Linear beam dynamics and radiation damping

Transcription

1 Fourth nternational Accelerator School for Linear Colliders Beijing, Setember 9 Course A4: Daming Ring Design and Phsics ssues Lecture Review of Linear Beam Dnamics and Radiation Daming And Wolski niversit of Liverool and the Cockcroft nstitute Linear beam dnamics and radiation daming n this lecture, we shall discuss: The otical (lattice) functions: Twiss arameters and disersion. The momentum comaction factor. mittance, energ sread, bunch length. Snchrotron radiation: energ loss and daming times. Quantum ecitation: equilibrium emittance, energ sread and bunch length. Lecture : Linear Dnamics & Radiation Daming

2 Twiss arameters describe the local oscillation amlitudes of articles As a article moves along an accelerator beam line, it erforms transverse (betatron) oscillations. s The horiontal motion can be described in terms of the horiontal coordinate and the normalised horiontal momentum γm/ & P Alternativel, we can use action-angle variables: J, ϕ defined b: J γ + α + β tanϕ β α The Twiss arameters α, β and γ are defined so that J is a constant of the motion (called the betatron action), and ϕ (the betatron hase) increases with s according to: dϕ ds β 3 Lecture : Linear Dnamics & Radiation Daming Transfer matri for one eriodic cell The Twiss arameters are determined from the transfer matri for one eriodic cell. s s s M ( s s ) s ; s The transfer matri M is arameterised as: α M cosµ + sinµ γ α n transort through one eriodic cell, the action-angle variables transform as: ( s ) J ( s ) ϕ ( s ) ϕ ( s ) + J µ β 4 Lecture : Linear Dnamics & Radiation Daming

3 The Twiss arameters describe the shae of a hase sace ellise f we observe the coordinate and momentum of a article at the eit of each eriodic cell and lot these variables on a hase sace diagram, we construct an ellise, the shae of which is determined b the Twiss arameters. s s s s s s s Area of the ellise πj 5 Lecture : Linear Dnamics & Radiation Daming Smlecticit f we ignore radiation (and some other) effects, then article transort along a beamline is smlectic. Mathematicall, this means that an transfer matri M satisfies: T M S M S where the antismmetric matri S is given b: S The smlectic condition on the transfer matri imoses a constraint on the Twiss arameters, which can be written: β γ α Phsicall, smlecticit means that the area of the hase sace ellise described b the motion of a article is fied. n other words, the amlitude J of the betatron oscillations is constant. 6 Lecture : Linear Dnamics & Radiation Daming

4 The Twiss arameters give the local oscillation amlitude The action J is constant for a article moving along the beamline, and the betatron hase ϕ increases monotonicall. B writing the Cartesian variables and in terms of the action-angle variables, we can reresent the motion of the article as similar to that of a simle harmonic oscillator, but with local variations in amlitude and frequenc: J β cosϕ J β ( sinϕ + α cosϕ ) 7 Lecture : Linear Dnamics & Radiation Daming Disersion describes the change in trajector with energ The energ of a article is usuall secified in terms of the deviation from the reference energ (or the reference momentum P ): Pc β Particles with higher energ ( > ) are deflected b a smaller angle in diole magnets, comared to articles with the nominal energ. n a storage ring, articles with non-ero energ deviation have a different closed orbit from articles with the reference energ. The disersion function η is defined as change in closed orbit with resect to the energ deviation: X η CO 8 Lecture : Linear Dnamics & Radiation Daming

5 amle: beta functions and disersion in a FODO cell 9 Lecture : Linear Dnamics & Radiation Daming Momentum comaction factor The resence of disersion means that there is a change in the ath length for a article following a closed orbit, deending on the energ deviation. The change in ath length C for one comlete revolution is described b the momentum comaction factor α : C C + α + α () + 3 O( ) Lecture : Linear Dnamics & Radiation Daming

