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

Deuterium (D)-Tritium (T) nuclear fusion reaction has potential as an energy source in the future. In both magnetic confinement and inertial confinement fusion reactors, solid D–T will generally be supplied as fusion fuel. The efficiency of the nuclear fusion reaction depends on the quality of solid D–T fuel, which is related to the composition, homogeneity, helium-3 (3He) content, and so on. However, there is no technique for in-situ examination of solid D–T fuel. In this study, we consider a simple and precise method for the characterization of solid hydrogen isotopologues at cryogenic temperature using refractive index measurement, and evaluate the distribution of hydrogen isotopologue composition and homogeneity. To evaluate without the effect of tritium decay, the homogeneity of the hydrogen (H2)-deuterium (D2) mixture is measured at first. By the in-situ refractive index measurement at cryogenic temperature, the homogeneity of solid H2–D2 mixture is roughly quantified. The phase diagram of the H2–D2 mixture shows a solid solution type. D2-rich crystal first appears from the liquid phase as a primary crystal. The composition of D2 in liquid phase ias homogeneous, whereas it reduces by obeying the liquidus line in the phase diagram with the crystallization. On the other hand, the composition of the H2–D2 mixture in solid phase is inhomogeneous because the mobility of H2 and D2 in solid phase was too slow to be homogeneous and solid. The compositions of H2–D2 mixture in liquid and solid phases could be evaluated by the in-situ refractive index measurement in time. Consequently, the refractive index measurement shows great potential as an inspection method of solid D–T fuel in fusion reactors.


Introduction
As a fusion fuel, Deuterium (D)-Tritium (T) fuel has a large cross-section, considerable energy yield, and low ideal ignition temperature, making the D-T fusion reactor the most likely to first achieve self-reliant fusion [1]. To overcome the Coulomb barrier, the D-T fuel needs to be heated to a temperature at which it forms plasma. Magnetic Confinement Fusion (MCF) and Inertial Confinement Fusion (ICF) are two fundamental confinement technologies being investigated at present.
In ICF reactors, the solid fuel layer is contained in a tiny (few-millimeter diameter) hollow pellet, as shown in figure 1. Laser or heavy ion beams uniformly ablate the surface of the fuel pellet and produce the required compression and heating to implode the D-T fuel layer, leading to a fusion reaction. To accomplish the fusion reaction, the target pellet properties should be strictly controlled, including sphericity, surface roughness, thickness uniformity of multiple layers in the pellet, homogeneity of the solid D-T layer, etc. To date, formations of solid fuel layers have been studied for central and fast ignitions [2][3][4][5][6].
For MCF reactors, new D-T fuel should be continuously replenished into the reaction chamber as a puff of gas, stream of ions, or liquid or solid pellets. Fusion reaction is caused when magnetic and electric fields are used to heat and confine fuel as plasma. At present, pellet injectors are equipped on most tokamaks and stellarators because solid pellets can increase plasma density, and pellets can penetrate the plasma boundary before the molecules become ionized, adding fuel directly to the confined plasmas [7]. At cryogenic temperature, the pellet injectors typically generate numerous millimetersized solid D-T pellets with a cylindrical shape. Since the mass loss on the fuel surface occurs during transportation from the injector into the plasma [7], fuel homogeneity is critical to avoid distortion in the fuel component ratio.
Regardless, in MCF or ICF, high-quality solid fuel production is an important issue. Successful fusion reaction requires that the fuel meet many stringent specifications. During D-T fuel solidification, however, isotope effects and β-decay of T might affect the homogeneity and composition of the fuel, leading to a decrease in the fusion reaction efficiency. To maximize fusion reaction efficiency, D:T should be kept close to 50:50.
Thus, to realize the design and fabrication of high-quality solid D-T fuel and ensure the quality of the fuel entering the reactor, this study has demonstrated a method for characterizing the fuel homogeneity simply, accurately, and nondestructively using the optical measurement of the refractive index distribution of fuel. Firstly, for establishing an isotopologue distribution evaluation method in an environment without beta decay effects, we prepared the solid hydrogen (H 2 )-deuterium (D 2 ) mixture to measure the refractive index. It can also be used as a preparatory experiment for solid D-T measurement.

Method
Solid D-T fuel would be characterized by its composition and homogeneity prior to use in fusion reactors. The refractive index measurement can simultaneously determine the hydrogen isotopologue composition and its homogeneity. In this study, the optical cell for the refractive index measurement, which has a wedge-shaped gas filling chamber, was used as shown in figure 2. The cell was put in the cryostat. The system detail was described in our previous literature [8]. In this section, the theory of the refractive index measurement is described at first, and the measurement procedure including the H 2 -D 2 solidification is explained.

