PHYSICAL REVIEW A 111, 022809 (2025)
Elas ic and inelas ic elec on sca e ing c oss sec ions o ichlo o luo ome hane
M. Dinge ,1,2,*Y. Pa k ,3and W. Y. Baek 1
1Physikalisch-Technische Bundesans al , Bundesallee 100, 38116 B aunschweig, Ge many
2Rup ech -Ka ls-Uni e si ä Heidelbe g, G abengasse 1, 69117 Heidelbe g, Ge many
3Ins i u e o Plasma Technology, Ko ea Ins i u e o Fusion Ene gy, 37 Dongjangsan- o, Gunsan, Jeonbuk-do 54004, Republic o Ko ea
(Recei ed 7 Augus 2024; e ised 6 Decembe 2024; accep ed 3 Feb ua y 2025; published 18 Feb ua y 2025)
Di e en ial elas ic elec on sca e ing c oss sec ions o ichlo o luo ome hane (CCl3F) we e measu ed o e
a b oad ene gy ange spanning 30 o 800 eV in he angula ange o 20° o 150°. The expe imen al esul s we e
compa ed wi h calcula ions using he IAM-SCAR+I model. Sa is ac o y ag eemen s be ween bo h da a se s we e
ound o elec on ene gies abo e 200 eV wi hin expe imen al unce ain ies, whe eas signi ican de ia ions o
up o 100% we e obse ed a elec on ene gies below 60 eV. In addi ion o he measu emen s o di e en ial
elas ic-sca e ing c oss sec ions, o al inelas ic-sca e ing c oss sec ions o CCl3F we e calcula ed using he
sphe ical complex op ical po en ial model. These calcula ions closely ma ch expe imen al o al ioniza ion c oss
sec ions a ailable in he li e a u e o ene gies below 50 eV. The sum o he expe imen al o al elas ic-sca e ing
c oss sec ions and he heo e ical o al inelas ic-sca e ing c oss sec ions aligns e y well wi h he o al elec on
sca e ing c oss sec ions o CCl3F measu ed by o he g oups ac oss he en i e ene gy ange (30 o 800 eV),
demons a ing he consis ency among hese h ee c oss sec ions.
DOI: 10.1103/PhysRe A.111.022809
I. INTRODUCTION
Chlo o luo oca bons (CFCs) we e he p ima y choice o
e ige a ion and a ious indus ial p ocesses un il hey we e
banned by he Mon eal P o ocol due o hei ozone-deple ing
e ec s. Among hese, ichlo o luo ome hane (CCl3F, F eon-
11, CFC-11) has a pa icula no o ious impac on chemical
p ocesses in he uppe a mosphe e due o he p esence o h ee
chlo ine a oms, leading o a global wa ming po en ial se e al
o de s o magni ude highe han ha o CO2[1].
The e is clea e idence ha cosmic ay-d i en elec on-
induced molecula eac ions play an impo an ole in ozone
deple ion [2]. Elec on impac on CCl3F esul s in he p o-
duc ion o Cl adicals, which ca alyze ozone deple ion. Wi h
an a mosphe ic li e ime o 52 yea s and a high ozone deple-
ion po en ial [3], unde s anding he a mosphe ic chemis y o
CCl3F is c ucial. Accu a e simula ions o adia ion anspo
p ocesses in he uppe a mosphe e ely hea ily on he a ail-
abili y o da a se s on elec on-molecule collisions.
Hi he o, only a ew s udies on he elec on-impac c oss
sec ions o CCl3F ha e been conduc ed. The i s measu e-
men o he elec on in e ac ion c oss sec ions o CCl3F da es
back o 1986, when Jones [4] de e mined he o al elec on
sca e ing c oss sec ions (TCSs) wi h a ime-o - ligh elec on
ansmission spec ome e in he ene gy ange om 0.6 o
*Con ac au ho : ma eike.dinge @p b.de
Published by he Ame ican Physical Socie y unde he e ms o he
C ea i e Commons A ibu ion 4.0 In e na ional license. Fu he
dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)
and he published a icle’s i le, jou nal ci a ion, and DOI.
50 eV. A ew yea s la e , Zecca e al. [5] ex ended his ange by
p o iding expe imen al TCSs o elec on ene gies be ween
75 eV and 4 keV. Jiang e al. [6] la e calcula ed ioniza ion
c oss sec ions using he bina y-encoun e Be he model (BEB)
o ene gies om 10 eV o 1 keV. The e is only one expe i-
men al s udy on he o al ioniza ion c oss sec ions o CCl3F.
Sie a e al. [7] measu ed he pa ial ioniza ion c oss sec ions
o CCl3F by collec ing he agmen ions upon elec on impac
o p ima y ene gies be ween 20 and 85 eV. The o al ioniza-
ion c oss sec ion was hen ob ained by summing he pa ial
ioniza ion c oss sec ions. Ma inez e al. [8] con ibu ed o
his s udy by compa ing he expe imen al esul s wi h heo-
e ical calcula ions based on he BEB, he Deu sch and Mä k
o malism, and he modi ied addi i i y ule.
