Refractive index measurement of hydrogen isotopologue mixture and applicability for homogeneity of hydrogen solid at cryogenic temperature in fusion fuel system

When H₂–D₂ mixture is solidified by thermal conductivity, the H₂–D₂ mixture has a temperature gradient. Since the refractive index is a function of the density of hydrogen isotopologues, refractive index data for H₂ and D₂ in various conditions are indispensable. To calculate the refractive index of H₂ and D₂ at a given temperature, the empirical formula between density ρ and refractive index n, known as the Lorentz–Lorenz relation, was used:

$$\begin{align} \frac{M}{\rho } \cdot \frac{{{n^2} - 1}}{{{n^2} + 2}} = \frac{{4\pi }}{3}{N_{\text{A}}}\alpha \end{align} \tag{ 1 }$$

where $\alpha$ is the molecular polarizability, M is the molecular mass, and N_A is the Avogadro constant. The specific refraction ${r_\lambda }$ can be identified via the Lorentz–Lorenz relation

$$\begin{align} {r_\lambda } = \frac{1}{\rho } \cdot \frac{{{n^2} - 1}}{{{n^2} + 2}} .\end{align} \tag{ 2 }$$

The specific refraction is a function of wavelength. Cauchy's dispersion formula is an empirical expression giving an approximate relation,

$$\begin{align} {r_\lambda } = {r_\infty } + \frac{A}{{{\lambda ^2}}} + \frac{B}{{{\lambda ^4}}},\end{align} \tag{ 3 }$$

where A and B are coefficients for λ in angstroms and r in cm³ g⁻¹. The values of A and B depend on the medium. ${r_\infty }$ is the specific refraction at λ = $\infty$. the ${r_\infty }$ of hydrogen is related to molecular polarizability and is approximately independent of temperature and density over a wide range [9]. Based on equations (2) and (3), the refractive index n, dependence of the density ρ, and wavelength of incident light (λ) are adequately represented by

$$\begin{align} {n_i} = \sqrt {\frac{3}{{1 - \rho \left( {{r_\infty } + \frac{A}{{{\lambda ^2}}} + \frac{B}{{{\lambda ^4}}}} \right)}}} - 2 .\end{align} \tag{ 4 }$$

The ${r_\infty }$ coefficients A and B for H₂ and D₂ at STP conditions were reported by Childs and Diller [9]. These data were summarized in table 1.

Table 1. Parameters reported by Childs and Diller [9].

	A	B	${r_\infty }$
	${{{{\unicode{x00C5}}}}^2}{\text{cm}}\;{{\text{g}}^{ - {\text{1}}}}$	${{{{\unicode{x00C5}}}}^4}{\text{c}}{{\text{m}}^3}\;{{\text{g}}^{ - {\text{1}}}}$	${\text{c}}{{\text{m}}^3}\;{{\text{g}}^{ - {\text{1}}}}$
H2	$0.7799569 \times {10^6}$	$0.495126 \times {10^{12}}$	1.0050
D2	$0.3616378 \times {10^6}$	$0.432143 \times {10^{12}}$	0.4983

For H₂ and D₂ in solid and liquid states, the density is strongly affected by temperature. The empirical formulas of density or molar volume as a function of temperature T (K) are as follows.

Liquid H₂ [10]:

$$\begin{align} {\rho _{\text{m}}} = 0.0134572 + 0.0092597 \times {{\left( {32.984 - T} \right)}^{\left( {\frac{1}{3}} \right)}} \end{align} \tag{ 5 }$$

where ρ_m (mole cm⁻³) is mole density (ρ_m=ρ/M), M (g mol⁻¹) is molar mass, and ρ (g cm⁻³) is density. The maximum deviation of the estimated mole density is 0.015% of the mole density.

Solid H₂ [11]:

$$\begin{align} {V_{\text{m}}} = 22.7121856 - 0.03716186 \times T + 0.00532275 \times {T^{\,2}} \end{align} \tag{ 6 }$$

where V_m (cm³ mol⁻¹) is mole volume (V_m = M/ρ), The estimated V_m should be in error by no more than 0.4%.

Liquid D₂ [12]:

$$\begin{align} {V_{\text{m}}} = 20.188 + 0.03587 \times T + 0.006565 \times {T^{\,2}} \end{align} \tag{ 7 }$$

with V_m in cm³ mole⁻¹. The standard deviation from this equation is 0.01 cm³ mole⁻¹ (or 0.04% of ${V_{\text{m}}}$).

