L4LP-93-3~c. I

INERTIALCONFINEMENTFUSIONTECHNICAL REV IEW

w

LOS ALAMOSJanuary 1993

Volume 2

ABOUT THIS REPORTThis official electronic version was created by scanning the best available paper or microfiche copy of the original report at a 300 dpi resolution. Original color illustrations appear as black and white images.

For additional information or comments, contact: Library Without Walls Project Los Alamos National Laboratory Research LibraryLos Alamos, NM 87544 Phone: (505)667-4448 E-mail: lwwp@lanl.gov

Front CoverSimulated GXl Test images forsymmetry legendre analysis.

Policy on ES&H Implicit in all of our work is a respectfor the heaIth and safety of ourselvesand those around us and an appreciationfor the need to protect the environmentin which we are privileged to work.

r

CONTENTS:

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Volume 2

Neutron Time-of-Flight Ion TemperatureDiagnostic for Inertial Confinement FusionExperiments

. . . . . . . Page 1 ,

Radiation Field Effects on the SpectroscopicProperties of Cl and Fe Seeded Plasmas

. . . . . . . Page 12

Amplified Spontaneous EmissionProduced by Large KrF Amplifiers

. . . . . . . Page 32

Low Threshold Virtual Cathode Formationin a Large Area Electron Beam on anKrF Laser

. . . . . . . Page 58

For further informationon this subject contact:

R. E. Chrien

Los AlamosNational LaboratoryLos Alamos, NM 87545

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Neutron Time-of-FlightIonTemperatureDiagnostic forInertial Confinement FusionExperiments

I. Introduction

Neutron time-of-flight (TOF) detectors canprovide important information about the fuel-ionburn temperature in various inertial confinementfusion (ICF) target designs. Conventional current-mode neutron TOF detectors measure the timehistory of the total light output from many neutroninteractions in a scintillator. These detectors areuseful for neutron yields above 1010,but are limitedat lower yields by the finite emission time of thescintillator, the finite response time of the detector,and the statistics of neutron scattering and lightproduction in the scintillator.1

We are constructing an ion temperaturediagnostic based on individual timing measure-ments of single neutron interactions in manyscintillators. This diagnostic is designed for low-y-ieldtargets on the Nova ICF laser facility atLivermore. The diagnostic measures the neutronarrival time distribution using an array of 960scintillator-photomultiplier detectors with aboutl-ns time resolution and operated in the single-hitmode.zThe arrival time distribution is constructedfrom the results of 100 or more detector measure-ments. The diagnostic will be located outside theNova target chamber at a distance of about 28 mfrom the target.

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The ion temperature is determined from thespread in neutron energy as denoted by the rela-tionsA.Ell(keV)= C~~Ti(keV)llz‘hc~AE,lis the energyspread (FWHM) and C~~= 82.5 (for d-t reactionsthe coefficient is C~f= 176). The energy spread isrelated to the time spread by At/t~ - AE1l/EJl.Thequantities of interest are summarized in Table I.Corrections to the raw data must account for thesystem time resolution (about 1 ns) and target burntime (about 100 ps).

TableI.Characteristicsofc1-dandd-tneutron

times-of-flightforNova

Fusionreaction dd d-t

Inversespeed(ns/m) 46.1 19.2

Distance(m) 28 28

Time-of-flight(ns) 1291 538

AtforTj = 1 keV(ns) 21.7 3,4

1 ns distance(cm) 2.2 5.2

The neutron arrival times are detected byusing a photornultiplier tube (PMT) to observe thephotons produced by recoil protons in a plasticscintillator. The recoil proton energy is dependenton the proton recoil angle (l through the kinematicreIation Ep= EHcOs2e, where e is measured withrespect to the incident neutron direction. Since n-pscattering is isotropic in the center-of-mass frame,the proton recoil energy distribution is uniform upto the neutron energy. The pulse height containsno information about the incident neutron energy,but can be useful for monitoring the system gain orin some pulse pile-up rejection methods.

The dynamic range of the diagnostic coversdd neutron yields from 5 x 107to 109. This range

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can be obtained through variation of the number ofarray channel hits (from 100 to 500) and scintillatorvolume (from 0.8 cmqdown to 0.2 cmq. Operationat yields above 109can provide over-lap with re-sults obtained from current-mode time-of-flightdetectors] and below 5 x 107with estimates of Tiobtained from first-hit analysisl of data from theLarge Neutron Scintillator Array (LaNSA) onNova.

Another way to extend the dynamic range isby operating the array in the multiple hit regime.An acceptable fraction (10%) of the channels willprovide single hit data even when the averagenumber of hits per channel is three. Thus a dy-namic range of about 30 is possible provided thatchannels affected by pulse pile-up can be rejected.Data reduction based on first-hit analysisAcan alsobe considered.

Thedetecto~amenclosedina cylindricalsteelchamber(Fig.1)whichpmvidesmagneticshieldingforthePMTs.Atotalof 1020PMTs,includingspares,amsupportedbyfomandbackplanesmadeofblacknylon.AspringbehindeachPMTbasep~ eachPMTagainstitssdntillatorAsiliconepadisusedtoimpmvetheopticalcoupling.Gammasoums axeincludedforinsitucalibra-tion. The chamber is supportedbya standwhicha.Uowsittoberotatedforalignmentormaintenance.

TheelectronicsforeachchanneI(Fig.2)consistofthePMTvoltagedividerbase,a ckrimimto~ a multiplehittimetc@#alconverter (llX), andagatedchargesensitiveanalog-to-digitalconverter(AK). ThemultiplehitcapabilityoftheTDCisvaluableforobservingthegammasfromneutroninteractionswithmaterialsalongtheneutronflightpathandthustocormctfortimingdiffemncesbetweenchannels.~ mnbevaluableforrecodingthepulseheightdktributionoftherecoilprotonscintillationstomonitortheoveraIlhealthofthePMTarray

Fig. 1. Assembly drnzuingof the photomu!tiplitr tubechmber, showing thescintillators, siliconecoupling pads, PMTs,bases, mzdcnblesfor thePMT orrny.

Fig. 2. Schenmtie ofelectronics for a singledetector clmwcl.

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Fore Back ChamberPlane Plane /

4.25’

r

FI i

7uHV— 1879 —

TDC

CF– Discrim-

inator

IADC1 -1

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II. Scintillator

The scintillator volume is chosen to giveneutron sensitivity appropriate to the yield pre-dicted for a particular Nova shot. In all cases thescintillator thickness should be restricted to about1 cm to keep the d-d neutron transit time across thescintillator less than 0.5 ns (0.2 ns for d-t neutrons).With the above restriction, the number of neutroninteractions in each scintillator is given by N = Y(A/4nRz) (x/h)where Y is the neutron yield, A isthe detector area, R is the distance from the Novatarget to the scintillator (28 m), x is the detectorthickness, and L is the neutron mean free path forelastic scattering (7.7 cm). For the minimum neu-tron yield of 5 x 107and a minimum neutron num-ber of 0.1 (to obtain 96 measurements), thescintillator volume Ax is 0.8 cms. For the maxi-mum neutron yield of 1 x 109and a maximumneutron number of 0.5, the scintillator volumedrops to 0.2 cms. If the array is operated at a neu-tron number of 3, the dynamic range is increasedsix-fold at a given scintillator volume.

The scintillator itself should be relatively fastand bright, similar to BC400 or NE102. The lightoutput of this scintillator is about 649Z0of anthra-cene. The light production in NE102 for electronenergy deposition is 0.01 photons/eV.S The relativeproduction efficiency for protons in NE102 hasbeen measured’ and can be expressed as TE=0.95TP- 8.0{1 - exp(-0.10T#”90)}whereTp is the pro-ton energy (in MeV) and TEis the equivalent elec-tron energy (in MeV). The light production depen-dence on proton energy is illustrated in Fig. 3. Thenumber of photons varies from 110 for a 0.25 MeVrecoil proton up to a maximum of 7433 for a full-energy recoil. For comparison, a full-energy recoilfrom d-t neutrons produces 80,300 photons. A

6

Fig. 3. Light productionby protons in BC400plastic scintilkztoras afunction of proton energy

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I I 1 I I I I II I I I I I I I I I I I I I I I [ I , 450000–

10000-5000

[-’

/

1000I

ProtonEnergy(MeV)

useful ampIitude calibration can be provided by0.662 MeV gammas from ISPCS,which produce aCompton edge at two-thirds of the recoil protonedge from d-dneutrons.

Light coupling efficiency can be estimated bycomparing the pulse height spectra from neutronor gamma interactions in a scintillator with thesingle photoelectron spectrum from the PMT. Vari-ous coupling methods were evaluated using Ti02-painted l-cm cylindrical scintillators and a 3-cmend-viewing PMT. Coupling efficiency of abouttwo-thirds is obtained when using optical couplinggrease between the scintillator and the photocath-ode together with titanium dioxide paint on theuncoupled surfaces. However, the use of opticalcoupling grease is prohibited by the need to changethe scintillators between Nova shots. Instead,silicone coupling pads can be employed. Thesecoupling pads permit coupling efficiency close tothat of coupling grease (about 50%). Bare couplingefficiency is around one-third.

III. Photomultiplier Tube

The PMT converts scintillation photons intoelectrons and amplifies them in the electron multi-plier. Bialkali photocathodes provide a good

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match for plastic scintillator emission spectrahave quantum efficiency of about 25%. Thus

and10-5

photoelectrons can be expected from a 0.25 MeVrecoil proton. The multiplier structure should be alinear focused type to obtain good time resolution.Inexpensive side-viewing PMTs are unsuitablebecause of inefficient light coupling betweenscintillators and the recessed photocathode.

Time resolution of PMTs is quoted in termsof the single-photoelectron transit time spread(SPTTS). These SPTTS values are 2-3 ns for PMTsof the type considered. The time resolution im-proves inversely as the square root of the numberof photoelectrons compared with the SPTTS. Thisdependence indicates the importance of efficientlight coupling. An SPTTS of 3 ns or better isneeded to obtain time resolution of 1 ns for d-dmeasurements with as few as 10 photoelectrons.Better time resolution is obtained with brighter d-tscintillations.