6 Momentum comaction factor We can calculate the momentum comaction factor in a lattice from the disersion in the dioles. With a curved trajector, the ath followed b a article a horiontal distance from the reference trajector is: dc + ds f the offset is the result of an energ error: The total ath length is: η + η ds C Then the momentum comaction can be written: α C dc d C η ds Lecture : Linear Dnamics & Radiation Daming Momentum comaction factor and first snchrotron radiation integral f we define: η ds then we can write the momentum comaction factor: α C is called the first snchrotron radiation integral: it is an integral over the circumference of a function that involves the lattice arameters (the disersion) and the magnet arameters (the bending radius). We shall define other snchrotron radiation integrals when we come to calculate the radiation daming times and the equilibrium beam sies. Lecture : Linear Dnamics & Radiation Daming

7 Momentum comaction, hase sli, and transition The time taken for a article to comlete one revolution deends on: the ath length (a function of the energ); the seed of the article. The seed of the article does deend on the energ, but at ultra-relativistic velocities, the deendence is weak. The revolution eriod is eressed in terms of the energ deviation: T T T + η + η () + 3 O( ) where η is the hase sli factor. t can be shown that: η α γ η < α < or γ < /α Below transition: revolution frequenc increases with energ η γ /α At transition: revolution frequenc indeendent of energ η > γ > /α Above transition: revolution frequenc decreases with energ 3 Lecture : Linear Dnamics & Radiation Daming Snchrotron oscillations f the momentum comaction factor is not ero, and there are RF cavities in the storage ring, then articles will erform snchrotron oscillations. A article initiall at oint B in the RF hase and ero energ deviation, will gain energ each time it asses the RF cavit. The momentum comaction means that the article will take longer to go round the ring, and will start to move towards A. Similarl, a article at oint C gains insufficient energ from the RF to relace snchrotron radiation losses. Such a article will start to take less time to go around the ring, and will also move towards A. 4 Lecture : Linear Dnamics & Radiation Daming

8 Snchrotron oscillations The trajector of a article in longitudinal hase sace is an ellise. For + towards the head of a bunch, articles above transition move round the ellise anticlockwise. The snchrotron tune is the number of revolutions of the hase sace ellise made in one revolution of the storage ring. For most lattices, the snchrotron tune is small, ticall of order.. above transition below transition tail of bunch head of bunch 5 Lecture : Linear Dnamics & Radiation Daming Beam emittance The beam emittance is a measure of the hase sace area covered b articles in the beam. We define the Σ matri using the second-order moments of the beam distribution: Σ nder a hase sace transformation: a M the Σ matri transforms as: Σa M Σ M T So the matri roduct Σ S transforms as: Σ S a M Σ M where the last ste follows from the smlecticit of M. T S M Σ S M - 6 Lecture : Linear Dnamics & Radiation Daming

9 Beam emittance We have found that Σ S transforms as: Σ S a M Σ S M But the eigenvalues of Σ S are reserved b an similarit transformation of this te. The eigenvalues of Σ S are ±iε where: - ε ε is called the emittance of the beam: the emittance is conserved under smlectic transort along a beam line. 7 Lecture : Linear Dnamics & Radiation Daming Beam emittance Recall that the article coordinate and momentum are related to the actionangle variables b: J β cosϕ J β ( sinϕ + α cosϕ ) From these eressions, we find that: β γ α J J J t then follows that: ε β γ α J J 8 Lecture : Linear Dnamics & Radiation Daming

10 Beam emittance Thus, the beam emittance is just the average value of the action variables over all articles in the beam: ε J The fact that the action of each article is conserved under (smlectic) transort along the beam line is consistent with the fact that the emittance is conserved. The beam distribution at an oint can be written in terms of the emittance and the Twiss arameters: β ε Note that a beam has three emittances: a horiontal, a vertical and a longitudinal emittance. The analsis resented here assumes that there is no couling, i.e. that the three degrees of freedom are comletel indeendent. However, in the general (couled) case, we can still obtain the emittances as the eigenvalues of Σ S. γ ε α ε 9 Lecture : Linear Dnamics & Radiation Daming Daming of emittances Particle sources (articularl ositron sources) roduce beams with relativel large emittances. One of the jobs of the daming rings is to reduce the emittances, to make a beam that can be used to roduce luminosit. njected emittance tracted emittance Horiontal e + µm.8 nm Vertical e + µm. nm Longitudinal e + > 3 µm µm But if the emittance is conserved in transorting a beam along a beam line, how do the daming rings reduce the beam emittances? Lecture : Linear Dnamics & Radiation Daming

