LOWERING OF IONIZATION POTENTIAL AND CLUSTER IONS

139

Table 4.1 Estimates for critical parameters of cesium (Kraeft et al. 1985)

	Tc, K	pc, MPa	nec, cm−3
Experiment	1925	9.25		1021
Calculations by Redmer and R¨opke (1985)	2200		1.7 · 21
		22.0	1	· 1021
Calculations by Richert et al. (1984)	2600	180.0	3	· 10

4.3Lowering of ionization potential and cluster ions in weakly nonideal plasmas

In a moderately dense plasma, the interaction of electrons and ions with atoms and molecules results in two e ects, namely, lowering of the ionization potential and formation of molecular ions to be followed by heavier cluster ions. At moderate temperatures, these e ects show up most clearly in a metal plasma. Below, data on molecular ions of alkali metals are given and the ionization equilibrium in a multicomponent mixture is considered.

4.3.1Interaction between charged particles and neutrals

The first corrections to the free energy of the system, F , due to electron–atom and ion–atom interactions, can be calculated given Bae(T ) and Bai(T ), the virial coe cients of these interactions,

F = F0 − kT nenaBae − kT n+naBai.

Let us consider first the electron–atom interaction. The second virial coe - cient Bae(T ) should be calculated from the Bethe–Uhlenbeck formula (Landau and Lifshitz 1980),

B = (2√πλe)3			n	e−βEn + π		∞	l	(2l + 1) dE e−βE dE			= Bd + Bc, (4.25)
				1					dδl
						0

where λe = (2mkT )−1/2 is the electron thermal wavelength, δl is the partial phase of electron–atom scattering, and En is the energy of electron–atom bound states. The sums are taken over all states of the negative ion and over all scattering moments. A set of scattering phases enables one to determine the e ect of the interaction in the continuous spectrum while the bond energies provide the possibility of finding the second virial coe cient of the interaction in discrete spectra. Let us first discuss the interaction in the continuous spectrum.

Within the framework of scattering theory in an e ective radius approximation, the electron–atom interaction is characterized by only two parameters, namely, the scattering length L and atomic polarizability α (Smirnov 1968). If k is the electron wavenumber, then

δ0(k) −Lk − παk2/(3a0),

δl(k) παk2[(2l + 1)(2l + 3)(2l − 1)a0]−1, l 1.

By substituting δl(k) in Eq. (4.25) and performing the integration and summation over l, we obtain the virial coe cient of interaction in the continuous

140			WEAKLY IONIZED PLASMAS
Table 4.2 Length of electron scattering by inert gas atoms (in a0)
	He	Ne	Ar	Kr	Xe	Reference
	1.16	0.45	−1.63	−3.8	−6.8	Gus’kov et al. (1978)
	1.12	0.14	−1.40	−3.1	−5.7	Massey and Burhop (1969)

spectrum, Bc = −4πLλ2e, in a medium of molecules with a dipole moment, Bc = 2q(T ) (2mkT )−1/2, where q is the transport cross–section of electron– molecule scattering (Polishchuk 1985). This value should be substituted in the expression for the plasma free energy F = F0 − kT nenaBc, where F0 is the free energy of an ideal plasma. From this expression, Saha’s equation is derived with the potential of atomic ionization in the medium,

Im = I + ∆I, ∆I = 2π 2Lna/m.

The conditions of applicability of the e ective radius theory are

|L| λe, α/a0 λ2e .

This is the low–temperature region.

The lengths of electron scattering by inert gas atoms are listed in Table 4.2 and by atoms of alkali metals in Table 4.3.

Given the scattering length L of the order of ±a0 and T = 2000 K, we have |∆I| kT only at na 7·1021 cm−3. This can be a dense plasma of inert gas with an easily ionizable, such as alkali, seed. If L = −10a0, then |β∆I| 1 already at na 1020 cm−3 and T = 2000 K. Large absolute values of the scattering length are typical for atoms of alkali metals. However, the e ective radius theory is not applicable in this case for a number of other reasons. Therefore, one needs for calculations a set of experimental values of δl. We shall use the values calculated by Karule (1972) and Norcross (1974).

