Anisotropy of electrical properties of a layer of spherical particles located near a substrate
In this paper, we study the effects of a semi-infinite matrix disperse system on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a...
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| Date: | 2001 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine
2001
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| author | Grechko, L. G. Gozhenko, V. V. Whites, K. W. |
| author_facet | Grechko, L. G. Gozhenko, V. V. Whites, K. W. |
| author_institution_txt_mv | [
{
"author": "L. G. Grechko",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "V. V. Gozhenko",
"institution": "Інститут хімії поверхні НАН України"
},
{
"author": "K. W. Whites",
"institution": "University of Kentucky"
}
] |
| author_sort | Grechko, L. G. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-11-27T09:42:39Z |
| description | In this paper, we study the effects of a semi-infinite matrix disperse system on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a sphere accounting for the interaction between ambient particles and the substrate. The polarizability tensor and resonant frequencies of a single sphere show anisotropy due to the influence of a substrate. |
| first_indexed | 2025-09-24T17:25:23Z |
| format | Article |
| fulltext |
.:.datlon
lnd an
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::riity foт
з :ratеrials
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Aп[IsoTRoPY oF ЕLЕСТRIсAL PROPЕRTIЕS
oF A LAYЕR oF SP}IЕRICAL PARTIсLЕS
LOсATЕD NЕAR A SUBSTRATЕ
L.G. Gгесhkol, v.v. Gozhеnkol and к.w. Whitеs2
I Institutе of Suфcе Сhеmistrу Nаtionаl Асаdеrnу of Sсiеnсеs
I7 Gеп. Nаlmov str,, Kуiv-I64,03680, (]KRАINЕ:
2Dep,аrtmе-n,:};i::':;:;g:{f :1:{',ili::::,;':f Kentuсkу
453 Аndеrson Hаll, Lехitlgtotl, Ky 10506-0046, rJSА,, ,-.oil,' whitеs@е7у.ukу.еdu
Abstraсt
In this рapег, wе study thе еffесts of a sеmi-infinite matгix dispеrsе systеm on thе
extегnal elесtromagnеtiс radiation in the еlесtrostatic appгoximation. йith tь" ь"lp
of our pгейous tесhniquе, wе obtain gеneгal ехpгissions foг the multipo1Ь
ехpansion сoеfiiсiеnts of thе еlесtгiс potеntial for a sphеre aссounting for thе
intеraсtion bеtwееn ambiеnt paгtiсlеs and the substгatе. Тhе polaгizabilГty tеnsoг
and rеsonant fгеquеnсies ofa singlе sphеrе show anisotгopy duЬ to thе influenсе of
a substгatе,
1. Introduсtion
Intеrеst in matгiх dispersе systеrns (I\,Ds) is stimulatеd, fiгst of all' bу thе possibility of
manufaсtuгing matегials with prediсtеd optiсal pгopегtiеs. At thе samе timЬ' thе propепiй of
МDS maу stгongly diffег from thosе of thе matегials usеd for thе foгmation oгtr,шis [l1. In tье
thеoгеtiсal studiеs МDS aгe usually сonsidегеd as infinitе systеms'
In this woгk, wе takе into aссount thе effeсts of an МDS intегfaсe. Namely, thе МDS is
сon'sidегеd as a half spaое diеlесtгiс matгiх йth a planе intегГaсе sеpaгating ii aom anothег
half spaсе homogеnеous dielесtгiс. Thе matгiх is fillеd йth spheгiсal inсIusions of diffеrеnt
diamеtегs loсatеd nеar- thе substгatе foгming a layer of randoпrly oг гegularly aггangеd
particlеs. The геsults [2] obtained foгthe monolayеr of spheres on a dielесtгiс substrate.un b"
obtainеd fгom ouг modеl as a paгtiсulaг сase, Basiсallу, this woгk is a gеnегalization of [2,3,4]
2. Basiс equation
Wе сonsidег thе semi-infinitе МDS сonsisting of dielесtгiо sphегеs of diffегеnt diamеtегs
еmbeddеd in a homogenеous diеlесtriс (ambiеnt) as shown in Fig. 1. Алothеr half spaсe is
fillеd with anothег homogenеous diеlесtгiс (substгatе)' Thе systеm is plaсеd in thе еlесtгiс fiеld
proportional to е,*.Let е"(a),с,(a) and я,(rо) bе thе diеlесtгiс funсtions of thе ambiеnt,
substratе and thе l.h sphеге, rеspесtivеly, and .R, bе thе гadius ofthе iф sphеге.
