the hume-rothery rules for structurally complex alloy...
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Uichiro Mizutani
Toyota Physical & Chemical Research Institute, Japan
The Hume-Rothery rules for
Structurally Complex Alloy Phases
The 3rd Euroschool 2008, Ljubljana, SloveniaComplex Metallic Alloys: Surfaces and Coatings
May 29, 2008
OUTLINE
1. Prior fundamental knowledge for the discussion of phase competition inCMAs
2. What is the Hume-Rothery electron concentration rule?
3. Two different definitions of electron concentration: VEC versus e/a
4. Historical survey
5. Why do we need the first-principles FLAPW band calculations?
6. Why have we chosen a series of gamma-brasses?
7. Hume-Rothery stabilization mechanism
8. Different stabilization mechanisms in Al8V5 and Ag5Li8 gamma-brasses
9. Stability mechanism in the Al-TM-Cu-Si (TM=Fe and Ru) approximants
10. 2kF versus Kp condition. Issues on VEC versus e/a. Which is more criticalparameter to discuss the phase stability of CMA phases?
The curve !o(r ) represents the lowest energy of electrons with the wave vector k=0 while the curve!kin represents an average kinetic energy per electron. !I represents the ionization energy neededto remove the outermost 3s electron in a free Na metal to infinity and !c is the cohesive energy.The position of the minimum in the cohesive energy gives an equilibrium interatomic distance ro.
Concept of a cohesive energy in a solid
!c: cohesive energy
:ionization energy
RnAtPo
144
Bi
208
Pb
197
Tl
181
Hg
67
Au
365
Pt
565
Ir
670
Os
783
Re
782
W
837
Ta
781
Hf
611
La
434
Ba
179
Cs
80
Xe
15
I
107
Te
193
Sb
259
Sn
301
In
247
Cd
112
Ag
286
Pd
380
Rh
555
Ru
639
TcMo
658
Nb
720
Zr
610
Y
423
Sr
164
Rb
83
Kr
11
Br
118
Se
206
As
289
Ge
374
Ga
269
Zn
130
Cu
338
Ni
428
Co
424
Fe
414
Mn
288
Cr
396
V
511
Ti
469
Sc
379
Ca
176
K
91
Ar
7.7
Cl
135
S
277
P
332
Si
448
Al
322
Mg
148
Na
108
Ne
2.1
F
84
O
251
N
477
C
711
B
561
Be
322
Li
159
AcRaFr
Cohesive energies (kJ/mole) of elements in periodic table
Cohesive energy represent an energy needed to separate all atoms in a solid at absolute zero into theassembly of neural atoms. The values underlined refer to those either at 298.15 K or at the melting point.The values are in the units of kJ/mole converted from the values in the units of cal/mole. C.Kittel,“Introduction to Solid State Physics”, (Third Edition, John Wiley & Sons, New York, 1967), Chapter 3.
Ce 460
!
Pr 373
!
Nd 323
!
Pm
!
Sm 203
!
Eu 174
!
Gd 399
!
Tb 393
!
Dy 297
!
Ho 293
!
Er 322
!
Tm 247
!
Yb 151
!
Lu 427
!
Th 572
!
Pa 527
!
U 522
!
Np 440
!
Pu 385
!
Am 251
!
Cm
!
Bk
!
Cf
!
Es
!
Fm
!
Md
!
!
Lw
!
!
No Lr
The number of intermediate phases increases from unity up to five as the heat of mixing increases in a negativedirection from zero, -37 and -75 kJ/mole in!"#$!%&'()*!+*+,-./!0F.R.de Boer, R.Boom, W.C.M.Mattens, A.R.Miedemaand A.K.Niessen, “Cohesion in Metals”, (North-Holland 1988)]
The number of intermediate phases increases, as "H at xB=0.5 increases.
Neighboring phases are competing within +10kJ/mole.
