Improving the Feasibility of Economical Proton-Boron 11 Fusion via Alpha Channeling with a Hybrid Fast and Thermal Proton Scheme Ian E. Ochs Elijah J. Kolmes Mikhail E. Mlodik Tal Rubin and Nathaniel J. Fisch

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Improving the Feasibility of Economical Proton-Boron 11 Fusion via Alpha

Channeling with a Hybrid Fast and Thermal Proton Scheme

Ian E. Ochs, Elijah J. Kolmes, Mikhail E. Mlodik, Tal Rubin, and Nathaniel J. Fisch

Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08540, USA

(Dated: October 18, 2022)

The proton-Boron11 (p-B11) fusion reaction is much harder to harness for commercial power than

the easiest fusion reaction, namely the deuterium and tritium (DT) reaction. The p-B11 reaction

requires much higher temperatures, and, even at those higher temperatures, the cross section is much

smaller. However, as opposed to tritium, the reactants are both abundant and non-radioactive. It

is also an aneutronic reaction, thus avoiding radioactivity-inducing neutrons. Economical fusion

can only result, however, if the plasma is nearly ignited; in other words if the fusion power is

at least nearly equal to the power lost due to radiation and thermal conduction. Because the

required temperatures are so high, ignition is thought barely possible for p-B11, with fusion power

exceeding the bremsstrahlung power by only around 3%. We show that there is a high upside to

changing the natural ﬂow of power in the reactor, putting more power into protons, and less into

the electrons. This redirection can be done using waves, which tap the alpha particle power and

redirect it into protons through alpha channeling. Using a simple power balance model, we show

that such channeling could reduce the required energy conﬁnement time for ignition by a factor of

2.6 when energy is channeled into thermal protons, and a factor of 6.9 when channeled into fast

protons near the peak of the reactivity. Thus, alpha channeling could dramatically improve the

feasibility of economical p-B11 fusion energy.

I. INTRODUCTION

Historically, fusion energy research has focused primar-

ily on the deuterium-tritium (DT) reaction, due to its

high cross section at relatively low temperature. This fea-

ture means that the conﬁnement requirements for achiev-

ing (DT) fusion are much lower than for other fuels, mak-

ing it the most logical fuel to exploit in the near term.

However, there are several disadvantages to DT fusion.

First, tritium is radioactive. Second, it is not abundant,

and must be bred from lithium or other materials. Third,

the DT reaction produces fast neutrons. In addition to

the proliferation risk that these entail, magnets and sen-

sitive instruments must be shielded from these neutrons

using considerable shielding material, which signiﬁcantly

adds to the volume and cost of any conﬁnement device.

Over time, the neutrons break down this shielding, turn-

ing it into a structurally unsound, radioactive slab that

must be safely stored away for hundreds of years.

Such deﬁciencies of the DT reaction have lead to an

interest in aneutronic fuels. One of the most appealing

of these is the proton-Boron11 (p-B11) reaction, which

has the additional advantage of fuel abundance.

For a long time, it was thought that achieving a self-

sustaining thermonuclear fusion reaction (ignition) was

impossible for p-B11. This pessimism came from the fact

that the fusion cross section was too small, and occurred

at too high a temperature [1]. Thus, it seemed that the

bremsstrahlung power would always exceed the fusion

power, requiring external heating power to maintain the

reaction [2, 3]. This lead to a proliferation in interest

in nonthermal and nonequilibrium schemes [4–12], which

accept the requirement for signiﬁcant external heating

and seek to optimize the output energy given that con-

straint.

Fortuitously, recent results have shown that the cross

section for the p-B11 reaction is larger than previously

thought [13]. These larger cross sections, combined with

more detailed calculations of how the fusion-born alpha

particles damp on the protons, have resulted in a more

optimistic picture, showing that ignition is in fact possi-

ble for p-B11 in thermonuclear fusion plasmas [14]. This

realization has led to a revival in interest in thermonu-

clear p-B11 fusion [15].