119
Fig. 1. G.oЬ.try of thе sеmi.infinitе mаtriх dispегse system
Lеt thе wavеlеngth of thе ехtегпal еlесtгоmagnеtiс fiеld bе. muсh largег than гаdii of thе
sphегеs and thе distanсеs bеtwееn thеm' Ii othеr words, wе usе thе еlесtrostatiс
approхimation' in suсh a оasе rеsulting еlесtгiс fiеld is сausеd by the inteгaсtion ofthe eхtегпal
fiеld with thе МDS and thе substratе and its potеntial sаtisfiеs thе Laplaсе еquation
Arд(r) = o (l)
in thе геgions I. inside МDS (out of sphеrеs)' Il - insidе thе sphеrеs, III. inside thr substratе,
and standaгd boundary соnditions
( a,,,, a,у , \ Q)
(ttri=v ,)o,,.1 t,:it= t i1f t ', l dn. ,
О|1 ; )\ ' ' 'Crj
whеrе q is diеlесtriс funсtion of thе mattег filling out thе 16 regiоn (i:I,II' ПI)'
, is thе rеsulting fiеld potential in thе lff rеgion,
o , o"no,.. thе сommon bound suгfaсе of thе rеgions i and 1.
Using idеas of thе imagе and multipolе ехpansion mеthods of solving oГ еlесtrostatiс pгoblеms
;;;;; a solution of йe pгoblеm (l,2) in thе folloйng foгm:
,, =,,"', +Т'V!-,,,,,*iV',"o,,.' =_E.f +\А,,,I'.,,(F,)oZдi',,П.(pi) (3)
у, =|B,ьG^G) '
/'' = y'lj, +,/{ +|с,,.|-,.(F'),
V'', = -Eov = -(Е ** + Е,У + E".,\
,/у:, = _ E;i = -(nЕ o', + bЕ
",у
+ сЕ
",,)
wherе FL^(r)=r,'r,-(r); G'^(v)=r'Y,^(F);
p =7-f,, F,'=f -7,';
r,- is a radius-vесtor of thе сеntеr of thе lф sphеrе ; i,is a гadius.vеоtoг оf thе iф sphеre оentег
imagеandу,o,,tsaсonstantсontributiontothеpotеntia|уI]1геlatedwithaсhoiсеofradius-
vесtor origin point, Notе, that all thе individual tегms in (3,4'5) automatiсallу satisfu еquation
(l)' and (6) ехрrеsses.i. iJ.^ offo,." linеs rеfгaсtion on the boundary ofdiffеrent mеdia'
(4)
(s)
(6)
180
and using thе геlation [5]
Thе unknown сoеffiсiеntS А,.,,А,,^,,B1,',С1,,,а,b,с aге obtainеd aftег applying thе
boundary соnditions (2) to thе ехpansions (3, 4, 5)
3. Boundary сonditions on thr substrate surfaсе
1 Potential сontinuity сondition on thе suгГaс Q o t -пl take thе foгm
(AЬ - Eфr -'у{l *Z{А,ь,Л.(ф)+ Аi6F6(p)_с,t,'t',,(p,)| - 0ilnt or-!il
Diffеrеnt tеrms herе havе diffегеnt aгgumеnts. It pгoves to bе morе сonveniеnt to геduсе allthе tегms to a сommon aгgumеnt, e'g. to p,. Using thе faсt, that Гoг anу point at thе Ьoundагv
suгГaсе o, -,,,
-/^\pi =\pi,0i,pi)
p! = bi,ol,pi) = b,," - oi,ai)
Yь,(o _ 0,p) = (-t)l*^ r,^(o,сp)
wе obtain
ьс/7r,, * AЕorir -v{' *Тh6* (_ l)/t,, Аit. -C,мF,,,(F) =ilm o!_trl
whеге wе havе usеd dесomposition 7 =7ll +r-r and analogous to it for
obtainеd еquation is еquivalеnt to thе sеt
| ьz6, .7', =О.l oЕo' .Ft -v{t =o
|n,, * (_ l)l*", А,11. - C 6, = О
2. Potеntial deгivativе сontinuity сonditioп on the suгfaс Q o
t -tlt in viеw .f * = } ,uк"
thе foгm
(с€" _ сo)Еo, + €o},Аit.*',,,G)- с.|Аi1.*',,G)_ u,Е,,,,
}г,^(F,) o,=,,,o
Again, геduсing all thе tегms to aгgumеnt p, and using rеlation
*r,,,(o - o,a)= (- 1)/*--t
*o,.(t.r) .
