via QFT Effects from Parallel Electric and Magnetic Fields 22 April 2016 Gerald B Cleaver Head Early Universe Cosmology amp Strings Division Center for Astrophysics Space Physics amp Engineering Research ID: 602675
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Slide1
Matter-Antimatter Propulsion via QFT Effects fromParallel Electric and Magnetic Fields22 April 2016
Gerald B. CleaverHead, Early Universe Cosmology & Strings DivisionCenter for Astrophysics, Space Physics, & Engineering ResearchBaylor UniversityWaco, TexasSlide2
Matter-Antimatter:I. History:
Discovery & GenerationII. Proposed Method of Spacecraft PropulsionIII. Collection & Storage IV. In Situ Production (QFT-Basis) (i) From Laser Interactions(ii) From Parallel Electric and Magnetic Fields(iii) Enhancement from Pulsed Electric Fields
V. Derivation of Parallel E & B EnhancementSlide3
I. History🚀 Particle/anti-particle pair production (PAP PP)
from vacuum via strong E-fields investigated for century🚀 QM foundation developed by Fritz Sauter et al. in1930’s 🚀 Placed on sound QED basis by Julian Schwinger in1951🚀 Pair production occurs when electric field strength E
0 above critical value E
*, Schwinger limit (SL), at which becomes non-linear with
self-interactions Slide4
🚀 SL when energy w/in Compton wavelength = rest mass of an electron, E* = m2c3/(
eħ) = 1.3 x 1016 V/cm 🚀 Corresponding SL intensity I* = 2.1 x 1029 W/cm2
🚀
As energy density of x-ray lasers approach SL critical strength, feasibility/functionality
of electron-positron pair production gaining interest. 🚀 Laser
intensities
within
1 order
of magnitudeSlide5
🚀 Physical processes for lowering critical energy density below the SL (& enhancing pair production above SL) through additional QM effects being explored. 🚀 U. of Connecticut/U. of Duisburg-Essen
examining pulsation of inhomogeneous E fields within a carrier wave. 🚀 Or enhancement via QM effects w/ addition of a magnetic field B parallel to the electric field E. Slide6
🚀 Magnetic field enhancement to quark/anti-quark pair production through QCD chiral symmetry breaking effects investigated theoretically by John Preskill at Caltech in1980’s
🚀 Don Page at the U. of Alberta showed in 2007 B fields parallel to E fields also enhance electron/positron pair production via an analogous QED effect, w/ enhancement going predominantly as a linear function of B0/E0,Slide7
🚀 Particle/antiparticle pair production as a highly efficient fuel source for intra solar system and interstellar propulsion proposed by Devon Crow in 1983 🚀 Viability
of this method of propulsion reviewed, esp. from the parallel electric and magnetic field methodSlide8
🚀 Note: Particle/anti-particle pair production does not (and cannot) take energy from the spacetime vacuum. 🚀 Energy is drawn from the external electric (and magnetic) fields.
Process is somewhat analogous to particle production near the event horizon of a black hole, which reduces the mass of the black hole accordingly. 🚀 Primary difference is, while both particle and antiparticle are produced from virtual pair by the electric and magnetic fields, only one particle of an initially virtual pair escapes from a black hole (as Hawking radiation) and antiparticle is captured by black hole
.Slide9
9
Left-handed versions (spin in direction opposite of its motion)Right-handed versions (spin in same direction of its motion)= Left-handedAnti-Particle
Matter/Antimatter& ParitySlide10
10 🚀 1928: British physicist P.A.M. Dirac showed that Einstein's relativity implied every particle has corresponding antiparticle, same masses, but opposite electric charges.
🚀 1932: Carl Anderson at Caltech recorded positively charged electron (positron) passing through lead plate in cloud chamber (for which he received Nobel prize).Matter/AntimatterSlide11
11 🚀 1955: Antiproton experimentally confirmed at Berkeley by Emilio Segre and Owen Chamberlain (earning1959 Nobel prize). 🚀
1956: Antineutron discovered at the Bevatron at Lawrence Berkeley Nation Lab by Bruce Cork and colleagues.Slide12
🚀 1995: CERN researchers use Low Energy Antiproton Ring (LEAR) to slow down antiprotons. Managed to pair positrons and antiprotons together, producing nine hydrogen anti-atoms, each lasting a mere 40 nanoseconds.
