On the cohesion of small clusters of alkali metals

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J. Friedel

To cite this version:

2023

Short Communication

On the cohesion of small clusters of alkali metals

J. Friedel

Laboratoire de

_Physique

des

Solides,

Université Paris

XI,

91405

_Orsay

_Cedex,

France

(Received

on March

28,1990,

accepted

in

final form

on June

28, 1990)

Résumé. 2014 Des résultats récents concernant la cohésion de

_petits

amas neutres et iooisés de Na et K

sont

_{interprétés}

dans un modèle

_simple

de Hubbard. Ils conduisent à des

_intégrales

de transfert et de Coulomb de l’ordre de

_{0,5 à}

1 eV Ces valeurs

_justifient

l’utilisation d’un traitement de Hartree Fock des interactions

_{électroniques}

et sont cohérents avec les structures assez _compactes de ces amas.

Abstract. 2014 Recent results on the cohesion of small neutral and ionised clusters of Na and K are

interpreted

in a

_simple

Hubbard

_model,

_leading

to values of transfer and Coulomb

_integrals

of about 0.5 to 1 eV These values

_justify

the use of a Hartree Fock treatment of electron interactions and are

coherent with the

_fairly

_compact

structures of these _aggregates.

J.

_Phys.

France 51

₍₁₉₉₀₎

2023-2032 15 SEPTEMBRE _1990,

Classification

Physics

Abstracts 68.00 - 73.00

1. Introduction.

From recent

_experiments

on

jets, Leygnier et

al.

[1, 2J

have deduced the successive

energies

of

_evaporation

of atoms A and dimers

_A2

from ionised clusters

_AN.

These show the even odd

alternation characteristic of Fermi statistics

_{[3, 4]}

and

_steps

_ending

at

_’magic

numbers’ related to

the

_filling

of successive orbital states

_{[5] .}

Both effects are

expected

in a delocalised

picture

of the valence electrons

_{[6] .}

Such effects are however small

enough

for the cohesive

energies

_per

atom

EN

and

Et

of

neu-tral and ionised clusters to _vary in a

fairly

smooth _way with N

_{(Fig. 1).}

_Indeed,

as remarked

by

Leygnier,

EN

varies

_{approximately}

_{linearly with}

N-113 :

for 2 N

_50,

down to the smallest value of N. The

_extrapolated

value

_Eoo

is _very near to the

bulk energy of cohesion in

_{macroscopic phases}

of the

metals ;

and the

_slope

A is

_only

10 to 15 %

larger

than that deduced from the value of the

_macroscopic

surface tension _1. We shall use here the values of

_Eoo

_{and 1}

for the

_{crystal phases ;}

there are indeed but small differences with those of the

_{corresponding}

_{liquid phases.}

Another

_experimental

result is that

Et

varies

_nearly

as

EN

for small values of

N-113

but sat-urates at a

_nearly

constant value for N less than about 30.

Thèse results are somewhat

_surprising

a

priori. They

can be

explained,

with reasonable values

of the

_parameters,

_using

a

simple

Hubbard model which involves a transfer

integral

t and an

intraatomic Coulomb

_integral

_U,

_together

with a short

_range

interatomic

_{repulsion R}

_{[7 -}

_{9] .}

The

_approximate

linear variation of

EN

witlt

N- 1/3

for _{small N’s cannot, of} course, be

_explained

in terms of a

_{(macroscopic)}

surface tension.

_Indeed,

for N = 4 _or

5,

the most stable clusters _are

two-dimensional,

and not the most

_compact

three dimensional clusters that would minimise the

effect of a surface tension. The observed variation of

EN

(N-1/3)

just

fits with the structure of the clusters in their most stable

_{configuration.}

The

_{corresponding}

_{analysis gives}

estimates of t and

U,

together

with the relative

_steepness

of R. The

_apparent

saturation of

_EN

for small N’s results

from the

_increasing

_importance

of the Coulomb

_repulsion

between the unscreened

_nuclei,

which

is not

_easily

_{computed exactly ;}

it can

only

be stated that the behaviour observed is cohérent with a small value of U.

