Generation of GeV protons from 1 PW laser interaction with near critical density targets

Generation of GeV protons from1PW laser interaction with near critical density targets

Stepan S.Bulanov,1,2Valery Yu.Bychenkov,3Vladimir Chvykov,1Galina Kalinchenko,1

Dale William Litzenberg,4Takeshi Matsuoka,1Alexander G.R.Thomas,1

Louise Willingale,1Victor Yanovsky,1Karl Krushelnick,1and Anatoly Maksimchuk1

1FOCUS Center and Center for Ultrafast Optical Science,University of Michigan,Ann Arbor,

Michigan48109,USA

2Institute of Theoretical and Experimental Physics,Moscow117218,Russia

3P.N.Lebedev Physics Institute,Russian Academy of Sciences,Moscow119991,Russia

4Department of Radiation Oncology,University of Michigan,Ann Arbor,Michigan48109,USA

?Received23October2009;accepted5March2010;published online12April2010?

The propagation of ultraintense laser pulses through matter is connected with the generation of

strong moving magnetic?elds in the propagation channel as well as the formation of a thin ion

?lament along the axis of the channel.Upon exiting the plasma the magnetic?eld displaces the

electrons at the back of the target,generating a quasistatic electric?eld that accelerates and

collimates ions from the?lament.Two dimensional particle-in-cell simulations show that a1PW

laser pulse tightly focused on a near-critical density target is able to accelerate protons up to an

energy of1.3GeV.Scaling laws and optimal conditions for proton acceleration are established

considering the energy depletion of the laser pulse.?2010American Institute of Physics.

?doi:10.1063/1.3372840?

I.INTRODUCTION

The acceleration of charged particles from intense laser interactions with targets of different density and composition is considered to be one of the main applications of high power laser systems.In particular,the acceleration of ions attracted a lot of interest over the last few years.The accel-erated ions can potentially be used for fusion ignition,1had-ron therapy,2radiography of dense targets,3and injection into conventional accelerators.4Ion beams with a maximum en-ergy of tens of MeV were observed in previous experiments from laser interactions with solid and gaseous targets.5,6The generation of energetic ions beams has also been thoroughly studied using both two dimensional?2D?and three dimen-sional?3D?particle-in-cell?PIC?computer simulations.7,8 These simulations show that with laser systems capable of producing ultrashort pulses in the multiterawatt or even peta-watt power range it is possible to generate ion beams with energies of several hundreds of MeV or even several GeV by optimizing the parameters of the pulse and the target,and employing new regimes of acceleration.

There are several regimes of ion acceleration discussed in the literature:?i?target normal sheath acceleration ?TNSA?,9?ii?Coulomb explosion?CE?,10and?iii?radiation pressure or laser piston regime?LP?.11In these three regimes the laser pulse interacts with a thin foil of solid density. When the pulse is not intense enough to burn through the target,it launches hot electrons from the front surface through the target.Upon reaching the back of the target,they establish a sheath electrostatic?eld which accelerates ions ?TNSA?.When the laser pulse is suf?ciently intense,it ef-fectively removes all the electrons from the irradiated vol-ume and subsequently the bare ion core explodes due to the repulsion of noncompensated positive charges?CE?.In the radiation pressure regime,the laser pulse is able to push the foil as whole due to the fact that as it is re?ected it acts as a

?ying relativistic mirror?LP?.Recently several new schemes

were proposed,based on the enhancement and different com-

binations of these regimes:?i?Breakout afterburner,the ef-

fective combination of TNSA with the direct acceleration by

the burning through laser pulse,12?ii?the enhanced CE, where the ions are injected into a Coulomb?eld,13?iii?the directed CE,14,15which is achieved in laser double-layer foil

interactions.In the directed CE regime the electrons are ex-

pelled from the focal spot,the?rst layer of heavy ions is

accelerated by the radiation pressure,and then experiences a

CE,transforming into a positively charged cloud expanding

in the direction of laser pulse propagation.The second layer

ions are accelerated in the moving charge separation electric

?eld of this cloud.

In this paper we report on the study of a different and

potentially more ef?cient mechanism of ion acceleration.

Whereas in all the above mentioned schemes thin and ultra-

thin solid density targets are used,here we utilize a regime of

laser pulse interaction with near critical density targets.

These targets have the thickness which is larger than the

laser pulse length.When laser propagates through such tar-

gets,it forms a channel in both the electron and ion density.

A portion of the electrons is accelerated in the direction of

laser pulse propagation by the longitudinal electric?eld.The

motion of these electrons generates a magnetic?eld,which

is circulating in the channel around its axis.The region

where the magnetic?eld is present moves behind the pulse.

Upon exiting the channel the magnetic?eld expands into

vacuum and the electron current is dissipated.This?eld has

the form of a dipole in2D and a toroidal vortex in3D.The

magnetic?eld displaces the electron component of plasma

with regard to the ion component and a strong quasistatic

electric?eld is generated that can accelerate and collimate

ions.The accelerated ions originate from the thin ion?la-

PHYSICS OF PLASMAS17,043105?2010?