🚀 Within 3 years, CERN group producing approx. 2000 anti-hydrogen atoms per hour. 🚀 Production rates of antimatter at LHC have increased significantly since then (as did past production rates at Fermilab)
12
Slide13
🚀 Matter/Antimatter [MAM] is ideal rocket fuel because all of mass in MAM collisions is converted into energy.
🚀 MAM reactions produce 10 million times the energy produced by conventional chemical reactions used to fuel the space shuttle, 1,000 times more powerful than nuclear fission produced at a nuclear power plant & 300 times more powerful than the energy released by
nuclear fusion.
13
Slide14
🚀 Should an ample supply of antimatter be produced or collected, a secure means of storage (i.e., magnetic confinement) would then be devised; the antimatter
must be kept separate from matter until a spacecraft needs more power, unless stored as anti-hydrogen (positronium annihilates within minutes) and/or MAM created in situ and immediately emitted as propellant.
14
Slide15
Then why hasn’t MAM spacecraft propulsion systems been developed: 🚀 Antimatter remains most expensive substance on Earth. In 2000 it cost $62.5 trillion a microgram ($1.75 quadrillion an ounce) of electron/positron pairs. Fermilab was producing
about 15 nanograms a year.🚀 However, price drops with each advancement in particle accelerator intensity and efficiency, The LHC now produces about 1 microgram ~ 1021 electron/positron pairs per12 days at a cost of $200,000 or 1 milligram in about 12,000 days ~ 30 years at a cost ~ two-hundred-billion dollars. Slide16
🚀 To be commercially viable, this price would need to drop by about a factor of Ten-Thousand.” As quoted in ”Status of Antimatter,” NASA Glenn Research Ctr.www.nasa.gov/centers/glenn/technology/warp/
antistat.html (dated 14 July 2015). 🚀 This goal could be reached within a decade or two. This was the projected time scale expected by some back in the 1980’s.16 Slide17
🚀 Much more antimatter needed for interstellar mission (& for planet reconnaissance/landing mission, will need enough fuel to decelerate into target star system). 🚀 Starship with a 100-ton payload designed to cruising at 0.40 c estimated to require equivalent of 80 ocean supertankers full of antimatter fuel.
🚀 For cruise speed ~ 0.25 c, fuel requirements dramatically lower (news.discovery.com/space/ harvesting-antimatter-in-space-to-fuel-starships-120523.html#mkcpgn =rssnws1 )--R. Obousy, JBIS 64 (2011) 378.
17Slide18
Early Papers:🚀 R. Forward, Antiproton Annihilation Propulsion, USAF Rocket Propulsion Laboratory Report AFRPL, 1985.🚀 Devon Crowe, Laser Induced Pair Production as a
Matter-Antimatter Source, JBIS 36 (1983), 507.🚀 G. Schmidt et al., Antimatter Production for Near-tern Propulsion Applications, AIAA 99-2691, NASA Marshall Space Flight Center
18
Slide19
Early Papers:🚀 In 2000 NASA scientists announced early designs for an antimatter engine that might be capable of fueling a spaceship for a trip to Mars using only a milligram of MAM.
19 Slide20
🚀 2012: R. Keane and W.M. Zhang examined magnetic nozzle design for charged pion emission from quark-antiquark collisions. Showed effective exhaust speeds ~ 0.7 c feasible. Efficiency ~ 30% for quark/antiquark emission
> 30% for electron/positron emission🚀 They optimized geometry and field configuration of nozzle using a magnetic field on order of 10 T. --R. Keane and W.M. Zhang, “Beamed Core Antimatter Propulsion: Engine Design and Optimization”, arXIv:1205.2281v2.