11 Radiation daming Whenever charged articles undergo acceleration, the emit radiation (in the case of relativistic articles, this is called snchrotron radiation). n a storage ring, the radiation emission is dominated b the bending fields of the diole magnets. Snchrotron radiation is a non-smlectic effect; in some resects, it is analogous to a frictional force that will steadil dam the motion of an oscillator. n the case of articles in a storage ring, the combination of radiation emission in the dioles and the restoration of this energ in the RF cavities leads to gradual reduction in the emittances, towards some equilibrium values. Lecture : Linear Dnamics & Radiation Daming Radiation daming The majorit of hotons are emitted within a cone of angle /γ around the instantaneous direction of motion of the article. For ultra-relativistic articles, γ is ver large, so nearl all the radiation is emitted directl along the instantaneous direction of motion of the article. Particles lose longitudinal and transverse momentum in bending magnets. emitted radiation article trajector closed orbit bending magnet Neglecting disersion and chromaticit, the trajector of the article after emitting a hoton is the same as it would be if no hoton were emitted: however the vertical momentum is reduced. Lecture : Linear Dnamics & Radiation Daming

12 Radiation daming n an RF cavit, the article sees an accelerating electric field arallel to the closed orbit: the RF cavities in a storage ring restore the energ lost b snchrotron radiation. The increase in momentum of a article in an RF cavit is arallel to the closed orbit. This leads to a reduction in the amlitude of the betatron oscillations of the article: however, the vertical momentum is not changed. RF cavit article trajector closed orbit 3 Lecture : Linear Dnamics & Radiation Daming Daming of vertical emittance The vertical motion is usuall easier to deal with than the horiontal motion, because most storage rings have (in the absence of alignment or steering errors) ero vertical disersion. Consider a article travelling round a storage ring. When the article travels through bending magnets, it emits radiation within a cone of oening angle /γ around the instantaneous direction of motion of the article. For γ >>, the direction of the radiation is aroimatel along the direction of motion of the article, so the direction of motion is unchanged b the recoil. The total momentum of the article changes with the emission of radiation: d d P where d is the momentum of the radiation, and the total momentum of the article is close to the reference momentum P. Since the direction of the article is unchanged, the vertical emittance after emitting the radiation is: d P 4 Lecture : Linear Dnamics & Radiation Daming

13 Daming of vertical emittance After emission of radiation, the vertical momentum of the article is: d P Now we substitute this into the eression for the vertical betatron action: J γ + α + β to find the change in the action resulting from the emission of radiation: We average over all articles in the beam, to find: where we have used: dj ( β ) α + dj d dε ε P d P α ε γ ε and β γ α 5 Lecture : Linear Dnamics & Radiation Daming Daming of vertical emittance Finall, we integrate the loss in momentum around the ring. The emittance is conserved under smlectic transort; so if the non-smlectic effects (in this case, the radiation effects) are slow, we can write: dε dt ε T where T is the revolution eriod, and is the energ loss in one turn. We define the daming time τ : τ d P T ε T so the evolution of the emittance is: t ( ) ( ) ε t ε e τ Ticall, the daming time in a snchrotron storage ring is measured in tens of milliseconds (whereas the revolution eriod is measured in microseconds). 6 Lecture : Linear Dnamics & Radiation Daming

14 Snchrotron radiation energ loss To comlete our calculation of the vertical daming time, we need to find the energ lost b a article through snchrotron radiation on each turn through the storage ring. The ower radiated b a article of charge e and energ in a magnetic field B is given b: C 3 P γ γ c e B π C γ is a constant, given b: e Cγ 3ε ( mc ) m/gev A charged article in a magnetic field follows a circular trajector with radius, given b: B ec Hence the snchrotron radiation ower can be written: 4 Cγ P γ c π 3 7 Lecture : Linear Dnamics & Radiation Daming Snchrotron radiation energ loss For a article with the nominal energ, and travelling at (close to) the seed of light around the closed orbit, we can find the energ loss siml b integrating the radiation ower around the ring: Pγ dt Pγ sing the revious eression for P γ, we find: Cγ 4 π Conventionall, we define the second snchrotron radiation integral, : ds ds c n terms of, the energ loss er turn is written: Cγ 4 π ds 8 Lecture : Linear Dnamics & Radiation Daming