"H=0 kJ/mole
"H=-37 kJ/mole
"H=-75 kJ/mole
A B
He
at of m
ixin
g, "
H/ kJ
.mol-1
Construction of the Fermi sphere. The reciprocal space is quantized in units of 2#/L in the kx-,ky- and kz-directions and is made up of cubes with edge length 2#/L as indicated in the figure.Electrons of up and down spins occupy the corner of each cube or integer set (nx, ny, nz) inaccordance with the Pauli principle while making nx
2+ny2+nz
2 as low as possible. The spherewith radius kF represents the Fermi sphere.
2 ! /L
kx
ky
kz
kF
2 !
L2 !
L
2 !
L
Electron theory of metals is an essential ingredient to deepen
understanding of an alloy phase competition
Fermi sphere
The free electron model
!h2
2m
"
# $ %
& '2( = E(
E =h2
2m
!
" # $
% (kx2
+ ky2
+ kz2) =
h2k2
2m
!
" = eir k #
r r
kx=2!n
x
L(n
x= 0, ±1, ±2,±3 " " " ")
plane wave
!
kF = 3" 2e /a
Va
#
$ %
&
' (
)
* +
,
- .
1/ 3
e/a: electron concentration defined as an average valency of constituent
atoms in an alloy.
Departure from the free electron model
Nature tries to lower the electronic energy of a systemby expelling electrons at the Fermi level. This is nothingbut the formation of a true or pseudogap at the Fermilevel, as indicated above.
Three representations for the
description of the electronic structure
E-k relations (dispersion relation)
Fermi surface
Density of states (DOS)
kx
ky
kz
0
kF
E
kx0
EF
kF
EEF!"#$%
0
N(E
)
!
DOS(E) = " E
-50
-40
-30
-20
-10
0
Gain
in e
lect
ronic
ene
rgy,
"U
/kJ
.mole
-1
5004003002001000
Width of energy gap, "E/ meV
psuedogap immediately below EF
pseudogap near the bottom of the valence band
A gain in the electronic energy amounts to several 10 kJ/mol, which is large
enough to stabilize a given phase among competing phases
Examples to stabilize a system by formation of (pseudo)gap at EF
1. A structurally complex phase by increasing the number ofatoms in a unit cell
2. A superconductor by forming the superconducting gap
3. A quasi-one-dimensional organic molecular metal byintroducing new modulations through deformation of thelattice. This is known as the Peierls transition.
Kinetic energy increases in this directionand becomes the maximum at the Fermilevel.
EEF!"#$%
0 7
0.2N(EF)free
50~500 meV
DO
S
(eV)energy
Covalent bondings12rbital hybridization between neighboring atoms
Metallic bonding (FsBz interaction)13ong-range interaction throughout a crystal
!
b
<550>
2d~4 Å
$F~4 Å
Electrons having the wave length $F resonate with aset of lattice planes with lattice spacing 2d.
What about the mechanism for the formation of the pseudogap?
!
"F =2#
kF
$1
e /a( )1/ 3
Orbital hybridization effect would occurnot only in crystals but also in non-crystalline systems like liquid metalsand amorphous alloys.
Orbital hybridizations
The FsBz interaction is unique in well-orderedsystems like crystals and quasicrystals
Anti-Bonding energy level
Bonding energy level HOMO
LUMO
EFAl-3p Mn-3d
Fermi surface-Brillouin zone
(FsBz) interaction
The concept of the Brillouin zoneStationary waves are formed and an energy gap opens, when the electron wave length
matches the lattice periodicity.
a
a
x
x
!
sin "x /a( )[ ]2
!
cos "x /a( )[ ]2
The Brillouin zone is a polyhedron bounded byplanes, which are formed by perpendicularlybisecting the relevant reciprocal lattice vectorslike G=<110> in the reciprocal space. An energygap opens across each zone face.
