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Study on the specific heat of Y3Al5O12 between 129 K and 573 K

Yoichi Sato and Takunori Taira

Open Access Open Access

Abstract

We measured the isopiestic specific heat (CP) of Y3Al5O12 (YAG) by the differential scanning calorimetry aiming to obtain thermal parameters under cryogenic and room-temperature (RT) conditions. It was also found that the applicable temperature range of our numerical model for CP of YAG was updated to the range between 129 K and 573 K with below 3% error. Obtained parameters were verified by the comparative study with the first principles calculations. Discrepancy between the calculation value and measured value at 273 K was 0.017 J/gK in CP.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

For the stabilization of the output power from laser oscillators, it is necessary to optimize the heat management based on the precise thermal parameters of laser gain media. Recent developments in the extraction of high energy laser pulse often requires the cryogenic laser operation where the temperature of laser gain media is controlled to be from 150 K to 200K [1]. The reason is that thermal parameters of laser gain media in the cryogenic condition is superior to those in RT, which is well-known since the first lasing in cryogenic Yb:YAG [2].

We already proposed the temperature-dependent model for various thermal and spectroscopic parameters of laser gain media around the room temperature (RT) [3–5], and now we think it is necessary to extend our numerical models for these parameters to the cryogenic condition. Here we focus on the isopiestic specific heat (CP) of Y3Al5O12 (YAG) [3,6–12]. YAG is treated as the standard of laser gain media because it is the most useful solid-state laser host crystal not only in the field of the scientific research but also in the various industrial applications.

CP is important because it is essential to evaluate the thermal conductivity (κ) and the thermal shock parameter (RT). The relations of these parameters are expressed by

$${R_T} = \frac{{({1 - \nu } ){\sigma _{\max }}}}{{\alpha E}}\kappa = \frac{{({1 - \nu } ){\sigma _{\max }}}}{{\alpha E}}\rho D{C_P},$$
where ν, σmax, α, E, ρ, and D are the Poisson ratio, the maximum surface stress at which fracture occurs, the linear thermal expansion coefficient, the Young’s modulus, the density, and the thermal diffusivity measured by the flash method [13]. Moreover, CP relates directly to the heat capacity of laser gain media as shown in the heat conduction equation:
$$\rho {C_P}\frac{{\partial T}}{{\partial t}} = \kappa \Delta T + q,$$
where T and q are the temperature and the heat generated per unit volume. Although only κ is important under temperature stabilized condition, not only κ but also CP become important to manage thermal response for temperature control.

In this work, we report the availability of the numerical model for CP of YAG under the cryogenic condition by the comparative study between experimental evaluation with the differential scanning calorimetry (DSC) and the first principles calculations.

2. Experimental setups and computations

2.1 Cp measurement by DSC

CP of YAG was measured by 2 types of DSC equipment: DSC2500 (TA Instruments) for the range from −145 °C (129 K) to 200 °C (473 K) with the temperature elevation rate of +20 °C/min, and DSC204F1 (NETZSCH) from 0 °C (273 K) to 300 °C (573 K) with +10 °C/min. Measured samples were three undoped single crystals (Scientific Materials, Shandong Newphotons), two undoped YAG ceramics (World Lab, Konoshima Chemical), and one 1at.% Nd:YAG single crystal (Scientific Materials). These samples had a size of ϕ5 mm in diameter with the thickness of 1 mm.

2.2 Geometry optimizations for the first principles calculations

Before phonon calculations we optimized the crystal structure of YAG in Ref. [14] by the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. We used the variable-cell relaxation program in Quantum Espresso Suite v.6.5 [15] as a solver for self-consistent field (SCF) calculation by the projector-augmented wave method using pseudopotentials in PSliblary1.0.0 [16] with cut-off energy of 680 eV and SCF convergence threshold of 1.4×10−7 eV. YAG primitive cell was processed in BFGS procedure with 4×4×4 Monkhorst-Pack k-point sampling for 0.2 / Å resolution in the reciprocal lattice. In order to execute the structure relaxation, three kinds of correlation functions (Self-interaction correction to density-functional approximations for many-electron systems by Perdew-Zunger (PZ) [17], Generalized gradient approximation by Perdew-Burke-Ernzerhof (PBE) [18], and the revised PBE for solids and their surfaces (PBEsol) [19]) were examined where the ionic minimization convergence thresholds on the total energy and forces were 1.4×10−4 eV and 2.6 ×10−3 eV/Å, respectively.

2.3 Phonon calculations

Isovolumic specific heat (CV) was calculated by the frozen phonon method using Phonopy 2.6.1 [20]. Forces to atoms in YAG structure was obtained by SCF calculations using Quantum Espresso Suite on unit-cells with 3×3×3 Monkhorst-Pack k-point sampling on the unit-cell of YAG. Cut-off energy and SCF convergence threshold were the same as geometry optimizations.

