a study of high-frequency properties of plasma and...

A STUDY OF HIGH-FREQUENCY PROP ER TIES OFPLASMA AND THE IN FLU ENCE OF ELEC TRO MAG NETIC

RA DI A TION FROM IR TO XUV

by

Vladimir A. SRE]KOVI]1*, Desanka M. ŠULI] 2,Ljubinko M. IGNJATOVI] 1, and Milan S. DIMITRIJEVI] 3, 4

1In sti tute of Phys ics, Uni ver sity of Bel grade, Bel grade, Ser bia2 Un ion – Nikola Tesla Uni ver sity, Bel grade, Ser bia

3 As tro nom i cal Ob ser va tory, Bel grade, Ser bia 4 Paris Ob ser va tory, Meudon Cedex, France

Sci en tific pa perhttp://doi.org/10.2298/NTRP1703222S

On the ba sis of cal cu lated val ues for the con duc tiv ity in an ex ter nal elec tric field, we de ter -mined the high-fre quency char ac ter is tics of plas mas un der ex treme con di tions (e. g. denseplasma fo cus de vice). The ex am ined range of fre quen cies cov ers the IR, vis i ble, UV, XUV upto X re gions and the con sid ered elec tronic num ber den sity and tem per a ture are in the rangesof 1021 cm–3 £ Ne £ 5×1024 cm–3 and 2×104 K £ T £ 106 K, re spec tively. The data ob tained us ingthis method are im por tant for plasma fo cus re search, lab o ra tory plasma re search, in ves ti ga -tion of at mo sphere plas mas of as tro phys i cal ob jects like white dwarfs with dif fer ent at mo -spheric com po si tions.

Key word: plasma fo cus, plasma, dy namic prop erty, elec tro mag netic ra di a tion

INTRODUCTION

The prop er ties of dense plas mas un der ex tremecon di tions are of great in ter est in var i ous re searchfields like nu clear phys ics, as tro phys ics, ter res trialphys ics, plasma phys ics [1-10]. Ex plor ing and im -prov ing the new cal cu la tion pos si bil i ties, sim u la tiontech niques and the ex ten sion of nu mer ous mod els incon nec tion with the dy namic prop er ties in the phys icsof high en ergy are in the fo cus of in ves ti ga tors now a -days [11]. Fur ther in ves ti ga tion of strongly cor re latedplasma and in ves ti ga tions of its elec tronic prop er tiesre main an on go ing prob lem. For an ex am ple the o ret i -cal cal cu la tions and mea sure ments of re flec tivity areim por tant be cause of its pos si ble use as di ag nos tic tool in the area of high-den sity and high-en ergy phys ics [9,12].

De ter min ing high-fre quency (HF) plasma prop -er ties in the ho mo ge neous elec tri cal field has been asub ject of sub stan tial in ves ti ga tion re cently and is ex -pe ri enc ing a real boom these days. The rea son for thisis the rapid de vel op ment of ex per i men tal fa cil i ties inthe field of ex treme con di tions aimed for dem on strat -ing nu clear fu sion in the lab o ra tory [13, 14], for de vel -op ing in tense ra di a tion sources with spe cial prop er ties

[11, 15], and to in ves ti gate im por tant and fun da men talphys ics pro cesses in as tro phys ics [16]. Un for tu nately,the the o ret i cal work and the ex pla na tions of these ex -per i men tal mea sure ments lag be hind ex per i men tal re -search and do not fol low it. There fore, there was aneed for more se ri ous and even harder work in the field of the o ret i cal work. There are two ba sic ap proaches tothose the o retic stud ies: the gen er al ized Drude-Lo rentz model, [17], or the re view [18], and the method of mo -ments [19]. In ad di tion, we have been work ing on thedi rect ex ten sion of the mod i fied ran dom-phase ap -prox i ma tion for the cal cu la tion of the HF prop er ties[20]. It should also be noted here, that we have to takead van tage of grow ing high per for mance com put ingre sources in the area of sim u la tions and catch up andreach ex per i men tal research.