Despi e he impo ance o elec on collision c oss sec ions
o CCl3F o simula ing adia ion anspo p ocesses, only
one expe imen al da ase on he di e en ial elas ic elec on
sca e ing c oss sec ions (DCSs) o CCl3F is cu en ly a ail-
able. In his da ase , Hoshino e al. [9] no only p o ided
expe imen al DCSs in he angula ange o 15°–130°, bu also
discussed he ole o he chlo ine a oms by compa ing hei e-
sul s o hose o o he CFCs. Addi ionally, hey compa ed hei
expe imen al DCS o calcula ions pe o med using he IAM-
SCAR model and a modi ied phase shi analysis. Fo elec on
ene gies up o 30 eV, u he heo e ical DCS da a we e e-
po ed by Na alese e al. [10] and F ei as e al. [11], who
employed he Schwinge mul ichannel me hod o hei cal-
cula ions. Howe e , all a ailable li e a u e on DCSs ocuses
solely on ene gies below 100 eV. To p o ide a comp ehensi e
da ase o adia ion anspo calcula ions, we measu ed he
DCS dσel/do CCl3F o he i s ime o e a b oade
ene gy and angula ange, co e ing sca e ing angles θ om
20° o 150° and elec on ene gies Tbe ween 30 and 800 eV.
2469-9926/2025/111(2)/022809(9) 022809-1 Published by he Ame ican Physical Socie y
M. DINGER, Y. PARK, AND W. Y. BAEK PHYSICAL REVIEW A 111, 022809 (2025)
FIG. 1. Schema ic iew o he expe imen al se up. The elec on
gun and Fa aday cup we e ixed in posi ion, while he hemisphe ical
elec on ene gy analyze was moun ed on a u n able. The sca e ing
angle θo elec ons o be de ec ed was adjus ed by o a ing he
u n able.
The expe imen al esul s we e compa ed o calcula ions using
he IAM-SCAR+I model [12–14]. Based on he expe imen-
al DCS, he o al elas ic-sca e ing c oss sec ions (TECSs)
σel and he momen um ans e c oss sec ions (MTCSs) σm
we e de e mined and compa ed o he calcula ions wi h he
IAM-SCAR+I model and he close-coupling code POLYDCS
[15]. Addi ionally, he o al inelas ic-sca e ing c oss sec ion
(TICS) σinel o CCl3F was calcula ed using he sphe ical com-
plex op ical po en ial (SCOP) model [16] and compa ed o
he expe imen al o al ioniza ion c oss sec ions epo ed by
Sie a e al. [7]. To check he consis ency and eliabili y o
bo h he expe imen al and heo e ical c oss sec ions, he sum
o he TECS and TICS was compa ed o he expe imen al da a
o he TCS σ o o CCl3F.
II. EXPERIMENT
The DCS o CCl3F was measu ed using a c ossed-beam
se up, as depic ed in Fig. 1. Since he expe imen al se up
was ho oughly explained in ou p e ious wo ks [17], we
p o ide only a b ie summa y he e. In his c ossed-beam
se up, he p ima y elec on beam in e sec s pe pendicula ly
wi h a molecula beam c ea ed by an e usion nozzle. The
ene gy spec um o sca e ed elec ons is measu ed wi h a
hemisphe ical ene gy analyze moun ed on a u n able. The
sca e ing angle θis adjus ed by o a ing he u n able, while
he elec on gun emains ixed. Since he de ec ion solid an-
gle is nea ly cons an h oughou he in e ac ion zone in he
p esen expe imen al se up, he DCS can be ob ained om he
change in he elas ic coun a e ˙
Nel pe solid angle :
dσel
d(θ,T)=˙
Nel
(θ,T)−I0
enFη(T),(1)
whe e I0is he p ima y elec on cu en (nega i e in sign), eis
he elemen a y cha ge, nFis he numbe o molecules pe a ea
hi by he elec on beam, and η(T) is he de ec ion e iciency
o he ene gy analyze .
Di ec de e mina ion o nFand η(T) is challenging. In his
pape , his di icul y is ci cum en ed by using he ela i e low
echnique (RFT) [18], which u ilizes he well-known DCS o
a e e ence gas. To use he RFT, he molecula beams mus
be gene a ed in he molecula low egime, and he mean ee
pa hs o in e molecula collisions in bo h he e e ence gas
and he gas o in e es mus be compa able. I hese condi ions
a e me , he a io nF/ˆnFo he a ea numbe densi ies in he wo
beams is gi en by
≡nF/ˆnF=ˆ
F/F×M/ˆ
M,(2)
whe e Fis he mass low a e h ough he e usion nozzle
and Mis he mass o he molecule o in e es . In Eq. (2)
and he ollowing equa ions, he quan i ies wi h ha s deno e
hose ela ed o he e e ence gas. Wi h ,Eq.(1) can be
e o mula ed as
dσel
d(θ,T)=dˆσel
d(θ,T)×˙
Nel
˙
Nel
׈
I0
I0×−1.
(3)
In he p esen pape , ni ogen (N2) was used as he e e -
ence gas, wi h i s DCS alues de e mined in an independen
p io expe imen ha u ilized a di e en me hodology and
helium as he e e ence gas [17]. Al hough nume ous expe -
imen al DCS da ase s o N2ha e been published o da e
[19–26], hese da a a e es ic ed o speci ic angula and en-
e gy anges, and none co e he b oad ene gy and angula
anges conside ed in his pape . Mo eo e , he published da a
exhibi signi ican sca e , wi h de ia ions exceeding 50% in
some cases. To es ablish a sui able e e ence da ase o his
pape , he exis ing da a would need o be in e pola ed and
ex apola ed o encompass he sca e ing angles and p ima y
elec on ene gies ele an o his pape , in oducing addi ional
unce ain ies.
