Verified modeling of a low pressure hydrogen plasma generated by electron cyclotron resonance

A self-consistent fluid model has been successfully developed and employed to model an electron cyclotron resonance driven hydrogen plasma at low pressure. This model has enabled key insights to be made on the mutual interaction of microwave propagation, power density, plasma generation, and species transport at conditions where the critical plasma density is exceeded. The model has been verified by two experimental methods. Good agreement with the ion current density and floating potential—as measured by a retarding energy field analyzer—and excellent agreement with the atomic hydrogen density—as measured by two-photon absorption laser induced fluorescence—enables a high level of confidence in the validity of the simulation.


Introduction
Low pressure hydrogen plasmas are of great interest to a plethora of industrial sectors including multiple stages of semiconductor fabrication [1][2][3][4], diamond-like carbon film manufacturing [5,6], and for their usage as a hydrogen radical source [7]. Equally, the interaction between a hydrogen plasma and a surface is of prominent importance in a number of academic research fields such as negative ion generation for neutral beam injection [8][9][10] and ammonia synthesis [11][12][13].
Numerous mechanisms can be used to generate the hydrogen plasma, however, the focus of this study is on microwave driven plasmas, specifically through the use of an electron cyclotron resonance (ECR) source. Recent developments * Author to whom any correspondence should be addressed. 1 Current address: Oxford Instruments Plasma Technology, Yatton, Bristol, United Kingdom Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. include compact source designs with a static magnetic field; this provides a high power efficiency and importantly the ability to scale the plasma by employing matrix configurations of individual plasma sources [14][15][16][17].
ECR plasmas represent a very interesting subject of investigation and a challenge with respect to modeling because of the large variety of processes that need to be considered. In addition to the plasma description, the static magnetic field must be included as well as the propagation of the microwave field. All of these aspects are strongly coupled and require a selfconsistent approach that takes into account the non-isotropic transport of charge carriers and the damping of microwave propagation due to the plasma.
The first models of ECR discharges have been published in the 1990s and were based on Monte Carlo simulations [18] or fluid approaches [19]. A series of investigations are devoted to describe electron cyclotron resonance ion sources (ECRIS). Because of the very low pressure and the large electron energy (up to several keV) particle methods such as particle in cell (PIC) or Monte Carlo have been used [20][21][22]. Partly, the particle methods are applied to the ion component only and electrons are taken as a neutralizing background with a given mean energy [21]. Recently, very time-consuming 3D PIC simulations of the electron component assuming ions as background have been published [22]. Only first steps to a self-consistent coupling with Maxwell's equations could be reached in this study due to the demanding particle simulations.
On the other hand, ECR plasmas in the pressure range of several Pa have been successfully described by fluid models. A self-consistent fluid model to describe an ECR plasma in argon with an iterated time integration of both the microwave and plasma equation and a semi-implicit Poisson method was published by Hagelaar et al [23]. This work is widely accepted as a basic reference for later modeling of ECR plasmas of this type. The authors reported results obtained with electron densities below and above the cut-off value of n c = 7.4 × 10 16 m −3 and the resultant changing damping of microwave propagation. Further ECR discharges in hydrogen [24] and argon [25] with electron densities lower than n c were investigated by a fluid and hybrid model, respectively. The weak microwave damping enabled observations of the typical ECR behavior in these studies. Here, the maximum electron power gain occurs in the vicinity of the ECR surface, as characterized by the critical field strength B c = 875 G of the static magnetic field.
The current investigations aim to achieve an accurate description of the hydrogen plasma driven by a compact, commercially available ECR source. A self-consistent fluid model is applied to investigate an ECR hydrogen plasma for larger values of power and electron density as typically produced by the ECR source under study. In particular, the interactions between microwave propagation and plasma generation with electron densities exceeding the cutoff value are analyzed. The competing effects of microwaves to increase electron density while impeding microwave propagation lead to complex spatial power density structures. The Maxwell equations describing the microwave are efficiently solved in the frequency domain in contrast to the time-dependent solution used in reference [23]. The strict solution of Poisson's equation yields correct spatial profiles of the plasma potential caused by space charges mainly occurring in the sheath which is important for the determination of ion fluxes. To verify the model, the results are compared with those obtained by two experimental methods. Firstly, a retarding field energy analyzer (RFEA) was used to measure the ion current and the floating potential. Secondly, two-photon absorption laser induced fluorescence (TALIF) was used to measure the absolute ground state atomic hydrogen density.
The paper is organized as follows: the setup and the diagnostic methods are described in section 2, details of the model are explained in section 3, results are presented and discussed in section 4, and a summary is given in section 5.