20
Slide21
Alternative to MAM Generation:
🚀 2011: antiprotons discovered trapped by Earth's magnetic field by the international PAMELA (Payload for Antimatter/Matter Exploration and Light-nuclei Astrophysics) satellite. The Alpha Magnetic Spectrometer on ISS also able to detect, identify, and measure antiparticles in Earth orbit. 21Slide22
Alternative to MAM Generation:
🚀 Theoretical studies suggest that the magneto- spheres of much larger planets, like Jupiter, should have more antiprotons than Earth. 🚀 "If feasible, harvesting antimatter in space would completely bypass the obstacle of low energy efficiency when an accelerator is used to produce antimatter,” – R. Keane and W.M. Zhang 22Slide23
MAM Propulsion Systems:🚀 However, an ideal MAM propelled spacecraft should contain systems for both
collecting and generating MAM, with creation especially as an emergency option if stored MAM leaks out of magnetic containment chambers or is annihilated prematurely by matter leaking in. 23
Slide24
In Situ MAM GenerationSchwinger Pair production of spin-1/2
fermions from the vacuum through intense electric field quantum effect🚀 F. Sauter, Z. Phys 69 (1931) 742🚀 W. Heisenberg and H. Euler, Z. Physics 98 (1936) 714. 🚀 V. Weisskopf and K. Dan Vidensk, Selsk. Mat. Fys
.
Medd. XIV (1936) #6 🚀 J. Schwinger, Phys. Rev. 82 (1951)
(putting pair production on sound QED basis)
24
Slide25
🚀 For electron-positron production, the critical value of the electric field strength is above the Schwinger limit defined by E* = m
2c3/(eħ) = 1.3 x 1018 V/m. with m and e the mass and |charge| of the electron. At this scale the electromagnetic field (vacuum) becomes non-linear.
25Slide26
🚀 In vacuum, classical Maxwell's equations perfectly linear differential equations. This implies – by the superposition principle – that the sum of any two solutions to Maxwell's equations is yet another solution to Maxwell's
equations. E.g., two beams of light pointed toward each other should simply add together their electric fields and pass right through each other. 🚀 In QED, however, non-elastic photon–photon scattering becomes possible when the combined energy is large enough to create virtual electron–positron pairs spontaneously,
26Slide27
🚀 When the average strength of an electric field E0
is above Es = 1018 V/m , the pair production rate of charged particles per unit time and unit cross-section found from QM computation of probability of “tunnelling” of virtual pairs from Dirac sea (a.k.a., instanton calculations)
🚀
Strongest lasers produce electric fields ~ 1017
V/m.
27Slide28
Matter/Anti-Matter (MAM) Production🚀 X-ray free electron lasers from Linac Coherent Light Source at SLAC and TESLA at DESY approaching E
S .🚀 The Extreme Light Infrastructure (ELI) Ultra-High Field Facility (4th site) planned for Eastern Europe around 2020 should also reach E*. (Ten lasers concentrating 200 petawatts of power into a very narrow beam for around 10-12 s pulses.).
28Slide29
Matter/Anti-Matter (MAM) Production🚀 Via localized electric fields: S. Kim and D. Page,Phys. Rev. D75 (2007) 103517.
Consider a static plane-symmetric z-dependent electric field E(z) in the z-direction of maximum value E0 and of effective length L
such that E
0L = ½ ∫E(z)
dzallows pair production of a particle of mass m and charge q if
ε
= m/(qE
0
L) < 1
equivalently
E
0
> m/(
qL
)
(in natural units of
G
N
= c =
ħ
= 1
).
29Slide30
Matter/Anti-Matter (MAM) Production🚀Alternately, if we want a time varying field E(t) rather than spatially varying, replace
ε with εT , L with T, and dz with dt. In this case, pair production occurseven with εT > 1, but is suppressed.