1070-664X/2010/17?4?/043105/8/$30.00?2010American Institute of Physics

17,043105-1

ment that is formed along the axis of the propagation channel ?see Fig.1?.The mechanism under consideration was pro-posed in Ref.16and studied in a number of papers.17–19It was shown recently that a short pulse can accelerate ions up to the maximum energy of 18MeV per nucleon from cluster-gas targets.20In the case of long pulses the acceleration of helium ions up to 40MeV from underdense plasmas was observed on the VULCAN laser.6However the scaling and optimal conditions were not established.

We study the dependence of ion maximum energy on the target density,thickness,as well as on the focusing and power of the laser pulse in order to optimize the acceleration process.We should note here the maximum proton energy is obtained for slightly overcritical targets and since the laser pulse should be able to establish a channel that goes through the target,the laser should be tightly focused on the front surface of the target to ensure the penetration through the target.We show that it is not only important that the pulse penetrates the target,but also that it does not break into ?laments.The latter will immediately reduce the effective-ness of acceleration.It is therefore necessary to establish matching between the dimensions of the focal spot,the po-sition of the focus relative to the target boundary,and the diameter of the self-focusing channel for each target density and thickness in order to avoid ?lamentation,as was shown in Ref.21.The effectiveness of this mechanism depends on the ef?cient transfer of laser pulse energy into the energy of fast electrons that are accelerated along the propagation channel.Because of this,for each laser target con?guration there exists an optimum target thickness that maximizes the ion energy.We show how the optimum target thickness scales with peak intensity and pulse duration.The energies of protons produced in such interaction by multiterawatt and petawatt class lasers are of the order of several hundreds of MeV ,which is interesting for many applications that require ion beams.1–4

The paper is organized as follows.In Sec.II we present the results of 2D PIC simulations of intense tightly focused laser pulse interaction with near critical density targets of different thickness and discuss the mechanism of proton ac-celeration.The scaling of the optimal target thickness with laser pulse parameters,based on the optimal laser pulse en-

ergy depletion,is presented in Sec.III.We conclude in Sec.IV .

II.THE RESULTS OF 2D PIC SIMULATIONS

In this section we present the results of 2D PIC simula-tions of an intense laser pulse interaction with underdense and near-critical density targets.The simulations were per-formed using the Relativistic Electro Magnetic Particle code.22Space and time are measured in units of laser pulse wavelength,?,and wave period,T =2?c /?,correspond-ingly,?is the laser pulse frequency.The grid mesh spacing is ?/20,and the time step is T /40.The total number of particles in the simulation box is about 5?106.A laser pulse with Gaussian temporal and spatial pro?les is introduced at the left boundary.The pulse duration is ?=30fs or 10??1/e 2?and it is focused to a 1.5??full width at half maxi-mum ?FWHM ??spot size at a distance of 6?from the left boundary ?f /D =1.5?and the power is 1PW ?I ?1023W /cm 2?.It is worth mentioning here that the simu-lations we performed for different f/D show that the optimal acceleration conditions are reached for f /D =1.5?E p ?400TW,2.7n cr ?=480MeV ?,which we use through-out the paper,unless stated otherwise.For f /D =1the pulse immediately goes into ?lamentation reducing the ef?ciency of acceleration,E p ?400TW,2.7n cr ?=300MeV.For f /D =3the peak intensity drops leading to a reduction in maxi-mum proton energy,E p ?400TW,2.7n cr ?=370MeV,meaning that thinner targets should be used.

At intensities of about 1023W /cm 2the effects of radia-tion backreaction can become important and lead to the modi?cation of the laser-plasma interaction,as it was shown in Ref.23.Though we use peak intensities of about 1023W /cm 2in our 2D PIC simulations the effects of radia-tion backreaction are negligible in our case and are not taken into account.It is due to the fact that as it will be shown below the channel in the electron density is formed by the front of the pulse which expels the electron sideways.Thus when the maximum of the pulse arrives there is only a neg-ligible amount of electrons or no electrons at all to interact with.

The target is 40?wide,its left boundary is placed at x =6?.Thus the laser is focused at the front of the target.Focusing the pulse before the target or inside the target leads either to reduced propagation length in plasma or to ?lamen-tation,which are both not bene?cial to ef?cient proton ac-celeration.The target is composed of fully ionized hydrogen.The density is measured in units of the critical density,n cr =m e ?2/4?e 2,m e is the electron mass,and e is the unit charge.We vary the thickness and density of the target since these are two parameters the dependence on which we study.The highest density used in simulations is 16n cr .In view of grid mesh spacing ?/20,it gives ?ve points per plasma wavelength.Most of the simulations were performed at n e =3n cr ,which gives 12points per plasma wavelength.It is enough to resolve the minimum characteristic scale of the problem.

When a tightly focused high-intensity laser pulse inter-acts with a target of near-critical density,it forms a

channel

FIG.1.?Color online ?The principal scheme of the acceleration mechanism.