Process chamber
Experiments were conducted in a vacuum vessel with a volume of approximately 70 l. The chamber shape corresponds to a vertical cylinder with window flanges perpendicular to the axis, which provides diagnostic access for the RFEA and TALIF, respectively. Figure 1(a) shows schematically the 3D CAD model of the chamber.
Three flanges of the chamber are used for the application of TALIF. For further details, the reader is referred to section 2.3 below. When performing RFEA measurements the probe is positioned within the process chamber by means of a UHVmanipulator which is mounted on a modified back flange together with the ECR-source. The photo through the front flange in figure 1(b) illustrates the geometric arrangement of the RFEA in the chamber with respect to the ECR source, here presented in a moved-off position.
Base pressures of 2 × 10 −4 Pa are achieved through a combination of a turbo molecular pump and a floor pump.
Operational pressures are controlled through a fixed gas flow, 100 sccm of hydrogen with a gas purity of 6.0, controlled by a mass flow controller. A feedback loop controls a gate valve with respect to the pressure, as measured by a Baratron gauge, that ensures consistent operational pressures.
The plasma is excited by a commercial microwave ECR source, Aura Wave from Sairem, mounted on the back flange; this source consists of a solid state microwave generator and a compact microwave launcher with coaxial geometry [17]. The Aura Wave source allows up to 200 W of microwave power to be coupled into a plasma in a pressure range of 3 to 10 Pa. The excitation frequency is variable in the range from 2.4 to 2.5 GHz which ensures minimized reflected microwave power for a given operation condition.

Retarding field energy analyzer
A home-made four-grid retarding field energy analyser [26][27][28] was used to investigate the ionic species within the plasma. Given the established diagnostic method only a brief description will be provided here. However, for a more detailed description of the diagnostic and measurement apparatus please refer to Harhausen et al [29]. A separate bias can be applied to the front electrode to use the RFEA as a planar probe. The floating potential V f can be determined from the current voltage characteristics as measured by the front electrode. From this, an estimation for the ion current density at V f can be calculated; this is conducted by adopting a model for the ion saturation current. Since no ion velocity distributions were extracted from the RFEA data for comparison to the model, the determination of the ion saturation current was conducted in this work by using a simplified linear extrapolation of the far ion saturation branch at floating potential V f . Therefore, compared to the model calculations in section 4, the experimental ion current densities reflect an upper limit being overestimated by up to 30%, which is supported by the difference to a commonly used power law approach considering an expected sheath expansion according to e.g. [30]. Using such a model, the knowledge of the plasma potential is mandatory since the extrapolated ion current must vanish there. In contrast, the contribution of secondary electron emission to the ion current is expected to be in the range of a few percent of the measured value [26]. Furthermore, the uncertainty of the floating potential remains below the size of the symbols used in the figures. The housing of the RFEA probe has an edge length of 48 mm and a thickness of 20 mm. As such a macroscopic device it affects the plasma in its vicinity. Therefore, it is considered in the model by adopting a simplified cylindrical geometry.
For the measurements, the RFEA probe is positioned along the axis of symmetry of the ECR source at various distances in the range from 10 to 25 cm (z axis in figure 1(a)). Radial measurements, i.e. along the x axis as depicted in figure 1(a) are also possible but are not subject of the presented work here. For clarity, the position and size of the RFEA as taken into account in the model is drawn in the figures 2, 5, 7 and 9 as well. It is worth noting that the modeled current densities integrated over the entire front plate are comparable to those measured by the RFEA.