whiсh сan bе sееn from [5]
! 1 1 1, у,. 1o,,l] =|,!,*,')', . =',,'.1)
( ц - lr ) r, -, -сz, уr,, *,хТiз)-j l;_;J )уt-t,,@'tl)*
*Г ,, -,, 1i(u, -/ + 1 .)..
|(2t -|х2I -1)] [ы-;I )Yt-''^10,,p1
'
wе obtain еquation
(cе, _ сo)Еo, +||сoАiм + €o(l)/+.-l Аi1. - е,С;1,)*г,*(p) _ 0
oг equivalent set
U. oI-]I
:зcii of thе
:.есtrostatiс
::.э ехtегnal
(l)
. substrate,
(2)
пс ::oblеms
(з)
(4)
(s)
(6)
0,
лi=i,-i uo
(7)
l8l
(8)
( сE.-€o=О
i ^1r\o.А,и+td\_l/.
; ;;. solution of еq (7.8) is
| а=|
II D=r
!a
\ ч,1\ ^-:
l^
! vсi r- \tt9
,,,III -\ \ -t \Е .h"
1ч,0
[ с, )
' J' =
(_ l),-.
t
"
- €, А,,,
\''u. \ , €"'0,
| .v1 А-г
lLilи
_
^ -"I oo '"'
whеrе ho is ti.rе ьеigьt of thе global origin ovеr the substratе
4.Boundary:"1^ч:1"}",;,:};.i:Hlffi ::i:i:'!i;Ц.?{;J$!;ж
i ' on thе sufiaсe
_ A o
. i * L-,А, *, F,,,|p,) + | Аi,oi1;i;j Ц :,,^G'^(F,\
o'=-',,o'
Applуing rеpтеsеntatlo ^",,
=,, + p, aod b, --
-Рi _Q, -'i,\,wеli-knоwn
addition thеoгеm [6l
for sphегiсal haгmоniсs ( r < /l )
'u'
sP'.-*
- л)=
,\T,',,,.,r-tV
[Q}j/.,' (r) ,
-.l
(e)
|L+М!t=-М1
|=I+I'' lVI=m_Ш1 '
and takinginto aссount that дJ' бl. 1I.I
(л,.o, и,), wе obtain еquatlon
'yi'Ц(aj{/,ц.^Rj,,-,
itn,zу|I,\+,,ГL^,V'-r,\*4пfut(д,-l,).t-в,,'*.n}}=',.71+(Ёg'F)o1-11'
lй..' t / t\ Iп.(i-r,\,i*j
whегe R;\., - .'J = 1 o' '. i .....^,,,on fоr thе сoеffiсiеnts
Intегprеting this еquation as multiрolе ехpansion' wе сan obtain еxprе
t } ьу using standaгo.i,.йou,.
