Helicon waves are a category of right-hand polarized (RHP) plasma waves which propagate along DC magnetic fields in bounded systems. They are related to so-called whistler waves, which have been studied in atmospheric physics since the early twentieth century. Whistlers and helicon waves belong to the group of right-hand polarized (RHP) waves propagating parallel to a magnetic field, in the frequency range ω _ci ≪ ω ≪ ω _ce (where ω _ci is the ion cyclotron frequency and ω _ce is the electron cyclotron frequency), together with electron cyclotron waves. Figure 2 shows the location of whistlers and helicon plasma waves within a ω-k diagram representing RHP cold plasma waves.

A historical perspective for the first twenty years of helicon research has been given by Chen and Boswell [13, 14]. The following twenty-year period has been covered in more recent reviews by Chen [15] and Shinohara [1]. Theoretical treatments of the physics behind helicon waves have been produced, among others, by Klozenberg et al. [16], Chen [17], and Chen and Arnush [18, 19].

A basic dispersion relation can be obtained for helicon plasma waves from simplifying the Appleton-Hartree expression for quasi-longitudinal right-handed cold plasma waves [20, 21], propagating at an angle θ from an axial, static magnetic field B = B 0 e ̂ z , β = ω k n 0 e μ 0 B 0 (1) where β 2 = k 2 + k ⊥ 2 is the total wave number, k = β cos θ and k _⊥ are the parallel and perpendicular components of the wave number, and n ₀ is the plasma density. This expression, despite being a simplification, provides an intuitive insight on the relationship between the magnetic field B ₀, the density n ₀, the wave frequency ω, and the wave number β, and can be used as a starting point when designing or analyzing a HPS.

A more detailed description of helicon waves can be obtained from Maxwell’s equations by neglecting ion motions and the displacement current, as originally shown by Klozenberg et al. [16]. When the effects of electron inertia are retained within the analysis [14, 18, 22] two solutions are obtained for the dispersion relation, β 1,2 = k 2 δ 1 ∓ 1 − 4 δ k ω 2 k 2 1 / 2 (2) where δ = ω/ω _ce is the ratio between the wave frequency and the electron cyclotron frequency ω _ce = eB ₀/m _e , and k ω 2 = ω ω p 2 / ω c c 2 = ω n 0 e μ 0 / B 0 ≡ α k is the wavenumber for low-frequency whistler waves along B ₀ in free space, with α = β the wave number previously described in Eq. 1. ω _p is the electron plasma frequency at density n ₀. δ is neglected when the effects of the electron mass are omitted or for frequencies ω ≪ ω _ce .

Eq. 2 describes two solutions for the wave dispersion relation, which can be simplified as shown in Eq. 3. β 1,2 ≈ k 2 δ 1 ∓ 1 − 2 δ k ω 2 k 2 ≈ k ω 2 / k k / δ (3)

Solution β ₁ corresponds to the zero electron mass limit, and describes the helicon wave (“H”) from Eq. 1. The second solution β ₂ = β ₂  cos θω _ce /ω describes a wave with frequency ω = ω _ce   cos θ, which is an electron cyclotron wave propagating at an angle to the magnetic field. This is the Trivelpiece-Gould mode (“TG”), first described in bounded systems by Trivelpiece and Gould [23]. The TG mode co-exists with the H mode, and becomes relevant at lower values of B ₀. The TG mode is thought to play a relevant role in the damping mechanism of helicon plasma sources and to contribute to its high ionization efficiency via mode-conversion processes [24].

Eq. 3 describes the dispersion relation for both the H-mode and the TG mode. Expressions for the magnetic and electric fields (B, E) have been derived for different geometries as described in the early works on helicons [16, 25] as well as in more recent literature [14, 17, 22]. These expressions depend as well on the boundary conditions chosen for the analysis and on whether these boundaries are modelled as conductors or not [18]. Practical implementations of HPSs are typically linear devices implemented as cylindrical enclosures made of dielectric materials, as will be described in section 4.

The expressions obtained from Eqs. 1, 3, as well as the detailed derivations of the B and E fields that can be obtained for a particular configuration and geometry, can be used as an initial approximation to understand the regimes of H and TG modes that can be propagated in a given configuration, and establish a baseline estimation of the expected density distribution in a given HPS device.