15 Daming of horiontal emittance The effect of radiation on the horiontal emittance is comlicated b the resence of disersion. When a article emits some radiation (resulting in a change in energ deviation ) at a location where there is disersion, the closed orbit changes b: X η P X η This means that the coordinate and momentum (with resect to the disersive closed orbit) of a article after emitting radiation with momentum d are: d η P d d η P P 9 Lecture : Linear Dnamics & Radiation Daming Daming of horiontal emittance There are two further comlications that we need to consider when calculating the effects of snchrotron radiation on the horiontal motion. The first is that, because of the curvature of the trajector, the ath length of a article in a diole magnet is a function of the horiontal coordinate. Thus the integral of the ower loss: d P Pγ dt P c becomes, in terms of the ath length: d P dc P γ γ + P P c c P c ds c 3 Lecture : Linear Dnamics & Radiation Daming

16 Daming of horiontal emittance The second comlication for horiontal motion is that diole magnets are sometimes constructed with a gradient, so the vertical field changes as a function of the horiontal coordinate: B B B + Taking all these effects into account, we can essentiall roceed the same wa as we did for the vertical betatron motion; that is: Write down the changes in coordinate and momentum resulting from an emission of radiation with momentum d (taking into account the additional effects of disersion). Substitute eressions for the new coordinate and momentum into the eression for the horiontal betatron action, to find the change in action resulting from the radiation emission. Average over all articles in the beam, to find the change in the emittance resulting from radiation emission from each article. ntegrate around the ring (taking account of changes in ath length and field strength with in the bends) to find the change in emittance over one turn. 3 Lecture : Linear Dnamics & Radiation Daming Daming of horiontal emittance The algebra is rather more comlicated than for the case of the vertical emittance. The result is that the horiontal emittance evolves according to: dε ε dt τ t ε ( t) ε ( ) e τ where the horiontal daming time is given b: τ The horiontal daming artition number j is given b: j T j 4 where the fourth snchrotron radiation integral 4 contains the effects of the variation in ath length and field strength with : η 4 + k ds k e P B 3 Lecture : Linear Dnamics & Radiation Daming

17 Daming of snchrotron oscillations The change in energ deviation and longitudinal coordinate for a article in one turn around a storage ring are given b: ev RF ωrf sin ϕs c α C where V RF is the RF voltage and ω RF the RF frequenc, is the reference energ of the beam, ϕ s is the nominal RF hase, and is the energ lost b the article through snchrotron radiation. f the revolution eriod is T, then we can write the longitudinal equations of motion for the article: d ev dt T RF ωrf sin ϕs c T d dt α c 33 Lecture : Linear Dnamics & Radiation Daming Daming of snchrotron oscillations Let us assume that is small comared to the RF wavelength, i.e. ω RF /c <<. Also, the energ loss er turn is a function of the energ of the article (articles with higher energ radiate higher snchrotron radiation ower), so we can write (to first order in the energ deviation): d d + + d d Further, we assume that the RF hase ϕ s is set so that for, the RF cavit restores eactl the amount of energ lost b snchrotron radiation. The equations of motion then become: d ev dt T d dt α c RF ω cosϕs c RF d T d 34 Lecture : Linear Dnamics & Radiation Daming

18 Daming of snchrotron oscillations Combining these equations gives: d d + α + ω s dt dt This is the equation for a damed harmonic oscillator, with frequenc ω s and daming constant α given b: ev ωs RF ω cosϕs T RF α α T d d 35 Lecture : Linear Dnamics & Radiation Daming Daming of snchrotron oscillations f α Ε << ω s, the energ deviation and longitudinal coordinate dam as: ( t) ˆ e( α t) sin( ω t θ ) α c ( t) ˆ e( α t) cos( ωst θ ) ω s To find the daming constant α, we need to know how the energ loss er turn deends on the energ deviation s 36 Lecture : Linear Dnamics & Radiation Daming

