impact, heat the cavity wall and produce X-rays. Field emission scales exponentially with the electric eld and is capable of consuming inordinate amounts of power. Thermal breakdown generally results when a highly resistive defect on the rf surface causes a large fraction of the cavity to quench. An abrupt reduction of the cavity quality results. Thermal breakdown can also be initiated by the heat from bombarding eld emission electrons. Even at low eld levels (below an accelerating gradient of a few MeV/m) all cavities
13 xii Introduction display losses higher than theoretically expected. The anomalous losses are attributed to a temperature independent residual resistance. Its sources are impurities on the rf surface, adsorbed gases and residual magnetic ux that is trapped in the superconductor as it is cooled through the transition temperature. The residual resistance (R 0 ) always dominates the rf surface resistance at low temperatures. Known that for well prepared niobium a R 0 value as low as 5 nω is possible, it is pointless the necessity to cool cavities to temperatures for which residual resistance is the most important term. At 1.5 GHz, for example, the niobium R 0 begins to dominate at T 1.8 K. It becomes mandatory to try to nd new materials behaving much better than niobium at 4.2 K: this will give us the opportunity to avoid the high cryogenic costs rising from the superuid helium utilization (i.e. below 2.2 K). 0.2 Alternatives to bulk niobium Besides the attempt to improve the Nb sputtered on Cu accelerating structures performance, it is important to pursue research on new materials. As mentioned in the goal will be the achievement of superconducting cavities working better than the Nb ones at 4.2 K. Selection criteria In the theoretical description of superconducting state (BCS theory) three main microscopy parameters need to be used : g(ε F ): density of states at the Fermi energy l: electron mean free path (due to impurity scattering) V 0 : eective (phonon mediated) electron - electron interaction These represent the eective number of free electrons, their scattering rate, and their (phonon mediated) eective attraction respectively. They can be written in terms of directly measurable corresponding macroscopic parameters: γ: Sommerfeld constant = (π 2 /3) k 2 B g(ε F ) ρ n : residual resistivity (1/RRR) = e 2 /3 v F g(ε F ) T c : critical temperature = 1.14 Θ D exp[-1/(v 0 g(ε F ))] (being k B the Boltzmann constant, RRR the residual resistivity ratio, v F the Fermi velocity and Θ D the Deby temperature). In the same frame, for a type II superconductor in the dirty limit, the relevant quantities for rf applications can be in turn expressed in terms of
14 0.2 Alternatives to bulk niobium xiii the following macroscopic parameters (CGS units, T < T c /2): the BCS surface resistance (R BCS ), the penetration depth (λ), the critical elds (H c and H c1 ). ( ) 3 R BCS = R n ω 2 σ 1 = A ρ n 2 π σ n stc T e ( k B T 1 + e k B T ) 2 ω 2 ln ω (1) [ ] 1 ρn λ = B η T c (2) H c = C γ 1/2 η T c H c1 = D B η T c ρ n H sh = 0.75 H c (3) (being η = s/3.52 = strong coupling correction, A = , B = 10 2, C = 2.4, D = ) These approximated expressions clarify that for superconducting alloys and compounds, at a given operating temperature, the best rf performances (low surface resistance and λ, high relevant critical elds) are obtained for high T c and low ρ n materials . Among the possible candidates, A15 compounds appear to be the most promising. Of course this is only a rst approximation approach since many other considerations come into play: the presence of the residual surface resistance and problems related to the preparation technique are the most important . The A15 materials structure is classied as the W 3 O or Cr 3 Si and the stoichiometric composition is A 3 B: A is a transition metal, B can be a transition metal or not. B atoms forms a bcc crystal lattice, while A atoms arrange themselves in chains parallel to the three crystallographic directions , , . Superconducting parameters of most A15 compounds are strongly inuenced by the Long-Range crystallographic Order (LRO) degree, especially when the B atom is not a transition metal. All what can aect the LRO must be carefully taken into account. Composition and strain (or lattice distortions) are strictly connected to the critical parameters of the A15 phase. Optimum properties are generally obtained in the stoichiometric strain free state. Just a few of them could have a practical interest for rf applications: Nb 3 Sn, Nb 3 Al, Nb 3 Ga, Nb 3 Ge, V 3 Ga, V 3 Si, the Mo-Re system. Of these, V 3 Si, V 3 Ga and Nb 3 Sn have ranges of homogeneity that include the A 3 B stoichiometric composition and the maximum T c is readily obtainable in bulk samples. Nb 3 Al and Nb 3 Ga include the ideal composition only at temperatures so high that thermal disorder is excessive. Nb 3 Ge does not exist in equilibrium at the stoichiometric ratio.