Two-photon absorption laser induced fluorescence
TALIF was used to measure atomic hydrogen densities within the plasma. Given the established method [31][32][33][34], only a brief description of the diagnostic will be given here. The principle mechanism of TALIF is that two photons are used to excite a ground state species into an upper level that then relaxes to a lower level and releases a photon in the process. By detecting the fluorescence photons produced during this relaxation, combined with knowledge of the reduced optical branching ratio and various invariant experimental terms, the density of the excited state can be calculated. This excited state density is proportional to the ground state density. However, a calibration, using a noble gas with a similar two-photon excitation scheme, is required to determine absolute ground state densities [31,35].
Conducting TALIF to measure atomic hydrogen requires the creation of laser radiation with a wavelength of 205 nm. This was obtained through the combination of a frequency doubled Nd:YAG, Spectra Physics Quanta Ray, pump laser coupled into a Sirah Precision Scan SL dye laser. The 615 nm red light from the dye laser then underwent frequency doubling, through a BBO crystal, that was then remixed with residual red light from the dye laser in a second BBO crystal for sum-frequency generation to the required 205 nm radiation. The output of the laser is then attenuated via an attenuation scheme to ensure that no photo-ionisation events take place; saturation curve experiments showed that an energy per pulse of 370 and 19 μJ should be used for hydrogen and krypton, as the calibration gas, respectively. Effective lifetimes were measured to be 12 and 33 ns for hydrogen and krypton, respectively and were found to be invariant on any parametric variation conducted in this study.
The setup at the process chamber is depicted in figure 1(a). The injection of the UV laser pulses (L) was conducted along the x axis from the right flange to the left one both equipped with quartz windows. After transmission through the second flange window, the laser energy was diminished with the help of a beam dump. A small region of interest, approximately 10 by 2 mm, directly in front of the ECR source but with a separation of 115 mm from the source tip in z direction was chosen. For clarity, the position of the region of interest is depicted in figure 2 together with the RFEA probe. The detection of the fluorescence at 656 nm was performed in z direction as well using an Andor iStar DH734x iCCD camera combined with an appropriate spectral filter.

Model description
The model of the ECR plasma consists of three main parts: the description of (i) the static magnetic field caused by the permanent magnets, of (ii) the microwave and of (iii) the plasma. These parts are solved in an axially symmetric geometry which is shown in figure 2.

Static magnetic and microwave fields
The static magnetic flux density B s caused by the permanent magnets is determined using Maxwells equations according to where μ 0 , H s and M denote the vacuum permeability, the static magnetic field strength and the magnetization, respectively. The magnetization M is a specific material property of the permanent magnet which is related to the magnetic moment m of the magnet by m = V M dV. The static magnetic field B s is determined in advance because it is not influenced by the model parts (ii) and (iii). The microwave fields E and B are described by Maxwell's equations with the vacuum permittivity μ 0 , the vacuum permeability ε 0 and the relative permeability ε r . The total current density J is the sum of the displacement current density J d and the plasma current density J p . The latter can be approximated in the microwave time scale by the electron current density neglecting the slow motion of ions. Here, e, n e and v e denote the charge, density and mean velocity of electrons, respectively. The mean electron velocity is obtained from electron momentum equation according to where ν m denotes the frequency of electron momentum transfer in collisions with heavy particles. The equations (3)-(5) are solved in the frequency domain by replacing all vectors X ≡ E, B, J p , v e with X =X e iωt . A Fourier transformation of equation (7) yields the velocity amplitude asv This relation can be used to determine the conductivity σ according toJ The magnetic field B s leads to a tensorial conductivity σ = en e μ e with the related tensorial mobility μ e which depends in cylindric coordinates (r, φ, z) according to on the static magnetic field strength B s and on the complex mobilityμ e = e m e (ν m + iω) .
Equations (3)- (5) with related variables in the frequency domain can be combined to the relation which is used to determine the componentsĒ r ,Ē φ andĒ z of the microwave electric field amplitudeĒ. The powerQ transferred from the microwave to the electrons can be determined bȳ whereĒ * denotes the complex conjugate ofĒ.

