SAST Lecture - III. Linear Accelerators. P N Prakash. IUAC, New Delhi

Transcription

1 Lecture - III Linear Accelerators P N Prakash IUAC, New Delhi School on Accelerator Science and Technology (SAST ) Inter-University Accelerator Centre, New Delhi May 16-27,

2 Outline: Lecture - III Figures of Merit (continued) TEM Class Resonator Design P N Prakash - SAST-

3 we continue with Figures of Merit from where we left it off. P N Prakash - SAST-

4 Peak Electric Field In superconducting cavities the issue is not so much with electric breakdown or sparking as in DC accelerators, but rather it is with electron field emission. Field Emission (FE) arises at sufficiently high surface electric fields from micron or sub-micron size contaminant particles. When field emission begins, the cavity quality factor falls sharply. If the electron current from field emission becomes sufficiently intense, the heating caused by it can result in thermal breakdown (i.e. cavity going from Superconducting to normal state). So, it is important to reduce the Peak Surface Electric Field E Peak. In fact, the quantity to reduce in a cavity design is E Peak /E a. P N Prakash - SAST-

5 Peak Electric Field When field emission begins: quality factor Q falls sharply, heating caused by the current can result in thermal breakdown. So, it is important to reduce the Peak Surface Electric Field E Peak. In fact, The quantity to reduce in a cavity design is E Peak /E a. onset of field emission Q drop due to field emission Components which support high electric fields, such as the Drift Tube, must have sufficiently P N Prakash large - SAST- radii (no sharp corners).

6 Peak Electric Field E peak In order to properly compare different resonator geometries / models / designs, the peak electric field per unit accelerating gradient, E peak, is commonly used as a design parameter. E peak E peak _ total E a U 1J The peak surface electric field occurs at the open-end of the coaxial line, i.e. on the drift tube. The radius of curvature (or radii, if there are more than one) primarily decides (decide) the value of the peak electric field. P N Prakash - SAST-

7 Field Emission Fowler Nordheim model: The tunneling current density J in A/m 2 is given by: J E exp E where: is work function of the metal in ev E is the electric field in MV/m P N Prakash - SAST- The Fowler Nordheim theory has been experimentally verified for DC electric fields.

8 Field Emission For the tunneling current I (=JA e ) in RF cavities the formula has to be modified, and becomes: I e A E exp E whiskers or sharp projections enhance the electric field where β is the enhancement factor. Semi-ellipsoidal micro protrusion. Calculated geometric enhancement factor for the semi-ellipsoidal micro protrusion. P N Prakash - SAST-

9 Field Emission How to identify if a Cavity is suffering from FE? Field emission is always accompanied by X-Ray radiation!! onset of field emission onset of X-Ray radiation P N Prakash - SAST-

10 Field Emission Field emission is different from resonant electron multipacting, where it becomes impossible to increase the accelerating gradient even after pumping more power into the cavity, until the multipacting barrier is broken. Field emission is also different from high field Q- slope, where Q-drop is observed in the absence of any X-Ray radiation. Many field emitters in field emitting cavities can be processed through high power pulsed conditioning or through helium pulse conditioning. P N Prakash - SAST-

11 Peak Magnetic Field For superconducting niobium cavities the value of the critical magnetic field beyond which the material ceases to remain in the Meissner superconducting state, i.e. H C1, dictates the value of the peak surface magnetic field. Like the peak electric field, the quantity to be reduced in a cavity design is B Peak /E a. Recall that: TEM class resonators, which are usually designed around MHz frequency, typically operate at 4.2 K temperature (boiling point of LHe), so H C1 ~1350 G. The TM class elliptical cavities, designed for much higher frequencies, usually operate at 2 K, so H C1 ~1700 G. P N Prakash - SAST-

12 Peak Magnetic Field Here are some pointers: How to reduce B Peak? The Central Loading Element, also called Central Conductor, is usually tapered rather than keeping it cylindrical (it appears like the frustum of a cone). In elliptical cavities the cell is shaped at the equator to reduce the flux density. The diameter of the Outer Housing is increased in the high magnetic field region, which further reduces the value of the peak magnetic field. In addition, it also improves the Shunt Impedance and Geometry Factor of the structure. more on this later!!! P N Prakash - SAST-