One particular advantage of HPSs stemming from the fundamental physics of helicon waves is the ability of these devices to couple RF waves at the core of dense plasmas, enabled by the presence of the axial magnetic field and the propagation of the H-mode. This fact presents an advantage over other types of plasma sources, such as inductively-coupled plasmas (ICPs) where the penetration of RF waves into the plasma is limited by its skin-depth, or electron-cyclotron sources (ECR), where microwaves cannot propagate below the O-mode cutoff frequency (the electron plasma frequency ω _pe ).

An investigation on the mechanisms which enable the initiation of the high-density helicon mode (the H-mode), based on modeling and experimental work, has been carried out by Carter et al. [26], including indirect evidence of the deposition of RF power at the high-density core in a helicon plasma source.

Section 2.1 described helicon plasma waves and derived their dispersion relation in various scenarios. The general behavior of magnetized plasmas in cylindrical geometries will now be analyzed, which is relevant to the characterization of practical HPSs as described in section 4.

The problem of describing the bulk behavior of a plasma discharge has been addressed since the early stages of the development of plasma physics. In the classical paper by Tonks and Langmuir [27], expressions were derived for the distribution of the electric potential in an arc discharge, for various geometries including cylindrical coordinates. Scenarios were analyzed for different regimes of ion collisionality and ionization rates. This work also contains a treatment of the plasma-material boundary at the edge of the plasma discharge, pointing to the discontinuity of the bulk model within the plasma sheath.

In a later paper, Tonks [28] studied the effects of the magnetic field in an arc plasma. One of the cases described was the positive column plasma immersed within a longitudinal magnetic field, the same typical configuration applied nowadays to most helicon plasma sources. A radial model is developed based on classical diffusion theory. More recent models for cylindrical magnetized plasmas have been developed by Fruchtman et al. [29] and Sternberg et al. [30]. These works introduced the use of 2D fluid models in cylindrical coordinates (with the assumption of azimuthal symmetry), the separation of variables in order to decouple the expressions for the radial and axial coordinates, and the analysis of different degrees of magnetization. Differences between these authors rely on the assumptions chosen to simplify their models. The previous works were further adapted and extended by Ahedo et al. [31, 32], who developed a 2D model for cylindrical magnetized plasmas as part of their work on describing the plasma dynamics within helicon plasma thrusters. The properties of these models have been summarized in Table 1.

These descriptions of cylindrical magnetized plasmas can be used to approximate the distribution of key parameters within the discharge, such as the density distribution, the velocity of ions and electrons, and the plasma potential. As an example, the complete model developed by Ahedo et al. [31, 32] is described by a set of four radial equations and five more for the axial dimension. These take as inputs information regarding the ion species taking part in the discharge, collisional rates related to the ionization and interactions between ions, electrons and neutrals, and constant parameters such as the magnitude of the axial magnetic field B ₀ and the isothermal electron temperature T _e .

The dispersion relations found for helicon plasma waves in section 2.1 can be used to obtain reference values for parameters such as the peak density value in the discharge. This information can be coupled with the description obtained from a 2D fluid model in order to project the distribution of plasma density, kinetic energy, and plasma potential throughout the discharge. Understanding the values of these parameters at the boundaries of the system, where the plasma comes into contact with solid materials, is essential to describe the interaction phenomena taking place in this region. Figure 3 shows an example of the models from Refs. [31, 32] being used to estimate the 2D plasma distribution within a particular HPS, the VX-CR device. Data from these models can be used to obtain the plasma conditions at the radial (r → R) and axial (z → − L) boundaries, which then enable the analysis of the interaction between the plasma discharges and the physical confinement materials.

Plasma-surface interactions (PSIs) or plasma-material interactions (PMIs) comprehend the different phenomena that occur when ions, electrons, and neutrals within a plasma reach a material boundary. These interactions might produce effects on both the plasma itself as well as on the boundary. PSIs are essential in the field of plasma materials processing, and are also critical to the successful development of practical fusion devices [38, 39], as most designs include open magnetic flux surfaces where the plasma directly impinges the physical boundaries. They are also crucial in the advancement of electric propulsion technologies, where the lifetime of the thrusters directly depends on the erosion rate of those critical surfaces directly in contact with the plasma discharge or the plume of the thruster [40, 41].

