ADITIONS TO THE METHOD OF ELECTRON BEAM ENERGY MEASUREMENT USING RESONANT ABSORPTION OF LASER LIGHT IN A MAGNETIC FIELD.

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1 ADITIONS TO THE METHOD OF ELECTRON BEAM ENERGY MEASUREMENT USING RESONANT ABSORPTION OF LASER LIGHT IN A MAGNETIC FIELD. Melikia R.A. (YerPhI Yereva) 1. NEW CONDITION OF RESONANT ABSORPTION Below we ca see that the resoat coditio ca be preseted i more coveiet orm allowig to cosider process o absorptio i details ad to estimate the accuracy o measuremet o electro beam eergy. Electros ad photos are ijected i a magetic ield B uder small agles ϕ ad θ to the z-axis accordigly (Fig.1). e - beam - > B D j > ƒ > Z Laser q > Beam < L > Fig.1 I a magetic ield the electros have a discrete spectrum o eergy [] show o Fig.: ε ζ = [ m + Pz + eb( + 1+ ζ )] where =1 labels the electro eergy levels z-compoet o the electro mometum. Ñw Ñw È È È Ñw P Z is the Fig. (1) R.Melikia YerPhI

2 Ater icomig o electros i a magetic ield the compoets o their mometum ad Z are redistributed ad electros occupy the quatum levels. The photos o requecy ca be resoatly absorbed at trasitios o electros betwee levels o eergy. Usig the law o eergy-mometum coservatio or photo absorptio: ε + ω = ε ζ ζ ad spectrum o eergy absorptio: ω (1 V z ε ς P + ω cosθ = z P z ω () rom (1) we id the coditio o resoat ω (siθ ) cosθ ) + m = ωc ( ) We cosider trasitios betwee o electro eergy levels without o chage o electros spi directio ad trasitios o the mai harmoic ' - = 1. The secod member i the relatio (3) ca be eglected because hω 1 ωc (siθ ) < 1. mc ω The rom (3): ωc ωc ω = = (4) (1 V cosθ ) P z z cosθ m 1 I (4) the actor (1 cosθ ) describes Doppler's eect which i case V Z o relativistic electros has essetial iluece o a spectrum o absorptio o photos. Substitutig i (4) P oud rom (1) we receive the ew resoace absorptio coditio : Z where depeds rom ε ς hω cosθ 1 ( + 1+ ς ) = mc = ω c ω θ ad. (3) (5) R.Melikia YerPhI.11.5

3 Let s estimate the value o or the give ad From (5) we id: Usig (6) rom coditio From (6) we ca id = 1 hω 1 (1 + ς cos ) ω θ mc mc h (6) we id the restrictio o agle θ : 1 θ (7) takig ito accout that / θ = hω << mc at θ =. Neglectig i (6) by small member (1 + ς ) 1 we id approximately: Accordig to (6) we ca id the depedece ad (Fig.3). θ. 1 = hω mc 1 (8) or rom θ or ixed g= W= 1-3 g= g= 1 5 Fig We assume that N electros are allocated o quatum e q levels accordig to the Gaussia distributio law (Fig.4). R.Melikia YerPhI

4 g= 1 5 W=1-3 Fig.4 From Fig.4 it is clear that electros with ixed absorb photos i iterval o agles q q where accordig to (7) θ = ad ca resoatly Let's otice that the light beam o diameter D because o diractio diverges i limits o agle θ d λ D ( λ - legth o wave) aroud o a wave vector directio ad θ < θ d. For example electros at levels Δ i ca resoatly absorb photos i iterval o agles Δθ i Fig.4. I itesity o absorptio o photos by electros rom quatum levels is equal to ΔI abs i the itesity Δ i o absorptio by all electros will be: D i ÈÈ È Dq i I abs ΔI abs = i= i q! q (9) (1) q R.Melikia YerPhI

5 . DETERMINATION OF THE ELECTRON BEAM ENERGY. THE RELATIVE ACCURACY OF MEASUREMENT. The -actor o electro beam ca be oud rom resoat coditio (5) ± cosθ = hω (siθ ) (1 + ( + 1+ ς )) mc (siθ ) (11) Depedece o rom accordig to (11) or dieret agles θ ad quatum umbers is show o Fig.5. 5 g 4 3 = ; q= = ; q= =1; q =1-4 1 È È È = ; q= = ; q= Fig.5 From Fig. 5 we see that the behavior o depedece rom or parameters ad θ essetially diers rom behavior o resoat curves whe parameters had bee agles ϕ ad θ [1]. I case o parameters (whereas i ad θ the resoat curves are crossed i a poit case o ϕ ad θ these curves are ot crossed). W ( ) R.Melikia YerPhI

