Fusion Gain and Triple Product for the Sheared-Flow-Stabilized Z Pinch

Abstract Fusion gain $$Q$$Q and triple product $$nT\tau $$nTτ are derived for the sheared-flow-stabilized (SFS) Z pinch by including the input power associated with driving the plasma flow and the additional advective loss of thermal energy. Plasma impurities contribute to radiative power losses and to thermal power losses by increasing the electron population. The presence of impurities increases the required plasma parameters, characterized by the triple product, to achieve fusion gain. The analysis is applied to deuterium-tritium (D-T) fusion, though the methodology can be extended to other reactions. Since D-T fusion produces an alpha particle, the possibility exists of magnetically confining the alpha with sufficiently high magnetic fields, which are self-generated by the plasma pinch current. Confined alpha particles can heat the D-T fusion fuel, reduce the needed input power, and thereby amplify the fusion gain. However, ignition ($$Q \to \infty $$Q→∞) does not occur since the axial plasma flow must be externally driven. The impacts of alpha heating and impurity losses are considered on the fusion performance of the SFS Z pinch. Requirements, assumptions, and limitations are described that would justify a determination of “D-T equivalent $$Q = 1$$Q=1 conditions” in a D-D plasma. A minimum set of experimental measurements of plasma parameters is specified that can be compared to a plasma parameter map to facilitate a “$$Q = 1$$Q=1” claim, where $$Q$$Q is defined by instantaneous values of fusion power and input power. Corroborating measurements are also discussed that would further support extrapolation of plasma and fusion performance to D-T operation.


I. INTRODUCTION
Fusion gain Q is a metric that characterizes a fusion concept's viability as a power-producing reactor.The apparent simplicity of defining the ratio of output power to input power belies the importance of selecting the control volume and its associated surface across which power flows are measured.Setting the control volume to include all components associated with employing nuclear fusion to amplify input electrical power into output electrical power, such that the control surface is the wall plug, provides the most inclusive definition of fusion gain, called engineering gain, [1] Q eng ¼ ðP E out À P E in Þ=P E in .Input power is the electrical power drawn from a wall plug, and output power is the electrical power converted from the fusion plasma, so that the numerator is the net electrical power delivered to the grid.This aggregate definition of fusion gain includes the impact of system components that are not directly related to the fusion concept, e.g., the conversion efficiency of electrical power to plasma heating or the thermodynamic efficiency of converting the fusion heat to electricity.The efficiencies of these ancillary components offer opportunities for improvement, and their performance can obfuscate the ultimate potential gain of a specific fusion concept.
Isolating the fusion component leads to definitions of gain using smaller control volumes; for example, a volume that encompasses only the plasma is called the scientific gain Q sci .While not necessarily indicative of the overall gain, Q sci provides an upper limit and can be defined such that it characterizes the performance of a fusion concept.
The majority of terrestrial fusion applications aim to fuse deuterium and tritium nuclei, since it has the highest reaction probability.Experimental demonstrations of scientific fusion gain have been performed in a few devices [2][3][4] using deuterium-tritium (D-T) fuel.However, most experimental fusion devices operate by fusing deuterium (D-D): which avoids the complications associated with using radioactive tritium.The scientific fusion gain is then reported as a "D-T equivalent Q," [5][6][7] where the experimentally measured performance of a D plasma is extrapolated to the expected fusion performance for a D-T plasma by assuming, for example, the same plasma temperature, plasma density, and injected power.
The value of the scientific fusion gain for magnetic confinement devices is typically calculated using the peak fusion power [2] or by averaging the input and fusion powers over an interval of the total plasma pulse, effectively giving a maximum instantaneous value of Q.The method ignores the energy costs associated with forming the plasma, which may be recoverable for some fusion concepts.The input power can also be defined [8] using a plasma energy loss rate, indicated by the absorbed input power, and assuming self-heating from alpha particles produced by the fusion reaction.While these approaches rely on several assumptions, the resulting calculation of an instantaneous Q nevertheless provides an upper limit that facilitates meaningful comparisons among various fusion concepts.
The triple product of density, temperature, and energy confinement time nTτ is often used as a proxy measure to compare the performance of various fusion concepts [1] since a successful fusion configuration requires large values of each of the factors, though their product has no inherent physical significance.Wurzel and Hsu [1] provide a thorough discussion of triple product and fusion gain as well as a comparison of experimentally measured triple products for many fusion devices.
Given the nuances of defining Q sci for a particular fusion concept and its implications for the eventual Q eng , the measurements, inferences, and assumptions for calculating the fusion gain and the triple product are described for a shearedflow-stabilized (SFS) Z-pinch fusion device. [9]A machine drawing of the Fusion Z-Pinch Experiment (FuZE) SFS Z-pinch device is shown in Fig. 1.The calculation of fusion gain for the SFS Z pinch has been outlined previously [9,10] ; however, this paper presents the details of more general derivations of fusion gain and triple product for a flowing plasma that includes the effects of impurities and alpha heating.The assumptions associated with the calculations are explicitly specified to allow for accurate comparisons.In addition, experimental measurements are described that are necessary to determine fusion gain and triple product.This paper is structured as follows.Section II defines the scientific fusion gain for the SFS Z pinch, which includes the power needed to drive the axial flow.The expressions for the gain and corresponding triple product are derived using an effective energy confinement time that includes advective losses inherent to the SFS Z pinch.The possibility of confining alpha particles, their contribution to plasma heating, and the resulting impact on Q and nTτ are considered in Sec.III.Section IV discusses the experimental determination of "Q ¼ 1 conditions" in the SFS Z pinch and the assumptions underpinning the determination.Potential experimental measurements are presented that would facilitate and corroborate the "Q ¼ 1 conditions" determination, and the context establishing its validity and uncertainty is discussed.Section V provides concluding remarks.