-\аa
r;^ '
.:'..:.:T-
Б*i; s\:, *!а,'R,flr"h,i,}]
|^p-,'-...Р,Тf'*\д,,,Г',,V,-7,\-1,.Гu,(i'-1)i-в,,,.R;\=^l4тЦr,ooо-з,*,
1 Д,/'nt / t, ' - .J /-) '',' '
I'tim
Whilе dеriving last eхprеss'""Ё:tЫ;Ё;[.i}: }:l = 6{:,
182
d. Б = аb ."i,.,) =
1
oouЁ',',lУ,;,(i)
Рotеntial dегivativе сontlnurty сondititоn on thе suгГaсе оf 7ф sphеге in viеw of
=.( takе thе foгm
I{+tz' l|)ry\,А,,,^+с,l,),B,ц,,-"J,4'Zt,'[4,.ri'^,h-ф+,1,s,*0,,,-o)]|,,.(a,)=-,"rЙ.D,l
Applying to this ехpгеssiоn thе samе pгoсеduге as еaгliеr, wе obtain rеlation
с"(l,, +|)ц,'-,А,o^ +е,I,$-|В,,. _".l,4'ZI/#[цJьV' _r,)*A'f*h,-r,)]=_|o""q Яn6b; ]
whеге
( l0)геm [6]
''й
t - :1PilбI-1I .
сoеffrсiеnts
i. JnЕЕr]
(11);l
o,)
r l-l
," )l li- Ir t\.
, \t - lt,L),
А" ,
=* und Р,l',,,
J. two еquatlons obtainеd fгom thе boundary сonditions on thе suгГaсе of 7s sphегеfогm the full sеt dеfining unknоwn сoеffiсiеnts А,,. and B,,. (notе,that ехpliсit foгm oГ l,]-аs funсtion of А,,,, was found еaгliеr, sее еq (9)) А.fteг somе transfoгmations it сan Ьеrеduсеd to thе foгm
Г B*. = /(А,'^)
JI bfl + К,,!,i^,fuu^ =|,,,,., ,
| ]]Й
L ii0
К| k = o,,Щ t- уу_ rf
*е - о)},
.'' l,€,-(l,+7)с'..!
.' 4 -l /^' Г=
tr''1,,-'=1oo,,'ьop,у,.^|Е,)6::,=а,''ь',]}{J1'cos0o6,|,o^,+sin0oе,,o6},j _sin 0oе-,'"6,|,,.,
'
u. = (,.'' Еo,,Еo,)= to(sin 0ocosсpo,sin do sin po,cos0)
'
Thе ехpliсit foгm of the funсtion / in (l0) is not neеdеd for fuгthег сonsidеration.
Sесond еquation of (10) сan bе wгitten in thетnatгiх fo.'
[i - kр=i or 2=[*к}.й,
that allows us to intегpгеt thе matгiх Й =|i nк|. , *ы"ь .onn..t, ехtегnal рotentia] matгiх
Р,,1. &nd multipоle сoеffiсiеnts А,7., &S thе multipolе polarizability matгix of the МDs sphегеs.
5. A singlе sphеre nеаr thе substrаtе. Thе геsоnant frequеnсies
Foг thе singlе sphеre nеaг thе substгate, wе сan obtain thе polaгizability tensoг in thе dipolе-dipole appгoximation by using (10).
n (o,, o
^r / \а=_rffi-€"\€-'")l o Gtt
[о 0
rvhеrе o, =|'," + L,(с - 1t ^ ^ \
L, =;l r *r,-l;J\ €.+Е,.l
183
i1
,,=+ ];'t'=lо' [i'ri = rr
Й is thе distanсe bеtwеen thе sphеrе's оеntеr and thе substгatе.
Let us сonsid". tь" .u,. oil-Ь,.ni,'' diеlесtгiс funсtions and я, = 1 (vaсuum);
'''2 / \ ., 01"
с(оi=l= .т!*_ ,^ , Е,(аl)=1+;} _;т-i,aJl-| a;-o,-iТol Uo'
Thе rеsonant frеquenсy is obtainеd bу using thе condition o,\о,.,) _ со. In ouг сasе
to thе following algеbгaiс еquation with геspесt to thе frеquenсy:
whеrе
6!. + а,0)' + а,a, + a{D + ao = О,
а,=i(y +r,)
it rеduсes
(12)
(13)
o, =
_('r +со!, +\'l -?i,* ,)
а, = _t(y
"oi
+yсl!, +L,, *i -!''-)
аo _- a!a|" *\'i, i - у fo ;, *}tr - l,у, fu
,,,
A solution to (12) neglесting damping \, = ," = o) is
/ \' l( Г
Ь|"Y =lЬ'*yzt.JUr -у,)z ++l,у,
2'
a2- " d2n, t', tzr'
whеге )'t=ш3*}, l'z_oi,+_f' '''=i, ,
Рaгtiсulaгlу, foг а mеtaliiс sphеre on thе diеlесtгiс
::9^']:*.n"m
(13), using thе inеquality
^;;Б'..,t
'
*" obtain thе following approхimate еxprеsslons
(^
i .,I)i,,=Ч-,,Ч (l4)
IJL
L", . ^'
l@(,:l)' = t&, t rt -t)L?
