Emissivity measurements on noble metals: validation of a free-electron description including the anomalous skin effect

Eu . Phys. J. Plus (2024) 139:1043
h ps://doi.o g/10.1140/epjp/s13360-024-05819-3
Regula A icle
Emissi i y measu emen s on noble me als: alida ion o a ee-elec on desc ip ion
including he anomalous skin e ec
Iñigo González de A ie a1,a, Telmo Echániz2,b, Albe Adibekyan3,c, Ch is ian Mon e3,d, Jö g Holland 3,e,
Gab iel A. López1,
1Physics Depa men , Uni e si y o he Basque Coun y UPV/EHU, 48940 Leioa, Spain
2Applied Ma hema ics, Uni e si y o he Basque Coun y UPV/EHU, 48013 Bilbao, Spain
3Physikalisch-Technische Bundesans al (PTB), Be lin, Ge many
Recei ed: 23 July 2024 / Accep ed: 8 No embe 2024
© The Au ho (s) 2024
Abs ac The op ical p ope ies o noble me als (Cu, Ag, and Au) ha e been s udied in he mid-in a ed using emissi i y measu e-
men s a 473 K. Thei op ical esponses can be desc ibed using classical ee-elec on exp essions in he Mo –Zene app oxima ion
wi h an addi ional e m o accoun o he ex a emission due o he anomalous skin e ec , wi h no ee pa ame e s. Clea di e -
ences be ween he specula and di use su ace sca e ing egimes can be obse ed. These esul s cons i u e me ological-quali y
alida ions o he he mal adia i e p ope ies o hese impo an me als.
1 In oduc ion
The op ical and he mal adia i e p ope ies o me als a e de e mined by se e al elec on in e ac ions ha can ake complica ed
o ms, depending on he me al and he spec al ange [1,2]. In pa icula , heo e ical desc ip ions o he ole o su ace sca e ing
ha e been adi ionally complica ed, especially o high-conduc i i y noble me als. The op ical esponse o ee-elec on me als is
o en calcula ed by means o semiclassical anspo equa ions, such as he Bol zmann equa ion, by assuming a local ela ionship
be ween he elec ic ield and he cu en [1,3]. This is o en a good app oxima ion, as he elec ic ield is exponen ially damped when
pene a ing a me al su ace, a phenomenon desc ibed by he skin dep h δ(ω)c/√2πωσ(ω). The pic u e o a equency-dependen
local conduc i i y σ(ω) b eaks down o good conduc o s, when he elec on mean ee pa h l Fτ(wi h F he Fe mi eloci y
and τ he elaxa ion ime) becomes close o he alue o he skin dep h [4]. In his case, he ca ie s expe ience a ying alues o he
elec ic ield be ween sca e ing e en s, which leads o he cu en a each poin being a unc ion o he ield a p e ious loca ions.
The no ion o op ical cons an s hus becomes ill-de ined in his spec al ange [3]. In eg al o mula ions o he anspo equa ions
a e o en he way o achie e nume ical esul s o his complex p oblem [5–12].
The anomalous skin e ec is pa icula ly ele an o unde s anding he he mal adia i e p ope ies o noble me als, o which
e en mino con ibu ions o hei al eady low emissi i ies lead o signi ican a ia ions. Thus, a quan i a i e desc ip ion o he
emissi i y o noble me als and he quan i ica ion o he con ibu ion om he anomalous skin e ec a e ele an ques ions. The
di icul y o pe o ming eliable emissi i y measu emen s in hese ma e ials is e iden om he low signal emi ed by hese highly
e lec ing me als. The e o e, mos li e a u e da a is epo ed as o al hemisphe ical emissi i y alues, measu ed using calo ime ic
me hods [13–16]. Howe e , o analy ical pu poses, spec ally esol ed da a in he in a ed ange a e he mos use ul o di ec
e i ica ion o heo e ical p edic ions. Un o una ely, he exis ing li e a u e on hese measu emen s o noble me als epo s sca e ed
esul s and con adic o y in e p e a ions [2,3,11,17–24].
In his wo k, we p o ide new da a on he emissi i y o noble me als and use hem o alida e his ee-elec on amewo k,
pa icula ly in connec ion o he ole played by he anomalous skin e ec . This e ec is expec ed o be ele an no only o in a ed
op ics, bu also o i s connec ions o he gene al beha io o hese me als and hei applica ions [11,25–28].
ae-mail: [email p o ec ed] (co esponding au ho )
be-mail: [email p o ec ed]
ce-mail: albe .adibekyan@p b.de
de-mail: ch is ian.mon e@p b.de
ee-mail: joe g.holland @p b.de
e-mail: [email p o ec ed]
0123456789().: V,- ol 123
1043 Page 2 o 9 Eu . Phys. J. Plus (2024) 139:1043
2 Theo e ical backg ound
In p inciple, he mid-in a ed esponse o noble me als can be desc ibed in a he basic e ms wi hin he ee-elec on amewo k,
because o he simplici y o hei Fe mi su aces. Thei good elec ical conduc i i ies and associa ed la ge elaxa ion imes (e en
a mode a ely high empe a u es) allow a u he simpli ica ion in his equency ange: he Mo –Zene app oxima ion ωτ > 1[2,
29]. In his app oxima ion, he bulk emissi i y o he me al is desc ibed simply as 2/ω pτ,whe eωpis he plasma equency [2].