Several processes can occur at the physical boundaries of a helicon plasma. Positive ions traversing the sheath typically become neutralized, in a process that either produces an excited neutral, or a neutral plus the emission of a secondary electron (Auger emission [35]). Secondary electron emission has been found to play a role in the sheath dynamics of certain types of low-energy plasma discharges, such as capacitively-coupled plasmas [42].

Another fundamental process is sputtering, the removal of material from a solid surface material due to the impact of an energetic impinging particle, typically ions in the case of plasma discharges. It is one of the most relevant phenomena occurring at the boundary surfaces of plasma discharges, since it can be responsible for significant erosion of said surfaces if the adequate conditions are met. Figure 5 depicts the basic mechanisms behind the most relevant PSI phenomena encountered in the study of HPSs.

Theoretical treatments of the phenomenon of sputtering are provided by Sigmund [43], Bohdansky [44], Yamamura [45], Eckstein [46], and Behrisch et al. [47]. Most models describe the process as the result of collisional cascades in the surface layer of the target material, in which the momentum of the impacting ion is transferred to an atom in the target material’s lattice through elastic collisions. The random arrangement of the position of both particles implies that an oblique collision is likely. The impacted target atom, in turn, collides with other neighboring particles triggering the cascade. With sufficient energy in the original impacting ion, eventually the collisional cascade will provide one of the atoms in the surface layer with an energy level surpassing the surface binding energy of the material [48], and a momentum directed outside of the surface. The atom will then be sputtered from the surface.

Simulation of the sputtering process based on the first principles from classical mechanics is possible, by using the technique of Molecular Dynamics [49, 50]. Other popular simulation packages are based on the Monte Carlo statistical method, such as TRIM. SP [51] and SRIM [52]. Sputtering yield estimations obtained by the use of these software packages are strongly dependent on the chosen input parameters, and have been shown to differ from experimental values in certain ranges [53].

The fundamental parameter in sputtering models is the sputtering yield, Y _sputt , defined as the number of surface atoms sputtered off the surface per incident impacting ion. Y _sputt is mainly a function of the ion species and surface material, the ion energy, and the angle of incidence between the surface normal and the ion’s velocity vector. Below a particular threshold energy level, E _thr , ion impacts are not able to sputter surface atoms and Y _sputt = 0.

Several models have been developed to produce estimations for the sputtering yield, each particular to the species involved in the process, and the angle of incidence and energy E ₀. Lieberman and Lichtenberg [35] report expressions valid for large atomic species within certain boundaries of their atomic number ratio. Eckstein and Preuss [46] proposed the model shown on Eq. 6, which is valid for ions impacting the surface at a normal angle of incidence. Y E 0 = q s n K r C E 0 E 0 E t h r − 1 μ λ + E 0 E t h r − 1 μ (6) where the krypton-carbon interaction potential s n K r C [46, 54] is used as an adequate mean value for different participating species and describes the nuclear stopping cross section. This parameter is defined as follows, s n K r C ε = 0.5 ⁡ ln 1 + 1.228 8 ε ε + 0.172 8 ε + 0.008 ε 0.150 4 (7)

The reduced energy ɛ is obtained as follows, ε = E 0 M t M i + M t a L Z i Z t e 2 (8) where the subindexes i and t are used to describe the atomic numbers Z and atomic masses M of the projectile ion and target surface atoms, respectively. a _L is the Lindhard screening length [55], a L = 9 π 2 128 1 / 3 a B Z i 2 / 3 + Z t 2 / 3 − 1 / 2 (9) where a _B is the Bohr atomic radius.

The remaining free parameters q and λ from Eq. 6 can be found in [47] for a variety of impacting ions, target materials, and ion energies.

When ions impact on a boundary surface not in a perpendicular direction, but instead at an angle α with respect to the surface normal, the calculation of the sputtering yield needs to take this geometry into account. Eckstein and Preuss [46] proposed the formula in Eq. 10, Y E 0 , α = Y E 0 , 0 cos α α 0 π 2 c − f ⁡ exp b 1 − 1 cos α α 0 π 2 c (10) where α 0 = π − arccos 1 1 + E 0 / E s p ≥ π 2 (11)

E _sp is a binding energy characteristic of impacting ions, and c and f are fitting parameters. Behrisch and Eckstein [47] have compiled tables for these formulae for the most common ions and target materials.