043105-2Bulanov et al.Phys.Plasmas 17,043105?2010?

in both electron and ion density.The evolution of the laser ?eld inside the channel is shown in Fig.2.The pulse is tightly focused at the front surface of the target ?t =15?,after which it begins to diverge in the plasma and to expel the electrons creating a channel in electron density.The results of these simulations indicate that this process is connected with a swift rise of electron density on the walls of this channel.Soon the laser pulse divergence stops and the pulse propagates inside this channel ?t =35,t =55?,losing en-ergy,which is transformed into the energy of fast electrons,which are mainly accelerated in the transverse direction.At t =75the laser pulse exits the plasma and diverges ?t =90?.Comparing the pulse at t =15and t =90we see that a signi?-cant part of laser energy has been transferred to plasma elec-trons.The formation of the channel in electron density and a stream of accelerated electrons,which exit the plasma behind the pulse,along the laser propagation axis are shown in Figs.3?a ?and 3?b ?correspondingly.We notice here that the den-sity in the stream of accelerated electrons is substantially higher than the electron density in the ambient plasma.In the equilibrium the electron density inside the bunch can be ?e 2times greater than the ion density in the plasma ?e.g.,see Ref.24?.Here ?e is the electron bunch relativistic gamma-factor.

These electrons generate a magnetic ?eld,circulating in-side the channel around its axis.The region,where the mag-netic ?eld is generated,moves behind the pulse.Upon exit-ing the plasma the magnetic ?eld expands and the electron current is dissolved.Some of the electrons leave the target,other return back,helping to sustain the magnetic ?eld on the back of the target ?see Fig.3?b ??.This ?eld displaces the electron component of plasma with regard to the ion compo-nent thus generating a strong quasistatic electric ?eld that can accelerate and collimate ions.In order to illustrate the process of this electric ?eld generation,Fig.4shows the

longitudinal electric and z-component of magnetic ?eld for different moments of time.It can be clearly seen that the growth of the electric ?eld is connected with the expansion of the magnetic ?eld,which has a form of a dipole in 2D ?in 3D it will be a toroidal vortex ?.

This electric ?eld will accelerate and collimate ions from the thin ?lament,which is formed along the laser propaga-tion axis inside the channel in ion density.In Fig.5we present a series of proton density pro?les at different time intervals to illustrate the creation of a channel,the formation of a thin proton ?lament,and the proton acceleration from this ?lament by the longitudinal quasistatic electric ?eld.

In Fig.6?a ?we show a typical spectrum of ions for a 1PW laser pulse interacting with a 50?thick target having a density of 3.0n cr and producing protons with a maximum energy of 1.3GeV .The number of protons with energy above 1GeV is about 4?108.The dependence of ion maximum energy on time,shown in Fig.6?b ?,illustrates the accelera-tion mechanism.The steep rise in maximum energy begins at t =70,which corresponds to the formation of a quasistatic ?eld at the back of the target ?see Fig.4?.By t =120this ?eld is almost dissolved and the acceleration stops.In Fig.6?c ?we present the angular distribution of protons.Two large side peaks at ?=?/2correspond to the protons pushed out from the channel by the laser pulse in the transverse direction.A narrow peak ???=6°at FWHM ?at ?=0represents the pro-tons accelerated along the laser propagation axis.

It is evident that there should be a strong dependence of the acceleration effectiveness on the target thickness and density connected with a variation in energy transfer ef?-ciency from laser pulse to the electrons and protons.This is due to the fact that if the target is very thick the pulse will not penetrate the target and generate an accelerating ?eld,and if the target is too thin then the pulse will transfer a negligible amount of its energy into accelerating ?elds.For low densities,tight focusing of the pulse leads to ?lamenta-tion,causing a rapid drop in acceleration ef?ciency.For high densities the channel radius is reduced,so the generated ac-celerating ?elds are also reduced.Hence,the laser pulse properties should be properly matched to the target condi-tions to ensure optimal conditions for acceleration.By opti-mal conditions we mean that the laser pulse is able to go through the target without ?lamentation,and the laser pulse energy is almost all converted into the energy of fast elec-trons.

That is why we perform a two-parameter scan of

the

FIG.2.?Color online ?The evolution of the laser pulse electric ?eld for a 1PW laser pulse interacting with a 50?thick target with a density of 3.0n cr

.

FIG.3.?Color online ??a ?The formation of a channel in electron density.?b ?The stream of laser accelerated electrons exiting plasma behind the pulse for a 1PW laser pulse interacting with a 50?thick target with density of 3.0n cr .