II. SCIENTIFIC FUSION GAIN FOR A FLOWING PLASMA
As described in Sec.I, the definition of scientific fusion gain requires defining a control volume and evaluating the total power supplied to the control volume and the total power leaving the control volume, which is set as the magnetically confined plasma volume.Scientific fusion gain Q (without a subscript for convenience) typically considers only the power from the fusion reactions P F and the input power delivered to the fuel plasma P in .The output power leaving the plasma volume equals the sum of fusion power and input power.The gain Q is then defined as the ratio of fusion power to total input power to the plasma: The SFS Z-pinch concept forms a magnetically confined cylindrical plasma with a pinch radius a and length L. The plasma flows axially to provide stability, [11][12][13][14] such that the plasma is confined in a quasi-steady-state equilibrium with a lifetime greater than the plasma axial flow-through time.The plasma properties are axially uniform with a radial profile.For this analysis, the radial profiles are assumed to correspond to the "sharp pinch," [15] which has uniform values of species densities n s , species temperatures T s , and center-of-mass axial velocity v z up to the pinch radius.No plasma exists beyond the pinch radius.A surface current at r ¼ a produces a self-generated azimuthal magnetic field B θ for r � a that provides the radial equilibrium force balance, where j z is the axial current density and μ 0 is the permeability of free space.A schematic representation of the Z-pinch equilibrium is shown in Fig. 2.
The power released for the D-T fusion reaction in a Z pinch with the assumed equilibrium profiles is then Fig. 2. Schematic representation of the Z-pinch equilibrium.An axial current I is driven between two electrodes (gray) along a plasma column producing an azimuthal magnetic field B θ that radially compresses the plasma to a pinch radius a until its pressure gradient balances the Lorentz force.Current returns through a surrounding conducting wall located at radius r w .Typically, a � r w .Reproduced from Ref. [9].[U.Shumlak, "Z-pinch fusion," Journal of Applied Physics, 127, 20, 200901  (2020) Power must be supplied to the SFS Z-pinch plasma to compensate for losses of thermal energy and kinetic energy.Thermal energy is lost through conduction, advection, and radiation.As described below, thermal energy loss rates due to conduction and advection are specified as the total plasma thermal energy divided by a confinement time associated with each process.The total thermal energy loss rate is summed as thermal power P th .Radiative power losses are included separately as P rad .Flow power is the advection of kinetic energy.The input power is then expressed as the sum of thermal power, flow power, and radiative power: which represents the power needed to maintain the plasma at steady-state equilibrium conditions.Selfheating effects are not included in Eq. ( 7) but are considered in Sec.III.
The plasma thermal power is the rate of thermal energy loss and is defined by a characteristic energy confinement time τ E .The total thermal energy of the plasma is so the thermal power loss is expressed as which includes the energy for all species s = D, T, e.The SFS Z pinch loses energy through parallel processes: thermal conduction to a cold boundary and advection of thermal energy through the end of the Z-pinch plasma.The processes are characterized by a thermal conduction time τ 0 E and by an advection time, τ flow ¼ L=v z .The effective energy confinement time is then which is dominated by the shortest timescale.Plasmas confined by strong magnetic fields and surrounded by large vacuum regions, such as the SFS Z pinch, minimize perpendicular thermal losses, so τ 0 E � τ flow .The effective energy confinement time given by Eq. ( 10) accounts for all thermal energy leaving the Z pinch.Note that if the axial current results in an electron speed larger than v z , Eq. ( 9) could be expanded to separately account for heat diffusion, ion advection, and electron advection, [16] with an appropriate energy confinement time for each physical process.
Plasma impurities introduce additional electrons beyond the ones from the D-T fuel, such that n e > n.The contribution to electron density can be significant even if the impurity density is low since impurities can have high atomic numbers.Defining the mean ionization state, where Z ¼ 1 would correspond to no impurities, the plasma thermal power loss becomes where thermal equilibrium is assumed, The flow power represents the rate of kinetic energy supplied to the SFS Z-pinch plasma, which in the assumed steady state equals the rate of kinetic energy leaving the Z-pinch plasma: where • m is the mass flow rate.The atomic mass of species s is M s .The expression for P flow can be derived formally by taking the dot product of the multifluid momentum equation [17] with velocity.Since the electron
mass is much less than the ion mass, electrons contribute a negligible amount to the total kinetic energy and therefore are not included in the flow power.
The radiative power loss is assumed to include only bremsstrahlung radiation while still accounting for increased electron density from impurities: where the summation is over the ion species and C ¼ 1:69 � 10 À 38 W•m 3 •eV -1/2 , such that P rad is given in watts and T e is expressed in electron volts.Defining an effective ionization state, [18] which is the mean square ionization state divided by the mean ionization state, yields where n e ¼ Zn and T e ¼ T have been used.The summations in Eq. ( 17) are performed for all fuel and impurity ions.The fraction of fusion product ions is small.Substituting Eqs. ( 12), (13), and (19) into Eq.( 7) and then combining with Eq. ( 6) give an expression for the fusion gain from Eq. ( 3): where the plasma volume has canceled from the expression.
Following the procedure of procedure of Wurzel and Hsu, [1] the numerator and denominator of Eq. ( 20) are divided by n 2 T. Note that τ from Wurzel and Hsu [1] is defined as a plasma confinement time, but this parameter is later set to the energy confinement time set by thermal conduction, which is equivalent to τ 0 E .For the SFS Z pinch, the effective energy confinement time τ E should include the advection time, as defined in Eq. (10).The fusion gain is then Interestingly, the coefficient of the flow term is related to the square of a Mach number M 2 .

