Longitudinal Tension and Mechanical Stability of a Pressurized Straw Tube

Abstract

For the development of charged particle detectors based on straw tubes operating in vacuum, a special measurement technique is required for the evaluation of their mechanical properties. A summary of the known equations that govern straw behavior under internal pressure is provided, and a new experimental method of a strained pressurized straw tube study is presented in this paper. The Poisson’s ratio of the straw wall, which defines the stability conditions of a built-in tube, was measured for the NA62 spectrometer straw, and its minimum pre-tension was estimated.

1. Introduction

During the past few decades, a series of experiments in high-energy physics have been designed to investigate very rare decay modes of kaon (see, for example, [1,2]). In particular, the purpose of NA62 experiment at CERN SPS [1] is to measure $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ decay branching ratio of the order of ${10}^{-10}$ with an uncertainty of 10%. It will be a test of the Standard Model and a probe to possible new physics. This is a challenging task, requiring an unprecedented precision of ${\pi }^{+}$ momentum measurement to reject the dominating background using the evaluation of event missing mass. This require a precise measurement of the momentum and position of the charged particle.

Another example of a physical problem that requires very accurate tracking of charged particles is Mu2e experiment, which will measure the conversion of a negative muon into an electron in the field of a nucleus [3].

A minimal material budget requirement in such experiments stimulates the development of gaseous particle detectors based on straw tubes containing gas under atmospheric pressure and operating in vacuum. These tubes maintain an internal overpressure of 1 atm, but higher overpressure may be desirable for a slower drift of electrons in the gas.

If pressurized straw tubes are close-packed and glued together to planar multi-layers, they form a stable structure. This approach is implemented, for example, for PANDA experiment trackers [4,5]. However, a very small material budget can lead to a detector design with separate straws [1,2,3]. In this case, the problem of lateral stability of straws is important, and the minimum required tension applied to the straw should be evaluated. In particular, straws in the NA62 spectrometer are separated in space and strained [1]. They are fixed in one lateral direction by means of thin wires and light spacers.

From the mechanics of a column, it is known that, if a compressive axial load is applied to a clamped thin-walled tube without overpressure, the load critical value $F_{crit}$ for buckling is [6,7]

$$F_{crit}=4{\pi }^{2}EJ/L^2.$$

(1)

where L is the straw length, and h is the thickness of the tube wall. Further, E is the Young’s modulus and $J=\int \int y^{2}dxdy=\pi R^{3}h$ is a second moment of the cross-section of the tube wall (R is the straw radius). $EJ$ is known as the object flexural rigidity. For an NA62 straw tube, $F_{crit}\approx 0.5N$, which is a small value.

It should be considered that a large straw deviation may appear when this limit is approached [7]. A shift of a few millimeters of the tube axis with respect to the anode wire may cause an electric discharge and detector malfunction. Only axial tension guarantees the buckling prevention for a long straw tube, and it has been demonstrated earlier [7,8] that an internal overpressure changes this necessary minimum tension.

Moreover, pre-tension is required to maintain the curvature caused by gravity to be small for a straw placed horizontally. For the given straw geometry and overpressure, the only parameter for the curvature control is the preliminary stretching force applied to the straw.

Therefore, estimation of the minimum pre-tension of the straw is necessary for any design of a straw tracker based on separated straws operating in vacuum. However, the calculations based on the published material properties of the straw wall are not sufficiently precise, because these properties depend on the batch of material. Therefore, an experimental procedure for the evaluation of material properties of a straw under operating conditions may be very useful.

The main purpose of this article is to present a new experimental method to estimate the minimum preliminary stretching force required to prevent the pressurized straw buckling and to limit its gravitational curvature. The equations used, which determine the behavior of straw under internal pressure, are given in the Appendix A.

2. Poisson’s Ratio Measurement

It can be observed from Equation (A7) that the straw buckling limit for a given overpressure is defined by the tube radius R and the axial Poisson’s ratio ${\mu }_{\parallel }$. The Poisson’s ratio is typically not provided in the PET film specifications. The published independent measurements of $\mu$ are not related to the specific batch of material used in the straw tubes production for the given detector. Moreover, the straw production process may change some foil properties. Therefore, a ${\mu }_{\parallel }$ measurement procedure applicable to a welded straw is required to evaluate the minimum straw pre-tension for the specific detector design.

2.1. Straw Specimens

Two straw specimens were tested (see

Table 1. Properties of the tested straws.

