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Elektronische Interferenz und Elektronentransport
in Ubergangsmetall-Legierungen
H. Solbrig und T. Schmidt
am 5. Juli 2005
1. Ubergangsmetalle in Aluminium
2. Ausgangspunkt Spektraloperator
3. Atomare Netzwerke in Legierungen
4. Was wurde gezeigt ?
1. Ubergangsmetalle in Aluminium: Interferenz
• Elektronen treiben die Struktur zu konstruktiver elektronischer Interferenz
Hume-Rothery
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JONES
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kf
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PSG
DOS
εF↑
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TM-Valenz
negativ
analog: Peierls, Jahn -Teller, martensitischer Ubergang
• Interferenz von Pfaden uber diskretes und kontinuierliches SpektrumFano
Re
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diskret
1. Ubergangsmetalle in Aluminium: Hybridisierung 1
•Amorphe Legierungen (Experimente, Gruppe Haussler)
• Sonderrolle der NFE-Legierungen mit 1.8 Elektronen/Atom
• Al1−x(3dTM)x zeigen Extrema von ρ4K, TK, |d ln(ρ(T))/dT|, ... bei kritischen
Konzentrationen xTMc
• Plasma-Resonanz deutet auf gefullte d-TM-Orbitale
−8
−6
−4
−2
0
2
0.0 0.2 0.4 0.6 0.8 1.0TM-Anteil
ZT
Meff
(x)
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........................
•Sc
•Ti
•V
•Cr
•Mn
•Fe
•Co
•Ni
• xTM
c
•Cu
• Interpretation
• aktive Fermi-Kugel in Resonanz
mit Schalenstruktur
Kpe = 2 kaktivF
• alle Al-p-Elektronen d-typisch machen
ZTMeff (x) = 1 − (1−x)Nat
xNat= 2 − 1
x
↑ von 4s1
• kritische Konzentrationen xTMc
ZTMeff (xTM
c ) = 1 − (10 − nTMd ) 1)
1) s. auch G. Trambly de Laissardiere et al. cond-mat/0410513v2
1. Ubergangsmetalle in Aluminium: Neutrale Atomkugeln
Partielle Ladungen der Atomkugeln in 20 kristallinen und amorphen Legierungen
LMTO-ASA liefert: nTMd ≈ nfrei
d + 1frei 4s
⇒ Negative TM-Valenz bedeutet nicht das Fullen aller TM-d-Orbitale.
0
2
4
6
8
10
12
10 20 30
Kernladungszahl
•••••••••••••••••••••••••••••••••••••••••••
•••••••
••••••••••••••
•
•• ••
............................................................................................................................................................................................................................................................................................................. nfreis − 1
•••••••••••••••••••••••••••••••••••••••••••
•••
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Ele
ktr
onenzahl/
Ato
m
s p d total
1. Ubergangsmetalle in Aluminium: Neutrale Atomkugeln
Partielle Ladungen der Atomkugeln in kristallinen Legierungen (4d, 5d)
⇒ 4dTM und 5dTM zeigen Unterschiede im Vergleich zu den 3dTM.
0
2
4
6
8
10
12
40 50 60 70 80
Kernladungszahl
••• •
← nfreis
−
−
••• •
•••
•
← nfreid
−
−
•••
• ← nfreitot
−
−
Ele
ktr
onenzahl/
Ato
m
s p d total
1. Ubergangsmetalle in Aluminium: Hybridisierung 2
• Bedingung fur maximal konstruktive Interferenz
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FERMI
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......................................................
JONES
1.81 NFE-Bandzustande je Atom besetzt
• Mit welcher effektiven TM-Valenz bleiben genau1.8 NFE-Bandzustande je Atom besetzt ?
(1 − x)Nat3 + xNat(1 + nTMd ) ≡ ZatNat Bandzust. total
(1 − x)Nat3 + xNatZTMeff (x) ≡ 1.8 Nat
ZTMeff (x) = 3 − 1.2
x
−8
−6
−4
−2
0
0.0 0.2 0.4 0.6 0.8 1.0
x (TM-Anteil)
ZT
Meff
(x)
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2-1x
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..............................................