043105-3Generation of GeV protons from 1PW laser interaction …Phys.Plasmas 17,043105?2010?

maximum proton energy.In Fig.7?a ?we show the results of this scan,i.e.,the dependence of the maximum proton energy on target thickness and density.The parameter space was explored by a large number of simulations to generate the plot.The shape of the surface indicates that for every density there is an optimal target thickness that maximizes the proton energy.Moreover the shape suggests that the optimal target thickness for each value of target density can be determined from the condition n e b L =const,where b is some number.Fit-ting the results of 2D PIC simulations we obtain that b =0.65?see Fig.7?b ?,where the dependence of target thick-ness on target density for optimal proton acceleration is plot-ted along with its ?t by a function f /n e b ,f =442,b =0.65?.We should mention here that the data and the power law ?t are in good agreement for 1.6?n i /n cr ?10,as it can be seen from the inset Fig.7?b ?.The comparison of this scaling with the analytical results and the discussion of the scaling applica-bility range are carried out in the next section.According to the results presented in Fig.7?a ?the maximum proton energy of 1.3GeV in the interaction of a 1PW laser pulse with a target of near-critical density is obtained for a target thick-ness of 50?and density of 3.0n cr .

III.THE OPTIMAL THICKNESS AND ITS SCALING WITH INTERACTION PARAMETERS

As it was demonstrated in the previous section there ex-ists an optimal target thickness that maximizes the acceler-ated proton energy for given laser and target properties.In other words the optimal target thickness corresponds to most ef?cient transformation of laser pulse energy into the

energy

FIG.4.?Color online ?The evolution of the longitudinal electric and the z-component of the magnetic ?elds for a 1PW laser pulse interacting with a 50?thick target with density of 3.0n cr

.

FIG.5.?Color online ?The evolution of ion density for a 1PW laser pulse interacting with a 50?thick target with density of 3.0n cr .

043105-4Bulanov et al.Phys.Plasmas 17,043105?2010?

of protons,which are accelerated by the longitudinal quasi-static electric ?eld.As it was shown in the previous section this ?eld is generated due to the expansion of the magnetic ?eld as it exits the channel.The magnetic ?eld in its turn is connected with the electron current along the axis of the channel.These are the electrons which are accelerated by the laser pulse in the forward direction.The magnetic ?eld,in-duced by the electron current,grows as long as these elec-trons are accelerated.That is why the optimal target thick-ness should be equal to the electron acceleration length.For the targets of such density,as considered in this paper,the acceleration length is determined by the laser depletion length.In other words the target thickness should be equal to laser depletion length.We can make an estimate of the opti-mal target thickness from the condition that all the laser en-ergy ?W p ?is transferred to the energy of electrons ?W e ?,which were initially in the would-be propagation channel:W p =W e ,where W e =?R 2L ch n e am e c 2,R is the radius of the channel,L ch is the channel length,and a is the maximum

value of the laser pulse dimensionless vector-potential in the channel.Here we assumed that the electrons acquire an av-erage energy of am e c 2after being pushed out from the chan-nel in the transverse direction.We also assume that the thick-ness of the target is much larger than the pulse length,so the channel can be established.The energy of the pulse can be estimated as follows:W p =?R 2?a 2m e cn cr .Then

a ?n e n cr L ch

L p

,?1?

where L p =?/c is the length of the laser pulse.If we assume that the diameter of the plasma channel is about R =a 1/2c /?pe /2=??n e /n cr ?1/2a 1/2/2,and express a in terms of

laser pulse energy then n e 2/3

L ch =const.This relation agrees well with the results of 2D PIC simulations,which gives a power of 0.65for density in the scaling ?see Fig.7?b ??.

Let us utilize a simple model to more carefully estimate a constant of proportionality in Eq.?1?.In order to do this we need to calculate with accuracy better then above the energy of the laser pulse inside the self-generated channel.We will also use the results of 2D PIC simulations below to prove the assumption that the average energy of electrons that are ejected from the channel is am e c 2.Let us ?rst calculate the energy of the laser pulse.Since we are considering the inter-action of an intense laser pulse with a plasma of near-critical density,it is plausible to expect that the walls of the self-generated channel will have high density.The laser pulse will be contained inside the channel almost completely since it would not be able to penetrate these high density walls.That is why in order to estimate the energy of the laser pulse inside the channel we can use a well-known result for the behavior of the EM wave inside a waveguide.25In doing so,we neglect the laser pulse energy loss,while the channel is being established.Since,initially,the pulse has no longitudi-nal component of electric ?eld,and it is tightly focused on the front surface of the target,it is reasonable to assume that the mode with lowest transverse frequency will propagate through this self-generated plasma waveguide.This will be an H -wave ?E x =0?with

H x =AJ 1??r ?cos ??t ?kx ?,

?2?

where J 1is the Bessel function of the ?rst kind and ?=1.84/R ;here R is the radius of the waveguide.The trans-verse components of electric and magnetic ?eld are ex-pressed through the longitudinal one according to well-known formulae,

E r =

i ??2c ?H x

?r

,

H r =

??2c ?H x

?r

.

?3?

Since the amplitude of the transverse ?eld inside the wave-guide is E 0=mc ?a /e then the amplitude of the magnetic ?eld in Eq.?2?is A =?2c ?/i ???mc ?a /e ?.The energy of the EM wave traveling in such a waveguide per unit length

is

FIG.6.?a ?The spectrum of protons at t =140.?b ?The dependence of maxi-mum proton energy on time.?c ?Angular distribution of protons,n =3.0n cr ,P =1PW,f /D =1.5,L =50?

.

