Scholarly article on topic 'Continuum lowering – A new perspective'

Continuum lowering – A new perspective Academic research paper on "Physical sciences"

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Abstract of research paper on Physical sciences, author of scientific article — B.J.B. Crowley

Abstract What is meant by continuum lowering and ionization potential depression (IPD) in a Coulomb system depends very much upon precisely what question is being asked. It is shown that equilibrium (equation of state) phenomena and non-equilibrium dynamical processes like photoionization are characterized by different values of the IPD. In the former, the ionization potential of an atom embedded in matter is the difference in the free energy of the many-body system between states of thermodynamic equilibrium differing by the ionization state of just one atom. Typically, this energy is less than that required to ionize the same atom in vacuo. Probably, the best known example of this is the IPD given by Stewart and Pyatt (SP). However, it is a common misconception that this formula should apply directly to the energy of a photon causing photoionization, since this is a local adiabatic process that occurs in the absence of a response from the surrounding plasma. To achieve the prescribed final equilibrium state, in general, additional energy, in the form of heat and work, is transferred between the atom and its surroundings. This additional relaxation energy is sufficient to explain the discrepancy between recent spectroscopic measurements of IPD in dense plasmas and the predictions of the SP formula. This paper provides a detailed account of an analytical approach, based on SP, to calculating thermodynamic and spectroscopic (adiabatic) IPDs in multicomponent Coulomb systems of arbitrary coupling strength with T e ≠ T i. The ramifications for equilibrium Coulomb systems are examined in order to elucidate the roles of the various forms of the IPD and any possible connection with the plasma microfield. The formulation embodies an analytical equation of state (EoS) that is thermodynamically self-consistent, provided that the bound and free electrons are dynamically separable, meaning that the system is not undergoing pressure ionization. Apart from this restriction, the model is applicable in all coupling regimes. The Saha equation, which is generally considered to apply to weakly-coupled non-pressure-ionizing systems, is found to depend on the Thermodynamic IPD (TIPD), a form of the IPD which takes account of entropy changes. The average Static Continuum Lowering (SCL) of SP relates to changes in potential energy alone and features in EoS formulas that depend on the variation of the mean ionization state with respect to changes in volume or temperature. Of the various proposed formulas, the Spectroscopic (adiabatic) IPD (SIPD) gives the most consistent agreement with spectroscopic measurements.

Academic research paper on topic "Continuum lowering – A new perspective"

High Energy Density Physics xxx (2014) 1—20

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Contents lists available at ScienceDirect

High Energy Density Physics

journal homepage: www.elsevier.com/locate/hedp

Continuum lowering — A new perspective

Q3 B.J.B. Crowley a,b, *

a Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK b AWE PLC, Reading RG7 4PR, UK

ARTICLE INFO

Article history: Received 9 December 2013 Received in revised form 25 March 2014 Accepted 16 April 2014 Available online xxx

52.25.-b

52.25.Jm

52.25.Kn

32.80.-t

32.80.Fb

Other relevant categories:

05.30.-d

05.30.Fk

05.70.-a

05.70.Ce

32.10.Hq

52.27.Aj

52.27.Cm

64.10.+h

71.10.-w

71.10.Ca

Keywords:

Coulomb systems

Ionization potential depression

Continuum lowering

Photoionization

Plasma equation of state

ABSTRACT

What is meant by continuum lowering and ionization potential depression (IPD) in a Coulomb system depends very much upon precisely what question is being asked. It is shown that equilibrium (equation of state) phenomena and non-equilibrium dynamical processes like photoionization are characterized by different values of the IPD. In the former, the ionization potential of an atom embedded in matter is the difference in the free energy of the many-body system between states of thermodynamic equilibrium differing by the ionization state of just one atom. Typically, this energy is less than that required to ionize the same atom in vacuo. Probably, the best known example of this is the IPD given by Stewart and Pyatt (SP). However, it is a common misconception that this formula should apply directly to the energy of a photon causing photoionization, since this is a local adiabatic process that occurs in the absence of a response from the surrounding plasma. To achieve the prescribed final equilibrium state, in general, additional energy, in the form of heat and work, is transferred between the atom and its surroundings. This additional relaxation energy is sufficient to explain the discrepancy between recent spectroscopic measurements of IPD in dense plasmas and the predictions of the SP formula. This paper provides a detailed account of an analytical approach, based on SP, to calculate thermodynamic and spectroscopic (adiabatic) IPDs in multicomponent Coulomb systems of arbitrary coupling strength with Te s Ti. The ramifications for equilibrium Coulomb systems are examined in order to elucidate the roles of the various forms of the IPD and any possible connection with the plasma microfield. The formulation embodies an analytical equation of state (EoS) that is thermodynamically self-consistent, provided that the bound and free electrons are dynamically separable, meaning that the system is not undergoing pressure ionization. Apart from this restriction, the model is applicable in all coupling regimes. The Saha equation, which is generally considered to apply to weakly-coupled non-pressure-ionizing systems, is found to depend on the Thermodynamic IPD (TIPD), a form of the IPD which takes account of entropy changes. The average Static Continuum Lowering (SCL) of SP relates to changes in potential energy alone and features in EoS formulas that depend on the variation of the mean ionization state with respect to changes in volume or temperature. Of the various proposed formulas, the Spectroscopic (adiabatic) IPD (SIPD) gives the most consistent agreement with spectroscopic measurements.

© 2014 Published by Elsevier B.V.

1. Introduction

1.1. Background

The fact that electrons bound to atoms in plasmas and metals require less energy to liberate them into the continuum than from equivalent states in isolated atoms was, until recently, generally thought to be reasonably well understood to the extent that it could be described in terms of a simple model, despite a lack of sound

* Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK. E-mail address: basil.crowley@physics.ox.ac.uk.

http://dx.doi.org/10.1016/j.hedp.2014.04.003 1574-1818/® 2014 Published by Elsevier B.V.

experimental validation of any such model. Direct spectroscopic observation of ionization potential depression, or continuum lowering as it is sometimes called, is generally frustrated by the Inglis—Teller effect [1] whereby the "true" bound-free edge is obscured through becoming merged with nearby bound—bound transitions. Indirect methods have generally been too imprecise to discriminate between possible alternative models.

Interest in the phenomenon has been revived by some recent spectroscopic measurements [2—4] exploiting new facilities, of dense plasmas, that claim to have circumvented the Inglis—Teller effect to yield good quantitative data. However, rather than confirming the generally accepted thinking, as embodied in the well-known Stewart—Pyatt model [5], for example, they have exposed

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inconsistencies and deficiencies in some well-established current models, and thereby in prior understanding of this phenomenon, while raising deeper questions about the underlying concepts.

In one type of experiment [2,3], a tuneable X-ray laser (FEL) is used to ionize the K-shell in solid-state aluminium. Whether ionization occurs or not is a direct function of the laser energy and is diagnosed by measuring the subsequent Ka emission. The experiment is thus a clean measurement of the spectroscopic ionization potential that does not depend on any underlying model of the subject system. The results of this experiment are illustrated in Fig. 1, in which the observed ionization depression for various ionization states of aluminium is compared with different theoretical predictions. It turns out that the results of this experiment significantly disagree with the predictions of Stewart and Pyatt [5] and are best described by an old model proposed by Ecker and Kroll [6]. This conclusion has raised concerns that the hitherto widely favored model of Stewart and Pyatt is at fault raising concerns over the validity of the large amount of data derived using it.

In another recent experiment [4] spectroscopic measurements are carried out on laser-shocked Aluminium and the presence or absence of the 1—3 lines as a function of temperature and density used as a diagnostic of the continuum lowering. The results of this experiment, and comparisons with various theories, are given in Table 2. While the interpretation of this experiment does depend, to some extent, on modeling of the in situ n = 3 atomic levels to represent the effect of the various continuum-lowering models, the results appear conclusive and are consistent with a simple ion-sphere model, which is much closer to Stewart and Pyatt than Ecker and Kroll.

Both experiments claim to be able to discriminate between different models of the ionization potential depression with the FEL direct ionization measurement apparently supporting Ecker and Kroll while the laser driven shock measurements are presented as being more consistent with Stewart and Pyatt. Neither model is capable of fitting both experiments.

The Stewart—Pyatt has the virtue of possessing a physics-based derivation, albeit a far from exact one, and incorporates the ion-sphere and Debye—Huckel models in its limits. Simple alternatives, such as Ecker—Kroll, are more ad hoc in nature, and/or are of more limited validity, so it is logical that Stewart—Pyatt should carry favor over them. So why experiment should take a contrary view and, in certain circumstances, favor a less well justifiable alternative models seems difficult to understand. Ecker—Kroll depends upon an ad hoc assumption, which, even in hindsight, remains unsupported. The application of the ion-sphere model to the

Table 1

Values of the force constant C for various lattices.

Ion sphere fcc/hcp bcc Sc

9/10 0.99025 1.01875 1.09189

laser driven shock experiment does not appear to be justified either, due to the ion coupling being insufficiently strong. Moreover, since all of these models, Ecker—Kroll, Stewart—Pyatt and ion-sphere, claim to model the same thing, any inconsistencies are indicative only of deficiencies in one or more of them. Which model should be used is certainly not a matter of arbitrary choice or preference. While it may be that, of the various models considered, only Stewart—Pyatt appears to be rationally supportable, it is undeniable that both sets of experiments clearly demonstrate that the spectroscopically-determined ionization potential depression in dense matter is significantly greater than that predicted by this model.

This is unfortunate. It is not just that a simple formula, like Stewart and Pyatt's, is too useful to discard lightly. While it is true that a detailed atomic physics calculation, using a many-body implementation of density functional theory, for example, that captures the essential physics, might be expected to reproduce observational data, this is not always feasible. This capability is recent and, even now, not all plasma regimes are accessible to such calculations. The formula is incorporated or is implicit in many atomic physics codes still in use or which have been sources of currently available atomic data. So the failure of experiment to support this model is of considerable concern and raises two immediate questions: What is wrong with the model? and Can it be fixed?

This is our starting position. A first step is to review the basis of the Stewart—Pyatt and closely-related formulas to ascertain why they may not yield the results expected of them. Theoretical treatments of continuum lowering typically approach the problem from the point of view of thermodynamic equilibrium. It is true that neither of the experiments is characterized by full thermodynamic equilibrium, but this does not in itself offer a satisfactory or useful explanation for the discrepancies. Continuum lowering features in non-equilibrium situations. In strongly-coupled plasmas, it is largely determined by the potential energy, which is dependent on the spatial configuration of the system independently of whether the system is in thermal equilibrium. Nevertheless, it is the presumed connection with equilibrium that turns out to be very much at the heart of the matter.

Fig. 1. Calculations of the ionization potential depression for various ion charge states in solid density aluminium compared with the measurements of Ciricosta et al. [2]. Stewart— Pyatt is equation (12), Ecker—Kröll is (169) and "This Work" is equation (81).