Just because ignition is theoretically possible, however,

does not mean that it is particularly feasible. As we

show later in this paper, the ignition window identiﬁed by

Putvinski [14] requires achieving an energy conﬁnement

time of around 500 seconds at ion densities of 1014 cm−3–

an enormous technological hurdle. Thus, it is important

to examine processes which might reduce these extreme

requirements.

Much of the reason for these extremely large conﬁne-

ment times required for ignition is that the fusion power

only exceeds the bremsstrahlung power by a few percent.

Widening this gap between fusion and bremsstrahlung

power to even 20% thus has the potential to produce a

7x improvement in the required conﬁnement time. To do

this, one must try to redirect power from the electrons

(which produce radiation) to the protons (which produce

fusion).

To redirect the power from the alpha particles into the

protons, one can make use of waves, in a process known as

alpha channeling [16–18]. This possibility was explored

by Hay [19], but crucially, that paper ignored thermal

conduction losses. As can be shown by a simple analytic

model [20], much of the aid alpha channeling provides is

in dramatically decreasing the conﬁnement time required

to achieve ignition.

In this paper, we delve more deeply into examining

arXiv:2210.08076v1 [physics.plasm-ph] 14 Oct 2022

the potential improvements to p-B11 fusion provided by

alpha channeling. We discuss the important key met-

rics in achieving economical fusion energy, focusing in

on the importance of the minimum energy conﬁnement

time to achieve ignition τ∗

E. We then provide a simple 0D

computational power balance model to evaluate this con-

ﬁnement time, which accounts for collisional and wave-

based energy exchanges between the diﬀerent species in

the plasma. As this model shows, using alpha channel-

ing to put power directly into the protons can lower the

required conﬁnement time to achieve ignition by a factor

of around 2.6. Furthermore, alpha channeling improves

the robustness of the reaction to contamination by fusion

ash.

With the use of waves, however, it is no longer nec-

essary to put the energy into thermal protons–instead,

the energy can be put directly into maintaining protons

near the peak of the fusion reactivity at 650 keV. Such a

reaction can be seen as a hybrid between beam and ther-

monuclear fusion, as it incorporates large populations of

both fast and thermal protons. By allowing for the pres-

ence of separate fast ion population in our power balance

model, we show that this hybrid scheme improves the

conﬁnement time by a further factor of three, resulting

in a total factor-of-6.9 reduction in the required conﬁne-

ment time for ignition relative to purely thermonuclear

p-B11 fusion. These results broadly match those in [20],

now shown with a more full optimization and a more

accurate power balance model.

To demonstrate these promising results, we begin in

Section II with a discussion of the power ﬂow in a fusion

reactor, explaining the rationale for conﬁnement time as

a performance metric for high-performance fusion plas-

mas, and why the p-B11 reaction is particularly chal-

lenging. In Section III, we introduce the power balance

model itself, which captures collisional exchange of en-

ergy between fast protons, thermal protons, boron, and

electrons, heating by alpha particles, bremsstrahlung ra-

diation, and alpha channeling. In Section IV, we describe

how to optimize the conﬁnement time given diﬀerent as-

sumptions for the alpha channeling. We then numerically

perform this optimization, showing how alpha channel-

ing results in much lower required conﬁnement times for

ignition.

In Section V, we consider the eﬀect of poisoning by al-

pha particle ash, the product of the fusion reaction. Such

ash increases the bremsstrahlung power, and without al-

pha channeling a very small quantity of ash (<2%) can

preclude ignition, even when assuming perfect conﬁne-

ment. We show that alpha channeling allows for ignition

at much higher ash concentrations, even when allowing

for non-perfect conﬁnement.

The core analysis of the paper is contained in Sections

II-V. In the subsequent sections, we brieﬂy discuss other

considerations in designing a reactor. In Section VI, we

discuss why the optimal ion mix for achieving ignition

contains a mix of fast and thermal protons, rather than

simply a beam of fast protons–a topic also covered in

[20]. In Section VII, we brieﬂy go over how consider-

ation of energy recycling in the full reactor power bal-

ance might lead to even lower required conﬁnement times.