Plasma description
The plasma equations describe the temporal relaxation of species densities and the electrostatic potential φ which is related to the electrostatic field E s according to (14) in timescales beyond the microwave period. The plasma equations comprise continuity equations for all species densities n l with the source term S l , the electron energy balance equation with the energy source term S to determine the electron mean energy density n = n e u m and Poisson's equation with q l being the charge of species l. The particle fluxes Γ l and the electron energy flux Γ are driven according to flux conditions for all species (Γ l ) and the electron energy flux Γ ε by drift from the electrostatic field (first term) and diffusion (second term). The electron mobility μ e is a real-valued tensor with components depending on the scalar mobility μ 0 = e/(m e ν m ) and the static magnetic field B s which reads similar to equation (10) replacingμ e by μ 0 . The electron diffusivity tensor D e is determined using Einstein's relation according to D e = k B T e e μ e . The corresponding electron transport coefficients for the energy transport are determined by the relations μ = 5μ e /3 and D = 5D e /3 [36]. The representations (18) and (19) are obtained from the momentum balance equations neglecting the inertia of the particles. While this assumption is justified for electrons, the approximation is limited for ionic species at low pressures. The authors are aware of possible limitations of the model at pressures of 5 Pa and below. Nevertheless, this assumption is required to prepare a numerically solvable model for the complex plasma situation.
The diffusion coefficients for the heavy species have been taken from literature [37] and Einstein's relation was used to determine the ion mobilities.

Reaction kinetics
The plasma model comprises electrons and the hydrogen species H, H + , H + 2 and H + 3 . The 34 reactions are listed with related references in table 3. The rate coefficients of the electron reactions are determined as functions of the mean electron energy u m by solving the 0D Boltzmann equation with BOLSIG+ in version 03/2016 [36]. Furthermore, it is used to determine the electron momentum transfer frequency ν m as function of the mean electron energy which is needed to prepare the plasma conductivity σ and the transport tensors μ e and D e .
Finally, the pumping of the gas that sustains the pressure is included as global loss rates S flow

Boundary conditions
The solution domains in the axially symmetric geometry are shown in figure 2. The different model parts are solved on different domains as follows: The equation for the static magnetic field B s is solved on the extended region A B C D (green shaded plus inner domains), the microwave fieldĒ is determined on the domain EFGHIJKCDE (blue and red domains) and the plasma equations are solved on the red shaded domain EFGJKCDLMNOE. The RFEA probe (LMNO) is situated at a distance of z p ≡ |OE| from the tip E of the source. The boundary conditions are listed in table 1. The microwave field is prescribed at boundary HI in TEM mode. Its power P port is adjusted iteratively to fulfill the condition P = VQ dV where P is the total deposited power used as parameter for comparison with the experiment. The surface charge σ s at the lower dielectric part (LMN) of the RFMA probe is determined by The floating potential V f is adjusted so that the total electron and ion current to the metallic probe surface (NO) becomes zero. The boundary conditions for the species densities incorporate the thermal velocity v th l = (8k B T l )/(πm l ) of respective species. The flux conditions for the electron particle flux n · Γ e = 1 2 v th e n e − l γ sec l (n · Γ l ) and for the electron energy flux n · Γ ε = 5 6 v th e n ε − l γ sec l u sec (n · Γ l ) include secondary electron emission with the coefficients γ sec l . A mean energy of u sec = 5 eV is assumed for the emitted electrons, however, the results of this model have been found to be relatively insensitive to this chosen value. For heavy particles, the condition n · Γ l = 1 4 γ f 1−γ f /2 v th l n l + n l μ l · E s is used which includes contributions from surface reactions described by sticking coefficients γ f as well as migrative flux due to the static electric field. The surface reactions and related coefficients are listed in table 2.
As shown in figure 2, the regions for the RFEA and the TALIF measurements overlap. Therefore, the RFEA probe is moved to the farthest position z p = 25 cm to enable a correct comparison with the TALIF data.

Numerical solution
Comsol multiphysics with the modules Plasma, RF and AC/DC is used to solve all equations of the model. For this purpose, an unstructured mesh with approximately 97 000 elements is prepared with refinements in the vicinity of the source and the probe. The solution of the model describes a temporal relaxation of the system over a period of 10 s. The interactions of different parts in the model, equations and variables as well as their dependencies are illustrated in figure 3. The stationary spatially homogeneous electron Boltzmann equation is solved once to determine the rate coefficients k of reactions between electrons and heavy particle as functions of the mean energy u m . The static magnetic field B s is determined for each geometric configuration (parameter z p ) in advance according to equations (1) and (2). The remaining equations describing the microwave (12) and the plasma (14)- (16), (20) are solved in a coupled system where the electron density n e and the absorbed powerQ determine the coupling between the microwave and plasma parts. Newton iterations are used to treat the nonlinear dependencies in each timestep as implemented internally in the time-dependent Comsol solver.

