New Photoacoustic Cell Design for Solid Samples †

: We have developed a new design for a photoacoustic (PA) cell particularly suited for quartz-enhanced photoacoustic spectroscopy (QEPAS), where a quartz tuning fork (QTF) is used as a sound detector for the PA signal. The cell is designed for the investigation of solid and semi-solid samples and represents a unilateral open cylinder. The antinode of the sound pressure of the fundamental longitudinal mode of the half-open cylinder occurs directly at the sample, where a measurement is difficult. Therefore, the first harmonic is used. A small hole in the resonator wall at the location of the pressure antinode allows signal detection outside the cylinder without (or only minimally) changing the resonance conditions. This design is particularly simple and easy to manufacture. A finite element (FE) simulation is applied to determine the optimal cell length for the given frequency and the location of the pressure maximum. One difficulty is that the open end dramatically changes the acoustic sound field. We answer the following research questions: where is the sound pressure maximum located and do simple analytical equations agree with the results of the FE simulation?


Introduction
In photoacoustic spectroscopy, a sound pressure wave is generated by modulated laser radiation.The molecules absorb the light energy by converting it into bending and stretching oscillations and rotations or electronic transitions.The associated temperature variation pulsates due to the modulated excitation, which leads to the development of sound waves.Over a wide range, the sound pressure amplitude is proportional to the concentration of the molecules under investigation.In contrast to gaseous samples, where the photoacoustic signal is often generated over the length of the laser beam within the cell, the signal of solid samples is generated directly at the surface of the sample or even slightly below it [1,2].
Often, acoustic resonances of the sample cell are exploited to improve the signal-tonoise ratio (SNR) of the PAS measurement.Special cells have been developed for the photoacoustic investigation of solid samples, e.g., in a cuboid [3,4] or a T shape [5,6].The latter was also modelled using Finite Element Methods (FEM) and optimised concerning the signal amplitude [7].
Recently, quartz tuning forks (QTF) with resonance frequencies beginning in the kHz region are used to detect the acoustic signal.QTFs possess a high specificity for separating noise in other frequency ranges from the signal.Further advantages are a high frequency stability over a wide temperature range, insensitivity to magnetic fields and a high resonance quality factor.Their low costs and small size make them suitable for small sensors in mass production [8][9][10].In this article, we present the design of a half-open cylindrical resonator for the photoacoustic investigation of solid samples particularly suited for QEPAS.One end of the tube is tightly sealed by the sample.The laser beam enters at the opening on the other side.The cell is considerably simpler to manufacture than a cuboid or T-shaped cell.The T cell consists of two different cylinders coupled together.Often, the larger one is referred to as the cavity cylinder and the one leading to the acoustic sensor as the resonance cylinder.The sample is attached to the cavity cylinder and the sound generated there has to couple into the second, geometrically much narrower resonance cylinder, which is accompanied by an additional damping because impedance discontinuities between large and small pipes usually lead to considerable energy losses [11].It is expected that the signal of the new cell will be significantly higher compared to the T cell.
The following section briefly comprises the theoretical background of sound propagation in a resonant cell in the case of the PA signal of a solid sample.Chapter 3 describes the finite element (FE) model and the results of the simulation.Chapter 4 contains a discussion and the study's outlook.

Acoustically Resonant PA Cell
PAS experiments with solid samples have already been carried out by Alexander Bell [12].The absorption spectrum of solid samples can be observed more easily with PAS than with conventional spectroscopy techniques since scattered light does not play an important role in the PA signal.The primary source of the PA signal is the result of periodic heat flow from the solid to the surrounding gas [13].
The SNR of the PA signal is intended to be enlarged by the resonator described here.In the cylinder, a standing wave is developing due to reflections at the tube's ends and the superposition of the waves.If the radius a of the tube is much smaller than the wavelength, the acoustic wave is basically longitudinal.At open ends, the sound wave does not undergo a phase shift, whereas closed ends lead to a phase shift of 180 degrees.In this application of a half-open cylinder, the tube will be in resonance if its length l ′ and the frequency f m are connected by the following equation [14] Here, l ′ = l + ∆l, where l is the geometric length of the cylinder and ∆l is called the end correction.At the open end, it can be assumed that the reflection takes place at an imaginary point ∆l away from the plane defined by the open end of the cylinder.The end correction is an effect resulting from a mismatch between the essentially one-dimensional acoustic field inside the pipe and the three-dimensional field radiated by its open end [14].The corrected overall length l ′ is also called the effective length of the cylinder.The end correction can be calculated from [15] ∆l = 0.6133•a. ( Due to loss effects, the true resonance frequencies are slightly lower than the eigenfrequencies.Therefore, the real end correction is slightly larger than the calculated value according to Equation (2).In order to obtain the highest possible photoacoustic signal, the radiation should be modulated with a frequency corresponding to the most intense acoustic resonance.Since the intensity decreases with increasing mode order [16] and the fundamental longitudinal resonance leads to a sound pressure maximum directly at the sample, the first harmonic of the longitudinal resonance is used.The according sound pressure distribution is schematically shown in Figure 1.
In upcoming experiments, a commercial quartz tuning fork will serve as a sound detector for the photoacoustic signal [17].Its resonance frequency is f R = 32.768kHz.The placement of the tuning fork in the so-called off-beam configuration is shown in Figure 2. In upcoming experiments, a commercial quartz tuning fork will serve as a sound detector for the photoacoustic signal [17].Its resonance frequency is  = 32.768kHz.The placement of the tuning fork in the so-called off-beam configuration is shown in Figure 2. Since the sample is stimulated by a laser beam, the diameter of the cylinder has to be larger than the beam width.The laser beam must not hit the wall of the tube.Otherwise, this results in a wall signal that interferes with the wanted PA signal of the sample.In this investigation, the radius of the tube is Combining the end correction formula Equation ( 2) the resonance frequency of the first harmonic  matching the QTF resonance frequency,  , leads to a geometric length of the pipe  = 6.6240 mm. ( where, for the speed of sound, the value 343 m s ⁄ has been used.

