Proca equation for laser pulses interaction with matter

Transcription

1 Proca euaion or laser pulses ineracion wi maer Janina Marciak- Kozłowska Mirosław Kozłowski* Insiue o Elecron Tecnology Al. Loników 3/ Warsaw Polan *Corresponing auor

2 Absrac In is paper e ineracion o ulrasor laser pulses wi maer is invesigae. Te scaering an poenial moion o ea carriers as well as e eernal orce are consiere. I is sown a e ea ranspor is escribe by e Proca euaion. For ermal Heisenberg ype relaion V τ ~ τ is e relaaion ime an V is e poenial e soluion o e Proca euaion PR are e isorionless ampe wave euaion. Key wors: Ulrasor laser pulses; Quanum ea ranspor euaion; Proca ermal euaion.

3 Inroucion Dynamical processes are commonly invesigae using laser pump-probe eperimens wi a pump pulse eciing e sysem o ineres an a secon probe pulse racking is emporal evoluion. As e ime resoluion aainable in suc eperimens epens on e emporal einiion o e laser pulse pulse compression o e aosecon omain is a recen promising evelopmen. Aer e sanars o ime an space were eine e laws o classical pysics relaing suc parameers as isance ime velociy emperaure are assume o be inepenen o accuracy wi wic ese parameers can be measure. I soul be noe a is assumpion oes no ener eplicily ino e ormulaion o classical pysics. I implies a ogeer wi e assumpion o e eisence o an objec an really inepenenly o any measuremens in classical pysics i was acily assume a ere was a possibiliy o an unlimie increase in e accuracy o measuremens. Bearing in min e aomiciy o ime i.e. consiering e smalles ime perio e Planck ime e above saemen is obviously no rue. Aosecon laser pulses we are a e limi o laser ime resoluion. Wi aosecon laser pulses belong o a new Nano-Worl were size becomes comparable o aomic imensions were ranspor penomena ollow ieren laws rom a in e macro worl. Te irs sage o miniaurizaion rom -3 m o -6 m is over an e new one rom -6 o -9 jus beginning. Te Nano-Worl is a uanum worl wi all e preicable an non-preicable ye eaures. In is paper we evelop an solve e uanum relaivisic ea ranspor euaion or Nano-Worl ranspor penomena were eernal orces eis. Tis is e generalisaion o e resuls o paper [] in wic e uanum relaivisic yperbolic euaion was propose an solve. In is paper we escribe e Proca ermal euaion or an ulra sor laser pulse ineracion wi maer. A soluion o e Proca euaion is obaine or e Caucy bounary coniions. Proca ermal euaion Elecromagneic penomena in vacuum are caracerise by wo ree imensional vecor iels e elecric an magneic iels E an B wic are subjec o Mawell s euaions an wic can also be oug o as e classical limi o a uanum mecanical escripion in erms o poons. Te poon mass is orinarily assume o be eacly zero in Mawell s elecromagneic iel eory wic is base on gauge invariance. Mawell s euaions mus be replace by e Proca euaion [5 6] or a poon wi a res mass. Te Proca euaion reuces o in ree space or a vecor elecromagneic poenial o A µ.. µ Aµ = c = Tis is essenially e Klein Goron euaion or massive poons. Te parameer µ can be inerpree as e poon res mass m wi. µ m =. c 3