Solid D₂ [13]:

$$\begin{align} \rho = 0.20459 - 0.000026337 \times {T^{\,2}} \end{align} \tag{ 8 }$$

with ρ in g cm⁻³. The standard deviation of the estimated density is 0.00156.

In this experiment, the wavelength is a fixed value (λ = 543 nm). Thus, we can calculate the refractive index for a given temperature using equations (4)–(8). The standard deviation of the refractive index can be calculated by the propagation of uncertainty. The standard deviations of the refractive index are shown in table 2.

However, there are few reports about the refractive index of solid H₂, liquid D₂, and solid D₂. There is not enough valid data to support the reliability of the empirical formula. The estimated values of solid D₂ in particular have considerable standard deviations. To improve the accuracy of the equation, the temperature dependence of the refractive index of H₂ and D₂ in the liquid and solid phases were measured by the same method as previous research [8].

In this experiment, H₂–D₂ mixture was used instead of D₂–T₂ mixture. Our previous experiment measured the refractive index of D₂–T₂, whereas H₂–D₂ mixture was used to observe the solidification behavior of hydrogen isotopologue without the effect of decay heat of tritium.

The H₂–D₂ mixture was made from the commercially available hydrogen gases where these purities were 99.999% for H₂ and 99.5% for D₂, and the gas composition was adjusted to be H₂:D₂ = 1:1. The solidification procedure is as follows.

• (1)  
  The temperature of the cell (see figure 2) was cooled down to 25.000 K; it was above the triple point of H₂ and D₂. Subsequently, gaseous H₂–D₂ was supplied to the cell.
• (2)  
  The bottom of the cell was gradually cooled, and the H₂–D₂ mixture was liquefied.
• (3)  
  The H₂–D₂ liquid became solid when the bottom of the cell was 16.074 K. A vapor bubble was observed at the top of the cell as shown in figure 3.
• (4)  
  The refractive index of liquid and solid H₂–D₂ was measured at optically flat points in the cell.

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The refractive index of the H₂–D₂ mixture depends on the ratio of H₂ and D₂. If the refractive index is simultaneously measured at many points on the H₂–D₂ mixture, the distribution of H₂–D₂ would be known. Thus, the distribution of H₂ and D₂ can be determined by the refractive index distribution. A vertical refractive index distribution measurement system was constructed, as shown in figure 4. In this system, the point of the laser light as a primary incident light was dispersed into the line laser light by the cylindrical lens. The measurement procedure is as follows.

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• (1)  
  The laser pattern of the incident laser as the reference position on the screen was determined by the laser passing through the prism cell in a vacuum.
• (2)  
  Subsequently, the H₂–D₂ mixture was loaded into the wedge-shaped optical cell. The H₂–D₂ mixture was liquefied and solidified in the cell as previously described.
• (3)  
  The line laser was refracted by passing the H₂–D₂ mixture in the cell. The position of the refracted laser on the screen was recorded. Figure 5 shows the typical photograph of laser patterns on the screen.
• (4)  
  The brightness of each pixel in the photograph was analyzed. To achieve the peak position of the laser pattern for the vertical direction, the brightness distribution of the laser pattern for the horizontal direction was fitted by the Gauss function.
• (5)  
  The distance of the peak position between the incident and refracted lasers, L1 in figure 4(b ), was calculated. The refracted angle by H₂–D₂, β (P) was calculated from

  $$\begin{align} \beta \left( P \right) = \arctan \left[ {\frac{{{L_1}}}{{{L_2}}}} \right] + \alpha .\end{align} \tag{ 9 }$$

  The refractive index, n_HD(P) is calculated by Snell's law,

  $$\begin{align} {n_{{\text{HD}}}}\left( P \right) \cdot {\text{sin}}\alpha = {n_0} \cdot \sin \beta \left( P \right) .\end{align} \tag{ 10 }$$