The aim of this work is de ter mi na tion of HFchar ac ter is tics of non-ideal plas mas. All the pre sentedval ues, like the elec tri cal per me abil ity and the co ef fi -cients of refractivity and re flec tivity, as well as elec tro -mag netic (EM) field pen e tra tion depth, are ob tainedby the help of the nu mer i cally cal cu lated val ues for thedense plasma dy namic con duc tiv ity in an ex ter nal HFelec tric field. The de ter mi na tion of the dy namic con -duc tiv ity of a highly ion ized plasma in a HF ex ter nalelec tric field, as well as ob tain ing the other plasma pa -

V. A. Sre}kovi}, et al.: A Study of High-Fre quency Prop er ties of Plasma and the ...222 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2017, Vol. 32, No. 3, pp. 222-228

* Cor re spond ing au thor; e-mail: [email protected]

ram e ters re lies on the method pre sented in [20]. Themethod it self is based on the self-con sis tent field the -ory de vel oped for the static plasma trans port co ef fi -cients cal cu la tion, such as static plasma elec tric con -duc tiv ity, etc. Af ter the ex per i men tally proven va lid ity of the pre sented the ory in the in ves ti gated do main ofthe elec tron den si ties and tem per a tures, the methodwas tested in the do main of more dense, non-idealplasma i. e. was ap plied for cal cu la tion of con duc tiv ity of De gen er ate B-type (DB) white dwarf at mo spheres[21, 22] in wide area of elec tron den si ties and tem per a -tures.

This work fo cuses on the de ter mi na tion of dy -namic char ac ter is tics of dense plas mas as well as onin tro duc ing us able form of the pre sen ta tion of cal cu -lated data (fit ting for mula) im por tant for lab o ra toryap pli ca tions, e. g. dense plasma fo cus de vice (DPF), as well as for as tro phys ics mod el ing. All the data pre -sented here will very soon be ac ces si ble throughhttp://servo.aob.rs as web ser vice and da ta base.

USED MODEL METHOD

This pa per con sid ers a highly ion ized plasma in a ho mog e nous and mono chro matic ex ter nal elec tricfield

r rE E( ) exp( )t i t= -0 w . Ac cord ing to [21] and [23],

the dy namic elec tric con duc tiv ity of a strongly cou -pled plasma

s w t r

s w s

w( ) ( )( )

( )

( )Re

= -é

ëêù

ûú=

= +

ò4

3

2

0

e

mE

w E

EE E E

i

4 d

dd

lm ( )w (1)

is pre sented by the ex pres sions

s wt

wtrRe ( )

( )

[ ( )]

( )( )=

+-

é

ëêù

ûúò

4

3 1

2

20

e

m E

w E

EE E E

E4 d

dd

(2)

s wwt

wtrlm 2(E)]

d

d( )

( )

[

( )( )=

+-

é

ëêù

ûúò

4

3 1

2 2

0

e

m

E w E

EE E

4

dE

(3)Here e, m, and E are the charge, mass, and en ergy

of the free elec tron, r(E) is the one-elec tron states den -sity in the en ergy space, w E E( ) [exp( ) ]= - + -b bm 1 1– the Fermi-Dirac dis tri bu tion func tion, m – the chem i calpo ten tial of the ideal gas of the free elec trons with theden sity ne and the tem per a ture T, b = -( )k TB

1 , and there lax ation time t t wtw ( ) ( ) / ( ( ))E E i E= -1 .. In up perex pres sions t(E) is the “static” re lax ation time and themethod of de ter mi na tion of t(E) is de scribed in the pre -vi ous pa pers [21, 23, 24] in de tail.

