Conduction Processes in Concentrated Metal-Ammonia Solutions

8 Conduction Processes in Concentrated Metal-Ammonia Solutions J. C. THOMPSON

Downloaded by UNIV OF NEW ENGLAND on February 11, 2017 | http://pubs.acs.org Publication Date: January 1, 1965 | doi: 10.1021/ba-1965-0050.ch008

Department of Physics, University of Texas, Austin,

Tex. 78712

The metallic nature of concentrated metal— ammonia solutions is usually called "well known." However, few detailed studies of this system have been aimed at correlating the properties of the solution with theories of the liquid metallic state. The role of the solvated electron in the metallic conduction processes is not yet established. Recent measurements of optical reflectivity and Hall coefficient provide direct determinations of electron density and mobility. Electronic properties of the solution, including electrical and thermal conductivities, Hall effect, thermoelectric power, and magnetic susceptibility, can be compared with recent models of the metallic state.

J h e conœntrated (>0.4Af) metal-ammonia solutions were first called " m e t a l l i c " b y K r a u s in 1921 (7). O n several recent occasions the term "semiconductor" has implicitly been substituted for " m e t a l " in interpreting various data (2, 2). Since Kyser and Thompson (8) have established the truly metallic nature of the solutions b y measuring the free carrier concentration, it is worthwhile to re-examine the relative data a n d interpret it i n terms of liquid metal theory. T h e first conclusion is that solvated electrons no longer exist i n appreciable quantities. T h e most direct determination has been b y Beckman and Pitzer (2) who do not find the famous 1.5μ peak at con centrations above 1M. Indeed, it would be difficult to understand how the solvation layer might be maintained around electrons'with the high thermal velocities characteristic of a degenerate electron gas. One may also note that at the concentrations in question the mole ratio ri (= moles solvent /mole solute) is rapidly approaching unity, so that fewer solvent molecules are available for the solvation process. A t the onset of the 96

Hart; Solvated Electron Advances in Chemistry; American Chemical Society: Washington, DC, 1965.

8.

THOMPSON

97

Conduction Processus

metallic state r is still ~ 5 0 . T h e solvated electron disappears, not by a gradual erosion of solvating layers, but i n a rapid reaction which may be represented b y : t


(e-U

-+e~

+ nNH

3

(1)

where η is the number of solvating ammonia molecules. T h e process is not that simple, however; at concentrations below the metallic range there is evidence for at least two classes of bound electrons. T h e spectral evidence of Beckman and Pitzer implies the usual electron-in-a-cavity to be present; at least the 1.5μ absorption is still present. A t the same time the conductance data imply that electrons exist, bound b y only 0.2 e.v. (2). T h e latter class, of course, is responsible for the conductivity since thermal energies are inadequate to release electrons from the rel atively immobile solvated state. T h e analysis to be presented is based on the results of K y s e r and Thompson (8), which are summarized briefly, then discussed i n turn with the other properties of the solutions. K y s e r and Thompson, using the H a l l effect, measured the free electron concentration directly. T h e y find, at concentrations above 4 mole % i n sodium or lithium solutions, that all metal valence electrons are free. Furthermore, the numbers of free electrons do not vary with temperature, though there is a slight temperature variation of number density (number of electrons/cc.) owing to the thermal expansion of the liquid as a whole. Hence, we must use the metal concentration and the electron concentration as identical, within the precision of their experi ment. T h e precision puts a limit of one i n 100 electrons per metal atom in any bound state—e.g., monomer, dimer, or other cluster. T h e bound electrons, if any, are bound b y energies so great that a 30 degree tem perature rise does not free a measurable number. A saturated solution is i n this sense no different from any other solution i n the concentration range above the metallic transition. T h e final model is just that proposed b y K . S. Pitzer i n 1958 (12). W e must consider a form of " l i q u i d so d i u m , " or other metal, as simple as the pure metal, but with a widely varying ionic and electronic density, and imbedded i n a dielectric medium. Electronic

Properties

T h e electrical conductivities of several alkali metals dissolved i n liquid ammonia are shown i n Figure 1 (7, 22, 15). T h e strong variation of the conductivity, σ, with concentration has been most difficult to explain. T h i s difficulty can be assessed b y referring to a simple model of conductance, the Thomas-Fermi model of a screened Coulomb potential (19). T h i s model has been used i n describing semiconductors as well as i n theories of metal-ammonia solutions (2). T h e scattering center is the metal ion, the Coulomb potential of which is screened i n the manner of the Debye-Huckel theory of weak electrolytes. T h e screened potential has the form (2)


SOLVATED ELECTRON


98

6

8

10

12

14

Metal Concentration (mole per cent) Figure 2.

