Bulk Plasmon-Assisted Ion Neutralization
at Metal Surfaes
Abdalaziz A. Almulhem
Department ofPhysis,KingFaisalUniversity
P.O.Box1759,Alahssa31982 SaudiArabia
Reeivedon18January,2002
Themehanismfor ionneutralizationat metalsurfaes viasurfaeplasmonexitation was
previ-ouslysuggested. Itwasshownthatthis mehanismisof omparableprobabilityas theothertwo
wellstudied mehanisms, namely,Augerand resonane tunnelingneutralizations. Inthepresent
workwe studythe probability ofexiting abulkratherthansurfaeplasmonduring the
neutral-izationproess. Thealulations show that theneutralizationratedepends exponentially onthe
iondistanefromthesurfae. Anditisfoundtobeimportantatsmalldistanesfromthesurfae.
Thetheoryis applied to the sattering ofprotons from aluminum surfae. Comparison between
thetransitionrateforbulkplasmon-assistedandsurfaeplasmon-assistedionneutralizationshows
that the transition rateislowerbytwoorders ofmagnitudeinthease ofbulkplasmon-assisted
neutralizationaswasexpeted.
I Introdution
It was reently disovered that the neutralization of
ionssatteredfrommetalsurfaesbysurfaeplasmons
isanimportanteletrontransferproessespeiallyfor
ionsarryinghigh potentialenergy[1-3℄. This
meha-nism wasoriginally suggestedbythe author asa
pos-sible mehanism of neutralization in addition to the
fullystudiedresonaneandAugerneutralization
meh-anisms[4℄. Theexperimentallymeasuredtimefor
neu-tralizationofaprotonsatteredfromametalsurfaeis
about10 15
seonds. Foraluminum, forexample,the
energyofthesurfaeplasmonisabout10:6eVandthat
ofabulkplasmonisabout15eV.Thisimpliesaperiod
of osillationof about 10 16
se. It anbeonluded
that a olletive response from the metal by exiting
a surfae plasmon or a bulk plasmon is quiet
possi-ble. Alsoplasmonsan beexitedbeausethevalene
eletronsannotrespondinstantaneouslyto sreenthe
moving harge. Reently a good number of
theoreti-alpapershavebeenpublishedonpotentialexitations
ofsurfaeplasmonduringsurfaeneutralization[5-10℄.
Thetransitionratefortheproessisompetitivewith
thatfortheothertwoproesses[6,11-12℄. Itwasseen
that surfae plasmon mediated ion neutralization at
metalsurfaes is veryimportant. Notheoretial work
wasreportedforbulkplasmon-mediatedneutralization
asfar as the author knows. Sineplasmons detetion
in experiment is indiret, relying on the observation
of the ejeted eletrons released from plasmon deay
[13℄,mostoftheworkinthiseldisdonetheoretially.
energy distribution of eletronsejeted by ionimpat
onrmedthe reality ofthemehanismof ion
neutral-ization by plasmonexitation. Inthe aseof He ions
onMg theeletronstruturedue to plasmondeayis
moreimportantthanthatofAugerneutralization[1℄.
Thepurpose ofthis work isto analyze anew
neu-tralizationhannelwhosephysialpitureisasfollows.
The ion approahes the surfaeof the metal, its eld
attratingeletronsinthemetal. Theseeletronsmove
toward the region losest to the ion, leaving
unom-pensatedpositivehargebehind. Thisreatesa
restor-ing fore on thedisplaed eletronsleadingto plasma
exitation. During the proess of exiting a plasmon
(takenin thisworkto bebulkplasmon), theionpiks
upaneletron,beomingneutralized. Thebinding
en-ergytherebyreleasedisarriedawaybyabulkplasmon
wave,whihexpandsfromthepointofimpatlike
rip-plesonapond.
TheHamiltonianfortheproessiswrittendownin
termsofeletronandplasmonannihilationandreation
operators. In the alulation a anonial
transforma-tionisusedinordertoabletoworkwiththe
Hamilto-nianoftheinterationbetweenthehargedpartileand
thepotentialofthebulkplasmons. Thistransformation
separates the plasmoni neutralization hannel from
other sattering and reation hannels. It also
intro-dues appropriate ontinuum-bound state
orthogonal-ityorretionsinto thematrixelements. This method
wasusedbeforeinthealulationsofionneutralization
viathethreedierentmehanismsnamely: resonane,
II Model
TheseondquantizedHamiltonianofourmodelis
b
H =
Z
d~r b
y
(~r)T(~r) b
(~r) + 1
2 Z
d~rd~r 0
b y
(~r 0
)V 0
el el (~r;~r
0
) b
(~r 0
) b
(~r)
+ Z
d~r b
y
(~r)[V
b
(~r) j~r ~sj 1
b
(~r)+ X
!
