Analyses of the thermal shifts of spectral lines in Nd³⁺-doped LiYF₄ laser crystal

Abstract

The red-shift of spectral line E ₁(R ₁ → Y ₂) and blue-shift of line E ₂(R ₁ → X ₅) with temperature in Nd³⁺-doped LiYF₄ laser crystal are studied by considering both the static contribution due to lattice thermal expansion and vibrational contribution due to electron–phonon interaction. The study is based on the analyses of pressure and temperature dependences of these spectral lines. It is found that for both lines, the static and vibrational contributions result in the blue- and red-shift, respectively. So, the observed red-shift of line E ₁ and blue-shift of line E ₂ are due respectively to the static contribution being smaller and larger than the vibrational one. Also, we infer that the thermal shifts of lines E ₃(R ₁ → Y ₅) and E ₄(R ₂ → Y ₅) are very small because both contributions may be approximately canceled. When both the contributions are contained, whether the red-shift or blue-shift of a spectral line can be fitted with the almost same theoretical expression as that by including only the vibrational contribution used in red-shift in the previous papers if we change the expression concerning electron–phonon interaction coefficient.

1 Introduction

LiYF₄:Nd³⁺ is an important high-power laser crystal because it presents the following advantages: the large thermal conductivity (which helps to extract the unwanted heat from the system during the optical pumping), the large natural birefringence (which can eliminate the thermal depolarization) and the weak thermal lens effect [1–3]. Since the thermal broadening and shift of laser lines are closely concerned with the output frequency stability, light amplification gain and thermal tunability of the laser [4–7], the effects of temperature (or thermal effects) on the sharp emission lines of LiYF₄:Nd³⁺ have attracted attention. Sardar and Yow [7] measured the crystal field energy levels (note: the sources on the physics of crystal field energy levels and spectra of rare earth ions in crystals can be found in Refs. [8, 9]) as well as the thermal effects on the linewidths and shifts of the inter-Stark transitions of lines E ₁(R ₁ → Y ₂) and E ₂(R ₁ → X ₅) within the respective inter-manifold transitions of ⁴ F _3/2 → ⁴ I _11/2 and ⁴ F _3/2 → ⁴ I _9/2 of LiYF₄:Nd³⁺ (the energy level diagrams and the values of crystal field energy levels related to these inter-manifold transitions are given in Figs. 3 and 4 in ref. [7]). The measurements [7] show that the spectral line E ₁ shifts toward the longer wavelengths (i.e., red-shift), while line E ₂ toward the shorter wavelengths (i.e., blue-shift) with the increasing temperature. Sardar and Yow [7] thought that the red-shift of line E ₁ was caused by the vibrational contribution due to the electron–phonon interaction and could be fitted well with the theoretical expression based on the dominant Raman’s two-phonon processes concerning Deybe model [7], i.e.,

$$ \Updelta E_{1} (T) = \Updelta E_{0} - \alpha \left( {\frac{T}{{\theta_{\text{D}} }}} \right)^{4} \int\limits_{0}^{{\theta_{\text{D} } /T}} {\frac{{x^{3} }}{{e^{x} - 1}}\text{d} x} $$

(1)

where ΔE ₀ = E ₁(0 K) − E ₁(10 K) and E ₁(0 K) is obtained by extrapolating the observed line position data at 0 K, θ _D is the Debye temperature and α is the electron–phonon interaction coefficient. For the blue-shift of line E ₂, Sardar and Yow [7] thought that it could not be fitted with the available theoretical expression. Similar opinion about the explanation of blue-shift of spectral line in Nd³⁺-doped laser crystals can be found in other papers [5, 10–12]. In fact, the thermal shifts of spectral lines arise not only from the explicit or vibrational contribution ΔE ^vib(T), but also from the implicit or static contribution ΔE ^st(T) due to lattice thermal expansion [4, 13–15]. Due to the difficulty in the determination of the relative importance of both contributions, in the theoretical studies of thermal shifts of spectral lines in many crystals, including the LiYF₄:Nd³⁺ considered, only the vibrational contribution is taken into account [5–7, 10–13]. The neglect of the static contribution to the thermal shifts for some crystals (in particular, the “soft” crystals [4, 15]) and some spectral lines (whose position is sensitive to the change of the geometry of active ion centers [15]) is not reasonable. In this paper, we study the thermal shifts of spectral lines E ₁ and E ₂ in LiYF₄:Nd³⁺ laser crystal by considering both the static and vibrational contributions. The results are discussed and the causes for the red-shift of line E ₁ and blue-shift of line E ₂ are explained.