The anode output must be compatible withthe discriminator threshold. Assuming a minimumof 10 photoelectrons, a gain of 106,and a triangularanode pulse with FWHM of 5 ns, the peak anodecurrent is 320 pA. This represents 16 mV into a50 Q load, which provides a reasonable minimumvoltage for the discriminators. In addition, thePMT output should remain linear up to the recoilproton edge (about 20 rnA for d-dneutrons).

A variety of small 10-stage end-viewingPMTs were considered for this diagnostic. ThePhilips XP1911 19 mm tube was selected, primarilybased on cost. This PMT provides the requiredgain (at a nominal voltage of 1600 V) and linearcurrent (up to 130 mA).

The time resolution was measured by com-paring the anode pulses from two XP1911 PMTs

Fig. 4. Schcnmticofelectronics used fornwwring single-pho~oclectronfransit timespread by k twophoto nndtiplicl-tubemethod.

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viewing the same scintillator (Fig. 4). Neutraldensity filters between the scintiIIator and eachPMT provide an attenuation of 200, ensuring thateach scintillation produces at most a single photo-electron pulse. Ortec VT120C linear preamplifiersprovide an amplification of 10 for each anode sig-nal. A 20 ns coax Iine was used to delay one of thesignals. Ortec 473A constant-fraction discrimina-tors and an Ortec 567 13me-to-AmplitudeCon-verter were used to obtain the raw timing spectrumshown in Fig. 5. The time spread Atcan be ex-pressed as Af2= At;+ At;+ At~inwhere AtA and At~refer to the SPTl_’Sof the two PMTs and M~Ci~refersto the F’WHMemission time of the scintiIIator(1.3 ns). For identical A and B channeIs, the SPTTSof each tube is 3.1 ns. Measurements with reducedneutral density filtering confirm the expected im-provement of time resolution with light level.

n

+=

PMT

$amp ND2.3 $ampCFD “ TAC CFD

IrPHAVolurne2

Fig. 5. Transit timespectrum obtained with thetwo pho~omultiplier~ubemethod using XP1911tubes and a BC418scintillator. Thecalibration fflctor is 155clmnnelsper ns. Thefullwidth at half maximumcorresponds to 4.5 ns.

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1’ ‘ ‘ [ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ 1‘ ‘ ‘ ‘ ‘ ‘i

1500 2000 2500 3000 3500 4000 4500Channel

The PMT voltage divider is conservativelydesigned for bleeder current equal to the averageanode current rating of the tube. The manufactur-er’s recommended string for maximum linearcurrent output is employed. Capacitive stabiliza-tion of the latter dynode stages is provided to limitvoltage changes to less than IYofor 10 full-energyproton recoil scintillations. Positive hi h voltage is

%su plied to the anode to permit the cat ode region!to e grounded. Anode output is capacitively

coupled to the discriminator.

Iv. Discriminator, ADC, TDC, and DataAcquisition

The anode signals are connected to novelconstant-fraction-like discriminators based on asimple RC network and a fast voltage comparator.TNo walk correction is required with this discrimi-nator. An updating output of 10 ns minimumduration is provided to allow pulse pile-up to bedetected by the TDC. This discriminator has lowpower dissipation and can be made quite compact.

The discriminator provides a buffered ver-sion of the anode signal for connecting to a LeCroy1885F 15-bit charge ADC. This is a 96-channel -

TechnicalReview

FASTBUS module. The 1885F can be gated for atime as short as 50 ns and can be used to measurethe amplitude of the d-d neutron pulses with asensitivity of 50 fc per count.

The discriminator output is connected toLeCroy 1879 96-channel multihit TDC FASTBUSmodules. These are 2 ns TDCS with double-edgeresolution of 10 ns. Up to 16 edge timing measure-ments per channel can be obtained. For d-t mea-surements, the TDCS will be upgraded to the 0.5-ns 1LeCroy 1877 when that product becomes available.

1

A DEC MicroVax 11computer is used tocontrol and read out the FASTBUS modules and toanalyze the data. A CAMAC-based Starburst LSI-11 mediates the control functions through aCAMAC-FASTBUS interface. The software neededfor this diagnostic has already been developed forthe LaNSA array.

v. Testing

The two general areas of testing are ampli-tude and timing tests. The overall gain of thescintillator-PMT system must be monitored rou-tinely,perhaps prior to each Nova shot, and thePMT voltage or discriminator threshold adjusted tocorrect for gain shifts or changes in PMT couplingefficiency. Small ISTCSradioactive sources are usedfor this calibration. Timing tests will be performedusing a l-ns, blue, unfocused CRT light pulserdeveloped at EG&G.

Once in operation, correct system perfor-mance will be checked by monitoring the recordeddata. Timing can be routinely checked by observ-ing x-ray generating targets or by using the gammafiducial provided by neutron integrationsin thetarget. Spare channels are included in the array.

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PMT failures can be discovered by detecting achange in the divider string current, the appear-ance of excessive noise, or by the absence of signalin a particular channel for a series of shots. Dis-criminator and TDC problems will be detected bycomparing the average detection probability ineach channel with the average detection probabilityin the whole array. An overall alignment check canbe performed by computing the average arrivaltime as a function of two orthogonal dimensionsacross the detector array. Individual channel tim-ing variations can also be detected by comparisonwith the average arrival time.

Iv. Acknowledgement

This work was done in collaboration withDavid F. Simmons and Dale L. Holmberg, EG&GEnergy Measurements, Las Vegas, Nevada.

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12

J. AbdalIah, Jr.,

R. E. H. Clark

Radiation Field Effects on theSpectroscopicProperties of

For further informationon this subject contact: Cl and Fe Seeded Plasmas

I. Introduction

Comparisonsof theoretical spectral simula-tions and experimental observations often provideuseful information concerning the state of a Labora-tory plasma. The electron temperature and densitycan be estimated from various spectral features, i.e.,Iine ratios, Iine widths, background continuum,comparison with synthetic spectra, etc. These tech-niques are useful for a variety of applications in-cluding magnetic fusion, inertial confinementfusion, opacity determination, x-ray lasers, andastrophysics. These methods are discussed atlength in several articles.1+

The calculation of accurate ionization bal-ance and level populations is of fundamental im-portance to any spectral simulation. Populationscan be calculated using a variety of methods whosevalidity depends upon p~asmaconditions. CoronaIequilibrium, in which ions exist in ground states

Los Alamosonly, is valid at low electron densities depending

National Laboratoryon the elements Local thermodynamic equilibrium

Los Alamos, NM 87545 (LTE) is valid at high electron densities, or whenradiation is in equilibrium with thermal electrons.Collisional-radiative steady-state which includeexcited states explicitly bridge the gap betweencoronal equilibrium and LTE. Time dependentkinetics is required for transient plasmas. Thecalculation of level populations is also directly

Technical dependent on the energy levels and cross sections

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13

that are used to describe the atomic processeswhich occur. These control the rate of populationand depopulation of individual atomic levels.

In principle, the coronal and collisional-radiative treatments should include radiation ef-fects when fields are present. Photon inducedprocesses should be accounted for in the rate equa-tions. However, usually only electron collisionsand spontaneous process are considered. Theradiation field introduces a competition betweenelectrons and photons for atomic excitation andionization processes. The competition varies as afunction of electron density, electron temperature,and the applied radiation field strength.

The purpose of the current work is to assessthe effect of an intense background radiation fieldon the spectroscopic properties of a plasma as afunction of electron temperature, density, andradiation temperature. A steady-state kineticsapproach is used. Plasma conditions are chosensuch that K-shell emissions from Cl and Fe ionswith 1-3 bound electrons are prevalent. The radia-tion field is taken to be black body with tempera-tures of several hundred electron volts.

The effect of radiation on various line ratioswill be studied. Line ratios are intensity ratios ofone line to another as a function of electron densityand temperature. Line ratios are useful becausethey can provide a rough diagnostic of plasmaconditions without a complete analysis of a compli-cated spectrum. They also provide insight into thevarious processes which drive transitions.

The computational method will be discussedin Sec. II, results will be discussed in Sec. III, andconclusions will be presented in Sec. IV.

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II. Computational Method

The kinetic approach used here has beenapplied previously.7 The net production rate oflevel p of ion stage i is given by

(1)

where NiPis the number density of ions in levelp of ionicity i (i = 1 represents the neutral atom,i = 2 represents the singly charged species, etc.), ~istime, {Nj~}represents the set of all levels that areinvolved in reactions with level ip, ZVcis the elec-tron density, T~is the temperature which describesthe electron energy distribution through theMaxwellian function, and Tris the radiation tem-perature that describes the photon energy distribu-tion through the I?lanckianblack body function.Electron and photon distributions are restricted toMaxwellians and Planckians, respectively, for thepurpose of this paper. The function F has the form

FiP= ~ NjqQajqip - ~ NiqQaipjq , (3ajq ajq

where the first sum contains the contributions fromall processes which populate level ip, and the sec-ond sum contains the contribution from all pro-cesses which depopulate level ip. Qaipjq representsthe rate coefficient for process a which act on anion in level ip and leaves it in a final leveljq. Therate coefficient is essentially the integrated crosssection for a process weighted by the appropriateelectron or photon distribution function. The Q’s.include the appropriate factors of Nefor processesinitiated by electron collisions. They also dependon Tefor processes initiated by electron collisions,Trfor processes initiated by photon collisions, andon both Teand Trfor processes like stimulated

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radiative recombination which involve both pho-tons and free electrons. Rate coefficients are inde-pendent of distribution functions for spontaneousprocesses such as autoionization and spontaneousemission. Other processes, such as ion-ion colli-sions are beyond the scope of the present paper.

The steady-state, or large time constantpopulation solution of Eq. (1) is given by

FiP= O . (3)

If the free electron density Neis specified thenEq. (3) forms a system of linear algebraic equationswhich is solved assuming charge conservation, thatis,

~ (j- l)Njq=Ne . (4)jq

The resulting set of Njqmay then be renormalizedto any desired total ion density N.

