Chapter 2 CHARACTERISTICS OF SYNCHROTRON RADIATION

Transcription

1 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION Chapte CHARACTERISTICS OF SYNCHROTRON RADIATION. Intoduction The adiation is chaacteized in geneal b the following tems: spectal ange, photon flu, photon flu densit, billiance, and the polaization. The photon flu is the oveall flu collected b an epeiment and eaching the sample, the photon flu densit is the flu pe aea at the sample and the billiance is the flu pe aea and opening angle. In the following chapte the fomulas fo the calculation of these tems of the snchoton adiation emitted fom a stoed beam in the bending magnet, wiggle and undulato ae compiled. Man authos have established the theo of snchoton adiation. Toda most of the calculations ae using the esults of the Schwinge theo []. A elativistic electon unning aound an obit emits a adiation chaacteistic as given in figue (.): Figue.: Chaacteistics of the snchoton adiation emitted b a elativistic electon moving on a cicle. The shape and intensit within the adiation cone accoding to the Schwinge Theo is given b: Φ 3 d α ω I X γ ( + X ) K ( ) + / 3 ξ K/ 3( ξ ) (.) dθdψ 4π ω e + X whee: photon flu (numbe of photons pe second). obsevation angle in the hoizontal plane. obsevation angle in the vetical plane. fine stuctue constant (/37). -

2 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME I e beam cuent. electon chage.60*0-9 coulomb. s, c E B speed of light (.9979*08 m/s). electon beam eneg. magnetic field stength. (+X)3//. The subscipted K s in equation (.) ae modified Bessel functions of the second kind. Equation (.) is the basic fomula fo the calculation of the of the chaacteistics of the snchoton adiation. The polaization is given b the two tems within the squae backets.. Radiation fom a Bending Magnet The photon flu of the snchoton adiation fom the bending magnet is given b equation (.), if integated ove the whole vetical angle. In the hoizontal plane the emitted hoizontal plane: dφ( ) 3 Photons ε ( E / GeV ) ( I / A) ( θ / mad ) G ( ) (.) dθ s 0.% BW mad ε c Accoding to Equation (.), the photon flu is popotional to the beam cuent, the eneg and the nomalized function G c) which depends onl fom the citical photon eneg. The equation fo the citical photon eneg is given in (.3). ε c 0.655keV ( E / GeV ) ( B / T ) (.3) The flues emitted within the bending magnet fom a stoed electon beam at diffeent enegies ae pesented in figue (.). The photon flu inceases accoding to equation (.) with the eneg and it is shifted to highe photon enegies because of the lage citical eneg. The photon flu densit fom the bending magnets fo the same electon beam is pesented in figue (.3). Because the coss section of the beam deceases with the eneg, the flu densit inceases with smalle enegies. The intensit of the snchoton adiation in the middle of the 0) is given b the following fomula (cental intensit): d Φ( ).36 0 dθdψ 3 Photons ( E / Gev) s 0.% madθ madψ ( I / A) H ε ( ) ε Because the adiation cone is getting naowe with highe eneg, the cental intensit is popotional the squae of the eneg. The spectal dependenc is given b the nomalized snchoton function H () H c). Fom the definition of the flu (equation (.)) and the cental intensit (equation (.4)) the vetical opening angle of the snchoton adiation is given b: c (.4) -

3 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION -3 Figue.: The photon flu emitted within the bending magnets of SESAME fom electon beams with diffeent enegies. Figue.3: The photon flu densit emitted within the bending magnet of SESAME fom electon beams with diffeent enegies. ) ( ) ( / ) ( ) ( 3 0) ( H G GeV E mad H G d d d d d Φ Φ γ π ψ ψ θ θ π σ ψ (.5) The opening angle of the snchoton adiation fom the bending magnet of SESAME fo a stoed beam with diffeent enegies is pesented in figue (.4):

4 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME Figue.4: Opening angle of the snchoton adiation emitted fom an electon beam with enegies fom.0 to.5 GeV. c) is, accoding to equation (.5): σ ψ ( ) 0.33mad ( E / GeV ) (.6) Fo a.5 GeV machine the coesponding angle is 0.3 mad. The billiance of the snchoton adiation fom a bending magnet is given b the cental intensit divided b the coss section of the beam: d Φ ( ψ 0) dθdψ B (.7) π whee with [ ε β + η σ + σ ] / E and ε + ε γ ε β + σ + σ ψ σ ) is the electon beam emittance in the hoizontal (vetical) plane, is the electon beam beta function in the hoizontal (vetical) plane, is the dispesion function in the hoizontal plane, E is the ms value of the elative eneg spead, is a Twiss paamete in the vetical plane, is the ms value of the adiation opening angle, ) is the diffaction limited souce size, is the obseved photon wavelength. At a photon eneg of 0 kev the coesponding photon wavelength is 0.4 nm and the opening angle is smalle than 0. mad (see figue (.4)). Both figues esult in a diffaction has a value between *0-3 and 4*0-3. These factos ae at least one ode of magnitude smalle, hence the oveall coss sections in equation (.8) educes fo the stoage ing SESAME to: / (.8) -4

