2. Characteristics of Synchrotron Radiation

Transcription

1 . Chaacteistics of Schoto Radiatio. Itoductio The adiatio i geeal is chaacteized b the followig tems: spectal age, photo flu, photo flu desit, billiace, ad the polaizatio. The photo flu is the oveall flu collected b a epeimet ad eachig the sample, the photo flu desit is the flu pe aea at the sample ad the billiace is the flu pe aea ad opeig agle of the souce. I the followig chapte the fomulas fo the calculatio of these tems of the schoto adiatio emitted fom a stoed beam i the bedig maget, wiggle ad udulato ae compiled. Ma authos have established the theo of schoto adiatio. Toda most of the calculatios ae usig the esults of the Schwige theo. Accodig to this theo the shape ad itesit withi the adiatio coe emitted b a adial acceleated elativistic electo beam is give b: whee: d Φ α = γ 4π ω I ω e ) K X ( ξ ) + + X 3 ( + X / 3 K / 3 ( ξ ) photo flu (umbe of photos pe secod) obsevatio agle i the hoizotal plae obsevatio agle i the vetical plae fie stuctue costat = (/37) electo eeg / m e c (me = electo mass, c= velocit of light) agula fequec of photos ( I = beam cuet e = electo chage =.6* -9 coulomb = c = / c ( c 3 c = 3 adius of istataeous cuvatue of electo tajecto = E/ecB (.) c = speed of light ( =.9979* 8 m/s) E = electo beam eeg B = magetic field stegth c = c c(kev) =.665*(E/GeV) *(B/T)] X = = (+X ) 3/ / tical plae) The subscipted K s ae modified Bessel fuctios of the secod kid. Equatio (.) is the basic fomula fo the calculatio of the of the chaacteistics of the schoto adiatio. The polaizatio is give b the two tems withi the squae backets.. Radiatio fom a Bedig Maget The photo flu of the schoto adiatio fom the bedig maget is give b the itegatio of Equatio (.) ove the whole vetical agle. I the hoizotal plae the emitted coe is costat ad theefoe the photo flu i plae: dφ( ) =.458 dθ 3 Photos s.% BW mad ( E / GeV ) ( I / A) ( θ / mad ) G ( ) 3 c (.)

2 Accodig to Equatio (.), the photo flu is popotioal to the beam cuet, the eeg ad the omalized fuctio G c) which depeds ol fom the citical photo eeg. This fuctio is give i Figue (.) ad is illustatig that the spectum of the schoto adiatio flu is a cotiuous oe with a maimum at /3 of the citical photo eeg. G ( ) Schoto Radiatio Fuctio G Reihe = Epsilo / Epsilo c Figue.: The omalized schoto adiatio fuctio G. The flu of the schoto adiatio fom the bedig maget is popotioal to this fuctio. The flues emitted fom a stoed beam i a GeV/.87Tesla ad GeV/.35Tesla stoage ig ae peseted i Figue (3.3). Accodig to the highe eeg, the flu emitted fom a GeV c=.4 (GeV/.87T) ad 3.59 KeV (GeV/.35T). The spectum fo the GeV stoage ig is oughl oe ode of magitude boade, although the citical photo eeg of the machies diffe ol b a facto 3. is give b the followig fomula (cetal itesit): d Φ( ) =.36 3 Photos ( E / Gev) s.% madθ madψ ( I / A) H ( ) Because the adiatio coe is gettig aowe with highe eeg, the cetal itesit is popotioal the squae of the eeg. The spectal depedec is give b the omalized schoto fuctio H () = H c). This fuctio is peseted i Figue (.). It is also cotiuous ad has a maimum ea the citical photo eeg. c (.3) 4

3 Schoto Radiatio Fuctio H().6.4. H() Epsilo / Epsilo c Figue.: The omalized schoto adiatio fuctio H. The billiace of the schoto adiatio fom the bedig maget is popotioal to this fuctio. Fom the defiitio of the flu (Equatio (.)) ad the cetal itesit (Equatio (.3)) the vetical opeig agle of the schoto adiatio is give b: dφ d π G ( ) G ( ) = θ ψ ( ψ = ) = =.7395mad (.4) π d Φ 3 γ H ( ) E / GeV H ( ) The opeig agle of the schoto adiatio fo a GeV electo beam is peseted i Figue (.3): Opeig Agle of Schoto Radiatio Opeig Agle (mad) Photo Eeg ( KeV ) Figue.3: Opeig agle of the schoto adiatio emitted fom a GeV electo beam The opeig agle of the adiatio fom ad GeV machies ae give i Figue (3.4). 5