Property	9.8-mm Straw	18-mm Straw
Diameter, mm	$9.8\pm 0.1$	$18.0\pm 0.1$
Length, m	$2.10\pm 0.01$	$1.90\pm 0.01$
Material density, g/cm${}^3$	$1.4\pm 0.1$	$1.4\pm 0.1$
Wall thickness, $\mathsf{\mu }$m	$36\pm 1$	$54.4\pm 0.8$
Tube linear density, g/m	$1.55\pm 0.12$	$4.31\pm 0.06$
$E_{\parallel }$, N/mm${}^2$	$4500\pm 500$	$4000\pm 500$
$E_{\perp }$, N/mm${}^2$	$5000\pm 500$	$5500\pm 500$

). The tubes were made of PET Hostaphan^® foils using longitudinal welding [9]. The parameters of the foils can be found in the manufacturer specifications [10].

The first specimen (9.8-mm straw) was obtained from a party of approximately 7000 straws produced at the Joint Institute for Nuclear Research (Dubna) for NA62 spectrometer [9]. This 9.8-mm straw was coated inside the tube with two thin metal layers ($0.05\mathsf{\mu }$m of Cu and $0.02\mathsf{\mu }$m of Au) to provide electrical conductivity on the cathode and to improve the impermeability of the straw tube. The wall material volume density [10] was used to estimate the linear density of the 9.8-mm straw. The contributions of metal layers and the air mass inside the tube to linear density were negligible.

The second specimen (18-mm straw) was made of a thicker Hostaphan^® film. In this case, the linear density of the tube was measured by weighing, since the wall thickness was not strictly defined in the manufacturer specifications.

2.2. Test Bench for Studies of a Strained Straw under Pressure

A special test bench (see Figure 1) was manufactured to study the properties of a built-in straw with an initial pre-tension and an inner overpressure applied subsequently. The longitudinal force applied to the straw end $T_P=T-P\pi R^2$ was measured using a tensometer $Tm$ based on a single-point aluminum load cell (Tedea-Huntleigh, model 1022) [11].

A straw specimen was closed on the tensometer side by the end plug $e1$ with the end cap C using glue TRA-BOND 2115 [12] for rigid sealing. A longitudinal straw seam was located on a side to reproduce the conditions of the NA62 spectrometer for horizontal straws. The seam can change the measured values, but the violation of axial symmetry was considered insignificant when long straw is strained. The end cap C was connected to the tensometer $Tm$ using two flexible joints and a rigid rod. The other straw end contained a plug $e2$ with a gas supply channel. This straw end was glued into the solid support, which may be moved along the rigid basement B and fixed at a specific place to create a preliminary straw tension prior to the test. Subsequently, the pressure supply was opened, and the changing pressure values P were recorded together with the corresponding results of the tensometer measurement $T_P$ (see points in Figure 2 and Figure 3A).

Straw oscillations were studied using the optical coupler O [9], which emits constant intensity infrared radiation and registers the radiation reflected from the straw wall. When the straw oscillations were mechanically excited, the registered radiation intensity was modulated by the changing distance to the straw wall. The obtained signal was amplified and sent to the oscilloscope with a fast Fourier transform function. The lowest frequency peak position was registered as the lowest frequency of the straw (points in Figure 3B).

The end cap C was slightly shifted down owing to its weight of 10–20 gf. However, for the horizontal force $T_P>300$ gf (assuming the cap weight of 25 gf), the vertical shift of the cup was less than 4 mm. It led to the relative elongation of 2-m straw of $\approx {10}^{-4}$, which is much less than the minimum straw elongation caused by the tested preliminary tension (${10}^{-3}$). Thus, the straw ends could be considered to be fixed during the test.

2.3. Effective Tension and Poisson’s Ratio Measurement

It is known that any material becomes nonlinear for a large stress, whereas the linear properties of the material are defined for zero-stress limit. However, on the built test bench, precision measurement becomes problematic for a low effective tension $T_P$. Therefore, we must consider the possible nonlinearity in such a way that the resulting ${\mu }_{\parallel }$ could be easily extracted for $T\to 0$.

Accordingly, we postulated a weak linear dependence of Poisson’s ratio as a function of axial stress. For the fits of our experimental results, we used the tension-dependent value

$$\nu ={\mu }_{\parallel }-\frac{kT}{2\pi Rh}$$

(2)

instead of only ${\mu }_{\parallel }$. Apart from the material nonlinearity, the coefficient k also absorbs the effect of the set-up deformation under tension and the next-order effects ignored in the formulae. Equation (2) provides a physically motivated interpolation and extrapolation of the measurement results, whereas to compare the resulting Poisson’s ratio with the other measurements, we could consider the measured ${\mu }_{\parallel }$ and ignore the stress-dependent term.