Electrical conductivity vs. concentration for several alkali metals dissolved in liquid ammonia at 33°C.

Data points have been omitted for clarity. The inset shows data for con centrated lithium-ammonia solutions. where λ is the screening constant, the reciprocal of the screening length, and € is the permittivity. F o r such a potential the scattering crosssection is easily calculated to be (25) 2me /(*) = \4ireh2(k* + λ ) 2

(3)

2

for an electron wave with wave number k. is then (18) - = 2wn

s

Λ

Jo

T h e mean free path, A,

(1-cos 0)/(0)sin0d0


(4)

8.

THOMPSON

99

Conduction Processes

where n is the number density of scatterers. T h e factor (1-cos Θ) enters because small-angle scattering does not remove momentum from the stream of particles. It is customary to assume elastic scattering so that 8

(Ο

2(l-cos Θ)

(5)

where k is the Fermi wave number 0

k =

(3ΤΓ 71Ο) 2

0

(6)

1/3


Now, the mean free path is found to be 1 _

n m>e* Γ

Λ "

8™*ft'*o« L

s

/4V \ λ

\ + V

2

4V/X

1

2

1 + 4*o /X J

"

2

2

U

)

A s i n standard kinetic theory the conductivity, σ, is

°

~

ηοβ Λ 2

_

mVns

Γ

" 24τΓ € Λ3η L 3

2

0

2

/4*o

\λ

Λ

4* /λ 0

1

2

~ 1 + 4^o A J

/

2

2

2

2

{

>

for degenerate statistics. T h i s very fine result is meaningless until we know λ. T h e primary particle involved i n the screening process is the mobile electron. One has then the problem of a self-consistent calculation of the charge distribution i n the neighborhood of a test charge. T h e Thomas-Fermi approach to this problem is the analog of the DebyeHiickel calculation wherein allowance has been made for the Pauli ex clusion principle. F r o m any standard text one can obtain the Poisson equation (19) VV

=

e p/e = - (n -

-

n) 0

en

0

c where ρ = e(n — n ) is the charge density; n is the uniform charge density i n the absence of the test charge; η = n(r) is the nonuniform charge density produced b y the test charge; K is the dielectric constant (e = K €o); and f is the F e r m i energy (or electrochemical potential) of the degenerate electron gas and is given by 0

0

e

e

0

no = |f, (2mf ) . on* 0

T h e symbols h, e, m have their usual meaning. equation for ψ by assuming: /çA Uo/

(10)

3/2

One now linearizes the

2

«

1


(ID

100

SOLVATED ELECTRON

gives: 2φ

= ™°(Μ) c \2fo/

ν

(12)

which if φ = ψ(τ) only, takes the form: W

2 ^ _ 3 r t M

+

dr*

r dr

( 1 3 )

2«Γο


and is easily solved to yield: φ =

(14)

where, using (10), λ'

^ 4 re (2m)" fo i

î

2

(15) I/î

W e define the Fermi wave number, k by 0

hk = (2mf ) , 0

(16)

1/J

0

and obtain, finally, λ· =

(16a)

e mko 2

A l l of this is based on the linearization procedure in Equation 11. If one examines this condition, the restriction turns out to be quite tight. We may approximate ψ by e/4Ter , where r is the ionic spacing (29). T h e n Equation 11 takes the form: 0

0

(eV47T€r fo) « 1

(Ha)

2

0

but r = (n ) ~ and on substituting from Equation 10 we find, ignoring the overall square, 0

0

1 / 3

7 7 1 6 2 2

3

7

« 1

1

2·3 > .ιτ / Λ € no 3

2

(lib)

1/3

We may now put i n such constants as are available and find r « 0

2.53 10 "

8

m*/m

cm.

(11c)

where m*/m is the ratio of effective electron mass to free electron mass. In the simplest case, K = m*/m = 1; hence r 2.5 A . In Figure 2 are plotted several simple parameters of an electron gas, including r . One sees at once that Equation 11c is not satisfied unless K /(m*/m) > 10, and the situation worsens as the electron concentration decreases. One may parenthetically note that i n the very dilute ( 1 0 M ) electron gas, the Thomas-Fermi result goes over into the Debye-Huckel formula e

0

0

e

_ 4


8.

THOMPSON

101


which is then justified on the basis that the thermal energy, kT, is to exceed e /4ircr , which condition obtains at high dilutions.


2

0

n (cm"') 0

Figure 2.