13 Efficiency figures of merit P N Prakash - SAST-

14 Quality Factor Apart from the accelerating electric field (gradient) the Quality Factor is perhaps the most important figure of merit in terms of the operational performance of the resonator. The efficiency with which an accelerating structure stores (or retains) the energy is given by the figure of merit Quality Factor, usually denoted by Q (or Q 0 ). Energy stored Energy stored U Q=2 2 f Energy dissipated per cycle Power dissipated P where ω is the angular frequency, U is the stored energy in the cavity and P is the power dissipated in the cavity walls. P N Prakash - SAST-

15 Quality Factor The total energy in the cavity oscillates between the electric and magnetic fields (remember, the two standing waves have a phase difference of π/2). The time averaged stored energy in the electric field is equal to that in the magnetic field: U= 0 H dv= v 0 E dv 2 2 v The power loss due to Joule heating is defined through the power dissipated per unit area as: dp 1 R s H ds 2 2 P N Prakash - SAST- where R s is the surface resistance.

16 Quality Factor Assuming R s to be constant over the entire surface, we get: 1 2 P= Rs H ds 2 s The quality factor Q then becomes: U 0 H dv Q= v P 2 R H ds s s 2 or QR = 0 v s 2 s 2 H dv H ds P N Prakash - SAST-

17 Geometry Factor The Geometry Factor, G (=QRs), of the resonator is given by: Geometry Factor G: G= 0 v 2 s does not depend on surface resistance, so, it doesn t matter whether the cavity is in the normal state or in superconducting state. 2 H dv H ds Any figure of merit which does not depend on the surface resistance is going to be extremely useful for comparing cavities of different designs, or different models / versions of the same design. P N Prakash - SAST-

18 Shunt Impedance The Shunt Impedance is a figure of merit that measures how efficiently the cavity transfers its stored energy to the beam passing through it. The Shunt Impedance R sh is defined as: V 2 acc R sh= P The Shunt Impedance must be made as large as possible in order to maximize the accelerating voltage V acc for a given power P (or conversely minimize P for a given V acc ). P N Prakash - SAST-

19 Geometric Shunt Impedance The quality factor Q is given by: Q U P but U U E 2 0 a where U 0 is the stored energy at unit accelerating electric field and E a is the accelerating electric field (gradient). (the stored energy at unit accelerating electric field U 0 comes from bead pull measurement) but E a V L acc eff where L eff is the effective acceleration length. P N Prakash - SAST-

20 Geometric Shunt Impedance Using these equations, we get the Geometric Shunt Impedance R Q: sh R Q sh = 2 Leff U 0 The Geometric Shunt Impedance depends only on the geometric design of the resonator and is independent of the surface resistance of the material or its state (normal or superconducting). P N Prakash - SAST-

21 A word on Effective Length, L eff P N Prakash - SAST-

22 Effective Length L eff Often there is considerable confusion on what L eff really is!! L eff may not be the length over which the accelerating electric field extends. Beam Port Remember that the field penetrates into the beam ports and tapers off. But the value of the field inside the ports is generally not high. The geometry of the resonator along with the port size decides the extent of field penetration. L eff?? L eff?? An approximate thumb rule is that P N Prakash the - field SAST- falls 20 db every r BP.

23 Effective Length L eff Few important points that designers may remember are: - The definition of L eff is quite clear in the TM class cavities; it is the length of the cell. - In TEM class structures many definitions have been used by different groups. - Currently the most accepted definition of L eff in TEM 2 class resonators is 3. - When designing accelerating structures, however, it is important to retain the same definition of L eff in all the design models. Otherwise comparing them can become tricky, difficult and may be wrong. P N Prakash - SAST-

24 Effective Length L eff - You may, for example, choose L eff for the design calculations to be from the inside of the beam port at one end to the inside of the beam port at the other end. This can help in uniformly comparing the various design models. - Remember that L eff affects E a - This can in turn affect U 0 acc 2 1 E. - So, the definition of L eff must be consistent. a V L. eff P N Prakash - SAST-

25 Other important parameters P N Prakash - SAST-

26 Accelerator Cost Accelerators are expensive instruments to build. A good portion of the project cost and effort goes into civil construction. Acquiring land can also be difficult. It is always the endevour of accelerator designers to obtain maximum beam energy from the system for a given amount of power over a given length of the accelerator. Much of the R&D in accelerating structures like cavities, resonators, DTLs etc. is aimed at maximizing the energy gain over the shortest length. P N Prakash - SAST-