FIG.7.?a ?The distribution of proton maximum energy in plane ?n e ,L ?;the black dot marks the optimal target for proton acceleration by a 1PW pulse:n e =3.0n cr and L =50?,which gives a maximum energy of 1.3GeV .?b ?The dependence of target thickness on target density corresponding to maximum

accelerated proton energy ?2?and its ?t by a function f /n e b

,f =442,b =0.65,?1?;the log-log plot of L ?n i ?is shown in the inset.

043105-5Generation of GeV protons from 1PW laser interaction …Phys.Plasmas 17,043105?2010?

w =

?2

8??2c 2

?

?H x ?2df

=?R 22

n cr a 2m e c 2??J 1??R ??2?J 0??R ?J 2??R ??.?4?

If we take into account the ?nite duration of the pulse and assume that it is Gaussian,then the integration over time will produce the following result:

???

+??exp ??4t 2

?

2

??2

dt =

?

?8

?.?5?

Then the energy of the laser pulse inside the self-generated plasma waveguide is given by the following formula:

W p =?R 2?a 2m e cn cr K ,

?6?

where K =??/32?J 1??R ?2?J 0??R ?J 2??R ??.Above we as-sumed that the average energy of electrons accelerated by the laser pulse is am e c 2,and the total energy of these electrons is

W e =?R 2

L ch n e am e c 2

.

?7?

Let us test this assumption against the results of 2D PIC simulations.The typical spectrum of electrons is presented in Fig.8.It is obvious that the electron energy does not follow the scaling a 2?=40000,or 20GeV electron energy ?,since the maximum electron energy is equal to 1GeV .Let us es-timate the average energy of accelerated electrons.We as-sume that all the energetic electrons come from the volume occupied by the laser produced channel,and de?ne their number as N ch .Then from the electron spectrum we can determine the threshold energy,E th ,i.e.,the minimum energy that the electron initially from the channel can have

N th =

?

E th

E max

N ?E ?dE .?8?

The average energy of these electrons then can be de?ned as

E ˉ=1N ch

?

E th

E max

EN ?E ?dE .?9?

For the spectrum,shown in Fig.8,the average energy is E ˉ=93MeV.This means that if E ˉ=am e c 2,as we assumed,then a =186.Such value of a coincides well with the value obtained in 2D PIC simulations for the amplitude of vector potential in the propagation channel.This supports the as-sumption that the bulk electrons which are accelerated by the laser gain energy that scales as a .

Then

L ch =a n cr

n e

L p K ,?10?

or

a =

1K n e n cr L ch

L p

.

If we consider a propagation of the electromagnetic pulse in a 2D waveguide then in the above calculation only factor K will change,which is directly connected to the dimension of the problem and exact form of the laser ?eld in the wave-guide.In the 2D case K =1/10?instead of K =1/13.5in 3D ?.For the parameters,used in 2D PIC simulations,the maxi-mum value of the vector potential inside the channel a =200,n e =3.0n cr ,and L p =10?,the optimal target thickness is L ch =50?,which is in good agreement with the prediction of condition ?10?.

The scaling ?10?can be rewritten in terms of laser pulse energy.Taking into account that the radius of the channel is R =?/2?n e /n cr ?1/2a 1/2and that a can be determined from the expression for the energy of the laser pulse,we obtain

n e 2/3L ch

W p

=C ,?11?

where the constant C is

C =?4K 2n cr L p

2??2m e c

2

?

1/3

.

As we can see from the comparison of results of 2D PIC simulation and the simple analytical model,the scaling fails at low and high densities,while it works well for the inter-mediate ones ?1.6?n i /n cr ?10,see Fig.7?b ??.In the case of low densities,n i ?1.6n cr ,the ?lamentation of the laser pulse and Langmuir wave generation due to self-modulation de-crease the optimal target thickness by adding new sources of laser pulse energy depletion not accounted for in the scaling ?11?.In the case of high densities,n i ?10n cr ,other mecha-nisms of ion acceleration come into play and their effective-ness is no longer determined by the depletion of the laser pulse.Moreover the optimal target thickness that follow from the result of 2D PIC simulations for high density targets is no longer much larger than the pulse length,whereas

the

FIG.8.The spectrum of electrons for a 1PW laser pulse interacting with a 50?thick target with density of 3.0n cr .

043105-6Bulanov et al.Phys.Plasmas 17,043105?2010?

condition of the theoretical model applicability is L ch?L p.

In contrast with the results of the present paper the scal-ing of the optimal target thickness for ion acceleration from thin foils is determined from the condition of the relativistic transparency of the foil.This condition stems out of the re-quirement that the electric?eld of the laser is equal to the electric?eld of the ion core,stripped of all the electrons,and which is as follows:26

a=?n e

n cr

l

?,?12?

where l is the foil thickness,and?is the laser wavelength.

This scaling was thoroughly studied in Ref.27in a series of

2D PIC simulations,where the ion acceleration from thin

double-layer foils was considered.The difference in scaling

is due to different targets and thus to different interaction

properties.The effect of relativistic transparency on en-

hanced ion acceleration schemes was studied experimentally

in Ref.28.