Finite Element Simulation
To obtain an independent and presumably more accurate result for , we performed a simulation using the FE tool COMSOL Multiphysics ® .Also, we aimed to obtain information on the location of the maximum sound pressure.
The simulation numerically solves the homogenous Helmholtz equation For the acoustic pressure field   ⃗ with the wave number .At the sound hard walls (the cylinder walls, the closed end and a plane surface defined by the flange at the  In upcoming experiments, a commercial quartz tuning fork will serve as a sound detector for the photoacoustic signal [17].Its resonance frequency is  = 32.768kHz.The placement of the tuning fork in the so-called off-beam configuration is shown in Figure 2. Since the sample is stimulated by a laser beam, the diameter of the cylinder has to be larger than the beam width.The laser beam must not hit the wall of the tube.Otherwise, this results in a wall signal that interferes with the wanted PA signal of the sample.In this investigation, the radius of the tube is  = 2 mm. ( Combining the end correction formula Equation (2) with the resonance frequency of the first harmonic  matching the QTF resonance frequency,  , leads to a geometric length of the pipe  = 6.6240 mm. ( where, for the speed of sound, the value 343 m s ⁄ has been used.

Finite Element Simulation
To obtain an independent and presumably more accurate result for , we performed a simulation using the FE tool COMSOL Multiphysics ® .Also, we aimed to obtain information on the location of the maximum sound pressure.
The simulation numerically solves the homogenous Helmholtz equation For the acoustic pressure field   ⃗ with the wave number .At the sound hard walls (the cylinder walls, the closed end and a plane surface defined by the flange at the Since the sample is stimulated by a laser beam, the diameter of the cylinder has to be larger than the beam width.The laser beam must not hit the wall of the tube.Otherwise, this results in a wall signal that interferes with the wanted PA signal of the sample.In this investigation, the radius of the tube is a = 2 mm. ( Combining the end correction formula Equation ( 2) with the resonance frequency of the first harmonic f 2 matching the QTF resonance frequency, f R , leads to a geometric length of the pipe l = 6.6240 mm.
where, for the speed of sound, the value 343 m /s has been used.