3-1.2x
≈ Experiment 2)
•Sc
•Ti
•V
•Cr
•Mn
•Fe
•Co
•Ni
•Cu
• xTM
c
i-AlCuFe Al3V AlRe
• Jedes 3d-TM hat sein xTMc
ZTMeff (xTM
c ) = 1 - (10 - nTMd )
• Wie viele non-NFE-Bandzust. entstehen bei xT Mc ?
Nnon−NFE = Nat (Zat − 1.8)
= xTMc Nat
“
1 + nTMd − ZTM
eff (xTMc )
”
= xTMc Nat 10
2) J. L. Barzola, Dissertation 2003 und M. Stiehler Diplomarbeit 2004 http://archiv.tu-chemnitz.de
1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung
• Al3V (6 Al, 2 V)Pearson’s Handbook, No. 139,
LMTO-ASA, special k-points, Kubo,
spez. Widerstand ρ(ε), Zustandsdichte n(ε),
Diffusivitat D(ε), z ist lange Seite
XCRYSDEN
1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung
Bandzustande unterhalb εF stark (V-d)-(Al-sp)-korreliert, oberhalb εF neuer Typ
Al3V, No. 139,
ASA-LMTO, spez. 5-set,
Anteile an Norm 1,
oben:
sp , Al , V
unten:
Al-sp , V-sp
1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung
• Cockayne-Modell (80 Al, 32 Cu, 16 Fe)LMTO-ASA, special k-points, Kubo,
spez. Widerstand ρ(ε), Zustandsdichte n(ε),
Diffusivitat D(ε)
< Psp >, 40-meV Mittelwert des sp-Anteils
XCRYSDEN
1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung
Bandzustande unterhalb εF nahezu fixierte (Fe-d)- und (Al-sp)-Anteile,
oberhalb εF lineare Al-sp-Zunahme, also anders zusammengesetzte Bandzustande
i-AlCuFeASA-LMTO,
(spez. 3-set),
Anteile an Norm 1,
oben:
sp , Al , Cu , Fe
unten:
Al-sp , Cu-sp , Fe-sp
1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung
• AlRe (2 Al, 2 Re)Pearson’s Handbook, No. 123,
LMTO-ASA, special k-points, Kubo,
spez. Widerstand ρ(ε), Zustandsdichte n(ε),
Diffusivitat D(ε), z ist lange Seite
XCRYSDEN
1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung
Uberall Re-d-dominiert, kein deutlicher Wandel der Zusammensetzung bei εF
AlRe, No. 123ASA-LMTO (okt 6-set)
Anteile an Norm 1,
oben:
sp , Al , Re
unten:
Al-sp , Re-sp
1. Ubergangsmetalle in Aluminium: Was wurde gezeigt ?
Fur isotrope amorphe Legierungen von Al mit 3d-TM bei den kritischen
TM-Konzentrationen xTMc wurde eine Verteilung des von den freien Atomen
eingebrachten sp-Gewichts auf zwei Gruppen von Bandzustanden vorgeschlagen.
Die Gruppe der NFE-Bandzustande ist mit 1.8 besetzten Bandzustanden je Atom so
bemessen, daß Ubergange an der Oberflache der zugeordneten Fermikugel durch
konstruktive Interferenz vieler schwacher Streuwellen aus dem mittleren
Ordnungsbereich maximal unterstutzt werden (Hume-Rothery).
Die zweite Gruppe besteht aus 10 non-NFE-Bandzustanden je TM-Atom. Sie
enthalt das gesamte resonante TM-d-Gewicht. Das totale sp-Gewicht dieser
non-NFE-Bandzustande bildet die negative TM-d-Valenz.
Stochiometrische Trends einer Reihe von Meßgroßen (LIT 1,2) befinden sich im
Einklang mit diesem Konzept.