4 I is uie ineresing a e Proca ype euaion can be obaine or ermal penomena. In e ollowing saring wi e yperbolic ea iusion euaion e Proca euaion or ermal processes will be evelope an solve. In paper [] e relaivisic yperbolic ranspor euaion was evelope: T m T = T. 3 In euaion 3 is e velociy o ea waves m is e mass o ea carrier an - e Lorenz acor =. As was sown in paper [] e ea energy eaon emperaure T can be eine as ollows: c Consiering a e ermal wave velociy euals [] T = m. 4 = αc 5 were α is e coupling consan or e ineracions wic generae e ermal wave α = /37 an α =.5 or elecromagneic an srong orces respecively Te eaon emperaure is eual o m α c T =. 6 α Base on euaion 6 one conclues a e eaon emperaure is a linear uncion o e mass m o e ea carrier. I is ineresing o observe a e proporionaliy o T an e ea carrier mass m was observe or e irs ime in ulraig energy eavy ion reacions measure a CERN []. In paper [] i was sown a e emperaure o pions kaons an proons prouce in PbPb SS reacions are proporional o e mass o paricles. Recenly a e Rueror Appleon Laboraory RAL e VULCAN laser was use o prouce e elemenary paricles: elecrons an pions [3]. In e presen paper e orce relaivisic ea ranspor euaion will be suie an solve. In paper [4] e ampe ermal wave euaion was evelope: T m T Vm T T =. 7 Te relaivisic generalizaion o euaion 7 is uie obvious: m Vm T T T T =. 8 I is worwile noing a in orer o obain a non-relaivisic euaion we pu =. Wen an eernal orce is presen F e orce ampe ea ranspor is obaine insea o euaion 8 in e one imensional case: T m T Vm T T = F. 9 4

5 Te yperbolic relaivisic uanum ea ranspor euaion escribes e orce moion m T o ea carriers wic unergo scaering erm an are inluence by e poenial Vm erm T. Euaion can be wrien as Vm = m T T. = F We seek e soluion o euaion in e orm τ T = e u were τ = / mv is e relaaion ime. Aer subsiuing euaion in euaion we obain a new euaion an τ u = e F Vm mv = 3 m = m 4 In ree space i.e. wen F euaion 3 reuces o u = 5 wic is essenially e ree Proca euaion compare euaion. Te Proca euaion escribes e ineracion o e laser pulse wi e maer. As was sown in paper[ ] e uanisaion o e emperaure iel leas o e eaons uana o ermal energy wi a mass m = [ ] were τ is e relaaion ime an is e inie velociy τ or ea propagaion. For i. e. or c m. I can be conclue a in nonrelaivisic approimaion c = ininie e Proca euaion is e iusion euaion or mass less poons an eaons. 3. Soluion o e Proca ermal euaion For e iniial Caucy coniion: u = u = g 6 e soluion o e Proca euaion as e orm or > [7] 5

6 6 u = 7 [ ] [ ] [ ]. ' ' ' ' ' J G J J g were F e G τ =. Wen < soluion o Proca euaion as e orm: u = 8 [ ] [ ] [ ] '. ' ' ' ' I G I I g Wen = euaion 4 is e orce ermal euaion. G u u = 9 On e oer an one can say a euaion is e isorion less yperbolic euaion. Te coniion = can be rewrien as:. 8 = τ V Te euaion is e analogous o e Heisenberg uncerainy relaions. Consiering euaion 4 euaion can be wrien as: 8 T V =. T V < I can be sae a isorion-less waves can be generae only i T > V. For T < V i.e. wen e Heisenberg rule is broken e sape o e ermal waves is cange.

7 3 Conclusions In is paper we evelope e relaivisic ermal ranspor euaion or an aosecon laser pulse ineracion wi maer. I is sown a e euaion obaine is e Proca euaion well known in relaivisic elecroynamics or massive poons. As e eaons are massive paricles e analogy is well oune. Consiering a or an aosecon laser pulse e ampe erm in E. ens o e ranspor penomena are well escribe by e Proca euaion. 7

8 Reerences [] J. Marciak-Kozłowska M. Kozłowski Lasers in Engineering p. 59. [] I. G. Bearen e al. Pys. Rev. Le p. 8. [3] K. W. D. Leingam an P. A. Norreys Conemporary Pysics p [4] M. Kozłowski J. Marciak-Kozłowska Lasers in Engineering p.. [5] A.Proca Comp. Ren p [6] Liang Ceng Tu e al. Rep. Prog. Pys p. 77 [7] E. Zauerer Parial Dierenial Euaion o Applie Maemaics Secon Eiion Wiley