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B.J.B. Crowley / High Energy Density Physics xxx (2014) 1—20 3

1 Table 2 66

2 Results from the experimental observations of shocked aluminium plasmas at Orion compared with various models for the putative plasma conditions given in Ref. [4]. 67

3 Plasma state Experiment Model predictions for presence of 1 —3 lines. 68

4 - - --69

Density (g/cc) Temperature (eV) 1—3 lines Observed? This work Eq. (81) Ion sphere Eq. (170) Stewart—Pyatt Eq. (14) Ecker—Kroll Eq. (169)

6 1.2 550 Yes Yes Yes Yes Yes 71 _ 2.5 650 Yes Yes Yes Yes No

7 4.0 700 Yes Yes Yes Yes No 72

8 5.5 550 Yes Yes (Lyb) Yes No No 73

9 9.0 700 No No Marginal Yes No 74

10 11.6 700 NA No No No No 75

13 The modeling of plasmas in equilibrium typically treats ioniza- However, since the application of any electric field to an atom 78

14 tion as a quasi-static transition between states of thermodynamic seemingly lowers the ionization potential irrespective of the di- 79

15 equilibrium. In equations that model equilibrium, such as the Saha rection of the field, it would appear that the average ionization 80

16 equation or the Gibbs distribution of the Canonical Ensemble, the potential should be lowered by the microfield. That is to say, for 81

17 continuum lowering appears as a correction to the free energy. We adiabatic processes, the microfield has the potential to increase the 82

18 shall refer to this as the thermodynamic ionization potential spectroscopic ionization potential depression relative to the ther- 83

19 depression (TIPD). Stewart and Pyatt, however, in the derivation of modynamic value. This question is examined in detail in Section 6.2 84

20 their formula, consider the effect in terms of the average electro- where it is concluded that this argument is spurious and that the 85

21 static potential experienced by the electrons in an atom, or, microfield effects are distinct from the continuum lowering and 86

22 equivalently, the self-energy of the ion-electron system, which is should be treated separately. 87

23 consistent with the approach taken by average-atom models, while In the following, we examine these assertions from the general 88

24 disregarding the effect of fluctuations that would be associated perspective of a Coulomb system in which the ions behave classi- 89

25 with the entropy term in the free energy. We refer to the depression cally. As a framework for this, we use the static continuum- 90

26 of the ionization potential in the average electrostatic potential as lowering model developed previously by the author. This re- 91

27 the static continuum lowering (SCL). However spectroscopy probes produces the Stewart—Pyatt model while incorporating the effect 92

28 dynamical process occurring between plasma microstates, in which of a non-uniform free-electron distribution induced by their po- 93

29 the changes of state of individual electrons/atoms are observed on larization in the field of the ion. For fast (adiabatic) processes, an 94

30 timescales that might not allow a response from the surrounding additional term is postulated, representing the subsequent relax- 95

31 plasma to each individual transition. ation energy that needs to be subtracted out. In conjunction with 96

32 In the linear regime of single-photon interactions, the active the SCL, this yields a more correct form of the spectroscopic IPD 97

33 electron remains close to the atom during the spectroscopic pro- (SIPD), one which provides reasonably good agreement with both 98

34 cess, specifically within the range of its initial wavefunction. The sets of experiments. 99

35 electron hole created by the ionization process, which occurs on a 100

36 timescale > 1/u where u is the photon frequency, does not become 2. Theory 101

37 visible to the surrounding plasma until the electron moves a dis- 102

38 tance comparable to the scale length of the plasma (such as may be 2.1. The ionization potential and static configuration lowering 103

39 represented by the Debye length or the mean ionic separation 104

40 distance). This happens after the spectroscopic interaction has We consider an electrically neutral plasma comprising a 105

41 occurred on timescales determined by the inverse of the electron Coulomb system of electrons and atomic ions in thermodynamic 106

42 plasma frequency. The response of the ions is much slower still, equilibrium. The ions comprise a fixed number Ni of immutable 107

43 occurring on timescales determined by the inverse of the ion atomic nuclei and variable numbers of bound electrons, which can 108

44 plasma frequency. Spectroscopic observations therefore see atomic be exchanged with the surrounding plasma. For added generality 109

45 transitions as being effectively uncorrelated with changes in the (and in order to model the experiments) the ions and electrons 110

46 plasma microstate. No energy, in the form of either heat or work, is (including bound electrons) are considered to have different tem- 111

47 exchanged with the surrounding plasma during the spectroscopic peratures, Ti and Te respectively. (Such a temperature separation 112

48 process itself, the only energy exchange being that between the can be expected occur when the ion and electron subsystems are 113

49 atom and the probe photon. Such a process is adiabatic, in the sense weakly coupled on the dynamical timescales controlling the ioni- 114

50 of preserving entropy, as well as occurring at constant volume. The zation, to which a contributory factor is the smallness of the elec- 115

51 energy of the ionizing photon depends only upon its ability to tron mass compared to the masses of the ions.) We start by defining 116

52 excite a bound electron into a continuum state, one in which the the ionization potential of a plasma in equilibrium to be the total 117

53 electron is able to migrate away from the ion, while the sur- energy required to change the charge state of a single ion in the 118

54 rounding plasma does not suffer any immediate change. This con- plasma, through excitation of an electron from a bound state in that 119

55 tinuum threshold may differ from that which applies after the ion to the plasma continuum while requiring that the electronic 120

56 plasma has relaxed in response to the changed charge state of the chemical potential me and the temperature(s) Te, Ti of the plasma 121

57 ion. The continuum lowering seen in spectroscopic measurements should be maintained precisely at their initial equilibrium values. 122

58 is therefore not the same as either the SCL or TIPD, a point which We shall presume, at this stage, that we know what is meant by a 123

59 seems to have originally been made by Ecker and Weizel in 1956 [7] "bound state" and the "continuum". Because ionizations are 124

60 and reiterated by Ecker and Kroll [6]. considered as occurring one ion at a time, individual ionization 125

61 The difference between the plasma environment for a micro- processes are conceived as being dynamically independent of each 126

62 state of the plasma and that due to the fully (space and time) other, and, importantly, that each process does not directly influ- 127

63 averaged equilibrium state is commonly known as the microfield ence the equilibrium ionization state of other ions, or of the plasma 128

64 [8]. As the microfield represents departures from the idealized as a whole, which would be the case if me and Te were allowed to 129

65 equilibrium state, it is, by definition, zero for the equilibrium state. change. The ionization process is thereby considered as a quasi- 130

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static transition between two equilibrium states of the plasma during which the plasma is considered to be in contact with the appropriate heat baths (at temperatures Te, Ti) and electron reservoir (at chemical potential me). Maintaining me and Te, for fixed numbers of the nuclear species, is equivalent to maintaining ne and Te, where ne = Zni is the free-electron density. It will be shown that these constraints are equivalent to considering the ionization process to be occurring, in the closed system, at constant pressure and temperature. These conditions are therefore sufficient for the process to maintain thermal and mechanical equilibrium and hence be considered to be quasi-static.

The ionization potential defined this way is the total energy change of the plasma given by

DEja - fja + DUj

where fja > 0 is the ionization potential of the electron state a in an isolated ion, j; e = e(me, Te) is the mean (kinetic) energy of a free electron in the surrounding plasma (which appears by virtue of Te being maintained) and DUja < 0 is the contribution to the potential energy of the bound electron from the surrounding plasma, taken as the static average. For sufficiently localized, ie deeply bound, states DUja is independent of the state, a and so DUja = DUj, which is the static continuum lowering.

The static continuum lowering is that which is associated with the static average-atom potential as seen by a test charge and is what is generally considered to be given by the Stewart—Pyatt formula [5] and the limiting (ion-sphere and Debye—Huckel) forms in their respective regimes of applicability, which is in accord with the derivations (see also Ref. [9]). In reality, this potential is modified by the microfield, which represents the spatial- and time-dependent fluctuations, around the average, of the electrostatic field experienced by individual electrons. A consequence of this is that some electronic states that are bound in the static average potential may, when subject to the microfield, not be bound to a single ion but rather exist in transient localized states.

It is significant perhaps that these are the same states that persistently oscillate between the continuum and the bound levels during explicit iteration of an average-atom calculation. Calculations that force convergence by placing these states in the continuum (coarse convergence) define a different continuum to those that determine the true static potential by carefully controlling the convergence process (fine convergence), while it is possible that some detailed configuration accounting (DCA) calculations include the microfield at the outset. These factors need to be borne in mind when comparing calculations between different codes since they determine where these codes place the continuum as well as influencing how well they might agree with experiments.

In a formal many-body theory description, the total potential energy associated with an ion is usually referred to as its self-energy, for which formal expressions are provided in terms of response functions [10]. Let the ionization process be considered to be a change of state of an ion-electron system embedded in the plasma and let the initial self-energy of the bound electron-ion system, considered to be effectively at rest, be S0. In the final equilibrium state, in which the emitted electron is absorbed by the plasma so as to maintain a constant electron density and temperature, neglecting any change in motion of the ion, the total energy is S0 + DS + fja + e in which DS is the change in the self-energy of the ion due to the quasi-static response of the plasma to the change of the charge state of the ion. Comparison with (1) shows that DU = DS so that the static continuum lowering is synonymous with the change in the self-energy of the ion during the complete ionization process. This connection with formal many-body theory will be revisited in Section 6.

As will be shown, the static continuum lowering provides a complete description of the physical ionization depression only in the strongly-coupled limit (high densities, low temperatures), which is also when ion microfield fluctuations become negligible. At finite coupling strengths, where the interaction energy depends upon temperature, the role of entropy needs to be considered as well.

2.2. Static continuum lowering

A suitably general reference model of the static continuum lowering in a multicomponent Coulomb system is that presented elsewhere by this author [9]. This uses the same ion pair correlation function gj(r) incorporated in Stewart and Pyatt's method, where, for some rj, which we call the ion core radius

gj(r) = -1 (r < j p.

gj(r) = --j exp ((r - r) /D) (r > j

where D is the total plasma screening length, while allowing for electron screening, which is the polarization of the electron density in the electrostatic potential of the ion. The static continuum lowering provided by this model is represented by the following formula [9]:

DU = j ZpXj

3 kTi G X2 - 2 ~ZpP

a2 - 1

in which the first term is due to the ions and the second term gives the electron contribution, and where

£Z2 2_

is the effective plasma perturber charge, which turns out to be essentially the same as z* as originally defined by Stewart and Pyatt [5], which note is a property of the plasma as a whole; Zj is the charge state of a particular ion j; Xj is the positive real root of

X3 + 3aXjYi + 3XjYj2 - 1

which yields the ion core radius

in terms of the ion-sphere radius

Rj = (4ZL

j * 4m,

and a is a number >1, which represents the screening effect of the electrons according to

a = ID

which is the ratio of the ion screening length Di that is deemed applicable in the regime r > rj, to the plasma screening length, D; and

Y - Rj

g = ZjZPe2 j 4pe0RjkTi

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The ion screening length is taken to be given generally by the classical Debye formula

Zpnee2 e0kTi

in terms of which the total screening length is given, in the Thomas—Fermi approximation, by

1 1 nee2 ¡1 =2 (ße/kTe)

D2 = D + D22 = D2 + ë0kTe ¡1 /2(ße/kTe)

in which ¡j(x) = / y/1 + exp(y

x)dy denotes the Fermi func-

tion, and Ij its derivative, where, for sake of argument, the free-electron screening is taken to be given by the finite-temperature Thomas—Fermi model.