A he sca e ing angles and ene gies s udied in he p esen
pape , ou e e ence da a gene ally align wi h he mean
li e a u e alues wi hin ou expe imen al unce ain ies o ap-
p oxima ely 15% and he s anda d de ia ions o he a e age
li e a u e da a. When only wo o ewe da ase s we e a ail-
able, we also conside ed he s a ed expe imen al unce ain ies
om he li e a u e. A no able de ia ion o 114% was ound a
135° o T=60 eV be ween ou e e ence DCS and he li e -
a u e da a, which canno be solely a ibu ed o expe imen al
unce ain ies. This disc epancy o igina es om he da ase o
S i as a a e al. [22], which is no only highe han ou own
e e ence da a bu also exceeds he only o he a ailable da ase
a his ene gy (Nickel e al. [25]). Apa om his ou lie ,
ou e e ence N2da a a e consis en wi h bo h he a e aged
li e a u e alues [19–25] and he compiled e e ence da ase o
Song e al. [26] wi hin expe imen al unce ain ies. The e o e,
he use o ou N2DCS measu emen s as he e e ence da ase
is jus i ied and app op ia e o his pape .
The numbe densi y o molecules in he gas beam, p o-
duced ia an e usion nozzle wi h a 0.3-mm-diame e exi
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ELASTIC AND INELASTIC ELECTRON SCATTERING … PHYSICAL REVIEW A 111, 022809 (2025)
ape u e, was app oxima ely 5 ×1013 cm−3. Conside ing he
gas kine ic diame e s o N2(0.364 nm [27]) and CCl3F (0.618
nm [28]), he mean ee pa hs (λ) o in e molecula collisions
in bo h gases exceed 4 mm. This is signi ican ly la ge han
he nozzle diame e , ensu ing he Knudsen condi ion is well
sa is ied. Consequen ly, he gas e usion occu s in he molec-
ula low egime, allowing o he applica ion o he RFT. To
main ain equal λin bo h gas beams, a necessa y condi ion
o RFT, he d i ing p essu e in he gas ese oi abo e he
e usion nozzle was adjus ed acco ding o he squa e o he
gas kine ic diame e s o bo h molecules. Gi en ha he o al
elec on sca e ing c oss sec ions o N2and CCl3F o he en-
e gies o in e es a e lowe han 6 ×10−15 cm2, he mean ee
pa h o elec ons in bo h gas beams wi h he a o emen ioned
numbe densi y exceeded 3 cm, which was much la ge han
he diame e o he molecula beam. Consequen ly, he single
collision condi ion was sa is ied.
The p ima y elec on-beam cu en , measu ed using he
Fa aday cup loca ed beyond he molecula beam, a ied be-
ween 0.2 and 0.5 µA. The angula esolu ion o he appa a us
was app oxima ely 2°. The o e all ene gy esolu ion o he
appa a us was 1.5 eV a T=800 eV, imp o ing o 0.7 eV
a T=30 eV. I should be no ed ha his ene gy esolu ion
is no su icien o esol e o a ional and ce ain ib a ional
exci a ions om elas ic sca e ing. The sca e ing chambe
was su ounded by h ee o hogonal pai s o Helmhol z coils
o compensa e o he Ea h’s magne ic ield. On he sca e ing
plane, he esidual magne ic ield was lowe han 2 µT. The
de lec ion o elec ons in his ield was negligible compa ed
o he angula esolu ion o he appa a us and was he e o e
dis ega ded.
III. UNCERTAINTY ANALYSIS
The s anda d unce ain y o he expe imen al DCS was
es ima ed ollowing he Guide o he Exp ession o Unce -
ain y in Measu emen [29]. The majo unce ain y sou ce
is he DCS o he e e ence gas N2, which had 15% unce -
ain y. Ano he unce ain y a ose om he empo al d i o
he p ima y elec on-beam cu en , which was co ec ed wi h
an unce ain y o 4%. The a io o he low a es o he CCl3F
and N2gas beams, de e mined om he empo al dec ease
o he p essu e in hei gas ese oi s, was associa ed wi h
a 5% unce ain y. Finally, he s a is ical unce ain ies due o
he sca e ed elec on coun s unde he elas ic peaks we e
lowe han 5%. The o e all s anda d unce ain y, calcula ed as
he posi i e squa e oo o he sum o he squa ed indi idual
unce ain ies, was 17%.
IV. THEORETICAL METHODS
Unless o he wise s a ed, he s a ic Vs , exchange Vex, and
co ela ion-pola iza ion po en ial Vcp as well as he elec on
densi ies needed in he ollowing heo ies we e ob ained in
hei single-cen e expansion using he SCELIB4.0 lib a y [30].
The molecula wa e unc ions used as inpu in he SCELIB4.0
lib a y we e compu ed using he Ha ee-Fock me hod wi h
he GAUSSIAN09 [31] p og am sui e. A 6–311++G basis
se , supplemen ed wi h addi ional (2d,p) pola iza ion unc-
ions, was employed o hese calcula ions. The op imized
TABLE I. Molecula geome y and pola izabili y o CCl3Fde-
e mined h ough Ha ee-Fock op imiza ion using he GAUSSIAN09
p og am. The a omic coo dina es x,y,andza e gi en in uni s o
Å and he pola izabili y enso elemen s αij a e p o ided in a omic
uni s. The cen e o mass was chosen as he o igin.
xy z α
C 0.000000 0.000000 0.251350 αxx 54.043
Cl 0.000000 1.672954 −0.305277 αyy 54.043
Cl 1.448821 0.836477 −0.305277 αzz 39.470
Cl 1.448821 0.836477 −0.305277 αxz,αyz,αxy 0.000
F 0.000000 0.000000 1.562337
s uc u e and pola izabili ies a e summa ized in Table I.