The solution of Eq. (3) provides level popula-tions for given N,, Te,and T, These level popula-tions may then be used to predict the spectroscopicproperties of the plasma. For example, the inten-sity of a spectral line maybe expressed ass

iPq=NiPAiPqEiPq,I (5)

where NiPrepresents the population of level ip,Aipqis the Einstein A coefficient for spontaneous emis-sion from level ip tolevel iq,and Eipqis the energyseparation between levels ip and iq. The quantitiesAip~and Eip~are readily available from the atoficphysics data and they are independent of IVe,Te,andTr. It is often useful to form line ratios,

Rlpquv = Iipq/IjUv, (6)

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which give the relative intensity of one line relativeto another. Hence, a line ratio can be extractedfrom an experimental measurement and comparedto theoretical predictions to estimate the plasmaconditions. A ratio which is approximately inde-pendent of electron density is a good temperatureindicator, and vice versa.

The processes included in Eq. (3) for thepresent study are photoexcitation, spontaneousdecay,stimulated decay, electron collisional excita-tion, and de-excitation, photoionization, radiativerecombination, stimulated radiative recombination,collisional ionization, three body recombination,autoionization, and dielectronic recombination.The cross sections for excitation and ionization areevaluated from the detailed atomic physics calcula-tions discussed below. The rate coefficients for thereverse reactions are calculated using the principleof detailed balance. For Tr= O,the rate coefficientsfor photoexcitation, photoionization, and the con-tributions due to stimulated decay and stimulatedradiative recombination are zero. All processes arefollowed individually, therefore, no net effectiverate coefficients are introduced. For example, innershell excitation, autoionization, and radiative decayare all included as separate processes.

Table I summarizes the electron configura-tions which were used to model each of the ionstages included in the kinetics. Note that the calcu-lations have been restricted to the Be-like, Li-like,He-like, and H-like ion stages, since those are theions of interest for the density and temperaturedomain of the current study. Detailed atomic struc-ture calculations using the method of CowangJOwere performed to generate the set of fine structureenergy levels corresponding to the configurationsof Table I. The structure calculations include

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Table 1

ls2[2s2p]2Be-like ls[2s2p]3

lsz[nl]Li-like ls[2s2p]bZl 2H5

He-likels[nl]l[2s2p]2 2n5

H-like [nl]l ln5

Configurations used to generate the level structurefor each of the ion sta es included in the atomicmodel. The notation h w means all possible dis-tributionsofzuelectronsintheshellsspecifiedinsidethe brackets. All possible allowed values of theorbitalangularmomentum quantum 1 are included.

spinorbit coupling and configuration interaction.Each resulting level is denoted by total orbitalangular momentum L, total spin S, and total angu-lar momentum J. The energy levels and wavefunctions produced by these calculations are usedconsistently to generate the required cross sections.

Oscillator strengths were calculated for allpossible level-to-level dipole allowed transitions.The oscillator strengths are used to evaluate theradiative excitation and de-excitation rate coeffi-cients. Plane-Wave-Born (PWB) collisionstrengths9 were calculated for all possible pairs oflevel-to-level transitions. These strengths are usedto calculate rate coefficients for electron collisionalexcitation and de-excitation. Distorted wave (DW)collision strengths were calculatedlyfor the mostimportant transitions. These include all 1s + nl inthe H-like ion, all 1s2+lSTZZand 1s21+1s21’ in theHe-like ion and all 1s221+1s22 Z’,1s221+ 1s2121’,and 1s2Z2Z’+ 1s2121”in the Li-like ion. The moreaccurate DW cross sections are required to yieldmore reliable spectroscopic results.

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Photoionization and autoionization crosssections were calculated for all possible level-to-level transitions between adjacent ion stagesusing the atomic structure wave functions dis-cussed above. Level-to-level collisional ioniza-tion cross sections were calculated using thecalculated fine structure energies and fits toscaled hydrogenic theory.12

III. Results and Comparisons

The methods described in the previoussection were applied to chlorine and ironseeded plasmas. The effect of varying radiationfields on level populations was studied byexamining the ionization balance and assortedline ratios as functions of electron temperatureand/or density. Results were randomlychecked using the RATION codeg which werein general agreement with the present calcula-tions (this code uses less detailed atomic phys-ics).

A. Ionization BalanceFigures (la-d) show the ionization bal-

ance for chlorine as a function of electron tem-perature for electron densities of 1019, 1020,IOZ1,and IOZZ cnz-3,respectively. Each figureshows the ionization balance for I“r= O,200,and 300 eV. The fraction for an ion species i isgiven by

fi = ~ (NiqM . (7)4

The figures show that the ionization balance forlow T, is not very sensitive to Neand that the larg-est effect of radiation is at low electron densities,where photon collisions are more dominant. Forexample, at N~= 1019 CM-3,the qoo-ev” radiation

Fig. 1 Ion fraction as a functionof electron temperature forvarious cases, l-a through l-dcorrespond to chlorine atelectrodensities of IO~g,ZOIo,1021,and 10II cm-J, and l-ecorresponds to iron at anelectron density of 1019cm-J.me solid line corresponds toTr = OeV, the dotted linecorresponds to Tr = 200 eV,and the dashed line correspondsto Tr = 300 eV. The curveslabeled by FS correspond to thetally stripped bare nucleus.

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1.0I“” ”’’’ ’” ”’”’ ””’ ”” ””’ ”” ””1’’’” 1”..,. .1-...

i

0.0 ~ - .. —-------------...0 600 1000 1600 2000 2600 3000 3600 4000

electron temperature ( eV )

1.0~ .....-..,..:-. t i 1 r 1 I

o.a

= O.o.-g

*c0 0.4.-

0.2

0.0 ~o

-,.

c1--------... -----------...-e-.

‘4~- .,. ,.. L

/ \/ \

‘.,.,

t

He-Ii e .“:—J. t!:ke ‘%<

A=H--ii-k%==”%--------600 1000 1600 2000 2600 3000 3600 4000

electron temperature ( eV )

20

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1.0

0.8

0.8

0.4

0.2

0.0

1.0

O.B

0.6

0.4

0 600 1000 1600 2000 2600 3000 3600 4000electron temperature ( eV )

1 I t 1 1 I #

0.2

0.00 600 1000 1600 2000 2600 3000 3600 4000

electron temperature ( eV )

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field strips most chlorine nuclei of all boundelectrons. As Ne is increased, electron collisionsbecome more dominant, and the ionizationbalance approaches the T~= Oresult. At Ne =10ZZ,the 7’r= 200-eV ionization balance is essen-tially the same as the T~= Oresult. Also notethat the stronger radiation fields produce moreionization, as expected. Figure le shows theeffect of radiation on ionization balance for ironat Ne = 10I9 cm-s. The figure shows that theradiation is ineffective for ionizing the K-shellelectrons of iron, which have a larger bindingenergy than chlorine. The figure also showsthat the He-like ion stage dominates over mostof the temperature range. As electron densityis increased (not shown), the Tr >0 ionizationbalance approaches the T~ = Oresult, justas inchlorine.

co.-Emsc0.-

1.0

0.8

0.6

0.4

0.2

no

1?/ –-----------

N----------------------------‘-- ---”----------.=.=. m.m-

. .. . . Be-1ike -\,.,. e-like -+..... .>%..%.

Li4ike

..., .... -.-..

---0 600 1000 1600 2000 2600 3000 3600 4000

electron temperature ( eV )

22

Fig. 2, The 3-1/2-1 line ratio ojHe-1ikechlorine ns nfunction o,radiation temperatureforvarious electron densities at aneZecfronfenzperafureof 1000 eiThe curves ore k7beledbylog~oNe.

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0.-.+-ruL

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

r ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘‘s‘ ‘ ‘‘ ‘ ‘ ‘ “ ‘ 08 ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ “ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘-i

[illl.li,,i#f,,!,,,,,ll,i,,,,i,,,,,,,,.,,:o 50 100 150 200 250 300 350 400 450 500

radiation temperature ( eV )

B. Electron Temperature Sensitive Line Ratios

Figure 2 shows the effect of radiation on thels3p IP1 - 1s2ISO/ls2p~Pl-1s21S0 (3-1/2-1) lineratio of He-like chlorine (Cl XVI) for various densi-ties at Te= 1000 eV. This ratio is mildly sensitive toelectron temperature and provides useful informa-tion about the populations of different n levels forthe same ion stage. The ratios start out constant atvalues of Trnear zero. At low electron density, asthe radiation temperature is increased, the mani-fold of n = 2 to n = 3 transitions are pumped byphotons and cause the ratio to increase by a factorof 3. The ratio begins to drop as n = 1 to n = 2transitions from the ground state become possible.As the radiation temperature increases even fur-ther, the ratio begins to increase again due to directexcitation from n = 1 to n = 3. As electron densityincreases, higher radiation temperatures must beattained to obtain these excitations because ofincreased photon-electron competition, and thecurves are displaced toward the right. Also note

Fig. 3. The 3-1/2-1 line ratio ofHe-like iron as a function ofradiation temperature forvarious electron densities at anelectron temperature of 1000 eVThe curves are labeled byIoglo N,.

Fig. 4. The3-1/2-l line ratio ofHe-like chlorine as a function ofelectron temperature for variousradiation temperatures at Ne =1019cm-J. Each curve inlabeled by the radiationtemperature in eV.

TechnicalReview

volume2

23

that the variation of the curves decreases withincreasing density. At N~= 10~, the effect n = 2 ton = 3 photo-pumping is negligible. Figure 3 showsthe results for iron under the same conditions asFig. 2. The Fe curves are displaced toward higherradiation temperatures than in the chlorine casebecause of larger energy level separations.

0.36

0.30

0.26

0.20

0.16

0.10

0.05

0.000 50 100 150 200 250 300 350 400 4S0 600

radiation temperature ( eV )

0.36

0.30

0.26

0.20

0.15

0.10

0.05

0.00

1’‘r‘‘ ‘ ‘ ‘ s‘ ‘ ‘ ‘ ‘ ‘ “‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ ‘ “ ‘ ‘ ‘ ‘ ‘ ‘ ‘‘‘ ‘‘ “ ‘ - ‘ ‘ ‘ 1‘ ‘ ““J!.

