Engineering: Open Access
Review Article
A Review of Lawson Criterion for Nuclear Fusion
Les G. Miklosy*
Software to Spec
Corresponding Author
*
Les G. Miklosy, Software to Spec
Submitted: 2024, Feb 23; Accepted: 2024, Mar 05; Published: 2024, Apr 26
Citation: Miklosy, L. G. (2024). A Review of Lawson Criterion for Nuclear Fusion. Eng OA, 2(2), 01-13.
Abstract
In the pursuit for abundant clean energy from nuclear fusion, the Lawson Criterion is cited by fusion researchers and experimentalists
as the condition to exceed for fusion of atomic nuclei to occur. In his 1955 paper J.D. Lawson analyzed a fusion plasma conforming
to thermodynamic principles at steadystate but with stated omissions and simplifications. Modern fusion researchers developed
many incantations of the Lawson Criterion urging their brand leads to fusion ignition, necessarily to sell a rational fusion
hypothesis and attract research funding. Others have shown how confined fusion in the laboratory cannot satisfy a positive energy
balance due to energy losses from the system. A literature search for "fusion energy balance" shows a suspicious absence of this
sanity-check to verify the energy condition was met. This paper applies conservation of energy from classical thermodynamics to
a fusion plasma and summarizes eleven modern Lawson-like interpretations in a uniform way, doing so shows the requirements
for fusion ignition are not met. Indeed despite billions of speculative dollars spent, sustained laboratory fusion has not been
demonstrated in any experimental apparatus built and tested to date worldwide. Heed the warning, in 1955 John D. Lawson
wrote: “To conclude we emphasize that these conditions, though necessary are far from sufficient. The working cycle that has
been assumed is very optimistic.”
The Fusion Record
The first demonstration of a large terrestrial fusion reaction was
the thermonuclear detonation of Ivy Mike in the Enewetak Attol,
Pacific Ocean November 1 1952. A common measure of power
yield from a nuclear process is the gain factor ratio Q, the power
produced to the power input. The best performing Magnetic
Confinement Fusion (MCF) reactor to date is the JET Tokamak
in the United Kingdom in 1997 with a gain Q = 0.67 (16 MW/24
MW). For Inertial Confinement Fusion (ICF) the National
Ignition Facility (NIF) reported a shot test on 28 September 2013
exceeding the breakeven gain factor of Q = 1. A fusion process in
our sun reliably fuses Hydrogen nuclei producing 1366 W/m2, the
accepted value of the Solar Constant at 1AU (Astronomical Unit)
[1].
Despite billions of dollars in public funds spent on fusion facilities
super computers and experimentation and more venture capital
infusion start-ups, sustained fusion has defied experimenters in any
laboratory setting on the planet. In full view of the record, fusion
pioneers like the Lawrence Livermore National Laboratory NIF
exaggerate results in order to mask the failure record of confined
nuclear fusioni The NIF project director boasted that at 5:15 a.m.
on 28 September 2013 the facility achieved an energy gain of 1.4
later disputed by astute observers as actually 0.0077 [2].
For the first time in the history of controlled fusion research, a DT
Eng OA, 2024
plasma has produced more energy (14 kJ) than was supplied to it
(10 kJ). That achievement has led to the first-ever positive fuelenergy balance achieved in a laboratory (defined as fusion energy
exceeding input energy to the fuel) [3].
A new renaissance in physics questions the validity of theoretical
physics in astronomy, cosmology and quantum foundations
when researchers “act based on the conviction that mathematical
structures don’t correspond to measurable entities” [4]. At
her blog Backreaction Sabine Hossenfelder begs a return to
foundations in quantum mechanics leaving shut-up and calculate
for religious evangelists of beautiful math.v In this paper we
review popular fusion success criteria and their disagreement
with well established fundamentals in both thermodynamics and
plasma physics using quantifiable calculations. This paper is not
about modeling the transport properties of physical phenomena
such as laser preheating, Rayleigh- Taylor and other instabilities,
or the hydrodynamics of multi-state phenomena. We simply seek
what fundamentals of themodynamic principles say about fusion
ignition for an ICF target.
Gain Factor Q
A fusion reactor containing a fusion plasma must recirculate the
products of reaction in a thermal process to produce useful net
energy. For fusion energy to be useful as a power generating
resource, the reaction rate must be high enough and the process
Volume 2 | Issue 2 | 1
recirculation requirement low enough to maintain the fusion
reaction. The process must also sustain the reaction at steadystate, when the energy gain rate dW/dt is zero and still drive turbo
machinery. Q is the gain factor for the fusion reaction, in terms
of power generated Q = power produced / power input >> 1 . We
will return to the gain factor but first it is helpful to identify these
energy ratios from a early analysis of a fusion plasma process.
energy required to heat the gaseous fuel is 3nkT where n is the
nuclei count density for electrons and ions uniformly and accounts
for the 3 factor rather than 3/2, k the Boltzmann constant, T is
the absolute temperature [K]. For a confinement time 't' Lawson
argued the power to heat the confined plasma is 3nkT/t. As used in
this paper in consistent units
L. G. Miklosy
Manuscript for Sub
W = 3nkT [MeV/cc].
Software to Spec
J.D. Lawson Criterion
L. G. Miklosy
Manuscript
for Submission
30 April 2020
To understand
the Lawson Criterion it is useful to revisit
the origin
Bremsstrahlung Energy
Software
Bremsstrahlung
of the
term. IntohisSpec
classified 1955 paper J.D. Lawson derived The energy
lost by radiation is Energy
given by the power radiated per
conditions for sustaining nuclear fusion through the simplified unit volume as
The energy
by
radiation
is given by the power radiate
1/2
power
balanceofofa fusion
a fusionplasma
plasma assumed
assumed atatsteady-state
[6]. vi Lawson
n2 Tcomponents
[Watts/cc],
PB =lost
1.4
10-34
balance
steady-state.
calculated
thex energy
of the
Lawson
calculated
theand
energy
components
of the power
balance
power
balance
specified
the internal
energy
as a function of confinement time,
particle
density
and
2 1/2
PB count
= 1.4density,
x 10-34Tnis
Tabsolute
[Watts/cc],
and specified the internal energy as a function of confinement time, n is the nuclei
temperature [K] with
temperature. Lawson reasoned two cases, the plasma in a star where the reactants and disintegration
particle density and temperature. Lawson reasoned two cases, the care given to units conversion.
products
dowhere
not escape,
or inand
a proposed
"pulsed
system" where plasma fuel reactants instantaneously
plasma
in a star
the reactants
disintegration
products
n is the nuclei count density, T is absolute temperature [K
riseescape,
in temperature
from an
external
energy
source
disintegration
products
do not
or in a proposed
"pulsed
system"
where
plasmaandSteady-State
Energy
Balance escape the system as
Lawson
then
developedfrom
a criteria
for fusion
occur in and
a "useful
reactor"are
hosting
DD infers
fuel energy
reactantslosses.
instantaneously
rise
in temperature
an external
After to
assumptions
simplifications
made, aLawson
Steady-State
Energy
Balance
energy
source
and disintegration
productshot-dense
escape theplasma.
system as the energy balance as steady-state. Lawson does not compute the
or DT
(Deuterium
and Tritium)
energy losses. Lawson then developed a criteria for fusion to occur internalAfter
energy
from an energy
but rather are
specifies
theLawson
assumptions
andbalance
simplifications
made,
in a "useful reactor" hosting a DD or DT (Deuterium and Tritium) plasmaLawson
internal energy
as
3nkT,
PR
is
the
fusion
power
due
to an en
does not compute
theasinternal
energy from
During this development Lawson states his assumptions and makes deliberate
omissions
follows.
hot-dense plasma.
fusion products, PB is the power lost by Bremsstrahlung radiation.
internal energy as 3nkT, PR is the fusion
power due to fus
He makes explicit assumptions:
From Lawson's paper in the original nomenclature
and consistent
Bremsstrahlung
radiation.
From
Lawson's
paper in the or
•
Radiation
is
lost
in
Bremsstrahlung
radiation
During this development Lawson states his assumptions and makes units
• omissions
Ion andaselectron
temperatures
the same
deliberate
follows. He
makes explicitare
assumptions:
3nkT
[MeV/cc]
• Radiation
in Bremsstrahlung
radiation at steady-state
• isAlost
fixed
number of particles
2
• IonSome
and electron
temperatures
are
the
same
P
=
1/4
n
<σv>(T)
E
[MeV/cc-s]
deliberate omissions are made for simplification:
R
-34 2 1/2
• A fixed•number
of
particles
at
steady-state
PB = 1.4 x 10 n T [MeV/cc-s].
Thermal conduction through the plasma or apparatus
Some deliberate omissions are made for simplification:
• Neutron energy is not reabsorbed but lost from the apparatus
• Thermal conduction through the plasma or apparatus
Since neutrons are lost from the system, he accounts for alpha
Since neutrons are lost from the system, he accounts for a
Some
poweris terms
are neglected
altogether:
• Neutron energy
not reabsorbed
but lost from
the apparatus
products alone ( 3.5 MeV) in PR, t is the confinement time and
confinement
time and repeated here
•
Power
inputs
to
the
sample
from
external
energy
input
repeated
here
Some power terms are neglected altogether:
Losses
from mechanical
• Power •inputs
to the sample
from external PdV
energywork
inputdone by the sample gas
=BtP
• Losses from mechanical PdV work done by the sample gas
3nkT = tP3nkT
. R – tPB.