Results
Model calculations have been done for the reference case at a pressure of 7 Pa and an absorbed power of 100 W. The probe is situated at the position z p = 10 cm. Variations have been performed with respect to the probe position z p , the pressure and the power. The flow rate is kept constant at a value of Q flow = 100 sccm.

Model results of the reference case
The magnetic field caused by the permanent magnets is shown in figure 4. The ECR condition for ω/(2π) = 2.45 GHz is fulfilled along the white iso curve with B c = ωm e /e 0 = 875 G. The comparison of this iso curve with that published for the Aura Wave source [40] allowed to determine the value for the magnetisation in equation (1)  The magnitude of the electric microwave field throughout the overall reactor volume is shown in figure 5 in logarithmic scale. The strong interaction of the plasma with the microwave causes the latter to reach only approximately 6 cm and 4 cm in the axial and radial directions, respectively. A strong damping by several orders of magnitude is obtained in the remote regions of the reactor.
A more detailed representation of the microwave field magnitude in the vicinity of the source is given by figure 6(a). This figure should be analyzed together with figures 6(b) and (c) which show the particle densities of electrons and H +  with H + 3 ions. However, electrons are accumulated there because of a saddle point of the electrostatic potential and the resulting electron migration flux. This density maximum also influences the propagation of the microwave field as shown in figure 6(a).
The saddle point is a detail which is marked in figure 6(d). This figure shows the electrostatic potential in the whole reactor and iso curves of the electron density at the base level. A pronounced peak of the potential is obtained near the microwave source which is caused by the large charge carrier density in that region and the resultant enhanced space charge density near the wall. Furthermore, the figure demonstrates the modification of the electrostatic potential due to the RFEA probe situated at z p = 10 cm.
The interplay between electron density, power gain, ionization rate and mean electron energy is depicted in figure 7. Results for the reference case (100 W) are compared with those obtained for an absorbed power of 50 W. The white contours mark the regions where the electron density exceeds the cut-off value. The power is transferred to the electrons primarily in regions where the overlap of the microwave field strength and electron density reaches a maximum. The power gain causes an increased mean electron energy (figures 7(c) and (d)) and ionization rate (black contours in figures 7(a) and (b)). The transport of electron energy along the static magnetic field lines leads to pronounced spatial structures which coincide with the magnetic field lines as shown in figure 4. The comparison of both cases demonstrates the broadening of those regions where the electron density exceeds the cut-off value with increasing power. These regions represent voids for the propagation of the microwave and the related power gain for electrons. Figure 8 shows the densities of the dominant species along the axial cut at r = 12 mm which is marked in figure 6 by white lines. For all species, a density decay of more than three orders of magnitude can be observed from the active region near the source at z > −2 cm in axial direction toward the reactor wall at z ∼ −53 cm. The RFEA probe causes the depletion of all density profiles around its position (z p = 10 cm). H + 3 represents the dominant ion species in the remote regions of the reactor. This species is generated mostly due to the conversion reaction R28 (H + 2 + H 2 → H + 3 + H) which also causes the rapid depletion of the H + 2 ions in the active region at z ∼ −2.5 cm. The H + 3 ions are then transported mainly due to ambipolar diffusion from the active to the remote regions of the reactor. The H + ions are generated mainly due to the ionizing reactions R3 (e + H 2 → 2e + H + H + ) and R27 (e + H → 2e + H + ) and get rapidly converted to H + 2 according to reaction R29 (H 2 + H + → H + 2 + H). The pronounced structure of electron density, power gain and mean energy as shown in figures 6 and 7 as well as the various reactions lead to multiple maxima of different species in the active region and considerable discrepancies in the density profiles across the axial direction. Figure 9 summaries the key charged species particle flux onto the RFEA probe. The ion current density in the vicinity of the probe is shown in figure 9(a) by a spectral and an arrow plot. The magnitude of the ion flux becomes large at the metallic corner of the probe due to the increased electric field strength. This enhancement up to 230 μA m −2 can also be observed in figure 9(b) that shows the flux of charge carriers onto the surface of the RFEA probe. The ion flux is mainly driven by the dominant H + 3 ions. The electron flux has an opposite radial profile because it is driven due to diffusion against the retarding electrostatic field.