27 Real Estate Gradient It is important to realize that merely achieving large voltage gains alone from the accelerating structures is not sufficient. It is also important to consider over what physical length such energy gain is achieved by a group of accelerating structures (e.g. in a cryomodule), since it has direct bearing on the building size, and therefore the cost of the project. The Real Estate Gradient E REG is given by: E REG V L TOTAL ACC_LEN P N Prakash - SAST-

28 Real Estate Gradient What does E REG depend on? E REG V L TOTAL ACC_LEN To some extent E REG depends upon the resonator design (remember that the design can be optimized to produce large energy gains over short lengths). To a slightly larger extent E REG depends on the packing of the resonators. P N Prakash - SAST- pack em up!!!

29 Electron Multipacting The word Multipacting comes from the words multiple and impact. Electron Multipacting is a resonant process in which a large number of electrons build up in the resonator, absorbing all the power, thus making it impossible to increase the accelerating electric field in it by increasing the input power. The electron multipacting process starts when an electron present in the resonator gets accelerated by the RF field and impacts on the walls thereby producing secondary electrons. The secondary electrons in turn get accelerated by the fields and produce more electrons by again impacting on the cavity walls. P N Prakash - SAST-

30 Electron Multipacting If the number of secondary electrons produced per impact is more than the number of primary electrons, the electron current grows exponentially. The secondary electron emission coefficient (SEC), δ, of a material gives the number of secondary electrons produced per primary electron impacting the surface. Material δ max K max (ev) K 1 (ev) K 2 (ev) Copper Niobium The upper and lower crossovers for which δ>1 are given by K 1 and K 2. K max is the electron energy for which δ is maximum. SEC also depends on the state of the material. P N Prakash - SAST-

31 Electron Multipacting Although the secondary electron emission coefficient is primarily a function of the kinetic energy of the impacting electron, it also depends on the state of the material. R. Calder et al., CERN Secondary electron emission coefficient as a function of electron kinetic energy for niobium material P N following Prakash - SAST- different treatments.

32 Electron Multipacting Two kinds of electron multipacting are possible in RF cavities: One point multipacting Two point multipacting One point (or sometimes called single point) multipacting is predominant in TM class elliptical type cavities. Cyclotron frequency: H c g n n 0He m m n c e e 0 n (order) is an integer g e P N Prakash - SAST- K ee m 2 2 e 2 g

33 Electron Multipacting Two kinds of electron multipacting are possible in RF cavities: One point multipacting Two point multipacting Two point multipacting is predominant in TEM class resonators where the combination of RF frequency and gap length results in the involvement of two impact sites. The time period between the impacts must be half multiples of the RF time period T. (2n 1) T 2 where n is an integer. Drift Tube voltage V n : Kinetic energy of electron K n : V n 2 2 d me (2n 1) e K 2e V m d 2 2 n n 2 2 where d is the gap between the impacting sites. P N Prakash - SAST- K f d (2n 1) n where f is in MHz and d in cm

34 Tackling Multipacting In TM class cavities, electron multipacting is tackled by appropriately shaping those regions where it dominates (mostly near the equator). BEAM H E E Pill Box with Beam Ports Single Cell Cavity Use multipacting simulation codes for detailed study. Some of the codes available are: MultiPac, MUPAC, MULTIP, XING, TRACK-3D etc. P N Prakash - SAST-

35 Tackling Multipacting In TEM class resonators, electron multipacting is tackled by avoiding combinations of frequency and gap lengths which can result in electron energies close to δ max. For n=1: K f d Avoid cylindrical symmetry. Often the central loading element in TEM class resonator is tapered (albeit not to reduce multipacting), but it breaks the symmetry and reduces electron multipacting. If combinations of f and d which result in K being in the multipacting range cannot be avoided, then attempts must be made to reduce the area over which multipacting is likely to occur. P N Prakash - SAST-

36 Operating Temperature Critical Temperature T C of Niobium Material is ~9.2K. Resonators made out of niobium material must be cooled below 9.2K to make them superconducting. The boiling point of liquid helium (LHe) is 4.2K, which is ideally suited for cooling down niobium resonators. Helium liquefiers are commercially available for producing liquid helium. The surface resistance, R S, of superconducting niobium is: Rs RBCS Rres where R BCS is the BCS resistance and R res is the residual resistance. P N Prakash - SAST-