{002}
{111}
II. fcc, N=4, truncated octahedron with 14 zone faces
G2=3 or 4
{110} N
HP!
I. bcc, N=2, dodecahedron with 12 zone faces
G2=2
G2=2
The Brillouin zone of Three Complex Alloy Phases
III. Quasicrystal, N is infinite, polyhedron with 60 planes
Inte
nsi
ty (
arb.u
nit
s)
807060504030202! (deg.)
111000
1/2
(311111)
111100
211111 221001
322101
322111 332002
The larger the number of atoms in the unit cell, the more the number of zone faces, across
which an energy gap opens. This in turn results in a deeper pseudogap on the DOS. As a
result, the electronic energy can be more efficiently lowered by increasing the number of
atoms in the unit cell.
I. gamma-brass, N=52, polyhedron with 36 planes
{411}
{330}
G2=18
(710)
(710)
(710)
(710)
(550)
(550)
(550)
(550)
(543)(543)
(543)
(543)
(543)
(543)
(543)(543)
(543)
(543)
(543)(543)
(543) (543)
(543)
(710)
(543)
(543)
II. 1/1-1/1-1/1 approximant, N=160, polyhedron with 84 planes
G2=50
Hume-Rothery (1926) pointed out a tendency for a definite crystal structure to occur at aparticular electron concentration e/a. Mott and Jones (1936) made its first interpretation interms of the Fermi surface-Brillouin zone interaction on the basis of the nearly free electronmodel.
What is the Hume-Rothery electron concentration rule!
{411}
{330}
H.Jones (1905-1986)N.F.Mott (1905-1996)
William Hume-Rothery (1899-1968)
There exist two different definitions for electron concentration:
e/a versus VEC
Slater-Pauling curve"Saturation magnetization against VEC
Matthias ruleSuperconducting transition temperature against VEC
5.9 5.95 6 6.05 6.1
-100
-50
0
50
100
Fe2(V1-xTix)AlFe2(V1-xZrx)Al
(Fe1-xCox)2VAl(Fe1-xPtx)2VAl
Fe2V(Al1-xSix)Fe2V(Al1-xGex)
-150
300 K
Fe2VAl
Fe2(V1-xMox)AlFe2(V1-xWx)Al
!"#$!%
&"#$!%
electron concentration VEC
Se
eb
eck
co
eff
icie
nt
(µV
/K)
Y.Nishino and U.Mizutani (2005)
p- and n-type thermoelectric alloys
Fe2VAl doped with various elements
!
VEC =8"2+5"1+ 3"1
4= 6.0
Various physical properties fall on a universal curve, when plotted
against the VEC (number of electrons per atom in the valence band)
Electron concentration in the Hume-Rothery electron concentration rule is not
VEC but should be e/a (averaged number of valencies of constituent atoms).
!
e /a(Cu5Zn
8) =1"5+2"8
13=21
13
e /a(Cu9Al
4) =1"9+ 3" 4
13=21
13
!
VEC(Cu5Zn
8) =11"5+12"8
13=151
13
VEC(Cu9Al
4) =11"9+ 3" 4
13=111
13
e/a
VECZnII
CdII
AlIII
GaIII
SnIV
1CuI
AgI
AuI3/2
7/41.4
21/13
(b)
N
H
P%
H!
(110)
(a)
L
X%
W K
!
(111)
!
(002)
Model of Jones for the interpretation of the Hume-Rotheryrule on "/# phase transformation in Cu-Zn system
Brillouin zone of fcc
Brillouin zone of bcc
(c)
Ufc
c-U
bc
c (1
02
eV
ato
m-1
)
e/a
H.Jones, Proc.Phys.Soc. A49 (1937) 250
Difficulties in model of Jones
Discovery of the neck in the Fermi surface of fcc-Cu by Pippard in 1957
Ignorance of the Cu-3d band
Hume-Rothery commented in 1961 that “the work of the last ten years has made the theory of alloystructures appear less satisfactory than was the case twenty-five years ago. It is now definitelyestablished that the assumption of a spherical Fermi surface for pure Cu is quite unjustified…”.