The values obtained for Na and K are of course

only

orders of

magnitude.

The low ratio

U/t ££

1 to 2

_justifies

however the

_neglect

of corrélation

_{effects ;}

it fits the condition for the

_stability

versus

dimerisation of these

_fairly

_compact

clusters.

z

2.

_{Computed énergies}

of formation of neutral clusters.

In a

simple

Hubbard

model

for valence

(ns)

electrons,

the structure is assumed to contain

_only

equidistant

bonds between nearest

_neighbours,

with transfer

_integrals

of absolute

_(positive)

value

TN.

We assume the intraatomic Coulomb interaction to be small

_enough

to be treated in

per-turbation. A short _{range Coulomb}

_repulsion

_RN

between atomic

_neighbours

is added. All other

parameters

are

neglected,

such as

overlap

integrals

between different atomic

orbitals,

interatomic Coulomb and

_exchange

_{interactions,}

virtual excitations to

_higher

orbital states. Then :

2

if nN is the

average

number of nearest

_neighbours

_per

atom.

Thble 1

_gives

the values of the

_positive

_parameters

aN,

8N,

lN for the most stable small clusters

with N atoms

[10].

The value

_QN

=

1/4

is obtained when each atom is

neutral,

which occurs

exactly

for alternate structures

[11],

such as for N = 2. For

increasing

values of

N,

QN

oscillâtes

around this value

_(cf.

_Appendix

_A)

but tends

_rapidly

towards

_it,

so that this can be taken as a

good

approximation

for N » 3. The values

_{Of -fN}

are

already large

[11]

for N = 2 or

3,

and

_expected

to increase for

_larger

_{N’s ;}

_they

have therefore not be

_computed

for N > 3.

For N

_large,

the

_{quasicontinuous}

distribution of the valence states can be

_schematized,

to a

good

approximation,

by

a continuous band of a width w and constant

_density

of states

which

fits

the 3 first moments of the real distribution. This

_leads,

with a further

_good

approximation

for the

(small)

correlation term in

U2@

to

[12]

2025

Thbîe I. -

Computed

values of

parameters.

one obtains at

_equilibrium,

in the

_{approximation}

where all bond

_lengths

are

_kept

equal,

with

We assume the correlation term in

U2

to be

_negligible.

We can then write

For

_large

clusters,

where aN oc

nN1/2,

this reduces to

_{[7 - 9]}

A

_development

₍₁₎

linear in

N- 1/3

can then be made in this limit. For

large

quasispherical

clusters

of radius

_R,,,

with

_Ng

surface atoms with ns bonds

per

atom, and

Nv

= N -

_Ns

volume atoms, with _ny = _noo bonds

_per

atom, _{one can} write

with

if

b3

is the atomic volume. A

_development

of

₍₂₎

in

_{Ns/ N, using (5), gives (1)}

with

and [12]

where

_again

the small term in

U2

has been omitted. A is related to the

_macroscopic

surface tension

-y

per

surface atom

_by

11

It

_might

be

_pointed

out that the factor

_{( 1- p/q - p)}

in A takes into _account, in an

approximate

_way,

the variation of bond

_length

with the size of the

_sample

in a

crystal

phase.

The same measurement

would

_appear

in the relation of _{y to} the

_{supersaturation}

of

_vapour

in contact with a

spherical

crystal

phase.

This factor however

_depends

on the

_geometry

of the

sample,

and would not

_appear

necessarily

with the same value for other

techniques

measuring

_1.

3.

_Comparison

with

_experiment.

A quantitative comparison with experiment requires

a

knowledge

of coefficients _p

and q.

Following

Joyes

[6] ,

we can deduce for Na from bond

_length

and

_{compressibility}

of

_{macroscopic phases}

Values of _nN for small N’s are

reported

in table

I,

as deduced from the stable

configurations

[3,7].

Figure

2 then

_plots

the variations with

N-1/3

of aN

tN jtoo.

