#
Hydrogen-, Helium-, and Lithium-like Bound States in Classical and Quantum Plasmas^{ †}

^{1}

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## Abstract

**:**

## 1. Introduction: Effective Interactions

- (1)
- Chemical forces are short-range and show strict saturation. For example, a hydrogen atom may bind with second one by chemical binding forming the H${}_{2}$ molecule, but there is no way to bind a third one with the same-strength force.
- (2)
- Coulomb forces are long-range and are strictly additive; this is limited only by the tendency to form neutral configurations.

- (i)
- The discrete states near to the continuum edge are not stable due to thermal collisions and screening effects;
- (ii)
- the contribution of these states is compensated by contributions of nearby states in the continuum;
- (iii)
- the discrete states near to the continuum edge are also destroyed if the wave functions are larger than the Debye length.

## 2. Physical and Chemical Level of Descriptions

#### 2.1. Virial Expansions and Mass Action Constants

#### 2.2. Mass Action Functions for Coulombic Pairs

**Pair association**: In earlier works, Falkenhagen and coworkers [2,9] proposed for the pair formation of charged hard spheres with diameter R and Bjerrum parameter $b=\ell /R$ the mass action constant [2,7,9]

**Quantum theory of pair association**: As first shown by Planck, Brillouin, Larkin and others [1,15,16], the quantum statistics of pair formation like in hydrogen plasmas leads to quite similar expressions, where the PBL partition function $\sigma $ appears [15]. In quantum statistics the partition function is expressed by a power series in the interaction parameter [15] ${\xi}_{\pm}=-{Z}_{i}{Z}_{j}\ell /{\lambda}_{\pm}$. This series for $\sigma \left(\xi \right)$-function is in full agreement with the early result by Max Planck [1]. There is also a structural analogy to the series $m\left(\xi \right)$ for the classical case. The only difference is that we have to use other coefficients and other definitions of the parameter $\xi $ [15]. The virial function for the quantum case ${Q}_{4}\left(x\right)$ has more advanced coefficients expressed by Riemann’s known Zeta and Euler’s Gamma functions, well known from mathematical physics [15,16]. The hydrogenic partition function may be written as a function of the interaction parameter including the elactronic spin states ${s}_{e}=1/2$ in the form

#### 2.3. Mass Action Functions for Coulombic Triples and Quadruples

## 3. Classical Osmotic Pressure and Chemical Potential including Association

#### 3.1. Semi-Chemical Description of Associating Systems

#### 3.2. Applications to $CaC{l}_{2}$, $LaC{l}_{3}$ and a Seawater-Model

## 4. Quantum Bound States of He and Li Plasmas Including Excited States

- (i)
- in simplest case the reduced uncropped ${K}^{0}\left(T\right)$ where only the ground state ${e}^{-{E}_{0}/{k}_{B}T}$ is taken for the intrinsic partition function, all excited states and the low-order terms with respect to the interaction, $-1+{E}_{0}/{k}_{B}T$, are neglected;
- (ii)
- the reduced cropped mass action constant ${K}^{\mathrm{cropped}}\left(T\right)$ where only the ground state contribution ${e}^{-{E}_{0}/{k}_{B}T}-1+{E}_{0}/{k}_{B}T$ to the intrinsic partition function is taken and the summation over all excited states is neglected;
- (iii)
- the full cropped expression $K\left(T\right)$ given above, which contains the summation over all excited states.

## 5. Pressure of Quantum Plasmas including Bound State Effects

#### 5.1. The Valley of Bound States in the Relative Pressure and the Concatenation Method

#### 5.2. Applications to Hydrogen, Helium and Lithium Plasmas

**Hydrogen plasmas:**For simplicity, we have treated only approximately the formation of H${}_{2}$-molecules, which dominate at temperatures below 10,000 K [19]. The shoulder in Figure 6 reflects the formation of atoms and the minimum reflects the formation of atomic and molecular bound states. In the temperature range studied here ($10,000$–$50,000$ K), no first order phase transitions were observed in our approximation.

**Helium and lithium plasmas:**Except for the need to include higher bound states, the details of the calculations for these plasmas are analogous to hydrogen [21]. Explicitly, for helium and lithium plasmas, we obtain expressions of the form (note that the ${z}_{e},{z}_{i}$ are fugacities [15,20,21]

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Typical factors in cluster integrals representing.

**Left**panel: For repulsive ions exponential function (green), same function subtracting first two expansion terms (red), only second-order term (blue line).

**Right**panel: exponential term subtracting the lowest power up to ${e}^{4}$ (in red) and the bare term of order $\left(O\left({\xi}_{ij}^{4}\right)\right)$ gives indeed a qualitatively correct overall shape. Note that all functions are multiplied with the spatial pre-factor ${x}^{2}$.

**Figure 2.**

**Left**panel: Relative pressure for the ions Na${}^{+}$ (above) and Cl${}^{-}$ (below) as well as the mean for a NaCl-solution.

**Right**panel: Relative (individual) osmotic pressures for a mixture of six ions mimicking seawater including association in dependence on salinity S (gram salt per liter). We are sorting from below looking at the values at salinity $S=5$. The lowest curve represents the ion SO${}_{4}^{2-}$, then follow the curves for Mg${}^{2+}$, K${}^{+}$, Ca${}^{2+}$, Cl${}^{-}$, Na${}^{+}$.