Die LMTO-Bandzustande einiger kristalliner Legierungen abseits der xTMc zeigen
trotz ihrer von xTMc abweichenden Konzentrationen deutliche Korrelationen der
Al-sp- und TM-d-Gewichte.
2. Ausgangspunkt Spektraloperator: Grundgleichungen
Spektraloperator ρ(ε) = 2X
p
|φp〉δ(ε − εp)〈φp| = − 2
πImGout(ε)
DOS n(ε) = 2X
p
〈φp|ρ(ε)|φp〉, Ne =
εFZ
dε n(ε)
Bandenergie Eband =
εFZ
dε ε n(ε)
Kubo σ(ε) = 2e2π~
Ω
X
p
〈φp|ρ(ε)vρ(ε)v|φp〉
Einstein σ(ε) =e2
Ωn(ε) D(ε)
Diffusionskonstante D(ε) = π~
X
p
〈vφp|ρ(ε)|vφp〉
Mott σ(ε) =e2π~
2 Ωn(ε)2
X
pq
|〈φp|v|φq〉|2ε
Große Leitfahigkeiten auf der Energieschale,
falls die Wellenfunktionen in Transportrichung verschiebbar sind.
2. Ausgangspunkt Spektraloperator: bond order Θ12
|Λ〉 Zustandsraum des Systems, neue Basis im Unterraum (|Λ1〉, |Λ2〉):
|±〉 = (|Λ1〉 ± |Λ2〉)/√
2
n(ε) = 2
2
4X
Λ6=Λ1,Λ2
〈Λ|ρ(ε)|Λ〉 + ρ++(ε) + ρ−−(ε)
3
5
ρ++(ε) =1
2(ρ11(ε) + ρ22(ε)) + Re ρ12(ε)
ρ−−(ε) =1
2(ρ11(ε) + ρ22(ε)) − Re ρ12(ε)
1
2
`ρ++(ε) − ρ−−(ε)
´= Re ρ12(ε)
Θ12 =
εFZ
dε Re ρ12(ε)
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ρ11 + ρ22
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Re ρ12
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....................................................................................................................................................................................................................................................................
ρ++...............................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
ρ−−spektrale bond order Re ρ12(ε)
analysiertBeteiligung symm./antisymm.
Uberlagerungen
2. Ausgangspunkt Spektraloperator: Atomkugel-Basis
n(ε) = 2X
Λ
〈Λ|ρ(ε)|Λ〉
|Λ〉 vollstandige, orthonormierte Basis des gesamten Systems
atomare Polyeder Atomkugeln Teilsysteme
|Λ〉 = |sλµ〉 |Λ〉 = |slm〉 |Λ〉 = |Netz〉, |Umgebung〉
•
•
••••
•
•• •
•••
••
•
•
••
•
• •
• •
•
•
..............................................................................................................................................................................................................................................................................................................
s′L
′
..............................................................................................................................................................................................................................................................................................................
sL..............................................................................................................................................................................................................................................................................................................
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ρ(ε) =X
sL,s′L′
|sL〉q
n0sL(ε) n0
s′L′(ε)“
δss′δLL′ + ThsL,s′L′
”
〈s′L′|
2. Spektraloperator als Ausgangspunkt: Was wurde gezeigt ?
Bei Vorgabe des Spektraloperators (ρ(ε), hier fur T = 0) lassen sich alle
interessierenden elektronischen Eigenschaften des Systems berechnen.
Wir gehen davon aus, daß ρ(ε) durch eine Superzellen-Methode mit angemessener
k-Summation berechnet wurde und in Atomort-Drehimpuls-Darstellung vorliegt.
Partielle Zustandsdichten (Diagonalelemente) und bond-order
(Nichtdiagonalelemente) liefern eine atomgestutzte Analyse des Zustandsspektrums.
Der Vortrag geht einen Schritt weiter, indem das System in zwei Teile, Umgebung
und Netz, zerlegt wird. Die Umgebung ist vom NFE-Typ, wahrend im Netz
Kovalenz und TM-d-Hybride eine fuhrende Rolle spielen.
