temperature-dependence of magnetism of free fe...
TRANSCRIPT
Temperature-dependenceof magnetism of free Fe clusters
O. Sipr1, S. Bornemann2, J. Minar2, S. Polesya2, H. Ebert2
1 Institute of Physics, Academy of Sciences CR, Prague, Czech Republic2 Universitat Munchen, Department Chemie, Munchen, Germany
– p.1/19
Outline
Introducing finite-temperature magnetism (a bit of theory)
Exchange coupling in clusters
Dependence of mean magnetization of clusters on temperatureand on cluster size
Critical (“Curie”) temperature Tc of clusters
Temperature-dependence of magnetic profiles of clusters
– p.2/19
Outline
Introducing finite-temperature magnetism (a bit of theory)
Exchange coupling in clusters
Dependence of mean magnetization of clusters on temperatureand on cluster size
Critical (“Curie”) temperature Tc of clusters
Temperature-dependence of magnetic profiles of clusters
– p.2/19
Outline
Introducing finite-temperature magnetism (a bit of theory)
Exchange coupling in clusters
Dependence of mean magnetization of clusters on temperatureand on cluster size
Critical (“Curie”) temperature Tc of clusters
Temperature-dependence of magnetic profiles of clusters
– p.2/19
Outline
Introducing finite-temperature magnetism (a bit of theory)
Exchange coupling in clusters
Dependence of mean magnetization of clusters on temperatureand on cluster size
Critical (“Curie”) temperature Tc of clusters
Temperature-dependence of magnetic profiles of clusters
– p.2/19
Outline
Introducing finite-temperature magnetism (a bit of theory)
Exchange coupling in clusters
Dependence of mean magnetization of clusters on temperatureand on cluster size
Critical (“Curie”) temperature Tc of clusters
Temperature-dependence of magnetic profiles of clusters
– p.2/19
Motivation
Magnetism is about ordering of magnetic moments
Rule of thumb for studying different geometries:
follow the coordination number !
Lowering the coordination number→ increase of magnetic moments → enhanced magnetization→ decrease of coupling → reduced magnetization
When comparing surfaces with bulk:
competition between enhancement of magnetic momentsand reduction of their coupling
– p.3/19
Motivation
Magnetism is about ordering of magnetic moments
Rule of thumb for studying different geometries:
follow the coordination number !
Lowering the coordination number→ increase of magnetic moments → enhanced magnetization→ decrease of coupling → reduced magnetization
When comparing surfaces with bulk:
competition between enhancement of magnetic momentsand reduction of their coupling
– p.3/19
Motivation
Magnetism is about ordering of magnetic moments
Rule of thumb for studying different geometries:
follow the coordination number !
Lowering the coordination number→ increase of magnetic moments → enhanced magnetization→ decrease of coupling → reduced magnetization
When comparing surfaces with bulk:
competition between enhancement of magnetic momentsand reduction of their coupling
– p.3/19
Motivation
Magnetism is about ordering of magnetic moments
Rule of thumb for studying different geometries:
follow the coordination number !
Lowering the coordination number→ increase of magnetic moments → enhanced magnetization→ decrease of coupling → reduced magnetization
When comparing surfaces with bulk:
competition between enhancement of magnetic momentsand reduction of their coupling
– p.3/19
Bulk, surfaces, clusters, . . .
Clusters contain a large portion of surface atoms
However, their properties cannot be expressed as a mere linearcombination of surface and bulk contributions
Finite cluster dimensions and associated quantum size effects aresignificant factors
⇒ intuition has a limited role
⇒ one has to perform calculations for the real stuff
– p.4/19
Bulk, surfaces, clusters, . . .
Clusters contain a large portion of surface atoms
However, their properties cannot be expressed as a mere linearcombination of surface and bulk contributions
Finite cluster dimensions and associated quantum size effects aresignificant factors
⇒ intuition has a limited role
⇒ one has to perform calculations for the real stuff
– p.4/19
Bulk, surfaces, clusters, . . .