The ion core radius rj defines the radius of the core region that characterizes the local environment of the ion j, within which, according to (2), there are no other ions. In this region, the electrons are dominated by the strong central field of the ion. The region Rj >r > rj is an intermediate region containing both ions and electrons in which the correlations with the central ion are respected according to the second part of (2). The region r [ Rj is the external "collective" region occupied by the rest of the plasma, within which no individual ion is considered to have a dominant influence.

Equation (9) introduces the ion coupling parameter Tj, which, in the form given, is a measure of the relative strength of the electrostatic potential energy of the ion to its thermal kinetic energy. In the strong-coupling limit, Tj [ 1, which implies rj ~ Rj; while, in the weak-coupling limit, Tj < 1 implying rj ~ e2ZjZp/4pe0kTi, which corresponds to the Landau length for a plasma perturber ion in the vicinity of the subject ion. Equation (5) expresses the condition that the derivative of the potential (the radial electric field) is continuous at r = rj. Conceptually, rj corresponds to the separation radius of Ecker and Kroll [6] who give a different, overtly ad hoc, formula for it, and the core radius of Stewart and Pyatt [5], whose treatment is basically similar to the above.

With the aid of (5), the formula (3) can be rendered in the much more elegant form

kTL 2Zp

(1 + j2/3 - 1

L = XL Lj = Y3 = (3Gj)

Xj = a( 1 + (a2 - 1)X3) >

In the special case when a = 1, which applies when the electrons are uniformly distributed, the above reduces to Stewart and Pyatt's well-known analytical formula [5,9,11,12]

DUSP = JZ- ((1 + L)2/3 - 1)

which, note, depends on the properties of the ions alone. (Note also that here it is Zp that appears in the denominator, rather than Zp + 1,

as in Stewart and Pyatt's original formula [5].1) Equation (12) is however applicable to multicomponent plasmas, in which electron screening can also contribute to the continuum lowering. In the strong-coupling limit (Tj [ 1 0 Lj [ 1, Xj = 1, X = a3)

3 Z,e2

TkTi~ 2 4pe0RjkTi~ 2 4pe0Rj 'ykTi ' ZpkTB where

T TI1/2(me/kTe)

Tb = TeI1 /^(me/kTe) (16)

If the electrons are non-degenerate and Te = Ti = T, this yields

3 (Zp +1)

4pe0R,kT

Note well that the ion-sphere radius is distinct from the Wigner— Seitz radius

y4pni y to which it is related by

fZj\ 1/3

Rj = w Rws

The former is a property of the plasma, while the latter is a property of an individual ion. In terms of the Wigner—Seitz radius, equation (17) is

_3 (Zp + 1 Z2/3Z1/3u0

DU, = 2 Zp Zj Z "WS

which expresses the different scalings with respect to Zp = (Z2 )/{Z), Z = <Z> and Zj, and where

4pe0Rws

is the Wigner—Seitz energy, which is defined independently of the ion charges.

In the weak-coupling limit (Tj < 1 0 Lj < 1, Xj < 1, X = a) equation (12) gives

kT. 7p2

DUX - 3Zp Xj - a 4jïïi

Zje2 4pe0D

-ZjUDH (22)

which is the Debye limit of the static continuum lowering, with the electronic screening included and where

DH 4pe0D is the Debye—Hückel electrostatic interaction energy.

1 This is due to an inconsistency in Stewart and Pyatt's argument in relation to the assumption of uniformly distributed free electrons. The origin of the Zp + 1 in Stewart and Pyatt's formula is the embedded relation between the classical Debye lengths, Zp + 1 = ZpD2 =D2 = Zp (1 + D2/Dj), which does not apply in this case. See also Section 2.3.

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2.3. Electrostatic potential energy

Let us proceed by first considering the above in situations when the electron screening is negligible compared to the effect of the ions, ie when a = 1. In this case the electrostatic potential energy, or self-energy, of the ion is given by

j "¿h<4>

h(A) = (1 + A)2/3 - 1

For fixed Zj, Zp

ne( VL

-wiTt = 2 3

while, for fixed Zp, ne, Ti, which are regarded as properties of the surrounding plasma, the derivative with respect to the ion charge is

The static continuum lowering represents the change in the electrostatic potential energy per unit charge d = 1, of the whole plasma, when a single ion undergoes a transition Zf / (Zj + d)+ + de-, without affecting Zp, ne, Ti. So, if the total Coulomb energy U is assumed to be given to be in the form

wr X Zjg<Aj)

the static continuum lowering is yielded as

~kTi = -2Zp 9Zjyjvv" 2Zp

9 <Zjg<Aj)) =-2r <g<Aj) + Ajg0<Aj))

Hence, upon comparing with (24)

h(L) = ^ (Lg(L))

which can be integrated to yield

g(A) = A j h(1)d1 = 1 g ((1 + A)5/3 - 1) - A 0

Combining equations (27) and (30) yields the electrostatic potential energy as

-S XzA [3 (<1 + Aj)5=3 - 0 - A,

which is the Coulomb energy in the Stewart—Pyatt (aka Generalized Ion Cell [9,12,13]) approximation, and which yields U < 0 for all possible Zj > 0, Zp > 0. In the strong-coupling limit,

G [ 1 0 L [ 1, g(L) ~ 3/5L2/3 = 9/5G and so

TwTis= * kTi x j Gj = X

is 10 1 j 10 4

' 4pe0R,

which is recognized as the Coulomb energy in the well-known ion-sphere approximation [14].

In the weak-coupling limit, G < 1 0 A < 1, g(A,) ~ 1/3A, = GjR,/ Di and so

T~Tdh = -1 X

4pe0Di

i 8pe0Di

which is the Debye—Hückel electrostatic energy due to the ions [10]. We observe that the electrostatic energy (31) generally satisfies both the Lieb—Narnhofer [15,16], (T > Tis) and Mermin [17] (T > TDH) bounds. In terms of the elementary Wigner—Seitz and Debye—Hückel energies, (21) and (23) respectively

Tis = -10 Ni(z5/3) (Z)1=3Uws

Tdh = -nà(z2) udh

which are indicative of the different dependences on the chargestate distribution (CSD).

The Stewart—Pyatt and Generalized Ion Cell models are presented as being applicable, at some level of approximation, in regimes of arbitrary plasma coupling. Importantly, equation (27) can be generalized to accommodate different, potentially more accurate, parameterizations of the potential energy, examples of which are the fits to hypernetted chain (HNC) and Monte-Carlo calculations of the one-component plasma (OCP) fluid [18] as well as pa-rameterizations more applicable to metallic solids.

According to the virial theorem, the pressure contribution from the Coulomb energy is 1/3 U/V.

2.4. Generalizations

We now extend above model to account for the electron contribution to the self-energy, which is the result of screening due to polarization of the electrons in the monopole electric field of the ion. We are able to do this, while also taking account of different possible geometric arrangements of the ions, as in a crystal lattice. The continuum lowering, with electron polarization included, is given by (12), in which, in the limit of strong coupling, Xj ~ a3 while, in the weak-coupling limit, Xj ~ a, neither of which, it should be noted, depend upon the properties of the individual ion. In general

9a, Z—j Z 9Z,

= ~ 1 +

'Xj 9 j

= a, + Aj

- ^ ZjvZj (35)

in which, according to (8)

a2 - != DLJL

" 1 D2 ZpTB

which is, in highly ionized and/or degenerate plasmas, typically small. Solving (5) forXj with a = 1 yieldsXj = (1 + x)1/3 - x1/3 where x = 1/Aj, from which the logarithmic derivative with respect to Zj can be calculated according to

9X3 _ Z 9j 9L 9_± = _x

j 9Z, j 9x 9 A, 9Z, 9x

y(1 - y)2(1 + y)

1 + y + y2

where use has been made of (26) and where

y= (râ)1/3 = <1 + Aj)-1/3

The expression on the right hand side of (37) has a maximum value of 0.137521 in the range 0 < y < 1 corresponding to y — 0.306

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(which corresponds to Lj = 33.9 and Gj = 3.49). Therefore, since

Xj > 1

/ -, \ Zi VA i 1 + 0.13752Ía2 - 1) >-J- > 1 V ) A- VZ-

The approximation

A- VZ-

is therefore reasonable in virtually all circumstances and means that the electronic contribution to the static continuum lowering and the Coulomb energy can be represented by replacing Lj by Lj as defined by (13) in which Xj is regarded as possessing a negligible derivative with respect to Zj. Hence, as well as the continuum lowering being given by (12), the total electrostatic energy given by (27) becomes

Ez-S(A-)

This now offers the possibility of further generalization, whereby the formula for Xj is extended to account for the ions being arranged on a regular close-packed lattice, by means of the introduction of a constant, Cj, which is assumed to be of O(1 ), according to

X = a( 1 + ( a2 ( 10C-

The Coulomb energy, in the strong-coupling limit, then becomes

U~ - a2J2 C

J4pe0Ri

NiCa2Z1/\Z5/3) uWs

where a2 is given by (36), uWS is the Wigner—Seitz energy (21) given in terms of the Wigner—Seitz radius (18) and

C Z5/3

In the weak coupling however, the energy becomes independent of C, which is consistent with any regular close-packed structure disappearing in this limit. In the ion-sphere approximation, C = 9/ 10, which is considered to apply to fluid-like systems with no discrete symmetry. For close packing, C = 1/2aM/41/3 where 4 is the packing fraction and aM is the appropriate Madelung constant. Taking values of the Madelung constants from Ref. [19] yields values of C for various close-packed lattices, as given in the q1 following table (Table 1).

In all these cases, C > 9/10, which preserves the property Xj > 1. (The dependence of Cj on the ion species incorporates the possibility of different ion species being arranged on different interpenetrating lattices.)

3. Photoionization

3.1. Spectroscopic ionization potential depression

So far, we have treated ionization as a quasi-static process connecting two states of thermodynamic equilibrium. We now consider ionization to be a dynamical process, in particular photo-ionization, in which radiation in the form of a single photon ionizes

an atom in a discrete event, during which no changes to the surrounding plasma are induced. This can be because the electron remains within or very close to the atom during the photon interaction process or because the process occurs on a timescale short enough to be considered instantaneous. Either way, the plasma does not respond to the changed state of the ion until the electron has moved a significant distance into the plasma, by which time the photon interaction has ceased, and the immediately resulting state of the system cannot be considered to be in local equilibrium. If the system is constrained at fixed ne, Te (by contact with electron and thermal reservoirs) it subsequently relaxes to equilibrium during which process energy, hereinafter referred to as the relaxation energy, is implicitly exchanged with these surroundings. The total energy supplied to the system in attaining the final equilibrium state is then Zu + Ax, where Zu is the photon energy and Ax is the relaxation energy, and is, by definition, equal to the ionization potential (1), whereupon fa + AUj + e = Zu + Ax. Writing Zu = Zu0 + ZAwj + Ae where Zu0 = *fja is the ionization potential for the isolated ion, yields the spectroscopic ionization potential depression (SIPD)

DU- - Dc + e - De

In Section 5.4, it is shown that the quantity AUj - Ax + e is the adiabatic IPD. The extra term, Ae in (45), is an offset introduced so that the SIPD corresponds to the photoionization threshold, which is how it is generally conceived, and will be explained later. For the moment, it is sufficient to note that such a term, with Ae = 3/2kTe, is necessary to cancel e in the low-density limit.