The density–temperature diagram for cesium shown in Fig. 4.5 represents the naBd = 1 and naBc = 1 curves. These curves separate the regions of strong and weak nonideality due to the electron–atom interaction in discrete and continuous spectra. At low temperatures Bd Bc and, with an increase of temperature, the di erence between them becomes smaller. It has been shown above that the contribution to the free energy by the term kT nenaBc can be interpreted as a shift of the continuous spectrum boundary, that is, as a reduction of the ionization potential. In turn, the interaction in the discrete spectrum can be taken into account by introducing negative ions into the plasma composition. Reduction of the ionization potential is a minor correction and becomes important at high densities.

The ion–atom interaction in the continuous spectrum can be regarded as classical. Therefore, we employ a quasiclassical approximation for scattering phases,

δl(E) =	M/ 2		dr[	E − V (r) − 2l2/M r2 −			E − 2l2	/M r2	], (4.26)

Fig. 4.5.

LOWERING OF IONIZATION POTENTIAL AND CLUSTER IONS

141

Table 4.3 Singlet, L+, and triplet, L−, length of electron scattering by atoms of alkali metals (in a0) (Karule 1972)

Scattering length	Li	Na	K		Cs
L+	3.65	4.23	0.45		4.04
L−	−5.66	−5.91	−15		−25.3

na, cm−3

1022

1021

5

4

3

1020

6

2

1

1019

1000

1500

2000

2500

T, K

Density–temperature diagram for cesium (Khrapak and Yakubov 1981): Curves naBc = 1 and naBd = 1 are for electron–atom (1, 2) and ion–atom (3, 4) interaction, respectively; 5 — vapor–liquid coexistence curve; 6 — isobar at 2 MPa.

where V (r) is the ion–atom interaction potential, E is the energy of colliding particles, and M is the ion mass equal to the atomic mass. By substituting Eq. (4.26) in Eq. (4.25), we replace summation by integration. The integration is performed in a classically accessible region where the expressions under the square roots in Eq. (4.26) are positive. In calculation, it is convenient to integrate first over l, and then over E. The result is (Hill 1956),

∞

Γ(3/2)

−

0

Bc = 4π

r2 dr

e−βV

Γ(3/2,

−βV )

1

,

(4.27)

∞

Γ(3/2) = √

Γ(3/2, −βV ) =

x1/2e−xdx,

π/2.

−βV

142	WEAKLY IONIZED PLASMAS

The obtained expression resembles the classical virial coe cient, but deviates from the latter by the factor with the gamma function. This di erence is due to the contribution of bound states, that is, the states of diatomic molecular ions that contribute to Bd. The expression for Bd reduces to the constant of dissociation equilibrium in the reaction A+2 A++ A,

√

Bd = (2 πλa)3(Σ+2 /ΣΣ+) exp(βD2+),

where Σ+2 , Σ, and Σ+ denote the internal partition functions of ion A+2 , atom A, and ion A+, respectively, and λa = (M kT )−1/2. Since small distances in the integral (4.27) yield no substantial contribution, one can use for V (r) the polarization potential. Then (Likal’ter 1969),

Bc = 4πC(αe2β/2)3/4, C 1.61.

Plotted in Fig. 4.5 are the naBc = 1 and naBd = 1 curves for the ion– atom interaction. The nonideality of cesium vapors in the case of the ion–atom interaction occurs earlier than in the case of the electron–atom interaction.

Thus, in a moderately dense plasma, the nonideality due to the charge– neutral interaction is reflected by the emergence of negative and positive molecular ions and by the lowering of the ionization potential by

∆I = 2π 2Lna/m − 4πCnakT (αe2/2kT )3/4.

(4.28)

4.3.2Molecular and cluster ions

Now we consider the region where naBd > 1. This means that the concentration of molecular ions A+2 exceeds that of atomic ions A+. In other words, the interaction between ions and atoms becomes strong. The heavier ions A+3 should be taken into account. One can expect that, with an increase of density or decrease of temperature, the concentrations of heavier ions will grow.