Clusters contain a large portion of surface atoms
However, their properties cannot be expressed as a mere linearcombination of surface and bulk contributions
Finite cluster dimensions and associated quantum size effects aresignificant factors
⇒ intuition has a limited role
⇒ one has to perform calculations for the real stuff
– p.4/19
Bulk, surfaces, clusters, . . .
Clusters contain a large portion of surface atoms
However, their properties cannot be expressed as a mere linearcombination of surface and bulk contributions
Finite cluster dimensions and associated quantum size effects aresignificant factors
⇒ intuition has a limited role
⇒ one has to perform calculations for the real stuff
– p.4/19
Bulk, surfaces, clusters, . . .
Clusters contain a large portion of surface atoms
However, their properties cannot be expressed as a mere linearcombination of surface and bulk contributions
Finite cluster dimensions and associated quantum size effects aresignificant factors
⇒ intuition has a limited role
⇒ one has to perform calculations for the real stuff
– p.4/19
Systems to be studied
Free spherical Fe clusters, geometry taken as if cut from a bcc Fecrystal
Cluster sizes: between 9 atoms (1 coordination shell) and89 atoms (7 coordinations shells)
center
1st shell
2nd shell
3rd shell
Fe cluster 27 atoms
view in Z direction
shells atoms radius [a.u.]
1 9 4.702 15 5.423 27 7.674 51 8.995 59 9.396 65 10.857 89 11.82
– p.5/19
Calculations for T =0
Ab-initio within LDA framework (material specific)
Scalar-relativistic real-space spin-polarized multiple-scatteringformalism
Atomic sphere approximation (ASA)
Using SPRKKR codehttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR
– p.6/19
Calculations for T 6=0
For localized moments, finite-temperature magnetism can be describedby a classical Heisenberg hamiltonian
Heff = −∑
i6=j
Jij ei · ej ei ej
– p.7/19
Mapping DFT onto Heisenberg
Coupling constants Jij can be obtained from ground-state electronicproperties:
Jij = −1
4πIm
∫ EF
dE Tr[
(t−1
i↑ − t−1
i↓ ) τij↑ (t−1
j↑ − t−1
j↓ ) τji↓
]
[Liechtenstein et al. JMMM 67, 65 (1987)]
Valid only if magnetism can be described by localized magneticmoments (fine for Fe)
– p.8/19
Mapping DFT onto Heisenberg
Coupling constants Jij can be obtained from ground-state electronicproperties:
Jij = −1
4πIm
∫ EF
dE Tr[
(t−1
i↑ − t−1
i↓ ) τij↑ (t−1
j↑ − t−1
j↓ ) τji↓
]
[Liechtenstein et al. JMMM 67, 65 (1987)]
Valid only if magnetism can be described by localized magneticmoments (fine for Fe)
– p.8/19
From Jij to M(T )
Mean magnetization M(T ) of a system described by a classicalHeisenberg hamiltonian is
M(T ) =
∑
k Mk exp(− Ek
kBT)
∑
k exp(− Ek
kBT)
Mk is the magnetization of the system for a particularconfiguration k of the directions of spins
Ek is the energy of configuration k
Practical evaluation: Monte Carlo method with the importancesampling Metropolis algorithm
For bulk Fe, this procedure yields finite-temperature results that arein a good agreement with experiment [Pajda et al. PRB 64, 174402(2001)]
– p.9/19
From Jij to M(T )
Mean magnetization M(T ) of a system described by a classicalHeisenberg hamiltonian is
M(T ) =
∑
k Mk exp(− Ek
kBT)
∑
k exp(− Ek
kBT)
Mk is the magnetization of the system for a particularconfiguration k of the directions of spins
Ek is the energy of configuration k
Practical evaluation: Monte Carlo method with the importancesampling Metropolis algorithm
For bulk Fe, this procedure yields finite-temperature results that arein a good agreement with experiment [Pajda et al. PRB 64, 174402(2001)]
– p.9/19
From Jij to M(T )
Mean magnetization M(T ) of a system described by a classicalHeisenberg hamiltonian is
M(T ) =
∑
k Mk exp(− Ek
kBT)
∑
k exp(− Ek
kBT)
Mk is the magnetization of the system for a particularconfiguration k of the directions of spins
Ek is the energy of configuration k
Practical evaluation: Monte Carlo method with the importancesampling Metropolis algorithm
For bulk Fe, this procedure yields finite-temperature results that arein a good agreement with experiment [Pajda et al. PRB 64, 174402(2001)]
– p.9/19
Fe clusters at T =0
0 20 40 60 80 100cluster size (number of atoms)
2.0
2.2
2.4
2.6
2.8
3.0
3.2m
ag.m
omen
tper
atom
[B]
bulk
Average mag. moments
Average magnetic moment of clusters oscillates with cluster size
Local magnetic moments increase when going from the centeroutwards in an oscillatory way
Further reading: O. Šipr et al. Phys. Rev. B 70, 174423 (2004)
– p.10/19
Fe clusters at T =0
0 20 40 60 80 100cluster size (number of atoms)
2.0
2.2
2.4
2.6
2.8
3.0
3.2m
ag.m
omen
tper
atom
[B]
bulk
Average mag. moments
0 2 4 6 8 10 12distance from center [a.u.]