Wi hin SCELIB4.0, he co ela ion-pola iza ion po en ial Vcp
was calcula ed using he modi ied ee-elec on gas model,
and he exchange po en ial was ob ained using he Ha a
ee-elec on gas exchange model, wi h de ailed in o ma ion
p o ided in Re . [30].
Figu e 2illus a es hese po en ials o T=100 eV. While
Vs and Vcp a e independen o elec on ene gy, Vex inc eases
as he elec on ene gy dec eases. The peak a =0.475 (in
a omic uni s) a ises om he ca bon a om, whe eas he one
a ound =3.0 is associa ed wi h he chlo ine and luo ine
a oms.
A. Elas ic sca e ing
In i s o de , he elas ic DCS o molecules can be de-
e mined using he independen a om model wi h addi i i y
ule (IAM-AR). In his app oach, he molecula DCS is
ob ained by cohe en ly summing he squa ed sca e ing am-
pli udes i(θ) o each a omic cons i uen (C, Cl, F). The
elas ic-sca e ing ampli udes a e de i ed om he in e ac ion
po en ials o he a omic cons i uen s, which include (i) a
s a ic e m, (ii) an exchange e m, and (iii) a e m cha ac-
e izing long- ange in e ac ions a ising om he molecula
FIG. 2. Po en ials used o he calcula ion in he IAM-SCAR+I
and SCOP model in a omic uni s o T=100 eV: Vs (—), Vex (- - -),
Vcp (−·−).
022809-3
M. DINGER, Y. PARK, AND W. Y. BAEK PHYSICAL REVIEW A 111, 022809 (2025)
pola izabili y. While he IAM-AR p o ides a use ul app oxi-
ma ion a high ene gies, i neglec s sc eening and in e e ence
e ec s a ising om di e en sca e ing cen e s in polya omic
molecules, which become inc easingly impo an a lowe
elec on ene gies.
Blanco and Ga cía u he imp o ed his app oach in hei
IAM-SCAR+I model [12–14,32] by inco po a ing sc eening
and in e e ence e ec s based on he molecula geome y:
dσel
ddi ec
=
N
i
s2
i| i(θ)|2+
N
i=j
ijsisj i(θ) ∗
j(θ)
×sin(qxij)
qxij
,(4)
whe e siand ij a e he sc eening and in e e ence coe icien s
de i ed om he spa ial a angemen o he cons i uen a oms.
The e ms qan xij desc ibe he momen um ans e and dis-
ance be ween a oms iand j, espec i ely. In addi ion o di ec
sca e ing desc ibed by Eq. (4), he IAM-SCAR+I model
accoun s o edispe sed con ibu ions a ising om mul iple-
sca e ing p ocesses. Fo de ailed de i a ions and discussions
o all e ms, we e e eade s o he o iginal wo ks [12–14,32].
In calcula ing he DCS wi h he IAM-SCAR+I model, he
sca e ing ampli udes o he a omic and pola iza ion po en ials
we e de e mined ollowing me hods desc ibed in an ea lie
wo k [17], u ilizing he Fe mi model o he exchange po-
en ial and he modi ied ee-elec on gas model [30] o he
pola iza ion po en ial.
Ro a ional c oss sec ions and TECSs we e de e mined us-
ing he close-coupling code POLYDCS [15] wi h he equi ed
Kma ix ob ained h ough he VOLSCAT package [33]. In he
VOLSCAT code, he Kma ix is calcula ed by nume ically
in eg a ing he in eg odi e en ial sca e ing equa ion, wi h he
p ocedu e go e ned by inpu pa ame e s de ining he in e ac-
ion po en ials, he p ojec ile p ope ies, and in eg a ion g ids.
As p e iously men ioned, he in e ac ion po en ials we e de-
i ed using he SCELIB4.0 lib a y. The nume ical in eg a ion
was pe o med on a g id wi h 32 pola and azimu hal poin s
and 256 poin s along he adial coo dina e, ex ending up o
250 a omic uni s. The Kma ix was compu ed up o a maximal
angula momen um quan um numbe o 25, achie ing a bal-
ance be ween compu a ional e iciency and desi ed accu acy
o he esul s. Using his Kma ix, he TECSs, including
o a ional exci a ion c oss sec ions o CCl3F, we e ob ained
ia he POLYDCS code.
The main inpu pa ame e s o POLYDCS include o a ional
cons an s, he molecula dipole momen , o a ional s a e quan-
um numbe s, and he maximum angula momen um quan um
numbe . The o a ional cons an s along he p incipal axes
a e 0.08203, 0.08203, and 0.05738 cm−1[34], while he
dipole momen o CCl3Fis0.46D[35]. The o al o a ional
exci a ion c oss sec ions we e de e mined by summing he
indi idual o a ional exci a ion c oss sec ions o ansi ions
om he g ound s a e up o he nin h exci ed s a e, wi h he
molecule modeled as an asymme ic igid o o . Di e en ial
elas ic-sca e ing c oss sec ions in he POLYDCS code a e ex-
p essed as Legend e expansions, wi h he maximum numbe
o e ms se o 50 o his pape .