30Slide31
Matter/Anti-Matter (MAM) Propulsion Systems:🚀 In each process when E0
is above the minimumvalue, PPR of charged particles per unit time andunit cross-section found from tunnelling of virtual pairs from the Dirac sea where instantonsdetermine the QM tunnelling probabilities. To leading WKB order:For “Sauter” electric field E(z) = E0
sech[2(z/L)], the PPR is
N = (qE0
)5/2L (1-ε2
)
5/4
exp[-Z{
1
-(
1
-ε
2
)
1/2
}]
/(4 π
3
m )
~
(qE
0
)
5/2
L/(
4 π
3
m
) as
ε
0
with
ε
= m/(qE
0
L)
and
Z = 2πqE
0
L
2
31Slide32
🚀 Can lower minimum value of E0
significantly by adding constant magnetic field B parallel to E0: PPR of charged particles per unit time and unit cross-section is modified (as derived by Don Page) to (Note: in natural units [B] = [E], since c = 1 rather
than [B] = E/c). In these matching units,
B0
> E*
realistically possible,
N
B
= (B/
E
0
)(qE
0
)
5/2
L (1-
ε
2
)
3/4
exp[-Z{1-(1-ε
2
)
1/2
}]
*
coth
[πB/E
0
(1-ε
2
)
1/2
]/(4 π
2
m)
~
(B/
E
0
)
(qE
0
)
5/2
L
coth[πB/E
0
]/
(4 π
2
m)
(
as
ε
-> 0
)
= (π
B/
E
0
)
coth
[πB/E
0
] N
Slide33
🚀 In the limit of B -> 0, NB reduces to N
. Allows significantly weaker E0. [Idea of B || E first presented in J. Preskill, 1987 lecture notes; Proposed for Spacecraft Propulsion by Cleaver in 100 YSS Proceedings]
33Slide34
34Orders of magnitude for Magnetic Fields to Enhance Rates
B ~ Es/c = (2 ・ 1018
V/M)/(3 ・
10
8 m/s) = 109 to
10
T
Or find from using B-field to reduce effective mass of electron (quark)
(2n
+ 1
– g/2)
ħ
e c
2
B
+
(mc
2
)
2
= 0
With n = 0, get
for electron/positron pair production:
B = (0.5
・
10
6
eV)
2
/[0.001 x 4
・
10
-15
eV s e (3 ・ 10
8
m/s)
2
] = 10
12
T
for quark/anti-quark
pair production
:
B =
(2.4
・ 10
6
eV)
2
/[0.4 x
4 ・ 10
-15
eV s e (3 ・ 10
8
m/s)
2
] =
10
11
T
(*Note: Actually pion/anti-pion production rather than quark/antiquark raises to above B for e/p.)
So order of magnitude estimates indicate B ~ 10
9 to 12
T ~ magnetic field strength of a
magnetar
!Slide35
35List of orders of magnitude for magnetic fieldsFactor (tesla)
Value (SI units) Item10−15 2 fT SQUID magnetometers on Gravity Probe B gyros measure fields 10−12 1
pT Human brain magnetic field
10−9
10 nT Magnetic field strength in the heliosphere
10
−6
24
µT Strength
of
magnetic
taoe
near tape head
10
−5
31
µT
Strength of Earth’s magnetic field at equator
10
−3
0.5
mT
The
suggested exposure limit
for cardiac pacemakers
5
mT
The
strength of a typical
refrigerator magnet
10
−1
0.15
T
The
magnetic field strength of
a sunspot
10
0
2.4
T
Coil
gap of a
typical loudspeaker magnet
9.4
T
Modern
high resolution research
magnetic resonance imaging
16
T
Strength
used to levitate a
frog
45
T
Strongest
continuous magnetic field yet produced in a
laboratory (FSU)
10
2
300
T
Strongest
pulsed non-destructive magnetic field yet produced in a
lab, LANL
730
T
Strongest
pulsed magnetic field yet obtained in a laboratory, destroying the
equipment used
(Inst. for Solid State Physics, Tokyo)
10
3
2.8
kT
Strongest
(pulsed) magnetic field ever obtained (
w/
explosives) in a
lab,
10
6
100
MT
Strength
of
a neutron star
10
9 to 12
0.1-100
GT
Strength
of
a
magnetar
10
53
2・10
29
YT Planck magnetic field strength Slide36
🚀 PPR can also be enhanced by a factor of orders 10 to 100 or greater if the electric field (laser in general) is pulsed with internal modulation.
36For the details of this type of PPR enhancement, see for exampleR. Schutshold, H. Giles, and G. Dunne, Phys. Rev. Lett. 101 (2008) 130404; Ibid., Int. J. Mod. Phys. A25 (2010) 2373; C. Schneider and R. Schutzhold, arXiv:1407.3584. S. Kim, H. Lee, and R. Ruffini, arXiv:1207.5213.Slide37
Matter/Anti-Matter (MAM) Production Enhancement:🚀 PPR could be strongly enhanced by simultaneous combination of a pulsed (laser) electric field E with internal modulation combined with parallel magnetic field B.