We also discuss how recycling might lead to a very dif-

ferent optimal mix of thermal and fast protons, if one

can achieve high recycling eﬃciencies from direct con-

version. Finally, in Section VIII, we discuss additional

power loss mechanisms due to the conﬁnement systems

and electron-cyclotron radiation, and how they might af-

fect the design of a fusion reactor.

II. POWER FLOW AND PERFORMANCE

METRICS

To consider the potential advantage of altering the en-

ergy ﬂow from the alpha particles, it is necessary to con-

sider the power ﬂow of an eventual fusion reactor. Here,

we consider a steady-state reactor, so that the initial in-

vestment of power during the startup process contributes

negligibly to the overall eﬃciency. Such a power ﬂow is

shown in Fig. 1. Electrical power Pin consists both of

power used to heat (PH,e) and conﬁne (PC,e) the plasma.

With some conversion eﬃciency ηH, the electrical heating

power is delivered to the plasma as heat PH=ηHPH,e.

As a result, the plasma produces some amount of fusion

power PF. Meanwhile, power exits the plasma primarily

through two possible mechanisms: bremsstrahlung radi-

ation PB, or thermal conduction loss PL. (We neglect for

now other forms of radiation, such as electron cyclotron

radiation, that depend on the magnetic ﬁeld. We also

assume that bremsstrahlung is not reabsorbed, which is

a safe assumption in the relatively low-density plasmas

typical of steady-state reactors.) In steady state:

PH+PF=PB+PL.(1)

The relative balance between these terms is determined

by the physics within the reactor. Finally, the power

that exits the plasma is converted back to into electri-

cal power, with in general diﬀerent eﬃciencies ηBand

ηLfor bremsstrahlung and thermal conduction loss re-

spectively, resulting in a ﬁnal output electrical power

Pout =ηBPB+ηLPL. Economical fusion energy requires

that Pout exceed Pin, preferably by a large margin.

The power ﬂow here closely resembles that used in

Wurzel and Hsu’s recent analysis of progress towards fu-

sion energy [21]. There are three main diﬀerences here.

First, we have simpliﬁed the analysis of the heating en-

ergy by considering only a single conversion eﬃciency.

Second, we have explicitly separated out the electrical

energy required for conﬁnement. Third, we have divided

the output power into two streams with diﬀerent electri-

cal conversion eﬃciencies. This last change reﬂects the

fact that the aneutronic p-B11 reaction produces charged

products, allowing for direct conversion of energy from

lost particles, which has the potential to be much more ef-

ﬁcient than the thermal processes likely required for con-

version of bremsstrahlung energy. Thus, keeping track of

PH,e

PC,e

Pout

ηH

ηB

ηL

FIG. 1. Simpliﬁed power ﬂow model for a fusion power plant. Electrical power Pin is split, with a portion PC,e supporting

the conﬁnement, and a portion PH,e going to heating. The electrical heating power is delivered with some eﬃciency ηHto

the plasma, resulting in PHof delivered heating power. This results in fusion power PF. The hot plasma sustains power

losses through thermal conduction losses PLand bremsstrahlung PB, which are converted into output electrical power Pout

with eﬃciencies ηLand ηBrespectively. The plasma and nuclear physics deﬁne a relationship between PH,PF,PL, and PB.

Engineering and technological considerations determine the various power conversion eﬃciencies η’s, as well as the power used

for conﬁnement PC,e. For a successful power plant, Pout > Pin .

how power leaves the plasma is important to the overall

energetic analysis.

The power leaving the reaction due to lost particles

(i.e. thermal conduction) is generally written in terms of

the conﬁned kinetic energy density UKand the energy

conﬁnement time τE:

τE≡UK

.(2)

Note that the power used to calculate this conﬁnement

time does not include the bremsstrahlung radiation PB.