Equation (45) shows that the SIPD and the static continuum lowering are generally different. Moreover, the SIPD relates to all adiabatic dynamical processes that change the ionization state of individual atoms, including collisional ionization and recombination. A purely kinetic model that describes the time evolution of a plasma in terms of microscopic physical processes at the atomic level will thus involve only the SIPD. According to this picture, the static continuum lowering is an emergent property of the plasma as a whole that does not relate to any individual atom or electronic state.

Following a discrete ionization process whereby a single electron is promoted to the continuum with sufficient energy to put it in thermal equilibrium with the other free electrons, the plasma is considered to undergo relaxation, through contact with the surroundings, to a new thermodynamic state in which the tempera-ture(s) of the electrons and ions and the free-electron density remain at their original values. Since AUj has been defined to give the energy change at constant ne, the extra continuum electron means that the system must expand by an amount AV = 1/ne.

The energy AE transferred to a general system during an isothermal incremental volume change AV is given by the First Energy Equation of Thermodynamics [20], according to which

ti IX - p> dv

where P is the pressure, and in which the first term represents heat transfer and the second, the work done. In this case, we have AE = Ax and AV = V/NiZ, where Ni is the total number of atomic nuclei (ions). The energy deficit following an adiabatic ionization process is therefore

Dc = T-(l V (PV

c Z\ dT \N{T

which must be evaluated for fixed Ni, since the number of ions is fixed, and for fixed ionization, since both the unrelaxed and relaxed

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states of the system are defined to differ from the initial state by the ionization state of a single ion. Equation (47) shows that this relaxation energy results from departures of the equation of state (of a fixed number of ions and free electrons) from a perfect gas. These departures are predominantly due to the Coulomb energy and electron degeneracy. The plasma equation of state can be written as

NkTi + 2 +1T

where U is the electrostatic potential energy and he — me/kTe. In situations when the direct electron contribution to the continuum lowering can be ignored (a = 1), U is given by (31) and depends only upon the ion temperature. Equations (47) and (48) then give the relaxation energy as the sum of two terms

DC — Da + DXe

which comprise the contribution from the ion subsystem

DC- (4 (&))

and the electron contribution

22 3 k'edfi

(¡3/2 (he)\ 1*1/2 (Ve)J

We now consider these two contributions in more detail.

3.2. Electrostatic contribution to the relaxation energy

Substituting for U according to (27) into (50) and making use of (25) and (29), yields

DXi - _L_T± (vZigU)] —L y- Zig'(A-

6NiZ 9Ti Zj

1 Z < ) 1 Z < < ) < ))

4NZjZg'<Aj)Aj = INZji <ha - g<j

where the functions h(A) and g(A) are given by (24) and (30) respectively.

Equation (52) applies only when the electron component of the plasma is negligibly polarized, ie a = 1. When a s 1, the potential energy depends non-trivially on both electron and ion temperatures and the electron and ion subsystems are not thermodynam-ically decoupled vis a vis equation (46). However we can still apply this equation if the temperature ratio Te/Ti is non-vanishing and a function of volume (or density) alone (the special situation of Te = 0 having been already been addressed above) which embraces thermal equilibrium (Te/Ti = 1). Then, referring to (36) and making use of equations in Appendix A

li~dT]

& = i ("2 - 0 ('t -1)

where 1T is the electronic isothermal bulk modulus pressure coefficient (as defined in Appendix A). In the case of non-degenerate electrons, lT 1, while, in the limit of extreme degeneracy, 'T = 5/3. Therefore, in the non-degenerate limit, da2/dT = 0 while, at

temperature derivative of a therefore vanishes in both limits and, since 1 < 1T < 5/3, remains close to unity, is small enough to ignore in all degeneracy regimes. (For he = 0, 3/2(a2 - 1)(1t - 1) = 1/4.) Considering the temperature derivative of Xj3, using the argument given in Section 2.4, gives, analogously to equation (37), making use of (25)

9Xj3 9x 9Aj 3.

T_L = t__— _ _- = —x

i 9Ti i 9x 9A, 9Ti 2 9x

3 y(1 -y)2(1 + y) 2 1 + y + y2

where 0 - y — 1 is given by (38). This yields X3

-0.20628 < Ti-L < 0 < 9Ti <

Combining these arguments, it follows that the temperature derivative of Xj, as defined by either (13) or (42), is virtually always negligible, which then allows the electron polarization screening to be treated by the simple device of replacing Aj everywhere with Aj — XjAj, and treating Xj as if it were constant. The result is that the relaxation energy (52) is generalized to

t = 4NZ ?! <h(a') - g<L»

which yields, in the strong-coupling limit, F [ 1 0 A [ 1,

h(Aj) -g(Aj) ~4/3Ca2F,

Da = 1 C*mr-e2-

Z2/3 \4Pe0Rwsy

A 1C<Z5/3Vi+ Ti V

) = 3 s^1 + ZppTBJUws

where RWS is the Wigner—Seitz radius (18), and uWS is the Wigner— Seitz energy (21); while, for weak coupling, F < 1 0 A < 1, h(Aj) -

g(Aj)~1/3A, = FjR,/D whereupon

Aci---= V

4NiZ ■ 4pe0D 16pe0D 4

(52) where uDH is the Debye—Hückel interaction energy (23).

3.3. Electron degeneracy contribution to the relaxation energy

According to (51), the contribution to the relaxation energy made by electron degeneracy is, for a free-electron gas, (Appendix A)

ACe = kT'2 I 1 - 2

¡3/21!/2A ¡2/2 )

2 _£ 3 kTB

£ - 3 kTB

which is equivalent to

Ace = -ne©Te

extreme degeneracy (Ti < TB), T(ôa2/ôT) =

1 w 0. The

According to (45), the total electronic contribution to the SIPD is then

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DCe - e = -2 kTB

which, in the limit of extreme degeneracy, is -kTF, according to which the spectroscopic ionization potential is raised by precisely the Fermi energy. This expresses the known result that, in an adiabatic isochoric ionization process in a fully degenerate system, the electron must be elevated, in accordance with the Pauli principle, to at least the energy of the Fermi surface, there being no available states of lower energy. The fact that the theory makes this adjustment automatically is reassuring and means that, for degenerate systems, a separate adjustment for the Fermi energy does not need to be made. It is also an example of a previously well understood circumstance when the static continuum lowering, which gives the bottom of the Fermi continuum, differs from the spectroscopic ionization potential, which corresponds to the Fermi surface. In the non-degenerate limit, (61) reduces to Axe - e = -3/2kTe which is just the average energy of a free electron.

Ionization potential depression is often thought of in terms of a change in the threshold energy, that being the minimum photon energy deemed to be required to cause ionization. In partially degenerate systems, this is not well-defined, because the photo-ionization edge is blurred by the thermal distribution. This has not been an issue thus far, because the ionization potential has been defined in terms of well-defined initial and final thermodynamic states of the plasma. However spectroscopic observation looks for thresholds, such as those relating to bound-free edges or the existence or non-existence of lines. These thresholds may not be sharply defined in terms of photon energy, resulting in some indefiniteness in how the SIPD is defined. In the non-degenerate and fully degenerate limits, this is not a problem: the ionization threshold energies are Zw0 + AUj - Axi and Zw0 + AUj - Axi + kTF respectively. At arbitrary electron temperatures (partial degeneracy) a reasonable definition of an effective photoionization threshold that interpolates between these limits is given by (45) with the reference energy offset given by

2kTB - kTeW(he) = 3kTe(lV2(Ve) j'1 /2(he) - |w(he)

which is everywhere O(kTe) and is a constant in the context of the problem (since Te and re both relate to the defined initial state of the plasma). This leads to the effective photoionization threshold

hu- = hu0 + DU- - Dci + kTew(he)

which defines the electron degeneracy-related contribution to the SIPD entirely in terms of the function w(r) whose properties are that it is monotone, positive definite and possesses the following behavior:

w(h)~h - 2

w(h)~0

h»2 h < 2

where the effective half-width of the Fermi surface is taken to be 2Te (corresponding to the intercepts of the tangent at e = m). The first of equation (64) places the threshold at me - 2kTe in the regime of he [ 2; When h < 2, the Fermi surface lies, fully or partially, below the continuum sufficient for there to be no degeneracy shift in the threshold energy. A suitable simple function complying with these limits is

where H(x) is the Heaviside function. Note that, defining Ae = 3/ 2kTe, as formerly proposed, leads to kTew(he) = 3/2k(TB - Te), which although possessing the correct extreme limits, vanishes too slowly with temperature in the non-degenerate regime at high densities, when the electrons are compressed to within separation distances of about a Bohr radius. The leading term in the high-temperature expansion (T [ TF) gives

3.,_ 1 /2\1/2kTf3/2 3p r-3

2KTb - Te=

4pe0De

where aN is the Bohr radius. At solid density (see Section 3.4) aNne~Z which leads to 3/2k(TB - Te)~Ze2/4pe0De which is of classical proportions, despite being quantum-mechanical in origin, showing that quantum effects in dense non-ideal plasmas can persist even at quite high temperatures. The persistence of this offset term in (63), would have led to unreasonably large corrections to the IPD at moderately high temperatures.

3.4. Relaxation energy in cold condensed matter

In the special case of cold condensed matter, the total pressure is zero and (47) yields Ax = 0. However, in a metal, the spectroscopic continuum must nevertheless start at the top of the Fermi surface, which implies Axe = e - kTF. This means

Da = Dc - Dce

-kTB - ex—kTi

which is the electron degeneracy pressure. The vanishing of the relaxation energy in this regime is just an expression of the fact that the repulsive electron degeneracy pressure is balanced by the attractive Coulomb bonding forces. The SIPD is then given, according to (45), by

hDu- = DU- - Da + 2 kTB - De

DU, + e - De

Equating the Coulomb relaxation energy Axi, (57), in the low temperature limit, to the electron degeneracy pressure given by (67) and using the standard relationship between the Fermi temperature and the electron density, yields the following estimate of the Brueckner parameter for solid metal at ambient

and hence

_ 2/3 ¿/A 4

, x 1/3 n1/3 _ (Z\ 1 /10.

a~nr = (2) (9Pa >025Z

w(h) = (h - 2)H(h - 2)

in which n¡ is both comparable with and greater than the Mott density [10], in line with expectation.