Then, he in luence o he anomalous skin e ec can be modeled as an addi ional abso p ion by elec on-su ace sca e ing, which
can be desc ibed by means o an empi ical elec onic specula i y pa ame e , p. The e o e, o a i s app oxima ion, he emissi i y in
his high- equency egion can be calcula ed as ollows [2]:
ε2
ωpτ+(1−p)3 F
4c.(1)
The calcula ion me hod is based on he Mo –Zene app oxima ion desc ibed abo e, wi h no addi ional abso p ion mechanisms
being conside ed. Thus, he emissi i y o each me al is comple ely de e mined by in insic ee-elec on p ope ies, which can be
de e mined by non-op ical means, as well as he phenomenological elec onic specula i y pa ame e p. This me hod is no applicable
o me als wi h complex Fe mi su aces. In he case o simple me als unde condi ions which iola e he Mo –Zene condi ion, mo e
complex exp essions can be used o e ie e he emissi i y and he in luence o he anomalous skin e ec [2,21].
Howe e , addi ional s eps ha e been in oduced o gene alize he emissi i y calcula ion p ocedu e. In he ee-elec on model,
only wo pa ame e s, he elec on densi y and elaxa ion ime, a e equi ed o a comple e desc ip ion o he op ical esponse o
he me al. The e o e, a gene aliza ion o Eq. (1) ha depends only on di ec ly measu able non-op ical pa ame e s is gi en by he
ollowing ela ions [2]:
ε5.263 ·10−6( s/a0)3/2T (T), (2)
(T)2411
( s/a0)3
ρ∗
μ(T)
T+p1995
( s/a0)5/2T,(3)
whe e (T) is a gene aliza ion o he elaxa ion equency ha includes bo h he bulk and su ace sca e ing e ms, a0is he Boh
adius, sis he ee-elec on adius o each me al, and ρ∗
μis he e ec i e elec ical esis i i y in μ·cm.
The la e pa ame e mus be in oduced o ake in o accoun he di e ence be ween he pa ame e s de ined a DC and hose in
he in a ed, and is gi en by he ollowing exp ession:
ρ∗(T)ρ(T)me
m∗1/2τDC
τIR ,(4)
whe emes ands o heba eelec onmassandm∗ o hee ec i eop icalmass.The elaxa ion- imequo ien a ises omaniso opic
sca e ing a he Fe mi su ace and can be de e mined om nume ical in eg als [2,13,18]. Typically, he wo quo ien s compensa e
each o he o gi e a numbe on he o de o 1 (co ec ion ac o s in Table 1).
The e o e, he emissi i y o hese me als is di ec ly compu ed om a ailable da a, wi h he necessa y pa ame e s gi en in Table 1.