B. Inelas ic sca e ing
In he SCOP model, elec on sca e ing is ep esen ed by
a complex in e ac ion po en ial V(
) consis ing o bo h a eal
and an imagina y pa :
V(
)=VR(
)+iVabs(
),(5)
whe e he eal pa VR(
) accoun s o elas ic sca e ing, and
he imagina y componen Vabs(
) desc ibes he abso p ion o
inciden elec on lux in o in e ac ion channels, leading o
inelas ic-sca e ing e en s.
The main ea u e o he SCOP model is he use o sphe i-
cally symme ic po en ials o desc ibe he elec on-molecule
in e ac ion. In his pape , he sphe ically symme ic op ical
po en ial Vop ( ) was ob ained by ully a e aging V(
)gi en
by Eq. (5) o e all possible molecula o ien a ions. To acil-
i a e his a e aging, V(
) was expanded a ound he cen e o
mass o he molecule using symme y-adap ed unc ions wi h
he SCELIB4.0 lib a y:
V(
)=
lm
Vlm( )XA1
lm (θ,ϕ).(6)
In Eq. (6), is he dis ance om he cen e o mass o
he molecule and XA1
lm is he symme y-adap ed unc ion o
he o ally symme ic i educible ep esen a ion (IR) A1 o
he angula momen um land i s componen m. The la e
can be ep esen ed as a linea combina ion o eal sphe ical
ha monics Slm(θ,ϕ):
XA1
lm (θ,ϕ)=
m
l=−m
bA1
lmSlm(θ,ϕ),(7)
whe e he coe icien s bA1
lm can be ob ained om he cha ac e
able o he IR A1. The a e aging o V(
) was pe o med by
in eg a ing he igh -hand side o Eq. (6) o e he h ee Eule
angles and hen di iding i by 8π2. Due o he o hogonali y o
sphe ical ha monics o e he su ace o a sphe e, he in eg al
o a single eal sphe ical ha monic o e Eule angles anishes
o l= 0, so ha Vop ( )isgi enbyVop ( )=V00/√4π
(wi h bA1
00 =1).
The eal pa VR(
) o he in e ac ion po en ial comp ises
he s a ic Vs , exchange Vex, and co ela ion-pola iza ion Vcp
po en ials men ioned in he beginning and he abso p ion po-
en ial Vabs(
) was gene a ed using he quasi ee-sca e ing
model desc ibed in de ail by S aszewska e al. [36,37]:
Vabs(
)=−1
2(
)2(T−VSE)4π
5k3
FTH(γ)(Z1+Z2+Z3).
(8)
He e, (
) is he elec on densi y pe uni olume, VSE
is he sum o he s a ic and exchange po en ials, kFis he
Fe mi momen um gi en by kF=(3π2)1/3, and H(γ)is he
Hea iside s ep unc ion wi h γ=k2+k2
F−α−β, whe e kis
he momen um o inciden elec ons. Among di e en models
o αand β, we chose α=k2
F+2I−VSE and β=k2
F−VSE,
whe e I=11.73 eV [38] is he ioniza ion po en ial o he
022809-4
ELASTIC AND INELASTIC ELECTRON SCATTERING … PHYSICAL REVIEW A 111, 022809 (2025)
TABLE II. P esen expe imen al esul s o he DCS o CCl3F as a unc ion o he sca e ing angle θ o di e en elec on ene gies T,
exp essed in uni s o 10−16 cm2/s . The o e all unce ain ies o he DCS a e 17%. Addi ionally, he TECS σel,MTCSσm, TICS σinel,and TCS
σ o a e gi en in uni s o 10−16 cm2. The TICS was calcula ed using he SCOP model, and σ o was ob ained by summing σel and σinel.
θ/T30 eV 40 eV 60 eV 80 eV
20° 28.4 20.2 15.1 11.5
30° 13.5 9.09 3.50 3.61
45° 2.10 2.53 0.653 0.705
60° 0.908 0.642 0.297 0.366
75° 0.494 0.615 0.298 0.324
90° 0.589 0.843 0.348 0.370
105° 0.700 0.849 0.372 0.326
120° 0.539 0.676 0.345 0.224
135° 0.408 0.636 0.431 0.405
150° 1.09 0.963 0.709 1.46
σel 40.54 ±7.09 31.75 ±5.37 22.08 ±3.48 20.09 ±3.46
σm11.58 ±2.03 12.25 ±2.07 7.41 ±1.17 8.24 ±1.42
σinel 7.21 9.16 10.43 10.35
σ o 47.75 40.91 32.51 30.44
θ/T100 eV 200 eV 400 eV 800 eV
20° 10.1 5.91 5.25 4.05
30° 2.67 1.20 1.30 0.698
45° 0.973 0.566 0.365 0.143
60° 0.444 0.354 0.162 0.0520
75° 0.318 0.226 0.0770 0.0403
90° 0.328 0.133 0.0840 0.0291
105° 0.308 0.103 0.0882 0.0348
120° 0.249 0.139 0.110 0.0299
135° 0.461 0.250 0.139 0.0340
150° 1.03 0.601 0.190 0.0545
σel 18.15 ±3.63 10.51 ±2.24 7.56 ±1.23 5.49 ±1.49
σm7.16 ±1.43 4.07 ±0.87 2.05 ±0.33 0.77 ±0.21
σinel 10.89 9.57 7.41 5.14
σ o 29.04 20.08 14.97 10.63
molecule. The e ms Z1,Z2, and Z3a e gi en by
Z1=5k3
F
α−k2
F
,
Z2=−k3
F5(k2−β)+2k2
F
(k2−β)2,
Z3=H(˜γ)2˜γ5/2
(k2−β)2(9)
wi h ˜γ=α+β−k2.