Using Phonopy results on relaxed unit-cells that have lattice constants modulated within ±3% from the optimized geometry, temperature dependences of CP was calculated under Rose-Vinet equation of state [21] by the phonopy-qha program in Phonopy.

3. Results

Figure 1 shows the reproducible error in DSC measurement by DSC2500 and DSC204F1 were ±1% and ±2%, respectively. Thus, difference between measured CP of 1at.% Nd:YAG and of undoped YAG was smaller than the reproducible error. The discrepancy between of CP measured by DSC2500 and DSC204F1 was below 0.012 J/gK, which was comparable to the reproducible error. Figure 2 shows the measured CP of undoped YAG ceramics synthesized by World Lab, where difference between DSC2500 and DSC204F1 is negligible. As shown in Fig. 2, the difference between CP of YAG measured by DSC and that evaluated by the first principles calculation was below 2%.

 figure: Fig. 1.

Fig. 1. Reproducible errors in experimental CP evaluation by DSC from 300 K to 460 K. (a) CP measured by DSC2500 with error bar of ±1% (0.006 J/gK). (b) CP measured by DSC204F1 with error bar of ±2% (0.012 J/gK).

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 figure: Fig. 2.

Fig. 2. Measured CP of undoped YAG ceramics synthesized by World Lab and calculated specific heats.

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The lattice constant of relaxed crystal structures of YAG derived from first principles calculation by use of various kinds of correlation functions are summarized in Table 1.

Tables Icon

Table 1. Lattice constant and density of relaxed crystal structures of YAG

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4. Discussions

4.1 Updating of the numerical model for CP of YAG

We already reported that CP of YAG within the range between 273 K and 473 K is given as a function of temperature T by the following equation using Debye temperature ΘD of 795 K:

$${C_P} = \textrm{2}\textrm{.521}{\left( {\frac{T}{{{\Theta _D}}}} \right)^3}\int_0^{\frac{{{\Theta _D}}}{T}} {\frac{{{z^4}{e^z}}}{{{{({{e^z} - 1} )}^2}}}dz} ,$$
where the unit of CP was J/gK [3]. Strictly speaking, the Eq. (3) is an expression which gives not the CP but CV. However, it can produce an accurate approximation of CP in a limited temperature range when we set Debye temperature to well-effective value. When we set ΘD to 800 K, Eq. (3) can give enough accurate CP of YAG from 190 K to 573 K with the error lower than 2%, which is shown in Fig. 3. On the contrary, CP from 129 K to 190 K can be reproduced by use of ΘD = 759 K, where the comparison between experimental value of CP and the value given by Eq. (3) is below 3%. In addition, Eq. (3) with ΘD of 795 K gives the approximation for CP of YAG with the error below 3% from 178 K to 573 K, 9% at 159 K, and 15% at 129 K. Applicable temperature ranges of numerical model for CP of YAG are summarized in Table 2.

 figure: Fig. 3.

Fig. 3. Comparison between experimental value of CP and the value given by Eq. (3).

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Tables Icon

Table 2. Applicable temperature range of numerical model for CP of YAG

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We also confirmed that Eq. (3) can be applicable to both YAG ceramics and single crystals, because the difference between CP of YAG ceramics and single crystal was negligible as opposite to several-% difference in the thermal conductivity.

4.2 Accuracy of the first principles calculations on CP of YAG

As shown in Fig. 2 it was found that the first principles calculation of CP could reproduce the experimental value well. Difference between measured CP and the first principles calculation was below 0.02 J/gK. In other words, the value of CP by the first principles calculation was experimentally proved within the temperature range from 129 K to 573 K. This suggests that the first principles calculation is applicable to evaluate CP in laser gain media as one of design parameters for high-power laser devices. The mean error of CP measured by DSC is below 2% according to the specification of DSC2500 and DSC204F1, thus we assumed the standard uncertainty u of our measurement was 0.015 J/gK. The minimum and maximum CP at 300 K in Refs. 6–12 were 0.585 J/gK and 0.628 J/gK, respectively, and they were measured by DSC. Therefore, the value of reported CP at 300 K was well reproduced by our first principles calculation as 0.604 ± 1.6 u J/gK. The difference between calculated CP and CV was 0.02 J/gK at 600 K, and it was nearly proportional to T. It was also found that the temperature dependence of α is quite similar to CV as shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. Temperature dependence of thermal expansion coefficient [5] and CV in YAG.

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From Grüneisen relation expressed by [23]

$$\alpha = \frac{\gamma }{{3K}}{C_\textrm{V}},$$
where γ and K are the Grüneisen parameter and the bulk modulus, respectively, K of YAG can be treated as independent on T around RT, because γ is a constant except under cryogenic conditions. It also implies the temperature independent characteristics of ΘD around RT since ΘD can be derived from the elastic constant. This is the reason why we can use Eq. (3) as a numerical model of CP.