II.A. Conventional Power Gain
Conventional power gain in electronic or microwave circuits, for example, is defined as the ratio of output power to input power, [19] even when considering components or subcircuits, e.g., amplifiers.Unity gain indicates an electronic circuit with an output power equal to the input power.This power gain contrasts with the definition of fusion gain Q given by Eq. ( 3), which subtracts the input power from the output power in the numerator.The input power that leaves the plasma volume is expected to be recaptured.Radiative power would be absorbed by the surrounding blanket of working fluid in a power reactor.Similarly, the power associated with thermal energy and kinetic energy leaves the plasma volume and may also be recovered by heating the blanket or be extracted as kinetic energy for direct energy conversion or space propulsion applications.A scientific power gain for a fusion core that aligns with the definition of conventional power gain could be expressed as where the control volume is still set as the magnetically confined plasma.For a system that completely recovers input power, P recover ¼ P in , power gain is related to fusion gain as Q power ¼ Q þ 1, such that any Q > 0 produces a net power increase over the input power.

II.B. Triple Product for a Flowing Plasma
The triple product for the SFS Z pinch can be found by manipulating Eq. ( 21) to give which can then be solved for nτ E .Multiplying the resulting expression by temperature gives the triple product: For the limiting case of no flow, v z ¼ 0, and no impurities, Z eff ¼ Z ¼ 1, Eq. ( 23) reduces to Eq. ( 23) of Wurzel and Hsu [1] if alpha heating does not contribute to the plasma energy.Suitable expressions for the fusion reactivity hσvi DT as a function of temperature exist, for example, as given in Bosch and Hale. [20]In the same limiting case of no flow, no impurities, and no alpha heating, the minimum triple product value from Eq. ( 23) that satisfies Q ¼ 1 is which occurs at T � 14 keV.