2.0
2.2
2.4
2.6
2.8
3.0
3.2
mag
.mom
enta
tsite
[B]
89-atom cluster
Average magnetic moment of clusters oscillates with cluster size
Local magnetic moments increase when going from the centeroutwards in an oscillatory way
Further reading: O. Šipr et al. Phys. Rev. B 70, 174423 (2004)
– p.10/19
Jij in bulk and in clusters
4 6 8 10 12distance between atoms i and j [a.u.]
-10
0
10
20
30
40
50
coup
ling
para
met
erJ i
j[m
eV]
bulk
Exchange coupling between atoms
crystalbulk
Atom i is fixed, atom j scans coordination shells around i
Oscillatory decay of Jij with distance
Jij for a given distance differs a lot between clusters and crystals
– p.11/19
Jij in bulk and in clusters
4 6 8 10 12distance between atoms i and j [a.u.]
-10
0
10
20
30
40
50
coup
ling
para
met
erJ i
j[m
eV]
bulk
central
Exchange coupling between atoms
89-atom cluster, central atomcrystal
bulk
central
Atom i is fixed, atom j scans coordination shells around i
Oscillatory decay of Jij with distance
Jij for a given distance differs a lot between clusters and crystals
– p.11/19
Jij in bulk and in clusters
4 6 8 10 12distance between atoms i and j [a.u.]
-10
0
10
20
30
40
50
coup
ling
para
met
erJ i
j[m
eV]
bulkoutermost
central
Exchange coupling between atoms
89-atom cluster, outermost atom89-atom cluster, central atomcrystal
bulk
central
outermost
Atom i is fixed, atom j scans coordination shells around i
Oscillatory decay of Jij with distance
Jij for a given distance differs a lot between clusters and crystals
– p.11/19
Site-dependence of∑
j Jij
Total strength with which one spin (at site i) is held in its direction:
Energy needed to flip the spin of atom i while keeping all the remainingspins collinear:
Ji =∑
j 6=i
Jij
Mild differences in Jij translate into large differences in∑
j Jij
No systematics in cluster size,no systematics in the position of atom within a cluster
– p.12/19
Site-dependence of∑
j Jij
0 2 4 6 8 10 12distance of atom i from the center of the cluster [a.u.]
100
200
300
sum
ofal
lcou
plin
gpa
ram
eter
sJ i
=jJ i
j[m
eV]
15
cluster of 15 atoms
Total coupling of a particular atom
Mild differences in Jij translate into large differences in∑
j Jij
No systematics in cluster size,no systematics in the position of atom within a cluster
– p.12/19
Site-dependence of∑
j Jij
0 2 4 6 8 10 12distance of atom i from the center of the cluster [a.u.]
