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A Remark on the Method of Electron Beam Energy Measurement Using Laser Light Resonance Abso

A Remark on the Method of Electron Beam Energy Measurement Using Laser Light Resonance Abso
A Remark on the Method of Electron Beam Energy Measurement Using Laser Light Resonance Abso

a r X i v :h e p -p h /0503004v 1 1 M a r 2005A REMARK ON THE METHOD OF ELECTRON BEAM ENERGY MEASUREMENT USING LASER LIGHT RESONANCE ABSORPTION.N.B.Skachkov,Joint Institute for Nuclear Research,Dubna,141980,Russia E-mail:skachkov@jinr.ru,skachkov@https://www.wendangku.net/doc/7211070788.html, February 2,2008Abstract The problem of measuring of the electron beam energy by help of the laser light interaction with the electrons is discussed.It is shown that the orthogonal orien-tation of the laser beam with respect to the electron one,proposed in the present Note,may allow to perform this measurement in accordance with the physical na-ture of a formation of an electron quantum levels in a magnetic ?eld.In result,the ?nal formula,that expresses the beam energy through the strength of a magnetic ?eld and the energy of the laser photon,gets a transparent physical meaning and do contain a less number of parameters (what may lead to an increase of the precision of the measurement).Some other sequences from this proposal,like the change of the geometry of the experimental set-up and the necessity of a new additional

detector to register the products of the Compton scattering for monitoring of the beam energy measurements,are discussed also.

1Introduction.

An application of the resonance absorption (RA)of the light by electron [1]-[6]for the beam energy measurements was discussed in a number of talks given at di?erent conferences [5],[6].

The idea of this method is based on the theoretical formula which was obtained for the ?rst time in the framework of the non-relativistic quantum mechanics (see [7])and latter on was generalized for the relativistic case basing upon the Dirac and the Klien-Gordon equations (see,for instance [8]).These formulae give the energy eigenvalues and the wave functions of the particle which moves in the homogeneous magnetic ?eld.The analogous exact solutions were found also for a more complicated cases when a particle moves in

1

the combined?eld,composed of an electromagnetic and a homogeneous magnetic?elds (Volkov solution,see also[9],[10]).Recently this study was continued by?nding out the exact solution of the Dirac equation for a case of the superposition of a homogeneous magnetic?eld and a circularly polarized electromagnetic wave[4]with an account of the electron anomalous magnetic moment.Its was shown that taking account of the electron anomalous magnetic moment removes the spin degeneracy of the energy levels.

According to the results of all these papers the particle which moves in a constant homogeneous magnetic?eld should have the quantized values of the energy,connected with its motion in the plane perpendicular to its velocity vector.In[1]and latter on in[4]it was proposed to use the transition from the lower levels of quantized transverse energy to the higher ones,caused by the laser photons absorption,for getting out the information about the initial electron beam energy(let us mention that in these papers the process of laser photon interaction with the beam electron was described within the Classical Electrodynamics framework).

In the present Note some new points of views on this measurement are proposed. First,the suggestion of a new orthogonal disposition of the laser beam with respect to the electron one is presented.Such a geometry of the measurement is di?erent to the previously considered case,see[1]and[6],where the laser beam was supposed to be injected at a small angle to the electron beam direction(taken as z-axis).New proposal is based on the arguments which follow from the physical origin of the quantized levels of the electron energy in a magnetic?eld and would be discussed below in Section2.The second point is connected with the usage of Quantum Electrodynamics(QED)approach for describing the laser photon interaction with the beam electron(see Section3).The importance of the measurement of the Compton scattering(as the process that may happen when the experimental parameters would not meet the conditions necessary for RA)for monitoring of the beam energy measurement is discussed in the Sections4and 5.

2The energy levels and the wave function of the elec-tron in the static magnetic?eld.

Here(having in mind the aim of a further application for the beam energy measurements at ILC,1i.e.at the ultra-relativistic energies of electron)the solution of the Dirac equation would be used(taken from the widely used text book[11]),derived for a case of the electron motion in the homogeneous magnetic?eld( B=rot A),and with the choice of the Landau gauge for the4-vector of the electromagnetic potential Aμ:A0=A x= A z=0,A y=B0x.Thus, A=B0x e y,where e y is the unit vector along the y-axis,and B=(0,0,B

).

z

The energy of the electron p0,which is the fourth component of the electron4-momentum vector Pμ=(p0,p x,p y,p z),is de?ned in this case by formula([11]):

(p0)2≡E2λ(n,p z)=E2z+E2T,λ(n),(1) where the?rst term

E2z=m2e c4+p2z c2(2) is the square of the relativistic energy of a free electron that moves in a beam(taken as the z-axis).The second term

E2T,λ(n)=h(

eB0

m e =ωc is the cyclotron frequency.