R – tP
Fusion Energy
Fusion Energy
Lawson
gave
the release
energyfrom
release
Lawson
gave the
energy
fusionfrom
as fusion as
PR = n1n2 <σv> (T) E [MeV/cc-s]
Original
Lawson Criterion
Original
Lawson Criterion
Lawson reasoned a useful energy parameter 'R' to be “energy
Lawson
reasoned
usefulsupplied”
energy parameter
to be “ene
released
in the hot
gas to thea energy
meaning the'R'
ratio
of alpha
fusion
products
to
energy
supplied
from
internal
plasma
supplied” meaning the ratio of alpha fusion products to e
energy plus
plus Bremsstrahlung
radiation:
Bremsstrahlung
radiation:
and equivalent to modern derivationsvii where viithe nuclei count
andn1
equivalent
to modern derivations where the nuclei count density n1 R= =n2P=/n/2,
and for DT n1n2 =
(3nkT + tPB)
density
= n2 = n/2, and for DT n1n2 =n2/4. The product of
2
t R
/
(3nkT
+
tP
R
=
tP
n
/4.
The
product
of
nuclei
relative
velocity
and
cross
section
<σv>
(T)
averaged
over
the
R is
B)
nuclei relative velocity and cross section <σv> (T) is averaged
where
velocity
absolute
temperature
T [K], E is the energy released by one
where
overMaxwellian
the Maxwellian
velocitydistribution
distribution atatabsolute
temperature
plasma
energy
reaction.
Lawson
cites by
theone
total
energy
density
to a 3nkT=
fusion
source
17.6 MeV,
for alpha
internal
energyisinternal
T [K],
E is the energy
released
reaction.
Lawson
citescontributing
the 3nkT= plasma
PR = alpha
fusion
product
power
totalnuclei
energy 3.5
density
contributing
to a fusion
is 17.6
MeV, for
PRpercent
= alpha
fusion
product
power
MeV,
for neutrons
14.1source
MeV.
Neutrons
comprise
eighty
of
the fusion
energy
but
PB
=
Bremsstrahlung
radiation
power
alpha
nuclei
3.5
MeV,
for
neutrons
14.1
MeV.
Neutrons
comprise
power
escape the system, only the alphas contribute to plasma heating aPknown
condition andradiation
accounted
for in
B = Bremsstrahlung
t = confinement
time
[s]p79). viiitime [s]
eighty
percent of the
fusion
energy but
escape
system,
t =2004,
confinement
calculations
since
Lawson's
time
(see the
Atzeni
andonly
Meyer-Ter-Vehn,
the alphas contribute to plasma heating a known condition and n = nuclei count density [1/cc]
n = nuclei count density [1/cc]
accounted for in calculations since Lawson's time (see Atzeni and k = Boltzman constant
= Boltzman
Plasma
Internal
Energy
T = absolutektemperature
[K].constant
Meyer-Ter-Vehn, 2004, p79) [8].
T = absolute temperature [K].
In a pulsed fusion energy system Lawson reasoned the heating energy required to heat the gaseous fuel
‘R’ is NOT the "Lawson Criterion" so often found in simplifications
Plasma Internal Energy
is 3nkT where n is the nuclei count density for electrons andterm,
ions and
uniformly
and accounts
for the 3factor ‘Q’ the
notthe
the "Lawson
definition of
fusion gain
In a pulsed fusion energy system Lawson reasoned the heating of the ‘R’
is NOT
Criterion"
so often found in sim
factor rather than 3/2, k the Boltzmann constant, T is the absolute temperature [K]. For a confinement
of fusion gain factor ‘Q’ the quotient of power produced
't' Lawson argued the power to heat the confined plasma is 3nkT/t. As used in this paper
in 2 | Issue 2 | 2
Volume
Eng time
OA, 2024
the fusion reaction does not heat
the gas
directly noting "
consistent units
the gas initially must be heated through energy recirculat
W = 3nkT [MeV/cc].
available as energy production. Lawson chose ŋ = 1/3 as a plausible value the system apparatus could
achieve to recirculate energy to sustain the plasma, values below this ratio are desirable, values above it
undesirable. As Lawson defined it the energy recirculated is the plasma internal energy plus the
Bremsstrahlung radiation loss ‘3nkT + tPB’.
In 1955
Lawson
cautioned
all input.
these In
conditions
though necessary are far from sufficient where he wrote
quotient
of power
produced
to power
1955 Lawson
1/(R+1)
< ŋ or the
Lawson Criterion
1 < ŋ(R+1).
recognized
fusion reaction
does nothas
heatbeen
the gas
directly noting
"The the
working
cycle which
assumed
is very optimistic".
Lawson
reasoned
the quotient
of the
"disintegration
products
escape".
He
reasoned
the
gas
initially
energy supplied to the total energy of the system is 1/(R+1). The Lawson Criterion says this ratio must
must be heated through energy recirculation by the experimental Lawson develops the dimensionless energy ratio ‘R’ in terms of
be less than the heating efficiency η:
confinement parameter ‘nt’ and absolute temperature T or R(nt, T),
apparatus.
he charts the value of R for DT DD reactions over temperatures
1/(R+1)
ŋ orathe
Lawson
Criterion
1 as
< ŋ(R+1).
In 1955 John
Lawson<defined
heating
“efficiency”
ŋ (Eta)
the and confinement values:
portion of total energy recirculated to heat the plasma. Lawson
2 < Log10
(T) < 6 [eV]
reasoned
the more
energy required
to heat the plasma
in proportion
Lawson
develops
the dimensionless
energy
ratio ‘R’ in terms of confinement
parameter
‘nt’ and
14
<
Log10
(nt)
<
17
[s/cc].
to theabsolute
total system
energy,
the
less
favorable
fusion
conditions
will
temperature T or R(nt, T), he charts the value of R for DT DD reactions over temperatures and
be. Aconfinement
high value of ŋvalues:
means the fusion process demands a large
portion of the total system energy to sustain a fusion reaction, a As a familiar starting point the original physical properties and
low value means little of the total system energy is necessary to notation in the 1955 paperix are used to compute Lawson’s original
< Log10
< 6useful
[eV]energy available as energy chart for energy ratio R, and the corresponding heating efficiency
sustain the2reaction
and(T)
more
14
<
Log10
(nt)
<
17
production. Lawson chose ŋ = 1/3 as [s/cc].
a plausible value the system ŋ. The charts match Lawson's original confirming the units used
apparatus could achieve to recirculate energy to sustain the plasma, and computations are the same as those in Lawson's 1955 paper. In
valuesAs
below
this ratio
are desirable,
above itphysical
undesirable.
1955 Lawson
considered
alpha products
reaction
a familiar
starting
point values
the original
properties
and notation
in only
the 1955
paperix in
aretheused
to value
As Lawson
defined
it
the
energy
recirculated
is
the
plasma
internal
for
'E',
the
matching
charts
confirm
this.
Like
Lawson's
compute Lawson’s original chart for energy ratio R, and the corresponding heating efficiency ŋ. The charts
energy
plus the
Bremsstrahlung
‘3nkT + tPBthe
’. units used
these and
showcomputations
ratio R and efficiency
(Eta) as
for those
Deuterium/Tritium
charts
match
Lawson'sradiation
originalloss
confirming
are theηsame
in
(DT) on Log scales for values of confinement parameter (nt). For
Lawson's 1955 paper. In 1955 Lawson considered only alpha products in the reaction value for 'E', the
In 1955 Lawson cautioned all these conditions though necessary Log10(T) = 6 and Log10(nt) = 17 the chart shows ‘R’ the ratio of
matching
chartswhere
confirm
this."The
Like
Lawson's
show
ratiotoRenergy
and efficiency
η (Eta) for
are far
from sufficient
he wrote
working
cycle charts
which these
energy
released
supplied approaches
Log10(R) = 2 or
Deuterium/Tritium
on Log
scalesreasoned
for values
confinement
parameter
(nt).eV)
Forand
Log10(T)
= 6 parameter
has been
assumed is very(DT)
optimistic".
Lawson
theof 100.
When Log10(T)
> 4 (1E4
confinement
andofLog10(nt)
17 thetochart
shows
‘R’ofthe
of energy
released
energy
approaches
quotient
the energy =
supplied
the total
energy
theratio
system
Log10(nt)
> 15 to
(1E15
s/cc)supplied
energy ratio
R shows the energy
is 1/(R+1).
The Lawson
says this
ratio must>be
than
exceeds the energy
supplied.
Log10(R)
= 2 orCriterion
100. When
Log10(T)
4 less
(1E4
eV) released
and confinement
parameter
Log10(nt) > 15
the heating
efficiency
η:
(1E15 s/cc) energy ratio R shows the energy released exceeds the energy supplied.