For the case of surface materials consisting of alloys or compounds of different elements, the sputtering yield will be different for each different species present in the target surface. For the steady state with a sufficiently high flux of incident ions, the sputtering yields will distribute according to the stochiometric concentration of each species within the target compound. However, this distribution is not kept for small fluences of impinging ions, and the phenomenon of preferential sputtering occurs.

For binary target materials, containing two elemental species i and j, the sputter preferentiality δ can be defined [47] as a ratio of the elemental sputtering yields Y _i , Y _j and their stochiometric concentrations c _i , c _j , δ = Y i Y j c j c i (12)

δ can also be estimated as follows, δ = M j M i 2 m U j U i 1 − 2 m (13) where M _i , M _j are the atomic masses, U _i , U _j the surface binding energies, and m is a power exponent describing the interaction potential.

When a plasma encounters a solid surface, such as at the boundaries provided by the containment surfaces of a HPS, a sheath will be formed and ions will be accelerated according to the potential present at the wall. If the ions are able to increase their energy beyond the threshold energy E _thr , sputtering will occur and the surface will be modified. Combining this information with the density distribution obtained through experimental measurements or simulations, such as the fluid models described in section 2.2, an etch rate or erosion rate can be calculated for the surface. This value can be used to project the behavior of the HPS and establish limits to its useful lifetime in a particular practical application.

In practical applications, the etch rate E of a surface bombarded with energetic ions, measured as a ratio of the etch depth per unit of time, is calculated as a function of the incident ion flux Γ _i , the particular sputtering yield Y, and the mass density of the target surface ρ _t as shown in Eq. 14, E = Γ i Y M m , t ρ t N A (14) where M _m,t is the molar mass of the target surface and N _A is Avogadro’s constant. The calculation of the sputtering yield would take into account all the considerations discussed in this section. The incident ion flux Γ _i is determined by the particular conditions of the plasma discharge near the surface; for example, it can be approximated by applying the Bohm Sheath Criterion and specifying that Γ _i = n _s u _B where n _s is the ion density at the entrance of the sheath and u _B the ion Bohm velocity.

Materials used for the construction of HPSs must comply with a number of often conflicting properties. RF-transparent materials are commonly used to manufacture the cylindrical tube, allowing for the efficient transmission of the RF waves produced by the external antenna to the plasma medium. This requires materials with a low dissipation of RF energy, which is usually measured in terms of the loss tangent (tan δ). The amount of thermal energy dissipated by the boundary material is directly proportional to this loss tangent parameter, which is in itself proportional to the material temperature [57]. This can potentially create a positive feedback loop of RF energy losses within the boundary material, showing the importance of the material selection in practical HPSs.

From a practical engineering point of view, HPS materials should feature a high thermal conductivity, enabling the distribution and extraction of the heat loads produced by the inherent inefficiencies of the RF transmission and the ionization process within the source. Materials with a high thermal conductivity will allow the heat loads present in the material to spread axially and azimuthally, promoting the creation of a more even temperature distribution and reducing the appearance of thermal hotspots. This in turn contributes to the reduction of the amount of thermal energy dissipated as the RF energy traverses the boundary material. Thermal management of HPSs is a critical issue in practical implementations [58–62] and is essential for the development of high-power systems relying on HPSs, such as the VASIMR engine [63], the Proto-MPEX PMI research device [64], and the PISCES-RF steady-state helicon device [2].

De Faoite et al. [65] compiled a thorough review of the available data on the most relevant thermal and mechanical properties of dielectric technical ceramics commonly used in HPSs, focusing on those aspects relevant to the thermal management issues described above. The materials included in the analysis included alumina, aluminum nitride, berylia, quartz, sialon, and silicon nitride. A later work [66] presents linear regressions of these properties as a function of temperature, where adequate fits were found for some of them while also highlighting the limits of the publicly available data sets.

In order to assess the reliability of these dielectric materials under the boundary conditions present in inner confinement surfaces of HPSs, their sputtering parameters would have to be evaluated under similar conditions, using the models and techniques discussed in section 3.2. As an example, Figure 7 compiles experimental and simulated data for the sputtering yields of singly charged argon ions impacting some of these dielectric materials commonly used in HPSs, as a function of the impacting ion energy and at normal incidence. These choices are typical for the materials used in the VX-CR research HPS [61].