{111}
{110}
!
"U =U fcc #Ubcc = {N(E) fcc # N(E)bcc}EdE0
EF$
Current understanding of fcc/bcc phase transformation of Cu on the
basis of the first-principles FLAPW electronic structure calculations in
combination with density functional theory
!
U = "i
i
# $1
2
n(r)n( % r )
r $ % r drd % r && + n(r) "
XCn(r)( )$µ
XCn(r)( )[ ]& dr
Total energy of a system at absolute zeroW.Kohn and L.J.Sham, Phys.Rev. 140 (1965) A1133
One-electron band structureenergy due to both valence and
core electrons
exchange-correlation energyaverage interaction energy
of electrons known as the
Hartree term
!
"h2
2m#2 +veff (r)
$
% &
'
( ) *i(r) = +i*i(r)
!
"i
is the solution of the effective one-electron Schrödinger equation of a system given by
!
Uvalence
= D(E)(E " E0)dE
Ebottom
EF
# This is a quantity evaluated by Jones and
corresponds to the contribution from the
valence electrons in the first term.
electron-electron interaction term
FLAPW valence band structure and van-Hove singularities in fcc-Cu
due to contact with {111} zone planes
due to contact with {200} zone planes
due to overlap across{111} zone planes
Fcc-Cu
Evidence for the neck
against {111} zone planes
Bcc-Cu
The first-principles band calculations prove that the
ignorance of the Cu-3d band was a vital failure in
the model of Jones. This led him to a wrong location
of van-Hove singularities and overestimation of
their sizes in both fcc and bcc Cu.
!
"U = Nbcc (E)(E #Eo#bcc )dE # N fcc (E)(E #Eo# fcc )dEEo
EF
$Eo
EF
$
You can see that the van-Hove singularities are negligibly small in both fcc- and bcc-Cu.The band structure energy difference arises essentially from the difference in Cu-3dbands and amounts to about 10 kJ/mole in favor of the bcc structure.
+41.19-1857872.647-1857913.837Uelectron-electron (kJ/mole)
-7.37-2483807.338-2483799.964core electron energy Ucore
(kJ/mole)
-31.455226.47695257.9289valence-band structureenergy Ubs (kJ/mole)
+2.37-4336453.49-4336455.852total energy U (kJ/mole)
2.86393.6048lattice constant (A)
"U=Ubcc-Ufcc
(kJ/mole)bcc-Cufcc-Cu
!
U = "i
i
# $1
2
n(r)n( % r )
r $ % r drd % r && + n(r) "
XCn(r)( )$µ
XCn(r)( )[ ]& dr
' ( )
* + ,
=Ubs +Ucore
+Uelectron$electron
"U =(-31.45)+(-7.37) +41.19=+2.37 kJ/mole
The stability of fcc Cu is correctly predicted, only if the electron-electron term is
properly evaluated.
!
Ubcc
>Ufcc
!
Ubcc
<Ufcc
!
Ubcc
>Ufcc
We believe the situation in complex alloy phases
to be different from that in fcc and bcc Cu,
where van-Hove singularities are too small to
affect the total energy.
As is clear from the argument above, the interpretation of &/'phase transformation in the Cu-Zn system is still far from our goal.
---First-principles LMTO band calculations---The origin of the pseudogap was attributed to the Mn-3d/Al-3p orbital hybridizations
A pseudogap across the Fermi level plays a key role in
stabilizing a complex alloy phase
T.Fujiwara, Phys.Rev.B 40 (1989) 942
Al-Mn approximant
Mn-3d/Al-3p bonding states
Mn-3d/Al-3p anti-bonding states
EF
!
b
<550>
2d~4 Å
$F~4 Å
Resonance of electrons having the wave length $F
with a set of lattice planes with spacing 2d.