**Figure 3.**We show the activity coefficients of several ions and electrolytes. From above: Cl in CaCl${}_{2}$ (red) and Cl in LaCl${}_{3}$ (green), CaCl${}_{2}$ (blue), LaCl${}_{3}$ (magenta), Ca${}^{2+}$ (turquoise), La${}^{3+}$ (black). The activities were calculated in the semi-chemical approach neglecting so far quadruples, what could be the reason for too low values for the activities of CaCl${}_{2}$ and LaCl${}_{3}$, Ca${}^{2+}$ and La${}^{3+}$. The points denote data measured by Wilczek-Vera et al. [34] for Cl${}^{-}$ and La${}^{3+}$ in LaCl${}_{3}$.

**Figure 4.**

**Left**panel: Degrees association for ions in seawater. The curves describe (looking at mean salinities from above): Mg${}^{2+}$, SO${}_{4}^{2-}$, Ca${}^{2+}$, Na${}^{+}$, K${}^{+}$, Cl${}^{-}$, as a function of salinity.

**Right**panel: Degree of triple association in seawater for the triples MgCl${}_{2}$ (red) and Na${}_{2}$SO${}_{4}$ (green) which is according to our estimates rather low. Typical is the maximum at finite concentrations/salinities.

**Figure 5.**Mass action constants for the He${}^{+}$ ion. The ratio ${K}^{\mathrm{cropped}}\left(T\right)/{K}^{0}\left(T\right)$ shows the effect of subtracting the low-order terms with respect to the interaction, $-1+{E}_{0}/{k}_{B}T$, for the reduced intrinsic partition function where only the ground state is considered. The ratio $K\left(T\right)/{K}^{\mathrm{cropped}}\left(T\right)$ compares the fully cropped intrinsic partition function with the reduced cropped partition function to show the effect of including excited states.

**Figure 6.**Hydrogen plasmas: We demonstrate the method of smooth concatenation for a hydrogen plasma at $T=20,000$ K. After identifying a crossing point near to ${n}_{i}\simeq {10}^{23}\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{-3}$ we construct a smooth concatenation of the branches on the left (red) and right (green) sides of the intersection of the low-density branch (red) with the high-density branch (green). The blue curve shows the result of the concatenation using complementary $\mathrm{tanh}$-functions ($a\simeq 1;y={n}_{e}{\mathsf{\Lambda}}_{e}^{3}/2$).

**Figure 7.**Helium and Lithium plasmas: Low and high-density approximations are concatenated by complementary tanh(x) functions at the crossing point. Left panel: Helium plasmas at 30,000 K (red), 50,000 K (green) and 80,000 K (blue curve) as functions of the total ion density. Right panel: Lithium plasmas at temperatures 80,000 K (red), 120,000 K (green), 200,000 K (blue). The minima are due to the formation of bound states.

**Table 1.**Table of contact distances for several ion pairs including alkaline earth metal ions, sulfate ions and adapted “ideal” seawater ions according to [18]. Further we give for quantum plasmas some typical values of the De Broglie thermal wavelength at $T={10}^{4}$ K (all length given in pm).

$\mathit{i}-\mathit{j}$ | ${\mathit{R}}_{\pm}$ | ${\mathit{R}}_{++}$ | ${\mathit{R}}_{--}$ | $\mathit{i}-\mathit{j}$ | ${\mathit{\lambda}}_{\pm}$ | ${\mathit{\lambda}}_{++}$ | ${\mathit{\lambda}}_{--}$ |
---|---|---|---|---|---|---|---|

$Na-Cl$ | 350 | 470 | 360 | ${H}^{+}-e$ | 204 | 7.4 | 297 |

$K-Cl$ | 320 | 400 | 360 | $H{e}^{2+}-e$ | 203 | 5.3 | 297 |

$Mg-Cl$ | 400 | 280 | 360 | $L{i}^{3+}-e$ | 202 | 4.3 | 297 |

$Ca-Cl$ | 500 | 320 | 360 | ||||

$K-S{O}_{4}$ | 340 | 400 | 300 | ||||

$Mg-S{O}_{4}$ | 290 | 400 | 300 | ||||

$La-Cl$ | 270 | 430 | 360 |

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**MDPI and ACS Style**

Ebeling, W.; Röpke, G.
Hydrogen-, Helium-, and Lithium-like Bound States in Classical and Quantum Plasmas. *Plasma* **2023**, *6*, 1-26.
https://doi.org/10.3390/plasma6010001

**AMA Style**

Ebeling W, Röpke G.
Hydrogen-, Helium-, and Lithium-like Bound States in Classical and Quantum Plasmas. *Plasma*. 2023; 6(1):1-26.
https://doi.org/10.3390/plasma6010001

**Chicago/Turabian Style**

Ebeling, Werner, and Gerd Röpke.
2023. "Hydrogen-, Helium-, and Lithium-like Bound States in Classical and Quantum Plasmas" *Plasma* 6, no. 1: 1-26.
https://doi.org/10.3390/plasma6010001