4 The opeig agle at the citical photo eeg ( c) is, accodig to Equatio (.4): ψ ( = ) =.33mad (.5) ( E / GeV ) Fo a GeV machie the coespodig agle is.66 mad. The billiace of the schoto adiatio fom a bedig maget is give b the cetal itesit divided b the coss sectio of the beam: whee ad [ β + η + ] / = E, d Φ ( ψ = ) B = π = (.6) + γ β + + ψ ) is the electo beam emittace i the hoizotal (vetical) plae, is the electo beam beta fuctio i the hoizotal (vetical) plae, is the dispesio fuctio i the hoizotal plae, E is the ms value of the elative eeg spead, is a Twiss paamete i the vetical plae, is the ms value of the adiatio opeig agle, ) is the diffactio limited souce size, is the obseved photo wavelegth At a photo eeg of KeV the coespodig photo wavelegth is m ad the opeig agle is smalle tha. mad (see Figue (.3)). Both figues esult i a diffactio limited has a value betwee * -3 ad 4* -3. These factos ae at least oe ode of magitude smalle tha the beam, hece the oveall coss sectios i Equatio (.7) educes fo the stoage ig SESAME to: = [ ] / β + η E, = [ ] / β / (.8) ad the billiace of the schoto adiatio fom the bedig maget of a o diffactio limited light souce is give b: (.7) B Maget d φ d Φ ( ψ = ) ( ψ = ) = = (.9) π [ β + η ] / [ β ] / π E The billiace of a ad GeV beam (4 ma) ae give i the Figues (3.7) ad (3.8). The citical photo eegies of the diffeet vesios ae: SE I =.4 KeV, SE I = 4. KeV, SE_3_I = 4. KeV ad SE_4_I = 3.6 KeV. Fom both figues it follows that the maimum billiace is aoud the citical photo eeg. Because of the highe eeg the billiaces of the GeV beam ae oe ode of magitude boade tha those fo a GeV beam. Accodig to the 6

5 smalle coss sectios (emittaces) the billiaces of the vesios SE_3_I ad SE_4_I ae of a facto up to 5 highe tha those fom the GeV beam (vesio SE I )..3 Radiatio fom a Wiggle The wiggle is a special maget with alteatig diectios of the magetic field ad the tajecto of a electo beam though a wiggle is like a sake as show i Figue (.4), it is a siusoidal oscillatio. The tajecto is detemied b the maimum slope X ad b the maimum amplitude X. Both epessios ae give b Equatio (.). X 5 K 8.3, = λ p = Kλ p, X = K / γ, K =.934 ( B / T ) ( λ p / cm) (. ) π γ ( E / GeV ) I the Gee Book ad i this Poposal diffeet wiggles ae foesee: I the Gee Book a 7.5 Tesla supe coductig wiggle ad withi this poposal a omal coductig oe with a field of.5 Tesla. The data of these devices ae summaized i the Table (.): Table.: Data s of the wiggles foesee i the Gee Book ad i this Poposal B W N W L K X X Gee Book 7.5 T 4 mm m 98.. mm 5 mad Poposal.5 T 8 mm 3.4 m mm 4.3 mad Tajecto i a Wiggle Seies3 Amplitude Leght (cm) Figue.4: Tajecto of a electo beam i a wiggle with a peiod legth of 9 cm The photo flu as well as the cetal itesit of the adiatio emitted b the wiggle is the same as fom the bedig maget but b a facto N p moe itesive, whee N p is the umbe of poles withi the wiggle. The photo flu emitted fom the wiggles beams fo the Gee Book ad this Poposal ae peseted i the Figues (3.9) ad (3.). Both wiggles have oughl the c c(poposal) = 6. KeV) ad theefoe the spectum of the flu is oughl the same. Fo the itesit of the photo flu the amplitudes X of the beam oscillatios withi the wiggles have to be cosideed (see Table (.)). Because of the amplitude of.mm i the Gee Book desig the spot sizes i the wiggles have a diffeece of. mm ad it is ot possible to collect both souces withi oe beam lie. Theefoe the useable flu fom the wiggle fo a epeimet is ol popotioal to half of the umbe of the poles. Fo the wiggle i this 7