s ^ y
q ^
q +
Z
d~r b
y
(~r) b
s (~r)
b
(~r)
+ X
!
p ^ a y
q ^ a
q +
Z
d~r b
y
(~r) b
p (~r)
b
(~r) (1)
d
IntheHamiltonian, b
and b
y
aretheeletron
anni-hilationandreationoperators,T(~r)istheeletron
ki-netienergy,V
b
(~r)isthepotentialofthepositive
bak-ground, and V 0
el el
(~r)is the eletron-eletron
intera-tion. Inthe equation aboveb
q and b
y
q
are thesurfae
plasmonannihilationandreationoperators.Here sis
thepositionoftheproton, b
s
(~r)istheseondquantized
potentialof the surfae plasmons. Its expliit form is
givenin relatedwork in apreviouspaper[4℄. The
po-tential b
(~r) is the seond quantized potential of the
bulkplasmonsgiveby
b
p (~r)==
X
g
p e
qjzj
(e i~q
~
R
b a
q +e
i~q ~
R
b a y
q ) (2)
Theouplingonstantg
p
isgivenby
g
p =
s
2q
z
!
p V
(3)
where!
p
isthebulkplasmonenergyandq
z
isthe
plas-mon wavevetorin thez-diretion.
Theinterationtermin theHamiltonianthatgives
risetothemehanismbeingstudiedisgivenbythelast
termin (1),namely,
Z
d~r b
y
(~r) b
p (~r)
b
(~r) (4)
Intheequationsaboveba
q andba
y
q
arethebulkplasmon
annihilation andreationoperators. Theprime onthe
summationimpliesq<q
whereq
istheplasmon
ut-owavevetor(maximumplasmonq). Inthe
Hamilto-nian(1),T(~r)istheeletronkinetienergy,V
b
(~r)isthe
potential of the positive bakground, V 0
el el (~r;~r
0
) the
eletron-eletron interation and b
(~r) and b
( ) d
ag(~r)
are the eletron annihilation and reation operators.
The z-axisis perpendiular to the metal surfae, the
half plane z < 0 onstitutes the jellium metal and
z > 0 onstitutes the exterior region. The eletron
position vetor(~r) in (1) may be either inside or
out-sidethemetal. Thisallowsfortunnelingoftheeletron.
Throughoutthispaperatomiunitswillbeused.
The eletron eld operator will be expanded in
termsof theomplete orthonormalset of orbitals and
orrespondingannihilationoperators b
k :
b
(~r)= X
k (~r)
b
k
(5)
Notethat b
(~r) isaaneletroneldoperator,
k (~r)
represents eletron orbital, while b
k
represents the
orresponding annihilation operator. The metal
ele-tronswavefuntionswouldbetakenassolutionsofthe
Shrodinger equation with a potential V(z) whih is
onstant inside and outside the metal with a step of
heightV
0
atthesurfae(z = 0)
V(z) = V
0
#(z) (6)
Thestepfuntion#(z)hasavalueofV
0
forz 0,
anda valueof0 forz > 0. Here V
0
= F + W with
F theFermi energyand W theworkfuntion,and
en-ergiesaremeasuredfromthebottomoftheondution
band. The orresponding orthonormal eigenfuntions
k (~r)are
k (~r) =
1
k
(V)
1=2 n
e i
~
kg~r
[(k 0
z +ik
z
)exp(ik 0
z
z)+ (k 0
z ik
z
)exp( ik 0
z z)℄
o
z < 0
=
1
k
(V)
1=2 f2k
z e
i ~
kg~r
e k
z z
whereV isthevolume ofthemetaland andk are de-nedby (k 0 z ) 2
= 2E 0
k ; k
2
z = 2(V
0 E
0
k ); k
2 +(k 0 z ) 2
= 2V
e 0 with E 0 k = E k (1=2)K 2 E k
is theeigenvalueof
k and
~
K is theomponentof
~
k parallel tothe surfae. Within theondution band
E
k
< F onehas
0 < E 0
k <E
k
< F < V
0
The wave funtions will be osillatory inside the
metalanddeayinto z diretion outside. Thesameis
truefortheunlledlevelswithF < E
k < V 0 + 1 2 K 2 .