For a fusion process with temperature Log10(T) = 4 and constant within the hot central core and the cold surrounding fuel,
confinement parameter Log10(nt) =14 the heating efficiency η is but both pressure and temperature jump at the hot-cold boundary
80% of the total system energy. For fusion to occur the energy 7 (Atzeni et al., 2004, Figure 4.1). This is the isochoric assembly
recirculated must be 80% of the total energy, in this example discussed in references on fusion ignition (Pfalzner, 2006).
the requirement is more severe than the 33% value suggested Applying first-law fundamentals to a fusion plasma we have a
by Lawson. Note for a confinement parameter Log10(nt) > 15 power balance at the hot central core shown in Figure 1, all terms
the recirculated energy meets Lawson’s 33% and useful energy are power densities in units of [MeV/cc-s] (Eq. 4.1, Atzeni et al.,
returned is much higher, the prospect for fusion is greater. This is 2004).
the original Lawson criterion. To summarize:
dW/dt = Pr + Pe – Pb – Pl
Eqs. 4.1.
1/(R+1) is (energy supplied) / (total energy) and
1/(R+1) < ŋ this ratio is less than heating efficiency η.
The nomenclature becomes:
dW/dt = plasma internal energy
The Fusion Power Balance
Pa = alpha fusion product power
For the remainder of this paper the notation is expanded to a Pn = neutron fusion product power
complete power balance for an ICF equimolar DT target. A fast Pr = Pa + Pn
ignition approach is taken where the target fuel density remains Pe = external input power
Eng OA, 2024
Volume 2 | Issue 2 | 3
Software to Spec
falzner, 2006). Applying first-law fundamentals to a fusion plasma we have a
G.
Miklosy
Manuscript
Submission
L. L.
G.
Miklosy
Manuscript
forfor
Submission
hot central core shown in Figure 1, all terms are power
densities
in unitswith
of Lawson's
For
consistency
work
the fusion energy sou
Software
Spec
Software
to to
Spec
Lawson defined them in 1955, the power losses are develo
Atzeni et al., 2004).
L. G. Miklosy
Manuscript for Submission
30 April
Pb = Bremsstrahlung radiation power
= powerEqs.
losses 4.1.
e – Pb –Pl Pl
Pp = nkT/t
t = confinement time (seconds)
comes:
For
consistency
with
Lawson's
work
fusion
energy
source
cross
section
Friedberg
or
Atzeni.
Here
are
power
losses
developed
by A
consistency
with
Lawson's
work
thethe
fusion
energy
source
andand
cross
sections
Software to Spec FornLawson
= nuclei
count
density
(cc)thethe
defined
them
1955,
power
losses
developed
references
Lawson
defined
them
in in
1955,
power
losses
areare
developed
in in
references
onon
nu
kFriedberg
= Boltzman
constant
Atzeni.
Here
power
losses
developed
Atzeni
Meyer-Te
Friedberg
or or
Atzeni.
Here
areare
power
losses
byby
Atzeni
andand
Meyer-Ter-V
7/2
2 developed
For consistency withTLawson's
work
source
cross sections
are the same value
Pc
= the
3cefusion
AeTh energy
/ lnΛR
Conduction
h and Thermal
= absolute
temperature.
Lawson defined them in 1955,
the
power
losses
are
developed
in
references
on
nuclear
by
7/2
2
2 h Th u Thermal
/Rh Conduction
Mechanical Workfusion
B hρ
= 3c
Ahe7/2
T=h/ 3
/Γ
lnΛR
Conduction
(Eq.
4.10
h
PcPc
= 3c
APw
lnΛR
Thermal
(Eq.
4.10)
eeT
Friedberg or Atzeni. HerePw
are=epower
losses
developed
by
Atzeni
and
Meyer-Ter-Vehn:
u h/Rh
Mechanical
Work
(Eq.
4.13
Pw = 3 Γ3B ΓρBh ρThh T
u h/R
Mechanical
Work
(Eq.
4.13).
2
Conduction
Pc = 3ceAeTh7/2 /Thermal
lnΛRThermal
Thermal
Conduction
(Eq. 4.10)
h
Conduction
Thermal
Conduction
Pw = 3 ΓB ρh Th u /Rh
Mechanical Work
(Eq. 4.13).
topower
evaluate
power
losses
due
to thermal
conduction,
s
Now
to
evaluate
power
losses
thermal
conduction,
specifically
electron
Now
to Now
evaluate
losses
duedue
to to
thermal
conduction,
specifically
electron
dif
gradT
where
χ
is
target
core.
The
power
flowing
through
a
unit
surface
is
-χ
target
core.
The
power
flowing
through
a
unit
surface
is
-χ
e
e
e
Thermal Conduction
target core. The power flowing through a unit surface is -χe gradTe where χe is the
conductivity,
gradT
isgradT
the
gradient
of
electron
temperature
hot
core
e the
conductivity,
gradT
gradient
thethe
electron
temperature
onon
thethe
hot
core
su
e is
conductivity,
gradient
of
the diffusion
electron
temper
e isofthe
Now to evaluate power
losses
due to thermal
conduction,
specifically
electron
atto
the
hot
sign
direction
of
power
flow.
The
heat
flux
is shown
proportional
the
gra
sign
thethe
direction
of
power
flow.
The
heat
flux
is
shown
proportional
to
the
gradi
sign
the
direction
of power
heat
is U(T)
shown pr
gradT
where
χtemperature
the electron
target core. The power
flowing
through
a unit
surface
is -χas
e flow.
e The
e is flux
energy
density
itself
approximated
a function
te
energy
density
U,U,
itself
approximated
as a function
of of
temperature
U(T) andand
temp
conductivity, gradTe197
is the
gradient
of
the
electron
temperature
on
the
hot
core
surface
andofthetemp
neg
energy
density
U,
itself
approximated
as
a
function
reference)
197 reference)
sign the direction of power
flow.
The heat flux is shown proportional to the gradient of the internal
197
reference)
energy density U, itself approximated
as a function of temperature U(T) and temperature as T(r, t) (
= -χ(T)
gradT
q =q -χ(T)
gradT
e. e.
197 reference)
internal energy
n product power
sion product power
put power
hlung radiation power
es
q = -χ(T) gradT .
e
tempera
The
reference
shows
ideal
electrons
with
velocity
av and
temperatur
The
reference
shows
forfor
an an
ideal
gasgas
of of
electrons
with
velocity
vavvand
q = -χ(T) gradTthe
e.
Coulomb
cross-section
conductivity
takes
form
χe χ=e0χTe0e T5/2e .5/2When
. Wh
c the
the Coulomb
cross-section
σc σthe
conductivity
takes
thethe
form
χe =
=
T
R
is
the
radius
of
the
hot
core
.
For
argument
is
made,
gradT
The
reference
shows
for
an
ideal
gas
of
electrons
with
velo
eT
hh/R
h where
h the
=
/R
where
R
is
radius
of
the
hot
core
.
For
a ca
argument
is
made,
gradT
e
h
h
time (seconds)
and-1 temperature
due
The reference shows=for
an 5/2
ideal
gaswhere
of electrons
with
velocity
v-1av cm
19
-7/2 -7/2, T is dependence
/ lnΛ
A=e 9.5
= 9.5
x 19
10
ergs
keV
the
electron
tem
e / lnΛ
= Ae ATee T5/2
where Aecross-section
x 10
ergs
sc-1 sthe
cm-1 conductivity
keV
electron
tempe
5/2 , Te ise the takes
the
Coulomb
σ
the
fo
the
Coulomb
cross-section
σ
the
conductivity
takes
the
form
χ
=
χ
T
.
When
a
dimensional
c
e
e0
e
t density
(cc)
Coulomb
logarithm.
For
a
surface
area
S
and
volume
V
the
thermal
conduction
Coulomb
logarithm.
For
a
surface
area
S
and
volume
V
the
thermal
conduction
thermal
conduction
power
takes
the
form
The charged fusion products released consist of alpha
nuclei
and
is made,
Thhot
/Rhcore
where
is the radius
op
= Th/Rh where
Rh is thegradT
radius eof=the
. ForRah classical
DT plasm
argument is made, gradTeargument
5/2
19
-1
-1
-7/2
5/2
19
-1
-1
-7/2
onstant neutrons Pr = Pa + Pn. The power losses in such
a
system
are
= Ae Te / lnΛ where AePc
==
9.5
10 /ergs
s ~
cm
keV
Te9.5
is the2 electron
temperature,
lnΛ is the
7/2e, =
T
lnΛ
where
10[MeV/cc-s]
ergs
s cm keV
,
χxeegradT
TA
/ lnΛ
[MeV/cc-s]
e S/V
h/ lnΛ
h
~ 3c
AeeATeh7/2
/ R/h2Rx
Pc =
= -A
χe-e gradT
e S/V
e3c
accounted for as Pl = Pn + Pw + Pc where
Coulomb logarithm. For aCoulomb
surface
area
S
and
volume
V
the
thermal
conduction
power
takes
the
for
mperature.
logarithm. For a surface area S and volume V the
where
cce is
aa coefficient
close to
tounity
unityand
andthe
theCoulomb
Coulomblogarithm
logarithm
where
iscoefficient
coefficient
close
is given by
ea
close
where
ce is
7/2
2 to unity and the Coulomb logarithm is given by
~ 3cby
[MeV/cc-s]
Pc =variable
- χe gradTeisS/V
eAe Th / lnΛ / Rh
Pw = PdV work rate in isochoric (constant density,
given
7/2
2
[MeV/c
=- -0.5
χe gradT
e S/V ~ 3ceAe Th / lnΛ / Rh
temperature)
expansion
roductspressure,
released
consist
of alpha nuclei and neutrons Pr =
Pa
+=Pc7.1
Pn.