100
200
300
sum
ofal
lcou
plin
gpa
ram
eter
sJ i
=jJ i
j[m
eV]
51
15
cluster of 15 atomscluster of 51 atoms
Total coupling of a particular atom
Mild differences in Jij translate into large differences in∑
j Jij
No systematics in cluster size,no systematics in the position of atom within a cluster
– p.12/19
Site-dependence of∑
j Jij
0 2 4 6 8 10 12distance of atom i from the center of the cluster [a.u.]
100
200
300
sum
ofal
lcou
plin
gpa
ram
eter
sJ i
=jJ i
j[m
eV]
89
51
15
cluster of 15 atomscluster of 51 atomscluster of 89 atoms
Total coupling of a particular atom
Mild differences in Jij translate into large differences in∑
j Jij
No systematics in cluster size,no systematics in the position of atom within a cluster
– p.12/19
Site-dependence of∑
j Jij
0 2 4 6 8 10 12distance of atom i from the center of the cluster [a.u.]
100
200
300
sum
ofal
lcou
plin
gpa
ram
eter
sJ i
=jJ i
j[m
eV]
89
51
15
bulk
cluster of 15 atomscluster of 51 atomscluster of 89 atomscrystal
Total coupling of a particular atom
Mild differences in Jij translate into large differences in∑
j Jij
No systematics in cluster size,no systematics in the position of atom within a cluster
– p.12/19
Decay of magnetization with T
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
89
bulk
crystalcluster of 89 atoms
Temperature-dependence of magnetization
[Bulk M(T ) curvewas extrapolatedto calculated TC ]
M(T ) curves are more shallow in clusters than in bulk
Small clusters — more shallow M(T ) curves than large clusters
– p.13/19
Decay of magnetization with T
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
89 59
bulk
crystalcluster of 89 atomscluster of 59 atoms
Temperature-dependence of magnetization
[Bulk M(T ) curvewas extrapolatedto calculated TC ]
M(T ) curves are more shallow in clusters than in bulk
Small clusters — more shallow M(T ) curves than large clusters
– p.13/19
Decay of magnetization with T
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
89 59
27
bulk
crystalcluster of 89 atomscluster of 59 atomscluster of 27 atoms
Temperature-dependence of magnetization
[Bulk M(T ) curvewas extrapolatedto calculated TC ]
M(T ) curves are more shallow in clusters than in bulk
Small clusters — more shallow M(T ) curves than large clusters
– p.13/19
Decay of magnetization with T
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
89 59
27
15bulk
crystalcluster of 89 atomscluster of 59 atomscluster of 27 atomscluster of 15 atoms
Temperature-dependence of magnetization
[Bulk M(T ) curvewas extrapolatedto calculated TC ]
M(T ) curves are more shallow in clusters than in bulk
Small clusters — more shallow M(T ) curves than large clusters
– p.13/19
Experimental and theoretical M(T )
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
50-60 82-92
59
8989 atoms, theory59 atoms, theory82-92 atoms, experiment50-60 atoms, experiment
M(T) by experiment and by theory
Experiment from Billas et al.PRL 71, 4067 (1993)
Caution:Each experimental curve corresponds to a range of cluster sizes(it represents an average over several M(T ) curves which may differquite a lot one from another)
– p.14/19
M as function of cluster size
0 20 40 60 80 100cluster size (number of atoms)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
0 K
Magnetization as function of cluster size
Dependence of M on cluster size does not really vary with T forlow (“experimental”) temperatures
For large T , magnetization of large clusters is significantly reduced
– p.15/19
M as function of cluster size
0 20 40 60 80 100cluster size (number of atoms)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
0 K90 K
290 K
Magnetization as function of cluster size
Dependence of M on cluster size does not really vary with T forlow (“experimental”) temperatures
For large T , magnetization of large clusters is significantly reduced
– p.15/19
M as function of cluster size
0 20 40 60 80 100cluster size (number of atoms)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
0 K90 K
290 K
610 K
810 K
1130 K
Magnetization as function of cluster size
Dependence of M on cluster size does not really vary with T forlow (“experimental”) temperatures
For large T , magnetization of large clusters is significantly reduced
– p.15/19
Dependence of Tc on cluster size
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
59
8989 atoms, theory59 atoms, theory
M(T) and its derivative