37 Operating Temperature T For T<T C /2, a convenient expression for R BCS (nω) is: R BCS f T 4 2 e T (where f in GHz & T in K) Frequency f R BCS (T=4.2 K) R BCS (T=2K) 100 MHz 3.2 n 0.06 n 300 MHz 28.4 n 0.6 n 1300 MHz 533 n 11 n TEM class resonators are usually designed around MHz frequency (essentially to keep its size neither too small nor too big). At this frequency R BCS 3 nω at 4K. P N Prakash - SAST-

38 Operating Temperature T A reasonable value for the Geometry Factor G (=QR S ) for a TEM class resonator ~15. With R BCS 3 nω at 4.2K, the limiting quality factor will be Q (=G/R S ) , which is quite high. For Q (factor of 5 lower than what we have estimated above), the power, P, required to operate the resonator at E a =4 MV/m, given that f=100 MHz, U 0 =0.1 J/(MV/m) 2, will be ~1 W!! Note that I have not taken into account the Residual Resistance, R res. So, resonators designed around MHz can be very conveniently operated at 4.2K temperature (boiling point P N Prakash of - SAST- liquid helium).

39 Operating Temperature T Frequency f R BCS (T=4.2 K) R BCS (T=2K) 100 MHz 3.2 n 0.06 n 300 MHz 28.4 n 0.6 n 1300 MHz 533 n 11 n TM class elliptical cavities are generally designed to operate at ~ MHz frequencies. At these frequencies R BCS > 100 nω at 4.2K. Clearly R BCS has to be reduced to a more tolerable value. At 2K, R BCS ~5-10 nω; which is tolerable. TM class Elliptical Cavities are operated at 2K. P N Prakash - SAST-

40 Residual Resistance R res The residual resistance R res arises from inclusion of foreign materials, impurities, residues from chemical processing, formation of oxides and hydrides on the niobium surface and trapped magnetic flux during cool down. By taking precautions during resonator fabrication, processing and in handling the material, a carefully prepared niobium surface can reach R res <5 nω, which is acceptable. P N Prakash - SAST-

41 Some more Parameters There are some more parameters which are very important. These are: Frequency Tunability Frequency Sensitivity Structural Stability Mechanical Frequency (microphonics). Due to shortage of time, I can t discuss them. P N Prakash - SAST-

42 What are the Key Parameters for designing a Niobium Resonator? P N Prakash - SAST-

43 Key Parameters Key Parameters for designing a Superconducting Niobium Resonator: 1. Synchronous Velocity β 0 2. Resonator Frequency f 3. Energy Gain E Gain 4. Transit Time Factor TTF 5. Stored Energy U 0 6. Geometry Factor QR S 7. Peak Magnetic Field B peak 8. Peak Electric Field E peak 9. Geometric Shunt Impedance R sh /Q 10. Operating Temperature T 11. Real Estate Gradient 12. Electron Multipacting 13. Frequency Tunability Δf ST 14. Frequency Sensitivity 15. Mechanical Stability Remember, out of these, some parameters have to be selected while others need to be optimized. P N Prakash - SAST-

44 Key Parameters It is simply impossible to optimize all the key parameters in a resonator design since they are coupled to each other. Remember the Car example that I had given in the second lecture. It is impossible to optimize all the parameters simultaneously. Therefore only a Global Optimization in a resonator design is possible to optimize its parameters. P N Prakash - SAST-

45 Let us look at a couple of Key Parameters in detail for designing TEM Class Resonators. Due to shortage of time, I won t be able to go over all of them. P N Prakash - SAST-

46 Resonator Frequency f There are two competing requirements for frequency selection. Lower f 1. Fewer Resonators are needed to reach the final energy goal. For example, in a 2-gap QWR the center of the first gap to center of the second gap d is given by: d 2 Leff d g where g is the gap between the drift tube and the end cap. P N Prakash - SAST- Higher f 1. Length of the Loading element (also called the Central Conductor), L λ/2 or λ/4, reduces. This is a major advantage, especially in the Top Loading type cryomodule where the height of the beam hall (more so in existing beam halls) is a major constraint.