III. PLASMA HEATING FROM ENERGETIC ALPHA PARTICLES
The alpha particle from the D-T fusion reaction in Eq. ( 1) carries E α ¼ 3:5 MeV of the total energy released from each fusion reaction.Since the alpha particle is electrically charged, it is possible to magnetically confine the alpha particle, so it heats the D-T fuel by undergoing Coulomb collisions and depositing power P α in the plasma.Using the fusion energy to increase the plasma's thermal energy decreases the fusion output power that leaves the plasma volume but has the benefit of decreasing the input power needed to maintain the plasma at fusion conditions.
Particular to the SFS Z pinch, alpha heating can offset the thermal power lost through conduction, advection, and radiation; however, since alphas are emitted isotropically from the fusion reactions, they cannot drive the axial flow, which must be externally powered.Note that effects such as alpha channeling, [21] which may drive waves in a preferential direction, are not considered in the present analysis.As mentioned previously, the alpha heating power can offset the required input power such that P in ¼ P th þ P flow þ P rad À P α , with the caveat that the required P in cannot decrease below P flow , regardless of how large P α becomes relative to P th þ P rad .With the alpha heating fraction defined as f c ¼ P α =P F , i.e., the fraction of fusion power carried by charged products and deposited in the fuel plasma, the fusion gain definition is modified accordingly as to represent the fusion power leaving the plasma in the numerator and the externally supplied power in the denominator.Note the numerator is unchanged since the alpha heating that is removed from the output power is exactly balanced by the decrease in required input power.Since P α cannot supply P flow , the denominator never reaches zero, and Q remains bounded.Power must always be supplied to the plasma to drive the axial flow.In this sense, the SFS Z-pinch plasma never reaches ignition, With the addition of plasma heating from energetic alpha particles, Eq. ( 21) becomes Fusion gain as a function of density and temperature is plotted in Figs. 3, 4, and 5 by using the Bosch-Hale [20] temperature-dependent reactivities for D-T fusion and setting values for v z , Z eff , Z, and f c .Impurities depress the value of Q by increasing thermal and radiation power losses and requiring higher plasma parameters to achieve Q ¼ 1, as seen by comparing Figs. 3 and 4, which shows the effects of setting Z eff ¼ 2 and Z ¼ 1:2, corresponding to a 0.04 fully stripped carbon impurity fraction.Even partially confining and thermalizing the alphas such that 25% of their energy heats the plasma, f c ¼ 0:05, significantly increases the fusion gain beyond Q ¼ 1, as seen in Fig. 5.While Q becomes large, it remains bounded and does not approach infinity.
Manipulating Eq. ( 26) as performed in Sec.II.B and solving for the triple product yield which represents a generalization of the Lawson criterion for the SFS Z pinch that includes alpha heating.Equation (27) gives an expression for the triple product nTτ E for specified values of T, Z eff , Z, v z , Q, f c , and nτ flow .

III.A. Alpha Heating Fraction
Magnetically confining the alpha particle from the D-T fusion reaction requires that the 3.5-MeV alpha particle have a sufficiently small gyroradius compared to the Z-pinch plasma radius.Using the assumed plasma profiles, the maximum gyroradius of an alpha particle at peak magnetic field is which corresponds to a radial emission such that The form of the alpha heating fraction f c , which is bounded between zero and E α =E DT ¼ 1=5, should have a dependence on the ratio of the pinch radius to the alpha gyroradius a=r L α .This ratio is proportional to pinch current I, as given by Eq. ( 28), and reaches unity at approximately 1.4 MA.The expected operating current of a high-Q SFS Z pinch is greater than 1.5 MA, [9,22] so alpha heating could compensate for power losses and significantly increase Q. Formal analysis of alpha heating in cylindrical plasmas with uniform magnetization shows Fig. 4. Logarithm of fusion gain log 10 ðQÞ as a function of density and temperature in a D-T Z-pinch plasma with a 50-cm length and an axial flow velocity of 10 5 m/s.The plasma is assumed to have impurities, such that Z eff ¼ 2 and Z ¼ 1:2, and to be spatially uniform within the magnetically confined Z pinch.Alpha heating is neglected, f c ¼ 0:0.The solid black contour identifies Q ¼ 1. Impurities depress Q and require higher plasma parameters to achieve Q ¼ 1. a quadratic dependence on a=r L α . [23]The analysis shows approximately 10% of the maximum alpha energy deposited in the plasma when a=r L α ¼ 1 and approximately 100% when a=r L α ¼ 10.As shown by the adiabatic scaling relations derived for the SFS Z pinch, [9,10,24] plasma performance improves with pinch current.To illustrate the effect of alpha heating and the increase in Q with pinch current, a sharp pinch [15] equilibrium is assumed, which gives a fusion reaction rate that scales as I 3 hσvi.Increasing the current of a 50-cm plasma initially at I ¼ 50 kA with parameters of T ¼ 20 eV and n ¼ 7 � 10 21 m -3 as measured on the ZaP SFS Z-pinch device [25] increases Q as shown by the solid lines in Fig. 6.If the alpha heating fraction is assumed to be which is limited to a maximum value of 1=5, alpha heating produces a rapid increase in fusion gain beyond Q ¼ 1, as evident by the dashed lines in Fig. 6.Using the quadratic dependence on a=r L α [23] does not qualitatively change the behavior.