Temperature-dependent refractive index measurements of H 2 and D 2
When H 2 -D 2 mixture is solidified by thermal conductivity, the H 2 -D 2 mixture has a temperature gradient. Since the refractive index is a function of the density of hydrogen isotopologues, refractive index data for H 2 and D 2 in various conditions are indispensable. To calculate the refractive index of H 2 and D 2 at a given temperature, the empirical formula between density ρ and refractive index n, known as the Lorentz-Lorenz relation, was used: where α is the molecular polarizability, M is the molecular mass, and N A is the Avogadro constant. The specific refraction r λ can be identified via the Lorentz-Lorenz relation The specific refraction is a function of wavelength. Cauchy's dispersion formula is an empirical expression giving an approximate relation, where A and B are coefficients for λ in angstroms and r in cm 3 g −1 . The values of A and B depend on the medium. r ∞ is the specific refraction at λ = ∞. the r ∞ 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 The r ∞ coefficients A and B for H 2 and D 2 at STP conditions were reported by Childs and Diller [9]. These data were summarized in table 1.
For H 2 and D 2 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.
Solid H 2 [11]: The estimated V m should be in error by no more than 0.4%.
Liquid D 2 [12]: with V m in cm 3 mole −1 . The standard deviation from this equation is 0.01 cm 3 mole −1 (or 0.04% of V m ).
Solid D 2 [13]: 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 H 2 , liquid D 2 , and solid D 2 . There is not enough valid data to support the reliability of the empirical formula. The estimated values of solid D 2 in particular have considerable standard deviations. To improve the accuracy of the equation, the temperature dependence of the refractive index of H 2 and D 2 in the liquid and solid phases were measured by the same method as previous research [8].

Refractive index distribution measurements of H 2 -D 2
In this experiment, H 2 -D 2 mixture was used instead of D 2 -T 2 mixture. Our previous experiment measured the refractive index of D 2 -T 2 , whereas H 2 -D 2 mixture was used to observe the solidification behavior of hydrogen isotopologue without the effect of decay heat of tritium.
The H 2 -D 2 mixture was made from the commercially available hydrogen gases where these purities were 99.999% for H 2 and 99.5% for D 2 , and the gas composition was adjusted to be H 2 :D 2 = 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 H 2 and D 2 . Subsequently, gaseous H 2 -D 2 was supplied to the cell. The refractive index of the H 2 -D 2 mixture depends on the ratio of H 2 and D 2 . If the refractive index is simultaneously measured at many points on the H 2 -D 2 mixture, the distribution of H 2 -D 2 would be known. Thus, the distribution of H 2 and D 2 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.
(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 H 2 -D 2 mixture was loaded into the wedge-shaped optical cell. The H 2 -D 2 mixture was liquefied and solidified in the cell as previously described. 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 H 2 -D 2 , β (P) was calculated from The refractive index, n HD (P) is calculated by Snell's law, 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 x H + x D = 1 (13) where x H is the mole fraction and v H is the molar volume of H 2 . x D is the mole fraction and v D is the molar volume of D 2 .

Temperature dependence of the refractive index of H 2 and D 2
The refractive indices of liquid and solid H 2 and D 2 were measured in the range of 13.752 K-15.000 K for H 2 and 13.579 K-21.984 K for D 2 . The measurement uncertainty was calculated from the measurement error of the distance between the incident and refracted laser positions, L 1 , by the propagation of uncertainty. The measurement uncertainties are 0.0000596 for solid H 2, 0.0000515 for liquid D 2 , and 0.0000520 for solid D 2 . Figure 6 shows the refractive indices of H 2 and D 2 as a function of the temperature. As the temperature decreased, the refractive index increased. 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-H 2 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 H 2 was 1 h and solid D 2 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 H 2 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 D 2 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 D 2 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 D 2 was The standard deviation was about 0.000193.