Once he sphe ical complex op ical po en ial was ob ained,
he adial Sch ödinge equa ion was sol ed o de e mine he
phase shi o sca e ed elec ons by applying he a iable
phase app oach [39,40]. In his app oach, he eal pa εland
he imagina y pa ηlo he phase shi s a e ob ained om wo
coupled i s -o de di e en ial equa ions:
dεl
d =−1
kVop
R(X2−Y2)−2Vop
abs XY,
dχl
d =−1
kVop
abs (X2−Y2)+2Vop
RXY(10)
wi h
X=cosh χl[ηlsin εl−jlcos εl],
Y=sinh χl[ηlcos εl+jlsin εl],(11)
whe e jl(k ) and ηl(k ) a e he usual Ricca i-Bessel unc-
ions. Equa ion (10) was sol ed by in eg a ing i using he
ou h-o de Runge-Ku a me hod [41]. The phase shi s
εl(k ) and χl(k ) o →∞a e ela ed o he Sma ix ia
Sl=exp[2i(εl+iχl)], which is u he linked o he TICS
σinel h ough he equa ion
σinel(k)=π
k2
l
(2l+1)[1 −|Sl(k)|2].(12)
V. RESULTS AND DISCUSSION
The esul s o he p esen expe imen a e lis ed in Table II
and displayed in Fig. 3, whe e hey a e compa ed o bo h
he expe imen al da a o Hoshino e al. [9] and heo e ical
calcula ions. Since ichlo o luo ome hane is a pola molecule
wi h a pe manen dipole momen [27], i is p one o subs an ial
o a ional exci a ions. Due o he ini e ene gy esolu ion o
ou expe imen al se up, i was no possible o ully sepa a e
pu e elas ic DCSs om con ibu ions due o o a ional and
ib a ional exci a ions. To p o ide a mo e accu a e compa -
ison, he o a ional exci a ion c oss sec ions we e included in
he heo e ical models. The heo e ical alues shown in Fig. 3
include he DCS calcula ed using he IAM-SCAR+I model
022809-5
M. DINGER, Y. PARK, AND W. Y. BAEK PHYSICAL REVIEW A 111, 022809 (2025)
FIG. 3. P esen expe imen al esul s () o he DCS o CCl3F o di e en p ima y elec on ene gies compa ed o he elas ic DCS epo ed
by Hoshino e al. [9](). The solid line (—) ep esen s calcula ions using he IAM-SCAR+I model, he dash-do ed line (−−) depic s
calcula ions using he IAM-SCAR+I model plus he o a ional exci a ion c oss sec ions, and he dashed line (−−) desc ibes he esul s o
bes i s o Eq. (13) o he expe imen al da a. Fo 30 eV, Schwinge mul ichannel calcula ions wi h Bo n closu e om F ei as e al. [11] we e
also a ailable: s a ic exchange (−−) and s a ic exchange plus pola iza ion app oxima ions ().
wi h and wi hou he inclusion o o a ional exci a ion c oss
sec ions o he CCl3F molecule.
I is impo an o no e ha he DCS da a epo ed by
Hoshino e al. [9] ep esen elas ic sca e ing wi h only mino
ib a ional con ibu ions, due o he expe imen al ene gy eso-
lu ion o 35–40 meV, and a e he e o e expec ed o be sligh ly
lowe han bo h ou expe imen al esul s and he heo e ical
alues including o a ional exci a ion c oss sec ions.
The esul s o he p esen pape , as shown in Fig. 3, ag ee
sa is ac o ily wi h heo e ical p edic ions o elec on ene gies
abo e 200 eV. Howe e , o ene gies below 100 eV, no able
disc epancies we e obse ed when compa ing ou da a o bo h
he heo y and he expe imen al da a o Hoshino e al. [9].
These disc epancies a e pa icula ly e iden in he angula
ange o 60° o 130°, whe e he DCSs exhibi wo dis inc
minima a ound 60° and 120°. While hese minima a e clea ly
obse ed in he expe imen al da a o Hoshino e al. [9]as
well as in he IAM-SCAR+I model, hey a e only ain ly
disce nible in he p esen expe imen al da a.
A 30 eV, he p esen esul s a e gene ally lowe han he
DCS alues epo ed by Hoshino e al. [9], which, excep a
30°, a e easonably well ep oduced by he IAM-SCAR+I
model. The heo e ical da a epo ed by F ei as e al. [11]also
show a simila angula dependence bu signi ican ly o e es-
ima e he DCS a he posi ions o he wo minima. In e ms
o absolu e alues, he da a om his pape a 60 and 100 eV
appea o align mo e closely wi h he heo e ical alues com-
pa ed o he co esponding da a om Hoshino e al. [9]. As
no ed abo e, he esul s o Hoshino e al., howe e , show
be e ag eemen wi h he heo e ical p edic ions ega ding he
angula dependence o he DCS. I is impo an o men ion ha
he calcula ed alues ha e limi ed eliabili y a low ene gies
due o he app oxima ions used in he model, which a e only
oughly alid o elec on ene gies below 50 eV. This limi a-
ion may pa ly explain he disc epancies obse ed a lowe
ene gies.