The
+ ln
10eV)
(Eq.
10.136
r
e >=
ln ln
Λ
=e 7.1
- 0.5
ln ln
ne n+epower
ln
Te Te(Te(T>=
10eV)
(Eq.
10.136
refe
eΛ
is
a
coefficient
close
to
unity
and
the
Coulomb
logarithm
is
given
by
where
c
e
Pc = thermal conduction power loss.
= 9.2
- 0.5
+ 1.5
AkeV),
i <=
ln ln
Λi Λ
=i 9.2
- 0.5
ln ln
ne n+e 1.5
ln ln
Ti Ti(Ti(T
<=
1010
AkeV),
m are accounted
for as Pl = Pn + Pw + Pc where
where30ceApril
is a 2020
coefficient close to unity and the Coulomb lo
L. G. Miklosy
for Submission
Miklosy
Manuscript Manuscript
for Submission
30nApril
ln Λe = 7.1 - 0.5and
ln
Te (Te >= is10eV)
(Eq. the
10.136
reference)
e + ln2020
x x Table 10.4
reference
shows
compute
ion
conductivity
neglected.
For
consistency
with
Lawson's
work
the
fusion
energy
source
and
ion
conductivity
is
neglected.
Table 10.4 of of
the reference
shows
computed
to Spec
re to Software
Spec
ln Λi = 9.2 - 0.510.136,
ln ne + 1.5
ln
T
(T
i
i <= 10 AkeV),
these
values
are
used
in
the
computations
for
the
thermal
conduction
and
cross
sections
are
the
same
values
as
Lawson
defined
them
10.136,
these
values
are
used
in
the
computations
for
the
thermal
conduction
Pc.P
rate in isochoric (constant density, variable pressure, temperature)
= 7.1in-agreement
0.5 ln newith
+ lntheTisochoric
lnofΛdensity
e expansion
e (Te >= 10eV)
independent
case
in
development
of
independent
of
density
in
agreement
with
the
isochoric
case
in
development
of
m
in
1955,
the
power
losses
are
developed
in
references
on
nuclear
and
ion
conductivity
is
neglected.
x
Table
10.4
of
the
reference
x
For consistency
with
Lawson's
workenergy
the fusion
energy
source
andconductivity
cross
are
the ln
same
values
as- 0.5
nsistency
withpower
Lawson's
work
the fusion
source
and cross
are sections
the same
values
asΛTable
10.4
of the
reference
shows
from Eq.
andsections
ion
is neglected.
ln
n
ln
T
(T
10
AkeV),
nduction
loss.
i = 9.2
e + 1.5
icomputed
i <=values
fusion
by
Friedberg
or
Atzeni.
Here
are
power
losses
developed
by
shows
computed
values
from
Eq.
10.136,
these
values
are
used
Lawson
defined
them
in
1955,
the
power
losses
are
developed
in
references
on
nuclear
fusion
by
n defined them in 1955, the power losses are developed in references
nuclear
fusion
by in the computations for the thermal conduction Pc. The relation is
10.136, on
these
values
are used
Mechanical
Work for the thermal conduction Pc. The relation
Mechanical
Work
Atzeni
in
the
computations
orand
Atzeni.
Herelosses
are power
lossesby
developed
byindependent
Atzeni and Meyer-Ter-Vehn:
erg orFriedberg
Atzeni.
Here
areMeyer-Ter-Vehn:
power
developed
Atzeni and
Meyer-Ter-Vehn:
of density
in agreement
with the isochoric case in development
of mechanical work ne
x
and
ion conductivity
is energy
neglected.
Table
10.4
of
the refer
The
hot
spherical
exchanges
with
surrounding
fuel
through
me
is
independent
ofcore
density
in agreement
with
the
isochoric
case
in
The
hot
spherical
core
exchanges
energy
with
thethe
surrounding
fuel
through
mech
7/2
2 7/2
3ceAheT
/ Thermal
lnΛRh2 Conduction
Thermal Conduction Mechanical
4.10)
matter
at pressure
p on
a volume
changing
by
an
amount
contributes
energy
= 3ceAeTPc
/ lnΛR
(Eq. 4.10) (Eq.
h
are
usedbyin
computations
for
the th
h =
development
ofthese
work
next.
Work
matter
at10.136,
pressure
pmechanical
on
a values
volume
changing
anthe
amount
dVdV
contributes
energy
d
3 Γh B ρh Th uMechanical
/Rh
Mechanical
Work
4.13).
reference
shows
aof
homogeneous
fuel
sphere
contribution
to
power
w = 3 ΓB ρhPw
Th u= /R
Work
(Eq. 4.13). (Eq.reference
shows
forfor
a homogeneous
fuel
sphere
thethe
contribution
to
thethe
power
balb
independent
density
in
agreement
with
the
isochoric
cas
The hot spherical core
exchanges
energy
with
the surrounding
fuel
through
pdVvolume
work,
= h(p
(dV/dt)
= h(pS/V
where
ismechanical
the
surface
= (1/V)(dE
h/dt)
h/V)
h S/V
=Work
(p
/V)
(dV/dt)
= (p
)u )u
where
S/VS/V
is the
surface
to to
volume ra
= (1/V)(dE
h/dt)
Mechanical
matter
at
pressure
p
on
a
volume
changing
by
an
amount
dV
contributes
energy
dE
=
pdV.
The
Thermal
Conduction
velocity
the
sphere
surface.
If the
ideal
equation
state
=B ГρT
wh
mal Conduction
B ρT
velocity
of of
the
sphere
surface.
If the
ideal
gasgas
equation
of of
state
is pis=p Г
where
hot
spherical
core
exchanges
energy
the
surrounding
Thermal
Conduction
reference shows for The
a homogeneous
the DT),
contribution
towith
the
power
balance
from
work is
14fuel sphere
Work
8
=Mechanical
7.66
x 14
10erg/g
erg/g
keV
for
the
power
due
to
work
becomes
(Г
B 7.66
=
x
10
keV
for
DT),
the
power
due
to
work
becomes
(Г
B
NowNow
to evaluate
power
dueconduction,
to thermal
electron
diffusion
the
hot )u where
to
evaluate
power
losses
dueconduction,
to thermal
conduction,
through
mechanical
pdV
fuel matter
at pressure
o evaluate
power
losses
due tolosses
thermal
specifically
electron
diffusion
the(dV/dt)
hot at
= (pat
/V)
= (p
S/Vwork,
is the surface
to volume
ratio 3/Rph and u is t
=specifically
(1/V)(dE
h/dt)
hfuel
h S/V
core. flowing
The power
flowing
through
a unit
-χe gradT
where
χe is
electron
core. target
The specifically
power
through
a unit
surface
is hot
-χsurface
χe isof
e the
electron
diffusion
at the
targeteiswhere
core.
The
power
onthe
aPw
volume
changing
by
an
amount
dV
contributes
energy
dEthe
e gradT
velocity
theelectron
sphere
surface.
If
the
ideal
gas
equation
of
state
is
p
=
Г
Гthe
gas con
The
hot
spherical
core
exchanges
energy
with
surround
B ρT where
B is
3B ГρBhTρhhu/R
Thu/R
[MeV/cc-s].
Pw
=negative
3=Гand
h h [MeV/cc-s].
14 core
the
gradient
of
the
electronwhere
temperature
onelectron
the
hot
surface
thethenegative
gradT
tivity,conductivity,
gradT
gradient
of the
electron
temperature
on the
hot
core
surface
and
the
e is
e is the
=
7.66
x
10
erg/g
keV
for
DT),
power
due
to
work
becomes
(Г
flowing
through
a unit
surface
is -χ
gradT
χ
is
the
=
pdV.
The
reference
shows
for
a
homogeneous
fuel
sphere
the
B
matter
at
pressure p on a volume changing by an amount d
e
e
e
sign the
ofgradTe
power
The
heat of
flux
shown proportional
toon
the
gradient
of theto
internal
e direction
of direction
power flow.
The heat
flux
is shown
proportional
to
the gradient
of the
internal
conductivity,
isflow.
the
gradient
theis electron
temperature
contribution
the
powerfor
balance
from work is fuel
Pw =sphere
(1/V)(dE
/
h contrib
reference
a homogeneous
the
9 9
energy
density
U, surface
itself approximated
asofa temperature
function
of direction
temperature
temperature
as
T(r,shows
t)=(page
density
U,the
itself
as
a function
U(T) and
temperature
ashT(r,
(page
Pwof
= U(T)
3 ГB ρand
hThu/R
hotapproximated
core
and
the negative
sign the
power
dt)
= (pt)[MeV/cc-s].