  The distribution of refractive index for the vertical direction was obtained.
• (6)  
  Using refractive index mixing rules can accurately predict the volume fractions of the mixture component. Various empirical and semi-empirical relations have been reported [14]. Among them, the Lorentz–Lorenz equation is approximately valid for solids as well as liquids and gases. The refractive indices of hydrogen [15] and deuterium [8] are calculated. The Lorentz–Lorenz mixing rule is represented as

  $$\begin{align} \frac{{{n_{{\text{HD}}}}^2 - 1}}{{{n_{{\text{HD}}}}^2 + 2}} = \left( {\frac{{n_{\text{H}}^2 - 1}}{{n_{\text{H}}^2 + 2}}} \right){\phi _{\text{H}}} + \left( {\frac{{n_{\text{D}}^2 - 1}}{{n_{\text{D}}^2 + 2}}} \right){\phi _{\text{D}}} \end{align} \tag{ 11 }$$

  where n_HD, n_H, and n_D are refractive indices of the H₂–D₂ mixture, H₂, and D₂, respectively. Φ_H and Φ_D are volume fractions of H₂ and D₂. For the convenience of calculation, the ratio of H and D was represented by the mole fraction of H₂ and the homogeneity of the H₂–D₂ mixture can be evaluated by the distribution of the H₂ mole fraction. The mole fraction of H₂, x_H, is calculated from

  $$\begin{align} {\phi _{\text{H}}} = {x_{\text{H}}}{v_{\text{H}}}/\left( {{x_{\text{H}}}{v_{\text{H}}} + {x_{\text{D}}}{v_{\text{D}}}} \right) \end{align} \tag{ 12 }$$

  $$\begin{align} {x_{\text{H}}} + {x_{\text{D}}} = 1 \end{align} \tag{ 13 }$$

  where x_H is the mole fraction and v_H is the molar volume of H₂. x_D is the mole fraction and v_D is the molar volume of D₂.

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The refractive indices of liquid and solid H₂ and D₂ were measured in the range of 13.752 K–15.000 K for H₂ and 13.579 K–21.984 K for D₂. The measurement uncertainty was calculated from the measurement error of the distance between the incident and refracted laser positions, L₁, by the propagation of uncertainty. The measurement uncertainties are 0.0000596 for solid H_2, 0.0000515 for liquid D₂, and 0.0000520 for solid D₂. Figure 6 shows the refractive indices of H₂ and D₂ as a function of the temperature. As the temperature decreased, the refractive index increased.

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Although at cryogenic temperature, ortho-hydrogen tends to convert to para-hydrogen [16], and para-deuterium tends to convert to ortho-deuterium [17], the ortho–para conversion rate is slow without a catalyst or other facilitating mechanism. At 20.4 K, the temperature at 1 atmosphere pressure liquid normal-H₂ will be converted after several days and require weeks to approach equilibrium [18]. The ortho–para conversion rate of deuterium is slower than hydrogen [19]. The measurement time for solid H₂ was 1 h and solid D₂ was 2 h. In such a short period, the effect of ortho–para conversion was negligible [20]. Thus, in the study, the hydrogen is normal hydrogen (ortho–para ratio of 3:1), and the deuterium is normal deuterium (ortho–para ratio of 2:1).

The estimation values of refractive indices were calculated by analysis using equations (4)–(8). The measured values of refractive indices were compared with previous data and the estimation values. The refractive index obtained from liquid H₂ agrees with the values reported by Diller [15]. Other data also agreed with the estimation values. Namely, both the refractive index measurement and the analysis procedure were appropriate. However, the measured values of solid D₂ around the melting point were slightly higher than that estimated by equation (4), as seen in figure 6. The estimation of the refractive index of solid D₂ by equation (4) would be limited around the melting point. Thus, the relationship between the temperature and refractive index was fitted by the 2nd-order function. The function fitted to the refractive index of solid D₂ was

$$\begin{align} n = 1.148 + 0.0013349 \times T - 0.000048494 \times {T^{\,2}} .\end{align} \tag{ 14 }$$

The standard deviation was about 0.000193.

Indeed, the isotopic exchange will lead the H₂–D₂ mixture to reach chemical equilibrium between H₂, D₂, and Hydrogen Deuteride (HD); but at ambient temperatures, this exchange reaction is slow (of many weeks) [21]. And the equilibrium constant for H₂+D₂ ⇌ 2HD decreases with decreasing temperature [22]. The measurement time for solid H₂–D₂ mixture was 3 h. The HD generated by the isotopic exchange was negligible.

The refractive index vertical distribution of H₂–D₂ mixture at optically flat points in the cell was measured during solidification in the range of 13.415 K–16.498 K. Two typical results are shown in figure 7 for clarity. The blue curve shows the refractive index vertical distribution of the liquid H₂–D₂ mixture at 16.286 K. The red curve shows the refractive index vertical distribution at 15.598 K, the top part is liquid H₂–D_2, and the bottom part is solid H₂–D₂. The state of H₂–D₂ in the cell is shown in figure 3.