Several parameters of an electron gas as a function of the density of electrons. Ço is the Fermi energy; r is the interparticle spacing; k is the Fermi wave number; and λ is the Thomas-Fermi screening length. The dashed line shows the sodium concentration which yields n free electrons (as determined from the Hall coefficient); the dotted line is the metal concentration. 0

0

0

In applying these ideas to metal solutions in ammonia, the restrictions on the Thomas-Fermi screening approach have generally been ignored. A t metallic concentrations the concentration of free ammonia molecules is low, and the dielectric constant is near unity; the pure ammonia value of K is only applicable when solvent molecules are not polarized by the ions. If we choose m*/m less than unity, then fitting the theory to experimental data on conductivity becomes impossible. F o r the calculated conductivity to agree with the measured conductivity, we require e



102

SOLVATED ELECTRON

at saturation and much less than 10 below saturation. T h i s requirement contravenes the condition for a linear version of the Poisson equation. T h e failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (29). T h e metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E . Wigner showed i n 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an "electron crystal" wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. B u t one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity i n terms of, say, effective mass at present. T h e effective mass may be introduced to account for errors in the density of states—not in the electron correlations. Pohler and Thompson (13) have treated the ionic potential b y a n O P W formalism. I n that calculation, screening was computed with a Hartree-Fock formalism. T h e electron correlations are still not included properly. Further criticism may be aimed at their calculation. T h e y took the wave functions of the core electrons and renormalized them so that they were spread over the greater volume available to the ion i n solution. There is no guarantee that the procedure used properly allowed for the increased orbital angular momentum produced b y the spreading process. T h i s effect should be less on the, intrinsically larger, cesium ion than on the lighter alkali metals. T h u s , no meaningful conclusions can be drawn from their computation, either as to magnitude or trend. A n omission has been made i n deriving Equation 4. T h e relative positions of the scattering centers are important and may be included b y multiplying f(6) b y α(Θ) [= a(k)\ where α(Θ) is the Fourier transform of the radial distribution function of the solution (20). T h i s quantity is usually determined from x-ray diffraction data. A l l of the temperature dependence of σ depends upon a(k). W e cannot explain the temperature variation of σ b y density changes or b y a n alternation i n the scattering process. Only through changes i n the relative positions of the scattering centers does the temperature influence the conduction process. Thus, the fact that conductivity increases with temperature i n the range below 0 ° C., need not force us to assume a semiconductor-like description of the solution. E v e n pure zinc shows such behavior just above its melting point (3). These conclusions are independent of the details of the scat tering process—i.e., of /(0), and are not altered b y the failure of the theory of the screening. L e t us turn now to the other conclusions which can be based on free electron theory. T h e H a l l effect measurements of K y s e r and Thompson permitted the computation of the free electron concentration. T h e H a l l effect is produced b y a balance between the magnetic force (Lorentz force) on a current carrier and the electric force produced b y a displaced charge density within a conductor. F o r a charge, q, moving



8.

THOMPSON

103


Metal Concentration (mole percent) Figure 3.

with drift velocity, field, B, is

The Hall coefficient of lithium, sodium, and calcium solutions in ammonia. Ο Ο

1

χ

IB

180 Figure 4.

190 200 Temperature (degK)

210

The temperature dependence of the Hall coefficient of con centrated lithium-ammonia solutions.

included. T h e small discrepancy is assigned to a small error i n deter mining t. These results indicate that the bronze solutions are metallic in every sense of the word. Previous efforts to make the Thomas-



8.

THOMPSON

105


Fermi calculations agree with experiment by postulating a temperature variation to η are shown to be wrong. K y s e r and Thompson (9) have also searched for an effect of Β upon σ: magnetoresistance. None was found within the precision of measure ment and over the metallic concentration range. T h i s provides further evidence for the metallic nature of the solutions since no magnetoresistance is to be expected for a simple metal with a spherical F e r m i surface. T h a t is, if the electron energy has the same dependence upon wave number regardless of propagation direction, there should be no magnetoresistance. Such isotropy is, of course, expected i n a liquid. However, if tunneling or hopping were the charge transfer process i n stead of free electron drift, a magnetoresistance might occur. L e t us look now to the thermoelectric power. F o r a simple metal with a spherical Fermi surface the thermopower depends only on the electron concentration (18). T h e relationship is: 1Γ k*T = Ξ. * ± = 3 ef

Τ - V/deg. Γο

2

€

-2.45

Χ ΙΟ"

2

(22)

M

0

where f is the Fermi level (in e.v.), as given by Equation 10. We may calculate € using the electron concentration shown i n Figure 3. The results of this calculation are compared with the measurements of Dewald and Lepoutre (4) i n Figure 5. T h e discrepancy appearing there is of the same order as that found i n pure, solid sodium. One of two modes of analysis is customarily used when dealing with the thermo power of metal (10). If a small impurity is present, and if this impurity contributes a second scattering mechanism for the electrons, then one may apply Matthiessen's rule: ρ = pi + p . T h e net resistivity is p; the contributions from the host and impurity are pi and p . One then finds the thermopower, e, to be 0