Finite Element Simulation
To obtain an independent and presumably more accurate result for l, we performed a simulation using the FE tool COMSOL Multiphysics ® .Also, we aimed to obtain information on the location of the maximum sound pressure.
The simulation numerically solves the homogenous Helmholtz equation For the acoustic pressure field p For the Helmholtz Equation, the truncation of a simulation domain for the open cylinder end is non-trivial.Conventional or periodic boundary conditions generate artifacts.A well-established method to deal with this problem is to define a perfectly matched layer (PML).PMLs can be imagined as non-reflecting wave absorbers [18].
The solutions of the differential Equation ( 5) subject to the described boundary conditions comprise the modes p m → r and the corresponding value k m .The eigenfrequencies are given by where c is the velocity of sound [19].It is assumed that the sample is sound-hard, closes the tube soundproof and that vibrations of the resonator walls are negligible.
To determine the geometric length of the tube and the location of the antinode of the sound pressure near the open end of the cylinder, a 2D axially symmetric FE model is used.Initially, l is set to the estimate of Equation 4.Then, the cylinder length is gradually varied until the eigenfrequency matches with the resonance frequency of the QTF (trial and error).The result of this procedure is l FE = 6.6605 mm.
In comparison to the analytical result (Equation ( 4)), the result of the FE simulation is 0.55% (0.037 mm) higher.The difference between the two methods is remarkably small.The FEM is supposed to rely on fewer approximations and is therefore considered to be more accurate.
Figure 3a shows the resulting sound pressure distribution of the first harmonic longitudinal mode in the centre plane of the cylinder closed by the sound-hard sample at the lower end.The semi-circle-shaped domain attached to the upper end of the resonator represents the outer space.The PML is implemented in the semi annulus.In Figure 3b, the FE mesh used for the simulations is depicted.The mesh is the result of a convergence study.The number of degrees of freedom is approximately 4500.
resonator's open end), the boundary condition applies.The normal derivative of the pressure is zero at the boundary For the Helmholtz Equation, the truncation of a simulation domain for the open cylinder end is non-trivial.Conventional or periodic boundary conditions generate artifacts.A well-established method to deal with this problem is to define a perfectly matched layer (PML).PMLs can be imagined as non-reflecting wave absorbers [18].
The solutions of the differential Equation (5) subject to the described boundary conditions comprise the modes   ⃗ and the corresponding value  .The eigenfrequencies are given by where  is the velocity of sound [19].It is assumed that the sample is sound-hard, closes the tube soundproof and that vibrations of the resonator walls are negligible.
To determine the geometric length of the tube and the location of the antinode of the sound pressure near the open end of the cylinder, a 2D axially symmetric FE model is used.Initially,  is set to the estimate of Equation 4.Then, the cylinder length is gradually varied until the eigenfrequency matches with the resonance frequency of the QTF (trial and error).The result of this procedure is  = 6.6605 mm. ( In comparison to the analytical result (Equation ( 4)), the result of the FE simulation is 0.55% (0.037 mm) higher.The difference between the two methods is remarkably small.The FEM is supposed to rely on fewer approximations and is therefore considered to be more accurate.
Figure 3a shows the resulting sound pressure distribution of the first harmonic longitudinal mode in the centre plane of the cylinder closed by the sound-hard sample at the lower end.The semi-circle-shaped domain attached to the upper end of the resonator represents the outer space.The PML is implemented in the semi annulus.In Figure 3b, the FE mesh used for the simulations is depicted.The mesh is the result of a convergence study.The number of degrees of freedom is approximately 4500.As expected, the sound pressure of the mode spreads into the exterior space.The sound pressure maximum is located 1.37 mm from the open end.This is supposed to be As expected, the sound pressure of the mode spreads into the exterior space.The sound pressure maximum is located 1.37 mm from the open end.This is supposed to be the optimal location for the sound detector.In order to avoid a substantial distortion of the sound field, a small hole could be drilled and the QTF placed in front of it (see Figure 2).

Discussion and Outlook
A simple cylindrical cell design of a PAS cell for measurements with QEPAS and solid samples has been developed.For the first harmonic longitudinal resonance, the results for the end correction of an analytical estimate and a numerical calculation are similar.
It is expected that the properties of the QTF can optimally complement the resonance behaviour of the small cylinder.The simple and inexpensive sensor design is also suitable for a miniaturised mass product.Since the expectation for the signal is that the SNR is larger than that of a T-shaped cell, these will be compared in the following experimental test series.
The next step will be to consider the effect of the hole in the cylinder wall on the sound pressure field.In the experimental testing of the sensor, we plan to investigate the dependency of the PA signal on the distance between the sound detector and the outer tube wall.Furthermore, the sample attachment can be examined more closely.Since the signal is generated at the surface of the sample, it is essential that it has a flat and smooth surface and is aligned perpendicular and acoustically sealed to the cylinder end.

Figure 1 .
Figure 1.Sound pressure distribution of the first harmonic longitudinal resonance of a half-open cylinder.

Figure 2 .
Figure 2. Cell with hole (marked with arrow) in a QEPAS off-beam configuration.

Figure 1 .
Figure 1.Sound pressure distribution of the first harmonic longitudinal resonance of a half-open cylinder.

Figure 1 .
Figure 1.Sound pressure distribution of the first harmonic longitudinal resonance of a half-open cylinder.

Figure 2 .
Figure 2. Cell with hole (marked with arrow) in a QEPAS off-beam configuration.

Figure 2 .
Figure 2. Cell with hole (marked with arrow) in a QEPAS off-beam configuration.

→r
with the wave number k.At the sound hard walls (the cylinder walls, the closed end and a plane surface defined by the flange at the resonator's open end), the boundary condition applies.The normal derivative of the pressure is zero at the boundary ∂p ∂n

Figure 3 .
Figure 3. (a) Sound pressure distribution at the central plane of the new cell at  = 32.768kHz.Dark blue and dark red correspond to the highest pressure values, which are 180 degrees out of phase.The distance of the maximum sound pressure to the open end is marked.(b) FE mesh.The PML is defined on the outer semi annulus.

Figure 3 .
Figure 3. (a) Sound pressure distribution at the central plane of the new cell at f 1 = 32.768kHz.Dark blue and dark red correspond to the highest pressure values, which are 180 degrees out of phase.The distance of the maximum sound pressure to the open end is marked.(b) FE mesh.The PML is defined on the outer semi annulus.