19 Daming of snchrotron oscillations We can find the total energ lost b integrating over one revolution eriod: Pγ dt To convert this to an integral over the circumference, we should recall that the ath length deends on the energ deviation; so a article with a higher energ takes longer to travel round the lattice. dc dt c η dc + ds + ds η P ds c γ + 37 Lecture : Linear Dnamics & Radiation Daming Daming of snchrotron oscillations With the energ loss er turn given b: and the snchrotron radiation ower given b: we find, after some algebra: where: η P ds c γ + 4 C Cγ c π π γ 3 P γ c e B d d and 4 are the same snchrotron radiation integrals that we saw before: j Cγ 4 j + π η ds 4 + k ds 4 k e P B 38 Lecture : Linear Dnamics & Radiation Daming

20 Daming of snchrotron oscillations Finall, we can write the longitudinal daming time: τ α is the energ loss er turn for a article with the reference energ, following the reference trajector. t is given b: j Cγ 4 π j is the longitudinal daming artition number, given b: j + 4 T 39 Lecture : Linear Dnamics & Radiation Daming Daming of snchrotron oscillations The longitudinal emittance is given b a similar eression to the horiontal and vertical emittances: ε n most storage rings, the correlation is negligible, so the emittance becomes: ε σ σ Hence, the daming of the longitudinal emittance can be written: t ε ( t) ε ( ) e τ 4 Lecture : Linear Dnamics & Radiation Daming

21 Summar: snchrotron radiation daming The energ loss er turn is given b: C γ 4 5 Cγ π 3 m/gev The radiation daming times are given b: τ T τ T τ j j j T The daming artition numbers are: j 4 j j + The second and fourth snchrotron radiation integrals are: η ds 4 + k ds 4 4 Lecture : Linear Dnamics & Radiation Daming Quantum ecitation f radiation were a urel classical rocess, the emittances would dam to nearl ero. However radiation is emitted in discrete units (hotons), which induces some noise on the beam. The effect of the noise is to increase the emittance. The beam eventuall reaches an equilibrium determined b a balance between the radiation daming and the quantum ecitation. emitted hoton on-energ closed orbit off-energ closed orbit bending magnet article trajector 4 Lecture : Linear Dnamics & Radiation Daming

22 Quantum ecitation of betatron motion Consider a article that emits a hoton of energ u at a location with disersion. The horiontal coordinate and horiontal momentum after the hoton emission are: u η η u Substituting into the eression for the action, and averaging over all articles in the beam, we find the change in the emittance is: ε u H where the curl H function is: H γ η + α η η + β η 43 Lecture : Linear Dnamics & Radiation Daming Quantum ecitation of betatron motion ncluding both quantum ecitation and radiation daming, the horiontal emittance evolves as: dε u ds ε dt C N H τ where N is the number of hotons emitted er unit ath length. We quote the result (from snchrotron radiation theor): N u Cqγ P γ where the quantum constant C q is: C q 55 h mc 3 m 44 Lecture : Linear Dnamics & Radiation Daming

23 Quantum ecitation of betatron motion sing the eression for N u, and eressions used reviousl for the radiation ower P γ, we find: dε 5 Cqγ ε dt j τ τ where the fifth snchrotron radiation integral 5 is: H 5 ds 3 The equilibrium horiontal emittance, ε, is reached when the quantum ecitation balances the radiation daming, dε /dt : ε Cqγ 5 j 45 Lecture : Linear Dnamics & Radiation Daming Quantum ecitation of betatron motion The equilibrium horiontal emittance is given b: ε Cqγ ε is determined b the beam energ, the lattice functions (Twiss arameters and disersion) in the dioles, and the bending radius in the dioles. ε is sometimes called the natural emittance of the lattice, since it is the horiontal emittance that will be achieved in the limit of ero bunch charge: as the current is increased, interactions betweens articles in a bunch can increase the emittance above the equilibrium determined b radiation effects. n man storage rings, the vertical disersion in the absence of alignment, steering and couling errors is ero, so H. However, the equilibrium vertical emittance is larger than ero, because the vertical oening angle of the radiation ecites some vertical betatron oscillations. For more discussion about vertical emittance, see Lecture 5. 5 j 46 Lecture : Linear Dnamics & Radiation Daming