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Since the temperature difference between the top and bottom of the cell was small, and the temperature dropped slowly, the temperature gradient can be regarded as an approximately uniform distribution in the vertical direction of the cell. Therefore, while measuring the refractive index, the temperature of this measurement point was also calculated, and the relationship between refractive index and temperature was obtained. The refractive index of H₂ and D₂, n_H and n_D, at a given temperature can be corrected by equation (4). Finally, the molar fraction distribution of H₂ was calculated using equations (11)–(13) as shown in figure 8. The statistical dispersions are 0.0334 for liquid H₂–D₂ and 0.0197 for solid H₂–D₂.

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At 15.598 K, the mole fraction of H₂ in the solid phase gradually decreases approaching the bottom, meaning that the composition of H₂–D₂ in the primary crystal, which is the solid initially frozen, gave the higher D₂ composition rather than the original composition. This behavior can be explained by the H₂–D₂ phase diagram which is the solid solution type. This point will be described in detail later.

Additionally, at 16.286 K, the H₂ mole fraction of liquid H₂–D₂ was approximately homogeneous, which could be represented by the spatial average value, about 0.485. Theoretically, the H₂ and D₂ contents are conserved. At a different temperature, the spatial average of the mole fraction should be the same, but at 15.598 K, the spatial average of the H₂ mole fraction was about 0.527.

The reasons for this are twofold: (a) the data at 15.598 K are discontinuous around the interface between liquid and solid. Since on the inside of the cell, H₂–D₂ solidified faster than on the outside, the interface was not horizontal, as shown in figure 9. As the incident line laser passed through the interface, the path of laser was changed (refraction, reflection, or scattering), and this part of the laser did not project on the screen. That led to a partial data loss of solid H₂–D₂. The H₂ molar fraction of solid was small, and thus the loss of solid data resulted in a large H₂ molar fraction of the overall H₂–D₂.

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(b) On the inner surface of the copper cell, H₂–D₂ solidified faster, so there was more solid H₂–D₂ on both sides than in the middle (measurement position), as shown in figure 9. Similarly, since the content of D₂ in the solid was higher, the H₂ molar fraction of liquid H₂–D₂ increased at the measurement location.

This issue does reflect a limitation of our experiment, but it does not prevent us from observing inhomogeneity in the solid H₂–D₂.

The mole fraction vertical distribution of H₂ at various temperatures is summarized in figure 10. The mole fraction of H₂ in liquid phase was relatively uniform. On the other hand, in solid phase at the lower side of the cell, H₂ has a wider mole fraction. The composition of solid H₂–D₂ was inhomogeneous. The range of the mole fraction of H₂ was found to be from about 0.35–0.60. It is mentioned that the mole fraction of H₂ in solid increased approaching the interface between solid and liquid. The primary crystal that has higher D₂ content was formed at the bottom of the cell.

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The phase diagram of H₂–D₂ at cryogenic temperature is shown in figure 11 [23]. Solid H₂–D₂ mixture is a completely soluble solid solution (H₂ and D₂ are soluble at all concentrations). In an equilibrium condition, the mole fraction of H₂ in the H₂–D₂ solid becomes a constant and the solid is a single phase. The phase change in the equilibrium condition is described as follows. The numbers in figure 11 indicate the solidification process.

• (1)  
  As an example, a 50/50 H₂–D₂ mixture is used. The mixture is cooled down to the liquidus line. The mixture is a single liquid phase. During the solidification process of liquid H₂–D₂, the liquid and solid phases appear .
• (2)  
  The H₂–D₂ solid (primary crystal) appears when the temperature of H₂–D₂ drops below the liquidus line. Since D₂ has a higher freezing point than H₂, D₂ solidifies (crystallizes) faster. The H₂–D₂ solid (primary crystal) is more enriched in the D₂ component. In this case, the primary crystal is formed at 16.600 K and its composition is 0.36 of mole fraction of H₂. This composition means that the D₂ content of solid H₂–D₂ is higher than that of the original composition.
• (3)  
  Crystal formation continues, and the composition of H₂ in the liquid phase increases with decreasing temperature because D₂ is extracted into the solid phase. The composition of liquid will change along the liquidus line.
• (4)  
  Meanwhile, the crystals exchange material with the H₂-rich solution. The composition of the solid H₂ is moving down the solidus line. Finally, the composition of D₂ in the solid is recovered into the original composition.
• (5)  
  When the solidification is complete, the residual liquid is completely solidified, and the liquid phase disappears.
• (6)  
  The H₂ mole fraction in solid phase (terminal crystal) becomes a constant at different temperatures.