47 Mechanical Frequency Lowest mechanical vibration frequency of a hollow rod anchored at one end (approximate formula neglecting the inner radius r i ): where: mech ro Y L 2 L is the length of the rod, r i & r o are the inner and outer radii of the rod, Y is the Young s Modulus of the material ρ is the density of the material Response to a particular frequency of disturbance falls as So, it is logical to increase ν mech as high as possible. For the Tapered Conductor, to a good approx.: R R 2 S O r o, So, in addition to reducing the peak magnetic field B peak (as we shall see), tapering also increases ν mech r o tapered r o untapered r o R O 1 2 mech Note: The mechanical frequency also depends on the mass at the open-end, which should be kept small. P N Prakash - SAST-

48 Resonator Frequency f There are two competing requirements for frequency selection. Lower f Higher f 2. Energy & Time spreads of the beam are better controlled. E t q Eaf sin s A E A E t Phase Acceptance: =5 (±2.5º about s ); at low frequencies this is more. P N Prakash - SAST- 2. Increases ν mech : mech ro Y L 2 this reduces microphonic coupling, and hence the RF frequency jitter Δω shake 2 PC shakeu shakeu 0Ea

49 Summary - Resonator Frequency There is no sharp cutoff where one of the (conflicting) requirements begins to dominate the other. In the frequency range MHz the conflicts are largely reconciled and reasonably taken care of. The final choice of frequency is made keeping β 0 in mind to obtain reasonably long effective acceleration length (L eff ) while keeping other constraints in mind. TEM class resonators are usually designed around MHz frequency. P N Prakash - SAST-

50 Peak Magnetic Field B peak In order to properly compare different resonator geometries / models / designs, the peak magnetic field per unit accelerating gradient, B peak, is commonly used as a design parameter. B peak B E total a U 1J Since the peak magnetic field produced as a result of the applied RF must remain below H C1 of superconducting niobium, one of the main efforts in the resonator design is directed towards reducing B peak, so as to maximize the achievable accelerating gradient. P N Prakash - SAST-

51 Reducing B peak One way to reduce B peak in the resonator is by tapering the central conductor (loading element). Consider a quarter wave coaxial line (this is the simplest TEM class structure). Note that for simplicity, we are considering the coaxial line without a drift tube. B peak 0V0 2 R Z open 0 P N Prakash - SAST- where Z 0 is the characteristic impedance of the coaxial line and V 0 is the peak voltage at the openend of the coaxial line.

52 Reducing B peak Consider now a quarter wave coaxial line with a linearly tapering central conductor (loading element). The tapered central conductor may be approximated to a stepped coaxial line of two sections of radii R short & R open, each of length l/2. P N Prakash - SAST-

53 Reducing B peak B For such a stepped (tapered) coaxial line, it can be shown that: 0 0 peak 1 2 Rshort sin(2 ) 2 Z1 Z2 for / 4, It can be shown that: for R short >R open : V 1 sin(2 ) 1 ½ R short [Z 1 +Z 2 ]>R open Z 0 B P N Prakash - SAST- peak where Z 1 & Z 2 are the characteristic impedances of the two sections of the coaxial line, and V 0 is the peak voltage at the open-end. Therefore for the same V 0 : stepped B peak untapered

54 Reducing B peak Another way to reduce B peak is by increasing the diameter of the outer cylinder (housing). Characteristic Impedance Z 0 of a coaxial line is given by: Z 0 1 R 0 ln 2 R hsg 0 open where R CC is the radius of the loading element / central conductor, (which can be R short or R open in our case). For the straight (un-tapered) line: B peak 0V0 2 R Z open 0 Z 0, B peak Physically, there is more volume P N Prakash - for SAST- the magnetic flux to expand.

55 Reducing B peak New shapes of tapered outer housing further reduce B peak. This in turn increases the achievable accelerating gradient E a. P N Prakash - SAST- Peter Ostroumov & Michael Kelly Argonne National Lab

56 What I could not cover? Perhaps more than what I covered in the lectures!!! TM class cavity design Complete design aspects of TEM class resonators Simulation codes Issues connected with niobium material Niobium cavity fabrication Processing of superconducting niobium cavities Bead pull measurements Cavity testing (dressing, loading, cooldown, MP conditioning, high power & helium pulse conditioning etc.) Issues related to electronic control of cavities (phase & amplitude locking, microphonics, etc.) Operational issues (in actual Linacs) New developments and R&D in this field R&D on new materials Last, but not the least, P the N Prakash work - SAST- done at IUAC.

57 Thank you P N Prakash - SAST-