4. Continuum lowering and ionization potential depression for discrete processes

4.1. Static continuum lowering

The static continuum lowering, defined by (28) for example (in common with Stewart—Pyatt and related formulas) relates to the reversible excitation of an infinitesimally charged electron. More

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precisely, the continuum-lowering contribution to the ionization potential depression is given by

B.J.B. Crowley / High Energy Density Physics xxx (2014) 1—20 hDuj = DUj - Dci =

- Zj + T Zp + 1 uD

DUJ = U(ZJ + 1) - U(Zj)x§.

DU{Zj + 2

(71) 4.4. General formula for the spectroscopic IPD

which states that the continuum-lowering contribution to the IPD is, to an approximation, the static continuum lowering evaluated for the average charge state.

4.2. Continuum lowering in the strong-coupling limit

For strongly-coupled systems, the electrostatic energy is given by (43), which implies the static continuum lowering

AUj = -Ca2Z1/3((Zj + 1)5/3 - Z5/3^

5^,2v1/3

- 3 Ca2z' [ZJ + 2

IN 2/3

in which the error resulting from the final-stage approximation is <5% even in the worst case of Zj = 0. From (57), the corresponding relaxation energy is

DXi = ~ Ca2±

i(Z5/\o

Upon combining the above results, the total SIPD (45) is given by

hAwj = AUj - Axi + kTew(he)

Cu0 Z1/3 (7. + 1 )5=3 75/3 1 X" -CUwsZ \(zj +1 - Zj + 3Z iy-ZpTB

+ kTeW(he)x - 1 CuWsZ1/^^ Zj + +

n2/3 (z5/3> -J- 2) +-Z"

x I 1 + Z^j + kTeW(he)

4.3. Continuum lowering in the weak-coupling limit

In the limit of weak coupling, the plasma energy is given by

1 Zj2e2

— a > —--

2 J 4peoDi

aUDH = -2 NiZZpuDH

The static continuum lowering implied by (33) is now )2

DU = (Zj + 12 - Zj2)Udh

- Zj +1) Udh

(Zj + 2 e2 4pe0D

in obtaining which no (further) approximation is necessary. From (58), the corresponding relaxation energy is

4NiZ J 4peoDi

cPu DH

Upon combining the above results, in the weak-coupling limit, assuming non-degenerate electrons (w(he) = 0) the total SIPD (45) is given by

The formula for the SIPD of plasmas under regimes of arbitrary coupling and electron degeneracy results from a combination of equations (12), (49), (56), (61), (63) and (71), which yields

hDu, = -

kTL 2Zp

+ 2NZ XXJh(Lj0) -g{lJ)

+ kTeW(re)

Lv = XnLn

LJ XJ LJ

lo — jla0 L + Lj = Z L0; Lj

r _ ZpZe2

^lJ, LJ = (3G)3/2

4pejRwSkTi

in which v denotes the index 0 or + as defined above and Xj is given by (42) in terms of Xj, which is the positive real solution of (5) with Yj = (Lv)-1/3; Cj, which is the force constant (=9/10 for fluid systems) as discussed in Section 2.4; and a, which is the ratio of the screening lengths as expressed by (8). Equation (79) depends on the CSD. If this is not precisely known, or when a simpler result is required, (79) can be replaced with the more approximate formula

-kTi[ 2Z-h{ Lj) + 1 (h(L°o) - g(L°o)) ) + kTeW(re)

In the strong-coupling limit, (81) yields

5 ( Zj + 1fV/3

(74) hDuj = -1 Ca2uW^^Zj + 1) ' Z1/3 + Z^j + kTeW(he) (82)

which agrees with (74), subject to the approximation (Z5/3> ~ <Z2)/<Z>1/3 = ZpZ2/3. For sharply peaked CSDs likely to be encountered in strongly-coupled dense matter, the error is of the order of a few percent or less, and the approximation is consistently, albeit marginally, better than <Z5/3> ~ (Z)5/3.

In the limit of weak-coupling and weak-degeneracy, (81) leads directly to the result (78) without further approximation.

5. Thermodynamic treatment of ionization

5.1. Entropy and free energy

Important insight into the problem is gained by considering ionization fully from the perspective of a thermodynamic process in an electrically neutral plasma comprising a fixed number, Ni, of atomic nuclei of a single species. (The generalization to multiple nuclear species, while straightforward, is omitted here in order to maintain clarity.) Let E be the plasma energy, which is the energy residing in the degrees of freedom involving the component particles (ions and free electrons) including their mutual (Coulomb) interactions but excluding the energy contained in internal states of the atomic ions (electrons in bound states). Let z represent one or more internal microscopic configuration variables describing these

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internal states, and suppose that it is possible to vary z through the application of external influences, such as electromagnetic fields or radiation. In such circumstances, it is reasonable to promote z to the status of a thermodynamic variable, in which case we can define I to be the thermodynamic potential associated with z whereby a change dz > 0 in z is associated with some energy Idz being made available. For an infinitesimal reversible process in such a system

B.J.B. Crowley / High Energy Density Physics xxx (2014) 1 —20

J^ZkNk k

dE + PdV - TdS - ¡dz = 0

where S(V,T,z) is the entropy function, in terms of which the probability of z in a closed system in equilibrium at fixed V,T is given by the usual Gibbs distribution

= e-S(V;TZ)

which satisfies the condition ]T V(z) — 1 or f V(z)p(z)dz — 1,

where p(z) is the density of states represented by z, depending on whether z takes on discrete or continuous values. The distinction is unimportant, and, for sake of argument, we shall start by assuming the latter. The expectation value of z is

z = <z)h j P(z)zp(z)dz corresponding to the macroscopic entropy

S0(V, T) = -j P ln(P)P(z)dz = <S(V, T,z))

which follows from (84). Expanding S(V,T,z) about z = z and taking the average yields

<S(V, T,z)) = S(V, T,z)+±S"(V, T,z)(Az2) + ■■•

where S"(V, T,z) = (d2S(V, T,z)/dz2)VT normally distributed variate, then z=z

S(V, T, z) = S0(V, T)+2 ((z - z)2/ (Dz2) in which case

S0(V, T) = (S(V, T, z)) = S(V, T, z)+2 and, in particular 'd

and Az = z - z. If z is a

w, t , ?;.( ¿s(vj,z>)

which expresses the important property, which will be shown to hold generally, that the equilibrium values of the macroscopic thermodynamic coordinates are stationary with respect to the microscopic variables z.

Equation (83) implies the following additional Maxwell relations:

(§S)z; (9S)t

iap)V = (w)z ; (az)p = (aT)z

If ZK is the charge on an ion in the state K, and NK is the mean number of ions in that state, charge neutrality is expressed by

which, note, is also a statement about the average charge state of the plasma, for fixed

Maximizing the entropy subject to the constraints of particle numbers and total energy yields

T - VeNe -J2 VkNk + 3

where 1 /T, he, {Vkg, 3 are the Lagrange multipliers, with ln3 being the partition function and T3 the grand potential. We now make an important departure from the standard theory of equilibrium systems by generalizing to systems exhibiting weak electron-ion coupling by treating these as separate subsystems, with different temperatures Te s Ti. Writing

S = Si(V ,Te ,Ti, V ,z) + Se(V,Te ,z)

and maximizing the entropies independently yields E

Si = t^ -E VkNk + 3i

T K Ee

Se = - VeNe + 3e

in which we have made the assumption that the (free) electron dynamics are negligibly affected by the ions being at a different temperature (should this be so). Equation (83) then generalizes to

dE + PdV - TidSi - TedSe - ¡dz = 0

where P = Pe + Pi and E = Ei + Ee, with the respective temperatures given by

9E] - T [91

9c. I = T 'I 9Se

ej V ,z

These equations describe the ion and electron subsystems as being independently in equilibrium.

The Gibbs free energies for the ion and electron subsystems are defined in the usual way

Gi ^ Mk% k

Ge = meNe

where me — Tehe and mK — TihK are the electron and ion chemical potentials respectively. The chemical potentials are intensive quantities that are, for a given plasma composition, functions of the respective temperatures and pressures. The plasma composition is determined by chemical equilibrium between the electrons and the various ion states.

5.2. Chemical equilibrium

The general changes in the respective Gibbs free energies of the system are given by [20]

dGe = -SedTe + VdPe + medNe dGi = -SidTi + V dPi + £ m-K dNK

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which hold for any infinitesimal process involving a change in the plasma charge state. Chemical equilibrium, at constant pressure and temperature(s) depends upon the total Gibbs free energy G = Ge + Gi being minimized with respect to variations dNK in the composition, subject to the number of ions being fixed, whereupon

]Td% = 0

and charge neutrality

dNe = J2 Zi<dNK

The minimization condition

dG_ VNk

where hereafter, until otherwise indicated, T denotes Te and Ti severally, where these are distinct, implies that the chemical potentials of those species present in the system must satisfy

mK + ZK me = m0

for some fixed m0 that does not depend on the atomic configuration, and which corresponds, by inspection, to the chemical potential of a neutral atom. It is important to recognize that (104) is a condition for equilibrium and is not a constitutive relation. Consideration from the point of view of equilibrium at constant volume follows equivalent lines, except that it is then the Helmholtz free energy F = G - PV that is minimized. Since, for any internal configuration variable z

dzh,T V dP/z,Adzyv ,T

_v(dP\ hfdG

this also leads to (104).

Thus chemical equilibrium between the electron and ion subsystems generally depends upon the chemical potential differences

mj - mx + (Zj - ZK) me

vanishing for VJ,K. Whenever any Ajk s 0, the system is not in equilibrium, with AJK > 0 (or AJK < 0) implying a tendency for the reaction between the states J and K to proceed spontaneously in the direction J 0 K (or K 0 J).

Ionization of a single atom or ion in a plasma in thermodynamic equilibrium, under conditions when the electronic chemical potential me, and the temperatures are fixed, via the reaction J 0 K + e corresponds, using (99), to dGi = mK - mj, dGe = me and hence dG = 0, which, in a closed system, corresponds to an isobaric, isothermal process.