I mus be no ed ha his app oach is only alid o empe a u es highe han he Debye empe a u e, which is he case o his
s udy. F ee-elec on adius da a ha e been aken om Re . [30], while he empe a u e-dependen elec ical esis i i y da a ha e
been in e pola ed om he ecommended alues gi en in he c i ical e iew by Ma ula [31]. The only ee pa ame e le in his
app oach is he elec onic specula i y ac o p, which can ake alues om 0 o 1. This pa ame e accoun s o he na u e o elec on
e lec ion a he sample su ace: 0 co esponds o di use sca e ing and 1 o pe ec ly specula e lec ion. The combined a io o
he pa ame e s in Eq. (4) has been de e mined based on op ical da a and Fe mi su ace in eg a ion [13,18]. Simila calcula ions
ha e been made in ecen imes based on ab ini io me hods, including a la ge numbe o me als wi h mo e complex Fe mi su aces
[32]. I mus be no ed ha mos in eg al calcula ions o he anomalous skin e ec do no ake in o accoun such co ec ions [5,
21]. Because he in a ed elaxa ion alue ends o he DC alue a su icien ly long wa eleng hs [22], i is unclea how his e ec
should be inco po a ed in o a ull spec al ep esen a ion o he emissi i y in he egion o alidi y o he anomalous skin e ec . I
would be necessa y o inco po a e a equency-dependen elaxa ion ime, ins ead o a high- equency limi . In any case, a ailable
s udies poin o he po en ial c osso e poin aking place a wa enumbe s lowe han 400 cm−1[19,22]. Thus, u he s udies in
he a -in a ed would be necessa y o cla i y his poin , bu he p esen echnique, based on he mal emission, is e y di icul o
Table 1 Lis o ee-elec on
pa ame e s used in he
calcula ions. Elec ical esis i i y
alues in μ cm 10−8ma e
ob ained by linea in e pola ion o
ecommended alues
Ma e ial s/a0[30]ρμ(473 K) [31] Co ec ion ac o [13,18]
Ag 3.02 2.70 1.24
Cu 2.67 2.90 1.09
Au 3.01 3.74 1.43
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Table 2 Roughness cha ac e iza ion o he samples. Ras ands o he a e age oughness o he p o ile, Rq o he oo -mean-squa e oughness, Rz o he
mean heigh o he p o ile peaks, and RSm o he a e age dis ance be ween p o ile elemen s
Sample Ra(μm) Rq(μm) Rz(μm) RSm (μm)
Cu 0.062 0.087 0.548 285
Ag 0.048 0.062 0.347 186
Bulk Au 0.036 0.046 0.286 219
Au mi o 0.01∗–––
*da a epo ed by he manu ac u e
pe o m a low equencies o poo ly emi ing ma e ials. S udies in he a -in a ed and submillime e ange could be pe o med
using non-op ical echniques, such as impedance measu emen s in esonan ca i ies [33].
3 Ma e ials and me hods
High-pu i y Cu and Ag spu e ing a ge s om Good ellow®we e used (pu i ies: 99.99 mass%). Addi ionally, wo gold samples o
>99% pu i y, in he o m o a bulk disk and a p o ec ed y ia-coa ed mi o ilm, we e also s udied. Some o hese samples ha e
been used as e e ence ma e ials o es ing new emissome e s and in me ological in e compa isons [34]. The coppe and sil e
samples we e mechanically polished, whe eas he gold ones we e measu ed as ecei ed. The su ace oughnesses we e de e mined
using s anda d p o ilome y, wi h he esul ing pa ame e s being epo ed in Table 2. The only excep ion is he gold mi o , whose
su ace is no di ec ly accessible, due o he Y2O3coa ing. Ins ead, he a e age oughness alue p o ided by he manu ac u e is
gi en.
In a ed emissi i y measu emen s we e ca ied ou in he Reduced-Backg ound Calib a ion Facili y (RBCF2) o he Physikalisch-
Technische Bundesans al (PTB) [34,35]. This acili y is a ecen ly upg aded e sion o he p e ious RBCF and is now loca ed on
an ISO 5 clean oom. The RBCF2 sys em connec s a sou ce chambe o a de ec o chambe h ough a beamline cooled by liquid
ni ogen. The sou ce chambe con ains mul iple adia ion sou ces ha can be p ecisely posi ioned along he beamline’s op ical axis. A
empe a u e-con olledsphe ewi hin he sou ce chambe holds a sample o emissi i y measu emen s. Thede ec o chambe ea u es
in a ed de ec o s and an ellipsoidal mi o ha ocuses adia ion om he sou ce chambe on o a Fou ie ans o m spec ome e ia
he cooled beamline. This se up allows o e ical and ho izon al scanning o bo h sou ces and de ec o s, enabling he measu emen
o hei homogenei y. Fu he in o ma ion can be ound in Re . [35] and e e ences he ein.
Emissi i y measu emen s a e pe o med using a di ec adiome ic me hod, by compa ing he adiance emi ed by he sample o
hose o wo e e ence blackbodies o high me ological quali y. The sample is moun ed on a o a ing hea ing pla e, loca ed inside a
he mally con olled sphe ical enclosu e pain ed wi h a black pain o known emissi i y. Dis inc i e ea u es o his sys em include a
high acuum h oughou (10−6hPa), he use o a liquid-ni ogen-cooled beamline o connec he de ec o and sample chambe s, and
a liquid-ni ogen-cooled blackbody as a low- empe a u e e e ence. The high- empe a u e blackbody is kep a a empe a u e close o
he measu ed sample. The measu emen s we e pe o med in acuum in he mid-in a ed ange wi h a DLaTGS de ec o . The samples
we e hea ed by conduc ion, wi h Apiezon® he mal g ease ensu ing good he mal con ac o he hea ing pla e. The sample su ace
empe a u e was ob ained om a Mon e Ca lo simula ion o a hea ans e model, based on eadings by esis ance he mome e s
[36]. The dominan sou ces o unce ain y using his me hod a e signal epea abili y a high wa enumbe s and sample empe a u e
a low wa enumbe s. In he case o highly e lec ing ma e ials, howe e , hei exceedingly small emission signal mo i a es he use
o an enclosu e hea ed o 80 ◦C, in o de o inc ease he o al adia ion coming ou o he sample su ace o de ec able limi s [37].