Es werden separate Eigendarstellungen des Spektraloperators in beiden Teilen
ermittelt und als neue Basis des Zustandsraumes benutzt. Es interessiert die
Zusammensetzung der Eigenvektoren und die Umgebung-Netz-Kooperation (U-N
bond-order).
Bei Verwendung des k-gemittelten Spektraloperators zur Berechnung der
Leitfahigkeit ist zu beachten, daß damit eine Brechung der Translationssymmetrie
einbezogen ist, was eher fur amorphe Phasen zutrifft.
3. Ubergangsmetall-Netzwerk: Fe-Netz in i-AlCuFe
• 150-meV-Pseudogap entsteht im Fe-Netzwerk
alle 2000 Fe aus 16000-Atom-cluster i-AlCuFe
MT-SW-DOS eines Bergman-Fe-Atoms
0
2
4
6
8
−2 0 2
(ε − εF )/eV
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
......................................................................................................................................
.......................................................................................................................
MT-SW
state
s/eV
/ato
m
0
2
4
6
8
−2 0 2
(ε − εF )/eV
........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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LMTO
state
s/eV
/ato
m
Welche Rolle spielen die Al-Atome ?
3. Ubergangsmetall-Netzwerk: Al-Brucken in d-AlNiCo
• Valenzdichten
(Modell nach
Mihalkovic)
ABINIT
Al-Brucken
TM-Al-TM
Kovalenz
bei Co
PGPLOT
Co ⇒
Co ⇒
3. Ubergangsmetall-Netzwerk: Eigendarstellung 1
Umgebung und Netz separat diagonalisieren, |U〉, |N〉 neue Systembasis
ρ(ε) =
|sL〉
=
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
|U〉 |N〉 ρ
〈U|
〈N|
l ↔
• Diagonalblocke: ρUU(ε), ρNN(ε) sind |U〉- bzw. |N〉-projizierte Zustandsdichten.
Interferenzcharakter (konstruktiv / destruktiv) in den Eigenvektoren ?
• Nichtdiagonalblocke: |U〉-|N〉 bond order
|±〉 =1
2(|U〉 ± |N〉)
Re ρUN(ε) =1
2
`ρ++(ε) − ρ−−(ε)
´> 0,
falls symmetrische Uberlagerungen |+〉 in den Eigenzustanden dominieren.
3. Ubergangsmetall-Netzwerk: Eigenvektoren und Interferenz
• AlRe: große Eigenwerte durch konstruktive Interferenz, Σ Al-Re bond-order
zeigt: antisymmetrische |U〉-|N〉-Uberlagerungen binden
0
2
4
6
−10 −5 0
(ε − εF )/eV
state
s/eV
/Zelle
DOS
AlRe
........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
0.0
0.5
1.0
−10 −5 0
(ε − εF )/eV
state
s/eV
32 Eigenwerte
in der Umgebung
ρ32 > ρ31 > . . .
ρ32 ρ29
ρ25...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
........................................................................................................................................................................................................................................................................................................................................................
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............................................................
0.0
0.5
1.0
−10 −5 0
(ε − εF )/eV
state
s/eV
32 Eigenwerte
im Netz
ρ32 > ρ31 > . . .
ρ32 ρ29 ρ25
..........................................................................................................................................................................
...............................................................................................................................................................................................................................................................
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−0.5
0.0
0.5
−10 −5 0
(ε − εF )/eV
state
s/eV
.................................................................................................................................
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Interferenzanteil
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ohne Interferenzen
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ρ32
−0.5
0.0
0.5
−10 −5 0
(ε − εF )/eV
state
s/eV
..............................................................................................................................................................................................................................................................................................................................................................................................................
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Interferenzanteil
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ohne Interferenzen
.........................................................................................................................................................................................................................................................................................................................................................................................
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.................................................................................................................................................................................................................................................
ρ25
−0.1
0.0
0.1
−10 −5 0
(ε − εF )/eV
state
s/eV
Σ Al-Re bond-order
....................................................