Other HF plasma char ac ter is tics can be ex pressed in terms of the quan ti ties sRe(w) and slm(w). In this waythe plasma di elec tric per me abil ity is shown as

e wws w e w e w( ) ( ) ( ) ( )Re= + = +1

4i i

plm (4)

where e w w s wRe ( ) ( / ( )= -1 4p ) lm and e wlm ( ) == ( / ( )4p w s w) Re . The co ef fi cients of re frac tion n(w),and re flec tion R(w), are de ter mined as

n n in

Rn

n

( ) ( ) ( ) ( ),

( )( )

( )

Rew e w w w

ww

w

= = +

=-

+

lm

1

1

2 (5)

where, bear ing in mind that

e w e w e w( ) [ ( ) ( )]Re/= +2 2 1 2

lm

the real and imag i nary part of refractivity are given by

nRe Re/( ) [ . ( ( ) ( ))]w e w e w= +05 1 2

andnlm ( )w = -[ . ( ( ) ( ))]Re

/05 1 2e w e w .

From here the equa tion for the re flec tivity couldbe ex pressed as

R( )( ) ( ) ( )

( ) ( ) ( )

Re

Re

we w e w e w

e w e w e w=

+ - +

+ + +

1 2

1 2(6)

In such a way, we de ter mined the other pa ram e -ter of in ter est i. e. the pen e tra tion depth of elec tro mag -netic ra di a tion into plasma, D(w). This quan tity is theskin-layer width de ter mined as the in verse imag i narypart of the elec tro mag netic field wave num ber D(w) ==.(c/w)[1/nlm (w)] where c is the speed of light and nem

(w) is de fined above.Here, it is com mon to use a dimensionless cou -

pling pa ram e ter, the plasma non-ideality co ef fi cient G,that char ac ter izes the phys i cal prop er ties of theplasma. It is of es pe cial im por tance in de scrib ing ofdense, non-ideal plas mas, as ones con sid ered in thispa per. The men tioned pa ram e ter G = e akT2/ as suchchar ac ter izes the po ten tial en ergy of in ter ac tion at av -er age dis tance be tween par ti cles a = (3/4p.Ne)1/3 incom par i son with the ther mal en ergy. The well-knownBrueckner den sity pa ram e ter rs = a/aB is the ra tio ofWigner-Seitz and Bohr ra dius.

RESULTS AND DISCUSSION

As the con tin u a tion of pre vi ous in ves ti ga tion,that is of in ter est for DB white dwarf at mo spheres [21,22] and high en ergy re search, the cal cu la tions of theHF char ac ter is tics of plasma in wide area of elec tronden si ties and tem per a tures were con ducted. The in -ves ti gated ar eas are of in ter est for both of the lab o ra -tory ex per i ments, e. g. DPF, and at mo spheres of dif fer -ent stel lar types like de gen er ate a-type (DA), orde gen er ate c-type (DC) white dwarfs.

The elec tri cal per me abil ity and the co ef fi cientsof refractivity and re flec tivity of dense non-ideal plas -mas are de ter mined on the ba sis of nu mer i cally cal cu -lated val ues for the dense plasma dy namic con duc tiv -ity in an ex ter nal HF elec tric field. Here are con sid ered elec tronic num ber den sity and tem per a ture in the

V. A. Sre}kovi}, et al.: A Study of High-Fre quency Prop er ties of Plasma and the ...Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2017, Vol. 32, No. 3, pp. 222-228 223

ranges of 1021 cm–3 £ Ne £ 1024 cm–3 and 2×104 K £ T £106 K, re spec tively. The ex am ined range of fre quen -cies w cov ers the re gion 0 £ w £ wp, where wp ==.(4p.Ne.e2/m)1/2 is the plasma fre quency.

Within the con sid ered ranges of vari a tion ofelec tron den sity and tem per a ture, the be hav ior of theHF plasma quan ti ties with the in crease of the Ne and Tis as it is ex pected. This means that the be hav ior ofthose char ac ter is tics, with the in creas ing of Ne and T,re mains the same within the whole range of elec tronnum ber den si ties and tem per a tures 1021 cm–3 £ Ne ££.1024 cm–3. The same con clu sion is also re lated to allyielded pa ram e ters, in clud ing the pen e tra tion depthD (w). From the re sults it could be ob served that for thelong-wave length ra di a tion, i. e. fre quency tends tozero, there is in ter est ing de pend ence on tem per a tureand fre quency of skin-ef fect D at sur face plot in fig. 1.Also, the re flec tivity co ef fi cient tends to 1, min ing theplasma layer acts as a mir ror.