4.3 Density of relaxed crystal structures

When we execute first principles calculation using pseudopotentials, we have to choose the appropriate correlation function. Table 1 shows the variation in the stress-free crystal structure of YAG obtained by the BFGS procedure, which indicated it was not so easy to estimate the precise density of materials by first principles calculation.

The lattice constant obtained by BFGS procedure in Table 1 is of the relaxed unit cell under T = 0 K, while experimental value was measured around RT. Considering that thermal expansion coefficient of YAG is ca. 6×10−6 /K at 300K, the lattice constant at T = 0 K can be roughly 0.1% smaller than 12.01 Å. Therefore, in this work we used PBEsol-correlation function for the calculation of CP because this correlation function brought the crystal density of YAG nearly equal to experimental value.

4.4 Temperature dependence of Debye temperature

Although the difference between calculated CP and CV became smaller under lower temperature region, it was more difficult to express CP by the constant ΘD under extremely lower temperature region. It means ΘD has temperature dependence [24], whereas we mentioned that ΘD around RT can be treat as constant in Sect. 4.2.

Figure 5 shows ΘD calculated by use of CV and CP in Fig. 2. As previously indicated in Sect. 4.1 ΘD should be obtained from CV, and ΘD estimated from CV was nearly constant above RT. On the contrary, ΘD estimated from CP contained larger error at higher temperature. Moreover, during ΘD estimation by use of Eq. (3) suffers severe error from the small deviation of the specific heat. This indicates that it is difficult to obtain effective ΘD from the experimental value of CP at T > RT by use of Eq. (3), though Eq. (3) is quite useful as the numerical model for CP. We summarize results of experimental and theoretical value in Table 3.

 figure: Fig. 5.

Fig. 5. Temperature dependence of ΘD by Eq. (3) with specific heat in Fig. 2.

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Tables Icon

Table 3. Unit cell length and density of relaxed crystal structures of YAG

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Figure 5 also shows the reason why reported ΘD of YAG has been variated from 500 K to 900 K [25–27]. ΘD shows the dependence to gradually decrease T towards 0 K and to gradually increase ΘD over 800 K, and to gradually increase T over 600 K under the isopiestic condition and to gradually decrease ΘD toward below 500 K.

5. Conclusion

CP of Y3Al5O12 was evaluated by DSC and first principles calculations. CP calculated by the first principles calculation was experimentally proved within the temperature range from 129 K to 573 K with the accuracy of the reproducible error in CP measurement.

It was found that our numerical model for CP of YAG was useful not only RT but also cryogenic conditions: using ΘD of 800 K, 795 K, and 759 K, applicable from 190 K to 573 K below 2% error, from 178 K to 573 K below 3% error, and from 129 K to 190 K below 3%, respectively. The temperature dependence in ΘD of YAG was also confirmed.

We tabulated numerical results as the comprehensive database of the specific heat of YAG. This table will be useful for the design of thermal management in high power laser devices.

Acknowledgements

Authors thanks to Ms. M. Maeda (TA Instrument Japan Inc.) and Mr. Y. Shinoda (NETZSCH Japan K.K.) for their supports in the DSC measurements.

Disclosures

The authors declare no conflicts of interest.

References

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Figures (5)
Fig. 1.
Fig. 1. Reproducible errors in experimental CP evaluation by DSC from 300 K to 460 K. (a) CP measured by DSC2500 with error bar of ±1% (0.006 J/gK). (b) CP measured by DSC204F1 with error bar of ±2% (0.012 J/gK).

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Fig. 2.
Fig. 2. Measured CP of undoped YAG ceramics synthesized by World Lab and calculated specific heats.

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Fig. 3.
Fig. 3. Comparison between experimental value of CP and the value given by Eq. (3).

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Fig. 4.
Fig. 4. Temperature dependence of thermal expansion coefficient [5] and CV in YAG.

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Fig. 5.
Fig. 5. Temperature dependence of ΘD by Eq. (3) with specific heat in Fig. 2.

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Tables (3)

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Table 1. Lattice constant and density of relaxed crystal structures of YAG

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Tables Icon

Table 2. Applicable temperature range of numerical model for CP of YAG

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Table 3. Unit cell length and density of relaxed crystal structures of YAG

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

R T = ( 1 − ν ) σ max α E κ = ( 1 − ν ) σ max α E ρ D C P ,
ρ C P ∂ T ∂ t = κ Δ T + q ,
C P = 2 .521 ( T Θ D ) 3 ∫ 0 Θ D T z 4 e z ( e z − 1 ) 2 d z ,
α = γ 3 K C V ,
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