The Total Energy in the Interaction of X-Ray Photons with Capacitors

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1. Introduction

Our main research goal is to explore the interaction between electromagnetic (EM) waves and matter and to find a law capable of consistently and quantitatively explaining such interaction throughout the whole EM spectrum. Significant effort was spent in exploring the interaction of visible and infrared (IR) light with matter. Here we focus on the high frequency region of the EM spectrum, *i.e.* to X-rays, by applying and assessing in this region the same law of conservation of energy successfully tested in the visible and IR light regions.

Currently several questions remain without clear answer in considering phenomena related to the EM wave interaction with matter. How much is the total energy conserved in light-matter interaction? Usually, researchers respond adopting a microscopic point of view in which photons transfer their energy h*ν* (where *h* is Planck’s constant and *ν* is the photon’s frequency) to electrons or other charges. However, when EM waves interact with a macroscopic object such as a sensor, a detector, or an energy harvester, their energy *hν* does not directly answer the question stated previously. Specifically, through *hν* it becomes difficult to correlate the response (*i.e.* the voltage, current, or temperature change) of sensor, detector, or energy harvester, to the power, or the intensity, of the absorbed EM wave. Vice-versa, it is hard to associate the power or intensity of the EM wave to the magnitude of the electric currents producing them in an EM wave generator. This difficulty proposes several questions in diverse physical situations. For example, why, in response to dim light at an intensity of few mW∙m^{−2} (10^{−3} W∙m^{−2}), is the current generated in rod cells, or photoreceptors in the eye’s retina, of the order of pA (10^{−12} A) [1] ? Why, in the bulk photovoltaic effect with the absorption of IR light at 10.6 μm wavelength and 10 to 300 mW power, is the produced current of the order of μA (10^{−6} A) [2] ? More generally, why is sensitivity steadily around 0.045 mAW^{−1} for photocurrents generated in different devices with red and IR light in the 1 to 200 mW power range [2] [3] ? On the other hand, why, in free-electron-based light sources, do electrons need to move close to the speed of light to emit ultraviolet (UV) light at a power of the order of only pW to nW (10^{−9} W) [4] ? Finally, why, when illuminated by visible and IR light with power in the mW range, do capacitors with capacitance of the order of pF produce voltages in the mV range [5] ?

Recent efforts provide a strategy to correlate the response (*i.e.* the voltage, current, or temperature change) of sensors, detectors and energy harvesters, to the power, or the intensity, of the absorbed EM waves. In these efforts, the law of conservation of energy is called into play [5]. Specifically, in absorption phenomena, the EM wave’s total energy is matched to the energy transferred to the absorbing device (sensors, detectors, and energy harvesters). This match is effective when the total energy of the EM waves is expressed as the product of their average power *P* and their period *τ*, *i.e.* *Pτ*, as proven in the case of the absorption of IR and visible light by capacitors [5] [6]. Mathematical details are provided in the Appendix. Examples of the effectiveness of the approach based on *Pτ* as the total energy are: the absorption of IR light by thin films in IR spectra [7], of visible light by the retina in vertebrate’s vision [8], by molecules in photoredox catalysis reactions [9], in photo-thermoelectric devices [9], and by neurons under two-photon excitation [9]. Vice-versa, in emission phenomena, *Pτ* is effective in a number of cases. For example, one is high-order multiphoton Thomson scattering [10], where the collision of a near IR laser beam with relativistic electrons gives rise to X-rays. Another example is the quantification of the power of the anomalous microwave emission (AME) by spinning dust in the interstellar medium [11]. A third example is the estimation of the power emitted by radio waves produced in the interaction between a red dwarf and an exoplanet [12]. However, the effectiveness of *Pτ* and the law of conservation of energy, expected to be valid in all manifestations of light-matter interaction, whether in absorption or in emission phenomena, is not yet assessed in the high frequency region of the EM spectrum, e.g. with X-rays. This assessment is the objective of this research.

We investigate the interaction of X-rays with a capacitor adopting a method similar to that in previous research [5]. Capacitors respond to visible and IR light through the production of a voltage with magnitude constrained by light’s power *P* [5] [6]. The capacitors used in Refs. [5] and [6] are, actually, thermoelectric devices, whose multilayer structure mimics a capacitor with estimated capacitance *C* of the order of pF (10^{−12} F) [5] [6]. The ability of capacitors to respond to light is widely documented in nature. Indeed, natural capacitors, such as the retinal neuron cells, respond to visible light with voltage production depending upon light’s power P [8]. Specifically, retinal neuron cells are compared to capacitors because their membrane acts as the capacitor’s dielectric layer, while the intra- and extracellular conducting fluids, that flank the membrane, act as the capacitor’s electrodes. The membrane’s capacitance *C* is of the order of 6 - 8 pF [13]. Such order of magnitude of the capacitance is common: for instance, in the hippocampus of behaving mice, the magnitude of the neuron’s capacitance is around 50 pF [14].

