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 hν (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 ΔV(t) produced versus timet 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 ν (or with increasing period τ) of the EM waves [5]. We probe this trend of π_V with pulsed X-rays in a standard medical machine and continuous X-rays in a diffractometer. Finally, we discuss our findings.

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 cm². 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 × 10¹⁸ 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 × 10¹⁸ Hz, and average period τ~0.51 × 10⁻¹⁸ s. The beam size is 4 cm². 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 ΔV when exposed to a temperature difference ΔT between the illuminated (T_hot) and non-illuminated (T_cold) faces. This effect is the Seebeck effect: ΔV = SΔT, where S is the Seebeck coefficient and ΔT = T_hot − T_cold. Finally, each module is a multilayer of AlO-Cu-Bi₂Te₃-Cu-AlO, as shown in Figure 1. The multilayer structure

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 cm². The value ε of the dielectric constant is ε = 1.423, as computed from

C = ε ε 0 area h

where ε₀ = 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 ΔV(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 ΔV(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 ΔV(t) andT_hot(t) using LabView 2012 and a National Instruments PXI-1042q communications chassis. We used OriginPro 2018 for analyzing the data.

The trends in time of T_hot(t) and ΔV(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 ΔV(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 ΔV(t). We ascribe the rise of both T_hot(t) and ΔV 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

noise, in the data analysis process we subtract the background from the acquired T_hot(t) and ΔV(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 ΔV(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 ΔV(t). We observe that the 40 keV X-rays cause a blip in both T_hot(t) and ΔV(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 ΔT_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 ΔV 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 ΔV and ΔT_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 ΔV(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 ΔV(t). In all panels, ΔV(t) increase linearly versus time, without blips. However, in panel (f), ΔV(t) bends because it falls precisely where in the illumination interval ΔV(t) achieves the plateau, visible in the top-right corner of Figure 2(b). On the other hand, T_hot(t)

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 ΔV(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 τ = 1 2 C Δ V 2 + 1 2 q 2 C − Σ 0 Δ 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 ΔV 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 ΔT is the amplitude ΔT_hot of the blip of the temperature T_hot, and Σ₀ 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 ~ 1 2 C Δ 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

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 Σ₀ 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.

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 P = 2.96 mW on the 42 pF capacitor [5]. Indeed, predictions involving a broad range of EM waves indicate that the voltage responsivity π_V generated in capacitors decreases by several orders of magnitude under illumination with EM waves of increasing frequency ν (or decreasing period τ, or wavelength λ). Radio waves are expected to yield larger voltage responsivities than IR and visible light [5]. We thus expect the voltage responsivities generated by X-rays to be smaller than those generated by the other EM waves, in particular visible and IR light, and radio waves.

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

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 Pτ and the law of conservation of energy enables us to unveil the relationship between the average power P of the X-rays and the effective dose delivered on an organic sample. The effective dose is an important operating parameter in standard medical imaging machines employing X-rays.

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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Hassan, S.I. and Scarel, G. (2020) The Total Energy in the Interaction of X-Ray Photons with Capacitors. World Journal of Condensed Matter Physics, 10, 159-177. https://doi.org/10.4236/wjcmp.2020.104010

To justify the use of Pτ and the law of conservation of energy, we exploit the total differential dE of the energy transferred from the EM waves to the capacitor on an area A_cs with diameter D corresponding to the cross-section of the EM beam:

d E = q d Δ V + Δ V d q − Σ d Δ T − Δ T d Σ , (A1)

where q = σA_cs is charge, σ = q/A_cs is surface charge density, and Σ is entropy. The thermal component of Equation (A1) is preceeded by a negative sign to signify that part of the energy transferred to the capacitor as electrical energy is thermally dissipated. Hypothesizing that the capacitance C = q Δ V enables us to separate the terms containing q and ΔV, thus reducing Equation (A1) to:

d E = C Δ V d Δ V + q C d q − Σ d Δ T (A2)

where we assume dΣ to be negligible. In a generalized grand-canonical ensemble [19], the entropy Σ is the Legendre transformation of Σ₀, the entropy of the canonical ensemble, such that Σ = Σ 0 − k B β q Δ V [19], where k_B = 1.38 × 10⁻²³ J/K is Boltzmann’s constant, and β = 1/k_BT. Therefore:

d E = C Δ V d Δ V + q C d q − Σ 0 d Δ T + k B β q Δ V d Δ T (A3)

Assuming k_BβqΔV to give just a slight correction to Σ₀, dE is further reduced to:

d E ≈ C Δ V d Δ V + q C d q − Σ 0 d Δ 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)Δt. 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 ( t ) = − P 0 exp ( − t / τ p ) + P ssli (A5)

where P₀ ≈ 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 = PΔt. To obtain E(t) and E_c, we integrate the variables ΔV, E, q, and ΔT between their minima and maxima values:

∫ Δ E min Δ E MAX d E = C ∫ Δ V min Δ V MAX Δ V d Δ V + 1 C ∫ q min q MAX q d q − Σ 0 ∫ Δ T min Δ T MAX d Δ 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 ΔV_MAX = ΔV(t), ΔE_MAX = ΔE(t), q_MAX = q(t), and ΔT_MAX = ΔT(t). The integration leads to:

E ( t ) = P ( t ) Δ t = 1 2 C Δ V ( t ) 2 + 1 2 q ( t ) 2 C − Σ 0 Δ T ( t ) (A7a)

In particular, in the ssli the integration gives:

E c = P Δ t = 1 2 C Δ V 2 + 1 2 q ssli 2 C − Σ 0 Δ 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 Δt= τ, the period of the EM wave.