This formulation is convenient, as it generally leads to

a requirement on the (temperature-dependent) product

of density and conﬁnement time nτE, which is a use-

ful fundamental target for fusion technology. Achieving

Pout > Pin with a physically realizable nτEis the funda-

mental challenge of fusion energy science.

To measure the progress towards fusion, one generally

looks at the Qfactor. There are several relevant Qfac-

tors on the road towards economical fusion energy. The

ultimate goal is for a power plant to produce net power

on the grid, determined by condition on the engineering

Qeng:

Qeng ≡Pout −Pin

>0.(3)

The higher Qeng, the greater the ratio of power applied

to the grid to recirculating power in the reactor.

Since we are looking at fundamental limits of the fusion

eﬃciency, we will here consider a modiﬁed version of this

metric, where we neglect the power used for conﬁnement,

i.e. assume PC,e = 0. We denote this modiﬁed Qas Q∗

eng.

Then:

Q∗

eng = ¯η(Qfuel + 1) −1,(4)

where we have deﬁned a quality factor associated with

the fuel:

Qfuel ≡PF

,(5)

and the average power recycling eﬃciency:

¯η≡ηHηL

PL+PB

+ηB

PL+PB<1.(6)

High Qfuel is not a strictly necessary condition for net

electricity production, if there is high recycling eﬃciency

in the plasma. Inverting Eq. (4) and demanding Q∗

eng >0

shows that net electricity production only requires:

Qfuel >1

¯η−1,(7)

which can be small if the recycling eﬃciency is large, as

can be the case with eﬃcient direct conversion. Never-

theless, achieving large values of Q∗

eng generally requires

achieving even larger values of Qfuel, making Qfuel a use-

ful physics-based metric for the plasma performance.

A. High-Performance Plasmas

If we want to focus on very high-performing plasmas,

then, our goal is ultimately to obtain Qfuel → ∞. This

limit represents the state where the fusion reaction sus-

tains itself without the need for external heating, known

as burning plasma.

To look at what is necessary to achieve burning plasma,

we use Eqs. (1), (2), and (5) to rewrite Qfuel as:

Qfuel =PF

PB+UK/τE−PF

.(8)

Here, PF,PB, and UKare all determined immediately

by the plasma parameters (the densities nsand tempera-

tures Tsof the species present), while τEdepends on the

details of the reactor design. However, as a general rule,

greater niτEis harder to achieve.

Thus, as a general performance metric, we deﬁne τ∗

the minimum value of τE(at a ﬁxed ni) that is required

to achieve Qfuel → ∞. From Eq. (8), this is given by:

τ∗

E≡τEQfuel →∞ =UK

PF−PB

.(9)

B. The Dual Challenges of Thermonuclear p-B11

Fusion

The quantity τ∗

Esuccinctly captures two main chal-

lenges that make proton-Boron 11 thermonuclear fusion–

i.e. fusion with all species approximately Maxwellian–

comparatively diﬃcult.

First, we see from Eq. (9) that Qfuel → ∞ requires the

fusion power to exceed the bremsstrahlung power. This

has historically been a problem for p-B11 fusion, which

requires large ion temperatures (∼300 keV), and thus

produces substantial bremsstrahlung, leading some to

conclude that thermonuclear p-B11 fusion was infeasible

[2, 3]. However, recent studies have indicated that the p-

B11 cross section, particularly at high energies, is larger

than previously thought [13]. A full energetic analysis by

Putnvinski et al., considering collisional energy transfer

between the various plasma species, revealed that these

new cross sections opened up a small window where the

fusion power could slightly exceed the bremsstrahlung

power, making burning plasma theoretically achievable

[14]. However, the margin by which PFexceeds PBis

only a few percent at the most optimal parameters, mak-

ing the prospect of burning plasma very diﬃcult to envi-

sion with that energy balance.