The wave funtions with E
k > V 0 + 1 2 K 2 osillate
with z outsideas well as inside the metal. Theyalso
requireadierentnormalization, but thisis irrelevant
sinetheplasmonineutralizationmatrixelementwill
involveondution bandeletrons. Sine
k
onstitute
aomplete set,theboundatomiwavefuntion
at in
thenalstatesanbeexpended intermsofthem.
Inordertooverometheproblemoflakof
orthog-onalitybetweenthenalatomiwavefuntionandthe
initialondutionbandwavefuntion,aunitary
trans-formation to a new representation is being used. In
thisrepresentationtheatomisdesribedbyastate
or-thogonalto allondution band stateswasused. The
appropriateunitarytransformationisoftheform
b
U = e ( 2 ) b F (8)
wheretheappropriateunitaryoperatoris
b F = X ( b A y b b y b A ) (9) and b A y = Z d * r ( * r * s) b y ( * r) (10)
ThisunitarytransformationrotatestheFokspae
by =2into anew spae alled the idealspae. The
transformation being unitary preserves the matrix
el-ements and the Hermitiity of the Hamiltonian. The
problemof thenal atomistatebeingnotorthogonal
to the initial band funtion is solved automatially in
thisformalism. Thenalatomistateswillbetakenas
theunperturbedground1sstateofhydrogen
1s = 1 p 2 e * r * s (11)
Ating on the Hamiltonian in (1) a transformed
Hamiltonian in whih the matrixelementsof the
pos-sible reativehannels are orthogonalizedwill be
pro-dued. Eah matrix element will ontain two parts,
the rst being the usual matrix element of the
pro-ess, and theseond istheorthogonalization term. In
(10) A y
is the reation operator for an eletron in a
boundhydrogenorbital
(
*
r *
s)enteredonthe
pro-ton (positions) and stands for the atomi quantum
numbers = (nlm).
The physial states on whih the transformed
Hamiltonianatsareoftheform
j::: i = b U 1 b y ( * r) (12)
where j:::i is any standard Fok state represented in
termsofeletronreationoperators b
d
ag(~r)atingon
thevauumstatejoi
Using (8-10) and making use of the ommutation
rulesof b and, b ( *
r)theeletroneldoperators
trans-formasfollows
b U 1 b ( * r) b U = b ( * r) Z d * r 0 ( * r * s; * r 0 * s) b ( * r 0 ) + X ( * r * s) b (13) d
where (~r ~s; ~r 0
~s) is the hydrogen bound state
kernel ( * r * s; * r 0 *
s) = X ( * r * s) ( * r 0 *
s) (14)
TransformingtheHamiltonianin(1)usingthe
uni-tarytransformationin(8),itanbeseenthatall
han-nelsofsatteringarerepresented;inludingthoseof
re-ativesatteringthataresought. Transformingthelast
term in the Hamiltonian, the bulk plasmon-mediated
ionneutralizationhannelwouldarise.
Z d * r b U 1 b y ( * r) b p ( * r) b ( * r) b U (15)
This an be manipulated by inserting an identity
operator b
U 1
b
U in betweentheoperatorsin thisway
R d * r b U 1 b y ( * r) b U b U 1 b p ( * r) b U b U 1 b y ( * r) b U (16)
neutral-b
T
11 =
X Z
d *
r
(
*
r *
s) b
y
b
p (
*
r) b
( *
r) (17)
b
T
12 =
X ZZ
d *
rd *
r 0
(
*
r *
s)( *
r *
s; *
r 0
*
s) b
y
b
p (
*
r) b
( *
r) (18)
Inserting(7) in thetwoterms b
T
11 ,
b
T
12
onendsthe followingexpressions fortheperturbation b
H
in
leading to
thehannel ofneutralization. Forbulkplasmon-mediatedneutralization
b
H
in =
X
( ;qjH
4 jk)
b y
b a y
q b
k
(19)
where thematrixelementsaregivenby
( ;qjH
4
jk) = g
p R
d *
r
(
*
r *
s)
k (
*
r)e qjzj
e i
*
q *
r
RR
d *
rd *
r 0
(
*
r *
s)
( *
r *
s; *
r 0
*
s)
k (
*
r 0
)e qjzj
e i
*
q *
r
(20)
d
Thisform ofthematrixelementsshowslearlythe
advantageof using the unitary transformation. It
in-ludes (the seond term) an orthogonalization term
that omesoutautomatiallywiththetheory. The
or-thogonalizationterm takesareof orthogonalizingthe
metal orbitals to all bound atomi orbitals. The
ma-trix elements in (20) are orreted forms of the Born
approximationto theexatT-matrixelementsfor the
sattering proess, and due to the inlusion of the
or-thogonalizationtermitisabetterapproximation. This
term wasfoundto beimportant intheion
neutraliza-tionatsurfaes [4,11-12℄.