2

2

€ = € + 2

(

+ *) ^ Ρ

6l

(23)

Equation 23 goes by the name, Gorter-Nordheim rule; it indicates that a plot of e vs. σ( = 1/ρ) will be linear. O n the other hand, i f one has a single scattering mechanism and two groups of carriers (two bands), then the conductivities rather than resistivities are added: σ = σι + σ . In the two-band model the thermopower is: 2

€=6

2

+

(

€ ι

-

e) 2

l

σ

(24)

and one expects e vs. p( = l / σ ) to be linear. Neither of the two approaches is successful when attempted on the metallic sodium-ammonia solutions. T h a t is, the linear relation is not found i n either case. T h i s does not rule out the presence of a single carrier and a single scattering center. Rather the absence of multiple carriers or scatters is indicated. T h i s observation supports the contention that the material is basically a simple metal.


SOLVATED ELECTRON


106

10

1

3

5 7 9 it" Metal Concentration (mole percent)

Ï3

Figure 5. The thermoelectric power of concentrated sodium-ammonia solutions from experiment and as calculated for a free électron gas. The dotted lines to the right of the saturation line refer to pure solid sodium.

T h e magnetic susceptibility is also derived from the electron density. There are two contributions (19): the Pauli paramagnetism and the L a n d a u diamagnetism. T h e former is given by ( 2 5 )


8. THOMPSON

107


and the latter by X, = -

5 ^ 26:o

(26)

where μ is the Bohr magneton. In the free electron gas one has Xd = — l / 3 x ; the total susceptibility is: Β

p


χ

= χ , + χ, = » p

(27)

However the electron effective mass enters the two terms i n different ways. T h e spin magnetic moment is fixed; hence m*/m enters x only through the Fermi energy. T h e diamagnetism is the result of the force exerted on the electron by a n applied field; the angular velocity change derived from this force depends on m*/m. Combining these results gives: P

x

. . ^ ! [

8

Î » ! _ J ! L l

(28)

Combining Equations 27 and 28 one finds — = — [3m*/m X 2

m/m*]

(29)

A change of sign is possible for small m*/m. In particular, if the electrons are bound, the L a n d a u turn is dropped. If we turn now to the value of x„ χ = 1.47 X 1 0 - n 1 4

in cgs. units.

0

(30)

1 / 3

Converting to mass units: x =

1.47 X 10

(30a)

where ρ is the density. Finally, for the susceptibility per gram atom metal one multiplies by the molecular weight. I n a sodium solution the equivalent molecular weight is S = *il7.03 + x 2 3 2

and to obtain the sodium fraction, divide b y x : 2

X

, ^ X 1 0 - » 8 Ρ

W

B

,

(

3

0

b

)

*2

Equation 30b duplicates Yost and Russell (27), though we shall not apply it to such a wide concentration range. One may now construct χ as a function of JC . T h e graph of this function is shown i n Figure 6 along with the meagre data. T h e rapid rise of χ below 1.3% does not match the rapid drop found from Equation 30b. Caution is necessary in applying Equation 30b since the assembly is no longer degenerate, the 2


SOLVATED ELECTRON


108

ο

ι

1

I

I

I

I

I

I

0

2

4

6

8

10

12

14

ι

16

Metal Concentration (mole per cent) Figure 6.

The atomic susceptibilities of sodium in ammonia.

The dotted line shows the free electron theory. Note the absence of data between two and 12 mole %. L a n d a u derivation of the diamagnetism no longer applies, and contri butions from the orbital motion of the electrons may be expected. Here, as before, the free electron model only reproduces the broad trends of the data. B o t h model and the sparse data are essentially flat at concen trations above two mole % ; the free electron model should not be used at lower concentrations. T h e magnitudes agree as well as for the pure solid metal. Since the thermal conductivity is limited b y the same scattering mechanism as the electrical conductivity we are also unable to make any calculation. Nevertheless, if a mean free path exists, then the Wiedemann-Franz law (19) must hold:

£ - 1 . where χ is the thermal conductivity and L

(3D = 2.45

0

X 10 ~

8

\vatt—ohm — deg.*

is the Lorenz number. E v e n i n pure metals, deviations from L occur though the value of κ/σΤ is usually close to L . Figure 7 shows χ/σΤ as a function of concentration for lithium-ammonia solutions (14,16). T h e measured values all are low though there is an upward trend with increas ing concentration. Error bars appropriate to the thermal conductivity used i n the calculation would be about ± 0.2 X 10 ~ watt Ω/deg. in width. Q

0

8

2


8.