/V)
(dV/dt)
(p
S/V
)u
where
S/V
is
the
surface
to
volume
h
h
= (1/V)(dE
197 reference)
erence)
h/dt) = (ph/V) (dV/dt) = (ph S/V )u where S/V is th
flow. The heat flux is shown proportional to the gradient of the ratio 3/Rh and u is the
velocity of the sphere surface. If the ideal gas
velocity
ofisthe
surface.
thegas
ideal
gas equation
of s
internal energy
density U, itself approximated as a function of equation
of state
p =sphere
ГB ρT9 where
ГB isIfthe
constant
(ГB =
q = e-χ(T)
gradTe .
.
= -χ(T) gradT
14
L.
G.
Miklosy
Manuscript
for
Sub
14
10DT),
erg/g
keV for
thebecomes
power due to work
B = 7.66
temperature U(T) and temperature as T(r, t) (page197 reference)
7.66 x(Г10
keVx for
the power
dueDT),
to work
erg/g
Software to Spec
due to
The
reference
gas of
electrons
velocity
vav and temperature
erence
shows
for anshows
ideal for
gas an
of ideal
electrons
with
velocitywith
vav and
temperature
dependence dependence
due to
q =conductivity
-χ(T)
. χe =the
Pw = 3 ГB ρhThu/Rh
[MeV/cc-s].
Coulomb cross-section
σc the
= χe0 Tae 5/2
. When a dimensional
ulombthe
cross-section
σc the conductivity
takes gradT
the form
χe0form
Te 5/2 χ.eWhen
dimensional
etakes
is made,
gradT
Rh isofthe
of the
hota core
. ForDT
a classical
plasma χe
Rhh/Ris
the radius
theradius
hot core
. For
classical
plasma χDT
ent is argument
made, gradT
e = T
h where
e = Th/R
h where
e
19-1
-1
-1
5/2
Te 5/2 reference
/A
lnΛ
A19e =ergs
9.5an
ergs
sof
, Te isvelocity
the
electron
lnΛ is
the where
= Aewhere
The
for
ideal
gas-7/2
electrons
with
vav temperature,
For
isochoric
case
the pressure
in the
hot coreinis the
much
/ lnΛ
9.5 xshows
10
sx-1 10
cm
keV
,cm
T
the -7/2
electron
temperature,
lnΛ the
is the
e = where
e iskeV
For
the
isochoric
case
where the
pressure
hot core i
9
Coulomb
logarithm.
a surface
area
volume
V the
thermal power
conduction
power
takes
the
form
mb logarithm.
a surfaceFor
area
S and volume
Vthe
the
thermal
conduction
the shock
form
and For
temperature
dependence
dueStoand
Coulomb
cross-section
σc takes
higher
than
theis
surrounding
fuel,
a
shock
is
driven
into
the
cold
driven into the cold fuel, the velocity u for the m
5/2
the conductivity takes the form χe = χe0 Te . When a dimensional fuel, the velocity u for the material behind a shock gives
7/2
27/2
Pc
- χe~gradT
S/V
3ceA/ eR
/ lnΛ
/ Rh2 R is[MeV/cc-s]
3c
Ae T
/~lnΛ
[MeV/cc-s]
= - χe gradT
e=S/V
h gradT
argument
is emade,
=ThhTh/R
where
the radius of the hot
e
h
h
u = (3ph/4ρc) 1/2 = (3ГBTh ρh/ρc)1/2 .
core. For a classical DT plasma χe = Ae Te 5/2 / lnΛ where Ae = 9.5
coefficient
to
unity and the
Coulombislogarithm
is given by
c is aclose
coefficient
unity
and-7/2
the
logarithm
given by lnΛ
ce is awhere
-1 close
x 10e 19 ergs
s-1tocm
keV
, TeCoulomb
is the electron
temperature,
is Finally the power density for an isochoric ignition can be written
Finally
for an isochoric ignition can be
the Coulomb logarithm. For a surface area S and volume V the with Am
= 5.5 xthe
1022power
cm3 s-3 density
keV-3/2 as
0.5
ne +
Te (Te >= 10eV)
10.136 reference)
Λeln= n7.1
Λe = 7.1 -ln0.5
Te ln(T
>=ln10eV)
(Eq. 10.136 (Eq.
reference)
e + -ln
Λiln= n9.2
0.5lnlnTni e +(T1.5
ln10
TiAkeV),
(Ti <= 10 AkeV),
Λi = 9.2 -ln0.5
e + -1.5
i <=
Eng OA, 2024
Pw = Am ρhTh3/2/Rh.
x
x
Table
10.4 of the
reference
showsvalues
computed
and ion conductivity
Table 10.4 of
the reference
shows
computed
fromvalues
Eq. from Eq.
n conductivity
is neglected.is neglected.
10.136,
values
are computations
used in the computations
for the
thermal Pc.
conduction
Pc. The
relation is Energy Source
, these
valuesthese
are used
in the
for the thermal
conduction
The relation
is External
density inwith
agreement
with the
isochoric
case in development
of mechanical
ndentindependent
of density inofagreement
the isochoric
case
in development
of mechanical
work next. work next.
Volume 2 | Issue 2 | 4
[MeV/cc-s].
shock is driven into the cold fuel, the velocity u for the material behind a shock gives
u = (3ph/4ρc) 1/2 = (3ГBTh ρh/ρc)1/2 .
u = (3ph/4ρc) 1/2 = (3ГBTh ρh/ρc)1/2 .
Finally the power density for an isochoric ignition can be written with Am = 5.5 x 1022 cm3 s-3 keV-3/2 as
Finally the power density for an isochoric ignition can be written with Am = 5.5 x 1022 cm3 s-3 keV-3/2 as
Power Balance Components
Pw = Am ρhTh3/2/Rh.
[MeV/cc-s].
Each power component from the previous section are calculated
Pw = Am ρhTh3/2/Rh.
[MeV/cc-s].
separately, finally the power balance is computed from Eq. 4.1 and
this resultant is the new transient internal energy. To compare with
External
EnergyEnergy
Source Source
External
Energy input External
from an external
source
is now developed for the Lawson's original work the power values are normalized to the
Energy
Source
Energy
input
fromIfan
sourcesource
is now
for the
fusion
initiation
If an
external
in Lawson's
original
paper these
density
squared
(n2). Asprocess.
fusion
initiation
process.
an external
external energy
suchdeveloped
as an nuclei
Energy
input
from
an
external
source
is
now
developed
for
the
fusion
initiation
process.
an external
energy
such energy
as an ion
or laser
to the
of radius
Rc intemperature
shotIftime
power E
terms
areICF
now target
functions
of absolute
(T) and
ion beam
or source
laser delivers
E tobeam
the ICF
target delivers
of radius energy
energy
source
such
asdelivered
an
ion beam
or laser
delivers
E to the
target of radius Rc in shot time
confinement
parameter
(nt)ICF
alone.
Rc ints shot
ts , the
powerdelivered
density
assuming
100%
, thetime
power
density
assuming
100%
efficiency
isenergy
efficiency is ts , the power density delivered assuming 100% efficiency is
The next charts show the contribution of each power term relative
Pe = E/(4/3πRc3)/ts .
3
to others for two values of the confinement parameter Log(nt)
Pe = E/(4/3πRc )/ts .
= 14 and 17 and a convergence ratio of the target shell Rc/Rh =
a 1.8MJ
laser
shot to
delivered
to a 2mm
diameter
in 20ns of
thelosses
power
density
is PdV work and
30. Noticetarget
the magnitude
from
neutrons
For aFor
1.8MJ
laser shot
delivered
a 2mm diameter
spherical
target spherical
16
23
L.
G.
Miklosy
Manuscript
for
Submission
30
April
2020
For
a
1.8MJ
laser
shot
delivered
to
a
2mm
diameter
spherical
target
in
20ns
the
power
density
16
23
2.149x10
or 1.341x10
MeV/cc-s.
thermal conductivity, these dominate the energy balance is
yet are
in 20ns
the powerJoule/cc-s
density
is 2.149x10
Joule/cc-s
or 23
1.341x10
16Spec
Software
to
MeV/cc-s. 2.149x10 Joule/cc-s or 1.341x10 MeV/cc-s.completely omitted in steady-state assessments of the fusion
reaction in literature.
Power Balance Components
Power Balance Components
Each power component from the previous section are calculated separately, finally the power balance is
Each
power
component
the previous
are internal
calculated
separately,
finallywith
the power balance is
computed
from
Eq. 4.1
and this from
resultant
is the newsection
transient
energy.
To compare
2
computed
from
Eq.
4.1
and
this
resultant
is
the
new
transient
internal
energy.
To
compare
Lawson's original work the power values are normalized to the nuclei density squared (n ). As in with
2
Lawson's
workpower
the power
arefunctions
normalized
to the nuclei
density squared
Lawson's
originaloriginal
paper these
termsvalues
are now
of absolute
temperature
(T) and (n ). As in
Lawson's
original
these power terms are now functions of absolute temperature (T) and
confinement
parameter
(nt)paper
alone.
confinement parameter (nt) alone.