The general change in the total Gibbs free energy during an infinitesimal process of a system in chemical equilibrium is, making use of (101), (102) and (104)

dG = dGe + dGi

= -SedTe - SidTi + V(dPe + dPi) + medNe + ^ mKdNK

= -SedTe - SidTi + VdP + J2(mK + Zkme)dNK

= -SedTe - SidTi + VdP

which, when combined with (97), yields that, for a reversible process of the closed system

dG = dE - SedTe - TedSe - SidTi - TidSi + VdP + PdV - Idz = dE - d(TeSe) - d(TiSi) + d(PV) - Idz

which reveals that Idz is a total differential, ie Idz = dF where F is a function of the state variables, and moreover that I must depend on z alone. Equation (91) then imply that the first derivatives of P,V,T,S with respect to z all vanish, which indicates that, at equilibrium, these variables are all at extrema with respect to z, so, for any independent set of coordinates, eg S,V,T

0; fVV

V ,T \U^/S,T

Integrating (108) then yields G = E - TeSe - TiSi + PV - F and, upon referring to (96), the grand potential is

Ti3i + Te3e = PV - F

Since, at equilibrium, I does not depend on the macroscopic coordinates, it can depend only the internal coordinates, and the energy variation is given by dE(z) = dE(z) - dF, which yields F as the deviation from equilibrium of the total binding energy of the electrons in the atomic system configurations, when the atoms are completely isolated from each other

J2(Ec &

- Ec (z)) ^ (NJ - JEc (zj)

with z- = (z^z-fi,...) denoting the electronic configuration of the atom -, and where Ec(zj) < 0 is the total energy of the configuration J defined by zj = (zJa,zJb,...). Equation (112) is exact, being an irrefutable consequence of the thermodynamics. It means that any changes to the configuration energies due to interactions between ions are contained in the other thermodynamic terms. Note that, while the average of F vanishes identically, the fluctuations of this quantity nevertheless have an important role to play.

Let z denote some z-a, which is the occupancy of an energy level a in the atom, -, whose initial configuration is J. The ionization reaction J 0 K + e, whereZK = Zj + 1,and zk = (zja - 1,zj.)hZj -ba then corresponds to Dz = -1. The reaction can then be expressed by the differential relations

dz 1 dz

VNk dz

VNl dz

0, Lsj, K

~dz = -1

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and hence, from (112), using (113)

. 9F 9F „ < ) ,

1 = ~dz = = Ec(zK )-Ec ^ hfJ/K

where, for ba — zj - zK, fja — fj ^ K > 0 is the ionization potential from the level a, in an isolated ion in the configuration J, leading to the configuration K.

5.3. The thermodynamic ionization potential

Equations (100) and (105) in conjunction with (113) yield

m- - mj- me

The condition for ionization equilibrium, AKj = 0, VKJ, is therefore expressed by

9z'p,t \9zj v,t

where the chemical potentials are given by

9G \ _ 9F \

mjPj {m)v,T

KJ P,T

'K ) V , T

where, it should be noted, NK, Ne are the actual particle numbers, which are independent of the macroscopic thermodynamic variables P,V,T..., in contradistinction to their averages, which are generally presumed to be functions of the macroscopic thermo-dynamic variables.

Now, let the total Helmholtz free energy be expressed in the form

F = F0 + AF

where F0 — F0 + F0 is the free energy of a system comprising the same mixture (expressed in terms of {NK},Ne) of non-interacting particles at the same volume and temperature. The condition for equilibrium (117) then becomes

9F) = mK

■mf - m0 +

where mK(nK, Ti), m0(ne , Te) are the non-interacting ion and electron chemical potentials at the respective particle densities and temperatures, and where

F0 + PfV ^ m-NK F0 + Pe0v = m0Ne

are the Gibbs free energies of the non-interacting system at the pressure P0 = Pf + P° corresponding to the same particle densities. The equivalent decomposition of the Gibbs free energy, G = G0 + DG, on the other hand, leads to

mj0 - m0 +

which, by virtue of (117), is equivalent to (120), in that G0, and the associated chemical potentials ~K; are now those that correspond to the non-interacting system at the same total pressure, P

and temperature. The relationship between DF and AG is expressed by

F = F0 + AF where

G - PV = G0 + AG - PV

G0 = G0(P J) = F0 <V0 , T) + PV0 F0 = F0(V ,T) = G0<P0, T) - P0V

and where G0(P,T) and F0(V,T) are the Gibbs and Helmholtz functions respectively for the non-interacting particle systems having the same particle concentrations as the interacting system, and

0 _ (9G0(P, T) 9P

9F0(V J ) 9V

The chemical potentials of any particle species x are then found to be related by

0 À9G0(P, T) 9Nx

m°(V0 nx

For Boltzmann particles and non-degenerate electrons (Appendix A)

m>-T ) = T toUffî)

where Gx is the spin degeneracy of the species x. This yields i~K - mK = kTi ln(V/V0), which does not depend on the species type K, and so, from equations (120) and (122), we obtain the thermodynamic ionization potential

f+AW,-(£

m0 - m0 -

which is the change in the free energy associated with the hypothetical removal of an electron from a bound state within the ion and which defines DW as the thermodynamic IPD (TIPD). Substituting (128) into the equilibrium condition (120) yields

mK - mf - m0 = fj + AWj

which, with the aid of (127) (for non-degenerate electrons) with Te Ti T, becomes

njne _ 2gj fmekT\3/2

nk QK \2vh2

Equation (130) is the Saha equation and, importantly, demonstrates that it is the thermodynamic IPD, DW that features in this particular equation of state [11], rather than any of the other forms of the IPD. The SCL has a different role in the equation of state, as discussed in Appendix B.

The contribution DF to the free energy is associated with the effective interaction energy U [10], in which case

AF = T - TAS - F where

whereupon

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In the case of pure Coulomb interactions, the scaling laws arising from the virial theorem etc, imply that the interaction free energy can be expressed in terms of some function f(L) of the coupling parameters Lj defined by equation (13), in the manner of

DFi = -3Z- X Zjf (Lj,

in terms of which, making use of (25)

ui = dfi- tiwiDF

-W- XZjLjf'(Lj)

VDPi = -VvVDFi = -§- XZjLjf'(Lj) = 3 ui (136)

Comparison of (135) with (27) then yields

g(L) = Lf'(L) (137)

Hence, using (26), the thermodynamic ionization potential is

vr = fj - 3TP f L) + gj = fj + AWj (138)

in which DWj = DWj where

^ =-J^(f (Lj) + Ljf ' (Lj))

is the Thermodynamic Ionization Potential Depression (TIPD), where

f (L) = j g^dX

Equation (139) resembles, but is distinct from, the corresponding formula (28) for the static continuum lowering. Taking g(L) to be given by (30), yields [12]

,, , 9 2 3 1 + s 1 s-1

f (L)=T^s2-F^-2 +v 3arctan —=—-

w 10 51 + s+s2 s +1

3 (\ + s+s2\ 1

-2l\ 3 ) - 2

s(L) = (1 + L)1/3 (141)

In the strong-coupling limit, f(L) ~ 9/10L2/3, g(A)~3/5L2/3 and the TIPD reduces to the static continuum lowering AUjs, which is as given by (28) in the limit of large L, otherwise there are differences due to temperature-dependent terms associated with the change of entropy.

In the limit of weak coupling f(L) ~ g(L) ~ 1/3L, and the TIPD becomes AWj = -2/9kTi/ZpLj = 2/3AUj, which is two thirds of the static value. Equations (24), (29), (137)—(140) imply the following direct relation between the TIPD, AW(L), and the static continuum lowering, AU(L)

DW = h x 0

Let f be any real function of l in (0, n ) with the property thatf(0) = 0 and suppose that, for some value of n, d(1rT(1))/d1 > 0Vl e (0, n ). Integration of the non-negative definite function Xvd(X~nf(X))/dX by

parts from zero to L > 0, then implies that f (A) > n f (X)/1)d1.

Application of this lemma to (142) with f -that

|DU| > \DW\

DU and v = 2/3, implies

Moreover, by application of the inequalities (1 + l)v > 1 + vl, 1 + l > (1 + 1)n, which hold generally for l > 0, 1 > v > 0, and making use of (24), equation (142), yields that

\dw^2 l S

to which DW is asymptotic at L = 0, and

\DW\ < h2/3,ß 2 Zp

to which it is asymptotic at L = n .The equalities in (144) and (145) correspond respectively to the weak- and strong-coupling limits, as given above. The TIPD is thus distinct from the static continuum lowering, except in the strong-coupling limit, and is generally smaller in the sense of less lowering.

Nor is DWj the same as the averaged self-energy [10], which is given by UjlkTi = -(Zj/2Zp)g(Lj), from which it differs, in the weak-coupling limit, by a factor of 4/3.

5.4. The adiabatic ionization potential

It should by now be clear that neither the thermodynamic nor static IPDs apply directly to "fast" processes, such as photoioniza-tion and collisional ionization, which are more reasonably considered to be adiabatic, constant volume processes. Accordingly, we define the adiabatic ionization potential be the energy AE that must be provided to the system in order to increase the ionization of one atom by one unit of charge (dz = -1) while maintaining the volume and entropy of the system. According to (83) or (97), while recalling

that z = X;

DEja = fja - ( —

i-l \VZjaJ

The ionization potential for a quasi-static (isobaric) isothermal process (equation (1)), on the other hand, with the aid of (71) and (114), is

fja + DUj + ë = fja - (VZj

Expressing the energy as a function of Si, Se, V, T, z, the two derivatives are related by the chain rule

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B.J.B. Crowley / High Energy Density Physics xxx (2014) 1—20 15

1 E E Se E Si difference in the photoionization potentials. This only makes sense 66

2 V*z)ne T = \dSeJV t +vVSiJV t if the threshold states are deemed always to lie in the spectroscopic 67

3 " ve\ /gV\S f9E\ " continuum, in the sense that any spectroscopic measurement will 68

4 +(— ji — j +(— j (148) determine these states to be free continuum states. Threshold 69

5 V S z z ne T z S V states are therefore seriously problematic only if they contain 70

6 electrons in the ambient state. 71

7 in which, making use Maxwell's relations, while referring to (98) The apparent dichotomy about whether or not the states are 72

8 1 f9S \ f 9S N 1 f VP \ f 9S N bound can be resolved by observing that the static equilibrium 73

9 (§)n T = --L i+l= -A ivTej +l potential is a fiction in that it represents some equilibrium average 74

10 x /n"e \uv\ z,T V^/ v,t 'levies V V v,t of the potential in the vicinity of a fixed charge, and, in particular, 75

11 /as.\ 1 f 9Sa /9Sa 1 f 9Pa /9Sa applies only when the level corresponding to the threshold state is 76

12 w n„,t = -~ne \ T ^"tej VT = -ne VvT^j V ^ley V tT empty. The fact that a threshold state is apparently represented as 77

13 ' ' being bound actually means only that an infinitesimal test charge 78

14 iw)sz = -P would be bound in it. When the level is occupied by a discrete 79

15 1 electron with finite charge, as in the immediate post-ionization 80

16 (9V) n T =----phase, the ion is maximally screened resulting in the positive 81

17 e ne charges in the surrounding plasma being less repelled, resulting in 82

18 (149) increased continuum lowering, compared with the ultimate final 83

19 . ,...., equilibrium state, when the electron is "absorbed" into the sur- 84 where P Pe f Pi is the total pressure. Hence, for ionization about

20 e i rounding plasma. The electron is thus capable of being free while 85 the equilibrium state, making use of (109)

21 the local state is occupied, while leaving behind an ostensibly 86

22 f E\ 1 f f P\ \ f E\ f E\ bound level when it moves into the surroundings. Anyone used to 87

23 (az) = IT \ - \99Tl f 9z) = -Dc +(doing self-consistent atomic physics calculations will be aware of 88