T and R variables in Equation (2) depend on the Poisson’s ratio value. Therefore, we implemented an iterative procedure starting with a tension $T=T_0$, nominal straw radius $R=R_0$, and the starting Poisson’s ratio value of $\nu ={\mu }_{\parallel }$. In each iteration, new $T,\nu ,R$ values were calculated, and three iterations were sufficient for the precise calculation.

Two free parameters (${\mu }_{\parallel }$ and k) describe the measured tensions and pressures satisfactorily. Figure 2 and Figure 3A show the measured effective tensions for the NA62 straw along with the result of their fit with the formula

$$T_P=T_0-(1-2\nu )P\pi R^2.$$

(3)

The units of technical atmosphere $at$ were used for the overpressure values P. The MINUIT [13] package and ROOT [14] interface were used to obtain the resulting parameter values and their fit errors.

The fit to all the measured $T_P$ values for the given straw was performed with a common set of free parameters and the same assumed measurement error. The error ($\pm 3.6$ gf for the 9.8-mm straw and $\pm 20.47$ gf for the 18-mm straw) was defined in such a way that the resulting ${\chi }^2/ndf=1$, to estimate the parameter statistical uncertainties.

The fit results and systematic uncertainty contributions are shown in

Table 2. Poisson’s ratio measurement results.

	9.8 mm Straw			18 mm Straw
	k, $\frac{\mu m^2}{gf}$	${\mu }_{\parallel }$	${\mu }_{\perp }$	k, $\frac{\mu m^2}{gf}$	${\mu }_{\parallel }$	${\mu }_{\perp }$
Central value	3.98	0.3055	0.3394	−2.89	0.2960	0.4070
Radius correction	0.43	0.0005	0.0006	0.92	0.0005	0.0007
Radius value	0.14	0.0083	0.0092	0.59	0.0039	0.0054
$T_P$ scale	0.06	0.0099	0.0110	1.24	0.0088	0.0121
P scale	0.27	0.0094	0.0104	1.03	0.0084	0.0116
$\delta \frac{E_{\perp }}{E_{\parallel }}$	0	0	0.0306	0	0	0.0296
Systematic error	0.53	0.0160	0.0353	1.95	0.0128	0.0344
Statistical error	0.18	0.0005	0.0006	14.35	0.0150	0.0206
Total error	0.56	0.0160	0.0353	14.48	0.0197	0.0401

. For the second specimen, the uncertainty of the nonlinearity coefficient k was so large that its measured value was consistent with both zero and the result for the first sample.

The radius correction (Equation (A12)) was considered, but its effect was also completely included in the systematic errors as a “radius correction” contribution. The Gaussian width of the NA62 straw diameter distribution was approximately 0.03 mm, and the systematic shift of its central value from the nominal number was the same size [9]. Thus, considering a conservative value of the radius systematic uncertainty of 0.05 mm for the tested specimens, we obtained the “radius value” contribution to the systematic error shown in Table 2. Moreover, the effects of scale change by the factor of 1.05 for both the measured effective tension $T_P$ and measured overpressure P were considered as independent contributions (“$T_P$ scale” and “P scale”) to the systematic errors.

Considering the manufacturer’s information about Young’s moduli (see Table 1), the ${\mu }_{\perp }$ central values shown in Table 2 were extracted as ${\mu }_{\perp }={\mu }_{\parallel }\frac{E_{\perp }}{E_{\parallel }}$. Their uncertainties depend on the error of the modules ratio, which may be approximately estimated from the significant digits of the provided values as $\delta \frac{E_{\perp }}{E_{\parallel }}\approx 0.1$.

Typically, the reported Poisson’s ratio values for the oriented PET films are 0.37–0.44 [15], and thus, the obtained ${\mu }_{\perp }$ values are consistent with them, given the uncertainties of our measurements. However, the resulting ${\mu }_{\parallel }$ was approximately $0.3$, which may be a specific property of Hostaphan^® foil or the consequence of the production of straw tube using longitudinal welding [9].

3. Oscillation Frequency Measurements

The oscillation frequency measurement was the last test of the straw tensions performed on the assembled NA62 spectrometer modules [9]. The same test was repeated on the present test bench for a qualitative verification of the pressure effect.

