A Full Analysis Including Both the Static and Dynamic Factors for the Thermal Shift of ⁷D₀ ⟶ ⁵F₀ Fluorescence Line in SrB₄O₇:Sm²⁺Crystal

1 Introduction

Strontium tetraborate (SrB₄O₇) possesses many good properties, such as great nonlinear optical coefficients and high optical damage threshold and can, therefore, be applied as non-linear optical materials [1], [2]. Furthermore, it, doped with Sm²⁺ ions is widely used in the areas of phosphor, light conversion agent and optical pressure sensor (in particular, at high pressure and high temperature) [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. These applications are closely connected with the spectroscopic properties of SrB₄O₇:Sm²⁺. So, many investigations on the spectral properties (including those under pressure and temperature) for SrB₄O₇:Sm²⁺were performed [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. For instance, the pressure and temperature dependences of the fluorescence line ⁷D₀⟶⁵F₀ (called E₀ line) in SrB₄O₇:Sm²⁺ at near or higher than room temperature were measured by several groups [5], [6], [7], [8], [9], [10], [11], [12]. It is found that the E₀ line position varies approximately linearly with the increase of pressure or temperature [5], [6], [7], [8], [9], [10], [11], [12]. The pressure leads the E₀ line to red shift and the pressure coefficient (or pressure dependence) dE₀/dP ≈ −5.3 cm⁻¹/GPa [5], [6], [8], [11], whereas the temperature results in a very small blue shift of E₀ line and the temperature dependence (or temperature coefficient) dE₀/dT ≈ 0.2 × 10⁻² cm⁻¹/K [6], [11], [12]. Compared with the R₁-line in ruby, a common optical pressure tensor crystal, one can find that the pressure dependence dE/dP (≈−7.8 cm⁻¹/GPa [14]) of R₁-line in sign and in magnitude is similar to that of E₀ line in SrB₄O₇:Sm²⁺, but its temperature dependence dE/dT (≈−15.1 × 10⁻² cm⁻¹/K) [15] around room temperature in magnitude is much larger than that of E₀ line in SrB₄O₇:Sm²⁺ crystal. Until now, one cannot find the theoretical interpretation and analysis on the very small thermal shift (or temperature dependence) of E₀ line for SrB₄O₇:Sm²⁺ crystal in literature. In fact, differing from the pressure shift (or pressure dependence) caused mainly by the static factor owing to the lattice compression [16], [17], the thermal shift of a spectral line position originates from both the static and dynamic factors, the former comes from the lattice thermal expansion and the latter stems from electron-phonon interaction [18], [19], [20], [21], [22]. In this article, we carry out a theoretical analysis and interpretation for the thermal shift of E₀ line in SrB₄O₇:Sm²⁺ crystal by dint of a full formula consisting of both the static and dynamic factors. Based on this, the static parameter A (characterising the static factor) and the electron-phonon coupling parameter α′ (characterising the dynamic factor) are estimated with an approximate processing even if there is no the observed thermal shift curve from low temperature (<30 K) to near and higher than room temperature. The outcomes are discussed.

2 Calculation

As both the dynamic and static factors of thermal shift from low temperature (T < 30 K) to near and higher than room temperature for a spectral line of transition metal or rare earth ions in crystals are closely connected with the Debye model, to the full formula of thermal shift ΔE(T) including both the static and dynamic factors is derived as [22]

where T_D is the Debye temperature. The parameters (A + α′) and T_D can be evaluated by fitting the experimental or total thermal shift curve from low temperature to near and higher than room temperature with (1) [22], [23] (noting that in many cases, people often take into account only the dynamic factor and so the static parameter A = 0 in (1). Thus, the estimated parameter α is the apparent electron-phonon coupling parameter [15], [19], [24]. For the E₀ line in SrB₄O₇:Sm²⁺ crystal under investigation, however, one can find only the observed linear thermal shift at near and higher than room temperature and the thermal shift curve from low temperature to near and higher than room temperature is not reported. So, other way should be considered for estimating the value of (A + α′). Here we evaluate the value of (A + α′) by the aid of an approximate processing.

At near and higher than room temperature, x=ℏω/kT is very little, then ex≈1+x and so (1) becomes

Equation (2) explains why the observed thermal shifts ΔE(T) of E₀ line in SrB₄O₇:Sm²⁺ and many other spectral lines in crystals at near and higher than room temperature vary roughly linearly with the increasing temperature. So, (2) can be regarded as suitable. Based on (2), we have the observed temperature dependence of line position:

Similar to the thermal shift ΔE(T), the observed temperature dependence consists of both the static and dynamic factors, i.e.

where the static factor can be computed by means of the pressure dependence (dEdP) of line position, namely,

in which αth=(1VdVdT) and β=(1VdVdP) are the thermal expansion coefficient and compressibility of the studied system. In consideration of the Gruneisen relation αth/β=rCν/V [25], the ratio αth/β in an impurity centre of crystal is similar to that in the host crystal. Here we have α_th ≈ 2.4 × 10⁻⁵/K [26] and β ≈ −5.9 × 10⁻³/GPa [12] for the host SrB₄O₇ crystal. Thus, from the observed pressure dependence dE0/dP≈−5.3cm−1/GPa [5], [6], [8], [11], we have

hence, from the observed (dEdT)obs.≈0.2×10−2cm−1/K [6], [11], [12],

The ratio t = A/α′ means the relative importance of static factor. For the E₀ line in SrB₄O₇:Sm²⁺, from (1), it can be given as

The Debye temperature of SrB₄O₇ is 1180 K [27], so, we find from the above values,

Then

3 Discussion

It can be seen from (1) to (3) that the positive signs of parameters A, α′ and the temperature dependence (dEdT)j (j = obs., st., dyn.) mean the thermal shift being blue shift, whereas the negative signs of them imply the thermal red shift. For the E₀ line in SrB₄O₇:Sm²⁺ studied, the static factor results in the thermal blue shift because of the positive values of A and (dEdT)st., but the dynamic factor gives rise to the thermal red shift owing to the negative values of α′ and (dEdT)dyn.. This point is in accord with the spectral lines of rare earth ions in crystals [23], [28]. In fact, not only the sign, but also the order of magnitude (∼10 cm⁻¹) of the estimated parameters A and α′ in SrB₄O₇:Sm²⁺ is almost the same as those of rare earth ions in crystals obtained by fitting the thermal shift curve from low temperature to near and higher than room temperature [23], [28]. Specially, for the same ion Sm²⁺ in SrFCl crystal, we have the static parameters A ≈ 57, 60 and 51 cm⁻¹, the electron-phonon coupling parameters α′ ≈ −21, −38 and −19 cm⁻¹ for ⁵D₀⟶⁷F₀, ⁵D₀⟶⁷F₁ and ⁵D₁⟶⁷F₀ spectral lines, respectively [23]. They in sign and in order of magnitude are identical with those of E₀ line in SrB₄O₇:Sm²⁺. So, our evaluated values of A and α′ for SrB₄O₇:Sm²⁺ are proper and the thermal shift parameters A and α′ can be acquired by an approximate processing even if only the thermal shifts at near and higher than room temperature are observed.

The observed thermal shift is due to the emulation between the static factor and the dynamic one. As the static factor (characterised by parameter A) in sign is contrary to and magnitude is slightly larger than the dynamic factor (characterised by parameter α′), the observed very small thermal blue shift for the E₀ line of SrB₄O₇:Sm²⁺ can be understood and explained.