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The temperature and refractive index of each measurement point was obtained, and the relationship between refractive index and temperature is shown in figure 12. We can observe the change in the H mole fraction during solidification.

• (1)  
  The H₂–D₂ mixture was cooled down. The ratio of H₂ and D₂ was about 48:52.
• (2)  
  The primary crystal of H₂–D₂ appeared at 16.100 K. The liquid H₂–D₂ reached supercooling state.
• (3)  
  The composition of liquid H₂–D₂ changed along the direction of the liquidus line.
• (4)  
  As solidification continued, the composition of solid H₂–D₂ did not change along the solidus line. This could be explained by the fact that the D₂-rich crystals continually precipitated at the bottom of the cell, and the later-formed crystals covered the previous ones (see figure 3) and prevented the material exchange between the previously formed crystals and solution.
• (5)  
  On the other hand, as the composition of H₂ in the liquid phase increased with continuing removal of crystals, the successively formed crystals became continuously more H₂; thus, the terminal crystal had higher H₂ content. The diffusion of H₂ and D₂ in solid phase was too slow to form a homogeneous solid within the measurement period (around 3 h). Finally, each crystal in the solid had a different composition.

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Each point on this chart (figure 12) is the H₂ mole fraction at a different temperature. Because the scale of this chart is too large, it looks like a straight line.

Very few studies about solid D₂–T₂ mixture have been reported. In McKenty et al [24], the distribution of D₂–T₂ in a solid is simulated with H₂–D₂ data experimentally obtained. The distribution of H₂–D₂ is obtained by the IR transmission coefficient of H₂–D₂ data. The experimental results show a 10.5% variation in deuterium fraction across the cryogenically formed layer of the solid H₂–D₂ sample. In our experiment, the range of the H₂ mole fraction was from 0.396 to 0.599 in solid H₂–D₂ (terminal crystal). The variation in deuterium fraction is about 20%. McKenty et al [24] also supported our observation that the inhomogeneity of the mole fraction in solid appeared during the solidification. However, the variation in deuterium fraction is larger than the literature data. There are two possible reasons: (a) the solidification process proceeded so fast that the later-formed crystals quickly covered the previously formed crystals, preventing the material exchange between the previously formed crystals and the solution. (b) The experiment time was insufficient. The content of D₂ in the solid phase did not reach an equilibrium state during the measurement period. In figure 12, the range of the H₂ mole fraction was 0.384–0.617 at 13.612 K; the range of the H₂ mole fraction was 0.396–0.599 at 13.612 K. The range narrowed; if it took more time, the scope might be narrowed down a bit more.

On the other hand, the decay heat of T would help the D₂–T₂ re-arrangement to be homogeneous in the solid. However, we have no result for the D₂–T₂ rearrangement in ice. This point is an important matter for D₂–T₂ solid in fusion fuel production. Because of isotopic exchange, D₂–T₂ takes place among D₂, Deuterium Tritide (DT), and T₂. The tritium decay heat can accelerate the rate of exchange reaction [25]. The half-time for this reaction has been estimated to be about 18 h in the 21 K liquid D₂–T₂ [26]. The Lorentz–Lorenz mixing rule is also applicable to the calculation of refractive index in ternary mixture [27]

$$\begin{align} \frac{{{n_{{\text{D}} - {\text{T}}}}^2 - 1}}{{{n_{{\text{D}} - {\text{T}}}}^2 + 2}} = \sum\limits_{i = {\text{D}},{\text{ DT}},{\text{ T}}} {\left( {\frac{{n_i^2 - 1}}{{n_i^2 + 2}}} \right)} \cdot {\phi _i} .\end{align} \tag{ 15 }$$

For the analysis of the D₂–T₂ phase diagram, a ternary phase diagram will be used [25]. In addition, tritium decays to helium-3, which has a boiling point of 4.2 K. Gaseous ³He will escape from the liquid or solid D₂–T₂ and will not affect the refractive index of D₂–T₂.