We are interested in e/a-dependent phase stability
!
"F =2#
kF
$1
e /a( )1/ 3
Orbital hybridization effect would occurnot only in crystals but also in non-crystalline systems like liquid metalsand amorphous alloys.
Orbital hybridizations
FsBz interaction is unique in well-orderedsystems like crystals and quasicrystals
Anti-Bonding energy level
Bonding energy level HOMO
LUMO
EFAl-3p Mn-3d
FsBz interaction
An increase in the number of atoms in unit cell accompanies an increase in
Brillouin zone planes
V. Quasicrystal, N is infinite, polyhedron with 60 planes
The gamma-brass is complex enough to producea sizeable pseudogap at the Fermi level but isstill simple enough to perform the FLAPW bandcalculations with an efficient speed.
{110} N
HP!
I. bcc, N=2, dodecahedron with 12 planes
G2=2
{002}
{111}
II. fcc, N=4, truncated octahedron with 14 planes
G2=3 or 4
III. gamma-brass, N=52, polyhedron with 36 planes
{411}
{330}
G2=18
(710)
(710)
(710)
(710)
(550)
(550)
(550)
(550)
(543)(543)
(543)
(543)
(543)
(543)
(543)(543)
(543)
(543)
(543)(543)
(543) (543)
(543)
(710)
(543)
(543)
IV. 1/1-1/1-1/1 approximant, N=160, polyhedron with 84 planes
G2=50
Inner Tetrahedron (IT)
Outer Tetrahedron (OT)
Octahedron (OH)
Cubo Octahedron (CO)
(b)
(a)
26-atom cluster
The 26-atom cluster forms either bcc or CsCl-structure and contains 52 atoms in the unit cell.
Gamma-brasses with the space group P43m and I43m
ZnII
CdII
AlIII
GaIII
InIII
SnIV
1CuI
AgI
AuI21/13
Our objective is to explore if all gamma-brasses listed here
are stabilized via the same mechanism at e/a=21/13
In literature, all isostructural gamma-brassesabove had been implicitly assumed to bestabilized at e/a=21/13 or the valency of thepartner element to be forced to take a valueto fulfill the total e/a equal to 21/13.
Empirical Hume-Rothery e/a rule forgamma-brasses in the past
1.6
1.6
1.6
1.6
1.46
DOS of Cu9Al4 gamma-brass
The FsBz interaction is really responsible for the formation of the pseudogap?
3.0
2.5
2.0
1.5
1.0
0.5
0.0
DO
S (
sta
tes/e
V.a
tom
)
-10 -8 -6 -4 -2 0 2 4
Energy (eV)
Cu9Al4 gamma-brass
EF
pseudogap
Cu-3d
!
"k (E, r) = C(k+Ghkl )#k+Ghkl(E
Ghkl
$ ,r)
!
"k+Ghkl(E,r) : FLAPW basis function
Principle for the extraction of the FsBz interaction from the first-principles
FLAPW (Full-potential Linearized Augmented Plane Wave method) band calculations
!
r " a #k(r) = Almk
ul(El, r) + Blmk ˙ u l(El, r)[ ]Ylm($, %)
lm
&
!
r " a #k(r) = exp(ik $r)
Adoption of spherically symmetric muffin-tin potential
!
"k (E, r) = C(k +Ghkl )exp i(k +Ghkl ) #r[ ]Ghkl
$
EF
!
1
21,1, 0( )
(1)
FLAPW-Fourier Analysis
N
%
{110}
H
k
We extract the electronic state k+Ghkl having the largest Fourier coefficient in wavefunction (1) at symmetry point N with the energy eigen-value near the Fermi level. Thisis nothing but the extraction of the set of lattice planes resonating with electron waves.
Cu5Zn8 and Cu9Al4 gamma-brasses in group 1
?Ir2Zn1121/13Cu9Ga4
21/13?