III Calulations
The evaluation of the matrix elements will be arried
outusing(20). ThetransitionrateofthesatteringP
isgivenby
P = 2 X
(1sjH
in jk)
2
Æ("
i "
f
) (21)
wherethenalhydrogenstatehasbeentakentobethe
1sstate. Theprime onthe summation signindiates
therestritionthattheksumisovertheinteriorofthe
lled Fermi sea only. Thematrix elements M for the
neutralizationhannel are now taken from the atual
evaluations,thusgiving
P = 2 X
jg
p j
2
jMj 2
Æ( 1
2 k
2
E(1s) !
p
) (22)
Changingthesumintoanintegralweget
P = 4V
2
(2) 6
Z
d *
kd *
q jg
p j
2
jMj 2
Æ( 1
2 k
2
E(1s) !
p )
(23)
Here "
i =
1
2 k
2
is themetal eletronenergy, !
p is
thebulkplasmonenergyandE(1s)istheatomi
ele-tronenergy, V is the volume and 2 is for the double
spinoftheeletron.
P(s) = V
!
p
2 ZZZ Z
d *
Qdq
z d
*
Kdk
z *
Q *
Kq
z jMj
2
Æ("
i "
f
) (24)
d
Intheintegralabove *
Qistheplasmonwavevetor
in theparalleltothesurfaediretion. Thedelta
fun-tionisusednowto evaluatetheintegralover *
Kwhere
~
K istheomponentof *
kparallel tothesurfae.
P(s) = V
!
p
2 ZZZ
d *
Kdk
z dq
z *
z jMj
2
(25)
withthematrixelementevaluatedat
1
2 K
2
= "
f 1
2 k
2
0
= E(1s) +!
p 1
2 k
2
z
(26)
It should be stated that the atomi energies are
shiftedupward by Vsine energies are measuredfrom
thebottomoftheondutionband. Henethevalueof
E(1s) shouldbegivenby
Figure 1. Transition rate P as a funtionof the distane
s from the metal surfae for a proton satteredfrom
alu-minumsurfae (solidurve for thease whena bulk
plas-mon is exitedwhile the dashedurve represents thease
whenasurfaeplasmonisexited).
Transition rate P as a funtion of the distane s
fromthemetalsurfaeforaprotonsatteredfrom
alu-minum surfae (solid urve for the ase when a bulk
plasmon is exited while the dashed urve represents
theasewhenasurfaeplasmonisexited).
IV Results and Disussion
Toapplythetheorydevelopedhereweassumethe
sat-teringsystem
H +e (Almetal)! H(1s) (28)
Thealuminumishosenbeauseitbestsatisesthe
assumptions madein thetheory. First, itan be well
approximated by a jellium model. Seond, its Fermi
surfaeisverylosetothefreeeletronsurfaeforafae
entered ubi monatomi Bravais lattie with three
ondution eletrons per atom. Third, plasmons are
welldenedforaluminumandtheirexistenehasbeen
demonstratedexperimentally. Fourth,theexisteneof
experimental work onthis system[15℄. This is in
ad-dition to the theoretial work for the dierent
meh-anisms of neutralization [16-22℄.The parameters used
for aluminum are: 0:9261 for the Fermi wave vetor
k;0:5862forthesurfaepotentialV. Thegroundstate
in the hydrogen atom H(1s) lies energetially within
theondution band ofaluminum. All otherstateslie
abovetheFermilevel.
Equation(23)isusedtoalulatethetransitionrate
P asafuntionofthedistanesoftheprotonfromthe
surfae. The integration over k is alulated
numeri-ally. Allotheralulationsaredoneanalytiallyusing
theinverseFouriertransformintegralsandmakinguse
ThegureshowstheTransitionratePasafuntion
of the distane s from the metal surfaefor a proton
sattered from aluminum surfae (solid urve for the
asewhen abulkplasmon is exitedwhile thedashed
urve represents the ase when a surfae plasmon is
exited). From the gure it is seen that the
neutral-ization ratedepends exponentiallyon theiondistane
from thesurfae. And it is found to beimportant at
smalldistanes fromthesurfae. Comparison between
the transition rate for bulk plasmon-assisted and
sur-faeplasmon-assistedionneutralizationshowsthatthe
transition rateislowerbytwoordersofmagnitudesin
theaseofbulkplasmon-assisted neutralizationas
ex-peted. Thereasonislearlysimple,sineeletronson
the surfae are more easily exited to form a surfae
plasmonthanthosein thebulkofthemetal.
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