I 1 I I I 1 1 1 1

0 50 100 150 200 250 300 350 400 4S0 S00radiation temperature ( eV )

24

Fig. 5. The 3-1/2-1 line ratio ofHe-1ikeirorzm a function ofelectron temperature for variousradiation temperatures at IVe=IOlg cm-s. Each curve in labeledby the radintion temperature ineV.

TechnicalReview

Volume 2

Figum4showstheHe3-1/2-1ratiofore.ldorineasa functionofelectrontemperatureforTr= 0,100,200,and300eVatN~=1019ems.Thedrivingphotmxcitationpmmssesareapparentby comparisonwithFig.2 AtT~=3(KI,thelevelpopulationsbecomeindependentofelectrontemperatum;indicatingthatphotonsdominatetheexcita-tionprocesses.Figure5showstheIineratioforthistransi-tioninimnat thesamedensity.Notethatthe100-eVdiation fieldhasnoeffectonthelevelpopulationswhichisConsistentwithFig.3. Asradiationtemperatumisin-, thepopulationofthen= 3 levelsamenhancedbyracbtion thmughn=2ton=3photowcitations. TheCUIVSarestronglydependentonelectronternperatsmbecauseelectroncollisionsm nemssarytopopuIatethen =2 levels. NotethatatevenTr= 300eV,pumpinghornthegroundstatedoesnotplaya significantmleasitdidinchloxine.Aselectrondensityisinmased (notshown),electrondlisionslxmme moreimportant,andthedevia-tionfmmTr= Odecreases.SimilarresultsareobtairmiforthecornqxmdinglineratiosoftheH-likeions.

030

0.25

0.20

0.-~ 0.15

0.10

0.05

0.00

[“’’’’’’”’’’’’’r’” “’’’’’’’’’’’’”J100

700 d

I 1 I I I f500 1000 1500 2000 2500 3000 3500 4000

electron temperature ( eV )

Fig. 6. The chlorine jkl/He 2-1line ratio as a function ofelectron temperature at variouselectron densities. The solidcurves correspond to Tr = Oat1019,1020,10~1,and IO~zandare diflicult to distinguish andunlabeled. The dashed curvescorrespond to Tr = 200 eVlabeled by the Zoglolie.

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Volume 2

0.40

0.35

0.30

0.25

0.-+~ 0.20

0.15

0.10

0.05

0.00

25

1’‘‘‘‘ ‘ ‘‘‘‘r‘ ‘‘‘ ‘ ‘ ‘ ‘r‘ ‘ ‘ q‘ ‘ ‘~“‘‘;

500 1000 1500 2000 2500 3000 3500 4000electron temperature ( eV )

Figure 6 shows how a perfectly good tem-perature diagnostic is affected by a radiation field.AFor this example, the jkZ/ls2p l.P1- 1s2ISO(jkl/He2-1) line ratio of Cl is considered. The notationjklis used to specify the ls2pz ZD- lsz2p 2P transitionof the Li-like ion. The four solid curves representthe line ratios at 1019,1020,1021,and 1022C# withTT= O. Note that these curves are essentially inde-pendent of electron density, all of them practicallylay on top of each other and hence provide anexcellent temperature diagnostic. When a 200-eVradiation field (dashes, curves) is applied thecurves become nondegenerate. This is caused byphoto-pumping of then = 2 level by the radiationfield. The effectis greatest for low electron densi-ties and temperature. As electron density is in-creased, electron collisionsbecome more dominant,and the effectof radiation decreases. Hence, theline ratio becomes an ineffective temperature diag-nostic. The same trend occurs, to a lesser extent,for the same line ratio in iron.

26

Fig. 7. The chlorine jkl/He 2-1line ratio as nfunction ofradiation tempcrntureforvarious electron densities atTe = 1000 eV. The curves arelabeled by the ZogloAlp.

Fig. 8. The iron jkllkle 2-1 lineratio as a function of radintiontemperature for various electrondensities at Tc = 1000 eV. Thecurves fire lnbelcd by the logloN,.

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Volume 2

016 1 ‘ ‘ ‘ 1“ “ “ 8 “ t ‘ “ “ ‘ 6 ‘ “ “ t “ ‘ “ “ t ‘ “ ‘ ‘ “ ‘ ‘ ‘ ‘ ‘ ‘ “ “ “ “4

014

0.12

010

008

0.06

004

0.02

0.000 50 100 150 200 250 300 350 400 450 500

radiation temperature ( ev )

4.6 1.’..,” ‘“1’ ’’’1 ’’’” 1’”””’ “’’1 ”’’” 1’’’’1’” ‘f’’”’l

4.0

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

,

0 50 100 150 200 250 300 360 400 450 600

radiation temperature ( ev )

Figures 7 and 8 show this line ratio as afunction of radiation temperature at various densi-ties for Cl and Fe, respectively. The electron tem-perature is fixed at Te= 1000 eV. The figures pre-dict the radiation temperatures where the 1 + 2photopumping begins to have an effect on the lineratio. This occurs when the ratio begins to fall offrapidly. Note at low density, at hl~- 10I9, a bumpoccurs before the 1-2 falloff. This enhancement isprobably due to 2s + 2p photoexcitations. The

2p2 ‘Dz+ls2p IP]

2p 2P +lspv(He sat/H2-1) line ratiols for Cl and Fe was alsostudied. The Tr= Oresult is a strong function oftemperature, and the variation of the line ratio with

Fig. 9. The chlorine He2-lfH2-1 line ratio as a function ofelectron temperature for variou;values of radiation temperatureat Te = IO~gcm-s. Thecurvesare labeled by radiationtemperature in eV.

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27

T, is similar to thejkZ/He2-1 ratio discussed above.The 1s + 2p photoexcitation causes the line ratio todecrease for Tr2200 eV in Cl. For Fe, the line ratiois rather insensitive to radiation.

Figure 9 shows the

ls2p 1P]-1s2Iso(8)

2p2P-IS2s

(1-Ie2-1/H 2-1)lineratiofor Cl as a function ofelectron temperature at Ne = 1019CVZ-S.This ratio isa strong function of temperature because it de-pends mostly on the relative amounts of He-likeions to H-like ions through the ionization balance.The Tr= Oand TT= 100 resuIts are similar becausethe ionization balance is essentially unchanged bythe radiation field. As Tris increased, the ioniza-tion balance is affected and these differences be-come apparent in the line ratio. The result for iron(notshown) shows very little sensitivity (-20%) toradiation, even at Tr= 300 eV. The ratio has valuesgreater than 100 below about T,= 2000 eV due tothe pre-dominance of He-like ions (see Fig. le).

10’

104

l(f

~ Id~

10’

10°

10-’

10-2

~“ ‘-10

3

3

300

I I 1 I 1 I I 10 500 1000 1500 2000 2500 3000 3500 4000

electron temperature ( eV )

Volume 2

28

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Volume 2

The result for the line ratio involving the n = 3upper level in the numerator and denominator ofEq. (8) is nearly identical to the n = 2 result sinceionization balance is the determining factor. It issometimes advantageous to use the n = 3 line ratiosas a diagnostic because of smaller optical deptheffects.

c. Electron Density Sensitive Line Ratios

Figure 10 shows the

ls2p3P]- 1s2IS()ls2p IPl - ls~ Iscl

(IC/He 2-1) line ratio for He-like Cl as a function ofelectron density at TC= 1000 eV. The transition inthe numerator is often referred to as the inter-combination (IC) line. The ratio is normally con-sidered to be density sensitive for Tr= O. Since theoscillator strength from the ls2p 3P1to the groundlevel is small (the transition is spin forbidden),population builds up in the 3P multiplet throughelectron collisions. As the electron density is in-creased, further electron collisions depopulate the3P,causing the line ratio to fall. The T’= Ocurve kFig. 10 shows this behavior and the T, = 100 eVshows littIe difference. However, for Tr= 200 eV,the line ratio at low density is significantly de-creased because of photoexcitations from theground state to the IP1 Ievel. As electron density isincreased from IO~g,the line ratio gets Iarger be-cause a rising of SF’state population due to electroncollisions. At about Ne= IOZ1the depopulationmechanism takes hold and the line ratio decreasescausing a maximum in the curve. At Tr= 300 eV,the line ratio is almost completely independent ofelectron density. Note that all curves approach thesame limit as Neget large where collision processesdominate.

Fig. 10. The chlorine HeICJH2-1 line ratio as nfunction ofelectrondensityfor variousofradiation temperatures at Te =1019 ~*-3, ~~e cu~es are

labeled by radiation temperaturein eV.

Fig. 11. The iron He IC/H2-lline ratio as a function ofelectron density for variousradiation temperatures at Te =1019~m-3. me curves are

labeZedby radiation ternperatuin eV.

);J

29

0.5 1 8 I 1 1 I 1I II * 1 T 1 1I 1111 T 1 1I I I

~

0.4

0.3

0.2

0.1

) 8 I I I 1 111I0.0 1 , 1 1 1I 11I

1 1 1 ) I I

10” 1020 10” Id’electron density ( cm-3 )

0.7

0.6

[1 1 I 1 I 1 I 1[ 1 I I 1 I 1 I 1 I 1 1 1 1 1 1 1 1

0.6

0.4

0.3

0.2lo” d“ 10” 10’2

electron density ( cm-3 )

Volume 2

—

30

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Volume 2

Figure 11 considers the intercombination lineratio for iron. The Tr= Oresult indicates that inter-combination line ratio is not sensitive to electron den-sity in this range. Unlike Cl, the oscillator strength forthe SP1transition to the ground state increased by afactor of 10 due to the larger spin-orbit interaction.The more rapid spontaneous decay to the ground statenegates the low density populating mechanism dis-cussed above for chlorine. It aIso shows that Iineratios do not necessarily obey simple scaling rulesfrom one nuclear charge to another. The T, >0 resultsdeviate from the T~= Ocase according to the amount ofphotoexcitation from the ground level to the preferredIPI IeveI. The largestdifference is about a factor of 3for T]= 300 and low density. The curves approacheachother at lower electron densities than for chlorine.Note that the intercombination line of iron is muchless sensitive to 200 eV radiation than chlorine.