To assess the effectiveness of *Pτ* and the law of conservation of energy in the absorption of X-rays by a capacitor, we first measure the voltage Δ*V*(*t*) produced versus time*t* by the capacitor, and then estimate the average power *P* of the X-rays through *Pτ*. Contemporarily, we collect the temperature *T*_{hot}(*t*) generated on the capacitor’s face exposed to the X-rays. This procedure is the same as that adopted in Ref. [5]. We then assess the value of the average power *P* by following two routes. The first consists of determining if the value of P complies with the performance of a standard X-ray medical machine and of an X-ray diffractometer. The second route consists of computing the voltage responsivity π_{V} of capacitors and comparing it to the value expected from the model based on *Pτ*. The voltage responsivity π* _{V}* measures the effectiveness of the capacitor’s response to the EM waves, and is expected to increase with decreasing frequency

2. Experimental Details

*The* *X-ray* *sources*: The main X-ray source we used in our experiment was originally designed for medical imaging. Therefore, we expect the performance of the X-ray source to be standard for medical imaging X-ray machines. The X-rays used in our experiment are 40 keV in energy, and are produced with a 10 mA source current. We used 40 kV to maximize the amount of absorbed beam. Indeed, at higher energies, much more of the X-ray energy would pass through the capacitor, whereas at lower energy nearly all of it should be absorbed and the observed signal be above the background. In our set-up, the distance between the X-ray source and the capacitor is 90 cm, and the beam size is 4 cm^{2}. The X-rays have a broad spectrum, however, given their 40 keV energy, we expect the X-rays to have a frequency *ν* centered around 9.65 × 10^{18} Hz, and a period *τ* = *ν*^{−1} centered around 0.104 × 10^{−18} s. The average power of the X-rays is unknown, and estimating its value is one of the objectives of this research. We directed these X-rays normally to the capacitor’s surface.

We also used Cu K-*α* and -*β* X-rays obtained from a powder X-ray diffractometer (PXRD) (X’pert PANalytical) with average wavelength *λ*~1.5406 × 10^{−10} m, average frequency *ν*~1.947 × 10^{18} Hz, and average period *τ*~0.51 × 10^{−18} s. The beam size is 4 cm^{2}. We directed also these X-rays normally to the capacitor’s surface.

*The* *Capacitors*: The main capacitor used to test the interaction with X-rays is a stack of three Custom Thermoelectric 07111-9L31-04B modules. Each module has a length of ≈23 mm along the x- and the y-axes, and a height *h* = 3 mm. Moreover, each module is designed to produce a voltage difference Δ*V* when exposed to a temperature difference Δ*T* between the illuminated (*T*_{hot}) and non-illuminated (*T*_{cold}) faces. This effect is the Seebeck effect: Δ*V* = *S*Δ*T*, where *S* is the Seebeck coefficient and Δ*T* = *T*_{hot} − *T*_{cold}. Finally, each module is a multilayer of AlO-Cu-Bi_{2}Te_{3}-Cu-AlO, as shown in Figure 1. The multilayer structure

Figure 1. (a) Cross section of one simple capacitor, a single Custom Thermoelectric 07111-9L31-04B module, consisting of a layer of Bi_{2}Te_{3} semiconducting elements embedded between alumina (AlO) and Cu plates. This module features a capacitance C = 459 pF; (b) Illustration of the capacitor used in this research and consisting of three Custom Thermoelectric 07111-9L31-04B modules in series with insulating tape (IT) on the face exposed to the X-rays, which enables C = 42 pF. The IT consists of heavy cotton-cloth pressure sensitive tape with strong adhesive and tensile properties. In both panels (a) and (b) the capacitance C is evaluated for the area of the capacitor exposed to the X-rays.

of each module, and the in-series combination of three of them, makes the overall device comparable to a capacitor [5]. In addition, placing an insulating layer on the face directly exposed to the X-rays, enables an equivalent capacitance *C* of ~42 pF within the X-ray’s beam size of 4 cm^{2}. The value *ε* of the dielectric constant is *ε* = 1.423, as computed from

$C=\epsilon {\epsilon}_{0}\frac{\text{area}}{h}$

where *ε*_{0} = 8.854 × 10^{−12} Fm^{−1} is the permittivity in vacuum.

We also used a 89 pF capacitor, which we built by assembling two Custom Thermoelectric 07111-9L31-04B modules and a 07111-5L31-03CJ module. The value *ε* of the dielectric constant in this case is *ε* ≈ 2.