15 xiv Introduction Two of them were chosen for the preparation of this work: V 3 Si that has a really high RRR value and Nb 3 Sn that is the only A15 compound already used for a resonant accelerating structure  GHz cavities The rf characterization of the obtained A15 samples would be an useful diagnostic tool to accurately investigate local properties of the grown superconducting lms. However, a common limitation of systems used for this, often consists in the diculty of scaling the measured results to the real resonator. The rf performance testing of a sample and its extrapolation to the frequency of a cavity is and will always remain an indirect way of measuring superconducting rf properties. Obviously the most direct way to test rf properties would be the use of cavities but 1.5 GHz resonant structures would be too onerous both for the material cost and the cryogenic expense. The idea was to build microcavities completely equal in shape to the real scale model. They are produced by spinning and they don't need any electron beam welding (neither for anges). It becomes feasible to produce small resonators in a short period of time, with a negligible cost and in large quantity. Furthermore we developed the cryogenic infrastructure necessary for the 6 GHz rf tests: it can be directly inserted in a liquid helium dewar and it is easy to perform more then one measurement a day. The process parameters optimization necessary to improve the A15 phase superconducting properties, crystal structure and morphology can go on through the small samples production. This thesis fundamental message is we really need an easy and performing system to produce and test the A15 cavities: it consists in 6 GHz resonators that become our cavity shaped samples. 0.4 Organization of the dissertation The following section gives a brief outline of the electrodynamics of cavities. In the second part I shall introduce some qualitative features of rf superconductivity needed to understand intrinsic cavity characteristics such as the nite surface resistance. I am going to explain the fundamental magnetic eld limitation of superconducting cavities too. Chapter 2 expands upon the A15 materials properties and the reasons why some of them are promising for rf superconducting cavity application. I will also point out the problems in the preparation of such a kind of intermetallic compounds. Chapter 3 represents a brief review of some of the methods developed to measure superconductors surface resistance. Being them quite complicate, and in the most of the cases indirect we tried to nd an alternative. We are perfectly aware that having satisfactory results with A15 samples doesn't mean
16 0.4 Organization of the dissertation xv obtaining good superconducting cavities with ease. That's why we decided to look for the best way to overcome this problem. Our solution is to work directly with cavities. The idea is to build 6 GHz resonators (Chapter 4). The following parts are about the A15 materials we decided to study: Nb 3 Sn (Chapter 5 and 7) and V 3 Si (Chapter 6 and 8). I shall list the specic characteristics, I'll describe the way to produce them and the main results we gained. The methods we chose and the experimental apparatus built for our purpose have been chosen to guarantee us the possibility to process samples or 6 GHz cavities, introducing slight modications. Nb 3 Sn have been obtained using the liquid tin diusion process. A bulk Nb sample is introduced into molten Sn for a certain period of time. Here the diusion of Sn into bulk Nb begins to take place: it is extremely important, as shown by the Nb 3 Sn phase diagram, the temperature to be higher than 930 C, to avoid the formation of spurious low T c phases. After this rst step (dipping) it is necessary to perform the second one (annealing): it consists in a heat treatment outside the Sn bath. In this way the residual Sn, still wetting the substrate, is leaden to diuse completely. The idea is to obtain satisfactory superconducting properties (high critical temperature and a sharp transition curve), an homogeneous and compact lm and a sucient surface nishing (both in terms of composition and roughness). To reach this scope, we decided to modify progressively the experimental procedure trying to control the tin percentage of our lms. The so called "hybrid" process seems to be extremely promising: the sample annealing is partly performed in Sn vapor, partly in vacuum. At the same time we worked on the bulk Nb 6 GHz cavities internal surface polishing: they have to be free from defects, smooth and shining. In this way we eliminate all the possible error sources connected to the substrate preparation. The rst Nb 3 Sn 6 GHz resonators have been produced and tested. V 3 Si have been obtained through the high temperature V-Si interdiusion from gaseous SiH 4. Our bulk vanadium substrates are xed in the vacuum chamber: they are then heated at high T ( 800 C) using a set of lamps, left in a silane atmosphere for some hours and annealed in vacuum for a certain period of time (generally tens of hours). Part of our time has been dedicated to the V polishing in order to develop an eective recipe. Then the rst V 3 Si samples have been produced. SiH 4 pressure, process temperature and time have been progressively changed to get better superconducting properties. A special eort has been made to try to discriminate the experimental conditions necessary to avoid spurious phases formation (mainly V 5 Si 3 ) and the A15 compound growing supercially. The rst V 3 Si 6 GHz resonator has been produced and tested.