The double-minima s uc u e obse ed in he expe imen al
da a, which a ies wi h elec on ene gy, is p ima ily a ibu ed
o he p esence o he h ee chlo ine a oms in CCl3F. Fo
example, he minimum a ound 120° is closely linked o he
esonancelike sha p minima in he DCS o indi idual chlo-
ine a oms (e.g., a θ=118◦and T=109.5eV[42]). These
chlo ine a oms domina e he DCS o CCl3F wi h an es ima ed
con ibu ion o app oxima ely 75%. This phenomenon was
also epo ed by Hoshino e al. [9], who ound ha he min-
imum a ound 120° became inc easingly p onounced in he
DCS as he numbe o chlo ine a oms inc eased in CF4−xClx
molecules.
To ob ain he TECS and he MTCS, he expe imen al
DCS shown in Fig. 3was i ed using a se ies o Legend e
022809-6
ELASTIC AND INELASTIC ELECTRON SCATTERING … PHYSICAL REVIEW A 111, 022809 (2025)
FIG. 4. (a) TECS (σel)o CCl
3F om he p esen pape () compa ed wi h heo e ical and expe imen al esul s. Shown a e alues
compu ed using he IAM-SCAR+I model including o a ional con ibu ions (−−), and he close-coupling app oxima ion wi h he POLYDCS
code excluding (−−) and including o a ional con ibu ions (—). Expe imen al da a om Hoshino e al. ()[9] a e also included. The inse
displays he con ibu ions o he speci ic i educible ep esen a ions (IR) deno ed by E, A1,andA
2 o σel, as calcula ed using he POLYDCS code.
(b) TICS (σinel )o CCl
3F calcula ed using he SCOP model (−−) a e compa ed o he expe imen al o al ioniza ion c oss sec ions epo ed
by Sie a e al. (*) [7]. The sum o σel +σinel =σ o de e mined in his pape () is compa ed o he TCS o CCl3F measu ed by Jones ()[4]
and Zecca e al. ()[5].
unc ions. In gene al, he DCS o a polya omic molecule can
be exp essed by he ollowing equa ion [15]:
dσel
d=dσB
d+
LAL−AB
LPL(cosθ)≡
L
˜
ALPL(cosθ),
(13)
whe e he supe sc ip B deno es quan i ies de i ed om long-
ange elec on-dipole in e ac ions calcula ed using he Bo n
app oxima ion. The e m dσB/d, ep esen ing he so-called
Bo n closu e [43] is inco po a ed in o he coe icien s ˜
AL.Fo
sho - ange po en ials, he coe icien s ALcon e ge apidly. In
cases in ol ing long- ange elec on-dipole in e ac ions, whe e
highe pa ial wa es may be signi ican , employing he Bo n
closu e subs an ially educes he equi ed numbe o pa ial
wa es. Tes s wi h he DCSs o a ious polya omic molecules
a di e en ene gies showed ha hey can gene ally be i ed
using he o mula on he igh -hand side in Eq. (13) wi h
L=6, wi hin a 95% con idence in e al, p o ided ha he
DCS does no exhibi esonancelike s uc u es.
The dashed line in Fig. 3 ep esen s bes i s o Eq. (13)
wi h L=6 o he p esen expe imen al esul s. As shown, he
i esul s closely ep oduce he measu ed alues. Since he
in eg al o he Legend e unc ions o e he ull angula ange
(0° o 180°) anishes o L⩾1, he TECS and i s unce ain y
a e de e mined solely by he alue o he coe icien ˜
A0.The
TECS de i ed om ˜
A0is p esen ed in Table II. The MTCS
was calcula ed using all bes - i coe icien s ˜
AL,including
hose o L⩾1.
Figu e 4(a) compa es he expe imen al TECS wi h he e-
sul s ob ained using he POLYDCS code and he IAM-SCAR+I
model, including o a ional exci a ion c oss sec ions. As
expec ed, he a io o he o a ional exci a ion c oss sec ions
o he pu e elas ic-sca e ing c oss sec ion dec eases wi h in-
c easing elec on ene gy, s a ing a app oxima ely 17% a 30
eV and dec easing o 6% a 800 eV. O e all, while he expe i-
men al TECS is sligh ly lowe han he alues p edic ed by he
POLYDCS code, i aligns sa is ac o ily wi h he IAM-SCAR+I
esul s (including o a ional con ibu ions) wi hin expe imen-
al unce ain ies. The TECS alues epo ed by Hoshino e al.
[9] a e signi ican ly lowe han no only ou esul s bu also
he p edic ions by he IAM-SCAR+I model and POLYDCS
code, e en a T=100eV, whe e bo h models a e expec ed o
deli e ai ly accu a e esul s. As shown in Fig. 3, his disc ep-
ancy p ima ily s ems om he lowe DCS alues epo ed by
Hoshino e al. [9], which p opaga e di ec ly in o hei TECS
calcula ions. Fu he mo e, addi ional di e ences may esul
om a ia ions in he ex apola ion echniques used o he
angula in eg a ion om 0° o 180°.