for thе ,oul i,=,,,1 resonant fгеquеnсiеs. Notе that ',/Ь is wеll-known suгfaсе plasmoс
fгеquеnсу of a sphеrе and ',i/,в is onе of a substratе.
As wе sее, substratе сhangеs thе dipolе momеnt of а spherе 11 .'u.h
a waу, that thе fou:
rеsonant frеquеnсies.;l. i; d. absoгption 'p..i*"'
of a sphеге' What сauses arising оf suс:..
numbеr of thе гesonant fгеquеnсiеs? First, 31. n"i','"1'ч^o.t:::'"s is obsеrved whеn th.
fleld diгесtion is paгallеl to ihe substratе, whi1e anothеr onе _ when pегpеndiсular, and thеs:
two pairs сon,t сoinсidе in addition' In gеnеral сase field has both thе сomponеnts an]
absoгption speсtrum *'Ь. ьu' гesonant frЪquеnсies respeсtivеlу..
nаir of frequеnсies arisе
Seсond, uno.,."i"in fiеld dirесtion (ll or tto thе substratе) thе pair offrequеnсies artsе.
duе to an intеraсtion Б..й.n suгfaсе рtu,п1on,-oг thе sphегe and of the substratе. Unc..
inсrеasing of thе distanl..ь.i*..n ,pь.,i unа ,uйstгatе this intеraсtion vanishеs and wе obtа
184
} ] . juсеs
well-known геsult: a singlе sгhегe and a singlе half-infinitе substrate absоrb гаdiation at thеfгеquеnсiеs o,,,/- and . o, ,,/- геspe1tivеlv'
,, ^i,З ', J2
6. Conсlusion
Wе obtainеd the gеnегal ехpгеssion for thе rеsonant frеquеnсy of the modеl systеm, whiсhis a diеlесtriс spherе in vaсuum on a diеlесtгiс substratе..тье iattе. гesults in splitting andshifting of thе rеsonant frеquеnсy dеpеnding on a diгесtion of thе ехtегna] field aссording to(l3) This allows onеto suggеst that Layегs of small paгtiсles on a substгatе possess anisotropiсеlесtгodуnamiсal pгopепies,
Aсknorvlеdgеment
The authors aсknowlеdgе finanсial suрpoгt from thе National Sсienсе Foundation thгough thеFaсulty Еarlу Caгееr Devеlopmеnt (CАRЕЕR) Awaгd ЕСs'962448б and an Еasteгn ЕuгopеProgгam Supplеmеnt'
Rеferеnсes
1. C.F. Bohгеn and P R-. Huffrnan, Absoгption and Sсattегing of Light bу Small Paпiсles. NеwYoгk: John Wilеy & Sons, 1983,
2 М T Haаrmans and D Bеdеaux, Тhе pоIaгizability and thе optiсal pгopегtiеs of lattiсеs andrandom distгibutions of small mеtal sphеrеs on a substrate l/ Thin Solid Films, l993, vol224' pp \17-1зlr
з' L G. Gгесhko, A'Ya' Blank, V v' N{otгiсh, A o Pinсhuk, L'V. Gaгanina' Diеlесtгiсfunоtion of matгiх dispersе s},stеms with mеtalliс inсlusions: aссount of multipo1е
intегaсtion bеtwееn inсlusions // Radio Physiсs and Radio Astгоnomy, 1997, уo|.2, pp l9-27
4 L.G. Grесhko, V'N. Pustovit, K w. Whitеs, Diеlесtгiс funсtion of аggregatеs of smallmеtalliс paгtiсlеs еmbеddеd in host insulating matrix l/ Appl. Phys Lеtt., 2000' vоl. 7б,no. 14, pp. 1854-1856.