IV. DETERMINATION OF "Q = 1 CONDITIONS"
The fusion gain analysis presented in Secs.II and III assumes uniform properties for a 50-50 D-T plasma to compute instantaneous values of Q and nTτ.Experimental plasmas are more typically formed with 100% deuterium as the working gas to avoid the expense and complication of using radioactive tritium.Evaluating the "D-T equivalent Q" [5][6][7] relies on experimental measurements of a D plasma and the validity of specific assumptions necessary to extrapolate the fusion performance to a D-T plasma.A determination of "Q ¼ 1 conditions" indicates experimentally measured properties of a D plasma that would result in a unity scientific Q in a D-T plasma and includes the caveats associated with the assumptions used for the extrapolation.
This section describes the assumptions and discusses their limitations and implications.It also identifies candidate plasma properties that could be experimentally measured to evaluate "D-T equivalent Q" and to determine "Q ¼ 1 conditions" in the SFS Z pinch.

IV.A. Plasma Uniformity
Experimental plasma measurements are inherently limited.The measurements may be localized to a single spatial location, e.g., a small light-scattering volume, of the Z-pinch plasma at an instant during the quiescent period [25,26] when neutrons are observed. [27]Other measurements may be path-integrated [28] along a beam or through a conical collection volume.The spatially integrated measurements are then inverted to determine the radial profile of specific plasma properties [29] along the integrated path.Applying the fusion gain analysis based Fig. 5. Logarithm of fusion gain log 10 ðQÞ as a function of density and temperature in a D-T Z-pinch plasma with a 50-cm length and an axial flow velocity of 10 5 m/s.The plasma is assumed to have no impurities, Z eff ¼ Z ¼ 1, and to be spatially uniform within the magnetically confined Z pinch.Alpha heating compensates for power loss with f c ¼ 0:05.The solid black contour identifies Q ¼ 1. Contributions from alpha heating have little effect below Q ¼ 1 parameters but rapidly increase Q beyond unity toward ignition.However, Q remains bounded and does not approach infinity.on such measurements assumes that the extracted plasma properties are uniform along the entire Z-pinch plasma column and have a radial dependence that is well approximated by the sharp pinch profiles described in Sec.II.
Implicit in computing the maximum instantaneous value of Q [2] is that the value is lower during other times of the plasma pulse unless the plasma properties are maintained constant in time.The SFS Z pinch has demonstrated an ability to sustain a steady-state equilibrium [9,[24][25][26][27][30][31][32][33][34] for durations much longer than instability growth times for a static Z pinch.

IV.B. D-T Equivalent
The plasma properties and fusion reaction rates measured in a D plasma are related to a D-T plasma by assuming a D-T plasma would produce identical parameters, e.g., n, T i , T e , and v z , as are produced in a D plasma.Therefore, measuring the properties of the D plasma provides a simple extrapolation to the expected performance of a D-T plasma by accounting for the different temperature-dependent reactivities between the D+D and D+T fusion reactions.
Since a D-T plasma will have a higher mass density than a D plasma with the same ion number density, the plasma dynamics are likely to differ.The details of the plasma acceleration are known to depend on mass density, with higher densities requiring longer current rise times to prevent "blow-by" instabilities. 35Unstable magnetohydrodynamic (MHD) modes, such as the sausage and kink instabilities, have growth times that are inversely proportional to the Alfvén speed, [11,12,36] which decreases with the square root of the ion mass.Therefore, the most virulent MHD instabilities in the Z pinch are expected to grow more slowly in a D-T plasma than they would grow in a D plasma.The flow shear required to stabilize the Z pinch is proportional to the Alfvén speed, [11][12][13][14] so a D-T plasma could be stabilized with a lower axial flow.Improved plasma performance has been observed when operating with D-T compared to operating with D in other magnetically confined plasmas. [37]Nevertheless, differences in the plasma dynamics are ignored in asserting D-T equivalence.