The next charts show the contribution of each power term relative to others for two values of the
The next
charts Log(nt)
show the=contribution
of aeach
power term
relative
others
for R
two
confinement
parameter
14 and 17 and
convergence
ratio
of thetotarget
shell
c/Rvalues
h = 30. of the
confinement
Log(nt)
= 14 and
17work
and aand
convergence
ratio of the these
target dominate
shell Rc/Rthe
h = 30.
Notice the
magnitudeparameter
of losses from
neutrons
PdV
thermal conductivity,
Notice
the
magnitude
of
losses
from
neutrons
PdV
work
and
thermal
conductivity,
these
dominate
the
energy balance yet are completely omitted in steady-state assessments of the fusion reaction in
energy balance yet are completely omitted in steady-state assessments of the fusion reaction in
literature.
literature.
10
10
Eng OA, 2024
Volume 2 | Issue 2 | 5
The power balance is fully populated including an external source and losses from neutrons, thermal
conduction and mechanical work. In this form Pr is the total particle energy 17.6 MeV, Pn the neutron
loss 14.1 MeV is accounted for in system losses separately.
Transient Fusion Criterion
criteria. As a sanity check for conditions necessary and sufficient
The original
Lawson
Criterion
neglected
energy
losses energy
and assumed
fusion,
the beam)
internal isenergy
dW/dt event,
is calculated
Initiating
fusion
ignition
using
a pulsed
sourcefor(laser
or ion
a transient
the from the
simplifications
previously,
heresteady-state
we include those
omissions inModern
a complete
power relation
1. ignition stray far from
power balance
is not
and non-zero.
interpretations
of Eq.
fusion
powerfirst-law
balance for
a pulsed system
proposed
in Lawson's
day. computer algorithm is criteria enoughxi, in another
fundamentals
thatasin
one case
a converging
Here the modern pulsed system is laser ICF with a 2-mm spherical Lawson reasoned (previous section) the ratio R consisted of energy
counting neutrons is the fusion criteria. As a sanity check for conditions necessary and sufficient for
target (Rc=1mm) of DT fuel modeled with equimolar molecular components that a fusion apparatus could recirculate to heat the
fusion,
the internal
dW/dt
calculated
from the
complete
powerneutrons
relation
Eq. 1.
weight
5.03 [gm/Mol]
and energy
convergence
ratiois 30.
The complete
plasma,
he reasoned
escape
the system and losses were
power balance is
omitted from consideration. For a fixed volume where dW/dt is
Lawson reasoned (previous section) the ratio R consisted
of energy
components that a fusion apparatus
substituted
for 3nkT/t
dW/dtcould
= Pr +recirculate
Pe – Pb – Pl
(Eq.
1)
to heat the plasma, he reasoned neutrons escape the system and losses were omitted
Pl = Pn
+ Pw
+ Pc. For a (Eq.
2) volume where dW/dt Ris=substituted
Pa / (dW/dt +for
Pb)
(Eq. 3)
from
consideration.
fixed
3nkT/t
ŋ(R+1) > 1
(Eq. 4).
The power balance is fully populated including an external source
= Paneutrons,
/ (dW/dtthermal
+ Pb) conduction and mechanical
(Eq. 3)
and lossesRfrom
ŋ(R+1)
4).when omissions to Lawson’s power
work. In this
form Pr>is1the total particle energy 17.6 MeV, Pn the As the charts below(Eq.
show,
neutron loss 14.1 MeV is accounted for in system losses separately. balance and in particular the transient internal energy dW/dt is
calculated,
there is no
case
the power
released exceeds
As the charts below show, when omissions to Lawson’s
power balance
and
in where
particular
the transient
Initiating
fusionenergy
ignitiondW/dt
using a is
pulsed
energy source
ion where
power the
supplied
andreleased
R ~ 1. Theexceeds
fusion plasma
is supplied
loosing energy to
internal
calculated,
there(laser
is noorcase
power
power
beam)and
is aR
transient
event,
the plasma is loosing energy to losses,
losses,
neutrons and
fasterfaster
than the
fusion
~ 1. The
fusion
neutrons
andBremsstrahlung
Bremsstrahlung
than
thereaction
power balance is not steady-state and non-zero. Modern rate and external sources can replenish the plasma. The power
fusion reaction rate and external sources can replenish the plasma. The power balance is mostly
interpretations of fusion ignition stray far from first-law balance is mostly negative shown with dashed curves and fusion
negativethat
shown
curves
and fusion
under under
thesethese
conditions
is is
not
fundamentals
in onewith
casedashed
a converging
computer
algorithm
conditions
notviable.
viable.
is criteria enoughxi, in another counting neutrons is the fusion
L. G. Miklosy
Software to Spec
Manuscript for Submission
30 April 2020
The ignition criteria given in fusion references state that dW/dt is large and positive, a simple treatment
from classical thermodynamics show this condition is not met.xii The heating efficiency exceeds
Lawson’s suggested ŋ = 0.33 for any value of temperature or confinement parameter (nt), by this
measure the necessary conditions for fusion are both unlikely and insufficient.
Note: Efficiency
Eta 'η' incase
the Lawson
Criteria 'η(R+1)>1'
shown
fusion Pr
and include alland
losses
contributors
to the fusion
As a bounding
let us assume
all lossesisdue
to neutrons
Bremsstrahlung
PdVaswork
somehow
Note: Efficiency
Eta 'η' in the Lawson
Criteria 'η(R+1)>1' is shown
as 'n(R+1)>1' due to charting software limitations.
as 'n(R+1)>1'
due totocharting
process:
contribute
plasmasoftware
heatinglimitations.
regeneration, then replace
the alpha fusion Pa with total fusion Pr and
include all losses as contributors to the fusion process:
The ignition criteria given in fusion references state that dW/12R = (Pr) / (dW/dt + Pb + Pl)
(Eq. 5)
dt is large and positive, a simple treatment from classical ŋ(R+1) > 1.
(Eq. 6)
R = show
(Pr) /this
(dW/dt
+ Pbis+not
Pl)met [7]. The heating
(Eq. 5)
thermodynamics
condition
ŋ(R+1)
> 1. suggested ŋ = 0.33 for any value of The next charts show
(Eq.this
6) transient condition, note R becomes
efficiency exceeds
Lawson’s
temperature or confinement parameter (nt), by this measure the positive meaning the internal plasma energy is increasing
Theconditions
next charts
showare
thisboth
transient
note R suggesting
becomes positive
the internal
a viable meaning
fusion process.
Still theplasma
heating efficiency
necessary
for fusion
unlikelycondition,
and insufficient.
required
exceeds
50%,efficiency
meaning more
than half
the total system
As a bounding
let us assume
all losses
due fusion
to neutrons
energy iscase
increasing
suggesting
a viable
process.
Still the
heating
required
exceeds
is required
to sustain
fusionprocess.
process.
Bremsstrahlung
and PdV
workthan
somehow
to plasma
50%, meaning
more
half thecontribute
total system
energyenergy
is required
to sustain
thethe
fusion
heating regeneration, then replace the alpha fusion Pa with total
Eng OA, 2024
Volume 2 | Issue 2 | 6
The charts also show the equivalent values of R, ŋ calculated from from empirical values of Pτ
(pressure and confinement time) reported by the National Academy of Sciences, NAS Review of NIF
Campaign in 2013.xiii Comparing the empirical values to the first law constraints suggests the heating
efficiency attained by experiment is much better than our bounding case allows but doubtful.
The charts also show the equivalent values of R, ŋ calculated from from empirical values of Pτ
(pressure and confinement time) reported by the National Academy of Sciences, NAS Review of NIF
xiii
Comparing
empirical
valuesNext,
to the
firstforlaw
the heating
Campaign
in 2013.
The charts
also show
the equivalent
values the
of R,
ŋ calculated
charts
the constraints
original 1955suggests
Lawson heating
efficiency and
efficiency
attained
by
experiment
is
much
better
than
our
bounding
case
allows
but
doubtful.
from from empirical values of Pτ (pressure and confinement time) the chart including transient heating effect for the bounding case
reported by the National Academy of Sciences, NAS Review of compared side-by-side. The bounding case shows 50% of the total
NIF Campaign
in 2013.xiii
Comparing
the Lawson
empirical heating
values toefficiency
energy must
to sustain transient
the fusion heating
reaction but not
Next, charts
for the original
1955
andbe
therecycled
chart including
the first
law for
constraints
suggestscase
the heating
efficiency
attained The
achievable
with the
current
apparatus.
effect
the bounding
compared
side-by-side.
bounding
case
shows
50% of the total energy
by experiment is much better than our bounding case allows but
must be recycled to sustain the fusion reaction but not achievable with the current apparatus.
doubtful.
13"On the grounds of basic physics, a complete energy balance of
Agreement with Previous Research
In his 1995 thesis Todd Rider shows how a fusion plasma in magnetized and non-magnetized plasmas is presented for pulsed,
thermodynamic nonequilibrium cannot produce net power and the stationary and self-sustaining operations by taking into account the
recirculation efficiencies required for recycling enough power to energy release by reactions of light nuclei as well as different kinds
sustain fusion are not yet available[9]. Rider shows for a pulsed of diffusive (conduction) and radiative (bremsstrahlung, cyclotron
system not in equilibrium that fusion is not possible. From the or synchrotron radiation and excitation radiation) energy losses.