As an indicative example, erosion phenomena will be estimated for a typical HPS operating with an electron temperature of T _e ≈ 5 eV and a density n ≈ 2 × 10¹⁸ m⁻³ in the regions near the surface of a floating dielectric confinement wall [67]. Eq. 5 estimates that the wall potential becomes Φ _w = − 23.5 V. If the ions enter the sheath with negligible kinetic energy, it can be assumed this will be the incident energy at the wall, slightly larger than the corresponding threshold energy for sputtering E _thr ≈ 19 eV. If the wall material is alumina, Eq. 6 produces a value of Y ≈ 0.06 atoms/ion for the case of normal incidence to the surface and Eq. 14 produces an approximate etch rate of E = 17.62 nm/s. If the wall thickness of this material is t = 2.5 mm, this means it would take Δt = 141.9 × 10³ s = 39.4 h for the wall to erode (in a scenario where all conditions remain constant). If the confinement surface is made of quartz glass (silicon dioxide), the wall potential Φ _w would be below the threshold energy for sputtering for argon ions impinging on SiO₂, E ₀ < E _thr ≈ 35 eV, and no sputtering would occur.

If these conditions exist in the vicinity of the antenna straps of the HPS, where the RF energy is transmitted as a 13.56 MHz signal with a peak-to-peak voltage amplitude of V _pp = 1 kV (and therefore a peak voltage of V _p = 500 V), the methods described by Berisford et al. [68] can be used to estimate an average sputtering rate given the ion energy distribution function for low-frequency RF sheaths [35]. In this particular case, an average sputtering yield of Y _avg = 0.08 is obtained for the case of Argon ions impacting the alumina surface. The corresponding etch rate would then be E = 23.5 nm/s, and it would take Δt = 106,400 s = 29.56 h for the wall to erode. If the material is quartz, the RF sheath would be able to produce sputtering, with an average yield of Y _avg = 0.06, an etch rate of E = 18.85 nm/s, and the surface would be eroded in Δt = 132,600 s = 36.84 h. These are extremely simplified estimations, where conditions remain constant during the whole process, and no variations in the sputtering yield are introduced due to surface modification or deviations from normal incidence as the surface degrades.

HPSs have been used as part of plasma processing devices since early in their development [5, 10]), generating plasmas with the adequate parameters in order to modify the surfaces of samples or substrates subjected to their discharge. However, few studies have been conducted on the effects of the plasma discharge itself upon the inner confinement surfaces of HPSs.

Among these, Aanesland et al. [69] reported on the effects of an additional, floating copper antenna immersed within the discharge itself. They describe the sputtering of copper atoms from this additional antenna, which are then redeposited on the inner surface of the dielectric discharge tube. At high power levels, they describe how the areas in this dielectric tube located under the straps of the external helicon antenna remain clean due to the re-sputtering of the deposited copper layer. They suggest this is an effect of the RF sheath created on the plasma-surface boundary, as previously discussed in section 3.1.2.

This same mechanism was observed by Berisford et al. [60], when researching the power distribution and erosion within the dielectric tube of a linear helicon device. These authors developed expressions to estimate the etch rates observed at these regions under the straps of the extenal helicon antenna, modelling the sheath present in these areas as a low-frequency RF scenario (refer to section 3.1.2) and averaging the sputtering yield according to the ion energy distribution throughout the RF cycle [35]. These findings were validated through experimental observations of the actual erosion in the dielectric cylinder used in their experiment. These authors were able to estimate the required particle flux at the regions under the helicon antenna conductor from the measured etch rates, and also by analyzing IR thermal data measured at the same location; both estimations agreed within a factor of two.

Barada et al. [70] investigated this phenomenon more thoroughly, experimentally confirming the existence of an increased negative DC bias under the straps of the external antenna in the inner surfaces of a HPS, and investigating how this wall potential is affected by variations in the helicon discharge parameters. Infra-red (IR) thermography measurements taken on the inner surface of the dielectric ceramic window of the PISCES-RF device [2] also provided indirect evidence of this phenomena, showing increased values of the plasma heat flux under the straps of the helicon antenna, particularly the conductor connected to the live (non-grounded) terminal of the RF power supply.