In our experiments, we assume the value of the capacitance *C* to be constant both in time, space, and for all frequencies of the EM waves. During the illumination with the X-rays, the capacitor is kept within two expanded polystyrene (EPS) enclosures to shield it from background noise.

*Data* *Acquisition*: Using Keithley 2000 multi-meters, capable of reading direct current (DC) voltages from 100 nV to 1 kV, we acquired the voltage difference Δ*V*(*t*) and the temperature *T*_{hot}(*t*) as a function of time *t*. We measured *T*_{hot}(*t*) using OMEGA type J self-adhesive thermocouple probes. The data collection lasted 50 minutes. During the first 30 minutes, called the *relaxation* *interval*, the data points were collected at low rate (one per minute) to monitor the relaxation of the capacitor in the EPS enclosures without exposure to the X-rays. During the final 20 minutes, called the *illumination* *interval*, the data collection under X-ray bombardment occurred at an increased rate (two per second) to visualize the effects on Δ*V*(*t*) and *T*_{hot}(*t*) of five 120 s long X-ray pulses. We allowed a time interval of three minutes between one pulse and the next. In both the *relaxation* and the *illumination* *intervals*, we contemporarily collected Δ*V*(*t*) and*T _{hot}*(

3. Results

The trends in time of *T*_{hot}(*t*) and Δ*V*(*t*) collected versus time from the 42 pF capacitor with and without the 40 keV X-ray illumination are shown in Figure 2(a) and Figure 2(b), respectively. We identify two intervals in the graphs. The first corresponds to the *relaxation* *interval*, in which the 42 pF capacitor is not exposed to the 40 keV X-rays. The second interval corresponds to the *illumination* *interval*, which describes the evolution of *T*_{hot}(*t*) and Δ*V*(*t*) upon exposing the 42 pF capacitor to five 120 s long X-rays pulses.

We observe that, in the *relaxation* *interval*, approximately 20 minutes after starting the data collection, *T*_{hot}(*t*) rises, followed about 10 minutes later by the rise of Δ*V*(*t*). We ascribe the rise of both *T*_{hot}(*t*) and Δ*V* without X-rays to background noise, which affects the 42 pF capacitor probably due to the insufficient shielding offered by the EPS enclosures. To mitigate the influence of background

Figure 2. Trends versus time of (a) the temperature T_{hot}(t) on the face of the 42 pF capacitor facing the X-ray source, and of (b) the voltage difference ΔV(t) produced by the capacitor during the whole 50 minute long experiment. During the first 30 minutes, the *relaxation* *interval*, the 42 pF capacitor relaxes inside two expanded polystyrene (EPS) enclosures. This interval corresponds to less dense data points. In the following 20 minutes, the *illumination* *interval*, the 42 pF capacitor receives five 120 s long 40 keV X-ray pulses. This interval corresponds to more dense data points.

noise, in the data analysis process we subtract the background from the acquired *T*_{hot}(*t*) and Δ*V*(*t*) data.

Figure 3 focuses on three zones in the *illumination* *interval* exhibiting the response versus time t of the 42 pF capacitor to 120 s long exposures to the 40 keV X-rays. The responses of *T*_{hot}(*t*) and Δ*V*(*t*) are shown after background subtraction. The top row of Figure 3 (panels (a), (c) and (e)) reports the trend in time of *T*_{hot}(*t*), while the bottom row (panels (b), (d) and (f)) reports those for Δ*V*(*t*). We observe that the 40 keV X-rays cause a blip in both *T*_{hot}(*t*) and Δ*V*(*t*). In all three cases, the blip of *T*_{hot} is pronounced and relatively narrow, confined within the 120 s exposure time to the X-rays. The amplitude Δ*T*_{hot} of the blip in *T*_{hot} varies between 0.045˚C and 0.1˚C, and its average value is (0.065 ± 0.030)˚C. On the other hand, in all three cases, the blip in voltage Δ*V* is pronounced but rather broad, rising at the beginning of the illumination pulse and declining at the end of it. The amplitude of the blip in voltage ΔV varies between 2.5 μV and 5 μV, with an average value of (3.5 ± 1.3) μV. The ratio *R* between the amplitudes of the blips in Δ*V* and Δ*T*_{hot} is on average (0.055 ± 0.005) mV/˚C for the three examined pulses. This value is two orders of magnitude lower than the *R* value found in the interaction between capacitors and visible light (*R* = 3.7 ± 0.2 mV/˚C with green light) and IR light (*R *= 21.5 ± 3.8 mV/˚C) [5]. Since the *R* value seems to decrease with increasing frequency *ν* (decreasing period *τ*), we find perfectly acceptable the low value of *R* under X-ray illumination in our experiment.