In his case, he co ec ion o e lec ed enclosu e wall adia ion domina es he unce ain y budge a low wa enumbe s and leads o
co ela ed unce ain ies h oughou he spec al ange, which a ec he s a is ical p ope ies o he esul s.
4 Resul s and discussion
The no mal spec al emissi i ies o he ou samples a e shown in Figs. 1and 2, whe e all samples ea u e e y low and spec ally la
emissi i ies. The unce ain y dec eases wi h wa enumbe , as he weigh o e lec ed signals om o he in a ed sou ces dec eases
in in ensi y. Measu ing he spec al emissi i y o such low-emi ing ma e ials wi h a s anda d unce ain y below 0.03 is a e y
challenging ask, one o he main mo i a ions behind he de elopmen o he RBCF2 ins umen [38].
The heo e ical alues o he emissi i ies o he h ee me als unde s udy, calcula ed using Eq. (2), a e also shown in Figs. 1
and 2. The ag eemen be ween hese heo e ical ela ions and he expe imen al esul s is excellen , which p o es he eliabili y o
hese exp essions and allows discussion on he ole o he su ace sca e ing. On he one hand, bo h coppe and sil e could be
ep oduced accu a ely by he p0 app oxima ion, wi h he expe imen al unce ain ies being su icien ly low o ule ou he p1
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Fig. 1 No mal spec al
emissi i ies o coppe and sil e a
473 K, compa ed o he
p edic ions o Eq. (2) o ex eme
alues o p. Shaded egions
co espond o s anda d
unce ain ies (k1). No e ha
he coppe measu emen s ha e
been pe o med a a highe
esolu ion
Fig. 2 No mal spec al
emissi i ies o wo gold samples
(bulk and mi o ilm) a 473 K,
compa ed o he p edic ions o
Eq. (2) o ex eme alues o
p. Shaded egions co espond o
s anda d unce ain ies (k1)
possibili y. Gi en ha hese samples we e polished by he same me hod, i is no su p ising ha hey bo h ea u e simila su ace
pa ame e s. In con as , he pic u e o he gold samples was mo e complex: The solid sample could be accu a ely ep oduced by he
p1 app oxima ion, bu he ilm had a highe nea -no mal emissi i y, mo e consis en wi h he p0 app oxima ion. I mus be
no ed ha hese di e ences in specula i y e e only o sca e ing a he elec onic scale, as all samples ha e e y low oughnesses
compa ed o mid-in a ed wa eleng hs (as shown in Table 2). I is well known ha he emissi i ies o hin ilms can be highe han
hose o hei bulk coun e pa s, e en o well-p epa ed samples [17,18,39]. I is wo h no ing ha one o he i s expe imen al
s udies o he anomalous skin e ec in sil e ilms obse ed an impo an educ ion in hei elec ical conduc i i y, which inc eased
he emissi i y [17]. We should also ake in o accoun he di e ences in empe a u e dependence o noble me al ilms when compa ed
o hei bulk coun e pa s [40]. These di e ences can be aced back o s uc u al and mo phological p ope ies (i.e., g ain g ow h
and su ace oughness, among o he s), no only di e ences in elec ical p ope ies. Ne e heless, he bigges e ec s a e p edic ed
o nanome ic ilms o g ain sizes, which a e he mos suscep ible o s uc u al deg ada ion upon hea ing, so only hick samples
we e used in his s udy. Fu he mo e, he chosen empe a u e (473 K) is close o 0.3Tm o all me als, whe e Tmis he mel ing
poin . This empe a u e is ypical o annealing p ocedu es, bu is no high enough o induce i e e sible changes.
Addi ional s udies a oblique angles e eal no addi ional ea u es in any o he bulk spec a (Figs. 3,4,and5). In he case o
he coa ed gold ilm (no shown), spec al peaks co esponding o he ib a ional modes o he y ia coa ing a e appa en a high
angles. In con as wi h o he di ec ional measu emen s [20], he e is no spec al ea u e con adic ing he Mo –Zene pic u e o a
spec ally la emissi i y o he o he h ee samples. These measu emen s will be used below o calcula e he o al hemisphe ical
emissi i ies.