Comparison with experimental results
The average of the calculated ion flux over the metallic surface of the RFEA probe can be compared with corresponding results obtained experimentally with the RFEA probe shown in figure 1(b). This comparison is shown in figure 10. The RFEA probe is used at the floating potential in the experiment and in the model. Good agreement has been obtained with respect to the relative dependence of the current density on the distance z p between source tip and probe and with respect to the pressure. The differences in absolute values may be caused by the limitations of the applied drift-diffusion approximation at this low pressure. In addition, the deviation between the real quadratic shape of the probe and the necessarily cylindrical approximation in the model may lead to further discrepancies. Furthermore, as discussed in section 2.2, some uncertainties from the experimental side come from the extrapolation of the ion current to floating potential.
Both methods also deliver the potential of the floating probe. The comparison of its dependence on the position z p is presented in figure 11 showing a very good agreement. Only a slight dependence on the position has been obtained theoretically and experimentally. The relative deviation between both results amounts to approximately 27% and 7% at 5 and 7 Pa, respectively. The increasing deviation with decreasing pressure is seen to be attributed to the limitations of the model at low pressures because of the drift-diffusion approximation used for the ion momentum.
The calculated and measured dependence of the ion current on the power for a pressure of 7 Pa is shown in figure 12. Apart from the deviation in absolute values already discussed, the same dependence was obtained in both methods. Furthermore, the figure 12 gives information on the dependence of the contribution of H + 3 and H + ions to the total current. The current of both species onto the probe surface increase similarly with the   power keeping H + 3 as the dominant contribution which almost entirely compensates the electron current. The comparison of the calculated and measured density of hydrogen atoms on the axis at z = −13.35 cm is shown in figure 13. Note that the RFEA probe is moved to the farthest position z p = 25 cm for this comparison in the model. The relative deviation between both results amounts to approximately 27% and 8% at 5 and 7 Pa, respectively. This reflects the same increasing deviation between model and experiment for low pressures as in the above described findings on the floating potential. The slope of the power dependence obtained theoretically and experimentally deviate by approximately 10% from each other in both cases. The very good agreement with respect to absolute values and the dependence on power and pressure represents a convincing validation of model and experiment.

Summary
The complex physics involved in a low-pressure hydrogen discharge driven by a commercial ECR source has been investigated by a self-consistent fluid model and validated by experimental investigations using both RFEA and TALIF measurements. This plasma source is characterized in its operating range for pressure and power by locally extended regions where the electron density exceeds the cut-off density. The model includes equations for the permanent magnetic field and the microwave field and fluid equations to describe the plasma generation. The strict solution of Poisson's equation yields correct spatial profiles of the plasma potential which is important for the determination of ion fluxes. The strong coupling between the microwave propagation, the power supply to the plasma and the damping of the microwave by the plasma becomes obvious in the results of the model. It leads to highly structured spatial profiles of the microwave electric field, the electron power gain, mean electron energy, and the electrostatic potential. These structures are largely determined by the field lines of the static magnetic field and the resultant transport of the magnetized electrons and their energy which is included in the model by a tensorial representation of the electron transport coefficients. The model also takes into account the influence of an RFEA probe consisting of conducting and dielectric parts that allows the comparison of measured ion current densities and floating potential with the model results.
Good agreement was obtained with respect to the dependence of the ion flux on the probe position, pressure, and power, respectively. The deviation with respect to absolute numbers may be caused by the limitation of the model at low pressure because of the drift-diffusion approximation for the ion momentum. In addition, the deviation between the real quadratic shape of the probe and the necessarily cylindrical approximation in the model may lead to further discrepancies. Furthermore, an exceptional agreement was obtained in the second verification of the model through comparison of atomic hydrogen densities results with the TALIF measurements. This twice-fold experimental validation yields a high-level of confidence in the physical solutions of the model and in the selection of the relevant chemical interactions chosen.
Future work will focus on an incorporation of more complex chemical interactions, an expansion of the relevant surface chemistry, and further experimental comparisons to maintain the current high-level of confidence in the models validity. Table 3. Reactions included in the plasma model. The dimensions of the rate coefficients are (m 3 s −1 ) for two-body reactions and (m 6 s −1 ) for three-body reactions. The electron temperature T e = (2/3)u m and the heavy particle temperature T g are specified in (K).