🚀 The enhancement combination may provide for viable in situ electron/positron and/or quark/antiquark pair production from the vacuum for spacecraft propulsion. As such, it would be made possible by non-linear quantum field theory effects on the electromagnetic vacuum.37Slide38
Chiral Fermion Pair Production From Parallel Electric and Magnetic Fields🚀 While not envisioned as a propulsion source for spacecraft, this basic idea for MAM production viaparallel fields was discussed by John Preskill at
Caltech in the late 1980’s.🚀 The underlying physics behind MAMproduction from parallel electric and magnetic fields is associated with chiral symmetrybreaking (CSB). 38Slide39
Chiral Fermion Pair Production From Parallel Electric and Magnetic Fields🚀 CSB is an effect thatconnects left- and right-handed elementary particles (specifically for quarks) in the strong coupling limit of QCD and/or
distinguishes between lh & rh particles via B-field interaction effects in QEDWhy only the parallel components of the electric and magnetic fields are relevant to this effect will be worked out in the Hamiltonian formalism. 39Slide40
40🚀 Chirality = Handedness (Left-Handed and Right Handed)🚀 Chiral Symmetry = Left-Handed and Right-Handed versions of same particle (equivalently Left-Handed particle and anti-particle) are independent particles (Technically mean the phases of each are independent.)
🚀 Chiral Symmetry Breaking (CSB) = L-H and R-H particles are not independent (phases are correlated & exactly opposite) ~Slide41
41🚀 At high energies (much above a few GeV), when “strong force becomes weak”, quarks have Chiral Symmetry🚀
At low energies (below a few GeV), when “strong force is strong”, quarks experience Chiral Symmetry Breaking ~Slide42
42🚀 CSB allows an interaction term FμνF
μν between the field strength tensor F and its dual field strength tensor Fμν= εμνρσ Fρσ, where indices range over 0, 1, 2, 3🚀 For Electromagnetic force, the field strength tensor components are F01 = -F10
= Ex, F02
= -F20 = E
y, F03 = -F30 = Ez
,
F
12
= -F
21
= -B
z
, F
13
= -F
31
= B
x
, F
23
= -F
32
= B
x
,
🚀
For electromagnetics,
F
μν
F
μν
= E
x
B
x
+ E
y
B
y
+
E
z
B
z
=
E
B
.
(The dot product “
”
indicates that only the parallel components of
E
and
B
interact).
~
~
~
~ Slide43
43🚀 Why this term can result in particle/antiparticle pair production is interesting.To start, consider a spin S = ½ fermions of mass
m and electric charge e in a constant magnetic field B aligned along the z-axis, B = Bz. 🚀 The electromagnetic gauge field producing the physical magnetic field B can be chosen as A = B x y
. The square of the Hamiltonian for a fermion in this field is
H2 = (p
– e
A
)
2
+ m
2
– g e
B
S
= p
x
2
+ p
z
2
+ (
p
y
– e B x)
2
+ m
2
–
geB
S
z
w/
p
the fermion’s momentum (operator), and
g
its
gyromagnetic operator.
⌃
⌃Slide44
44The g gyromagnetic operator is very close to 2:
H2 = px2 + pz2 + (
py
– e B x)2
+ m2 – 2 e B S
z
.
p
y
and
p
z
are
constants
of motion (since they
commute with H). Hence these can be ignored
(eliminated from the Hamiltonian as
constants). Then note that the contribution
p
x
2
+ (eB)
2
( x
– x
0
)
2
, with
x
0
= -
p
y
/(
eB
)
has the form of a simple harmonic oscillator (SHO).
A QM SHO of this form always has stated quantized
e
nergy of the form
(2 n + 1) e B
, where
n
is an
integer. This yields
H
2
= p
z
2
+ p
y
2
+ (2n + 1 - 2
S
z
)e B + m
2
,
with
S
z
= +1/2
or
-1/2.Slide45
45 🚀 So, consider a RH particle (S
z = ½) and the ground state mode (n=0) with no motion in the y direction. Then the Hamiltonian simplifies to: H
2 = p
z2
+ m2
In the massless limit, this indicates a “zero mode”
which even very low energy parallel
E
(not on yet)
and
B
fields can excite, resulting in pair production.
Slide46
46 🚀 In reality, for massive fermion states (quarks,
electrons), the E field must be stronger. (Note that the Sz = -½ fermion has an increased effective mass, an indication of CSB. That is, RH Particle & LH antiparticle production rate increases w/
B, But, LH Particle &
RH antiparticle
production rate decreases w/
B
.
Slide47
47🚀 Turn on an E
field in the B direction, slowly (adiabatically) increasing its strength.