An impo an esul o his wo k is ha he e is no s aigh o wa d ela ion be ween he oughness pa ame e s shown in Table 2and
hei su ace specula i ies om an elec onic sca e ing iewpoin . This poin s o he complexi y o de ining a specula i y pa ame e
and quan i ying i s ela ion o measu able su ace p ope ies. Because i accoun s o elec on sca e ing, i s cha ac e is ic leng h
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Fig. 3 Di ec ional spec al
emissi i y o he coppe sample a
473 K
Fig. 4 Di ec ional emissi i y o
he sil e sample a 473 K
Fig. 5 Di ec ional emissi i y o
he bulk gold sample a 473 K
scale is no he ligh wa eleng h, bu he de B oglie wa eleng h o elec ons in hese ma e ials (3 Å o coppe and 5 Å o sil e
and gold) [18]. Based on di ac ion a gumen s, i seems ha he oughness needs o be almos a omically low in o de o ha e an
elec onic specula i y di e en han 0. In e es ingly, he e is expe imen al e idence ha somewha oughe su aces can appa en ly
s ill ha e impo an specula i ies [17,20], a poin which is suppo ed by some heo e ical esul s [41]. Some au ho s ha e also claimed
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1043 Page 6 o 9 Eu . Phys. J. Plus (2024) 139:1043
Fig. 6 To al hemisphe ical
emissi i y o coppe [14,15],
compa ed o he li e a u e da a and
heo e ical p edic ions (4/3 o he
alues p edic ed by Eq. (2)). The
e o ba co esponds o he
s anda d unce ain y (k1)
Fig. 7 To al hemisphe ical
emissi i y o sil e , compa ed o
he li e a u e da a [14,16,24]and
heo e ical p edic ions (4/3 o he
alues p edic ed by Eq. (2)). E o
ba s co espond o s anda d
unce ain ies (k1)
ha pa ial e ec i e specula i y is allowed o any sample su ace [9]. Along his line o a gumen , i is no able ha he bulk gold
sample does ep oduce he p1 exp ession, which migh be ela ed o he absence o a polishing p ocess p io o measu emen ,
e en i i s oughness is simila o o highe han hose o he o he samples. In any case, cla i ica ion o his poin has s ill no been
achie ed, no is i expec ed o be u he de eloped wi hou mo e sophis ica ed su ace science s udies. The poin o his wo k is
o p o ide p ecise da a on he basic he mal adia i e p ope ies o noble me als and alida e a simpli ied bu accu a e heo e ical
pic u e. Ne e heless, he in luence o his specula i y on he in a ed and isible p ope ies o noble me als is o g ea impo ance,
gi en hei low bulk emissi i ies, bu i is also c ucial when discussing he low- o-high- equency ansi ion egime [9,10]. A lowe
equencies, bo h su ace sca e ing egimes di e by only a 9/8 abso p ion ac o [9].
Finally, a compa ison o he li e a u e da a is shown o comple eness. To al hemisphe ical da a, ob ained by in eg a ion o he
spec al di ec ional esul s, a e compa ed o e e ence alues. The in eg a ion p ocedu e o e ie e his p ope y om angle- esol ed
da a is based on a i ing p ocedu e o F esnel’s ela ions [34,36]. Figu e 6shows he esul o coppe , oge he wi h he li e a u e da a
and heo e ical p edic ions ( o good conduc o s, εH4/3·ε)[2,13]. To al hemisphe ical measu emen s o coppe we e e iewed
in Re . [13] and ound o be ep oducible wi h ee-elec on heo ies accoun ing o su ace sca e ing and e ec i e esis i i ies. Da a
om Ramana han and Yen [14] a e desc ibed in ha e e ence and shown o be expe imen ally ep oducible wi hin he bounds se
by he heo y. Addi ional up- o-da e low- empe a u e da a on coppe ilms [15] a e shown, which poin owa d an in e media e alue
o he elec onic su ace specula i y. The da a poin epo ed in his wo k is consis en wi h he li e a u e alues and he heo e ical
esul s o a pe ec ly di use su ace a he elec onic scale (p0). Simila ag eemen s a e ound o sil e and gold (Figs. 7and
8), al hough some o he li e a u e da ase s epo alues highe han hose p esen ed in his wo k.
Despi e e idence o he quali y o he measu emen s, unambiguous compa isons o he heo e ical p edic ions a e complica ed
by he ela i ely la ge s anda d de ia ions, which a e una oidable in his kind o measu emen s. This complica ion is in insic
o an emission measu emen me hod when used o cha ac e ize highly e lec ing ma e ials. Howe e , because he heo y p edic s
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Fig. 8 To al hemisphe ical
emissi i y o gold, compa ed o
he li e a u e da a [15,24]and
heo e ical p edic ions (4/3 o he
alues p edic ed by Eq. (2)). E o
ba s co espond o s anda d
unce ain ies (k1)
Fig. 9 Resul s o he i ing o he
no mal spec al da a o a cons an
unc ion, compa ed o he
heo e ical bounds o each me al
(Eq. (2)). Da a poin s co espond
o he a e ages o he i ed alues
ob ained assuming ei he pe ec ly
co ela ed ( 1) o unco ela ed
( 0) spec al poin s. E o ba s
ep esen he s anda d de ia ions
o he co esponding ec angula
p obabili y dis ibu ions
spec ally cons an alues o ma e ials which obey he Mo –Zene app oxima ion, a solu ion can be o mula ed by pe o ming a
s a is ical compa ison o he no mal alues o a cons an unc ion (x) 0. The i ing was made by minimizing he ollowing
objec i e unc ion:
χ2 Tσ−1 ,(5)
whe e y− (x) is he ec o con aining he esidual a each poin , and σis he a iance–co a iance ma ix. Fo he pa icula
case o unco ela ed spec al da a, he o -diagonal e ms become 0.