Data on the bond energy and structure of molecular and cluster ions has been accumulating intensively in recent years (Lagar’kov and Yakubov 1980; Smirnov 1983, 2000a, 2000b). Listed in Table 4.4 are the parameters of alkali diatomic ions A+2 and triatomic ions A+3 , dissociation energies D2+ and D3+ (for the A+2 → A+ + A and A+3 → A+2 + A processes), equilibrium internuclear distances Re and vibrational frequencies ωe, as suggested by Gogoleva et al. (1984). Table 4.5 presents the same quantities for inert gas ions.

Table 4.6 gives D2 (dissociation energy of molecules A2), D2+, D3+, and q (heat of vaporization of metals per atom). The most important is the fact that D2+ > D2. For instance, at T = 1500 K the ratio n+2 /n+ for sodium and cesium is an order of magnitude larger than n2/n. It is known that the concentrations of trimers and dimers in alkali metal vapors are low. In spite of this, trimer and dimer ions can play an important role in ionization equilibrium.

LOWERING OF IONIZATION POTENTIAL AND CLUSTER IONS

143

Table 4.4 Parameters of alkali ions A+, A+ (Gogoleva et al. 1984)
					2	3
	Parameter	Li+		Na+	K+	Rb+	Cs+
			2	2	2	2	2
	D2+, eV	1.28		1.02	0.84	0.77	0.66
	Re, a0	5.91		6.69	7.9	8.5	9.1
	ωe, cm−1	273		119	69	43	31
	Parameter	Li+		Na+	K+	Rb+	Cs+
			3	3	3	3	3
	D3+, eV	1.37		1.12	0.95	0.88	0.79
	Re, a0	5.67		6.48	7.94	8.76	9.40
	ωe, cm−1		239	104	60.6	37.5	27.5
			342	149	86.7	53.7	39.4

Table 4.5 Parameters of molecular ions of inert gases A+, A+
					2		3
	Parameter	He+	Ar+	Kr+	Xe+
		2	2	2	2
	D2+, eV	2.23	1.33	1.15	1.03
	Re, 10−8 cm	1.08	2.50	2.75	3.22
	ωe, cm−1	1627	300	160	123
	Parameter	He+	Ar+	Kr+	Xe+
		3	3	3	3
	D3+, eV	0.17	0.20	0.27	0.27
	Re, 10−8 cm —		2.67	2.97	3.47

The inequality D2+ > D2 implies also that the ionization energy I (A2) of a

molecule is less than that of an atom. Indeed, from the energy conservation in the Born cycle, A2 → A+2 + e → A+ + A + e → A + A → A2, it follows that

I(A2) = I(A) − (D2+ − D2).

Another important conclusion that follows from Table 4.6 is that the values of D2+, D3+, and q are close to each other. Hence, one can consider the separation energy of an atom and heavy ion as almost constant and, in the first approximation, suppose that the value of Dm+ for heavier ions is known.

Very important information about heavy ions A+m was provided by experiments where ions A+m emerged upon passage of a molecular beam through an ionization chamber. Figure 4.6 shows the facility sketch by Forster et al. (1969).

Table 4.6 Dissociation energy and heat of vaporization, eV (Khrapak and Yakubov 1981)

Atom	D2	D+	D+	q
		2	3
Li	1.03	1.28	1.37	1.69
Na	0.73	1.02	1.12	1.13

K0.514 0.84 0.95 0.87

Rb	0.49	0.77	0.88	0.78
Cs	0.45	0.66	0.79	0.74

144	WEAKLY IONIZED PLASMAS

4

3

1 2

5

7 7 6

8	7	9

Fig. 4.6. Experimental setup of Forster et al. (1969): 1 — boiler; 2 — heater; 3 — xenon lamp; 4 — monochromator; 5 — ionization chamber; 6 — trap; 7 — evacuation; 8 — liquid metal; 9 — magnetic field region.