2 Calculation

The thermal shift ΔE(T) of the spectral line E can be written as:

$$ \Updelta E(T) = \int\limits_{0}^{T} {\left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)\text{d} T} $$

(2)

where $(\frac{{\text{d} E}}{{\text{d} T}})$ is the temperature dependence of line E. Obviously, for the red-shift of line E, $\frac{{\text{d} E}}{{\text{d} T}} < 0$ and for the blue-shift, $\frac{{\text{d} E}}{{\text{d} T}} > 0$. Similar to the value of ΔE(T), $\frac{{\text{d} E}}{{\text{d} T}}$ comes from the static and vibrational contributions, i.e.,

$$ \left( {\frac{{\text{d} E}}{{\text{d} T}}} \right) = \left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{st}} + \left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{vib}} $$

(3)

The static contribution, due to lattice thermal expansion can be given by the form:

$$ \left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{st}} = \left( {\frac{{\text{d} E}}{{\text{d} P}}} \right)_{T} \left( {\frac{{\text{d} P}}{{\text{d} \ln V}}} \right)_{T} \left( {\frac{{\text{d} \ln V}}{{\text{d} T}}} \right)_{P} = - \left( {\frac{{\text{d} E}}{{\text{d} P}}} \right)_{T} \frac{{\alpha_{\text{th}} }}{\beta } $$

(4)

in which $\alpha_{\text{th}} ( \approx \frac{1}{V}\frac{{\text{d} V}}{{\text{d} T}})$ and $\beta ( \approx - \frac{1}{V}\frac{{\text{d} V}}{{\text{d} P}})$ are the thermal expansion coefficient and bulk compressibility, respectively. $(\frac{{\text{d} E}}{{\text{d} P}})_{T}$ (hereafter simplified as $\frac{{\text{d} E}}{{\text{d} P}}$) is the pressure dependence of line E. Because of the Grüneisen relation [16],

$$ \alpha_{\text{th}} = \gamma C_{\text{v}} \beta /V $$

(5)

where γ and C _v are the Grüneisen constant and the heat capacity, respectively. Note that C _v is approximately constant above 200 K and that the ratio $\frac{{\alpha_{\text{th}} }}{\beta }$ in the rare earth ion clusters in crystals can be reasonably assumed to be similar to that in the host crystal [17, 18]. By means of the thermal expansion coefficient α _th(≈3.49 × 10⁻⁵/K [19]) and bulk compressibility β(≈1.23 × 10⁻³/GPa [19, 20]) of LiYF₄ crystal at room temperature and the observed pressure dependence $\frac{{\text{d} E_{i} }}{{\text{d} P}}$ (i = 1, 2, see Table 1) of LiYF₄:Nd³⁺ [21], we calculate the static contributions $(\frac{{\text{d} E_{i} }}{{\text{d} T}})_{st}$ from Eq. (4) and the results are shown in Table 1.

Table 1 The pressure and temperature dependences $\frac{{\text{d} E_{i} }}{{\text{d} P}}$ and $\frac{{\text{d} E_{i} }}{{\text{d} T}}$, the static and vibrational contributions $\left( {\frac{{\text{d} E_{i} }}{{\text{d} T}}} \right)_{\text{st}}$ and $\left( {\frac{{\text{d} E_{i} }}{{\text{d} T}}} \right)_{\text{vib}}$ of temperature dependences and their ratio t, the electron–phonon interaction coefficient α′ and the parameter A concerning the static contribution for the thermal shifts of spectral lines E ₁(R ₁ → Y ₂) and E ₂ (R ₁ → X ₅) in Nd³⁺-doped LiYF₄ laser crystal

The observed position of spectral line E _i changes linearly with the temperature at T > 200 K [7], so the experimental or total temperature dependence $\frac{{\text{d} E_{i} }}{{\text{d} T}}$ is a constant and given in Table 1. Thus, from Eq. (3), the vibrational contribution $(\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{vib}}$ at T > 200 K can be calculated and the results are also collected in Table 1.