The use o Eq. (5) is mo i a ed by he co a ian na u e o hese measu emen s. As men ioned abo e, unce ain ies co esponding
o he sample empe a u e and he co ec ion o adia ion e lec ed by he sample in oduce co ela ions be ween spec al da a
poin s, especially a low wa enumbe s [37]. These co ela ions a e a p io i unknown, bu hey signi ican ly a ec he i ing esul .
Thus, ollowing he p inciple o maximum en opy, we ha e pe o med wo i s o each me al, conside ing ex eme alues o he
co ela ion coe icien : ij 0o 1;i j,∀i,j. The a iance–co a iance ma ices we e cons uc ed in he usual manne om he
expe imen al unce ain ies and hese assumed alues o he co ela ion coe icien . Nega i e co ela ion was no conside ed, because
he e was no physical mechanism in he measu emen model ha could p o ide such beha io . The i ings we e made using he
cu e_ i me hod o SciPy 0.19, which allows speci ying an inpu co a iance ma ix. The s anda d unce ain y due o he i ing
p ocess i sel was negligible, bu di e ences be ween he esul s ob ained assuming no co ela ion o pe ec co ela ion be ween
wa enumbe s we e signi ican . The esul s a e shown in Fig. 9, whe e he a e age o he wo i s is ep esen ed o each me al, and
he s anda d de ia ion co esponds o ha o a ec angula dis ibu ion be ween bo h ex emes: u(a+−a−)/√12. This app oach
assumes ha all non-nega i e alues o he co ela ion coe icien a e equally p obable, and hus, i p oduces he bes es ima e o
he ue alue o he unde lying cons an emissi i y assumed by ou model.
As expec ed om inspec ion o he spec al cu es, h ee o he i ed emissi i ies a e consis en wi h he alues p edic ed by
Eq. (2)(p1 o bulk gold, p0 o sil e , and an in e media e alue o coppe ). The emissi i y o he gold mi o is sligh ly
highe han he maximum alue, al hough by less han one s anda d de ia ion. A possible explana ion o his de ia ion is he p esence
123
1043 Page 8 o 9 Eu . Phys. J. Plus (2024) 139:1043
o he p o ec i e y ia ilm, which can ac as an an i- e lec i e laye . Ne e heless, all esul s a e compa ible wi h he heo e ical
p edic ions wi h only one ee pa ame e . In summa y, his es se es as a obus s a is ical alida ion o he hypo hesis o he a icle,
aking all he knowledge abou he measu emen unce ain y in o accoun .
5 Conclusions
To sum up, his wo k epo s on good-quali y emissi i y da a ha can be accu a ely ep oduced using simple exp essions om
ee-elec on heo y in he Mo –Zene app oxima ion and pa ame e s de e mined by non-op ical me hods, wi h only he elec onic
specula i y ac o pas a a iable pa ame e . The need o accoun o di e ences be ween DC and op ical p ope ies o hese me als
is highligh ed. Ten a i e di e ences a e obse ed be ween a bulk sample o gold (which sugges s be e ag eemen o p1), and
sil e and coppe (which a e close o he p edic ed alues o p0). Mo eo e , di e ences be ween wo gold samples (bulk s
ilm) can be accoun ed o by di e en alues o he specula i y pa ame e . Despi e he la ge unce ain ies a low wa enumbe s, he
esul s ep oduce he heo e ical ela ions accu a ely and lend alidi y o i s ounda ions. Fu he da a a lowe wa enumbe s would
be in e es ing o alida e he model in a egion whe e he anomalous skin e ec heo y p edic s a depa u e om he esul s o his
wo k.
Acknowledgemen s The au ho s acknowledge inancial suppo om he Basque Go e nmen . I. González de A ie a also hanks he Physikalisch-
Technische Bundesans al (PTB) o in i ing him o hei labo a o y.
Au ho con ibu ions Iñigo González de A ie a, Telmo Echániz, and Gab iel A. López con ibu ed o he concep ualiza ion and design o he s udy.