6 From Fig. 5 we see whe the agle θ varies i limits rom θ up to θ the resoat poit is o the upper brach o curve whereas i iterval o agles θ up to θ = the resoat poit is o the lower brach. The choice o parameters ad θ allows to uderstad details o absorptio ad the possibility o measuremet o eergy with accuracy 4 ear to cetre o distributio over eergy o electro beam whe the eergy spread electro beam is Resoat lies ear to the poit whe o Fig.5 are show o Fig.6 ad the electro beam eergy spread is = ± Δ ± Δ. g +Dg g W -DW W +DW g -Dg Fig.6 From Fig.6 it is clear that electros would give the cotributio i process o absorptio rom arrow area o eergy ± δ i to use with small spread Δ. R.Melikia YerPhI

7 O Fig.7 it is schematically show the depedece o itesity o absorptio I abs rom. Numerical estimatios accordig to (11) 4 ad Fig.6 show that i ad Δ / = 1 the the mai Δ / = 1 4. = 1 3 cotributio to itesity o absorptio will be give by electros with Besides that it is clear the urther decreasig o Δ /. will lead to the decreasig o Δ/ I absi DW W Fig.7 About the cotributio to process o absorptio o photos with various agles θ gives represetatio the Fig g W= = = 1 = = ÈÈÈÈÈ q q Fig.8 R.Melikia YerPhI

8 From Fig. 8 we see that i process o absorptio give cotributio photos with agles rom θ = with = up to (8). Usig Fig.1 ad (1) takig ito accout that ϕ<<1 ad >> 1 we id approximately: hω P P = (siϕ) ϕ (1) or agle ϕ ϕ ad m m up to θ m (9) ad electros Accordig to (9) ad Fig. 3 with growth o icreases ad θ ad as a result icreases the itesity o photo absorptio. For the urther calculatios it is ecessary to id the relatio betwee agle o electro ijectio ϕ ad parameters θ. O the other had rom (6) we have: hω 1 θ. m (13) From (1) ad (13) we receive the relatio which we used earlier [1]: ϕ 1 θ + (14) We have oted [1] that itesity o photo absorptio is imal where 1 ϕ = (15) It is importat that the agle ϕ is parameter idepedet rom electro beam eergy. We ca use this property o beam or determiatio o electro beam eergy. R.Melikia YerPhI

9 For this purpose irst or kow eergy over imum o I abs id (Fig.9) ad the we ca calculate: Ater that or ukow eergy (Fig.9) ad the we ca calculate: F g 5 1 ϕ = (16) U over imum o I abs ϕ we id F 1 F = (17) U U we 4 g U g ÈWF 1 ÈW W Fig.9 Takig ito accout that ϕ = ϕ F rom (16) ad (17) we id U = F + F ( 1 1) (18) R.Melikia YerPhI

10 3. DIFFERENCE OF DIRECTIONS OF RADIATION AND ABSORPTION OF PHOTONS BY ELECTRONS Above we have see that absorptio o photos occurs withi o agles θ = up to θ ad the mai cotributio is give by photos with agles θ / (Fig.4 Fig.1). Radiatio o electro goes uder agle α 1/ cocerig o electro velocity V e. e - b e a m L a s e r b e a m Fig.1 I the electro beam has Gaussia distributio over agles rom up to ϕ the the mai cotributio to radiatio give electros with agles ϕ / (Fig.4). Thus overlappig o directios o radiatio ad absorptio does ot occur ad detector D (Fig. 1) ca register oly laser photos. ϕ ϕ = 4. EXPERIMENT Acceleratio o electros i a ield o electromagetic wave propagatig alog a costat homogeeous magetic ield has bee experimetally ivestigated i 1969 []. The experimets has coirmed opportuity o acceleratio o electros i these ields ad the theoretically predicted gai o eergy was i the reasoable agreemet with the measured value. R.Melikia YerPhI

11 I the circular cross-sectio cavity resoator (with TE-111 mode) the electro beam with low eergy 1-1 ev ad with a curret 1-1 mа was ijected (Fig.11). The magetic ield was with a value o 4. kg wavelegth 3cm ad 3 cm. It has bee measured that electros are accelerated up to 3 kev at iput power o wave 5 kw ad 46 kev at iput power o wave 46 kw. B ---> - -->e - beam Cavity Resoator ÈïWave quide Feed Fig CONCLUSIONS The choice o parameters ad θ allows to uderstad details o absorptio ad the possibility o measuremet o eergy with accuracy 1 4 (ad better) ear to cetre o eergy distributio o electro beam. Directios o radiatio ad absorptio are ot overlappig ad detector D registers oly laser photos. 6. REFERENCES 1. ew_aspects5_melikia.pd. H.R. Jory ad A.W. Trivelpiece J. o Applied Physics v.39 No.7 p R.Melikia YerPhI