Critical temperature Tc defined as the inflection point of M(T )
curves (no phase transition for finite systems)
Tc oscillates with cluster size
Proper cluster-adjusted Jij have to be taken into account
– p.16/19
Dependence of Tc on cluster size
0 50 100 150 200cluster size (number of atoms)
0
200
400
600
800
1000
1200
criti
calt
empe
ratu
reT
c[K
]
bulk
clusters
clusters, calculated with proper Jij
crystal
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
59
8989 atoms, theory59 atoms, theory
M(T) and its derivative
Critical temperature Tc defined as the inflection point of M(T )
curves (no phase transition for finite systems)
Tc oscillates with cluster size
Proper cluster-adjusted Jij have to be taken into account
– p.16/19
Dependence of Tc on cluster size
0 50 100 150 200cluster size (number of atoms)
0
200
400
600
800
1000
1200
criti
calt
empe
ratu
reT
c[K
]
bulk
proper Jij
bulk-like Jij
clusters, calculated with bulk-like Jij
clusters, calculated with proper Jij
crystal
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
mea
nm
agne
tizat
ion
M(T
)[
B]
59
8989 atoms, theory59 atoms, theory
M(T) and its derivative
Critical temperature Tc defined as the inflection point of M(T )
curves (no phase transition for finite systems)
Tc oscillates with cluster size
Proper cluster-adjusted Jij have to be taken into account
– p.16/19
Shell-resolved magnetization
Expectations:
outer shells have smaller coordination numbers than inner shells
⇒ M in outer shells should decay more quickly with T
than M of inner shells
Projecting M of a shell ontothe direction of the total M
of the 89-atom cluster
(Projections are normalizedto T =0)
Not monotonous in order of shells
Although M of outer shells usually decays faster than M of innershells, no systematics can be found.
– p.17/19
Shell-resolved magnetization
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.2
0.4
0.6
0.8
1.0
norm
aliz
edpr
ojec
tion
ofm
agne
tizat
ion
whole cluster7th shell
1st shell
Shell-resolved projection of M(T)for a 89-atom cluster
Projecting M of a shell ontothe direction of the total M
of the 89-atom cluster
(Projections are normalizedto T =0)
Not monotonous in order of shells
Although M of outer shells usually decays faster than M of innershells, no systematics can be found.
– p.17/19
Shell-resolved magnetization
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.2
0.4
0.6
0.8
1.0
norm
aliz
edpr
ojec
tion
ofm
agne
tizat
ion
whole cluster7th shell
2nd shell1st shell
Shell-resolved projection of M(T)for a 89-atom cluster
Projecting M of a shell ontothe direction of the total M
of the 89-atom cluster
(Projections are normalizedto T =0)
Not monotonous in order of shells
Although M of outer shells usually decays faster than M of innershells, no systematics can be found.
– p.17/19
Shell-resolved magnetization
0 200 400 600 800 1000 1200temperature T [K]
0.0
0.2
0.4
0.6
0.8
1.0
norm
aliz
edpr
ojec
tion
ofm
agne
tizat
ion
whole cluster7th shell3rd shell2nd shell1st shell
Shell-resolved projection of M(T)for a 89-atom cluster
Projecting M of a shell ontothe direction of the total M
of the 89-atom cluster
(Projections are normalizedto T =0)
Not monotonous in order of shells
Although M of outer shells usually decays faster than M of innershells, no systematics can be found.
– p.17/19
Magnetic profile for T 6=0
4 6 8 10 12distance from the center of the cluster [a.u.]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
proj
ectio
nof
shel
l-res
olve
dm
agne
tizat
ion
[B]
0 K
Cluster of 89 atoms(7 coordination shells)
Magnetic profile gets more flat if temperature increases
M of outermost layers is similar in magnitude to M of inner layerseven for large T (i.e., no drastic decrease of surface M )
– p.18/19
Magnetic profile for T 6=0
4 6 8 10 12distance from the center of the cluster [a.u.]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
proj
ectio
nof
shel
l-res
olve
dm
agne
tizat
ion
[B]
0 K90 K
290 K
610 K
810 K
1130 K
Magnetic profile of 89-atom clusterdepending on temperature
Cluster of 89 atoms(7 coordination shells)
Magnetic profile gets more flat if temperature increases
M of outermost layers is similar in magnitude to M of inner layerseven for large T (i.e., no drastic decrease of surface M )
– p.18/19
Magnetic profile for T 6=0
4 6 8 10 12distance from the center of the cluster [a.u.]