Figure 4 focuses on three 120 s long zones in the *illumination* *interval* in which the 42 pF capacitor is not exposed to the 40 keV X-rays. In all panels, the responses of *T*_{hot}(*t*) and Δ*V*(*t*) versus time *t* are presented without background subtraction. The top row of Figure 4 (panels (a), (c) and (e)) illustrates *T*_{hot}(*t*), while the bottom row (panels (b), (d) and (f)) displays Δ*V*(*t*). In all panels, Δ*V*(*t*) increase linearly versus time, without blips. However, in panel (f), Δ*V*(*t*) bends because it falls precisely where in the *illumination* *interval* Δ*V*(*t*) achieves the plateau, visible in the top-right corner of Figure 2(b). On the other hand, *T*_{hot}(*t*)

Figure 3. Zoom into three zones of T_{hot}(t) and ΔV(t) in the *illumination* *interval* of Figure 2, corresponding to exposure of the 42 pF capacitor to a 120 s long pulse of 40 keV X-ray (period *τ* = 0.104 10^{−18} s). Panels (a), (c) and (e) show, after background subtraction, the trend in time of T_{hot}(t) measured on the exposed face of the 42 pF capacitor. Panels (b), (d) and (f) illustrate, after background subtraction, the corresponding voltage difference ΔV(t) in the same time zones. For each pulse, we indicate the amplitudes of the blips in T_{hot}, ΔT_{hot}, and voltage ΔV, the estimated magnitude of the average power P of the X-rays, and the R ratio (*i.e.* the ratio between ΔV and ΔT_{hot}).

Figure 4. Zoom into three time zones of T_{hot}(t) and ΔV(t) in the *illumination* *interval* of Figure 2 in which the 42 pF capacitor is not exposed to X-ray pulses. Panels (a), (c) and (e) show, without background subtraction, the trends of T_{hot}(t). Panels (b), (d) and (f) illustrate, without background subtraction, the corresponding voltage difference ΔV(t) in the same time zones.

in panel (c) decreases linearly versus time, whereas it is flat in panel (e). Only panel (a) shows a blip of about 0.05˚C, *i.e.* of the same order of magnitude of the blips detected upon exposure of the 42 pF capacitor to the 40 keV X-rays in Figure 3. The origin of this blip in *T*_{hot}(*t*), visible in Figure 4(a), is for the moment, uncertain. Overall, the comparison between Figure 3 and Figure 4, considering that no background subtraction was applied to the data in Figure 4, suggests that the blips in Δ*V*(*t*) found in Figure 3(b), Figure 3(d) and Figure 3(f) are a consequence of the 120 s long exposure of the 42 pF capacitor to the 40 keV X-rays.

Previous research [5] used the law of conservation of energy to quantify the energy involved in the interaction between EM waves and a capacitor. The law of conservation of energy is expressed in the following equation:

$P\tau =\frac{1}{2}C\Delta {V}^{2}+\frac{1}{2}\frac{{q}^{2}}{C}-{\Sigma}_{0}\Delta T$ (1)

More details about the derivation of this equation are provided in the Appendix. In Equation (1), *Pτ* is the total energy conserved in the interaction between EM waves and a capacitor. The variables *C* and Δ*V* are, respectively, the capacitance of the capacitor, and the amplitude of the blip in voltage generated by the EM waves interacting with the capacitor. The variable *q*, in the second addend of Equation (1), is the amount of charge displaced by the EM waves upon their arrival on the surface of the capacitor [5]. Finally, in the third addend of Equation (1), the variable Δ*T* is the amplitude Δ*T*_{hot} of the blip of the temperature *T*_{hot}, and Σ_{0} is entropy. All addends in Equation (1) are required to be of the same order of magnitude of *Pτ*, and vice-versa, in order to be significant. Otherwise, if they are too small they are neglected, if they are too big they prevail. A similar use of constraints is employed by the authors of Ref. [12] to estimate the surface magnetic field of the red dwarf emitting radio waves.