24 z ne T ne T V z S V z S V this phenomenon: that the energy of a level generally varies ac- 89

25 (15°) cording to its occupancy and that indeed a level can be bound or 90

26 free depending on whether or not is occupied. This is similar, except 91

27 which, when combined with (147), yields that the effect is due to polarization of the surrounding plasma 92

28 f E\ ("plasma relaxation") rather than of the other bound electrons 93

29 -(— j = DUj - Dc + e (151) ("orbital relaxation" [21]). 94

30 \9zJS,V Ultimately, a complete resolution ofthis problem has to address 95

31 the fundamental limitations of the standard picture of ionization, at 96

32 according to which, the term Dc - DUj - e in the SIPD (45) is the least for strongly-coupled many-body systems. For modeling pur- 97

33 adiabatic ionization potential depression. poses, while it is a convenient notion, to consider that the electrons 98

34 The adiabatic IPD applies to discrete ionization processes that in a closely-coupled many-body system fall into one of just two 99

35 occur locally in such a manner that the surrounding plasma is categories: those that are bound and thereby localized in the vi- 100

36 unable to respond, eg photoionization and (fast) collisional ioni- cinity of individual atomic nuclei and those that are free in the 101

37 zation. The entropy of the surrounding plasma therefore remains sense of being entirely delocalized and virtually decoupled from 102

38 unchanged during the initial process. A prevailing assumption is the ions, is certainly naive. That there might be electronic states 103

39 that the system as a whole remains reasonably near to thermody- that fall, even approximately, into neither (or both) categories is not 104

40 namic equilibrium, an assumption which holds reasonably weU for only possible, but also necessarily so in the pressure ionization 105 4! the experiments considered in Section 8. However, any extrapola- regime when bound electrons are evidently interacting with the 106

42 tion of the results that follow to systems that are strongly driven boundaries of the system. Moreover, the division of any closely- 107

43 out of equilibrium, such as when the intensityis sufficient to ionize coupled dynamical system into subsystems according to the en- 108

44 a significant propOTdra of the atoms at the same time,or within the ergy of those systems does not accord with a proper Hamiltonian 109

45 same equilibration time frame, would not be justified. description. However, if there are sufficiently few electrons occu- 110

46 pying threshold states, then the subsystems can be considered to be 111

47 6. Phenomenological interpretations approximately dynamically separable. This approximation is 112

48 generally applicable to weakly-coupled systems in thermodynamic 113

49 6.1. Threshold states and plasma relaxation equilibrium. It is also applicable in some dense strongly-coupled 114

50 regimes when there is a large energy gap between the highest 115

51 A question that arises from this concerns the nature of those bound state and the continuum — a situation that prevails in typical 116

52 states, which we shall refer to as threshold states, that are bound in metals. In the presence of occupied threshold states, the system is 117

53 the static potential by energies less than ~D\. In what sense can not dynamically separable into bound and free-electron sub- 118

54 these states be described as either bound or free, and how should systems. This underlies many of the problems often encountered, 119

55 they be treated in model-based calculations? including discontinuous behavior and thermodynamic in- 120

56 First of all, if we consider only excitation of an electron from an consistencies, in treating plasmas in the high-density pressure 121

57 initial ambient state to a threshold state, there are no in- ionization regime. The explanation of threshold states given above 122

58 consistencies arising from making an a priori assumption as to fails in the pressure ionization regime, since these states are likely 123

59 whether the states are bound or free. It would then seem to be an to be already occupied in the ambient system. A different resolution 124

60 open choice whether the states are treated as bound, according to of the dichotomy has therefore to be sought and it is likely that 125

61 the static continuum lowering, or free, according to the spectro- these ambient threshold states possess properties characteristic of 126

62 scopic IPD. Inconsistencies do arise however when we try to both bound and free states, such as being semi-localized and 127

63 consider photoionization from such states. Koopman's theorem, contributing partially to the pressure, as would be implied by a 128

64 and energy conservation, imply that the transition energy for smooth equation of state. As already noted, the pressure ionization 129

65 photoexcitation between bound levels must be given by the regime is beyond the scope of any theory, like the one given here, 130

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that attempts to treat bound and free electrons entirely separately. However, treating the ambient threshold electron states as a separate intermediate group is suggestive of a possible ad hoc approach to bridging the pressure ionization discontinuity within the context of such a picture.

In non-self-confining systems, P > 0, the relaxation energy that accounts for the threshold states is due to deviations of the equation of state from perfect gas, which may be due to repulsive atomic cores, and inter-particle forces (Coulomb and exchange) and significantly, the presence of electrons in the threshold states themselves. The Coulomb contributions have already been considered. It is straightforward to show that a hard repulsive core does not contribute to the relaxation energy, by writing the equation of state in the Van der Waals form

P - 3V ) (V - 4Nvc)

where vc is the core volume, which yields

NkT = V - 4Nvc ' 3NkT

Referring to equation (47), it is clear that the volume-related term involving vc does not contribute to Ax, provided that vc is temperature-independent. Thus the relaxation energy, which is a measure of the width of the threshold band, is essentially determined by the finite-range interatomic forces both Coulomb (which acts to lower the continuum threshold) and exchange (which acts so as to raise it).

The other issue, the difference between the static and thermo-dynamic IPD's is associated with the change in entropy, as is apparent from the equation AU - AW = d/dZ(U - AF) = TdS/dZ. In regimes of moderate to weak coupling, increasing ionization reduces the entropy, with the result that AW > AU, tending to equality only in the strong-coupling limit. In Appendix B, a clear link is established between the average static continuum lowering and the equation of state in non-pressure-ionizing regimes (In a pressure-ionizing regime, the situation is less clear.) while the Saha equation, which holds for weak coupling, depends only on the thermodynamic IPD. The thermodynamic and static IPDs become equal in the strong-coupling limit, while the additional entropy-related thermodynamic lowering increases with decreasing plasma coupling.

The entropy connection is motivation for seeking an explanation of the IPD dichotomy in terms of the plasma microfield. This is considered in the following section.

6.2. Transient states and the microfield

A property of a system of charges that might be expected to have a bearing on the IPD and threshold states is the microfield [8]. The microfield, expressed in terms of the electric field fluctuation AE, can give rise to transient states, or hopping states [22], in which electrons would be only transiently bound to, or localized within, the vicinity of a particular ion. Such states are bound, by virtue of being at negative energies in the average potential, but would be spatially delocalized. By this mechanism, the microfield might be considered to give rise to a reduction in the ionization potential, from the average, by an amount ~e|AE|Rj, which would then appear as an apparent contribution to the observed spectroscopic IPD. The microfield is due to spontaneous fluctuations in the charge states in the surrounding plasma, while the spectroscopic IPD, as argued above, depends upon the response of the plasma to the changed charge state of the ion. The two processes, while

apparently quite separate, are in fact connected through the fluctuation dissipation theorem [23—25], which relates the charge— density correlation function, which characterizes the plasma fluctuations, to the imaginary part of the response function, which is expressed, in the spectral representation (k,u), by the dielectric function e(k,w). The variance of the electric microfield, (AE2), at an arbitrary location in the plasma, equivalent to the spatially-averaged mean microfield, is given by, [10]

(2p)4e0

e(k, u)

Zu 2kT

Applying this formula to the 'slow' ion component of the microfield through the classical approximation coth(Zw/2kT) = 2kT/Zu, which requires that the ion temperature be much greater than the ion plasma frequency, and carrying out the integral over u using the screening sum rule, then yields, for the quasi-static microfield

(DE2\x^L f ^ \ / pen ./ u

— N d3k (2p)3

f d3k ' (2P)3

ecStö)

e(k;u)

On the other hand, the dielectric function describes the response of the plasma to a change in the charge state of an ion within it (equivalent to the introduction of a test charge). The change in the self-energy of a stationary ion, due to the removal of one electron, is, [10]

(2Z+1) H

d3k J_ (2p)3 k2

e(k;0)

which is equivalent to the static continuum lowering. Equations (154)—(156) reveal a connection between the microfield and the continuum lowering in terms of a more general underlying theory.

For the classical one-component plasma (De[Di) the static dielectric function e(k,0) is the reciprocal of the static structure factor Sii(k), which is deemed to satisfy f (1 - Sii(k))d3k = (2p)3ni by virtue of the ion—ion pair distribution, for charged particles of the same sign, vanishing at zero separation. Equation (155) then yields

(DE2)x

njkTi ' e0

according to which, the classical electric microfield fluctuations in a weakly-coupled system of charged particles are equivalent to a single classical normal mode per particle, independently of the actual charges. This gives the energy associated with the microfield as V2NikTi, corresponding to the free energy

2NikTi ln(G)

where G = Z2e2/4pe0RWSkTihL2/3 is the ion coupling parameter. The microfield thus makes a contribution to the equation of state that is different in character and therefore supplementary to the normal quasi-static Coulomb part, such as described in Appendix B (cf equation (141). (Moreover, since the resulting pressure satisfies PmfV/NikTi = 1/6, the microfield (157) makes no contribution to the relaxation energy (47).)

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To understand the strongly-coupled limit, we consider a solidstate plasma where pointlike ions of charge Z are confined close to specific locations {r,} about which they collectively undergo small harmonic oscillations whereby the displacement of the ith ion is

Ar,(t) = Exk cos(k-r,' k

Ukt). The resulting micro field is

AE(t) = E^ Exk c°s(k-r* - Ukt) 0 i 'i k

from which an estimate of the mean square microfield at an ion site is

0) (2Rws)

■X <xk>

where nnn is the effective number of nearest neighbors, (xk) is the mean square displacement in the mode k. For a system of classical oscillators of total mass M, in equilibrium at temperature Ti,

(xk) = kTi/MQk; while, for acoustic modes, £ 1/Uk = 3Ni/U

where U0 is the upper limiting frequency, which we identify with the ion plasma frequency. Hence, combining these formulas with (160)

nnn )23nikTi

12J e0

which, apart from O(1) numerical factors, is the same as (157). Indeed, nnn = 12 is a reasonable choice meaning that the microfield is now associated with 3 degrees of freedom per particle, and moreover, these should be the same three degrees of freedom as are associated with the potential energy of the oscillators. The fact that two such similar equations as (157) and (161) arise in such different limits suggests that

with l = 1 in the case of a classical Coulomb fluid and l = 3 for a classical solid-state Coulomb plasma, describes the general case. In the solid, the microfield is just the effective oscillator field, and the equation of state is adequately described by that for a system comprising a collection of Ni classical oscillators (as per the classical phonon model) with no additional field-related terms. The transition from l = 1 to l = 3 then corresponds to a discrete phase transition, thus avoiding any need for l, along with the implied scalings, to take on intermediate values.

Expressing the result (162) in terms of the normal field E0 = Ze/4pe0RWS2 gives

(aE2) = 3'Eg/ F = 9'E2/a2

Note that this gives the spatially-averaged mean microfield. The (time-averaged) mean microfield at the center of a particular ion is generally what is considered appropriate in line-broadening theory, and is modified from the spatial-average by the correlations with neighboring ions. This is not necessarily what is relevant to the continuum lowering.