5 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION [ ε ] / β + η σ E, [ ε ] / β (.9) And the billiance of the snchoton adiation fom the bending magnet of a nondiffaction limited light souce is given b: B Magnet d φ d Φ ( ψ 0) ( ψ 0) dθdψ dθdψ (.0) π [ ε β + η σ ] / [ ε β ] / πσ σ E The billiance of the snchoton adiation emitted within the bending magnets at SESAME fo diffeent enegies ae pesented in figue (.5): Figue.5: The billiance emitted within the bending magnet of SESAME fom an electon beam with diffeent enegies. The citical photon enegies of the diffeent enegies ae:.5 GeV ε c 5.73keV,.0 GeV ε c.93 kev,.5 GeV ε c.4 kev and.0 GeV ε c 0.37 kev. Fom the above figues it follows that the billiance has its maimum aound the citical photon eneg and it has oughl the same value..3 Radiation fom a Wiggle The wiggle is a special magnet with altenating diections of the magnetic field and the tajecto of an electon beam though a wiggle is like a snake, it is a sinusoidal oscillation. The aangement of the magnets in a so-called Hbid Design (HYB) is given in figue (.6). The aows in figue (.6) smbolize the diection of the magnetic field in the diffeent mateials. The geen and the ellow blocks ae special pemanent magnets and the bown blocks ae made of magnetic steel. The tajecto of the electon beam in such a wiggle is pesented in figue -5

6 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME (.7) (blue line). The ed aows smbolize the emitted adiation and it follows that an ovelapping of the adiation cone will occu. Figue.6: The aangement of magnets within an wiggle. Geen and Yellow ae pemanent magnets and bown is magnetic steel. Figue.7: The tajecto of an electon beam within a wiggle (blue line). The ed aows smbolize the emitted snchoton adiation. The maimum slope X 0 and the maimum amplitude X 0 chaacteizes the tajecto of the electon beam. Both epessions ae given b equation (.). X 0 5 K 8.3 0, λ p Kλ p, X K / γ, K ( B / T ) ( λ p / cm) (.) π γ ( E / GeV ) The chaacteistics of the snchoton adiation fom the wiggle ae the same one as fom the bending magnet, and so, the citical eneg ε c detemines evething. To each the same spectal ange as fom the bending magnets (given b ε c (see equation.3)), the magnetic flu densit within the wiggles must be the same as within the bending magnets, o fo shifting the spectum to highe photon enegies, even highe. At CCRL, Daesbu Laboato, UK, fields up to.5 Tesla could be each with Hbid Design, highe fields ae possible with supe conducting devices, at MAXLAB, Lund, Sweden fields up to 3 Tesla could be eached. In table (.) the data s fo the wiggles foeseen fo SESAME ae summaized. Table.: Data s of the wiggles foeseen at SESAME. Tpe B 0 W N W L K X 0 X W-00.0 T 00 mm 4.4 m mm 3.8 mad W-0.5 T 0 mm 0.4 m mm 5.7 mad W T 60 mm 30.8 m mm 4.0 mad -6

7 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION Figue.8: The photon flu emitted fom wiggles SESAME. Figue.9: The photon flu densit emitted fom wiggles at SESAME. The photon flu as well as the cental intensit of the adiation emitted b the wiggle is the same as fom the bending magnet but moe intensive b a facto N p, whee N p is the numbe of poles within the wiggle. The numbe of N p is accoding to table (.) oughl 50 ( times N w ). The photon flu as well as the flu densit emitted fom these wiggles at SESAME is pesented in the figue (.8) and (.9). The compaison to the flu of the bending magnet shows, that the wiggle is oughl of a facto 50 moe intensive, fo the flu densit it is a facto 0 to 5. The calculation of the billiance of wiggles needs to take into account the depth-of-fields, i.e. the contibution to the appaent souce size fom diffeent poles. The epession fo the billiance of wiggles is: -7