Here we apply Equation (1) to the interaction between X-rays and a capacitor to determine the average power *P* of the X-rays. To this end, we exploit the constraint that all addends in Equation (1) must to be of the same order of magnitude of *Pτ*, and vice-versa, to be significant. This constraint enables us to estimate *P* as:

$P~\frac{1}{2}C\Delta {V}^{2}$ (2)

The values of *P* we found using Equation (2) for the three pulses in Figure 3 are summarized in Table 1. On average, the power of the 40 keV X-rays is

Table 1. Amplitude of three blips in voltage ΔV measured during the illumination of a 42 pF capacitor with a 2 minute long 40 keV X-rays pulse. The three blips are depicted in Figure 3, panels (b), (d), and (f). Average power P of the 40 keV X-rays pulse hitting the 42 pF capacitor estimated using Equation (2). The average value of P is (2.96 ± 2.40) mW. Finally, voltage responsivity π_{V} = ΔV/P estimated for the three blips in voltage ΔV depicted in Figure 3, panels (b), (d), and (f). The average value of π_{V} is (1.50 ± 0.56) 10^{−3} V/W.

*P* = (2.96 ± 2.40) mW. Interestingly, this value of power is within the range (from 3 μW to 20 mW) measured for synchrotron X-rays [15]. This observation suggests that the value we have found is reasonable. Knowing *P*, we estimate the average value of the total energy as *Pτ* = 0.31 × 10^{−21} J. Proceeding in the same way from Equation (1), and using the average value of the amplitude of the blip in *T*_{hot} (0.065˚C ± 0.030˚C), we determine the average value of the entropy Σ_{0} as ~4.7 10^{−21} J/˚C. Similarly, since the capacitance *C* of the capacitor is 42 pF, the displaced charge q in Equation (1) is estimated to be of the order of ~1.61 × 10^{−16}˚C.

4. Discussion

In our experiment, two routes help assessing whether *Pτ* is the total energy conserved in X-ray absorption. Both routes exploit the average power *P* = 2.96 mW that we estimated through *Pτ* and the law of conservation of energy. The first route verifies that *P* = 2.96 mW supports the performance of the standard medical X-ray imaging machine we used in our experiment. The second route compares with trends established for other EM waves, the voltage responsivityπ* _{V}* [5] imposed by

4.1. The X-Ray Power in a Standard X-Ray Imaging Medical Machine

Power-meters responsive to visible light easily allow measuring the power of an optical laser beam. Unfortunately, power meters responsive to X-rays are at the infancy of their development and, to date, are only in use for research purposes [15] [16]. Usually these power meters are large, heavy, operating at low temperatures and difficult to operate. Even if compact and easy-to-use X-ray power-meters are available [15], neither power nor intensity (power per area) are operating parameters in standard medical X-ray imaging machines. Rather, these machines are set to deliver an effective dose (Δ_{eff}, in units of Sv, Sievert, or J * kg^{−1}) of X-rays on a specific patient’s organ. The effective dose Δ_{eff} establishes the energy *E*_{abs} = Δ_{eff} m to be absorbed by the mass m within the organ to obtain the desired imaging or therapeutic goal. We call *E*_{abs} the transferred energy, which differs from the total energy *Pτ*. Specifically, the transferred energy *E*_{abs} is the amount of energy transferred to the mass m in a selected volume *V* independently of the exposure time Δ*t*. Instead, the total energy *Pτ* corresponds to the amount of energy transferred within the period *τ* of the EM wave in the volume of matter hit by the EM wave. Analogously to Equation (1), we hypothesize that the total energy *Pτ* and the transferred energy *E*_{abs} are related through the law of conservation of energy such that:

$P\tau =\tau {\Delta}_{\text{eff}}\frac{m}{\Delta t}-{\Sigma}_{0}\Delta T$

or

$P\tau ={E}_{\text{abs}}\frac{\tau}{\Delta t}-{\Sigma}_{0}\Delta T$ (3)

As in Equation (1), also in Equation (3) all addends, to be significant, are required to be of the same order of magnitude of *Pτ*. Therefore, we can approximate Equation (3) as follows:

$P\tau ~\tau {\Delta}_{\text{eff}}\frac{m}{\Delta t}={E}_{\text{abs}}\frac{\tau}{\Delta t}$ (4)

Knowing both the mass m and the exposure time Δ*t*, this approximation allows us to constrain the value of the average power *P* of the X-rays from the known value of the dose Δ_{eff}. For instance, for a typical mammogram, Δ_{eff} = 0.36 mSv or 0.36 mJ∙kg^{−1} [17]. The density of the breast tissue is, on average, *δ* = 0.985 g/cm^{3}. Therefore, a volume *V* = 9 cm^{3} (diameter *d* = 1.3 cm) of breast tissue has a mass m of about 8.86 × 10^{−3} kg. The transferred energy *E*_{abs} = Δ_{eff} m from the X-rays to the breast tissue, is therefore 3.19 × 10^{−6} J, and an exposure time Δ*t* = 1.1 ms would be compatible with the average power *P* = 2.96 mW found with P*τ* and the law of conservation of energy in Section 3. Similarly, with Δ*t* = 0.2 s, which is the exposure time normally adopted in mammogram screenings, and assuming an average power *P* = 2.96 mW, a breast tissue volume V with a diameter *d* = 12 cm could be imaged. Thus, either *V* = 9 cm^{3} (*d* = 1.3 cm) and Δ*t*= 1.1 ms, or *d* = 12 cm and Δ*t* = 0.2 s, represent realistic scenarios. From these examples, we conclude that the X-ray’s average power estimated as*P* = 2.96 mW satisfies the expectations on the performance of a standard medical X-ray imaging machine.

