# Laser-Ultrasound-Based Grain Size Gauge for the Hot Strip Mill
## 1 Introduction
### Grain-Size Measurement with Laser Ultrasonics  
Ultrasonic attenuation has been used to characterize microstructure in polycrystalline materials for a long time and there are numerous proposed methods. Some of the suggestions for analyzing laser ultrasound include monitoring the rise-time of the echoes [13]; the multiple echo method [14; 15], wherein the attenuation between two echoes are compared; and later the single echo method [16], which is an ingenious way of circumventing the diffraction effects and thus calculating the grain size rather accurately. In addition, more exotic methods have been proposed, such as backscattering amplitude analysis [17] and machine learning [18]. These methods have their individual strengths and weaknesses, and the conditions for which they are robust and valid depend on the experimental setup, as well as the material properties of the object being studied. Currently, there is no method that is universally supreme. Instead, one must select that method that fits best for the requirement of the application of interest and apply the appropriate assumptions/simplifications. In the present paper, the frequency-dependent attenuation is considered to calculate the grain size. Generally, four steps are involved to acquire the average through-thickness grain size, which is illustrated in Figure 1a-d below. Figure 1a displays a schematic of a sample cross-section where the ultrasound is generated by laser ablation on the surface, which creates a shockwave in the material. The ultrasonic response of a sample is acquired by recording the surface vibration amplitude as a function of time for several backwall echoes, illustrated by Figure 1b. The cross-section of the sample illustrates the ultrasonic wave as the first backwall echo reaches the surface, which corresponds to the second peak in the A-scan. In Figure 1c, a theoretical plot of the measured frequency-dependent ultrasonic attenuation for the echoes is displayed. Finally, the grain size is acquired with a calibration curve, which transfers the measured 3 rd order attenuation \(b\) to a grain size value, as illustrated in Figure 1d. The total ultrasonic attenuation \(\alpha\) can be described by the sum of three parts as:  
\[\alpha=\alpha_{abs}+\alpha_{diff}+\alpha_{scat} \tag{1}\]  
where \(\alpha_{abs}\) is the material absorption stemming from, e.g., internal friction, \(\alpha_{diff}\) is the frequency-dependent attenuation due to diffraction, and \(\alpha_{scat}\) is the frequency-dependent attenuation describing the average grain-scattering effects that in turn are dependent on the material properties, such as the anisotropy, texture and the average grain size. The LUS measurement of attenuation in the material is a sum of all of the scattering events at the grain boundaries, where the reflected/scattered/refracted waves reach the area of the detection laser. Scattering in polycrystalline materials has been described in the literature by, e.g., [19] and is simplified to the following relation between the average grain size \(\overline{D}\) and the attenuation due to scattering:  
\[\alpha_{scat}\ \sim K(T)\overline{D}^{n-1}f^{n}, \tag{2}\]  
where \(n\) is dependent on the scattering regime (Rayleigh n = 4, stochastic n = 2), and K(T) accounts for the temperature-dependent anisotropy. The frequency range considered here is roughly 1-40 MHz, which is in between the stochastic and the Rayleigh regimes. It has been shown that fixing the exponent to n = 3 gives a reasonable linearization of the problem [20; 21; 22; 23; 24]. Additionally, if only high enough frequencies are considered in the calculation, then the effects from diffraction are small in comparison and can be neglected. The result of this is that the total attenuation can be written as:  
\[\alpha=a+bf^{3}, \tag{3}\]where  
\[b=K(T)\overline{D}^{2}, \tag{4}\]  
and \(a\) is a constant offset which contains the material absorption and other constant contributions.