The length of the tube on the test bench was defined by the position of the movable support and the end cup C position. Unfortunately, when a straw tension was applied to the end cup fixed on the short rod with flexible joints, an effective oscillator was formed with a frequency close to the measured frequency of straw. Therefore, using this test bench, only a qualitative understanding of straw oscillations could be obtained using the simple frequency Equation (A11).

The results of the lowest frequency measurements are shown in Figure 3B along with the corresponding calculation results based on Equation (A11). The error bars show the frequency uncertainty of 1 Hz defined by the width of the observed spectrum peaks.

The prediction using Equation (A11) of the 9.8-mm straw satisfactorily described the measurement results in the vicinity of NA62 design parameters ($T_0=1500$ gf, $P=1$ at). However, the overall discrepancy reached 2 Hz, which is more than the measurement precision. Nevertheless, the obtained qualitative description of the straw vibration confirms that the lowest frequency of the straw was significantly diminished by the internal overpressure despite the increase in straw wall stress according to Equation (A5).

4. Minimum Straw Pre-Tension Evaluation

We could use the obtained Poisson’s ratio to evaluate the minimum straw pre-tension $T_0$ in the straw-based NA62 spectrometer operating in vacuum [9].

There are two requirements for $T_0$. The buckling limit $T_P>0$ (where $T_P$ is calculated from Equation (3) for $P=1$ at) is defined by the Poisson’s ratio and the straw radius. The flexural rigidity contribution (Equation (1)) to the buckling prevention is small for the long NA62 straw and may be included into the safety margin. Moreover, the experimental design requirements define the gravitational deviation (sagitta) limit for a horizontal straw. Sagitta $Sag=y_s\left(0\right)$ calculated from Equation (A9) depends on the effective tension, flexural rigidity, and straw length.

The sagitta requirement $Sag<100$ $\mathsf{\mu }$m of NA62 [9] could not be satisfied using a reasonable pre-tension for the complete 2.1-m straw. Hence, special supporting spacers dividing each straw into three equal parts were implemented in the NA62 spectrometer [9]. These spacers fixed the straw only in one of the lateral directions, and thus, they did not change the small buckling critical load value (Equation (1)) ignored in this study.

Straw symmetry near each spacer ensured that the boundary conditions of Equation (A9) are satisfied. Hence, we used this solution with $L=0.7$ m in order to calculate the sagitta.

The results of the sagitta calculation and buckling limit estimations are shown in Figure 4A for the 9.8-mm straw in the NA62 spectrometer. The corresponding results for the 18-mm straw, assuming the same straw length of 70 cm, are shown in Figure 4B for comparison.

Apart from the most probable sagitta values and buckling limits, the worst cases are also shown in Figure 4. The worst case for each straw corresponds to $\nu$ (Equation (2)) central value diminished by the tripled total uncertainty evaluated for each $T_0$ from the measured ${\mu }_{\parallel }$ and k considering their correlation. All other parameters for this worst-case scenario were considered at their uncertainty limits leading to the largest sagitta and the easiest buckling.

Figure 4 shows that, for the NA62 spectrometer, a straw pre-tension value above 900 gf guarantees $Sag<100$ $\mathsf{\mu }$m (NA62 requirement), whereas the worst case of the buckling limit is below 400 gf. Thus, both the NA62 nominal straw pre-tension of $1.5$ kgf and the factual minimum pre-tension ($\approx 1.2$ kgf) [9] have a good safety margin.

However, if the 18-mm straw were used in the same detector design, the minimum $T_0$ would be defined by the increased buckling limit above $1.15$ kgf with a small safety margin, whereas the gravitational sagitta would always be below 100 $\mathsf{\mu }$m for the pre-tension above the buckling limit.

5. Conclusions

A new technique for the study of mechanical properties of straw tubes subjected to inner pressure and longitudinal tension was tested using a specially built test bench. It included the measurement of Poisson’s ratio of a straw wall, which defines the buckling limit of a straw with a given radius under a definite inner overpressure.

The axial Poisson’s ratio ${\mu }_{\parallel }$ for Hostaphan^® foil was measured for two specimens under the conditions close to those of a detector operating in vacuum. The lateral Poisson’s ratio ${\mu }_{\perp }$ was evaluated using the elasticity moduli in two directions provided by the foil producer. The spectra measurements of straw oscillations qualitatively confirm the effective tension predictions based on the measured ${\mu }_{\parallel }$.

The minimum pre-tension requirement for the NA62 spectrometer was re-evaluated based on the measurement results. The obtained limit confirms the detector design pre-tension with a safety margin of approximately 600 gf. The tested technique can be used for the development of future straw trackers.