An additional example consists of using the X-rays of our standard medical machine to image wrist and hand bones. In this case, Δ_{eff} = 4 mSv [17], and *δ* = 1.92 g/cm^{3}. Thus, for *V* = 9 cm^{3} (*d* = 1.3 cm), m is 17.3 × 10^{−3} kg, which determines a transferred energy *E*_{abs} = 69.1 × 10^{−6} J. In this condition, an exposure time Δ*t* = 0.02 s is compatible with the average power *P* = 2.96 mW. If we instead choose Δ*t*= 0.2 s while *P* = 2.96 mW, we could image a wrist or hand bone volume of diameter *d* = 4 cm. Both scenarios are realistic, thus we conclude that also in imaging wrist and hand bones the average power *P* = 2.96 mW satisfies the expectations on the performance of a standard medical X-ray imaging machine.

Summarizing, in Section 3, using the total energy *Pτ* and the law of conservation of energy, we estimate the average power of the X-rays from a machine previously used for medical imaging to be *P* = (2.96 ± 2.40) mW. Subsequently, in Section 4.1, using the concept of transferred energy, we find that the average power *P* = 2.96 mW is compatible with the X-ray exposure time Δ*t* typically applied to image patient’s organs. This result validates our approach for estimating the power of an X-ray beam with *Pτ* and the law of conservation of energy.

4.2. The Responsivity π_{V} of the 42 pF Capacitor under X-Ray Illumination

Responsivity is a parameter used to characterize the performance of devices such as sensors, detectors and energy harvesters under the effect of EM waves [18]. Analytically, responsivity is the ratio between the change in voltage, or current, produced by the specific device, and the power *P* of the EM wave that illuminates the device. For example, when the device produces a voltage change Δ*V*, the voltage responsivity is π* _{V}* = Δ

Using *Pτ* and the law of conservation of energy, Boone *et* *al.* [5] found that the voltage responsivity π* _{V}*= Δ

In Figure 5 and Table 2 we illustrate the trend of the voltage responsivity π* _{V}* from X-rays (this research) to visible and IR light (Ref. [5] ) by showing π

Table 2. Comparison of ratio R (ratio between the amplitudes of the jumps of voltage ΔV and temperature ΔT_{hot}), average power P, energy P*τ*, entropy Σ_{0}, charge q, and voltage responsivity π_{V} involved in the interaction with a capacitor of 40 keV X-rays (this research), Cu K-*α* and -*β* X-rays (this research), visible and infrared (IR) light (Ref. [5] ). The trend of the voltage responsivity π_{V} over the range from X-rays to IR light is further illustrated in Figure 5.

Figure 5. Trend versus electromagnetic (EM) wave’s period *τ* of the voltage responsivity π_{V} = ΔV/P of capacitors illuminated by radio waves (preliminary data), infrared (IR) [5], visible [5], and X-rays (this research). The magnitudes of π_{V} are summarized in Table 2. The x-axis, in logarithmic scale, covers thirteen orders of magnitude of the period *τ*, while the y-axes, also in logarithmic scale, spans over thirteen orders of magnitude of the voltage responsivity π_{V}. The experimental data points are fitted with the function π_{V} ≈ 1 10^{14} *τ* C^{−1} (here C = Coulomb). Most of the data points were taken from illumination with continuous EM waves (cw). The datum generated by the 40 keV X-ray pulses is marked as “pulsed”.

detected in Figure 5, of the voltage responsivity π* _{V} *of capacitors exposed to X-rays also validates our approach to estimate the power of an X-ray beam with P