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3. Ubergangsmetall-Netzwerk: EVekt. und Ladungsdichten
Al3V: Elektronendichten zwischen den Atomkugeln (Flachen 0.04 e/A3).
tot. ValenzdichteABINIT
“Stabe“ entlang z
Transport bevorzugtentlang z imAluminium
partielle Dichte
|U〉-EVektormit maximal
konstruktiver Interferenz
⇐ ⇒mit ohne
Interferenz
partielle Dichte
|N〉-EVektormit maximal
konstruktiver Interferenz
⇐ ⇒mit ohne
Interferenz
XCRYSDEN
3. Ubergangsmetall-Netzwerk: Eigendarstellungen 2
Kooperation Umgebung - Netz in der Kubo-Leitfahigkeit
σ ∝ Tr(ρ v ρ v) = Tr([RρR†] v [RρR†] v) = Tr(ρ [R†vR] ρ [R†vR])
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
ρ ρR† R†R Rv v
ll ↔↔
| z | z
• Achtung: k-summierter Spektraloperator bricht k-Erhaltung ( ≈ amorph) !
• Beitrage der Eigenvektoren mit stark konstruktiver / destruktiver Interferenz ?
• Beitrage der Pfade ohne / mit Ubergangen zwischen Umgebung und Netz ?
3. Ubergangsmetall-Netzwerk: Eigenvektoren und Transport
• Al3V: Transportpfade an der Fermigrenze
−0.05
0.00
0.05
0.10
0.15
−10 −5 0
(ε - εF )/eV
σzz(ε
)/(µ
Ωcm
)−1
σzz
Al-AlV-VV-Al und Al-V
.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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−0.05
0.00
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0.15
−10 −5 0
(ε - εF )/eV
σyy(ε
)/(µ
Ωcm
)−1
σyy
Al-AlV-VV-Al und Al-V
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• trotz Brechung der k-Erhaltung (≈ amorph)
• Haupttransportpfade im Al (s. auch partielle Dichten)
• Wechsel zu V kann behindern (negativer Beitrag)
• Anisotropie bleibt im V-d-Band erhalten
3. Ubergangsmetall-Netzwerk: Eigenvektoren und Transport
• AlRe: Transportpfade an der Fermigrenze
−0.05
0.00
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0.10
0.15
−10 −5 0
(ε - εF )/eV
σzz(ε
)/(µ
Ωcm
)−1
σzz
Al-AlRe-ReRe-Al und Al-Re
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−0.05
0.00
0.05
0.10
0.15
−10 −5 0
(ε - εF )/eV
σyy(ε
)/(µ
Ωcm
)−1
σyy
Al-AlRe-ReRe-Al und Al-Re
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• trotz Brechung der k-Erhaltung (≈ amorph)
• Haupttransportpfade im Al (s. auch partielle Dichten)
• Wechsel zu Re kann behindern (negativer Beitrag)
• Anisotropie bleibt im Re-d-Band erhalten
4. Was wurde gezeigt ?
• Amorphe Legierungen von Al mit 3d-TM haben bei den kritischen
Konzentrationen xTMc
• die optimale Fermikugel mit 1.8 besetzten NFE-Bandzustanden je Atom
• dazu 10 besetzte non-NFE-Bandzustande je TM-Atom
• eine negative TM-Valenz im Umfang des sp-Anteils dieser
non-NFE-Bandzustande.
• Die Untersuchung von Wechselbeziehungen zwischen Teilen eines Systems
(strukturell, chemisch) ist eine aktuelle Fragestellung. Eigendarstellungen der
ensprechenden Teile des Spektraloperators liefern geeignete Basissysteme des
Zustandsraumes des Gesamtsystems.
• Die Eigenvektoren selbst enthalten alle Interferenzen im Teilsystem.
• Die mit Eigenvektoren zweier Teile berechnete bond-order beschreibt die
Kooperation der Teile.
• Das Verfahren gestattet es, Transportpfade in einer Teilmenge von
Eigenvektoren zu isolieren.