In what follows we shall use the expression for the di?erence of E2T,λ(n)for the two neighboring values of energy levels with n+1and n quantum numbers which stems from the formula(3):

?E2T,λ=E2T,λ(n+1)?E2T,λ(n)=2h(

eB0

2the ground state(i.e.,

n=0)would have,as it follows from(3),the value of the transversal energy been equal to zero:2

E2

T,λ=?1

2the expression for the total

energy of the electron in magnetic?eld do coincides with the energy of a free electron:

Eλ=?1m2e c4+p2z c2.(6) In a complete correspondence with the formula(1),that demonstrates the independent entrance of E2z and E2T terms into the expression of the square of the total energy E2n,λ,3 the wave functions of the Dirac equation,found as the exact solutions for the mentioned above di?erent combinations of external magnetic and electric?elds[1]-[11],also do factorize in two parts.

One part(an oscillator like solution)describes the motion in the plane transverse to the beam and along the x-axis(because the only nonzero component A y=B0x of the

2,would have the energy

E2

T,λ=+1

4-vector of the electromagnetic potential Aμis de?ned by the x-coordinate).The another one,which has an exponential form,is connected with the free motions along the y-and z-axis(see[7]-[11]).For example,the solution,that corresponds to the formula(1),has a form( r=(x,y,z))[11]:

ψ( r)=e i(p y y+p z z)f(x),(7) where

f(x)=Ce?ξ2/2H n(ξ);ξ=

(with the4-momentum p1)looks,in a general case,like(k2and p2are the?nal state momenta of the photon and the electron):

p1+k1=p2+k2.(9) If we shall pass to the components of the4-momentum:kμ=(k0, k); k=( k T,k z),then the equation(9)may be presented as a set of the independent conservation laws for each component separately:

for the energy,

p01+k01=p02+k02,(10) for the z-projection:

p z1+k z1=p z2+k z2,(11) and for the transversal one

p T

+ k T1= p T2+ k T2.(12)

1

Thus,if the laser photon would have some small angleθto the electron beam,as it was considered in[1]and[6](in such a case the k z1component of the photon momentum would dominate over his transverse one k T1),then the z-component of the electron would change,in general case,from p z1to some p z2due to the conservation law of the z-component of4-momentum.

Independently from the z-component,the transversal component of electron may also change after the interaction.There are two possible cases,which may be realized in the framework of QED.

The?rst one,most simple,may be realized when the k T1contribution to the electron transversal motion would not match the di?erence(4)of two quantized electron energy levels in a magnetic?eld.In this case,according to QED a pure Compton scattering would take a place.

The second one may happen when the value of| k T1|should match the equation(4).In this case,keeping in mind what was written before about the independent conservation of z-and transverse components of the total4-momentum of electron-photon system,we shall have the?nal state consisting of:

.the photon,radiated due to changing of the electron momentum z-component(and in accordance with(11)),

.and the electron,which| p T2|should be de?ned in this case by formula(3)for the (n+1)-th,or even higher level,if it was at the n-th orbit in the initial state.

Hence,in both cases,even if the resonance condition would be ful?lled and the trans-verse component k T1would exactly coincide with the energy necessary for moving from the lower orbit to the higher one(i.e.the equality p T2= p T1+ k T1would be ful?lled for the inde-pendent(!)motion in the transverse plane),the changing of the independent z-component

5

of the electron momentum would lead to the radiation from the electron.In QED it is an emission of the photon via the e2order Compton process.4

In other words,according to QED,in a case,when the laser beam would have a small angleθwith respect to the electron beam,the Compton process(with or without moving of an electron from the lower orbit to a higher one)would take place and the emission of the photon would happen.In such a case the whole situation becomes more complicated for the theoretical description.5

A possible simple way to avoid these problems and to use the theoretical prediction about the existence of the quantized energy levels(of the transverse motion of the particle in the magnetic?eld)for the electron beam measurement is given below.It is based on a special choice of the angle between the laser and the electron beams,mentioned already in the Introduction.