4.3. Additional Assessment through the Illumination of an 89 pF Capacitor with Cu K-*α* and -*β* X-Rays

To further assess the ability of our procedure to effectively measure the average power *P* of X-rays, we exposed a 89 pF capacitor to Cu K-*α* and -*β* X-rays obtained from a powder X-ray diffractometer (PXRD) (X’pert PANalytical). We built the 89 pF capacitor by assembling two Custom Thermoelectric 07111-9L31-04B modules and a 07111-5L31-03CJ module. The X-rays have an average wavelength *λ*~1.5406 × 10^{−10} m, average frequency *ν*~1.947 × 10^{18} Hz, and average period *τ*~0.51 × 10^{−18} s. We directed the X-rays normally to the capacitor’s surface for about 20 minutes. Due to the impossibility of contemporarily fit three multi-meters in the housing of the X’pert PANalytical X-ray diffractometer, we asynchronously measured the trends in time of the voltage Δ*V*(*t*) and temperature *T*_{hot}(*t*), shown in Figure 6. The jumps of Δ*V*(*t*) and *T*_{hot}(*t*) are well defined, and the data do not demand a background subtraction. We observe an amplitude of the jumps in voltage and temperature of Δ*V* = 0.41 mV and Δ*T*_{hot} = 0.52˚C, respectively. From these amplitudes, and using a procedure similar to that adopted in Section 3 and Section 4.1, we estimate the ratio *R* between the amplitudes of the jumps of Δ*V* and Δ*T*_{hot}, the average power *P*, the total energy *Pτ*, the entropy Σ_{0}, the displaced charge *q*, and the voltage responsivity π* _{V}*. These results are summarized in Table 2, where the quantities are compared to the corresponding ones obtained with the 40 keV X-rays (this research) and with visible and IR light (Ref. [5] ). From these comparisons, we observe that the values of

One striking characteristics of the energy *Pτ* emerging from Figure 6 needs further comments. The jump in voltage Δ*V* caused by the Cu K-*α* and -*β* X-rays on the 89 pF capacitor is of the same order of magnitude of that generated at

Figure 6. Trends versus time of (a) ΔV(t) and (b) T_{hot}(t) generated by the illumination of a 89 pF capacitor with Cu K-*α* and -*β* X-rays (period *τ* = 0.51 10^{−18} s). The trends were measured asynchronously, due to the impossibility to contemporarily fit three multi-meters in the housing of the X’pert PANalytical X-ray diffractometer used for the experiment. No background subtraction is necessary in these data sets. In each panel, we indicate the amplitudes of the jumps in voltage ΔV and temperature ΔT_{hot}, and the estimated magnitude of the average power P of the X-rays.

much lower average power *P* by green light from Ref. [5], and radio waves from our preliminary data. The result is depicted in Figure 7 and Table 3. The corresponding values of Pτ in the three cases turn out to be of the same order of magnitude. This observation suggests that waves with the same total energy *Pτ* produce similar effects, e.g. the same jump in voltage ΔV. Therefore, since X-rays have smaller values of the period *τ* than, e.g. visible light, it turns out that X-rays achieve the same energy *Pτ* of visible light with larger values of average power *P* than visible light. Accordingly, the effects of the average power *P*~3 mW of X-rays with period *τ* = 1 × 10^{−19} s produced by our standard medical X-ray imaging machine and generating a *Pτ* = 3 × 10^{−22} J, are the same as those due to green light at *τ* = 1.77 × 10^{−15} s with average power *P*~1.69 × 10^{−7} W!

5. Conclusions

In this research, we explore the interaction between X-rays and a capacitor. We achieve the following conclusions: 1) The product of the average power *P* times, the period *τ*, *i.e.* *Pτ*, is the total energy conserved in the absorption process also

Figure 7. Trends in time of the voltage difference ΔV(t) generated through the illumination of capacitors with various values of capacitance C, by Cu K-*α* and -*β* X-rays (a); green light (b); and radio waves (c). The period *τ* and the average power P of the various electromagnetic (EM) waves, and the values of the jumps in voltage ΔV are indicated in each panel. We highlight that the values of ΔV and P*τ* are of the same order of magnitude in the three cases (see Table 3 for numerical details).

Table 3. Summary of the similar experimentally measured jumps of voltage ΔV of about (0.51 ± 0.20) mV produced on a capacitor of capacitance C by electromagnetic (EM) waves of period *τ*, average power P, and total energy P*τ* (average value 18.3 ± 22.6 J). As EM waves for this summary, we have chosen Cu K-*α* and -*β* X-rays from this research, green light from Ref. [5], and radio waves from our preliminary data. The experimental trends of ΔV(t) under illumination of the capacitor by Cu K-*α* and -*β* X-rays, green light, and radio waves are summarized in Figure 7.

for X-rays, not just for visible and infrared (IR) lights. 2) The voltage responsivity of the capacitor decreases with increasing frequency *ν* (or decreases with decreasing period *τ*) of the electromagnetic (EM) waves. Indeed, the voltage responsivity π* _{V}* of the capacitor we used in our investigation is significantly smaller than that previously determined by illumination with visible and IR light of the same capacitor. 3) The formalism related to

In addition to these fundamental results, we achieve a significant practical result: the procedure we used in our investigation enables us to measure the average power *P* of the X-rays generated in two machines, one originally designed for usage in medical imaging, and the other designed for X-ray diffraction of powders. The power *P *is of the order of few mW for the standard X-ray medical imaging machine, and W for the X-ray diffractometer. Our procedure paves the ground for designing an X-ray power-meter, a tool presently missing in the market of X-ray characterization tools.