If we

again

neglect

the correlation terms in

UZ

and also

_neglect

the small différences of

_QN

with

_1/4,

we can write

Figure

2 shows

_that,

in these

_{conditions, EN}

varies

_{roughly linearly}

with

N -1/3

and

_extrapolates

roughly

to the value

Eoo ~

_{(1- p/q)O:ootoo - U/4 computed}

for

_macroscopic

_samples.

Thus

equa-tion

₍₁₎

is

_{approximately}

fulfilled,

with deviations near to those observed

_{experimentally}

_{(Fig. 1).}

From a

quantitative

point

of view,

the

coefficient A’

in

équation

(1) computed

from the

straight

line,

figure

2 is such that

6 z

2027

Fig.

1. -

Experimental

variations of

EN

and

Et

with

N-1/3

for Na. Ibe

straight

line

passes

through

Eoo

and

_E5.

where A is the

_slope

of

EN

(N-1/3)

for

_large

values of

_N,

as deduced from the surface

tension,

this

_gives

’1

With

_qlp

4.7 and also nv =

12,

this

gives

_a reasonable value

In other

_words,

for such a value of

(nv - ns) /nv,

the

_slope

of the

_straight

_line,

_figure

_2,

is about 10 %

_larger

than that

_computed

for

_large

values of

_N,

as observed.

This

_agreement

can be taken as a confirmation of the value taken for the relative

_steepness

q/p

of the short

_range

_repulsion.

Thus for

_{qlp 1/3,}

as in noble or transition metals

[6,12] .

EN

(N-1/3)

would be

_nearly

constant in the

_range

of

_figure

2.

Now the

_experimental

value of A’ is 1 eV for Na. This leads to

Also the

_experimental

value of

Eoo

is 1.15 eV in Na.

Fig.

2. -

_Computed

values of

_aNtN/toc

versus N-1/3.

Crosses :

_computed

values for N

_{small ;}

dot :

computed

value for

_{macroscopic sample.}

The

_straight

line passes

through

the

_point

for N = 5 and N = oo.

Similar estimates of

toc

and U can be made for K. In both cases, one checks

easily

that the

cor-relation corrections in

U2

are then

_negligible

in all formulae : the Hubbard model reduces to a

Hartree Fock

_(Slater)

scheme.

4.

_Energies

of formation of ionised clusters.

A

_straight

extension of the

_preceding

treatment

_gives

for the

_energy

_per

atom of a

singly

ionised

cluster

-For N

_small,

at, (3t¡

and

ijb

are

given

[3,11,12]

in table II

(cf.

Appendix

A).

For N

_large,

Appendix

B

_gives

2029

if we take

The alternation with the

_parity

N of the one électron term

(aN -

aN)

_tN,

which come from Fermi

statistics,

are then

partly compensated by

those of the Coulomb term

(PN - pt)

_U,

a

general

ef-fect

_pointed

out

_by

_Joyes

_[14].

The correlation term in

U2

is

_negligible,

and the main contribution

to

EN -

_EN

comes from the Coulomb term.

This estimate

₍₁₄₎

of U is the same order of

magnitude

as that

(11)

deduced from

EN (N).

It can be

argued

that

EN -

EN

is more sensitive to U than

EN itself,

so that

(14)

should be a better

measurement of U. But it must be stressed that the

_{analysis leading}

to

_{equation (12)}

is less self consistent than for

_equation

_(2).

_First,

the

_changes

in interatomic distances due to ionisation of the cluster have been

_{neglected ;}

then the

_long

_range Coulomb

_potential

of the

_positive

_charge

produces

a small

repulsion

between the atoms which has been

_neglected

in

_{Et :}

a self consistent

study

of this

_effect,

with the

_{complicated configurations}

of the

clusters,

is

_beyond

the

_scope

of this

paper.

5. Conclusions.

The

_approximate

linear variation of

_EN

versus

N- 1/3

observed

extrapolate

to the smallest

clus-ters the

_macroscopic

variation due to the surface tension effect. This can be

_explained,

with a

suitable choice of

_parameters,

_using

a

simple

Hubbard model of the valence électrons

together

with short _range

_{pair repulsions.}

This

_requires

a

large

relative

_steepness

q/p ::!