4What laser photon beam orientation meets better the physics of electron motion in magnetic?eld? The physical picture of the electron motion in the static magnetic?eld as been a sum of two independent motions(what was discussed in the previous Sections)as well as the 4-momentum conservation equations written above,do prompt(if not to say,dictate)an easy way to prepare the initial state to use the possibilities connected with existing of the quantized energy levels of a particle in the magnetic?eld.This possibility is based on the proper choice of the relative geometrical orientation of the laser and the electron beams.

Indeed,if we shall choose the laser beam to be orthogonal to the electron one,i.e. shall consider a case when k z1=0in equation(11),then the collision of the laser photon with the electron would not e?ect the z-component of the electron momentum at all.

If also,in addition to the orthogonal angle orientation,we shall adjust the laser photons energy(and the magnetic?eld strength B0)6to be equal to the di?erence of the two lowest quantized levels of energy,connected with the transversal motion of the electron in a magnetic?eld and de?ned by formula(4),then the laser photon would be absorbed.7In this case the whole initial photon energy,been prepared as of the transversal nature only, would transform into the energy of a its circular motion,but on a higher then previous one orbit in the plane transverse to electron beam.Therefore,in a case of the proposed

orthogonal orientation and the properly tuned value of the magnetic?eld strength B0in a way to have the di?erence of two energy levels of the electron in accordance with the energy of the laser photons there would be no any phase space left for the production of an additional photon in the?nal state!

In this special case of k z1=0and when also the laser energy would?t the di?erence of two E T for some n and n+1levels,the formula(11)shall take a very simple form:

p z1=p z2.(13) (This equation may be treated also as one of the conditions of RA e?ect.) The energy conservation law in the such a case ofθlaser=900should look like

p01+k01=p02,(14) where,due to our choice of the laser photon momentum4-vector in a form kμ1=(k01, k1); k

=( k1T,0),the?nal state electron transverse momentum p T2would be de?ned by the 1

the equation

p T

+ k T1= p T2,(15)

1

as the sum of the initial electron transverse momentum p T1and the laser photon transverse vector k T1which modulus is equal to the energy of the absorbed photon8.

The4-component equation(9)would take in such a case the following form:

p1+k1=p2.(16) Note that for the considered here process the right hand side of equation(14)may be represented like

p02=

8i.e.,k0

=| k1T|due to the relation k01=| k1|,which is valid for the massless photon.

1

7

2Eγlas Eλ(n,p z1)+(Eγlas)2=E2λ(n+1,p z1)?E2λ(n,p z1).(20) From formula(1)-(3)it is clear that the longitudinal contributions E2z would cancel in the right hand side of(20)9and only the di?erence of the transverse energies E2T,λ(n)would remain there:

E2λ(n+1,p z1)?E2λ(n,p z1)=E2T,λ(n+1)?E2T,λ(n).(21) The expression for this di?erence was already given by formula(4).So formula(20) could be presented in a form

2Eγlas Eλ(n,p z1)+(Eγlas)2=?E2T,λ,(22)

which does not contain any more the dependence on the beam energy in the right hand side of the equation.In such a case it is convenient to present(22)as an equation which includes the dependence on the electron beam energy(see(1))only in the left hand side:

Eλ(n,p z1)=

?E2T,λ?(Eγlas)2

2

),then the formula(23)(written as Eλ=?1m2e c4+p2z c2=E beam)gives,due to (6),the following expression for the beam energy E beam

E beam=?E2

T,λ=?1

2Eγlas

.(24)

Keeping in mind that according to formula(4)for the non spin-?ip case,i.e.when the value ofλdoes not change in the process of the beam electron interaction with the the laser photon,we have?E2T,λ=2hωc(m e c2)=2h(eB0/m e)(m e c2),we come to a very simple?nal formula

E beam=2h(eB0

2Eγlas

.(25)

The last equation does express the value of the beam energy through those values of the magnetic?eld strength B0and of the laser photon energy Eγlas,which were tuned to provide the conditions for the laser photon energy to be equal to the di?erence of two neighboring quantum energy levels in a case of the orthogonal relative orientation of the laser to the electron beams.

It is natural that the error of the measurement of the electron beam energy?E beam should be de?ned in this case by two errors only:

.the error of the magnetic?eld strength?B0measurement,and by

.the error of the measurement of the laser photon energy?Eγlas.