IV.C. Measurements Needed for "Q = 1 Conditions"
The expressions for fusion gain given by Eqs. ( 21) and ( 26) can be solved by setting Q equal to unity, which reproduces the solid black contours in Figs. 3, 4, and 5.Note that at unity gain, alpha heating does not modify the required plasma parameters, as is evident by the independence of nTτ from f c in Eq. ( 27) when Q ¼ 1.The resulting relation for nTτ defines the plasma parameters needed for Q ¼ 1 conditions.The effective and mean ionization states needed for the relation are difficult to accurately measure, but an ideal limiting case can be used, which assumes no impurities, namely, Z eff ¼ Z ¼ 1. Relaxing this assumption allows for an exploration of the sensitivity of the required plasma parameters to achieve Q ¼ 1 when impurities are present.At fusion conditions, the Z pinch has a large azimuthal magnetic field surrounding the plasma column and a large volume separating the plasma from the outer electrode, as shown in Fig. 2. The configuration reduces radial thermal losses such that heat diffusion can be assumed to be negligible, τ 0 E ¼ 1.Since the plasma radius is much smaller than the plasma length, a � L, axial thermal conduction is assumed small.The confinement time is then set by the axial advection through the Z pinch plasma, τ E ¼ τ flow ¼ L=v z , as defined in Sec.II.The triple product from Eq. ( 27) can then be expressed as Setting Q ¼ 1 and solving for density, gives an expression for the plasma density needed for Q ¼ 1 conditions as a function of temperature, flow velocity, ionization state, alpha heating fraction, and Z-pinch length.
In the temperature range of interest, fusion power increases with ion temperature, and radiative power loss increases with electron temperature.Producing plasmas with T i > T e would lower the minimum density required for Q ¼ 1. Radial compression is the primary heating mechanism in the SFS Z pinch, which heats ions and electrons.Radiation is the primary energy loss mechanism, which cools the electrons.As a result, T i , > T e is expected and has been observed. [25]igure 7 provides a parameter map that identifies "Q ¼ 1 conditions."If a 50-cm-long Z pinch has a plasma density and temperature that locate a point to the right of the curve corresponding to the plasma's axial velocity, then Q � 1, as defined above, has been achieved.Therefore, experimental measurements of plasma density, temperature, and axial velocity constitute a minimum set to determine if "Q ¼ 1 conditions" have been reached.
Since quasineutrality is a valid approximation for the SFS Z pinch, the density can be determined by measuring either the electron number density or the ion number density and using the relation of n ][51][52][53][54] The measurement must distinguish the plasma density confined in the Z pinch from the surrounding background plasma density.Hence, single-chord interferometry is inadequate without separate measures of pinch radius and of background plasma density.Though pinch radius does not appear in Eqs. ( 21) and (26) and is not required to calculate Q ¼ 1, pinch radius is required to apply neutron measurements to corroborate the Q determination, as discussed in Sec.IV.D.
7. Density required for Q ¼ 1 conditions in a D-T SFS Z-pinch plasma with a 50-cm length and flowing at various axial velocities v z .The plasma is assumed to have no impurities, Z eff ¼ Z ¼ 1, and to be spatially uniform within the magnetically confined Z pinch and to have no alpha heating.Impurities have a large impact on plasma performance.Measurements of Z eff can be made using spectroscopy [55][56][57] to record bremsstrahlung radiation intensity in the visible wavelengths of the spectrum where impurity emission lines are absent.Assuming thermal equilibrium between the electrons and ions, T e � T i , likely underestimates fusion performance as described above.However, measuring either electron or ion temperature would provide an adequate measurement of T to ensure Q ¼ 1. Analyzing the line shape of impurity emission line spectroscopy indicates the Doppler broadening due to thermal motion.][60] The ion Doppler measurements can be better localized using focusing optics [61] or through charge exchange recombination spectroscopy [62,63] by injecting a neutral atomic beam into the plasma.Incoherent Thomson scattering [25,52] provides localized measurements of T e in the Z pinch plasma volume.At a sufficiently large α parameter, α ; ðkλ D Þ À 1 1, collective Thomson scattering [50,51,53,54] can resolve the ion feature to provide local measurements of T i in the Z-pinch plasma.
1]64] The measurements can again be better localized using focusing optics [61] or through charge exchange recombination spectroscopy. [62,63]Since impurity emission line spectroscopy measures the Doppler shift of impurity ions, extracting v z requires that the impurity ions move at the same velocity as the bulk plasma ions, which has been verified for the SFS Z pinch. [31]By resolving the ion feature, collective Thomson scattering [50,51,53,54] provides a direct and local measurement of v z in the Z-pinch plasma.The axial flow velocity in the SFS Z pinch typically has a radial dependence, [9,10,[24][25][26][29][30][31]64] so a measurement of v z averaged over the pinch radius is appropriate and is consistent with the assumptions of Secs. II and III Using these diagnostic techniques or equivalent ones, the measured values of radially averaged plasma density, temperature, and axial velocity should be compared to Fig. 7 to determine if a claim of instantaneous Q ¼ 1 is justified.
The presence of plasma impurities increases the electron number density and the radiative power loss, which then requires higher plasma parameters to achieve Q ¼ 1.For comparison, the analysis is repeated assuming that plasma impurities result in Z eff ¼ 2 and Z ¼ 1:2.The corresponding parameters that produce Q ¼ 1 conditions are plotted in Fig. 8.