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
proj
ectio
nof
shel
l-res
olve
dm
agne
tizat
ion
[B]
0 K90 K
290 K
610 K
810 K
1130 K
Magnetic profile of 89-atom clusterdepending on temperature
Cluster of 89 atoms(7 coordination shells)
Magnetic profile gets more flat if temperature increases
M of outermost layers is similar in magnitude to M of inner layerseven for large T (i.e., no drastic decrease of surface M )
– p.18/19
Summary
Exchange coupling constants Jij in clusters differ from the bulk,with no obvious systematics
Magnetization M(T ) curves are more shallow for small clustersthan for large clusters
For large T , magnetization of large clusters is significantly reduced
Critical temperature Tc oscillates with cluster size
(Normalized) M of the outer shells usually decreases with T morequickly than M of inner shells
Simple models (such as taking Jij from bulk) do not work,systematical trends cannot be guessed beforehand
⇒ one really has to calculate all the quantities of interest
– p.19/19
Summary
Exchange coupling constants Jij in clusters differ from the bulk,with no obvious systematics
Magnetization M(T ) curves are more shallow for small clustersthan for large clusters
For large T , magnetization of large clusters is significantly reduced
Critical temperature Tc oscillates with cluster size
(Normalized) M of the outer shells usually decreases with T morequickly than M of inner shells
Simple models (such as taking Jij from bulk) do not work,systematical trends cannot be guessed beforehand
⇒ one really has to calculate all the quantities of interest
– p.19/19
Summary
Exchange coupling constants Jij in clusters differ from the bulk,with no obvious systematics
Magnetization M(T ) curves are more shallow for small clustersthan for large clusters
For large T , magnetization of large clusters is significantly reduced
Critical temperature Tc oscillates with cluster size
(Normalized) M of the outer shells usually decreases with T morequickly than M of inner shells
Simple models (such as taking Jij from bulk) do not work,systematical trends cannot be guessed beforehand
⇒ one really has to calculate all the quantities of interest
– p.19/19
Summary
Exchange coupling constants Jij in clusters differ from the bulk,with no obvious systematics
Magnetization M(T ) curves are more shallow for small clustersthan for large clusters
For large T , magnetization of large clusters is significantly reduced
Critical temperature Tc oscillates with cluster size
(Normalized) M of the outer shells usually decreases with T morequickly than M of inner shells
Simple models (such as taking Jij from bulk) do not work,systematical trends cannot be guessed beforehand
⇒ one really has to calculate all the quantities of interest
– p.19/19
Summary
Exchange coupling constants Jij in clusters differ from the bulk,with no obvious systematics
Magnetization M(T ) curves are more shallow for small clustersthan for large clusters
For large T , magnetization of large clusters is significantly reduced
Critical temperature Tc oscillates with cluster size
(Normalized) M of the outer shells usually decreases with T morequickly than M of inner shells
Simple models (such as taking Jij from bulk) do not work,systematical trends cannot be guessed beforehand
⇒ one really has to calculate all the quantities of interest
– p.19/19
Summary
Exchange coupling constants Jij in clusters differ from the bulk,with no obvious systematics
Magnetization M(T ) curves are more shallow for small clustersthan for large clusters
For large T , magnetization of large clusters is significantly reduced
Critical temperature Tc oscillates with cluster size
(Normalized) M of the outer shells usually decreases with T morequickly than M of inner shells
Simple models (such as taking Jij from bulk) do not work,systematical trends cannot be guessed beforehand
⇒ one really has to calculate all the quantities of interest
– p.19/19