Homogeneity of solid H 2 -D 2
Indeed, the isotopic exchange will lead the H 2 -D 2 mixture to reach chemical equilibrium between H 2 , D 2 , and Hydrogen Deuteride (HD); but at ambient temperatures, this exchange reaction is slow (of many weeks) [21]. And the equilibrium constant for H 2 +D 2 ⇌ 2HD decreases with decreasing temperature [22].   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 H 2 and D 2 , n H and n D , at a given temperature can be corrected by equation (4). Finally, the molar fraction distribution of H 2 was calculated using equations (11)-(13) as shown in figure 8. The statistical dispersions are 0.0334 for liquid H 2 -D 2 and 0.0197 for solid H 2 -D 2 .
At 15.598 K, the mole fraction of H 2 in the solid phase gradually decreases approaching the bottom, meaning that the composition of H 2 -D 2 in the primary crystal, which is the solid initially frozen, gave the higher D 2 composition rather than the original composition. This behavior can be explained by the H 2 -D 2 phase diagram which is the solid solution type. This point will be described in detail later.
Additionally, at 16.286 K, the H 2 mole fraction of liquid H 2 -D 2 was approximately homogeneous, which could be represented by the spatial average value, about 0.485. Theoretically, the H 2 and D 2 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 H 2 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, H 2 -D 2 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 H 2 -D 2 . The H 2 molar fraction of solid was small, and thus the loss of solid data resulted in a large H 2 molar fraction of the overall H 2 -D 2 .
(b) On the inner surface of the copper cell, H 2 -D 2 solidified faster, so there was more solid H 2 -D 2 on both sides than in the middle (measurement position), as shown in figure 9. Similarly, since the content of D 2 in the solid was higher, the H 2 molar fraction of liquid H 2 -D 2 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 H 2 -D 2 .
The mole fraction vertical distribution of H 2 at various temperatures is summarized in figure 10. The mole fraction of H 2 in liquid phase was relatively uniform. On the other hand, in solid phase at the lower side of the cell, H 2 has a wider mole fraction. The composition of solid H 2 -D 2 was inhomogeneous. The range of the mole fraction of H 2 was found to be from about 0.35-0.60. It is mentioned that the mole fraction of H 2 in solid increased approaching the interface between solid and liquid. The primary crystal that has higher D 2 content was formed at the bottom of the cell.
The phase diagram of H 2 -D 2 at cryogenic temperature is shown in figure 11 [23]. Solid H 2 -D 2 mixture is a completely soluble solid solution (H 2 and D 2 are soluble at all concentrations). In an equilibrium condition, the mole fraction of H 2 in the H 2 -D 2 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.  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. Very few studies about solid D 2 -T 2 mixture have been reported. In McKenty et al [24], the distribution of D 2 -T 2 in a solid is simulated with H 2 -D 2 data experimentally obtained. The distribution of H 2 -D 2 is obtained by the IR transmission coefficient of H 2 -D 2 data. The experimental results show a 10.5% variation in deuterium fraction across the cryogenically formed layer of the solid H 2 -D 2 sample. In our experiment, the range of the H 2 mole fraction was from 0.396 to 0.599 in solid H 2 -D 2 (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 D 2 in the solid phase did not reach an equilibrium state during the measurement period. In figure 12, the range of the H 2 mole fraction was 0.384-0.617 at 13.612 K; the range of the H 2 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 D 2 -T 2 re-arrangement to be homogeneous in the solid. However, we have no result for the D 2 -T 2 rearrangement in ice. This point is an important matter for D 2 -T 2 solid in fusion fuel production. Because of isotopic exchange, D 2 -T 2 takes place among D 2 , Deuterium Tritide (DT), and T 2 . 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 D 2 -T 2 [26]. The Lorentz-Lorenz mixing rule is also applicable to the calculation of refractive index in ternary mixture [27] n D−T 2 − 1 For the analysis of the D 2 -T 2 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 3 He will escape from the liquid or solid D 2 -T 2 and will not affect the refractive index of D 2 -T 2 .

Conclusion
This study demonstrated a method for the high-precision characterization of homogeneity using optical measurements of the refractive index distribution of solid hydrogen isotopologue. This technique has been demonstrated for liquid and solid H 2 -D 2 mixture for observing the hydrogenic isotopologue fractionation without the effects of beta decay. Before characterizing the solid D 2 -T 2 properties, as a quasi-experiment, we used a H 2 -D 2 mixture in which the 1:1 ratio of H and D, mimicked the D 2 -T 2 mixture situation. The homogeneity of solid H 2 -D 2 was measured.
The experiment results showed that H and D were spatially inhomogeneous to H 2 -D 2 during the solidification process. This can be explained by the H 2 -D 2 phase diagram and the molecular mobility in solid phase. Since D 2 has a higher freezing point, D 2 solidified from the solution faster than H, and the D-rich solid H 2 -D 2 precipitated at the bottom of the cell. As crystallization continued, H-rich crystals were produced. But diffusion was slow in the solid H 2 -D 2 , causing D-rich crystals and H-rich crystals to exist at the same time. Therefore, the spatial inhomogeneity of D 2 -T 2 can be anticipated during D-T fuel solidification. This is a step towards the characterization of D-T, but this will be complicated due to the presence of DT. At present, we have no experimental results of solid D 2 -T 2 homogeneity. This work is significant for the fabrication of D-T fuel pellet.
The experiments prove that the method of characterization of solid H 2 -D 2 using refractive index distribution is effective, and the same method can characterize solid D 2 -T 2 . This characterization method has the potential to be practical and usable in the characterization of solid D-T fuel pellets for fusion reactors in the future. The study of D 2 -T 2 properties will encourage the production of higher-quality D-T fuel pellets.