Ag5Li831.60?Co2Zn1121/13Cu9Al4
?Mn3In1.60?Fe2Zn1121/13Au5Zn8
1.46?Al8V51.60?Pd2Zn1121/13Au5Cd8
?Pt2Zn111.60?Ni2Zn11
2
21/13Cu5Cd8
?Mn2Zn1121/13Au9In421/13Ag5Zn8
?Ni2Cd1121/13Ag9In421/13Ag5Cd8
?Ni2Be11
2
21/13Cu9In4
1
21/13Cu5Zn8
1
e/agamma-
brasse/a
gamma-brass
e/agamma-brass
E-k dispersions and DOS derived from FLAPW for Cu5Zn8 gamma-brass
5 4 3 2 1 0-1
0-8
-6-4
-20
24
Energ
y (
eV
)
Cu5Zn8 gamma-brass
DOS (states/eV.atom)
EF
Cu-3d
Zn-3d
pseudogap
R.Asahi, H.Sato, T.Takeuchi and U.Mizutani, Phys.Rev. B 71 (2005) 165103
Energ
y (e
V)
E4E3
E2E1
EFEF
N
%
EF
Energ
y (e
V)
E6E5
E4E3
EF
E8E7
E2E1
E-k dispersions and DOS derived from FLAPW for Cu9Al4 gamma-brass
5 4 3 2 1 0-1
0-8
-6-4
-20
24
Energ
y (
eV
)
Cu9Al4 gamma-brass
DOS (states/eV.atom)
EF
Cu-3
dpseudogap
0.001
2
4
0.01
2
4
0.1
2
4
1
!|Ci G
hkl/2|2
16012080400
|G0|2=h
2+k
2+l
2
Cu9Al4 gamma-brass
E=-0.41 eV
2
3
4
6
7
8
9
0.001
2
4
0.01
2
4
0.1
2
4
1
!| Ci G
hkl/2|2
16012080400
|G0|2=h
2+k
2+l
2
E=-0.62 eV
2
3
4
6
8
Cu5Zn8 gamma-brass
1: {211}, 2: {310}, 3: {321}, 4: {411}+{330}, 5: {332}, 6: {510}+{431}, 7: {521}, 8: {530}+{433}, 9: {611}+{532}
!
IG2
=18/ Inext =123
!
IG2
=18/ I
next= 45
!
"k(E, r) = C(k
N+G
hkl)exp i(k
N+G
hkl) #r[ ]
Ghkl
$
18 18
N
The Fourier coefficient is extremely large at G2=18. This implies that electron waves near
the Fermi level exclusively resonate with the set of {411} and {330} lattice planes with
G2=18, resulting in the formation of the pseudogap.
FLAPW-Fourier spectra for the wave function outside the MT sphere at
the point N at energies immediately below the Fermi level
Evaluation of the e/a value by means of the FLAPW-Fourier method
---Hume-Rothery plot---
This technique is now extended to a whole valence band. We need to do this only in the irreduciblewedge of the Brillouin zone. The wedge is divided into 200 elements and the electronic state (ki+G)having the largest Fourier coefficient for the wave function at energy E in the i-th element isextracted. This is done for all elements over i=1 to 200 and an average value of (ki+G) av iscalculated. In this way, a new single-branch dispersion relation E-(ki+G) av for electrons extendingoutside the MT sphere is derived in the extended zone scheme.
We have so far limited ourselves to the extraction of the largest plane wave component of the FLAPW wavefunction only near the Fermi level and only at the symmetry point N.
!
k+GE" #
iki+G
Ei=1
N=200
$
Once the E-(k+G) is determined, the Fermi diameter 2kF is obtained from the value of 2(k+G) at EF.
The e/a is immediately calculated by inserting theFermi diameter 2kF into the relation below:
!
(e/a)total ="
3N2kF( )
3
The variance ( must be small to validate this approach.