The maximum energy of the accelerated protons can be

estimated from the fact that the acceleration itself takes place

in the region which is of the order of channel diameter.The

electric?eld,which accelerated protons,should be of the

order of the magnetic?eld generated by the accelerated elec-

tron bunch,i.e.,E?B?4?en e?e2R e,b,where R e,b ?a1/2c/?pe is the transverse size of the electron bunch.Here we take into attention the known fact that the relativistic

electron bunch density at the equilibrium due to the plasma

lensing effect is in a factor?e2higher than the background plasma density n e.Then the proton energy is given by

E p?e2n e?e2?a1/2c/?pe?2?a?e2m e c2.?13?The coef?cient of proportionality should be determined from PIC simulations.

It is easy to?nd a relationship between the dimension-

less amplitude a of laser beam inside the self-focusing chan-

nel of the radius a1/2c/?pe and the laser power P.It reads a=?8??P/P c??n e/n cr??1/3,?14?where P c=2m e2c5/e2=17GW.For P=1PW and n e/n cr=3, it yields a?102.As we can see,Eq.?13?gives1GeV proton energy for1PW laser if the bulk energy of fast electrons corresponds to?e?7.

IV.CONCLUSIONS

In this paper we studied,using2D PIC simulations,the mechanism of proton acceleration from near-critical density plasma.In this scheme the laser pulse burns through the target generating strong electric and magnetic?elds in the propagation channel.The magnetic?eld begins to expand along the transverse direction upon exiting the channel, pushing the electron component of plasma inside the target and generating the quasistatic electric?eld.This electric?eld accelerates and collimates ions from the thin?lament which is formed in the propagation channel.

The results of simulations indicate the existence of an optimal target thickness that maximizes the energy of accel-erated ions.Since the quasistatic?elds are generated by the electron current the optimal target thickness is connected with the maximum energy transfer from the laser pulse to the plasma electrons.A parameter scan using2D PIC simula-tions con?rmed a scaling law for proton acceleration from the near critical density targets,given by

L ch=a

n cr

n e

L p K,?15a?or

a=

1

K

n e

n cr

L ch

L p

,?15b?or in terms of laser pulse energy,

n e2/3L ch

W p

=constant,?15c?

where K is a numerical factor determined by the laser pulse pro?le and its propagation inside the self-generated plasma channel.This scaling comes from the requirement of optimal laser pulse depletion in plasma and the fact that the laser pulse should be able to burn through the https://www.sodocs.net/doc/1e16038040.html,paring the analytical scaling and the results of2D PIC simulations, we came to a conclusion that the scaling?11?has an appli-cability range of 1.6?n i/n cr?10.For high densities?n i ?10n cr?other mechanisms of ion acceleration come into play and their effectiveness does not depend on the laser pulse energy depletion.Thus the scaling?11?is no longer valid.For low densities?n i?1.6n cr?the scaling fails due to the fact that the?lamentation of the laser pulse and Lang-muir wave generation due to self-modulation decrease the optimal target thickness by adding new sources of laser pulse energy depletion not accounted for.

A parameter scan over different values of target density and thickness allowed us to determine the optimal conditions for proton acceleration by a1PW laser pulse focused to a 1?m focal spot from a near-critical density target:a thick-ness of50?and density of3.0n cr.The resulting protons have the maximum energy of 1.3GeV.This mechanism also works for lower intensities.For example,a100TW laser pulse with intensity of6?1021W/cm2interacting with 2.7n cr dense plasma produces40MeV maximum energy pro-tons.For a500TW laser pulse the maximum proton energy goes up to770MeV.

Let us brie?y mention that the protons accelerated by the studied in this paper mechanism can be of interest for appli-cations in hadron therapy.For a225TW laser pulse which interacts with a2.25n cr,60?plasma slab the maximum pro-ton energy is about300MeV.The number of protons accel-erated to the energy of250MeV with an energy spread of 1%is estimated to be about108.As a result these proton beam parameters make this acceleration regime an interest-ing potential candidate for proton therapy.

ACKNOWLEDGMENTS

This work was supported by the National Science Foun-dation through the Frontiers in Optical and Coherent Ul-trafast Science Center at the University of Michigan and by Grant No.R21CA120262-01from the National Institutes of

043105-7Generation of GeV protons from1PW laser interaction…Phys.Plasmas17,043105?2010?

Health and the International Science and Technology Center ?Project2289?.The authors would like to acknowledge fruit-ful discussions with A.Brantov.