To summarize shortly it is possible to say that in a case when the following conditions would be ful?lled:

*the angle between the laser and the electron beams would be chosen to beθ=900, (i.e.cosθ=0),

**the magnetic?eld strength B0and the laser photon energy Eγlas would be tuned to provide the conditions for the laser energy to be equal to the di?erence of two neighboring quantum energy levels of the electron in a homogeneous magnetic?eld,

then the photon may be absorbed by the quantum system of the quantized electron levels in a homogeneous magnetic?eld and

the laser photon collision with the beam electron may lead only to a change of the orbit number of the transverse motion of the electron.No changes of the longitudinal(i.e.,of z-)component of the electron momentum would happen.It means that no any additional radiation(like that one which take place in the Compton scattering process)would have su?cient phase space to appear.

5How it may look from the experimentalist point of view.

There are some points have to be discussed in a connection with a possible planning of the measurements if they should be done in the context of the described above picture of the laser photon interaction with the beam electron.

The obvious question needed to be asked for such a planning would be about the choice of the laser energy(i.e.,the wave length).The answer can be easily found from the solution of the quadratic equation(25)for the Eγlas.In a case of n=0andλ=?1

1+2h(eB0

E2beam

?1)=E beam(

E2beam?1),(26)

which expresses Eγlas through the value of the beam energy E beam and the strength of the

magnetic?eld B0.So,with the help of this formula one can?nd out(for a chosen value of E beam and an expected value of B0)what kind of a laser does have the wave length that corresponds to the energy de?ned by formula(27).

The next important sequence for planning of the experiment is the need to pass to a new geometry for the experimental set-up for such measuring of dropping of the laser light intensity due to the photon absorption.It is evident that the detector(D),proposed to measure the decrease of the laser light intensity after its interaction with the electron beam[6],and the laser should be placed on the same line,perpendicular to the electron beam.Such a geometrical disposition of the experimental setup looks very di?erent to what was discussed in[1]-[6],where it was proposed that the line which connects the laser and the detector(D)would have a small angleθto the electron beam line.

This changing of the position of the detector D may be important from the experimen-tal point of view.Really,in principle,there is a non zero probability that the electron, which was forced by the laser light to move to a higher orbit,would emit a photon and then return after some time back onto the ground state orbit.Here we have a direct analogy with the radiative transition from higher to lower orbits in atoms.Fortunately, in the case,proposed in the present Note,the electron,been moved onto any the higher orbit,would keep,as it was discussed in the previous Sections,the same value of the initial longitudinal momentum p z1(as it was discussed in the previous Sections),i.e.,it shall con-tinue to move with the same speed,practically equal to the speed of light.Hence,if the photon emission would happen at some moment,de?ned by the lifetime of the“n+1-th”excitation level,nevertheless,this photon have no chances to hit the detector D,because during this even small time the electron,that was pushed to the“n+1-th”orbit,would ?y already away from the position of the“laser-detector D”perpendicular line(to the electron beam line).To this reason the proposed here perpendicular orientation of the laser to electron beam is a preferable one as it is more safe from the view point of the possibility that the secondary radiative photon emission may reach the detector D.

Up to now we were discussing the aspects connected with the case of the resonance absorption within the situation of a new orthogonal disposition of both beams.Now let us turn to the situation when the energy of the laser photon would not?t to the the value of energy splitting between the main n=0and one of the next to the ground states levels.

As it was already discussed in the previous Sections,in a case when the resonance condition(25)would not be ful?lled(most probably,due to some variation of the electron beam energy)then according to QED the Compton process ofγ+e→γ+e scattering would take place.

The appearance of the Compton scattering products,i.e.of the?nal state photon and electron(with p z2=p z1)may be used as the signal of a dropout from the resonance condition(25).Therefore,the addition of experimental equipment,supposed for beam energy measurement,by a new detector which may identify the electrons and photons, produced in the Compton scattering,may be very helpful.It may be an electromagnetic calorimeter with the appropriate granulation for the registration of the photons and for de?ning of their position,accompanied by some electron tracker.11

It should be mentioned also that if we shall suppose that the e?ects of the electron beam energy variation would not be so large,then it would not be so di?cult to calculate the angle and energy distributions of the scattered photons and electrons which would be characteristic for the such possible variations of the electron beam energy in both directions from the position of the resonance value.

Thus,it may happen so that the control over the Compton process(by a comparison of the observed experimental distributions of the photon and the electron parameters with those predicted by QED)would turn out to be a useful guide for adjusting the magnetic ?eld strength B0during the beam energy measurement.In such a case the measuring of the characteristics of the Compton scattering within the region of parameters,close to the resonance position,may be used for the on-line tuning of the magnetic?eld strength to those values which would be an adequate to keep the resonance condition.The detailed study of this possibility would be a subject of the following publications.

6Summary.