Second, even in the absence of bremsstrahlung, the

small cross section and high temperatures required for

the reaction put a stringent limit on the conﬁnement

time. To see this, note that the typical temperature of

the reactants is around 300 keV (with around 150 keV for

the electrons), while the typical fusion power per density

product is:

Ufus hσpBvi= 4 ×10−9eV cm3/ s.(10)

Thus, even if bremsstrahlung were somehow suppressed,

the maximum allowable τ∗

Eat the optimal density

nB/ni= 0.15 where ni=np+nB, is:

τ∗

E=3

niTi+ (nBZB+np)Te

npnBUfus hσpBvi∼(16 s) 1014 cm−3

ni.

(11)

Thus, at typical ITER densities, even in the absence of

bremsstrahlung, the required energy conﬁnement time is

on the order of 16 seconds. Given the results of Putvin-

ski et al.[14], the presence of bremsstrahlung makes this

requirement ∼34 times more stringent, i.e. τE∼540

seconds.

III. INTERNAL POWER BALANCE

The stringent requirements for thermonuclear p-B11

fusion encourage a consideration of nonthermal plasmas.

To examine the potential advantage such plasmas pro-

vide, we explore a power balance model similar to Putvin-

ski et al.[14], incorporating collisional temperature equi-

libration between species, fusion power production, and

collisional transfer of alpha particle energy to the vari-

ous thermal species. However, to this balance of thermal

protons p, boron b, and electrons e, we add a beam of

monoenergetic fast protons f. These fast protons can be

maintained either by external energy input, or by using

alpha channeling to transfer alpha power directly to the

fast protons. The power balance model thus takes the

form:

dUf

dt =−KfpEf−KfbEf−KfeEf

−KF,f Ef+αfPα+PH(12)

dUp

dt =KfpEf+Kpb(Tb−Tp) + Kpe(Te−Tp)

−3

2KF,pTp+αpPα−γpPL(13)

dUb

dt =KfbEf+Kpb(Tp−Tb) + Kbe(Te−Tb)

−3

2(KF,f +KF,p)Tb+αbPα−γbPL(14)

dUe

dt =KfeEf+Kpe(Tp−Te) + Kbe(Tb−Te)

−PB+αePα.(15)

Here, we recognize the heating power PH, thermal con-

duction loss power PL, and bremsstrahlung power PB.

This last can be approximated as[14, 22]:

PB≈7.56 ×10−11n2

ex1/2Zeﬀ 1+1.78x1.34

+ 2.12x1+1.1x−1.25x2.5eV cm3/s, (16)

where x=Te/Erest,Zeﬀ =PiniZ2

i/PiniZi, and

Erest = 5.11 ×105eV is the electron rest energy.

In Eqs. (12-15), we have also deﬁned many new vari-

ables.

First, Kss0for s, s0∈ {f, p, b, e}represents the en-

ergy transfer rate between species sand s0. These rates

are standard, but for completeness are described in Ap-

pendix A.

Second, KF,f and KF,p represent the fusion rate from

fast and thermal protons respectively. These rates in-

clude a kinetic enhancement factor to agree with Putvin-

ski et. al.[14] in the appropriate limits, and are described

in Appendix B.

Third, in agreement with Putvinski, we assume that

power is lost through thermal conduction only from the

ions. Since we track both boron and proton tempera-

tures separately, we must choose how to partition this

loss, which is encoded in the parameters γpand γb. We

assume that thermal losses in each thermal ion species

occur proportionally to that species’ pressure:

γi≡niTi

PjnjTj

, i, j ∈ {p, b}.(17)

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ImprovingtheFeasibilityofEconomicalProton-Boron11FusionviaAlphaChannelingwithaHybridFastandThermalProtonSchemeIanE.Ochs,ElijahJ.Kolmes,MikhailE.Mlodik,TalRubin,andNathanielJ.FischDepartmentofAstrophysicalSciences,PrincetonUniversity,Princeton,NewJersey08540,USA(Dated:October18,2022)Theproton-Boron11...

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Improving the Feasibility of Economical Proton-Boron 11 Fusion via Alpha Channeling with a Hybrid Fast and Thermal Proton Scheme Ian E. Ochs Elijah J. Kolmes Mikhail E. Mlodik Tal Rubin and Nathaniel J. Fisch

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