17 xvi Introduction
18 1 Chapter 1 Superconducting Resonant Cavities In this chapter we give an overview of the basics of superconducting cavities. We start by discussing the electrodynamics of radiofrequency (rf) cavities, the accelerating mode and the general expressions used to describe power dissipation. Although later we focus exclusively on superconducting cavities, this section apply equally well to both normal and superconducting resonant structures. In the second part we introduce the rudiments of supercoductivity. In particular, we will illustrate why superconducting cavities dissipate a small, but nite amount of power despite the fact that superconductors carry dc currents without losses. We will also explain the fundamental magnetic eld limitation of superconducting cavities. This chapter is not designed to give a deep description of the theory of cavities: we will just emphasize the aspects needed to understand this report. For further information the reader is referred to numerous texts that give an excellent review of the subject (see, for example [2, 4]). 1.1 Cavity fundamentals and cavity elds Radio-frequency elds in cavities The rf eld in cavities are derived from the eigenvalue equation ( ) (E ) c 2 t 2 H = 0 (1.1) which is obtained by combining Maxwell's equations . It is subject to the boundary conditions n E = 0 (1.2) and n H = 0 (1.3) at the cavity walls. Here n is the unit normal to the rf surface, c is the speed of light and E and H are the electric and magnetic eld respectively. In cylindrically symmetric
19 2 Superconducting Resonant Cavities Figure 1.1: Schematic of a generic speed-of-light cavity. The electric eld is strongest near the symmetric axis, while the magnetic eld is concentrated in the equator region. cavities, such as the pillbox shape, the discrete mode spectrum given by 1.1 splits into two groups, transverse magnetic (TM) modes and transverse electric (TE) modes. For TM modes the magnetic eld is transverse to the cavity symmetry axis whereas for TE modes it is the electric one to be transverse. For accelerating cavities, therefore, only TM modes are useful. Modes are classied as T Mmnp, where integers m, n, and p count the number of sign changes of E z in the Φ, ϱ, and z directions respectively. Only T M 0np (n = 1, 2, 3...; p = 0, 1, 2...) modes have a non vanishing longitudinal electric eld on axis, and the TM 010 mode is used for acceleration in most cavities. The typical shape of speed of light cavities  is shown in Figure 1.1. The electric eld of the T M 010 mode is greatest at the irises and near the symmetry axis, while the magnetic led is concentrated in the equator region. The geometry of the cell and the addition of beam tubes make it very dicult to calculate the elds analytically, and one reverts to numerical simulations with codes such as SUPERFISH to obtain the eld proles . Although TM modes acquire a nite H z due to the perturbative eect of the beam tubes, the main characteristic of the TM modes are preserved, and one still uses the T Mmnp classication scheme to identify modes The accelerating eld The accelerating voltage (V acc ) of a cavity is determined by considering the motion of a charged particle along the beam axis. For a charge q, by denition, V acc = 1 max energy gain possible during transit q (1.4)
20 1.1 Cavity fundamentals and cavity elds 3 We use 6 GHz speed of light structures in our tests, and the accelerating voltage is therefore given by V acc = z=d z=0 E z (ρ = 0, z)e iω 0 z c dz (1.5) where d is the length of the cavity and and ω 0 is the eigenfrequency of the cavity mode under consideration. Frequently, one quotes the accelerating eld E acc rather than V acc. The two are related by E acc = V acc d (1.6) With single cell cavities the choice of d is somewhat ambiguous, since the beam tubes can be made arbitrarily long. hence E acc is not uniquely dened. Frequently one therefore calculates E acc for an equivalent innite periodic structure and quotes its E acc for the single cell Peak surface elds When considering the practical limitations of superconducting cavities, two elds are of particular importance: the peak electric surface eld (E pk ) and the peak magnetic surface eld (H pk ). In most cases these elds determine the maximum achievable accelerating gradient in cavities. In the ones we have (6 GHz speed of light structures), the surface electric eld peaks near the irises, and the surface magnetic eld is at its maximum near the equator. To maximize the potential cavity performance, it is important that the ratios of E pk /E acc and H pk /E acc be minimized. In an ideal pillbox cavity, the ratios are given by E pk E acc = π 2 H pk E acc = 30.5 = 1.6 (1.7) Oe MV/m (1.8) The addition of beam tubes increases these values. For example, the ratios of monocell TESLA-type cavities are E pk E acc = 1.83 (1.9) H pk E acc = 45 Oe MV/m (1.10) These values were obtained by solving for the elds in the TM010 mode numerically with the code SUPERFISH (, Chapter 4).
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