Rider thesis abstract:
Already the energy losses by radiation make the energy balance
negative. Hence, a fusion reactor ─ an energy producing device ─
"For virtually all possible types of fusion reactors in which the seems to be beyond the realms of realization."
major particle species are significantly non-Maxwellian or are
at radically different mean energies, this minimum recirculating As the department lead from the Princeton Plasma Physics
power is substantially larger than the fusion power. Barring the Laboratory, Daniel Jassby writes in Voodoo Fusion Energy:
discovery of methods for recycling the power at exceedingly high
efficiencies, grossly nonequilibrium reactors will not be able to "A tepid plasma of deuterium cannot produce measurable levels of
produce net power."
fusion neutrons because one or more of the ion temperature, ion
density or plasma volume is too small."
In the 2000 paper Energy Balance of Controlled Thermonuclear
Fusion by Hashmi and Staudenmaier the authors derive the fusion The present work agrees with these previous treatments of fusion
energy balance for both magnetized and non-magnetized fusion in Inertial Confinement Fusion, that a fusion plasma is unlikely
plasmas not from thermodynamics but from basic physics [10]. [11].
The authors carefully take account for binary interactions during
disassociation and reabsorbtion of bound-bound, ionized and free- Summary of Lawson-like Criterion
free transitions between particles in a fusion plasma process. They As we have shown, some ICF researchers stress the importance of
compute the probability of binary interactions accounting for the each term in an power balance, as this excerpt from researchers at
size of particle cross-section. They formulate the energy balance Los Alamos National Laboratory shows:
of fusion considering these interactions and show the energy losses
due to radiation make the energy balance negative. From the paper, “By focusing on the temperature rate equations and assuming that
power losses given by Prad = Pb + Pc + Pexc + Prec (Eqs 23) the electron and ion temperatures are the same in a homogeneous
are due to bremsstrahlung, cyclotron, and excitation radiation fusion fuel contained in a spherical shell, the net rate of temperature
respectively. Their results show for a nonmagnitized plasma change depends on the work rate (dV/dt) due to compression, the
such as ICF, the energy balance is already crippled by excitation fusion energy self-heating, the thermal conduction (both electron
radiation Pexc alone. From the paper Table IV shows the ratio and ion), the Bremsstrahlung (or free-free) emission, the inverse
of ionization N* that makes the energy balance negative, for the Compton cooling, the synchrotron radiation, and the re-radiation
temperature regime considered there is no temperature where the of the shell material [12].”
sustainable reaction criteria is satisfied.
Other researchers search for a new parameter space that seeks to
From the Hashmi and Staudenmaier abstract:
make ICF fusion ignition likely, like this excerpt from Lawrence
Eng OA, 2024
Volume 2 | Issue 2 | 7
Livermore:
14 < Log10(nt) < 17 nt in [1/cc-s].
"Such a survey revealed that there is a new region in the parameter
space for thermonuclear fusion, which exists at lower initial
density r0 and implosion velocity v0 than necessary for ICF [13]."
For each Lawson-like criterion reported by the researchers these
steps were followed:
1. Formulate the power balance (if applicable)
2. Identify energy released and energy supplied terms
3. Calculate the power ratio R = power produced / power input
4. Calculate 1/R+1 is the power input to system total power
5. Chart R and the Lawson Criterion ŋ(R+1) > 1.
ER
Again some charts also show the empirical value of R, ŋ achieved
by the premier National Ignition Facility as reported in the 2013
National Academy of Sciences Review [14]. The reported NAS
values in Pτ units (pressure and confinement time) are converted
by simple calculation to the energy ratio R.
PY
Given less than favorable conditions for fusion predicted by a
comprehensive power balance, some researchers re-interpret the
1955 Lawson Criterion to serve a new fusion hypothesis more
favorable to fusion and ignition. A Lawson-like criterion may
simplify interpretation for both lay-people and venture capitalists,
the true path to fusion remains cloaked in first-law fundamentals.
It is useful for analysts to check physics fundamentals for a sanitycheck using real numbers, this paper strives to achieve that.
PA
P
D
HE
IS
BL
SA
M
PL
Lawson Criterion 1955
R = Pa / (Pp + Pb)
E
CO
Comparison Methodology
Next, some recent and popular Lawson-like criteria are cast in We begin by repeating the original 1955 Lawson Criterion
a consistent way for comparison as power ratio R and heating revisited for an ICF target. In the Lawson paper a steady-state
efficiency ŋ. The energy terms and thus R, η are again functions of reaction was assumed for DD and DT fuels and the values for R
the fusion process temperature ‘T’ and the confinement parameter charted, only the result for DT fuel is reproduced here. Like the
Miklosy
Submission
30the
April
2020
Lawson paper these results suggest
energy
ratio R is
‘nt’. R L.
andG.
η are
graphed over the region of interest Manuscript fororiginal
Software to Spec
quite optimistic and efficiency necessary reasonable, particularly
at higher temperatures.
2 < Log10(T) < 6 T in [eV],
UN
PU
Lawson Criterion 1955 – Breakeven
R = Pa / Pp
Lawson Criterion 1955 – Alpha Self-heating (Pfalzner xx)
Eng OA, 2024R = Pa / Pb
Volume 2 | Issue 2 | 8
Lawson Criterion 1955 – Alpha Self-heating (Pfalzner xx)
R = Pa / Pb
L. G. Miklosy
Manuscript for Submission
Software to Spec
L. G. Miklosy
Manuscript for Submission
Software
to
Spec
Lawson Criterion – with Bremsstrahlung Self-heating1
30 April 2020
30 April 2020
R = Pa / (Pp - Pb)
1
Lawson
Criterion – with Bremsstrahlung
Self-heating
L.
G. Miklosy
Manuscript
for Submission
R = Pa
/ (Pp - Pb)
Software
to Spec
30 April 2020
Lawson Criterion – with Bremsstrahlung Self-heating1
R = Pa / (Pp - Pb)
16
Where the dominant term produces a negative ratio R, the absolute value is taken to produce a
logarithmic scale and the curve is dashed. To the degree Bremsstrahlung Pb could be captured for selfWhere
produces
negativeis ratio
R, thebut
absolute
value
is takenassumption
to produce when
a
heatingthe
thedominant
term Pb isterm
included,
the asolution
academic
suggests
an invalid
Pb
logarithmic
scale
and
the
curve
is
dashed.
To
the
degree
Bremsstrahlung
Pb
could
be
captured
for
selfexceeds the plasma internal energy Pp.
heating the term Pb is included, the solution is academic but suggests an invalid assumption when Pb
Where the exceeds
dominant term
produces
a negative
ratioPp.
R, the absolute value is taken to produce a logarithmic scale and the curve is dashed.
plasma
internal
energy
Where thethedominant
term produces
a negative ratio R, the absolute value is taken to produce a
To the degree Bremsstrahlung Pb could be captured
for selfheating the term Pb is included, the solution is academic but suggests an
xxi
logarithmic
scale
and the
curve
is dashed.
the degree
Bremsstrahlung Pb could be captured for selfLindl LLNL
Physics
Today
1992
invalid assumption
when
Pb exceeds
the
plasma
internal To
energy
Pp.
heating
theεPr
term
R=
/ PePb is included, thexxisolution is academic but suggests an invalid assumption when Pb
Lindl LLNL
Physics
Today energy
1992 Pp.
exceeds
the plasma
internal
R = εPr / Pe
Lindl LLNL Physics Today 1992 xxi
R = εPr / Pe
Betti/Tipton xxii xxiii
R = Pa / Pp
Betti/Tipton xxii xxiii
R = Pa / Pp
Betti/Tipton xxii xxiii
R = Pa / Pp
1
Like alpha self-heating assumes Bremsstrahlung used in regeneration, not a system loss.
Eng OA, 2024
1 Like alpha self-heating assumes Bremsstrahlung used in regeneration,
not a system loss.
17
17
1
Like alpha self-heating assumes Bremsstrahlung used in regeneration, not a system loss.
Volume 2 | Issue 2 | 9
L. G. Miklosy
Software to Spec
L. G. Miklosy
Software
toand
SpecKirkpatrick xxiv
Lindemuth
R = Pr / Pe
Lindemuth and Kirkpatrick xxiv
R = Pr / Pe
Manuscript for Submission
30 April 2020
Manuscript for Submission
30 April 2020
Atzeni and Meyer-Ter-Vehn xxv
R = Pr / (Pb + Pc + Pw)
Atzeni and Meyer-Ter-Vehn xxv
R = Pr / (Pb + Pc + Pw)
In this development by Atzeni and Meyer-Ter-Vehn a departure from the Lawson convention assumes
Bremsstrahlung
conduction and
mechanical
work
could
be recovered
if they
appear in the
In this development
by Atzenithermal
and Meyer-Ter-Vehn
a departure
from the
Lawson
convention
assumes
Bremsstrahlung
thermal conduction
In
this
development
by supplied.