The ratios of certain satellite lines of Li-like andHe-like ions have been used to determine electrondensity.q Usually, the upper level of one line is mainlypopulated by collisional excitation, and the other isfed by dielectronic capture, making the ratio sensitiveto electron density. However, these ratios were foundto have a small variation (less than a factor of 2) in thedensity range studied here, even in the presence ofradiation. These ratios become better diagnostics atelectron densities higher than those considered here.

v. Conclusions

An externally applied black body radiation fieldwith temperatures of a few hundred electron volts canhave a significant effect on the spectroscopic proper-ties of low to mid-Z plasmas in the electron tempera-ture and density range considered here. In general,the effect is greater for Cl than Fe at a given radiationtemperature, because the greater energy level separa-tions in Fe. A 300-eV radiation field can fully strip Cl

TechnicalReview

31

of all bound electrons, while Fe is stripped only tothe He-like stage. Radiation fields can have ad-verse effects on line ratios which are generallyconsidered to be diagnostics of electron tempera-ture and density in the absence of radiation. Someline ratios (such as the intercombination line ratio)are very different for Cl and Fe suggesting thatsimple Z-scaling arguments are not always valid.

VI. Acknowledgement

This work was done in collaborationC. J. Keane, T. D. Shepard, and L. J. Suter,Lawrence Livermore National Laboratory.

with

Volume 2

32

For further informationon this subject contact:

W.T. Leland

Los AlamosNational LaboratoryLos Alamos, NM 87545

TechnicalReview

Volume 2

Amplified SpontaneousEmission Produced byLarge KrF Amplifiers

I. Introduction

The excited KrF* molecules in large KrFamplifiers spontaneously emit photons uniformlyin all directions. The photons exhibit adistribution in frequency that depends on themixture of rovibration states of the KrF*molecules and details of the electronic transition.While escaping from the amplifier medium, thespontaneously emitted photons interact withother molecules in the media which may absorbthem or in the case of KrF* molecules, enhancetheir numbers by induced photon emission. Thetotality of photons involved is referred to asamplified spontaneous emission (ASE). Theamount of ASE exiting the amplifier media is thesubject of this note. Since induced emission andabsorption cross sections depend on photonfrequency, the frequency distribution of thespontaneously emitted photons must beconsidered. The formulation in Section II dealsfirst with the case where a single rovibrationlevel is involved. The generalization to caseswith several rovibrational states is thenconsidered. Appropriate values for someparameters are discussed in Section III. InSection IV calculations and experimental resultsare considered.

Fig. 1. Geometry for calculatingASE radiance at point Pindirection R. Contributions arisefrom medium locafed along linefrom P to P’.

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Volume 2

II. Calculation of External ASE 33

The characterization of external ASE thatwill be used herein is its radiance which is ascalar function of position and direction; its unitsare photons per second per steradian per unitarea normal to the direction considered. Forpropagation of photons in media devoid ofabsorption or gain, radiance in the direction ofpropagation is invariant along the propagationpath. For active media radiance is modified bythe amount of gain or loss experienced along thepropagation path. For a non-monoenergeticgroup of photons it is appropriate to considerspectral radiance which is defined as radianceper unit frequency interval. Radiance is thus theintegral of spectral radiance over theappropriate frequency range.

Calculation of ASE radiance at a givenpoint and direction on the boundary of anextended media can be achieved by summingcontributions resulting from spontaneousemission in each portion of the medium.Assuming straight line propagation of photons inthe medium, it is clear that contributions to theselected radiance direction can arise only fromthose portions lying along a line in the givendirection and passing through the chosen point.The geometry of the problem is illustrated-inFig. 1.

34

TechnicalReview

Volume 2

The spectral radiance 5R~x)arising from anelemental volume of area dA and length dx isgiven by the relation

where(1)

n(x) is the density of KrF* molecules at X, ~Spisthe spontaneous life time, and [v-vO)a normalizedfrequency distribution function for the emittedphotons. We have:

rfv-vO)dv= 1 .

-m

In its passage from point x to point P on theboundary this incremental radiance will bealtered by media gain and attenuation.

(2)

We have;

dRv(P,x)= 8RJx) exp /

\[@-VtiS)-u(s)]dsI

/

where g(v-vo,s) is the frequency dependent gaincoefficient at point s and a is the attenuationcoefficient at points.

The attenuation coefficient may exhibit afrequency dependence but for the calculationsherein considered it wilI be taken as frequencyindependent. To obtain RV(p)we must sumcontributions from all dx intervals from P to l?’.TOget R(P) we must firther sum (integrate) OVerfrequency.

For cases where the range of frequenciesinvolved is small relative to “line center” fre-quency, the frequency dependence of g is to good

(3)

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Volume 2

35

approximation identical to the function f(v-vO).We have:

g(v-vo)=g(o)~ .

The inversion density n is also expressable interms of the stimulated emission cross sectionand gain coefficient. We have:

g(v-vox)=n(x)o(v-vO).

The frequency v. can in principle be selectedarbitrarily. In the development that follows itwill be chosen as the “linecenter” frequency. Thischoice leads to more convenient expressions forf(v-v~).

Combining (l), (3), (4), and (5) and defining

RO= 14@o) z~pwe have:

dRv(Px)= ROg(ox)dxf(v-vO)exp0

x g(o,s)f(v-vo)f(o) -o(s)]ds

o

(4)

(5)

(6)

/

f(v-vo)

(0)

/[

xg(o,s)f(v-vo)~exp -Q(s)]ds\I

= ‘°KoBx f@-Vo) 4x) dx o (0)I

. (6a)—-—

(0) g(oJ)

From (6a) an integration by parts yields

\

L

R~P) =(4 )h( - )

dRv(P,x)= &f(o) ex jj(o)L!$&-~ L) h(o)o

36

TechnicalReview

VoIume 2

where we have defined

f(v-vo)qo)

h(v-vo~x)= f(v.vo) @x) ‘

f(o) g(o,x)

\

L

g(o)= ; g(o,x)dx ,0

(n

(8)

(9)

In obtaining (7) it has been assumed that f(v-vo)isindependent of x. This is certainly reasonable fora single rovibration line in KrF which is homo-geneously broadened and the broadening isrelatively independent of collision interactions.For multiple rovibration lines, however, theirrelative abundance may vary spatially andintroduce some variation in f(v-vo).

The function h depends on x principallythrough the ratio a(x)/g(o,x). If alpha is zero orif the ratio of a to g is constant there is no xdependence. One then has:

i-(v-vo)

Rv (P)= Rof(o)W::) ~[’xp(L~(o)(fl;;O) -a))-’l -

qo) ‘g

(7a)

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37

There are two results of interest that followimmediately from (7a).

First: Rv(P) and consequently R(p) dependsonly on the average gain and absorptioncoefficient along the path of length L from P to P’.

Second: For small Lao) the.ASE radiance isproportional to Lao) and independent of thenature of f(v-vO). Expansion of the exponent in a

[ 1-t-p-v.) ~

power series in L~(0) ~[o) - ~ yields the

result for small ~o)L. We obtain:

RV(P)SRO f(V-VO)Ljj(o)

Integration overgives

R(P)SRO L~(o)

frequency and use of (2)

(11)

(12)

For larger values of Lao), details of thefrequency dependence factor f(v-vO)come intoplay. To illustrate this further, three forms off(v-vO)are considered.

Case A: Square hat distribution with Avthe fullwidth of the square hat. For this case we have:

1 for A> ~ V-vo

38

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Volume 2

Case C: Gaussian distribution with Avthe fullwidth at half maximum. In this case:

“-’o’=:wex’[-’ (alal“Integration of RV(p)over frequency iS

straightforward in Case A. We obtain

RSH(P)=R. exp{@(o)(l - Wg)-l } .

1- Wg

(15)

(16)

To perform the integration over frequencyfor Cases B and C we first expand the ex-ponential function of (7a) in a power series” Weobtain

R,(P) = R. f(0)%yW%W’

u%)

m (L~o))nf@-Vo) ‘-’= Ro Ho) x ~ _._go(-:) ‘(*)”-’ k(”;)

nal .

()where ‘~1 (n-l)!= k! (n-k-l)! .Defining

f[ 1m%) i ~v ,

bi = flo) —- qo)

cni=(~)i (nil] ~

and

(17)

(18)

I

TechnicalReview

Volume 2

n-1

Sn= ~ Cn,kbn.kk=o

we obtain

To evaluate the b coefficients of (17) it isconvenient to make the substitutions

and

dv= ~dy ,

(19)

(20)

(21)

(22

For the Lorentzian Case B we then have

bi=ij’?dy(tir

2=—r ( )

dy~ix I+yz

o

or

bi= 1for i=l

bi H (~) (~) ... (~) for i~2 (23)

When a/g is zero Snis equalto b. as can beseen from examinationof (7b). Using (20) we canthen write for this special case:

39

40

TechnicalReview

R~P) = RO ~ ‘LE(:~)nbn“=1

m

= RO ~ (Lao)) annel

where al= 1 and & = ~2k(k+l)

For a Gaussian line shape using (15), (21), and(22) we obtain

bi=fif~[*dflexP(-Y21n2)Y.00

.

= 2vq

dy exp(-iy21n2) .‘z

o

Letting z=y ~, (25) becomes

bi = ~i-iii rdz exp-~’o

L.‘il-When a/g is zero, we obtain the formula:

(24)

(25)

(26)

00

= R ~ (LE(0))”G (27)n=]

where d] = I and dk+l= 1——rdk (n+l) % “

Volume 2

TechnicalReview

Note that the power series representation for 41both the Lorentzian and Gaussian line shapeconverges for all values of Lao). Note also thatthe values of RsI-I(P),RL.(P),and R@) do notdepend on the parameter Av(full width at halfmaximum) used in defining the line shape.However, if a line were somehow broadened thiswould imply lower gain for a given inversion andhence less ASE.