N=52:number of atoms in the unit cellirreducible wedge
E-{2(k+G)}2 relations for itinerant electrons for Cu5Zn8 and Cu9Al4 gamma-brasses
---Hume-Rothery plot---
(2kF)2=18.47 for Cu5Zn8
(2kF)2=18.45 for Cu9Al4
!
(e/a)total ="
3N2kF( )
3
N=52
30
25
20
15
10
5
0
!2 (
E)
-10 -8 -6 -4 -2 0 2 4Energy (eV)
EF
Zn-3d
Cu-3d
Zn-3d Cu-3de/a=1.60 for bothgamma-brasses
?Ir2Zn1121/13Cu9Ga4
21/13?
Ag5Li831.60?Co2Zn1121/13Cu9Al4
?Mn3In1.60?Fe2Zn1121/13Au5Zn8
1.46?Al8V51.60?Pd2Zn1121/13Au5Cd8
?Pt2Zn111.60?Ni2Zn11
2
21/13Cu5Cd8
?Mn2Zn1121/13Au9In421/13Ag5Zn8
?Ni2Cd1121/13Ag9In421/13Ag5Cd8
?Ni2Be11
2
21/13Cu9In4
1
21/13Cu5Zn8
1
e/agamma-
brasse/a
gamma-brass
e/agamma-brass
Ekman (1931) studied the TM-Zn gamma-brasses and proposed that they obey
the Hume-Rothery electron concentration rule with e/a=1.60, provided that the
valency of the TM element is zero.
W.Ekman, Z.Physik.Chem. B 12 (1931) 57
Studies of TM2Zn11 (TM=Ni, Pd, Fe, Co) gamma-brasses in group 2
FLAPW-derived E-k relations and DOS for Ni2Zn11 gamma-brass
E1
E2pseudogap
Ni-3d
Zn-3d
5 4 3 2 1 0-1
0-8
-6-4
-20
24
En
erg
y (
eV
)
Ni2Zn11 gamma-brass
EF
DOS (states/eV.atom)
EF
0.001
0.01
0.1
1
!|Chkl|2
16012080400
h2+k
2+l
2
E= -1.09 eV|G|2=18
5
4
3
2
1
0
DO
S (
sta
tes/e
V.a
tom
)
-10 -8 -6 -4 -2 0 2 4
Energy (eV)
Ni2Zn11 gamma-brass
EF
FLAPW-Fourier spectra just below and above the
pseudogap in Ni2Zn11 gamma-brass
G2=18 resonance is active inNi2Zn11 gamma-brass.
0.001
0.01
0.1
1
!| Chkl|2
16012080400
h2+k
2+l
2
E= +0.05 eV|G|2=18
FLAPW-derived E-k relations and DOS for Pd2Zn11 gamma-brass
pseudogap
Pd-4d
Zn-3d
5 4 3 2 1 0-1
0-8
-6-4
-20
24
En
erg
y (
eV
)
Pd2Zn11 gamma-brass
EF
DOS (states/eV.atom)
EF
FLAPW-Fourier spectra just below and above the
pseudogap in Pd2Zn11 gamma-brass
5
4
3
2
1
0
DO
S (
sta
tes/e
V.a
tom
)
-10 -8 -6 -4 -2 0 2 4Energy (eV)
Pd2Zn11 gamma-brass
EF
G2=18 resonance is active inPd2Zn11 gamma-brass.
0.001
0.01
0.1
1
!|Chkl|2
16012080400
h2+k
2+l
2
E=-0.11 eV|G|2=18
0.001
0.01
0.1
1
!|Chkl|2
16012080400|G|
2=h2+k2+l2
E=-1.14 eV|G|2=18
FLAPW-derived E-k relations and DOS for Co2Zn11 gamma-brass
Zn-3d
pseudogap
Co-3d
5 4 3 2 1 0-8
-40
4
En
erg
y (
eV
)
EF
Co2Zn11 gamma-brass
DOS (states/eV.atom)
EF
5
4
3
2
1
0
DO
S (
sta
tes/e
V.a
tom
)
-8 -4 0 4Energy (eV)
EF
Co2Zn11 gamma-brass
FLAPW-Fourier spectra just below and above the
pseudogap in Co2Zn11 gamma-brass
The G2=18 resonance still
survives only above the Fermi
level in Co2Zn11.