1M.Roth,T.E.Cowan,M.H.Key,S.P.Hatchett,C.Brown,W.Fountain, J.Johnson,D.M.Pennington,R.A.Snavely,S.C.Wilks,K.Yasuike,H. Ruhl,F.Pegoraro,S.V.Bulanov,E.M.Campbell,M.D.Perry,and H. Powell,Phys.Rev.Lett.86,436?2001?;V.Yu.Bychenkov,W.Rozmus, A.Maksimchuk,D.Umstadter,and C.E.Capjack,Plasma Phys.Rep.27, 1017?2001?;A.Macchi,A.Antonicci,S.Atzeni,D.Batani,F.Califano, F.Cornolti,J.J.Honrubia,T.V.Lisseikina,F.Pegoraro,and M.Temporal, Nucl.Fusion43,362?2003?;J.J.Honrubia,J.C.Fernández,M.Tempo-ral,B.M.Hegelich,and J.Meyer-ter-Vehn,Phys.Plasmas16,102701?2009?.

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043105-8Bulanov et al.Phys.Plasmas17,043105?2010?

潜水排污泵优点与缺点

潜水排污泵优点与缺点 潜水排污泵是一种泵与电机连体，并同时潜入液下工作的泵类产品，与一般卧式泵或立式污水泵相比，潜水排污泵明显具有以下几个方面的优点。 1.结构紧凑、占地面积小。潜水排污泵由于潜入液下工作，因此可直接安装于污水池内，无需建造专门的泵房用来安装泵及机，可以节省大量的土地及基建费用。 2.安装维修方便。小型的潜水排污泵可以自由安装，大型的潜水排污泵一般都配有自动藕合装置可以进行自动安装，安装及维修相当方便。 3.连续运转时间长。排污泵由于泵和电机同轴，轴短，转动部件重量轻，因此轴承上承受的载荷(径向)相对较小，寿命比一般泵要长得多。 4.不存在汽蚀破坏及灌引水等问题。特别是后一点给操作人员带来了很大的方便。 5.振动噪声小，电机温升低，对环境无污染。 正是由于上述优点，排污泵已越来越受到人们的重视，使用的范围也越来越广，由原来的单纯地用来输送清水到现在的可以输送各种生活污水、工业废水、建筑工地排水、液状饲料等等。 在市政工程、工业、医院、建筑、饭店、水利建设等各行各业中起着十分重要的作用。 但是任何事物都是一分为二的，对于排污泵来说最关键的问题是可靠性问题，因为排污泵的使用场合是在液下;输送的介质是一些含有固体物料的混合液体;泵与电机靠得很近;泵为立式布置，转动部件重量与叶轮承受水压力同向。这些问题都使得排污泵在密封、电机承载能力、轴承布置及选用等方面的要求比一般的污水泵要高。 为了提高排污泵的寿命，现在国内外大部分厂家都在泵的保护系统上想办法，即在泵发生泄漏、过载、超温等故障时能进行自动报警，并自动停机备修。可是我们认为，在排污泵中设

置保护系统很有必要的，它能有效地保护电泵的安全运行。 但这并不是问题的关键，保护系统只不过是在泵发生故障后的一种补救办法，是一种比较被动的办法。问题的关键应该是从根本着手，彻底解决泵在密封、过载等方面的问题，这才是一种较为主动的办法。为此我们把副叶轮流体动力密封技术及泵的无过载设计技术应用于潜水排污泵中来，较大提高了泵密封可靠性和承载能力，延长了泵的使用寿命。 为了提高潜水排污泵的寿命,国内外大部分厂家都在泵的保护系统上想办法,即在泵发生泄漏、过载、超温等故障时能进行自动报警,并自动停机备修。 可是我们认为,在排污泵中设置保护系统很有必要的,它能有效地保护电泵的安全运行。但这并不是问题的关键,保护系统只不过是在泵发生故障后的一种补救办法,是一种比较被动的办法。问题的关键应该是从根本着手,彻底解决泵在密封、过载等方面的问题,这才是一种较为主动的办法。 为此我们把副叶轮流体动力密封技术及泵的无过载设计技术应用于潜水排污泵中来,较大提高了泵密封可靠性和承载能力,延长了泵的使用寿命。 当泵工作时,副叶轮随泵主轴一起旋转,副叶轮中的液体也会一起旋转,转动的液体会产生一个向外的离心力,这个离心力一方面顶住流向机械密封处的液体,降低了机械密封处的压力。 另一方面阻止介质中的固体颗粒进入机械密封的摩擦副中,减少机械密封磨块的磨损,延长了其使用寿命。

The way常见用法

The way 的用法 Ⅰ常见用法： 1)the way+ that 2)the way + in which（最为正式的用法） 3)the way + 省略（最为自然的用法） 举例：I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法： 在当代美国英语中，the way用作为副词的对格，“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2）The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3）The way =how/ how much No one can imagine the way he missed her. 4）The way =because

文艺心理学第三章练习题

文艺心理学第三章练习题

《文艺心理学》第三章练习题答案 一、名词解释 1、审美相似律 审美相似律是以存在于原始思维中的那种广义相似性原理为基础的。它是艺术品外在形式生成的一条基本规律，也是人在审美活动中所遵循的独特心理规律，基本上是一种无意识的存在。它是使人将本来与人无关的外在事物变为主观存在，再由主观存在升华为新的客观存在的心理规律，标示着艺术创作主体与艺术形式之间的内在联系。 2、形象范式 是掩藏于现象形态之后，因而不易在创作与阅读中发现的一种艺术范式形态。形象范式是艺术的特殊存在方式和艺术形象特殊的内在结构，能体现出艺术独具的内在性质。形象范式既具有表象的抽象性，又具有情感性、充盈性和个异性。 3、显动机 创作显动机是从事创作的直接心理驱力。生活中，艺术家因各种物象、事件的触发，常发生心理波动，造成失衡，并引发适当强度的情感。宣