For large values of Lao) the power seriesrepresentations converge slowly. By using apower series in l/L~o)one can ascertain thebehavior for large Lao). The resulting(asymptotic) series do not converge for any valueof Lao) but nevertheless provide a goodrepresentation if the series is judiciouslytruncated and its use limited to values of Lao)exceeding an appropriate minimum value. Toarrive at the asymetric series we begin with Eq.(7a), factor out exp ~Lg(0)[l- ~]~and obtain

f(v-vo)

Upon the further substitution

z=f(o)Wo)

and integration over frequencywe obtain for theLorentzian line shape:

R~P) ❑~:

1R. ex~Lj!j(o)-et] 1+Z2Z2

& .__l-- -a1+ 7C2Z2 g (28)

Vohune2

42

TechnicalReview

Volume 2

The further substitution

x = m Z7Cgives:

R~P) =R\

1exp[L(jij(o)-a)] - ~x 1+~xz

‘0 n = - 1 -a1+ Axz g

(expia-ex’k?l)(29)where the parameter L equals u@(o). Expansionof the integrand in a power series in k provides

11-@/g)the required asymptotic series. The term exp ~J

yields the null series in k. The leading term yi~ld~the behavior for large Lg(o). We have for Lao)>>1 in the Lorentzian case

.&’’P[wNlmm (1*) (30)

For the Gaussian line shape, we obtain for theequivalent of (28):

F [1‘Xp. &

~P) = R. exp[I&(o) -u)] dy Lg(o)

4‘7 124.4-00 ‘x Lg(o) g( { L(o) 1, )0 ’31)exp L~(o) ex @ -1 ~-exp[I&(o)-a)]The further substitution

TechnicalReview

Volume 2

gives:

\

~P)=ROexp[I&(o)-cx)] ‘dx exp[-Zx2]YE- -w wh21-&

(exP&P[-Ax21-d]-exP[-y]) ~

Noting that

exJ-Lx2]-l =-AX2+Z2X4 +... .

The leading term in the k expansion yields

~p). ~. .@&(o) -~)1-2-

[m (l-(x/g)fi ~dx (yX2

= ~. exp[I.&(o) -a)]m (l-a/g)

If one invokes the Einstein equilibriumtheory relating the A and B coefficients oneobtains the relation

(32)

(33)

(w

where q is the refractive index of the medium.The quantity ROthen can be expressed as:

RO= 2 (35)Ao2Tlf(o)”

As noted earlier, the symbol zero does notnecessarily imply line center. f(o), & and ROallrelate to a selected wavelength and the functionf(v-vo) is to be constructed accordingly. Theassumptions of identical frequency dependenciesfor gain and spontaneous emission is equivalent

43

44

TechnicalReview

Volume 2

to neglecting the variation of v in the range ofinterest.

Quantum mechanical non equilibriumtheories yield similar results for regimes of timeand ASE intensities of interest.a

When several excited states emitspontaneously into the frequency range ofconcern, one can generalize Eq. (1) by summingthe contribution from each state. We have

where ni(x) is the population density of state i,~i,~Pis the spontaneous life time of state i, andfi(v-vo) is the normalized frequency distributionfunction for state i.

(36)

The states are presumed distinct and emitindependently. For propagation through anactive medium, however, all states contribute tothe amplification of photon flux originating fromdifferent states. The overall gain from thevarious states is given by

[

fi(V-VO)&JV-V.,X) = ~ hi(X) CYi(0)

fi(0)i 1

=Xgi(”,x) ‘i(v-vo)fi(0) “i

(37)

(37a)

The relation corresponding to (6) is:

dRv (p,x) =z

ni(X)fi(V-VO)dx 47CTi \

exp x [g. (v-vO,X) -CX]ds .i o

(38)

asee for example, “TheQuantum Theory of Light”, Chapter2,byRodney Loudon, Clarendon Press Oxford1982

TechnicalReview

45

Using (35) we obtain:

\

dRv(p,x) = 2 ~c(v-vO,x)~xp x ~gc(v-vO,x)-al d~ ~38a)dx ?b.2q o

as before an integration by parts and the assumption ofno x dependence of ratio of ctto gC(v-vO)yields thecounterpart to (7a)

()R, (P)= + 1a(x) {exp(L&(v-vo)-CL) -I] . (39)xo2111-gc(v-vo?x)

For small L(gC-a)

2 L&(v-vo)=R,(P) s — +Zfii(x) ai(o)fi(v-vo) (~~)Ao2?’1 L. ~ i fi(0) “

Integration over frequency gives

(41)

L z-- L gc(o)fij(X) = LN=— — (42)

4TCi ‘tj ——4m 47rcCTc(o)

where

N = ~ iii(x) , (43)i

(’w

(45)

V&une2

46

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Volume 2

For large values of gcL, Eq. (39) must beintegrated over frequencies. Defining

fc(v-vo)gc(v-vo) = gc(o) fc(o)

where

gc(o)= ~ nloi(o)= No(o) ,

and

\dv fC(v-vo)= 1

allows (39) to be written in the form

(46)

(47)

(48)

(49)

(50)

R,(P) =

fc(v-vo)f.(o)

fcwo) - ~fc(o) Ec(o)

1-[ 1fc(v-vo) 0. -1explLgJo)fc(o)-EC(0) (51)

which is identical to (7a) with quantities f, g and——a replaced by fC,gc, and a z. Integration of (51)over frequency requires specification of fc(v-vo).The result will be of the form:

R(P)= 1 - F4n ac(o) ‘r

(52)

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Volume 2

47where F depends on ~C(o)L,a, and the form offC(v-vO).In the limit of small jijC(o)L,F alwaysreverts to ~C(o)L.In general

fc(v-vo)

\F = fC(o) dv fc(o)

[ 1j - fc(v-vo) 0. Jfc(v-vo) exP L gc(o) fc(o) - gc(o) - / “Ifc(o) - -J&

III. The value of ~c(o)i

To proceed with the calculation of externalASE via Eq. 53, a value for =~oji mustbeassigned. It is evident from their definitions(Eq. 44 and 45) that the appropriate value forboth &(o) and ; will depend on the mixture ofrovibrational states invoIved. The mixture thatobtains in a given case will depend on kineticdetails involved in the formation of KrF* and itssubsequent interactions with other molecules inthe media. Tiee et aIl found substantial variationin the spectra of KrF* produced by photolysis ofKrF2 with radiation of different wave lengths.For the relatively low pressures and short timesinvolved in their experiment the measureddistribution would be sensitive to the mix ofrovibrational states formed in the photo-excitation reaction. Tamagake and Setserzfound similar variation when KrF* is formed byinteraction of metastable Kr with F2, NF3, andCF30F. Pumrner et als observed the timeevolution of rovibrational states by observing thetime evolution of emitted radiation at differentwave lengths. Thus the appropriate value of=c(o~ is clearly not a universal quantity. In manycases, however, the externally observedemissions from KrF* will be dominated by asemi-steady-state distribution of rovibrationalstates with a predominance of the low vibration

(53)

48

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states. Theoretical calculations predict thevalues of o(o) and z do not vary drastically forthe first three or four vibration levels.lQ Theproportion of molecules in higher vibrationallevels (20 or 30%) will not be a serious problem.This circumstance arises because the highervibration levels contribute little to the overallgain in the frequency region where the bulk ofspontaneously emitted photons are produced.

There are several ways in which theproduct & (o)7 can be obtained experimentally, In~rinciple at least, a measurement-of the emissionspectra from a “thin”source will provide theanswer via the formula

(54)

where S(V-VO)is the measured signal in unitsproportional to photons cm-zseel per unitfrequency interval and the integral extends overthe range of concern. This approach does notappear to have been used directly but rather inconjunction with reduction of data via appli-cation of theoretical models. By adjusting themixture of states and details of potential curves a“best fit” of the theoretical model to spectral datais effected. The theoretical model using adjustedquantities is then used to calculate quantities suchas & (o);. Tellingheusin et ala fitted their ob-served spectra with a “thermalized”360”Kdistribution of states and report a oz value of1.7t.2x10-zAcm~sec. This value is frequentlyquoted in the literature. It should be noted,however, that Tellingheusin’s fit to the experi-mental data was not very good. Patterson andHansons have analysed spectral data fromMurray and PowellGas well as the data fromTellingheusin. They used potential curves

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49

produced via ab initio calculation by Hay andDunning.p By adjusting the mixture of statesthey report a good fit to the data of Murray andPowell. The best fit state distribution was farfrom a thermalized one and gave a peak m valueof 1.2x1O-Z4cmz sec. A 300”K thermalizeddistribution applied to the Tellingheusin datagave a cn of 1.8x10-24cmz sec.

The Tellingheusin and Murray spectrawere produced by e-beam excitation in gasmixtures at “high”pressures. They do not statethe gL product of the media which they measuredthe spectra for and it is not possible to judgewhether the spectra qualify as due to a “thinsource”. To qualify for “thin”would require anegligible second term in the power seriesexpansion of (51). The two term expansion of(51) for Rv(p) gives:

R,(P) =4::?; 7(f’@-v”)+gc!)L f’@-v”)rc$30) -ad)“

(55)

For a negligibly altered spectra one would like:

+ 1gc(o)L fc(v-vo) ~

50

TechnicalReview

This method is straightforward but does notappear to have been used in published papersrelevant to the subject.

A third method for determining GJohinvolves measuring 7 and &-Jo)in separateexperiments. The usual questions pertainingto the distribution of rovibrational states arises.Experimental measurements of ~have beenreported by several investigators.l~slgIn allexperiments KrF* was produced by photolysis ofKrF2 and the time decay of spontaneous emissionobserved. Measurements were made at lowpressure and extrapolated to zero pressures.The reported values of z, however, range from6.89 t. 9.51 nanoseconds. patterson andHanson10 have suggested that “B-C statemixing” might have influenced the measure-ments. McCowanll examined the B-C statemixing question also. At the Iow pressures usedin the experiments it is difficult to believe therewould b; sufficient B-C mixing to compromisethe observed decay of the B state. Some vari-ation in the measurements may be traceable tovariation in rovibrational state mixtures.Theoretical calculationsZ~Gindicate a moderatedependence of life time on the vibrational stateinvolved. Both Tamagake et alz and Morgan etallz use the Hay and Dunningp potential curvesto calculate z. Morgan et al report calculatedvalues of z from 7.0 to 8.6 nanoseconds as thevibrational quantum number ranges from Oto 10.Dunning and Hayls report a calculated value of6.7 nanoseconds, presumably for v=O.