6 Poposal it is with a amplitude of 5 cosideed. All these agumets ae icluded i the Figues (3.9) ad (3.), with a esult, that the flu fom the GeV stoed beam is of a facto 8 highe tha that fom the GeV oe. The calculatio of the billiace of wiggles eeds to take ito accout the depth-of-fields, i.e. the cotibutio to the appaet souce size fom diffeet poles. The epessio fo the billiace of wiggles is: whee: B wi d Φ = ( ψ = ) dφdψ N / N / π ep, [ + z ] 8 ψ X + z + z = λ p + (.) 4, + z, / (.) ae the ms tasvese size ad agula divegece of the electo beam at = ). This meas that the billiace of the wiggle, calculated accodig to Equatio (.), is omalized to the middle of the staight sectio. The epoetial facto i Equatio (.) aises because the wiggles have two poits, sepaated b *X accodig to Equatio (.) ad idicated i Figue (.4). (The ifluece of this two souce poits upo the photo flu has bee discussed befoe.) The sum i Equatio (.) goes ove all poles of the wiggles. As alead discussed ude the adiatio of the bedig magets ad ca be eglected. The epessio z is the icease of the souce size fom the cete of the isetio device. Istead of omalizig the billiace to the cete of the staight sectio, the coss sectios of () at the positio of the diffeet poles ca be used, with the esult that the epessio fo the billiace will be simple: B Wig N d Φ = = ( ψ = ) = / X ep ( ) π ( ) ( ) (.3) The ivese of the domiato of Equatio (.3) fo diffeet beta fuctios is peseted i Figue (.5). Fo high beta fuctios (4 to m/ad) the coss sectio of the beam does t chage ve much ad the billiace is accodig to Figue (.5) popotioal to the legth of the wiggle. Fo small beta fuctios (.5 ad m/ad) the billiace of the wiggle will satuate, because the coss sectio of the beam i the oute pats of the wiggle gets high ad the cotibutio to the billiace is small ad ca theefoe be eglected. Accodig to Figue (.5) the optimized betato fuctios should be i the age of.6 to. m/ad ad the legth should be i the age of to 3 m. The billiace of the adiatio emitted fom the wiggles withi the Gee Book ad this Poposal ae peseted i the Figues (3.) ad (3.4). Figue (3.4) is that oe with the miibeta-sectios. Agai, because of the same citical photo eegies the emitted spectum coves the same age. Howeve, because of the diffeet coss sectios of the beam the itesit is diffeet. Fo the vesios SE_3_ ad SE_4_ the itesit is oughl the same, but i compaiso to vesio SE the have a facto of 4 highe itesit. The vesio SE is of a facto 5 moe itesive. The pictue chages completel b itoducig mii-beta-sectios. The billiace of the

7 wiggle adiatio fo this vesio is peseted i Figue (3.4), with the esult that the billiace of the vesio SE_4_ is of a facto 4 highe tha that fom the vesio SE. Optimizatio of Wiggle Legth Beta =.5 Beta =. Beta =. Beta = Beta =.6 Beta =. Beta =. Beta = 4. Beta =. Beta =. Popotioal to Billiaz Half Wiggle Legth (m) Figue.5: Depedec of the billiace of wiggle adiatio upo the beta fuctios i the middle of the wiggle. The popotioalit is give b the sum of all ivese coss sectios at the diffeet poles of the wiggle. I ode to optimize the billiace of the wiggle adiatio the beta fuctios i the middle of the staight sectios should be as small as possible (mii-beta-sectio) ad the legth of the wiggle has ol to be aoud m, because the oute egios of the wiggle do t have a sigificat cotibutio to the billiace..4 Radiatio of a Udulato of the schoto adiatio fom the bedig maget at the citical c deflectio agle i a wiggle is withi the opeig agle of the schoto adiatio. Fo this special case the adiatio fom diffeet peiods itefeeces coheetl, thus poducig shap peaks with the esult of completel diffeet chaacteistics. This adiatio is called udulato adiatio ad the coespodig isetio devices ae udulatos. The udulato emits adiatio ol at chaacteistics photo eegies: with the badwidth: whee: =.949KeV ( E / GeV ). (.4) ( λ / cm) ( + K / ) = N = Hamoic umbe ( =, 3, 5, 7,.. ) Ud 9 (.5)