Par tic u larly, here we give spe cial at ten tion toanal y sis of re flec tivity data be cause this quan tity isvery im por tant in high en ergy den sity plas mas (mea -sure ments as an im por tant di ag nos tic tool pro vide in -for ma tion about the den sity pro file). In fig. 3 we com -pare the be hav ior of re flec tivity co ef fi cient R,de ter mined in this pa per, with the be hav ior of the cor -re spond ing quan ti ties cal cu lated in [25] for rs = 5 andG = 0.1, G = 0.5, G =1 and also for G = 0.5 and rs = 0.4, rs

= 1 and rs = 2. Here rs and G are previous de finedBrueckner den sity and non-ideality pa ram e ters, re -

spec tively. The cor re spond ing curves show ing the o -ret i cal data from [25] are pre sented by full, dashed,and dot ted lines. Fig ure 3 shows a qual i ta tive agree -ment (the dif fer ences are less or close to 10-25percents) of our re sults with the re sults of [25] in thewhole re gion of w and for all val ues of andnon-ideality pa ram e ters. By an a lyz ing the be hav ior ofcor re spond ing curves in figs. 2 and 3 one can see theten dency that in crease in the den sity and cou pling pa -ram e ters leads to a de crease in the re flec tion in dex.Then, in fig. 3 it is seen that at fre quen cies w < wp there flec tion oc curs most ef fec tively and at w ® wp theplasma be comes much more trans par ent.

In fig. 4 is pre sented a sur face plot of the plasmare flec tivity for dif fer ent val ues of den sity as a func tionof tem per a ture and fre quency. With this fig ure it ispos si ble to map the re gion of high non-ideality. For agiven den sity the ap pro pri ate panel from fig. 4 can bese lected and the value of R found for the cho sen pair oftem per a ture and fre quency.

In or der to more sim plify the pre sented re sultsfor the fur ther use a fit ting pro ce dure is ex ploited to get the plasma re flec tivity from the tem per a ture and fre -quency for var i ous val ues of elec tron den sity. For allan a lyzed den si ties the ex pres sion is given by

R x y a bx cy dx ey

fxy gx hy ixy jx y

( , ) = + + + + +

+ + + + +

2 2

3 3 2 2(7)

where x = w/wp ra tio of fre quency and plasma fre -quency (0 £ w £ wp), y = T, tem per a ture and a-j are cal -cu lated pa ram e ters. These pa ram e ters were com puted


Fig ure 1. Left: sur face plot of theskin layer depth for Ne = 1024 cm–3

and 2×104 K £ T £ 106 K;right: same as in left but forNe = 1024 cm–3

Fig ure 2. Down: the plasma re flec tivity for Ne = 1024 cm–3

and 2×104 K £ T £ 106 K; up: same as in lower panel but forNe = 1023 cm–3

Fig ure 3. The plasma re flec tivity as a func tion of w/wp forvar i ous val ues of quan ti ties rs and G; pre sent re sults areshown to gether with re sults from [25]

from a stan dard 3-D fit ting pro ce dure, (best-fit value i.e. r-Square close to one) and their val ues as the func -tion of den sity are shown in tab. 1. We must em pha sizethat those ex pres sions give good re sults for val ues offre quen cies in the re gion 0 < w £ 0.3 wp al though theyhave to be ap plied with cau tion be yond this re gion.

Some HF quan ti ties: The be hav ior of some HFquan ti ties for var i ous plasma con di tions is il lus tratedhere al though it is not the main topic of our re search.The rea son of its rep re sen ta tion is that it en ables de ter -mi na tion of other trans port prop er ties. The elec tri calper me abil ity and the co ef fi cients of refractivity ofdense non-ideal plas mas are de ter mined on the ba sis of nu mer i cally cal cu lated val ues for the dense plasma dy -namic con duc tiv ity in an ex ter nal HF elec tric field.Here we will pres ent the most ex treme cases which wethink are the most in ter est ing for read ers (all other datacov er ing the ranges of 1021 cm £ Ne £ 1024 cm–3 and104 K < T < 106 K will be pre sented in da ta base).