Measured values of Gare obtained fromshort pulse energy extraction experiments forwhich the quantityhv/c is derived. Using pulseswith duration of a few picosecond, Szatmariand SchaferlA report a value of 2x1O-Sjou.les/cmzfor hv/cx Taylor et ails in a similar experiment

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Fig. 2. Deriving E~flfusingdiflerent models.

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find a value of 1.8x1O-Sjoules/cmZ.Bucksbaum etallGreport a value of 2.1x1O-Sjoules per cm2. Fora 2 nanosecond pulse Banic et allT report a valueof 3.2x1O-Sjoules/cmz. All the measurementsnoted above were made in discharge typeamplifier and are based on data reduction using aFrantz-Nodvikls formulation of output versusinput. The derived value is thus model de-pendent. Models modified to take into accountsemi overlapping states, absorption, etc. willresult in different derived values of G. This isillustrated in Fig. 2. Predictions obtained withthree different models are compared with theFrantz-Nodvik curve with E~.tset equal to 2xnillijoules/cmzand a small signal gain ofexp(l O). In all models the small signal “net”gainwas set equal to exp (10) and the value of E~~tad-justed to provide the same Ek at a net amplifiergain of 100. In case (1) a non saturable absorptionto gain coefficient ratio a/g of .07 was used.

105

104

“~103u

102

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\Cdg fs fn E~~t(m~

rontz Nodvik (0) — 0.0 0.O 0.O 2.0(1) - 0.07 0.00 0.0 2.56(2),— 0.0 0.1 0.3 2.45(3) --- NoMolecular 2.89

ReorientationI I I

-4 -3 -2 -1Log Ein

51

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TechnicalReview

In Case (2), two partially overlapping stateswere used. The secondary state was assumed tohave 30% as much population as the primarystate. The cross section for the secondary state atline center of the primary state was taken as 10%the cross section of the primary state. Case (3)assumes that no molecular reorientation occursand that the extracting beam is polarized.Szatmari and Schaferlg observed effects theyascribe to finite molecular reorientation times.The time constant appeared to be on the order of.9 picosecond with an amphfier pressure of 2.5bar. The models used are based on rate equationformulations. The possibility of coherent effectsmust also be considered for “veryshort” pulses.The situation in regard to KrF amplifiers hasbeen examined by Milonni et al.zo They find nosignificant impact on the energetic of pulseextraction in KrF amplifier with pulse lengths of1 picosecond.

Other values of cm,have been noted inthe literature. The group at AVCO report(1.6x10-24)21,and (1.75x10-zA)~for this quantitybut the details of how they arrived at the valueare not specified in the articles cited.

Besides the value of m, the frequencydistribution of gain and spontaneously emittedphotons have a bearing on the amount ofexternal ASE produced in high gain KrFamplifiers. The frequency spectra depends on thedistribution of rovibrational states. Milonniet alzosuggest that for the V=Oground state, anearly Gaussian shape should obtain. Thespectra from states with n>l are not Gaussian.sTheir contribution to the shape in the vicinity of“line center” tends to broaden the line.

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53

Iv. Experimental Measurements of ExternalASE

External ASE has often been used as adiagnostic to monitor KrF laser media. Whenobserving desired laser output it presentsunwanted background. Numerical analysisbased on formulations detailed above have beenapplied to several sets of experimental data.Within error limits inherent in the experimentalresults and parameters used, the analysesproduce predictions in agreement with measuredresults. It is nevertheless the case that nodefinitive experiment has yet been performed toallow a careful assessment of how precise thepredictions are.

Thomas et alzs made a quantitativemeasurement of the radiance from the LumonicsAurora front end laser. Two apertures were usedto define both an area and solid angle of ASEemerging from the pumped media. Time historyof the radiance was observed with a fast photo-diode while the time integrated intensity wasmeasured in a calibrated calorimeter. Themeasured radiance in watts/cmz/steradian isshown in Fig. 3. Using a = 1.8x1O-ZAcm%ec anda Lorentzian line shape, the calculated gl productis also shown in Fig. 3. No direct measurementwas made of small signaI gain coefficient. Thelength of amplifier media is 90 cm which thenyields a calculated maximum gain coefficient of12.6%/cm which is reasonable.

John Oertel et alzQs report measurementsand comparison with calculations for the Auroralarge aperture module.

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54

Fig. 3. Mensurcd rmlimzcffio~nLwnonics mnpifier ond cd-czdcftedgl.

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o 10 20 30 40Time (ns)

5

,0

4u)

5

In more recent measurements Czuchlewski etalzGmeasured the time integrated radiance from theAurora PA amplifier with a calorimeter and a de-fined geometry. PA amplifier measurements weremade with and without a mirror at the far end. Ananalysis of the results was performed by using themeasured (on an earlier experiment) time history ofside light to infer a time dependence of amplifier gain.

Fig. 4. Memwred and calculatedsingle pass external ASE fromthe PA mnpl~ier,along withcalculated gO.The modelincludesoptic transmissionlosses and losses due toabsorption in the amplifier.

Fig. 5. Me(zsuredand calculateddouble pass external ASE fromthe PA amplfier, along withcalculated gO. The modelincludes optic ~rmzsmissionlosses and losses due toabsorption in the amplifier.

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102

10

g301:==.—z

1o-’

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100(

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56

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Fig. 4 and 5 show calculated curves of energy(time-integrated intensity times area) versussmall signaI gain coefficient. The two curvesshown are for Lorentzian and Gaussian lineshapes respectively. The measured outputenergy is indicated by horizontal lines. Thecalculations are consistent with a small signalgain coefficient of about 3.3%/cm. This issomewhat higher than values measured earlieron the same amplifier prior to making repairs.Calculated values of the gain coefficient gOthatwould obtain if no internal ASE were present arealso shown.

External ASE contributes to prepulse intarget shooting systems. The analysis forsystems is relatively complex. It is necessary tospecify time dependent details of all amplifiersinvolved on an individual basis as well asincluding their relative timing. The presence ofan extracting beam and its timing must also beconsidered since it has major impact on systemgain. In multichannel systems such as Auroraand Mercury, the manner in which systemarchitecture affects timing in different channelsmust also be considered. InterampIifier andamplifier-to-target beam transport must also betaken into account. Two system studies havebeen made, one version of AurorazTand one for adownsized Aurora~. R. Krystalzgmeasured totalprepulse power for the Aurora system. Themeasured value was approximately 1/3 of thecalculated value. The calculated value was basedon “best estimate” values for amplifier per-formance and timing. A trivial change inassumed performance or timing could easilyeffect agreement with Krystal’s measurements.An often specified parameter in regard toprepulse tolerance is power density. CalculatedASE power densities based on “perfect optics”generally far exceed acceptable levels. Non

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perfect optics will lower the average powerdensity and most likely the peak power density aswell. For average power densities the ratio ofactual beam size on target to a diffraction limitedarea would seem to be a legitimate correctionfactor. Peak power densities are more difficult toaddress and are not likely to scale as an arearatio.

58

For further informationon this subject contact:

V.A. Thomas,

M. E. Jones

Los AlamosNational LaboratoryLos Alamos, NM 87545

TechnicalReview

Volume 2

Low Threshold VirtualCathode Formation in aLarge Area Electron Beamon an KrF Laser

I. Introduction

One possible scenario for laser fusion is touse large aperture krypton fluoride lasers.1~2Theselasers are pumped by large area (.1 to 2 mz) elec-tron beams in the 250 keV to 1 MeV ranges KrFlasers are not storage lasers, hence the maximumpeak intensity at the output of the laser is depen-dent on the pumping power of the electron beams.The power in the electron beam can be increased byincreasing the voltage or the current, howeverincreasing the voltage on the electron gun bothincreases the difficulty in the pulse power designand requires greater gas pressure for a fixed sizelaser aperture in order to stop the higher energyelectrons. The greater pressure increases the re-quired strength on the hibachi and gas foil struc-ture which separates the electron beam source fromthe laser of a volume, while also modifying thelaser kinetics. The Limitsfor increasing the currentare the difficulty in designing large low-impedancewater pulse-forming lines and the effects of the selfgenerated field in the electron beam. By the addi-tion of a guide magnetic field to the electron gun,the current can be increased over the self pinchlimit. The applied guide magnetic field is parallelto the electron flow. The self generated field addsto the applied field to result in a net angle between

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59

the accelerating electric field and the total magneticfield. The angle will be the greatest near the edgeof the electron beam. The guide field cannot bemade arbitrarily large due to cost constraints andthe large amount of energy stored in the guidefield. The applied magnetic field should be theminimum required to provide acceptable lossesnear the edges of the beam when the electronsstrike the hibachi due to the an Ie of attack deter-

?mined by the total magnetic fie d.Q

Recent research at NRL and simulationsperformed at Los Alamos indicate additional con-cerns when trying to reduce or eliminate the guid-ing magnetic field. The stability of the electronbeam as it is accelerated and drifts through thehibachi to enter the gas is important. Strong oscilla-tions will spread the electron energy distributionlargely to lower energies reducing the deliveredenergy to the gas, increasing the foil losses, andmaking it more difficult to get a smooth energydeposition in lasers which are pumped from twosides. A strong instability was observed in a laserat NRL when the guide magnetic field was reducedbelow a critical value. In order to examine thisresult, the experimental arrangement and the ex-perimental results will be described first. Thesimulation results will follow to confirm the basisfor the physical explanation.

II. The Experiment

Figure 1 shows the design of one of the elec-tron guns on the NIKE 20 cm driver amplifier. Theelectron beam is created by a cold cathode dis-charge which uses a 20 cm by 80 cm velveteenemitter with a parabolic curvature near the edgesto prevent the halo effect. The anode is a 85% opti-cally transmissive stainless steel screen 2.8 cmaway from the velveteen. DirectIy behind theanode screen is the hibachi structure. The hibachi is

60

a.