Acknowledgements

This work was supported by the JMU Department of Physics and Astronomy, and the JMU Center for Materials Science. The authors thank A. Banu and S. Pendleton for fruitful discussions and for technical support at the Madison Accelerator Laboratory (JMU). The authors also thank B. A. Reisner and X. Hu (JMU, Chemistry and Biochemistry) for technical support, and gratefully acknowledge the National Science Foundation (NSF) for instrumentation support (DMR-0315345) enabling the X-Ray measurement performed with the X-ray diffractometer (PXRD) (X’pert PANalytical). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.

Appendix

To justify the use of *Pτ* and the law of conservation of energy, we exploit the total differential d*E* of the energy transferred from the EM waves to the capacitor on an area *A _{cs}* with diameter

$\text{d}E=q\text{d}\Delta V+\Delta V\text{d}q-\Sigma \text{d}\Delta T-\Delta T\text{d}\Sigma $, (A1)

where *q* = *σA _{cs}* is charge,

$\text{d}E=C\Delta V\text{d}\Delta V+\frac{q}{C}\text{d}q-\Sigma \text{d}\Delta T$ (A2)

where we assume dΣ to be negligible. In a generalized grand-canonical ensemble [19], the entropy Σ is the Legendre transformation of Σ_{0}, the entropy of the canonical ensemble, such that
$\Sigma ={\Sigma}_{0}-{k}_{B}\beta q\Delta V$ [19], where *k _{B}* = 1.38 × 10

$\text{d}E=C\Delta V\text{d}\Delta V+\frac{q}{C}\text{d}q-{\Sigma}_{0}\text{d}\Delta T+{k}_{B}\beta q\Delta V\text{d}\Delta T$ (A3)

Assuming *k _{B}*

$\text{d}E\approx C\Delta V\text{d}\Delta V+\frac{q}{C}\text{d}q-{\Sigma}_{0}\text{d}\Delta T$ (A4)

At the start of the illumination of the capacitor with visible or IR light, the energy conserved in each instant of time *t* is *E*(*t*) = *P*(*t*)Δ*t*. Here *P*(*t*) is the exponential function that describes the rise in time of the EM wave’s power with a time constant *τ*_{P} such that:

$P\left(t\right)=-{P}_{0}\text{exp}\left(-t/{\tau}_{p}\right)+{P}_{\text{ssli}}$ (A5)

where *P*_{0} ≈ *P*_{ssli}≈ *P *and *P* is the average power in the steady state illumination (ssli) regime. In the ssli the energy conserved is *E _{c}* =

${\int}_{\Delta {E}_{\mathrm{min}}}^{\Delta {E}_{\text{MAX}}}\text{d}E}=C{\displaystyle {\int}_{\Delta {V}_{\mathrm{min}}}^{\Delta {V}_{\text{MAX}}}\Delta V\text{d}\Delta V}+\frac{1}{C}{\displaystyle {\int}_{{q}_{\mathrm{min}}}^{{q}_{\text{MAX}}}q\text{d}q}-{\Sigma}_{0}{\displaystyle {\int}_{\Delta {T}_{\mathrm{min}}}^{\Delta {T}_{\text{MAX}}}\text{d}\Delta T$ (A6)

We redefine the variables such that their minima are set at zero in their respective units and their maxima correspond to the values of the variables at an arbitrary instant of time *t* such that Δ*V*_{MAX} = Δ*V*(*t*), Δ*E*_{MAX} = Δ*E*(*t*), *q*_{MAX} = *q*(*t*), and Δ*T*_{MAX} = Δ*T*(*t*). The integration leads to:

$E\left(t\right)=P\left(t\right)\Delta t=\frac{1}{2}C\Delta V{\left(t\right)}^{2}+\frac{1}{2}\frac{q{\left(t\right)}^{2}}{C}-{\Sigma}_{0}\Delta T\left(t\right)$ (A7a)

In particular, in the ssli the integration gives:

${E}_{c}=P\Delta t=\frac{1}{2}C\Delta {V}^{2}+\frac{1}{2}\frac{{q}_{\text{ssli}}^{2}}{C}-{\Sigma}_{0}\Delta T$ (A7b)

where *q*_{ssli} is the surface charge in the ssli on the area *A _{cs}*. Simulations show that Equation (A7) is effective if Δ

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