THOMPSON

109


In pure metals the deviation from L is usually interpreted as evidence for inelastic scattering processes contributing to the resistivity. I n the absence of any valid calculation of the conductivity such a conclusion is hard to justify i n the present case. T h e optical data will not be discussed here. Beckman and Pitzer analyzed their reflectance data i n terms of thermally activated electrons; the H a l l effect data are inconsistent with this interpretation. A n effective mass could be introduced to bring their data into concert with the Drude (free electron) model, but would add little to our understanding. 0


2.5 Free Electron

Φ

TD

\ «·— σ

* 2.1 ε ο CO Ο 1.9 χ σ

or Ν

IT Li- Ν Η

σ

3

1.5

ε

Φ

"Ο

φ

* 13 Ι

Ι

+

I 4

I

6

I

I

•

I

I

I

8 10 12 14 16 Metal Concentration (mole per cent)

I

I

18

20

Figure 7. The Wiedemann-Franz ratio for solutions of lithium in ammonia at -33° C. The Lorenz number is 2.45 X 10~* watt Q/deg. 2

T h e futility of a n effective mass analysis, alluded to above, may be demonstrated by referring to the thermopower and susceptibility. Each of these quantities may be used to determine a n effective mass i f calcula tion and measurement are reconciled by varying m*/m. One finds that χ requires m*/m > 1, while e requires m*/m < 1. A more sophisticated approach is doubtless needed. Conclusions Simple models of electron behavior fail to describe quantitatively metal-ammonia solutions. Though the solutions show qualitatively the


SOLVATED ELECTRON

110


properties predicted by a free electron model and a simple scattering law, no single parameter has been found to bring quantitative agreement. T h e obvious, single, parameter—effective mass—shows opposite trends when adjusted i n a search for agreement. A n y prospective quantitative theory of the metallic state must include a proper treatment of the two (at least) types of bound electron levels present i n the nonmetallic state. I n addition, a proper theory of electron transport must be developed for situations wherein electronelectron interactions are important. Acknowledgment T h e author gratefully acknowledges the valuable conversations with many of his colleagues, including J . D . Gavenda, L . M . Falicov, D . S. Kyser, W . E . Millett, and C . M . Thompson. Literature

Cited

(1) Arnold, E., Patterson, Α., J. Chem. Phys. 41, 3089, 3098 (1964). (2) Beckman, Τ. Α., Pitzer, K . S., J. Phys. Chem. 65, 1527 (1961). (3) Cusack, Ν. E., Rept. Prog. Phys. 26, 361 (1963). (4) Dewald, J. F . , Lepoutre, G., J. Am. Chem. Soc. 76, 3369 (1954). (5) Huster, E., Ann. Physik 33, 477 (1938). (6) Hutchinson, C . Α., Pastor, R. C., J. Chem. Phys. 21, 1959 (1953). (7) Kraus, C. Α., J. Am. Chem. Soc. 43, 754 (1921); 43, 2533 (1921). (8) Kyser, D . S., Thompson, J. C., J. Am. Chem. Soc. 86, 4509 (1964). (9) Kyser, D . S., Thompson, J . C., J. Chem. Phys. 41, 1162 (1964). (10) MacDonald, D . K . C., "Thermoelectricity," p. 103, Wiley and Sons, New York, 1962. (11) Morgan, J . Α., private communication. (12) Pitzer, K . S., J. Am. Chem. Soc. 80, 5046 (1958). (13) Pohler, R. F . , Thompson, J . C., J. Chem. Phys. 40, 1449 (1964). (14) Schroeder, R. L . , private communication. (15) Schiff, L . I. "Quantum Mechanics," First ed., p. 168, McGraw-Hill, New York, 1949. (16) Varlashkin, P. G., Thompson, J . C., J. Chem. Phys. 38, 1974 (1963). (17) Yost, D . M., Russell, H . , "Systematic Inorganic Chemistry," p. 142, Pren tice Hall, Englewood, 1946. (18) Ziman, J . M., "Electrons and Phonons," Clarendon Press, Cambridge, 1960. (19) Ziman, J. M., "Principles of the Theory of Solids," Cambridge University Press, Cambridge, 1964. (20) Ziman, J. M., Phil. Mag. 6, 1013 (1961). R E C E I V E D May 3, 1965. A . Welch Foundation.

Supported by the Office of Naval Research and the R.


Conduction Processes in Concentrated Metal-Ammonia Solutions

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