泄情感，以恢复心理平衡，便是显动机的主要内容。 4、动机冲突 指一个动机簇内各种子动机的矛盾纠葛。 5、内在形式 又叫审美意象，它是艺术家的心理体验与艺术品的外在形式之间的一个中间环节。是指艺术创作或艺术欣赏以及其他审美活动中主体脑海里活跃着的包含丰富意蕴的形象。 5．同形性：这是格式塔心理学的基本观点，又称为异质同构论。这一理论认为在知觉活动中，在作为对象的物理现象与作为认知主体的人的大脑生理现象之间存在着某种同形关系。 6、潜动机 指艺术家从事创作时内心的某种无意识驱动力量。潜动机的主要特点是驱动性和潜在性。 7、内觉体验 “内觉”是美国心理学家阿瑞提的术语，指的是那些不能用形象、语词、思维和任何动作表达出来的“无定形认识”，亦即非语言的、无意识的或前意识的认识。内觉是一种深层心理状态，不

惠斯通电桥实验报告南昌大学

南昌大学物理实验报告 课程名称：_____________ 大学物理实验 实验名称：_______________ 惠斯通电桥 学院：___________ 专业班级： 学生姓名：_________ 学号： 实验地点：___________ 座位号： 实验时间：第11周星期4上午10点开始

、实验目的: 1. 掌握电桥测电阻的原理和方法 2. 了解减小测电阻误差的一般方法 、实验原理: (1) 惠斯通电桥原理 惠斯通电桥就是一种直流单臂电桥，适用于测中值电阻，其原理电路如图 7-4所示。若调节电阻到合适阻值时, 可使检流计 G 中无电流流过，即 B 、D 两点的电位相等，这时称为“电桥平衡”。电桥平衡，检流计中无电流通过， 相当于无BD 这一支路，故电源 E 与电阻R ,、R x 可看成一分压电路；电源和电阻 R 1 上面两式可得 R 2 桥达到平衡。故常将 R 、R 2所在桥臂叫做比例 臂,与R x 、R S 相应的桥臂分别叫做测量臂和比 较臂。 V B C 点为参考，贝y D 点的电位V D 与B 点的电位V B 分别为 R 2 R S R S V D R X 因电桥平V B V D 故解 R 2、R S 可看成另一分压电路。若以 R x 为 E 待测电阻，则有 R>< R X R S 上式叫做电桥的平衡条件，它说明电桥平衡时，四个臂的阻值间成比例关系。如果 1 10,10 1等)并固定不变，然后调节 金使电

(2)电桥的灵敏度

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系数比15口径的电动球阀小很多。 潜水泵的驱动是通过电磁线圈，比较容易被电压冲击损坏。相当于开关的作用，就是开和关2个作用。 污水泵的驱动一般是用电机，比较耐电压冲击。潜水泵是快开和快关的，一般用在小流量和小压力。 污水泵反之，污水泵阀的开度可以控制，状态有开、关、半开半关，可以控制管道中介质的流量，而潜水泵达不到这个要求。潜水泵一般断电可以复位，污水泵要这样的功能需要加复位装置。 潜水泵是代表该水泵可以潜在水里运行，好多潜水泵都是适用于清洁水；清洁的潜水泵不适用于污水。

污水泵叶轮一般是闭式的，主要适用于含杂质的水。开式叶轮的污水泵一般用于带颗粒的或者含纤维布条的水，脏的东西不容易卡住。污水泵也有不能放在水里的，比如卧室的污水泵。 潜水泵是指泵的安装形式，电机和泵体都装在水下。可以做污水泵，热水泵也可以做清水泵或者别的用途。污水泵看名字就知道，就是打污水的，可以用潜水泵，也可以用别的形式的泵。 以上就是潜水泵和污水泵的差别所在，希望能对大家有一定的帮助。想要了解更多有关潜水泵和污水泵的资讯，或者购买潜水泵或者污水泵，可以电话联系河北京潜泵业有限公司。

The way的用法及其含义(一)

The way的用法及其含义（一） 有这样一个句子：In 1770 the room was completed the way she wanted. 1770年，这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么？其意义如何？在阅读时，学生经常会碰到一些含有the way 的句子，如：No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中，the way之后的部分都是定语从句。第一句的意思是，“没人知道他是怎样发明这台机器的。”the way的意思相当于how；第二句的意思是，“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中，the way也是as的含义。随着现代英语的发展，the way的用法已越来越普遍了。下面，我们从the way的语法作用和意义等方面做一考查和分析： 一、the way作先行词，后接定语从句 以下3种表达都是正确的。例如：“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如：“火灾如何发生的，有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.