On fig. 5 the real and imag i nary part of re frac tion co ef fi cient for den sity Ne = 1024 cm–3 in tem per a turere gion 2×104 K £ T £ 106 K is pre sented as a func tion offre quen cies w. From this fig ure one can see that the be -hav iour of this HF quan tity is sim i lar to the in ves ti -gated ones in up per ma te rial.

Fur ther ana lys ing the pre sented re sults it couldbe ob served that for the long-wave length ra di a tion,i. e. fre quency tends to zero, even for most ex tremecases the imag i nary part of the di elec tric func tion di -verges, i. e., e w( )® 0 is much greater than real part of the di elec tric func tion (fig. 6).

As tro phys i cal rel e vance: As we noted in theabove text the in ves ti ga tions of HF char ac ter is tics is im por tant for re search of at mo sphere plas mas of as tro -phys i cal ob jects like white dwarfs with dif fer ent at mo -spheric com po si tions (DA, DC etc.), and some otherstars (M-type red dwarfs, Sun, etc). In or der to dem on -strate this as tro phys i cal rel e vance fig.7 is pre sented


Fig ure 4. Sur face plot of the plasma re flec tivity for dif fer ent val ues of elec tron den sity 1021 cm–3 £ Ne £ 1024 cm–3 as afunc tion of tem per a ture and fre quency

Ta ble 1. Pa ram e ters (from a to j) needed for R(x, y) fit given by eq. 7, as a func tion of elec tron den sity where x = w/wp and y = T

Fp/Ne 1021 [cm–3] 1022 [cm–3] 1023 [cm–3] 1024 [cm–3]

a 0.935052097 0.947234943 0.967203184 0.981546732

b –1.317415548 –1.44143753 –1.253135644 –0.863649267

c 7.29876E-07 4.12914E-07 9.29337E-08 –3.83153E-10

d 2.200777151 2.59112663 2.623576736 2.2174305

e –1.81598E-12 –9.04495E-13 –1.02346E-13 2.41917E-14

f 1.97343E-06 1.52641E-06 6.20522E-07 1.38124E-08

g –1.535152818 –1.760714503 –1.851589254 –1.69034055

h 1.15411E-18 5.43242E-19 3.83954E-20 –1.3866E-20

i –1.17436E-12 –6.97197E-13 –1.04538E-13 6.15697E-14

j –3.59279E-07 –4.84845E-07 –3.40011E-07 –6.11506E-08

here. In fig. 7(a) is il lus trated, by sur face plot, the con -duc tiv ity as a func tion of the log a rithm of Rosselandopac ity log t for DB white dwarf at mo sphere mod elswith log a rithm of sur face grav ity log g = 8 and var i ous

ef fec tive tem per a tures from Teff = 1.2×104 K to Teff = =.3×104 K. Fig ure 7(a) shows that in creas ing the whitedwarfs ef fec tive tem per a ture, at a fixed value of thelog a rithm of Rosseland opac ity, leads, as a con se -


Fig ure 6. HF di elec tricper me abil ity for den sityNe = 1023 cm–3 and1024 cm–3 in thetem per a ture re gion2×104 K £ T £ 106 K

Fig ure 5. Left: the real partof re frac tion co ef fi cient forden sity Ne = 1024 cm–3, and2×104 K £ T £ 106 K;right: imag i nary part ofre frac tion co ef fi cient for thesame val ues of T and Ne as in left

Fig ure 7(a). Sur face plot: onductivity as a func tion ofthe log a rithm of Rosselandopac ity log t for DB whitedwarf at mo sphere mod elswith log a rithm of sur facegrav ity log g = 8 andvar i ous ef fec tivetem per a tures Teff = 1.2×104