24 Quantum ecitation of snchrotron oscillations Quantum effects ecite longitudinal emittance as well as transverse emittance. Consider a article with longitudinal coordinate and energ deviation, which emits a hoton of energ u. θ ω α θ ω α θ θ ˆ cos cos ˆ ˆsin sin ˆ c c u 47 Lecture : Linear Dnamics & Radiation Daming Averaging over the bunch gives: u θ ω θ ω s s sin ˆ ˆ ˆ u u + θ ˆ σ σ where u Quantum ecitation of snchrotron oscillations ncluding the effects of radiation daming, the evolution of the energ sread is: From the revious eression for N u, we find: σ τ σ ds u C dt d N 3 σ τ τ γ σ q j C dt d 48 Lecture : Linear Dnamics & Radiation Daming We find the equilibrium energ sread from dσ /dt : The third snchrotron radiation integral 3 is given b: 3 j C qγ σ ds 3 3

25 Natural energ sread The equilibrium energ sread determined b radiation effects is: σ Cqγ This is often referred to as the natural energ sread, since collective effects can often lead to an increase in the energ sread with increasing bunch charge. The natural energ sread is determined essentiall b the beam energ and b the bending radii of the dioles. Note that the natural energ sread does not deend on the RF arameters (either voltage or frequenc). 3 j The corresonding equilibrium bunch length is: α c σ ω We can increase the snchrotron frequenc ω s, and hence reduce the bunch length, b increasing the RF voltage, or b increasing the RF frequenc. s σ 49 Lecture : Linear Dnamics & Radiation Daming Summar: radiation daming ncluding the effects of radiation daming and quantum ecitation, the emittances var as: The daming times are given b: ( ) ( ) t ( ) t ε t ε e + ε e τ τ j τ jτ jτ T The daming artition numbers are given b: The energ loss er turn is given b: j 4 j j + C γ 4 5 Cγ π 4 3 m/gev 5 Lecture : Linear Dnamics & Radiation Daming

26 Summar: equilibrium beam sies The natural emittance is: ε 5 3 Cqγ Cq 3.83 j The natural energ sread and bunch length are given b: 3 C qγ σ σ j ωs The momentum comaction factor is: α C σ α c The snchrotron frequenc and snchronous hase are given b: ω RF RF s α ϕs ϕs T ev ω cos sin ev m RF 5 Lecture : Linear Dnamics & Radiation Daming Summar: snchrotron radiation integrals The snchrotron radiation integrals are: η ds ds 3 3 ds 4 η + k ds k e P B 5 H 3 ds H γ η + α η η + β η 5 Lecture : Linear Dnamics & Radiation Daming

27 Daming times in the LC daming rings Let us consider the daming time that we need in the LC. The shortest daming time is set b the vertical emittance of the ositron beam. njected emittance tracted emittance Horiontal e + µm.8 nm Vertical e + µm. nm Longitudinal e + > 3 µm µm We must reduce the injected emittance to the etracted emittance in the store time of ms (set b the reetition rate of the main linac). sing: ( ) ( ) t ε t ε e τ we find that we need a vertical daming time of 3 ms. n ractice, the daming time must be less than this, to allow for a non-ero equilibrium. 53 Lecture : Linear Dnamics & Radiation Daming Daming times in the LC daming rings The LC daming rings are 6.7 km in circumference and have a beam energ of 5 GeV. The energ loss er turn is: Cγ 4 π f the onl diole fields are those that determine the ring geometr, and have field strength B, then we can write: Hence: B ds eb ds π B P C γ 3 ecb For a diole field of.5 T, we find 5 kev. 54 Lecture : Linear Dnamics & Radiation Daming

28 Daming times in the LC daming rings The beam energ is 5 GeV; the energ loss er turn from the dioles is 5 kev. This means that the vertical daming time is: τ T 45 ms We need a daming time of less than 3 ms; the radiation from the dioles rovides a daming time of 45 ms! To reduce the daming time, we need to increase the energ loss er turn. ncreasing the diole field can hel, but has adverse imact on other asects of the dnamics (the momentum comaction factor is reduced, which lowers some of the instabilit thresholds). The other otion is to use a daming wiggler, which consists of a sequence of dioles bending in oosite directions we will discuss wigglers used to enhance radiation daming in Lecture Lecture : Linear Dnamics & Radiation Daming