It is shown that the orthogonal orientation of the laser photon beam with respect to the electron beam allows to bring the idea of the use of the theoretical prediction about the existence of electron quantum levels(in the presence of the homogeneous magnetic?eld) in the correspondence with the physical origin of these resonance levels.

In result,the formulae that describe a process of a possible absorption of the laser light by the transverse degrees of electron motion in the constant magnetic?eld become much simpler because in this case the absorption process does not interfere with the longitudinal motion along the beam direction.Due to this no additional radiation would appear in a case when the laser photon energy would?t the value of the energy splitting of two electron levels in the magnetic?eld(i.e.in a case of ful?lling the resonance condition). In result,the physical picture of the process of measuring the electron beam energy by measuring the drop of the laser light intensity in the absorption process becomes more transparent.This new relative orientation of two beams means also passing to another geometry for experimental tools set-up.

In the opposite case when the energy of the laser photon would not be equal to the di?erence of two electron energy levels(in magnetic?eld)the Compton scattering process would take place.The characteristics of the scattered electron and photon(angles and energies)may be measured(it may require to enlarge the experimental equipment by the additional electromagnetic calorimeter and the tracker)and then used for an on-line monitoring of the process of tuning the magnetic?eld to keep the resonance photon absorption condition(25)for the electron beam energy measurement.

It is a pleasure for the author to express his gratitude to R.Melikian and especially to H.J.Schreiber for the useful discussions.

References

[1]I.A.Gilinski and K.A.Ryasantsev;Izv.Vuz.Radiophysics,5(1964)838;

[2]V.G.Bagrov and V.R.Khalilov;Izv.Vuz.Physics,2(1968)37;

[3]https://www.wendangku.net/doc/7211070788.html,ant’ev;https://www.wendangku.net/doc/7211070788.html,pekhi,40,1(1971)1;Uspekhi Fiz.Nauk,167,1(1997)37;

[4]R.A.Melikian and D.P.Barber,DESY report98-015(1998),hep-ph/9903007;

[5]R.A.Melikian and D.P.Barber Proc.13Intern.Symposium on High Energy Spin

Physics(Spin98),p.495,Protvino,Russia,8-12Sept.1998;

[6]D.P.Barber and R.A..Melikian,Procc.of EPAC2000,Vienna,Austria,p.996,

http://accelconf.web.cern.ch/accelconf/e00/PAPERS/MOP1B01.pdf;

[7]https://www.wendangku.net/doc/7211070788.html,ndau and E.M.Lifshitz,”Quantum Mechanics”,Nauka,Moscow,1989.

[8]A.A.Sokolov,I.M.Ternov and C.W.Kilminster”Radiation from Relativistic Elec-

trons”,American Institute of Physics(translation series),1986;Naika,Moscow,1983.

[9]P.J.Redmond,J.Math.Phys.6(1965)1163;

[10]J.Bergou,F.Ehlotzky,Phys.Rev A27(1983)2291;

[11]A.I.Akhiezer,V.B.Berestetskii”Quantum Electrodynamics,Fizmatgiz,Moscow,

1957.Interscience Publishes,New York,1965;

[12]V.B.Berestetskii,E.M.Lifshitz,L.P.Pitaevskii,”Quantum Electrodynamics”,Nauka,

Moscow,1980;Pergamon press(1982),Addison Wesley Publ.,1971;

[13]N.N.Bogoluibov,D.V.Shirkov“Introduction to the Theory of Quantized Fields”,Fiz-

matgiz,Moscow,1957;A Wiley-Interscience Publication;John Wiley and Sons,New York,1979;

[14]TESLA.Technical Design Report http://tesla.desy.de/tdr/Chapter IV,The Photon

Collider at TESLA http://tesla.desy.de/tdr/PartVI/append.html.

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一般过去式 时间状语:yesterday just now (刚刚) the day before three days ag0 a week ago in 1880 last month last year 1. I was in the classroom yesterday. I was not in the classroom yesterday. Were you in the classroom yesterday. 2. They went to see the film the day before. Did they go to see the film the day before. They did go to see the film the day before. 3. The man beat his wife yesterday. The man didn’t beat his wife yesterday. 4. I was a high student three years ago. 5. She became a teacher in 2009. 6. They began to study english a week ago 7. My mother brought a book from Canada last year. 8.My parents build a house to me four years ago . 9.He was husband ago. She was a cooker last mouth. My father was in the Xinjiang half a year ago. 10.My grandfather was a famer six years ago. 11.He burned in 1991

学生造句--Unit 1

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