At low temperature (T < 200 K), only the heat capacity C _v is sensitive to the temperature, thus, from Eqs. (2), (4) and (5), the static contribution ∆E ^st(T) to thermal shift is

$$ \Updelta E^{\text{st}} (T) = \int\limits_{0}^{T} {\left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{st}} \text{d} T} = -\left( {\frac{{\text{d} E}}{{\text{d} P}}} \right)\frac{\gamma }{V}\int\limits_{0}^{T} {C_{\text{v}} \text{d} T} $$

(6)

In terms of Debye model, one can obtain [16]:

$$ \int\limits_{0}^{T} {C{\text{v}}\text{d} T} = B\left( {\frac{T}{{\theta_{\text{D}} }}} \right)^{4} \int\limits_{0}^{{\theta_{\text{D}} /T}} {\frac{{x^{3} }}{{e^{x} - 1}}\text{d} x} $$

(7)

Consequently, the static contribution can be written as:

$$ \Updelta E^{\text{st}} (T) = A\left( {\frac{T}{{\theta_{\text{D}} }}} \right)^{4} \int\limits_{0}^{{\theta_{\text{D}} /T}} {\frac{{x^{3} }}{{e^{x} - 1}}\text{d} x} $$

(8)

where A and B are the parameters almost independent of temperature T. Thus, we get,

$$ \Updelta E(T) = \Updelta E_{0} + \Updelta E^{\text{st}} (T) + \Updelta E^{\text{vib}} (T) = \Updelta E_{0} + (A - \alpha^{\prime} )\left( {\frac{T}{{\theta_{\text{D}} }}} \right)^{4} \int\limits_{0}^{{\theta_{\text{D} } /T}} {\frac{{x^{3} }}{{e^{x} - 1}}\text{d} x} $$

(9)

where α′ is also the electron–phonon interaction coefficient in the case of including the static and vibrational contributions and A is a parameter related to the static contribution. Thus, even the static contribution is also considered, the theoretical expression of thermal shifts of spectral lines is almost the same as that [Eq. (1)] by including only the vibrational contribution used in the previous papers if we change the coefficient from −α to A − α′. This expression of Eq. (9) can be used to interpret whether red-shift or blue-shift of the spectral lines.

From the above formulas, we have:

$$ \frac{A}{{ - \alpha^{\prime}}} = \frac{{\Updelta E^{{_{\text{st}} }} (T)}}{{\Updelta E^{\text{vib}} (T)}} = {{\left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{st}} } \mathord{\left/ {\vphantom {{\left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{st}} } {\left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{vib}} }}} \right. \kern-\nulldelimiterspace} {\left( {\frac{{\text{d} E}}{{\text{d} T}}} \right)_{\text{vib}} }} \approx t $$

(10)

where t denotes the relative importance of static contribution to thermal shift. For spectral lines E ₁ and E ₂ of LiYF₄:Nd³⁺, the values of t are given in Table 1. Thus, one can get:

$$ A = - t\alpha^{\prime} ,\;A - \alpha^{\prime} = - \left( {t + 1} \right)\alpha^{\prime} = - \alpha $$

(11)

For the spectral line E ₁, the measured red-shifts ∆E ₁(T) were fitted to Eq. (1) with the fitting parameters ∆E ₀ ≈ 0, θ _D ≈ 460 K and α ≈ 23.5 cm⁻¹ [7]. Thus, the parameters A and α′ can be calculated from Eq. (11) and shown in Table 1. If we apply Eq. (1) to fit the blue shift of spectral line E ₂, we can obtain the same ∆E ₀ and θ _D as those in the red-shift and α ≈ −21.6 cm⁻¹. The parameters A and α′ in Eq. (9) are therefore obtained and shown in Table 1.