0.001
0.01
0.1
1
!|Chkl|2
12080400
h2+k
2+l
2
E= +0.91 eV|G|2=18
0.001
0.01
0.1
1
!|Chkl|2
12080400
|G|2=h
2+k
2+l
2
E= -0.20 eV|G|
2=18
FLAPW-derived E-k relations and DOS for Fe2Zn11 gamma-brass
5 4 3 2 1 0-1
0-8
-6-4
-20
24
Energ
y (
eV
)
Fe2Zn11 gamma-brass
EF
DOS (states/eV.atom)
pseudogap
Fe-3d
Zn-3d
EF
5
4
3
2
1
0
DO
S (
sta
tes/e
V.a
tom
)
-10 -8 -6 -4 -2 0 2 4Energy (eV)
Fe2Zn11 gamma-brass
EF
FLAPW-Fourier spectra just below and above the
pseudogap in Fe2Zn11 gamma-brass
0.001
0.01
0.1
1
!| Chkl|2
16012080400
h2+k
2+l
2
E=+0.025 eV|G|2=18
0.001
0.01
0.1
1
!| Chkl|2
16012080400
h2+k
2+l
2
E=+1.04 eV|G|
2=18
The G2=18 resonance still
survives only above the Fermi
level in Fe2Zn11.
(2kF)2=19.36 for Ni2Zn11
=19.27 for Pd2Zn11
(2kF)2=19.5 for Co2Zn11
=20.0 for Fe2Zn11
!
(e/a) total ="
3N2kF( )
3
N=52
e/aNi-Zn=1.72e/aPd-Zn=1.70
!
(e/a) total ="
3N2kF( )
3
N=52
e/aCo-Zn=1.73e/aFe-Zn=1.80
E-(k+G)2 relations for itinerant electrons for TM2Zn11 (TM=Ni, Pd, Co, Fe) gamma-brasses
---Hume-Rothery plot---
11.38511.53811.711.78.511.6VEC
0.700.260.070.150.970.96(e/a)TM
1.801.731.701.721.601.60(e/a)total
20.019.519.2719.3618.4518.47(2kF)2 deduced
form the H-R plot
!!18181818|G|2 deduced
from the FLAPW-Fourier method
XXA pseudogapformed below EF
Fe2Zn11Co2Zn11Pd2Zn11Ni2Zn11Cu9Al4Cu5Zn8
Gamma-brasses in group 1 obey the Hume-Rothery e/a law with e/a=21/13
The Hume-Rothery stabilization mechanism refers to the mechanism, in which apseudogap is formed across the Fermi level as a result of electron waves resonatingwith a particular set of lattice planes and thereby the particular e/a value is specified.
group (1) group (2)
Summary for gamma-brasses in groups 1 and 2
2?Ir2Zn1121/13Cu9Ga4
21/13?
Ag5Li831.73Co2Zn111.60Cu9Al4
?Mn3In1.80Fe2Zn1121/13Au5Zn8
1.46?Al8V51.70Pd2Zn1121/13Au5Cd8
?Pt2Zn111.72Ni2Zn1121/13Cu5Cd8
?Mn2Zn1121/13Au9In421/13Ag5Zn8
?Ni2Cd1121/13Ag9In421/13Ag5Cd8
?Ni2Be11
2
21/13Cu9In4
1
1.60Cu5Zn8
1
e/agamma-
brasse/a
gamma-brass
e/agamma-brass
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