IV.D. Corroborating Measurements
The analysis to determine the fusion gain in the SFS Z pinch has assumed perfect radial thermal energy confinement, τ 0 E ¼ 1, which is justified by the high magnetic field and large separation distance between the Z pinch plasma and the outer electrode, a � r w .According to Eq. ( 10), the flow time τ flow provides an upper bound on the energy confinement time τ E , though not a lower bound.For the expected plasma parameters, [9,22] calculations indeed show τ 0 E � τ flow .Alternative methods for calculating τ E involve measuring the instantaneous input power into the Z-pinch plasma, e.g., P in ¼ V pinch I pinch , and relating it to the measured thermal energy in the same plasma volume, Eq. ( 8), to give τ E � E th =P in .While the pinch current is routinely measured, the voltage across the Z pinch plasma is difficult to measure accurately due to electrical potential drops across the plasma sheaths at the electrodes.Using the voltage measured at the beginning of the acceleration region, as shown in Fig. 1, introduces an inconsistent control volume, namely, one that encompasses the plasma volume and the accelerator volume and, thereby, overestimates P in .Nevertheless, computing the thermal energy confinement time as τ E � E th =ðV gap I pinch Þ could provide a lower bound to bracket τ E .
The assumption of plasma uniformity could ostensibly be examined by directly measuring the spatial distributions of the plasma properties, nðr; zÞ; Tðr; zÞ; and v z ðr; zÞ.Diagnostics capable of measuring such detailed profile information are not feasible.However, the fusion reaction rate from a deuterium plasma can provide insight into plasma uniformity.The fusion reaction rate is directly related to the neutron production rate, which has been measured using plastic scintillators [32,34] on the FuZE device.Plasma density, temperature, and pinch radius measurements can be used to calculate the expected D-D fusion reaction rate if the plasma properties are uniform throughout the volume defined by the pinch radius and length of the Z pinch.Other fusion-relevant parameters such as the density-radius product can also be calculated.Observed fusion reaction rate measurements that corroborate the plasma density and temperature and the assumed uniformity would support the performance extrapolation to D-T operation.

IV.E. Toward Calculating an Engineering Gain
A power-producing reactor must operate such that it produces more power than it consumes.As described in Sec.I, such an analysis would set the control volume to encompass all components of the system and produce an engineering gain Q eng > 1, which is ultimately necessary for net positive power delivered to the electrical grid.
A useful intermediate measure of gain is the wall-plug gain, [1] Q wp ¼ P F =P E in , which includes the inefficiencies associated with converting electrical power into input power absorbed by the plasma and the energy supplied to the plasma when it is not at fusion conditions, e.g., during plasma initiation and termination.
Current experimental embodiments of the SFS Z pinch have operated in a long-pulse mode, where the pulse length is greater than the energy confinement time, which is the mode of operation of most magnetically confined fusion concepts.Calculating Q wp in such longpulsed systems is possible by computing the ratio of the fusion energy integrated over a pulse and dividing by the input electrical energy: where the fusion energy is determined from the measured neutron yield, which is directly proportional to the D-D fusion energy.With no auxiliary sources of heating, compression, or current drive, the only input power supplied to the SFS Z pinch is directly from a charged capacitor bank.The input electrical energy is the net decrease in capacitor bank energy-the difference between the initial charge energy and the remaining charge energy at the end of a pulse.
Such a measure of Q wp includes inefficiencies in the pulsed power driver system and in the plasma formation, acceleration, and compression processes.It also includes the initiation and termination energy that could be minimized by extending the relative duration of fusion reactions, and it ignores the possibility of recovering magnetic energy.Nevertheless, a calculation of Q wp provides a useful lower limit on overall fusion gain and highlights a path for optimization.Similar calculations of Q wp for other fusion concepts of any pulse length are possible using Eq. ( 32) and would prove valuable as fusion concepts continue to develop and approach the goal of net power production.