Atzeni
and appear
Meyer-Ter-Vehn
a departure
fromsupplied.
the Lawson convention assumes
denominator
as be
energy
and mechanical
work
could
recovered
if they
in the denominator
as energy
Bremsstrahlung thermal conduction and mechanical work could be recovered if they appear in the
denominator
Friedberg xxvi as energy supplied.
R = Pr / (Pb + Pc)
Friedberg xxvi
R = Pr / (Pb + Pc)
Again a departure from the Lawson convention of energy released to energy supplied, to the degree that Bremsstrahlung and thermal
conduction loss could be recovered those contributions appear in the denominator as energy supplied.
18
18
Eng OA, 2024
Volume 2 | Issue 2 | 10
Bremsstrahlung and thermal conduction loss could be recovered those contributions appear in the
denominator
as energy
Again a departure
fromsupplied.
the Lawson convention of energy released to energy supplied, to the degree that
Bremsstrahlung and thermal
conduction loss could be recovered those contributions appear in the
xxvii
Wikipedia:
Tokamak
denominatorJET
as energy
supplied.
Q = Pr/(Pe - Pp)
Wikipedia: JET Tokamak xxvii
Q = Pr/(Pe - Pp)
Here Gain factor Q and R are treated as equivalent for the sake of charting the ratio R and η. For the
confinement parameter Log10(nt) > 15 the results look quite promising except the equivalent power
Here Gainbalance
factor QPp
and= RPeare
treated
as equivalent for the= sake
of charting
the ratio R is
and
For the confinement parameter Log10(nt)
(power
net fusion
power
notη.meaningful.
Here Gain
factor–QPrand
R areaccumulated
treated as equivalent
for the
sakeinput)
of charting
the ratio R and η. For the
> 15 the results look quite promising except the equivalent power balance Pp = Pe – Pr (power accumulated = net fusion power input)
confinement parameter Log10(nt) > 15 the results look quite promising except the equivalent power
is not meaningful.
balance Pp = Pe – Pr (power
accumulated = net fusion power input) is not meaningful.
Wikipedia: Fusion Energy xxviii
R = Pr/ε(Pr - Pb – Pc) where ε = 48%
Wikipedia: Fusion Energy xxviii
R = Pr/ε(Pr - Pb – Pc) where ε = 48%
L. G. Miklosy
Software to Spec
Manuscript for Submission
30 April 2020
These curves show only the reduced domain where the power ratios remain positive and valid.
Interpreting
Results
confinement
parameter
(nt) case
and temperature
(10E17
heating efficiency
demands 50% of total system energy.
As a final
bounding
lets assume
both1/cc-s and
The fusion
criterion
proposed
from
different
researchers
remain
10E6
eV)
the
heating
efficiency
demands
50%
of
total system
These
curves
show
only
the
reduced
domain
where
the
power
ratios
remain
positive
and
valid.
Interpreting
Resultscan be recirculated to heat the plasma and a 1.8MJ laser external source
Bremsstrahlung
and neutrons
quite optimistic for the prospect of steady-state fusion particularly energy. As a final bounding case lets assume both Bremsstrahlung
used,
the
charts
belowproposed
show
case.
Again
the power
balance
isbe
negative
and
aheat
dashed
curve
is a 1.8MJ
The
fusion
criterion
from
different
researchers
quite
optimistic
fortothe
prospect
of and
at ishigher
particle
concentrations
and this
temperatures.
However
and remain
neutrons
can
recirculated
the plasma
Interpreting
Results
steady-state
fusion
particularly
at
higher
particle
concentrations
and
temperatures.
However
when
used.
when accounting for system losses and omissions the transition to laser external source is used, the charts below show this case.
accounting
system
losses
and
omissions
transition
steady-state
through
a transient
barrier
steady-state
through a for
transient
barrier
remains
which no process
Againtothe
power balance
is negative
a dashed
The
fusion criterion
proposed
from
differentthe
researchers
remain
quite optimistic
for
theand
prospect
ofcurve is used.
accounted
for
(here)
can
cross.
When
a
complete
power
balance
is
remains
which
no
process
accounted
for
(here)
can
cross.
When
a
complete
power
balance
is
taken
Bounding
Case: Bremsstrahlung
neutrons
recirculated
steady-state
fusion particularlyand
at higher
particle
concentrations and temperatures. However when and
taken andthe
the transient
transient conditions
leading
to fusion
are are
accounted
Bounding
Case:
Bremsstrahlung
and
recirculated
conditions
leading
to
fusion
accounted
for,
prospects
for fusion
areneutrons
quitebarrier
bleak.
R=
(Pr+Pe)/(dW/dt+Pb+Pn)
accounting
for system
losses
andEven
omissions
the transition
tothe
steady-state
through
a transient
for, the prospects
areconfinement
quite
bleak.
at the (nt)
highest
R= (Pr+Pe)/(dW/dt+Pb+Pn)
Even at for
the fusion
highest
parameter
and temperature
(10E17 1/cc-s and 10E6 eV) the
remains which no process accounted for (here) can cross. When a complete power balance is taken and
the transient conditions leading to fusion are accounted for, the prospects for fusion are quite bleak.
Even at the highest confinement parameter (nt) and19
temperature (10E17 1/cc-s and 10E6 eV) the
19
With neutrons included for regeneration the system losses remain so great the power ratio remains
negative and the heating efficiency must be 100%. In other interpretations of the Lawson criterion
researchers have ignored the power balance and defy first-law fundamentals but the computation
Volume 2 | Issue 2 | 11
Eng OA, 2024
models developed here reveal the transition barrier persists.
Conclusions
With neutrons included for regeneration the system losses remain
so great the power ratio remains negative and the heating efficiency
must be 100%. In other interpretations of the Lawson criterion
researchers have ignored the power balance and defy first-law
fundamentals but the computation models developed here reveal
the transition barrier persists.
Conclusions
These charts are a reminder that regardless of creative fusion criteria
for predicting fusion ignition, basic physics and fundamental
thermodynamics still apply and should guide any new fusion
postulate. Modern derivations of fusion ignition criteria assume
plasma internal energy accumulation is positive and much greater
than zero or dW/dt >> 0. Computations show the energy balance
is mostly negative, applying a Lawson-like criterion to this
energy state is inappropriate. Assuming a bounding case shows
the heating efficiency must be at best 50% of the total system
energy to sustain fusion, however this bounding case assumes
re-circulation of energies from neutrons, Bremsstrahlung, and
system losses not achievable with present experimental apparatus.
Computations applying basic thermodynamic principles show
the conditions required to initiate fusion remain severe and do
not predict sustained fusion is possible. Despite the plethora of
tortured Lawson-like criteria in technical literature and convergent
computer code schemes predicting fusion ignition, sustained
fusion has not happened in any Laboratory [15-28].
References
1. h t t p s : / / w w w. b b c . c o m / n e w s / s c i e n c e - e n v i r o n m e n t 24429621and
https://www.sciencemag.org/news/2013/10/
fusionbreakthrough-nif-uh-not-really .
2. https://www.sciencemag.org/news/2013/10/fusionbreakthrough-nif-uh-not-really .
3. RiccardoBetti, Physics Today, ttps://physicstoday.scitation.
org/do/10.1063/PT.5.2004/full/
4. Hossenfelder, S. (2018). Lost in math: How beauty leads
physics astray. Hachette UK.
5. http://backreaction.blogspot.com/2019/07/the-forgottensolution- superdeterminism.html#comment-form
6. Lawson, J. D. (1955). Some criteria for a useful thermonuclear
reactor. Atomic Energy Research Establishment.
7. Atzeni, S., & Meyer-ter-Vehn, J. (2004). The physics of
inertial fusion: beam plasma interaction, hydrodynamics, hot
dense matter (Vol. 125). OUP Oxford.
8. Atzeni, S., & Meyer-ter-Vehn, J. (2004). The physics of
inertial fusion: beam plasma interaction, hydrodynamics, hot
dense matter (Vol. 125). OUP Oxford.
9. Lawson, J. D. (1955). Some criteria for a useful thermonuclear
reactor. Atomic Energy Research Establishment.
10. Atzeni, S., & Meyer-ter-Vehn, J. (2004). The physics of
inertial fusion: beam plasma interaction, hydrodynamics, hot
dense matter (Vol. 125). OUP Oxford.
11. Atzeni, S., & Meyer-ter-Vehn, J. (2004). The physics of
inertial fusion: beam plasma interaction, hydrodynamics, hot
dense matter (Vol. 125). OUP Oxford.
Eng OA, 2024
12. Atzeni and Meyer-Ter-Vehn, The Physics of Inertial Fusion,
Eq. 4-1; Suzanne Pfalzner, An Introduction to Inertial
Confinement Fusion, Eq 7-5; Wikipedia: Lawson Criterion,
13. National Research Council, Division on Engineering,
Physical Sciences, Board on Energy, Environmental Systems,
Board on Physics, & Committee on the Prospects for Inertial
Confinement Fusion Energy Systems. (2013). An assessment
of the prospects for inertial fusion energy. National Academies
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