4.7 of the short

range

repulsions,

coherent with

macroscopic

properties.

The

slope

of

EN

(N-1/3)

then leads to

a transfer

integral

_too

~ 05 eV and the cohesive _energy of the

macroscopic

phase

Eoo

then leads

to a small value of U ^--’ 0.5 eV. Within the same kind of Hubbard

model,

the Coulomb term in U

plays

a

leading

role in

EN -

_EN,

from which one deduces a somewhat

larger

estimate U ~ 1 eV.

This however is not

_necessarily

more _accurate,

owing

to the

_larger

uncertainties of the model used in this case.

Values of U 1 eV are

definitely

smaller than that induced from atomic

data ;

a similar

re-duction in effective U is observed in transitional metals

_{[12] .}

The

_{corresponding}

ratio

_U/too

2 leads to

_negligible

correlation terms in

U2.

Indeed

_U/too

is much less than the critical value

Uc/too

above which dimerisation is stabilised

_by

corrélations : for

_qlp

=

4.7,

an

application

of the

dis-cussion

_given

in

_[6]

to

[9]

leads to

It can be stressed

that,

for small values of N where the _energy

EN

is

strongly

sensitive to the

topology

of

_clusters,

the

_agreement

found would

_disappear

for other

_topologies

than the ones

considered. Thus three-dimensional

_closepacking

would

_give,

_up

to N =

6,

much smaller

energies

EN

which would not

_extrapolate

_linearly

in

N-1/3

to

_Eoo,

as the actual flat clusters do. The

energy différences between such clusters with different

_topologies

are indeed much

larger

than

the thermal

_energies,

so that

only

the most stable

_topology

is observed for each value of N.

Contrary wise,

the

_simple

model used here is not

_perfectly

selfconsistent

_Thus,

for N =

3,

one

knows that a Jahn Uller effect distorts the

_{equilateral triangle}

assumed here

_{[11] .}

For N >

3,

all bonds are not

_{perfectly equal}

_{either ; and,}

for

_large

values of

_N,

one knows that the bonds

between surface and volume atoms are smaller than those between volume atoms

[7 - 9,12].

Also the electronic

_density

is not

_exactly

_uniform,

and this leads to small correction to the electronic structure and to the interatomic forces. However all these terms are

certainly negligible

with

Appendix

A

Computation

of

_{f3 N}

and

f3f¡

for small N’s.

This

_appendix

extends the Hartree Fock treatment

_given

in

_[11]

for N = 3.

’Iàking

le

> as a

_single

determinant of one electron states

_{[a(1, JL)}

_>,..., one can write

easily,

in the most stable

state,

etc

where V and I are the Coulomb and

exchange

interactions of the

respective

statcs.

Using

one

ôbtains

Expliciting

the

Vaa

and

_VA,

in term of

_U,

_{using explicit}

values

_{of la}

_{>, IP}

> etc..., one finds the

values

_given

table I. Thus if

where a are the N atomic

_orbitals,

When N

_increases,

the electronic

_density

of each

_occupied

one electron state becomes more

unifrom. In the limit of

_{complete uniformity,}

one would have

Then an extension of

preceeding

formulae

gives

2031

and

Finally

1 ,

Table 1 shows that actual values of

_QN

never deviate _very much from 0.25. Thblc II shows that the

asymptotic

formulae

_(Al)

are followed

approximately

already

for N = 6. For N _>

6,

the values

of f3N - f3t

given

between

_parentheses

are deduced from

(Al).

This

_justifies,

for an order of

magnitude,

the direct

_comparison

of

_figures

1 and 2.

’Iable II. -

Measured Et -

EN

Compared

with estimated terms

_(in

_eV).

Appendix

B

Computation

of

The Hubbard model for z and z’ electrons

_per

atom

_{respectively,}

leads to a difference in cohesive

energy per

atom in a cluster of N monovalent atoms

[9,12J

where w and w’ are the band widths for z and z’. Formulae

(15)

is obtained

using

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