3 Discussion

The above studies show that if only the vibrational contribution is taken into account, the calculated electron–phonon interaction coefficient α [using Eq. (1)] in the blue-shift ΔE ₂(T) of line E ₂ is negative. The opposite sign of coefficient α to that in the red-shift of line E ₁ is difficult to be understood in physics and so some authors [5, 7, 10–12] thought that the blue-shift of spectral lines for Nd³⁺ ions in crystals cannot be fitted with the available theoretical expression. However, if both the static and vibrational contributions are considered, the calculated electron–phonon interaction coefficient α′ [using Eq. (9)] in the blue-shift of line E ₂ in sign is the same as and in magnitude close to that in the red-shift of E ₁ in LiYF₄:Nd³⁺ crystal. So these coefficients α′ are reasonable and both the red- and blue- shifts can be fitted with the almost same theoretical expression [Eq. (9)] as that by including only the vibrational contribution [i.e., Eq. (1)], if we change the coefficient from −α to A − α.

For the thermal shifts of both spectral lines E ₁ and E ₂, the vibrational contributions always result in the red-shift (characterized by $(\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{vib}} < \, 0$ and α′ > 0, see Table 1), while the static contributions result in the blue-shift (characterized by $(\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{st}} > 0$ and A > 0, see Table 1). The static contributions depend strongly upon the pressure dependence of spectral lines E _i and can be quite different among diverse spectral lines. So the red-shift of spectral line E ₁ is due to the static contribution being smaller than the vibrational contribution (i.e., $\left| {(\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{st}} } \right| < (\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{vib}}$ and A < α′, see Table 1), and the blue-shift of spectral line E ₂ is caused by the larger static contribution in comparison with the vibrational one (i.e., $\left| {(\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{st}} } \right| > (\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{vib}}$ and A > α′, see Table 1). Even for the red-shift of line E ₁, when the neglected static contribution is also considered, the fitted electron–phonon interaction coefficient (≈37.2(2) cm⁻¹, see Table 1) should be larger than that (≈ 23.5 cm⁻¹ [7]) obtained by only including the vibration contribution. So, the reasonable and precise theoretical calculations of the thermal-shifts of spectral lines for Nd³⁺ (and other laser active ions) in crystals should take both the static and vibrational contributions into account.

It is noted that for the spectral lines E ₃(R ₁ → Y ₅) and E ₄(R ₂ → Y ₅) in Nd³⁺-doped LiYF₄ crystal, the observed pressure dependences are $(\frac{{\text{d} E_{3} }}{{\text{d} P}}) \approx - 5.76$(3) cm⁻¹/GPa and $(\frac{{\text{d} E_{4} }}{{\text{d} P}}) \approx - 5.86$(3) cm⁻¹/GPa [21]. From Eq. (4), we can calculate

$$ \left( {\frac{{\text{d} E_{3} }}{{\text{d} T}}} \right)_{\text{st}} \approx 16.3\left( 1 \right) \times 10^{ - 3} {\text{cm}}^{ - 1} /{\text{K}},\quad \left( {\frac{{\text{d} E_{4} }}{{\text{d} T}}} \right)_{\text{st}} \approx 16.6\left( 1 \right) \times 10^{ - 3} {\text{cm}}^{ - 1} /{\text{K}} $$

(12)

If we assume that the vibration contributions related to $(\frac{{\text{d} E_{i} }}{{\text{d} T}})_{\text{vib}}$ for the spectral lines E ₃ and E ₄ are close to that (≈− 20 × 10⁻³cm⁻¹/K, see Table 1) of the spectral lines E ₁ and E ₂, the total temperature dependences of spectral lines E _{3, 4}(R _{1, 2} → Y ₅) may be near −3×10⁻³cm⁻¹/K. This value is much smaller than the observed or total temperature dependences of spectral lines E ₁ and E ₂ (see Table 1). So, we infer that the thermal-shift of spectral line E _{3, 4} may be very small. Although the experimental thermal shift of E _{3, 4} (or E ₃ and E ₄) was not reported, the very small thermal-shift of the same spectral line E _{3, 4} observed for Nd³⁺ ion in Y₃Al₅O₁₂ (YAG) laser crystal [13] implies that the inference or assumption may be reasonable. This point remains to be checked by the further experimental study.