Boiler 1 was loaded with liquid metal 8. A source of metal vapors was provided by a heater. The molecular beam was generated as a result of molecular e usion through a hole of small diameter. The beam was collimated with the aid of diaphragms and entered the ionization chamber 5. A high–pressure xenon lamp 3 served as a UV–radiation source. By varying the wavelength of radiation coming from a monochromator 4, one could measure the photoionization thresholds. The emerging ions were pulled out to a mass-spectrometer 9. Herrmann et al. (1978) performed the experiments up to m = 14 for sodium, and Kappes et al. (1986) up to m = 66 for sodium, and up to m = 34 for potassium. The results are listed in Table 4.7. Similar data were obtained for clusters of lead, iron, nickel, and aluminum (see the list of references in Rademann et al. 1987).

Let us now discuss the dependence of the measured ionization potentials Im on the number of particles m in a complex at high m. In the case of m → ∞, the ionization potential must tend to the electron work function for a macroscopic metal sample. A good description of the measurements is provided by a simple macroscopic model. One can assume that a heavy ion is a metallic droplet with a radius R, the value of which depends on the number of atoms in the droplet, m. Then, the ionization potential Im is the electron work function for a drop with a finite radius I(R). It can be related to the electron work function for a plane surface, I(∞) (in the limit R → ∞), by the following relation:

I(R) = I(∞) + e2/2R.

(4.29)

Plotted in Fig. 4.7 are the experimental and calculated dependencies Im for sodium. The relation R = Rmm1/3 between the drop radius and the number

LOWERING OF IONIZATION POTENTIAL AND CLUSTER IONS

145

Table 4.7 Photoionization thresholds for sodium and potassium clusters, eV (Forster et al. 1969; Herrmann et al. 1978; Kappes et al. 1986)

Cluster	Im	Cluster	Im	Cluster	Im	Cluster	Im
Na	5.15	Na11	3.8 ± 0.1	K	4.34	K15	3.21 ± 0.10
Na2	4.934 ± 0.01	Na12	3.6 ± 0.1	K2	4.05 ± 0.05	K20	3.23 ± 0.10
Na3	3.97 ± 0.05	Na13	3.6 ± 0.1 K3		3.3 ± 0.1	K28	3.05 ± 0.10
Na4	4.97 ± 0.05	Na14	3.5 ± 0.1 K4		3.6 ± 0.1	K34	3.01 ± 0.10
Na5	4.5 ± 0.05	Na15	3.7 ± 0.1 K5		3.4 ± 0.1
Na6	4.12 ± 0.05	Na19	3.6 ± 0.1 K6		3.44 ± 0.10
Na7	4.04 ± 0.05	Na35	3.4 ± 0.1 K7		3.40 ± 0.10
Na8	4.10 ± 0.05	Na45	3.3 ± 0.1 K8		3.49 ± 0.10
Na9	4.0 ± 0.1	Na55	3.3 ± 0.1 K9		3.40 ± 0.10
Na10	3.9 ± 0.1	Na65	3.2 ± 0.1	K10	3.27 ± 0.10

Im, Em, eV

5

4

1

3

2

2

1

0

2

4

6

8

10

12

m

Fig. 4.7. Ionization potential Im (1) and electron a nity Em (2) of sodium clusters Nam :

•— experiment Nam; ◦ — Na− and Na−2 ; curves are calculations with Eqs. (4.29) and (4.30).

of particles was employed for this, where Rm = Rinf exp[−2kγ/(3Rmm1/3)] and Rinf is the atomic radius in a macroscopic metal sample, and k and γ denote the macroscopic compressibility and surface tension, respectively. Inclusion of the surface tension somewhat reduces the value of Rm for small clusters.

The experimental results reveal good stability of positively charged heavy cluster ions and provide some information on their structure, which apparently approaches the structure of liquid metals.

At low m, considerable fluctuations are observed in the dependence Im. These fluctuations are well described by the results of rather complicated quantum– mechanical calculations performed by a number of authors.