8 N = Numbe of peiods Ud = Peiod legth of the udulato K = Deflectio paamete (see Equatio (.)) The opeig agle of the udulato adiatio coe is: ( + K / ) = (.5) γ N Fo the illustatio of the chaacteistics of the udulato adiatio, the followig eample shall be used: Ud = 4 mm, K = ad N = 5 =.36KeV ad =.85 9 KeV = 6.3eV ad = ev () =. 443mad ad (9) =. 48mad Q-Fuctio fo the Udulato - Radiatio = =3 =5 =7 =9 Q-Value k-value Figue.6: The udulato adiatio fuctio Q (K) fo the calculatio of the adiatio flu The flu of the udulato adiatio withi the coe of the hamoics is give b (i pactical uits): 4 Φ (, K) =.43 N ( I / A) Q ( K) [Photos/(s.%BW)] (.6) Ud Ud Accodig to Equatio (.6) the flu of the udulato adiatio is popotioal to the umbe of peiods, the cuet ad the fuctio Q (K). It is idepedet of the eeg of the electos. The fuctio Q (K) is give i Figue (.6) fo the hamoics = to = 9. To each a high photo flu (also fo highe hamoics), the deflectio paamete K should be i the age of to 3.

9 Table.: Photo eegies ad flues of a udulato istalled at SESAME N c [KeV] Ud [Pho/s.BW.5* 5.3* 5 7.5* 4 4.9* 4.86* 4 The flues of the diffeet hamoics ad the coespodig photo eegies ae summaized i Table (.). I compaiso with a wiggle (see Figue (3.9)), ol the flues of the hamoics ad 3 ae highe tha that fom the wiggle. The photo eegies of these hamoics ae smalle tha KeV, hece the ca t be used fo had X-a epeimets. O ais the peak itesit of the th hamoic of the udulato adiatio is give b (i pactical uits [photos / ( s.%bw m )]): d ΦUd ( θ = ψ = ) = N Ud ( E / GeV ) ( I / A) F ( K ) (.7) The peak itesit is popotioal to the electo cuet, the adiatio fuctio F (K) ad the squae of the peiod umbe. The opeig agle of the adiatio coe is ivesel popotioal to the eeg, ad theefoe the peak itesit accodig to Equatio (.7) is popotioal to the squae of the eeg. The adiatio fuctio F (K) is peseted i Figue (.8). To each maimum cetal itesit, the k-value should be i the age of to 3.5 fo the highe hamoics. F-Fuctio fo the Ud.-Rad. = =3 =5 =7 =9 F-Value K-Value Figue.7: The udulato adiatio fuctio F (K) fo the calculatio of the cetal itesit of the udulato adiatio. The billiace of the udulato adiatio is give b: B Ud = Φ Ud ( π ) ( π ) ( θ = ψ = ) (.8)

10 whee = +, = +, = +, = + (.9) =,, =, / β,, = λl, = λ / L (.) 4π,, β, ae give b: λud ( + K / ) N ( + K / ), = 4 πγ = (.) γ N accodig to Equatio (.) ae summaized i the Table (.3). Table.3: Coss sectios ad divegeces of the udulato adiatio emitted at SESAME [ ad] Fo the elect = 46 accodig to Equatio (.9) ae summaized i Table (.4): Table.4: Oveall coss sectios ad divegeces fo the calculatio of the udulato billiace N [mm] [mm] [mm^].8e-.6e-.5e-.5e-.5e- [mad] [mad] * [mad^].9e-3.48e-3.5e-3.e-3 9.8E-4 The billiace of the diffeet hamoics ad the diffeet cotibutios ae give i Table (.5): Table.5: Paametes fo the calculatio of the udulato billiace [kev] Ud [Phot/s.BW].5E+5.3E+5 7.5E+4 4.9E+4.85E+4 Aea*Agle [mm^*mad^].4e-3 6.9E E-4 4.5E E-4 B (Ud) [Phot./s*mm^*ma^*.BW].8Ê+8.99E+8.5E+8.3E E+7 The billiace of the udulato is fo the st hamoics (.36 KeV) two odes ad fo the 9 th (.88 KeV) oe ode of magitude highe tha that fom the wiggle. The scietific case fo SESAME has to show if this is iteestig fo the uses. With the Equatios (.9) to (.) the billiace ca be calculated accodigl i pactical uits [photos/(s.%bw mm mad )]:

11 whee: A B Ud ( I / A) N ( E / GeV ) F ( K) 4 =.745 (.) A B = ( π ) ad B ( + / ) ( + / ) = (.3) 3