8 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME X 0 ep /, N d Φ ( 0) σ + znσ B wi ψ (.) / dφdψ N / π ε,, [ σ + znσ ] + σ + znσ σψ whee: zn λ p n + (.3) 4 ae the ms tansvese size and angula divegence of the electon beam 0). This means that the billiance of the wiggle, calculated accoding to equation (.), is nomalized to the middle of the staight section. The eponential facto in equation (.) aises because the wiggles have two points, sepaated b *X 0 accoding to equation (.). The sum in equation (.) goes ove all poles of the wiggles. As alead discussed unde the adiation of the bending magnets the facto and can be neglected. The epession z n is the incease of the souce size fom the cente of the insetion device. Instead of nomalizing the billiance to the cente of the staight section, the coss sections (n) at the position of the diffeent poles can be used, with the esult that the epession fo the billiance will be simple: X 0 ep n N / d Φ σ ( n) BWig ( ψ 0) (.4) dθdψ n π σ ( n) σ ( n) The billiance of the thee tpes of wiggles given in table (.) ae pesented in figue (.0). The billiance is detemined b the flu and the coss sections of the beam, because of the coss sections at the locations of the wiggles ae 3 o 4 times lage as in the bending magnet, the billiance of is of a facto 0 moe intensetive as of the bending magnet. The billiance of a wiggle is accoding to equation (.4) invesel popotional to the coss section of the beam and not to the emittance. The beam coss section can b manipulated with the beta function in the stoage ing. The so called mini beta sections ae leading to a small beam size. The dependenc of the wiggle billiance upon the beta function at the location of the wiggle is given in figue (.), which is the invese of the dominato of equation (.4) fo diffeent beta functions in the middle of the staight section. Fo high beta functions (4 to m/ad) the coss section of the beam doesn t change ve much and the billiance is accoding to figue (.) popotional to the length of the wiggle. Fo small beta functions (0.4 and 0.6 m/ad) the billiance of the wiggle will satuate, because the coss section of the beam in the oute pats of the wiggle gets high and the contibution to the billiance is small and can theefoe be neglected. -8

9 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION Figue.0: The billiance of the snchoton adiation emitted fom wiggles at SESAME. Figue.: Dependenc of the billiance of wiggle adiation upon the beta functions in the middle of the wiggle. The popotionalit is given b the sum of all invese coss sections at the diffeent poles of the wiggle. In ode to optimise the billiance of the wiggle adiation the beta functions in the middle of the staight sections should be as small as possible (mini-beta-section) and the length of the wiggle has onl to be aound to 3 m, because the oute egions of the wiggle don t have an significant contibution to the billiance. -9

10 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME.4 Radiation fom an Undulato of the snchoton adiation fom the bending magnet at the citical imum slope of c deflection angle in a wiggle is within the opening angle of the snchoton adiation. Fo this special case the adiation fom diffeent peiods intefeences coheentl, thus poducing shap peaks with the esult of completel diffeent chaacteistics. This adiation, as smbolized in figue (.), is called undulato adiation and the coesponding insetion devices ae undulatos. Figue.: The chaacteization of the undulato adiation. L is the peiod length of the undulato Undulatos ae insetion devices like the wiggles but with a smalle K-values (between and 3). A pinciple set up of an undulato is given in figue (.3). The blocks with the diffeent colous smbolize the pemanent magnets with the diection of the magnetic field. The peiod length is indicated with λ. Figue.3: The aangement of magnets within an undulato. Red, Geen, Blue and Yellow ae pemanent magnets. -0