E = E z = -
dA/dt
w/
A
z
=
E t
(gauge choice
A
0
= 0
)
.
Then,
H
2
= (
p
z
– E t)
2
+ m
2
🚀
Energy levels are discrete and, with increasing t,
move along a mass-shell hyperbola.
Slide48
48 🚀 While this process was originally considered by
John Preskill at CIT to produce a quark-antiquark pair, used to produce electron-antielectron pair. The latter is more likely since electron mass ~ 1/5 up quark mass ~ 1/270 pi meson mass
Slide49
49
We can understand what is happening physically by applying the famous “Dirac Sea” concept of bothpositive and negative energy states. In the ground state of a fermion system, all negative energy modes are filled and all positive energy modes areempty. Each mode can be assigned a helicity. Let all have
Sz
= +1/2 (RH).
Positive energy modes with positive (negative) momentum are RH (LH). Theopposite it true for filled negative energy modes.Slide50
50
For an electric field E with sufficient energy density,the negative energy quarks will “jump” across the 2m (~1 MeV for electrons, ~ 8 MeV for up quarks, ~14 MeV for down quarks) gaps separating the negative and positive energy states.The physical realization of this is chiral particle pair production: RH particle (filled energy state)
& LH antiparticle (negative energy state—that is, a hole).
Slide51
51 🚀 A quark/anti-quark pair will either form
uncharged pion state or multiple charged or uncharged pions if pair has sufficient kinetic energy to separate far enough for the potential energy from strong force interaction of the quarks to be greater than the mass of another quark pair. Then another quark/anti-quark pair will pop into existence and a net effect can be a pair of pions of opposite charge. Slide52
52 🚀
More likely, an electron/positron pair will pop into existence. Thus, parallel electric and magnetic fields could be used as MAM generator (a.k.a., chiral fermion pair production) via low energy effects allowed through chiral symmetry breaking or (more likely) through E-field spatial/temporal modulation.
🚀 The charged pion pairs and electron/positron
pairs can be directed by external magnetic fields to produce thrust for a manned or unmanned
spacecraft. Slide53
Conclusions:🚀 MAM production from strong electric field near or above Schwinger limit nearing feasibility.🚀
Enhancement of MAM PPR via inclusion of magnetic field parallel to electric field possibly in combination with enhancement from pulsed electric field with internal modulation. The latter may provides MAM-on-demand propulsion 53Slide54
Acknowledgements:🚀 Sauter, Heisenberg, Euler, Weisskopf, Dan, and Schwinger of course!🚀 R. Schutshold, H. Giles, G. Dunne, C. Schneider, and
S. Kim for series of pulsed E field pair production design & engineering papers.🚀 Richard Obousy, VARIES proposal in JBIS 64 (2011) 378.🚀 John Preskill for B||E idea and his related notes.🚀 Don Page and S. Kim for B||E system pair production rate calculation.
54Slide55
55References: [1] F. Sauter, Z. Phys 69 (1931) 742; W. Heisenberg and H. Euler, Z. Physics 98 (1936) 714; V. Weisskopf and K. Dan Vidensk, Selsk. Mat.
Fys. Medd. XIV (1936) #6.[2] J. Schwinger, Phys. Rev. 82 (1951) 664.[3] R. Schutshold, H. Giles, and G. Dunne, Phys. Rev. Lett. 101 (2008) 130404; Ibid., Int. J. Mod. Phys. A25 (2010) 2373; C. Schneider and R. Schutzhold, arXiv:1407.3584. S. Kim, H. Lee, and R. Ruffini, arXiv:1207.5213.[4] J. Preskill, 1987 QCD Lecture Notes, www.theory.caltech.edu/~preskill/notes.html [5] S. Pyo Kim and D. Page, Phys. Rev. D78 (2008) 103517.
Slide56
56[6] R. Forward, Antiproton Annihilation Propulsion
, USAF Rocket Propulsion Laboratory Report AFRPL, 1985; D. Crowe, JBIS 36 (1983) 507; G. Schmidt et al., Antimatter Production for Near-tern Propulsion Applications, AIAA 99-2691, NASA Marshall Space Flight Center[7] R. Keane and W.M. Zhang, Beamed Core Antimatter Propulsion: Engine Design and Optimization, arXIv:1205.2281v2.