Ma e ial p epa a ion, da a collec ion, and analysis we e pe o med by Iñigo González de A ie a and Albe Adibekyan, unde he supe ision o Ch is ian
Mon e and Jö g Holland . The i s d a o he manusc ip was w i en by Iñigo González de A ie a, and all au ho s commen ed on he p e ious e sions
o he manusc ip . All au ho s ead and app o ed he inal manusc ip .
Funding Open Access unding p o ided hanks o he CRUE-CSIC ag eemen wi h Sp inge Na u e. This wo k was suppo ed inancially by he Basque
Go e nmen (Eusko Jau la i za, G an Numbe IT-1714-22).
Da a a ailabili y s a emen Da acon ainedin hisa icle a ea ailable inZenodo(h ps://doi.o g/10.5281/zenodo.12801555).Themanusc ip has associa ed
da a in a da a eposi o y
Code a ailabili y The calcula ions epo ed can be ep oduced om he in o ma ion a ailable in he manusc ip .
Decla a ions
Con lic o in e es The au ho s ha e no ele an inancial o non- inancial in e es s o disclose.
Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap a ion, dis ibu ion
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Re e ences
1. F. Woo en, Op ical p ope ies o solids (Academic P ess, Camb idge, 1972)
2. A. Sie e s, The mal adia ion om me al su aces. J. Op . Soc. Am. 68(11), 1505–1516 (1978)
3. D.W. Lynch, W.R. Hun e , Commen s on he op ical cons an s o me als and an in oduc ion o he da a o se e al me als, in Handbook o op ical
cons an s o solids. ed. by E.D. Palik (Academic P ess, Camb idge, 1997), pp.275–367
4. A. Pippa d, The su ace impedance o supe conduc o s and no mal me als a high equencies II. The anomalous skin e ec in no mal me als. P oc.
Roy. Soc. A Ma h. Phys. 191(1026), 385–399 (1947)
5. G.E.H. Reu e , E.H. Sondheime , The heo y o he anomalous skin e ec in me als. P oc. Roy. Soc. A Ma h. Phys. 195(1042), 336–364 (1948)
6. R.B. Dingle, The anomalous skin e ec and he e lec i i y o me als I. Physica 19(1–12), 311–347 (1953)
7. D. Ma is, J. Ba deen, Theo y o he anomalous skin e ec in no mal and supe conduc ing me als. Phys. Re . 111(2), 412 (1958)
8. M.I. Kagano , G.Y. Lyuba skiy, A.G. Mi ina, The heo y and his o y o he anomalous skin e ec in no mal me als. Phys. Rep. 288(1–6), 291–304
(1997)
9. A. Jauho, H. Smi h, Anomalous skin e ec wi h gene al elas ic bounda y sca e ing. J. Phys. F: Me . Phys. 15(9), 1951 (1985)
10. P. Gilbe d, The anomalous skin e ec and he op ical p ope ies o me als. J. Phys. F: Me . Phys. 12(8), 1845 (1982)
11. E.A. Se o , V.V. Pa shin, G.M. Bubno , Re lec i i y o me als in he millime e wa eleng h ange a c yogenic empe a u es. IEEE T ans. Mic ow.
Theo y 64(11), 3828–3838 (2016)
12. S. Robe s, In e p e a ion o he op ical p ope ies o me al su aces. Phys. Re . 100(6), 1667 (1955)
13. R. Smalley, A. Sie e s, The o al hemisphe ical emissi i y o coppe . J. Op . Soc. Am. 68(11), 1516–1518 (1978)
14. K.G. Ramana han, S.H. Yen, High- empe a u e emissi i ies o coppe , aluminum, and sil e . J. Op . Soc. Am. 67(1), 32–38 (1977)
123
Eu . Phys. J. Plus (2024) 139:1043 Page 9 o 9 1043
15. J. F olec, T. K álík, V. Musilo á, P. Hanzelka, A. S nka, J. Jelínek, A da abase o me allic ma e ials emissi i ies and abso p i i ies o c yogenics.
C yogenics 97, 85–99 (2019)
16. S.I. Woods, T.M. Jung, D.R. Sea s, J. Yu, Emissi i y o sil e and s ainless s eel om 80 o 300 K: applica ion o ITER he mal shields. C yogenics 60,
44–48 (2014)
17. H.E. Benne , J.M. Benne , E.J. Ashley, R.J. Mo yka, Ve i ica ion o he anomalous-skin-e ec heo y o sil e in he in a ed. Phys. Re . 165(3), 755
(1968)
18. J.A. McKay, J.A. Rayne, Tempe a u e dependence o he in a ed abso p i i y o he noble me als. Phys. Re . B 13(2), 673 (1976)
19. L. Ha is, P. Fowle , Abso p ance o gold in he a in a ed. J. Op . Soc. Am. 51(2), 164–167 (1961)
20. T. Echániz,I.Se ién-Fe nández, R.Pé ez-Sáez,M.Tello,Expe imen al e i ica iono heanomalousskine ec in coppe usingemissi i ymeasu emen s.