11 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION The undulato emits adiation onl at chaacteistics photon enegies: n ε n 0.949KeV ( E / GeV ). ( λund / cm) ( + K / ) (.5) with the bandwidth: ε n ε nn n Und (.6) whee: n Hamonic numbe ( n, 3, 5, 7,.. ). N Und Numbe of peiods. Und Peiod length of the undulato. K Deflection paamete (see Equation (.0)). The opening angle of the undulato adiation cone as indicated in figue (.) is: ( + K / ) σ (.7) γ Nn Fo the illustation of the chaacteistics of the undulato adiation, the following eample shall be used: Und 40 mm, K and N 50. ε 0.36 kev and ε 9.85 kev. ε 6.3 ev and ε ev. σ γ () mad and σ γ (9) 0.048mad. The flu of the undulato adiation within the cone of the hamonics is given b (in pactical units): 4 Φ ( n, K).43 0 N ( I / A) Q ( K) [Photons/(s 0.%BW)] (.8) Und Und n Accoding to equation (.8) the flu of the undulato adiation is popotional to the numbe of peiods, the cuent and the function Q n (K). It is independent of the eneg of the electons. To each a high photon flu (also fo highe hamonics), the deflection paamete K should be in the ange of to 3. The photon spectum is accoding to equation (.5) popotional to the squae of the eneg and invesel popotional to the peiod length λ. In ode to each -a adiation the eneg has to be in the ange of 4 to 6 GeV with a peiod length of 40 mm. The gap would be mm and the magnets of the undulato ae outside the vacuum (figue (.3)). Smalle peiod lengths can be eached onl b educing the gap; this is possible b putting the undulato inside the vacuum. This tpe of undulato is called In Vacuum Undulato (figue (.5)). The smallest peiod lengths can be eached with supe conducting magnets. These tpes ae called Mini-Undulato (figue (.6)). The photon flu of these undulatos ae pesented in the figues (.4) (.6). The compaison of these figues show that the highest flu and the lagest spectum can be each with small gaps. -

12 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME Figue.4: The photon flu emitted fom an undulato at SESAME. The paametes ae: λ 40mm, gap mm, k.4.8, L.4 m, E.5 GeV, ε 3.7 nm, coupl. %. Figue.5: The photon flu emitted fom an undulato at SESAME. The paametes ae: λ 5 mm, gap 7mm, k.0.0, L.4 m, E.5 GeV, ε 3.7 nm, coupl. %. Figue.6: The photon flu emitted fom an undulato at SESAME. The paametes ae: λ 4 mm, gap 5mm, k.0.0, L.4 m, E.5 GeV, ε 3.7 nm, coupl. %. The peak intensit on ais of the nth hamonic of the undulato adiation is given b (in pactical units [photons / (s 0.%BW m )]): -

13 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION d ΦUnd ( θ ψ 0) dθdψ 4 N Und ( E / GeV ) ( I / A) F ( K ) n (.9) The peak intensit is popotional to the electon cuent, the adiation function F n (K) and the squae of the peiod numbe. The opening angle of the adiation cone is invesel popotional to the eneg, and theefoe the peak intensit accoding to Equation (.9) is popotional to the squae of the eneg. The billiance of the undulato adiation is given b: whee B Und Φ Und ( π σ σ ) ( π σ σ ) ( θ ψ 0) (.0) σ σ + σ, σ σ + σ, σ σ + σ, σ σ + σ (.) σ, σ, ε, / β,, σ λl, σ λ / L (.) 4π, ε, β, ae given b: λund ( + K / ) N ( + K / ) σ, 4 πγ n σ (.3) γ nn accoding to equation (.3) ae summaized in the table (.). Table.: Coss sections and divegences of the undulato adiation emitted at SESAME n σ can be neglected fo the calculation of the billiance. The billiance of the thee tpes of undulatos: U40, U5 (In Vacuum) and U4 (Mini- Undulato) ae pesented in figue (.7) (.9). The compaison of these figues show that the highest billiance can be eached with small gaps. At SESAME it is onl possible to each the kev photons with In Vacuum undulatos and the 0 kev with Mini-Undulatos. In figue (.0) the photon flues and in figue (.) the photon billiances emitted within the bending magnet, the wiggles and the undulatos have been compiled. -3

14 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME Figue.7: The billiance emitted fom an undulato at SESAME. The paametes ae: λ 40 mm, gap mm, k.4.8, L.4 m, E.5 GeV, ε 3.7 nm, coupl. %. Figue.8: The billiance emitted fom an undulato at SESAME. The paametes ae: λ 5 mm, gap 7mm, k.0.0, L.4 m, E.5 GeV, ε 3.7 nm, coupl. %. Figue.9: The photon flu emitted fom an undulato at SESAME. The paametes ae: λ 4 mm, gap 5mm, k.0.0, L.4 m, E.5 GeV, ε 3.7 nm, coupl. %. -4

15 SESAME : CHARACTERISTICS OF SYNCHROTRON RADIATION Figue.0: The photon flu emitted fom the bending magnet, wiggles and undulato at SESAME. The details of the insetion devices ae given within the tet. -5

16 : CHARACTERISTICS OF SYNCHROTRON RADIATION SESAME Figue.: The photon billiance emitted fom the bending magnet, wiggles and undulato at SESAME. The details of the insetion devices ae given within the tet. Refeence [] Schwinge Theo, Phs. Rev.75, 9 (949). -6