Let the notional microfield "contribution" Demf to the continuum lowering be the extra energy that an electron can gain from the microfield in moving from an initial location r0 within the bound-state orbital to the surface of the ion-sphere, where it is deemed to be ionized, ie

Aemf = e J AEdr r0

Schwarz's inequality for a randomly directed microfield, in conjunction with (157), then yields

fkTi Z + 1

for r0 < R0, where f = (Z + 1)e2/2r0, which is a measure of the initial binding energy of the electron. The inequality (165) is perhaps over-strict. In a weakly-coupled system, the mean microfield is relatively weakly correlated with any particular ion, in which case

A£mf xkTiV3F /Z

is a better estimate. A measure of the importance of the microfield in this regime is therefore

Aemf 4pe0DiA£mf

Fp3 A2

which demonstrates that the microfield is the dominant influence in weakly-coupled plasmas. A similar measure of the relative importance of the microfield in the strongly-coupled regime, referring to (163), is

which shows that the microfield can be expected to cease to dominate the continuum lowering for F = 4/3. However we should bear in mind that the microfield does not ionize but rather perturbs with the possibility of creating transient states, which spectro-scopically and thermodynamically have more in common with bound states.

Unlike threshold states, transient states are, by definition, at negative energies and so are not considered to be within the continuum. While they do represent a possible mechanism whereby an electron can be removed from an ion, the process resembles colli-sional charge exchange in which increased ionization of one ion is accompanied by an equal reduction in that of another with little or no change in the plasma potential energy. Moreover, since there is no change in the overall particle number, if the electron remains bound, there is no direct contribution to the pressure. This strongly suggests that this process is therefore better regarded as being separate from normal photoionization, while transient states and threshold states are evidently not the same things. Transient states are one-body bound states that are delocalized by interaction with neighboring ion(s) while threshold states are an emergent property of the many-body (electron f plasma) system. In moderately-coupled or weakly-coupled plasmas, transitions into transient states will merge with the continuum via the Inglis—Teller effect and can be properly described in those terms without invoking an additional continuum-lowering effect. In particular, transient states are spectroscopically bound while threshold states are spectro-scopically in the continuum.

7. Equation of state

The treatment of continuum lowering in multicomponent Coulomb system comprising ions and electrons in both free and bound states is intrinsically linked to a non-trivial equation of state

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model, which is developed in Appendix B. It is found that the (approximate) applicability of this model to real plasmas is apparently limited only by the inability of the model to treat pressure ionization (8V < 0) which is attributed to the approximation whereby the ions are treated as structureless point charges, and the lack of dynamical separability between electrons bound in threshold states and those in the true continuum. Nevertheless it represents an important enhancement over models that treat the component charges as being inert, one which can be considered to be approximately applicable in all regimes where pressure ioniza-tion is not an issue. The deficiency in the pressure ionization regime appears to reside in the equation of state rather than in the model of the IPD. Nevertheless this raises doubts about the general applicability of the Coulomb model and this is something that needs more careful examination, theoretically, using a model that treats pressure ionization, or by direct experimental observation.

8. Comparisons with experiment

The experiments that we have modeled fall into two categories: direct measurements using a tuneable FEL [2,3]; and measurements of 1—3 lines in shocked warm dense aluminium plasmas created using a high-power laser [4]. The former provide direct measurements of the ionization thresholds, and hence IPDs, that are virtually model independent but only explore ion configurations, at a fixed density, at the limit of strong coupling (Ti = 0). The results of these experiments are found to be remarkably consistent with the Ecker—Kroll formula

-(Zj + 1)(1 + Z)1/3uWs

The experimental results along with the results of various calculations are shown in Fig. 1, which clearly shows the inadequacy of the Stewart—Pyatt equilibrium model. Most importantly, the experiment is also found to be reasonably well explained by the theory described in this work. In these experiments, the ion plasma coupling parameter is estimated to be in the range 3000—50,000 putting these plasmas clearly in the solid-state regime where the microfield can have no effect on the ionization potential depression.

An alternative technique involves measuring the strengths of the 1 —3 lines as a function of temperature and density and to determine whether the IPD encompasses the 3p states, for example. This is able to explore higher ion temperatures as well as a range of densities, but depends upon some modeling, to determine where the n = 3 levels are expected to lie in high charge states as well as to infer the plasma conditions. Recent experimental results for aluminium from the Orion high-power laser [3], along with predictions of the various models for the putative plasma conditions are given in Table 2.

The temperature-density grid is quite coarse and the experiment is thus only able to bracket the IPD along the track of the measurements. The uncertainty in the density at the critical densities, 5.5 g/ cc and 9 g/cc is quoted as being around 10%, which translates to a 3% error in uWS. Nevertheless the measurements are able to discriminate between various models to the extent that it can be said that the results, shown in Fig. 2, are consistent with the model derived in this work, equation (81), as well as, as claimed by the authors of the experiment, to the modified ion-sphere (IS) formula

-2 (ZJ +1)

-\ 1/3

(cf equation (72) with C = 9/10) albeit applied in a regime of moderate coupling. On the other hand, they are inconsistent with

both Stewart—Pyatt (14) and Ecker—Kroll (169), with the former under-estimating the IPD and the latter considerably overestimating it.

In this experiment, the plasma coupling strength F is in the range 2—3, indicating a moderately-coupled fluid plasma and that the n = 3 bound states lying close to the continuum can be expected to be perturbed by the microfield. The observations are however not consistent with an additional microfield continuum lowering of the magnitude predicted by (168) confirming that affected states, whether transient or not, manifest themselves spectroscopically as bound states.

The critical measurements are those for 5.5 g/cc, from which n = 1 —3 emission lines are observed, and 9 g/cc, which is characterized by an absence of n = 1—3 lines. The plasma is determined to be predominantly mixtures of He-like, H-like and fully stripped ions under these conditions. Calculations, employing a simple screened hydrogenic model, without [-splitting, differ in that the IS model predicts that n = 3 levels should be spectroscopically bound in He-like Al at 5.5 g/cc, whereas the relaxation model proposed here does not. However n = 3 bound levels are found to be present in H-like Al at 5.5 g/cc using both models. At 9 g/cc, the relaxation model predicts that the n = 3 levels should be unbound in all ion states, and that no n = 1 —3 lines should be seen. Taking a realistic view, the experiment is probably unable to discriminate between the predictions for the SIPD given by equations (81) and (170) in this regime, on account of the uncertainties in the plasma conditions and those inherent in the atomic calculations upon which the interpretation may depend. Nevertheless, the new model proposed here does produce a slightly better fit by unequivocally removing the n = 1 —3 lines from the 9 g/cc case. On this basis we conclude that the new IPD model presented in this paper is fully consistent with the observations of this experiment, unlike any of the proposed alternatives. Stewart—Pyatt, as per (14), for example, predicts n = 1—3 lines at 9 g/cc, while the IS model is marginal under these conditions.

The fact that the new model is able to give a reasonably good account of both experiments is compelling. The alternative models, Ecker—Kroll and Ion-Sphere, fit the data only in the regimes of the FEL and laser-shock experiments respectively, and neither fits both experiments. Without an underlying explanation, these alternative formulas should be considered as being no more than fits to the data.

9. Conclusions

On the basis of a theoretical re-examination of the IPD problem, motivated and supported by observational evidence from recent experiments on very dense plasmas, we conclude that the Stewart— Pyatt (SP) model, provides only an incomplete description of the ionization potential depression (IPD) as one would define it in terms of a spectroscopic measurement, or even in some equation of state contexts. The SP model and its close derivatives provide only the static continuum lowering (SCL, here denoted by AU), which represents the average effect of the electrostatic field of the surrounding plasma on the electronic states. Closely related to the static continuum lowering, but distinct from it, is the thermody-namic ionization potential depression (TIPD, here denoted by AW), which is the change in the thermodynamic free energy associated with ionization. This accounts additionally for the entropy-related terms in the free energy, which arise when the average potential energy becomes temperature dependent. The TIPD is shown to be that which appears in the Saha equation thus demonstrating direct relevance of the TIPD to equation of state modeling. In the limit of strong coupling (high densities and/or low temperatures) when the electrostatic energy becomes independent of temperature (eg the

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Fig. 2. Comparison of different average-ion IPD calculations for the ORION IPD experiment of Ref. [4]. The legend is the same as Fig. 1 with the addition of Ion-Sphere, which is as per equation (170), which represents the experiment's authors' considered match to the results. Unlike the FEL experiment, there are no quantitative data. The experiment observes n = 3 lines up to and including 5.5 g/cc but not at the higher densities. Stewart—Pyatt predicts the presence of n = 3 lines up to 9 g/cc while Ecker—Kroll predicts their absence at 2.5 g/cc and above.

ion-sphere approximation) the SCL and the TIPD become synonymous. In general the TIPD is less, in the sense of less depression, than the SCL.

Spectroscopic and other dynamical processes may occur on timescales too fast for the surrounding plasma to respond or come into equilibrium, when they cannot be considered to be transitions between equilibrium states of the plasma. In the case of near-threshold ionization, the electron is deposited close to the parent ion and has to move away before the surrounding plasma has anything to respond to. Neither the TIPD nor the SCL is then a good measure of the ionization potential. The spectroscopic ionization potential depression (SIPD, denoted by ZDu) applies to such adia-batic processes in which energy is exchanged locally by the atomic system interacting only with the photon. The absence of any energy exchange with the plasma surroundings is accounted for by subtracting out an additional relaxation effect. In general, h\Du\ > |DU| > |DW|.

The SIPD model proposed in this paper, as represented most generally by equation (79) above, and by the approximate and limiting formulas, (78), (81) and (82), accounts reasonably well for the published observational data, over a wider range of conditions than any of the other simple models on offer. However, there is some doubt over the validity of the model in regimes of pressure ioniza-tion due to the underlying equation of state model lacking validity.

A closer look at the plasma equation of state reveals not only close links with the TIPD in the context of the Saha equation, but also a more formal link between the static continuum lowering DU and the correlations between the internal state of the ion and the potential energy due to the surrounding plasma (as expressed by equations (S.44) and (S.45) in Appendix B). The fundamental quantity linking the various quasi-static IPDs (SCL and TIPD) is the shift in the Helmholtz free energy, DF, use of which helps maintain thermodynamic consistency. Accordingly, it is possible to model the equation of state of a Coulomb system and derive equations describing the underlying charge-state distribution that are posed as being applicable in all regimes apart from those where pressure ionization is occurring.

Acknowledgments

The author acknowledges discussions with Justin Wark, Orlando Ciricosta and Dave Hoarty and is grateful to Dave Hoarty and Justin

Wark for making their data available. The author is indebted to Stephanie Hansen for a critical reading of the manuscript and for drawing attention to some early mistakes.

Appendix A. Supplementary data

Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.hedp.2014.04.003.

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