V. CONCLUSIONS
The SFS Z pinch is a novel approach to achieve thermonuclear fusion power.In addition to the magnetic confinement feature of other concepts, it incorporates a plasma flow that is externally driven.Including the flow power into the scientific fusion gain calculation produces a generalized formulation for Q and nTτ for a flowing plasma, as presented in Sec.II.The generalization also defines an effective energy confinement time that combines the parallel processes of conduction and advection.The SFS Z pinch plasma is confined by a large azimuthal magnetic field and is separated from a solid wall by a large radial distance.These features reduce the radial conduction such that for typical fusion plasma parameters, the energy confinement time is set by the axial flow time.The general formalism can be applied to treat other configurations where higher radial conduction may decrease the effective energy confinement.Fig. 8. Density required for Q ¼ 1 conditions in a D-T SFS Z-pinch plasma with a 50-cm length and flowing at various axial velocities v z .The plasma is assumed to have impurities such that Z eff ¼ 2 and Z ¼ 1:2.The plasma is assumed to be spatially uniform within the magnetically confined Z pinch and to have no alpha heating.Another consequence of the large azimuthal magnetic field is that energetic alpha particles emitted from the fusion reaction can be magnetically confined at expected operating currents, which avails the possibility of using the alpha energy to directly heat the fusion plasma and decrease the required input power.The fusion gain increases rapidly as a result of alpha heating for values of Q beyond unity.Since alpha heating cannot replace the power needed to drive the axial flow, a finite amount of input power is always required, so ignition (Q ! 1) is never reached.Nevertheless, even modest fractions of alpha heating can increase Q by many orders of magnitude, resulting in large values of Q that would be attractive for a commercial power-producing reactor.
Achieving Q ¼ 1 in any controlled thermonuclear fusion device represents a critical milestone.Analysis is presented in Sec.IV that describes a path to determine D-T equivalent Q ¼ 1 conditions from experimental measurements in a deuterium plasma.Plasma density, temperature, and axial velocity constitute a minimum set of plasma parameters needed to make the Q ¼ 1 determination.The assumptions on an extrapolation to a D-T plasma are identified and discussed in the context of limitations placed on the Q evaluation.
Corroborating measurements are discussed that would test the assumptions.The measurements include a direct measure of input power into the device to calculate an estimate for a lower bound on the energy confinement time.Neutron measurements to determine the fusion reaction rate can be compared to the calculations of expected fusion reaction rates for the measured density and temperature and would provide additional support for the Q ¼ 1 determination.

Fig. 1 .
Fig. 1.Machine drawing of the FuZE SFS Z-pinch experimental device showing the coupling of the coaxial acceleration region to the 50-cm pinch assembly region.A schematic representation of the Z-pinch plasma is shown for reference.Electrical power (current and voltage) is supplied to the device across the terminals on the left, where the voltage V gap is measured.Reproduced from Ref. [9].[U.Shumlak, "Z-pinch fusion," Journal of Applied Physics, 127, 20, 200901 (2020); https://doi.org/10.1063/5.0004228],with the permission of AIP Publishing.]

Fig. 3 . 7 FUSION
Fig.3.Logarithm of fusion gain log 10 ðQÞ as a function of density and temperature in a D-T Z-pinch plasma with a 50-cm length and an axial flow velocity of 10 5 m/s.The plasma is assumed to have no impurities, Z eff ¼ Z ¼ 1, and to be spatially uniform within the magnetically confined Z pinch.Alpha heating is neglected, f c ¼ 0:0.The solid black contour identifies Q ¼ 1.

Fig. 6 . 9 FUSION
Fig.6.Dependence of Q on pinch current I using adiabatic scaling relations for an equilibrium SFS Z pinch at various axial velocities v z .The solid lines assume no alpha heating, f c ¼ 0, and the dashed lines include alpha heating with an alpha heating fraction proportional to pinch current.Dotted black line indicates Q ¼ 1 for reference.