In Fig. 4(a), he ela i e con ibu ions o he molecula
o bi als (MOs) associa ed wi h he IRs A1,A
2, and E o σel
a e also displayed as a unc ion o he elec on ene gy T.To
a i s app oxima ion, he ela i e con ibu ion o each IR o
he TECS co esponds o he a io IR
MO, de ined as he numbe
o he MOs belonging o a speci ic IR di ided by he o al
numbe o MOs. A no iceable de ia ion be ween hese quan i-
ies is obse ed o he IRs A1and A2. While he con ibu ion
o A1 o he TECS is sligh ly lowe han expec ed based on
IR
MO (33%), he con ibu ion o A2is app oxima ely wice as
high (6%). This disc epancy is a ibu ed o he cha ac e is ics
o he MOs in ol ed. Speci ically, one o he wo o bi als
belonging o A2is he HOMO, which signi ican ly enhances
i s con ibu ion o elec on collision p ocesses compa ed
o A1.
022809-7
M. DINGER, Y. PARK, AND W. Y. BAEK PHYSICAL REVIEW A 111, 022809 (2025)
In addi ion o he measu emen o elas ic-sca e ing c oss
sec ions, he TICS o CCl3F was calcula ed using he SCOP
model [Fig. 4(b)]. The calcula ed alues align well wi h he
expe imen al o al ioniza ion c oss sec ions epo ed by Sie a
e al. [7] o T⩽80 eV, emaining wi hin he expe imen al
unce ain ies. I is impo an o no e ha , in he de ini ion o
he αpa ame e in he SCOP model, he h eshold exci a ion
ene gy was se equal o he ioniza ion po en ial o CCl3F.
Thus, σinel co esponds o he o al ioniza ion c oss sec ion o
CCl3F.
Fu he mo e, he TICS was combined wi h he TECS and
compa ed o he expe imen al TCS da a om Jones [4] and
Zecca e al. [5]. As shown in Fig. 4(b), he sum o he TECS
and TICS ag ees well wi h he expe imen al TCS σ o wi hin
unce ain ies. Addi ionally, i can be seen om Fig. 4 ha he
TICS o CCl3F app oaches he TECS abo e 400 eV, e lec ing
he balanced con ibu ion o elas ic and inelas ic sca e ing
o he TCS a high ene gies—a phenomenon commonly ob-
se ed in o he molecules [16].
VI. CONCLUSIONS
The expe imen al esul s o he DCS o CCl3F ag ee sa is-
ac o ily wi h he IAM-SCAR+I model wi hin expe imen al
unce ain ies o T⩾200 eV. This is expec ed, as he IAM-
SCAR+I model is known o eliably p edic he DCS o
polya omic molecules a high elec on ene gies. The poo es
ag eemen was ound in he angula ange o 60° o 130°
below 100 eV, whe e wo p onounced minima we e obse ed
in he expe imen al da a a elec on ene gies. A 30 and 60 eV,
di e ences o up o 100% we e obse ed be ween he p esen
measu emen s and he model p edic ions.
In his ene gy ange, ou esul s also show conside -
able de ia ions om he da a o Hoshino e al., which a e
cu en ly he only a ailable expe imen al da a. These de i-
a ions a e obse ed no only in he absolu e scale bu also
in he angula dependence. No ably, he da a o Hoshino
e al. closely ma ch he angula dependence p edic ed by
he IAM-SCAR+I model in con as o ou esul s. To al
elas ic-sca e ing c oss sec ions o CCl3F, de e mined om
he expe imen al DCS, a e la gely consis en wi h calcula-
ions pe o med wi h he IAM-SCAR+I model plus o a ional
exci a ion c oss sec ions, wi hin expe imen al unce ain ies.
Howe e , hey end o be sligh ly lowe han hose ob ained
om he close-coupling calcula ions wi h he POLYDCS code,
which also includes o a ional exci a ion c oss sec ions. The
la e con ibu es 17% o σel a 30 eV and dec eases o 6% a
800 eV. No ably, he TECSs epo ed by Hoshino e al. [9]a e
again lowe han ou expe imen al da a and he p edic ions o
he IAM-SCAR+I model.
To al inelas ic-sca e ing c oss sec ions o CCl3F, calcu-
la ed using he SCOP model, ep oduce he expe imen al o al
ioniza ion c oss sec ions a ailable in li e a u e sa is ac o ily
wi hin expe imen al unce ain ies below 80 eV. Th oughou
he en i e measu ed ene gy ange, he sum o he TICS and
TECS ag ees well wi h he expe imen al TCS measu ed by
o he g oups. This u he indica es ha he h ee ypes o he
sca e ing c oss sec ions σel,σinel, and σ o de e mined in he
p esen pape and by o he g oups a e consis en wi h each
o he .
ACKNOWLEDGMENTS
This esea ch was suppo ed by he join esea ch p ojec
BIOSPHERE. The p ojec 21GRD02 BIOSPHERE has e-
cei ed unding om he Eu opean Pa ne ship on Me ology,
co inanced by he Eu opean Union’s Ho izon Eu ope Re-
sea ch and Inno a ion P og amme and by he Pa icipa ing
S a es. The au ho s exp ess hei hanks o Heike Ni mann
and And eas Pausewang o hei assis ance and echnical sup-
po du ing he measu emen s. The au ho s also hank Felix
Lehne o his echnical suppo while se ing up he VOLSCAT
and POLYDCS code and calcula ions.
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