5. D A. Vaгshaloviсh, A.}i Мoskalyov, V.K. Kheгsonsф, Quantum Тhеory of Angular
Мomеntum. Lеningrad, Nauka, l975.
6. o.R. Cruzan, Tгаnslation addition thеoгеms Гor sphегiсal wavе funсtions // Quaгt. Appl.Мath, l962, vol. 20, pp. 33-40
r 13)
i .еquality
t
I
I t'o'
h
t
hй -. эlasmon
L
F
}, .. the fouт
t .,з ot suсn
Ь.- rvhеn thе
!':.. and thеse
iг:.nеnts and
I
ir..:.;ies aгisеs
3 --зlе' Undеr
t l. : rvе obtain
t85
|
| id | oai:ojs.pkp.sfu.ca:article-57 |
| institution | Surface |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-24T17:44:39Z |
| publishDate | 2001 |
| publisher | Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | surfacezbircomua/6c/8c04e6786643f49f58c267fbbcb99a6c.pdf |
| spelling | oai:ojs.pkp.sfu.ca:article-572018-11-27T09:42:39Z Anisotropy of electrical properties of a layer of spherical particles located near a substrate Anisotropy of electrical properties of a layer of spherical particles located near a substrate Anisotropy of electrical properties of a layer of spherical particles located near a substrate Grechko, L. G. Gozhenko, V. V. Whites, K. W. In this paper, we study the effects of a semi-infinite matrix disperse system on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a sphere accounting for the interaction between ambient particles and the substrate. The polarizability tensor and resonant frequencies of a single sphere show anisotropy due to the influence of a substrate. In this paper, we study the effects of a semi-infinite matrix disperse system on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a sphere accounting for the interaction between ambient particles and the substrate. The polarizability tensor and resonant frequencies of a single sphere show anisotropy due to the influence of a substrate. In this paper, we study the effects of a semi-infinite matrix disperse system on the external electromagnetic radiation in the electrostatic approximation. With the help of our previous technique, we obtain general expressions for the multipole expansion coefficients of the electric potential for a sphere accounting for the interaction between ambient particles and the substrate. The polarizability tensor and resonant frequencies of a single sphere show anisotropy due to the influence of a substrate. Chuiko Institute of Surface Chemistry National Academy of Sciences of Ukraine 2001-06-10 Article Article application/pdf https://surfacezbir.com.ua/index.php/surface/article/view/57 Surface; No. 4-6 (2001): Chemistry, Physics and Technology of Surface; 179-186 Поверхность; № 4-6 (2001): Химия, физика и технология поверхности; 179-186 Поверхня; № 4-6 (2001): Хімія, фізика та технологія поверхні; 179-186 3154-8091 3154-8083 en https://surfacezbir.com.ua/index.php/surface/article/view/57/56 Авторське право (c) 2001 G. Grechko, V.V. Gozhenko, K.W. Whites |
| spellingShingle | Grechko, L. G. Gozhenko, V. V. Whites, K. W. Anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| title | Anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| title_alt | Anisotropy of electrical properties of a layer of spherical particles located near a substrate Anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| title_full | Anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| title_fullStr | Anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| title_full_unstemmed | Anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| title_short | Anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| title_sort | anisotropy of electrical properties of a layer of spherical particles located near a substrate |
| url | https://surfacezbir.com.ua/index.php/surface/article/view/57 |
| work_keys_str_mv | AT grechkolg anisotropyofelectricalpropertiesofalayerofsphericalparticleslocatednearasubstrate AT gozhenkovv anisotropyofelectricalpropertiesofalayerofsphericalparticleslocatednearasubstrate AT whiteskw anisotropyofelectricalpropertiesofalayerofsphericalparticleslocatednearasubstrate |