LASER CELL I

-lL1~ VACUUM INSULATORz

E \\SWITCH SECTION2o= WATER PULSE FORMING LINE~

Fig. 1. a. Front view of the20 cm laser of the NIKEsystem. b. Top view.

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b. /r 7I

t

I

I

IJ- J.1

made up of stainless steel rectangular bars 3.2- mmthick and 1.3 cm deep spaced 2.9 cm apart. The barsare 22 cm tall where they are welded into a framewhich provides the vacuum seal to the gun box andsupports the gas foil. The open part of the hibachi islarger than the emitter area so all emitted electrons gointo the gas. The optical transmission of the hibachistructure alone is 9070. The gas foil is 50 pm ofuncoated Kapton. The gun vacuum is routinely lessthe 5x1O-5torr and up to 1400 torr of gas is used onthe other side of the foil in the laser cavity.The pres-sure on the Kapton causes the foil to bow inward sothe distance from the anode screen to the Kapton is1 cm on average.

Fig. 2. Gray scaZe imageof the center 20 cvzof theelecfron beam of oneelectron gun tnken just asthe electrons emerge intothe gas. No regulflrstrucfure ofher fhan theblockage from the hibachibars is visible. Mngneficfield slanf effecfs are visiblein fhe hibachi shadows onthe top and boftom.

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The electron beam has been diagnosed by alarge sheet of GAF Chromic Dosimetry Mediaplaced inside the laser cell up against the hibachi.One shot was fired with a gun voltage of 240 kVand 1 kG guide field. A gray scale image of thecenter portion of the beam is shown in Fig. 2. Thedistribution densitomitered on the media showed agenerally uniform distribution with a 30% slope topto bottom due to a error in the anode-cathode (AK)spacing. The size of the beam was only slightlylarger than the emitter area (21 x 81 cmz10% of peakenergy density). Assuming a uniform current distri-bution the geometric shape of the electron beamgives the following self generated magnetic field fornear the center on the top and bottom of the beamas well as the field near the center on the ends:

&(T)= 0.62X/(MA) (1)

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If there is no guide magnetic field, then anelectron emitted from the cathode near the edgewill be accelerated through the perpendicularmagnetic field. In Child-Langmuir limited currentflowz the electron’s initial velocity is very small,approximately the temperature of the cathodeplasma. Since the electron will become nearlyrelativistic during its traveI, a small relativisticcorrection can be approximated by a correction inm. Defining +Zas the direction of acceleration forthe electron and +y to be the short dimension of thecathode (up in the experiment) the self field at thetop of the electron beam is in the +x direction. Theelectrons z velocity is approximately

and $ is the cathode potential and E is the electricfield in the anode-cathode gap. The turning point

(UZ= O)for an electron would beat z = 2E/B~co;,.All of the above expressions are only valid in theanode cathode gap. Once the electron goes throughthe anode screen there is no accelerating electricfield. However, it is a good estimate to see if theelectron will significantly decelerate in the gap dueto the self generated magnetic field. This estimatedoes not attempt to deal with the radial componentof the space charge electric field . The effect of thespace charge field is probably not small but verydifficult to estimate given the two conductingsurfaces. The field will probably reduce the selfpinch. A more complete treatment of the beam inthe drift region shows it to be marginally stable.G

The instability measurements were madewith a gun voltage of 205 kV with a 2.8 cm anodecathode gap. The gun is driven by an 180 ns longco-axial water line with a laser triggered switch.The current was 60 kA and the maximum self

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Volume 2

generated magnetic field is 372 gauss. The turningpoint is z = 6.7 cm, well above the anode-cathodegap. Therefore, according to this simple estimate,at this operating point there should be no real needto apply an external magnetic field to prevent selfpinch.

Figure 3 shows a series of voltage traces andintegrated current traces for three values of guidemagnetic field. In the lowest field case the currenttrace is distorted and the voltage probe shows highfrequency oscillations on the voltage pulse. A B-dot probe was inserted just above the anode-cath-ode gap approximately 15 cm from center. Figure 4shows the oscillations for the more sensitive probe.No integrator was used with this probe to retainthe high frequency response. The expandedtimescale photograph shows the highest frequencyto be around 1.5 GHz., however the oscilloscopebandwidth is 1 GHz. and higher frequency compo-nents could be filtered out. The amplitude of theinstability vs the applied guide magnetic field isshown in Fig. 5. The critical field for the turn on ofthe instability appears to be very close to the esti-mated value of the self generated magnetic field. Itis not obvious why the electron beam should gounstable at this point.

If a virtual cathode were to develop in theshort drift region between the anode screen and thekapton foil\laser gas interface then the electronswould slow down as a beam-plasma instabilitywould develop. The growth rate of the instabilitycan be estimated by

~= n~virt.cath. (3)~= ~ &LoPbeam2: 3 n~beam

The growth rate will be 1/2 the beam elec-tron plasma frequency when the virtual cathodedensity is only 12% of the beam density. The seedfor the virtual cathode is probably backscatter from

63

64

Fig. 3. EleCh-OJZgulzz?oltage aid infcgrf?fedcurrellf frflces fo)’ (n.) 400gflllss, (b.)200 gfllfsst(c.) ogauss. The f7ZX’J”f7~t’ jk’f fopvolfnge is 205 kV and fheflzw.flgc /7f7ffop currmf is60 kA. The spike on fhcvolfflgc fmce is n laserfriggcr ~iducif71.

the Kapton and laser gas. Kapton transmission ofhigh energy electrons significantly degrades below160 keV. If the electrons have a significant angleaway from normal, then higher energy electronswiIl‘beabsorbed or backscattered from the Kapton.In particular, as the Kapton absorbs electrons itwould become charged, increasing the number ofelectrons which would reflect.

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ReviewVolume 2

Fig. 4. Unintegrated B-dotcurrent probe located 15 cmfrom fhe horizontal andverfical centerline in fhe A-K gap wifh a 70 gaussmagnefic guide field.

I

Fig. 5. Amplifude of theoscillations on fhe B-dofprobe as a function of fheguide field strength. Theoscillation strength vnriedby 50% for zero guide field.

TechnicalReview

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INSTABILITYAMPLITUDEVS.GUIDEFIELD●

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Additional evidence that the instability isoccurring in the drift region is obtained from thevoltage and current traces prior to onset of theinstability.The voltage and currents are virtuallyunchanged from Omagnetic field to 1 kG, indicat-ing no significant change in the Child-Langmuirlimited flow, hence no significant pinching effect orcharge build-up in the anode-cathode gap prior toonset of the instability. The plasma frequency forthe beam (~b~~~) in the vicinity of the anodeshould be 1 GHz., close to the frequency observed.In order to examine some of these issues quantita-tively,computer simulations are required.

III. Simulation Results

One dimensional steady state analysis ofelectron flow is commonly used to predict behaviorof electron beams in real devices. While this ap-proach is successful in many applications wherethe dynamics are not strictly one dimensional, itcannot be applied to this problem. One approachwhich can be used is to use a 2-d code and assumea infinitely long cathode. However, as will bedescribed, three dimensional analysis became ismore relevant. Simulations using the experimentalparameters were performed on the ISIS PIC code atLos Alamos.s The code solves the complete set ofMaxwell’s equations together with a particle in celltreatment of the particle dynamics. The pulsedpower is fed into the simulation via a TEM waveon the cathode side of the simulation region. Thecathode is treated as a perfect conductor with spacecharge limited emission. The anode is treated as aIossless conductor of zero thickness.

Simulations have been performed in bothtwo and three dimensions. The two dimensionalsimulations correspond to the center plane of thethree dimensional simulations, the long and thinapproximation. The simulations show that there isa virtual cathode instability for some parameters.

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67The threshold is lowest for long drift spaces and noapplied magnetic field. The threshold for the two(three) dimensional simulations with a 2 cm driftspace is a voltage of about 270KV (21OKV).Thethreshold for the two (three) dimensional simula-tions with a 1 cm drift space is about 400KV(320Kv). An applied magnetic field comparable tothe self magnetic field stabilizes the system. Notethat the nominal drift space is 2 cm. However, inactual operation gas pressure forces the foil towardthe anode screen to give an effective drift spacewhich is thought to be about 1 cm.

Results from three different simulations arepresented in order to eIucidate the instabilitymechanism. The simulations are three dimensional,have approximately 100,000 grid cells, with ap-proximately 100,000 particIes. Since the particIesare concentrated in a relatively small number ofcells, there are many particles per cell in those cellsthat are occupied. The cathode has an emitting areaof 20 cm by 80 cm cm, the AK gap is 2.8 cm, thedrift distance is 2.0 cm, the voltage is 300 kV, andthe resulting current is approximately 80 kA. Theself magnetic field was approximately 0.5 KG forall three cases. The three values of the guide fieldsimulated were (a) OKG, (b) 0.125 KG, and (c) 0.25KG. These values are in the range of the experi-mental study,although the voltage is somewhathigh. Figure 6 shows the magnetic field recorded ata point in the AK gap for cases (a), (b) and (c) andFig. 7 shows the magnetic field recorded at a pointin the drift space for the same cases. Magnetic fieldoscillations exist at lower magnetic fields, whereasthe oscillations are eliminated by using largerguide magnetic fields. The magnitude of the oscil-lations increases substantially in the drift region.Also evident in Fig. 7 are the presence of low fre-quency oscillations due to the gross pinching of theelectron beam.

68

Fig. 6. Traces of themagnetic field in thedirection of the cathode( the “three” direction )inside the AK gap butoutside of the beam. Theinstability in present inparts (a) and (b) but isabsent in part (c).

m

m

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Fig. 7, Traces of thenmgnetic field in the shortdirection of the cathode( the “three” direction )inside the drift region butoutside of the beam. Theinstability in present inparts (a) and (b) but isabsent in part (c). Noticethat the low frequencyoscillations persist in evenfor case (c). Theseoscillations are due to thebeam self pinching, whichperiodically changes theextent of the beam, thuschanging the strength ofthe magnetic fieZd at thepoint of observation.

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The mechanism for creation of the highfrequency oscillations is dire