The use of Faraday shields has been explored as a means to mitigate the effect of capacitive coupling within inductively-coupled plasmas (ICPs), and their application to HPSs has been suggested for the same purpose [71]. The Faraday shield has been implemented as a cylindrical jacket made of conducting material, installed between the dielectric plasma confinement surface and the helical antenna used in the ICP reactor [72]. Longitudinal slits have to be cut along this shield, to enable the inductive fields to penetrate the discharge. Specific experiments applying this technique to HPSs have yet to be performed. This method could potentially improve the performance of HPSs by reducing the erosion rate due to capacitive coupling under the antenna straps; however, its effects on other aspects of the source such as thermal management, and discharge efficiency, have to be investigated.

Recent experiments by Beers et al. [73, 74] describe the analysis of the helicon discharge section of the Proto-MPEX device, where they combined a finite-element model describing the helicon discharge, an ad-hoc sheath model, and a transport code in order to analyze the production of impurities due to sputtering at the material boundaries. Their results confirm the experimental findings of Berisford et al. [60] and Barada et al. [70], showing the importance of the electrostatic potentials near the helicon antenna straps as a source of energetic ions impacting the radial boundaries. They also showed the difference between the operation in non-magnetized and magnetized regimes, as was also discussed by Ahedo et al. [32].

The effect of the strength and geometry of the magnetic field on the performance of HPSs has also been researched. The magnetic field has an effect on the density profile within the source. Lafleur et al. [75] show that its intensity affects the peak value of the plasma density in the helicon mode, and they show the existence of optimal configurations for given values of input RF power and magnetic field intensity. The axial magnetic configuration is also able to modify the performance of an HPS. Takahashi et al. [76–78] have described the distribution of momentum transfer between the plasma and different elements of the source, its relationship with the magnetic field configuration, and how it can affect the total thrust of a helicon plasma thruster. These experiments describe how the ions are able to impart an axial momentum to the inner wall of the dielectric confinement material, due to the fact that their velocity vector is not completely normal to the wall surface [78]. This method could be used to indirectly estimate the incident angle with the confinement surface as the ions traverse the sheath, a critical factor in the calculation of the sputtering yield, although it is shown how the radial component is responsible of the energy transfer towards the wall.

The profile of the magnetic field within a HPS can be designed to mitigate the consequences of plasma-wall interactions within the source. Caneses et al. [79] describe experiments where two configurations of the magnetic field within the Proto-MPEX high-power helicon device were used to demonstrate the usefulness of controlling where the last uninterrupted magnetic flux surface (LUFS) makes contact with the inner confinement surfaces of the source. They relocated this contact point away from the dielectric ceramic window towards a purposely-designed stainless steel cylindrical limiter surface, an element with a function analog to that of divertors in fusion devices. This design change reduced the thermal heat loads under the dielectric window associated with direct impingement of the plasma, since the magnetic geometry maintains the LUFS at a minimum distance of approximately 1 cm away from the boundary surfaces. The plasma density decays rapidly beyond this point, as the magnetic lines intersect the material boundaries more often. This technique of magnetic field shaping allows the Proto-MPEX to reduce the heat loads on the dielectric window, but its effects on the sputtering and erosion related to plasma-surface interaction have not been thoroughly investigated. However, the careful design of magnetic geometries is commonly used for this purpose on electric propulsion devices [80, 81].

Figure 8 summarizes the findings of these experiments with regard to the appearance of sputtering phenomena within the internal dielectric confinement surfaces of HPSs. Region (1) in the figure represents areas within these internal surfaces in direct contact with the plasma, where a sheath forms and the dielectric surface obtains a negative electric potential Φ _w as described by Eq. 5. The positive ions are then accelerated towards the wall with a surface flux determined by the product of the bulk plasma density n ₀ and the Bohm velocity u _B they obtain when entering the sheath. The effect of the impinging ions on the dielectric surface can then be analyzed according to the sputtering models discussed in subsection 3.2, and effective surface etch rates may be computed. Region (2) in Figure 8 describes the particular phenomena observed by Berisford et al. [60], Aanesland et al. [69], Barada et al. [70], and Beers et al. [73, 74], where the creation of RF sheaths on the internal surfaces directly under the helical antenna straps may create the conditions for high-voltage DC sheaths in the negative part of the cycle. In this scenario, average sputtering yields can be computed through the ion energy distribution within the negative portion of the RF cycle [35], and hence etch rates can be computed as well.