Appl. Phys. Le . 102(24), 244106 (2013)
21. T. Echániz, R. Pé ez-Sáez, M. Tello, Op ical p ope ies o me als: in a ed emissi i y in he anomalous skin e ec spec al egion. J. Appl. Phys. 116(9),
093508 (2014)
22. H.U. Yang, J. D’A changel, M.L. Sundheime , E. Tucke , G.D. Bo eman, M.B. Raschke, Op ical dielec ic unc ion o sil e . Phys. Re . B 91(23),
235137 (2015)
23. R.L. Olmon, B. Slo ick, T.W. Johnson, D. Shel on, S.-H. Oh, G.D. Bo eman, M.B. Raschke, Op ical dielec ic unc ion o gold. Phys. Re . B 86(23),
235147 (2012)
24. W. Sabuga, R. Tod enhaup , E ec o oughness on he emissi i y o he p ecious me als sil e , gold, palladium, pla inum, hodium, and i idium. High
Temp High P ess 33(3), 261–269 (2001)
25. R.F. Boehm, G.A. Domo o, C.L. Tien, P edic ions o he o al emissi i y o me als a c yogenic empe a u es. Technical Repo NASA Repo No.
TS-68-5 (1968)
26. B. Podobedo , Resis i e wall wake ields in he ex eme anomalous skin e ec egime. Phys. Re . Spec. Top. Accel. 12(4), 044401 (2009)
27. K.L.F. Bane, J. Tu, G. S upako , Re lec i i y measu emen s o coppe and aluminum in he a in a ed and he esis i e wall impedance in he LCLS
undula o . In: Eu opean Pa icle Accele a o Con e ence (EPAC 06) (2006)
28. J.P. Elche , C yogenic mul ilaye insula ion heo y and an analysis o seams unde a a ie y o assump ions. The mal and Fluids Analysis Wo kshop
2018 (2018)
29. N.F. Mo , C. Zene , The op ical p ope ies o me als. Ma h. P oc. Camb. 30(2), 249–270 (1934)
30. N.W. Ashc o , N.D. Me min, Solid s a e physics (Saunde s College Publishing, Philadelphia, 1976)
31. R.A. Ma ula, Elec ical esis i i y o coppe , gold, palladium, and sil e . J. Phys. Chem. Re . Da a 8(4), 1147–1298 (1979)
32. D. Gall, Elec on mean ee pa h in elemen al me als. J. Appl. Phys. 119(8), 085101 (2016)
33. M.A. O dal, R.J. Bell, R.W. Alexande J ., L.A. Newquis , M.R. Que y, Op ical p ope ies o Al, Fe, Ti, Ta, W, and Mo a submillime e wa eleng hs.
Appl. Op . 27(6), 1203–1209 (1988)
34. A. Adibekyan, C. Mon e, M. Keh , B. Gu schwage , J. Holland , Emissi i y measu emen unde acuum om 4 μm o 100 μm and om −40◦C o
450 ◦C a PTB. In . J. The mophys. 36(2–3), 283–289 (2015)
35. I. Mülle , A. Adibekyan, K. Anhal , C. Bal uscha , B. Gu schwage , S. König, E. Kononogo a, C. Mon e, M. Reinige , S. Schille e al., Non-con ac
empe a u e measu emen a he Physikalisch-Technische Bundesans al (PTB). Quan . In aRed. The m. J. 18(3), 187–212 (2021)
36. C. Mon e, J. Holland , The de e mina ion o he unce ain ies o spec al emissi i y measu emen s in ai a he PTB. Me ologia 47(2), 172 (2010)
37. A. Adibekyan, High-accu acy spec al emissi i y measu emen o indus ial and emo e sensing applica ions. PhD hesis, Be gischen Uni e si ä
Wuppe al (2016)
38. A. Adibekyan, E. Kononogo a, J. Hameu y, M. Lauens ein, C. Mon e, J. Holland , Emissi i y measu emen s on e lec i e insula ion ma e ials. TM-
Technisches Messen 88(10), 617–625 (2021)
39. S. Edala pou , M. F ancoeu , Size e ec on he emissi i y o hin ilms. J. Quan . Spec osc. Radia . T ans. 118, 75–85 (2013)
40. J. T ollmann, A. Pucci, In a ed dielec ic unc ion o gold ilms in ela ion o hei mo phology. J. Phys. Chem. C 118(27), 15011–15018 (2014)
41. R. G eene, Bounda y condi ions o elec on dis ibu ions a c ys al su aces. Phys. Re . 141(2), 687 (1966)
123