K, ..., 3×104 K; (b) ex am ple ofHF con duc tiv ity for log t = 2for DB white dwarfat mo sphere mod elsTeff = 3×104 K and log g = 8.For the cal cu la tions ofplasma char ac ter is tics of DB white dwarf at mo spheres,the data from Koester(pri vate com mu ni ca tion)were used

quence, to an in crease of con duc tiv ity. Also, it is il lus -trated here in fig. 7(b) plot of HF con duc tiv ity for log t = 2 for DB white dwarf at mo sphere mod els Teff = 3×104 K and log g = 8 which cor re sponds to elec tron den -sity: ~1019 cm–3 and tem per a ture: ~8×104 K. From thepre sented fig. 7(a,b), it can be ob served a reg u lar be -hav iour of the con duc tiv ity con sid er ing the char ac ter -is tics of DB white dwarf at mo spheres.

CON CLU SIONS AND PER SPEC TIVES

In this pa per we cal cu lated the dy namic char ac -ter is tics of plas mas of in creased non-ideality un der the in flu ence of IR to XUV, elec tro mag netic ra di a tion.The pre sented data cov ers wide re gion of elec tron den -si ties and tem per a tures. These re sults, es pe cially fitfor mula, can be ap plied in the ex per i ments of DPF,high pres sure dis charge, shock waves etc., wherestrongly non-ideal plas mas, in clud ing ex tremelydense plas mas, are cre ated. The de vel oped methodpres ents a use ful tool for the re search of white dwarfswith dif fer ent at mo spheric com po si tions (DA, DCetc.), as well as for some other stars (M-type reddwarfs, Sun etc.). In the near fu ture we aim to fur therim prove anal y sis us ing GEANT code [26]. Also, ourplan is to pres ent the re sults ob tained dur ing this in ves -ti ga tion in da ta base which can be ac cessed di rectlythrough http://servo.aob.rs as a web ser vice sim i larlyto the ex ist ing MolD and E-MOL da ta baseshttp://servo.aob.rs/mold, http://servo.aob.rs/emol/[27, 28].

AC KNOWL EDGE MENT

The au thors are thank ful to the MESTD of RSfor sup port of this work within pro jects 176002 andIII44002.

AU THORS' CON TRI BU TIONS

The cal cu la tions and sim u la tions were car riedout by V. A. Sre}kovi}, who wrote the manu script. M.S. Dimitrijevi}, D. M. [uli}, and Lj. M. Ignjatovi} par -tic i pated in ed it ing and re vis ing of the manu script. Allau thors dis cussed the re sults and com mented on themanu script.

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Re ceived on June 22, 2017Ac cepted on July 17, 2017


Vladimir A. SRE]KOVI], Desanka M. [ULI],Qubinko M. IGWATOVI], Milan S. DIMITRIJEVI]

ISPITIVAWE VISOKOFREKVENCIJSKIH OSOBINA PLAZME I UTICAJ ELEKTROMAGNETNOG ZRAÊWA U DOMENU OD IR DO XUV

Na osnovu izra~unatih vrednosti provodnosti u spoqa{wem elektri~nom poqu,odre|ene su visokofrekvencijske karakteristike plazmi u ekstremnim uslovima (na primer, kodplazma fokusnih ure|aja). Ispitivani opseg frekvencija pokriva IR, vidqivu, UV, XUV sve do Xregiona i razmatrane su elektronske gustine i tem per a ture u rasponu 1021 cm–3 £ Ne £ 5×1024 cm–3 i2×104 K £ T £ 106 K, respektivno. Podaci dobijeni kori{}ewem ove metode va`ni su za istraìvawaplazma fokusnih ure|aja, za istraìvawa laboratorijske plazme, a tako|e i za istraìvawaplazme astrofizi~kih objekata, kao {to su beli patuqci sa razli~itim atmosferskimkompozicijama.

Kqu~ne re~i: plazma fokus, plazma, dinami~ko svojstvo, elektromagnetno zra~ewe

a study of high-frequency properties of plasma and...

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