Download Energy Scales of Physical Phenomena

Energy Scales of Physical Phenomena
energyOrder
scales:
pressure
of Magnitude
Astrophysics
Order of Magnitude Astrophysics
Pressure exerted by the a system of particles as the rate of momentum tran
Pressure exerted by the a system of particles as the rate of momentum transferred (per
exerted
by the
aofsystem
of !:
particles
as the rate of momentum transferred (pe
area)
from
particles
of energy
:
nitPressure
area) unit
from
particles
energy


nit area) from particles of energy !:
!
2
2
∞

<
n!
>
if
mc
!!
1
3
!
2
 2=< n! >
P 1= ∞
n(!)p(!)v(!)d!
if
mc
!!
3  1
3
2
O
P =
n(!)p(!)v(!)d!
=
<
n!
>
if
mc
"!
3
 1
3
2
O
3
< n! >
if mc " !
2
Rest
Mass
Energy:
We
can
associate
the
rest
mass
energy
mc
eachofparticle
2
est Mass Energy: We can associate the rest mass energy mc with eachwith
particle
mass m
 
2 m c2 ≈ 0.5MeV
 me
c
≈ 0.5MeV
e
2
2
mc =
mc =
 m
2
pc ≈
m1GeV
c2 ≈ 1GeV
p
the particles of the system have internal structure (molecular, atomic, nuclear, etc) then
f the particles of the system have internal structure (molecular, atomic, nuclea
articles of energy !:
area) from particles of energy
!:

1
=
3
!
∞
O

mc2 ! !
<
n!
>
if
scales: rest-mass
energy
! ∞=
 2 < n! >
n(!)p(!)v(!)d!
1
 1 < n! >
32
if
mc
"!
P =
n(!)p(!)v(!)d!
=
3
3
O
2
3

1
3
< n! >
if mc2 ! !
if mc2 " !
can associate
the rest
mass
energy mc2 with
particle
of mass
: We can We
associate
the rest
mass
energy
witheach
each
particle
of m.
mass m.
2
Mass Energy: We 
can
associate
the
rest
mass
energy
mc
with each particle
 me c2 ≈ 0.5MeV

mc2 =
 m c2 ≈ 1GeV
2
p
2
mc =
me c ≈ 0.5MeV
 m c2 ≈ 1GeV
p
the system have internal structure (molecular, atomic, nuclear, etc) then
ergy scales that are characteristic of the interactions. The simplest is
of the system
internal structure
(molecular,
atomic, nuclear,
etc) nuclear,
particlesIf the
of particles
the system
havehave
internal
structure
(molecular,
atomic,
we get further energy scales that are characteristic of the interactions. The
ng energy then
of
atoms
and molecules, which arises from the electromagnetic
simplest is the atomic binding energy of atoms and molecules, which arises from
et furthertheenergy
scalescoupling
that are
characteristic
of the interactions. The sim
electromagnetic
between
particles.
particles.
tomic binding energy of atoms and molecules, which arises from the electro
Energy: Size and energy of the ground state of a hydrogen atom:
ing between particles.2
!
−9
≈
5.2
×
10
cm
2
q and energy of the ground state of a hydrogen atom
ic Binding Energy:mSize
a0 =
energy scales: rest-mass
2
we
get
further
energy
scales
that
of the The
interactions.
mc
=are characteristic
getwefurther
energy
scales
that
are
characteristic
of the of
interactions.
simplest
get further
energy
scales
that
are
characteristic
the
interactions.
The sim

2
we
get
further
energy
scales
that
are
characteristic
of
the
interactions.
atomic binding energy of atoms and molecules,
arises from the electrom
mp cwhich
≈ 1GeV
atomic
binding
energy
of atoms
andofmolecules,
arises from
electromagne
the
atomic
binding
energy
atoms
andwhich
molecules,
whichthe
arises
from
th
the atomic
binding
energy
of
atoms
and
molecules,
which
arises
from
the
electro
energy
scales:
atomic
the atomic
binding energy of atoms and molecules, which arises from th
pling between
particles.
pling
between
particles.
coupling
between
particles.
coupling
between
particles.
between
particles.
Ifcoupling
theEnergy:
particles
of the
system
have
structure
atomic,
nuc
mic Binding
Size
and
energy
of internal
the ground
state (molecular,
of a hydrogen
atom:
mic
Binding
Energy:
Size
and
energy
theenergy
ground
ofstate
a hydrogen
atom:
Atomic
Energy:
Sizeenergy
and
ofstate
the ground
state
of
a hydrog
Size
andBinding
energy
of the
ground
state
ofof
a hydrogen
atom:
Atomic
Binding
Energy:
Size
and
of
the
ground
of
a
hydrogen
atom
weAtomic
get further
energy
scales
that
are
characteristic
of
the
interactions.
The
Binding Energy: Size2 and energy of the ground state of a hydrog
−9
2!
2−9
a0 = ! 2 !≈
10
cm
2 5.2 ×
!
2
the atomic binding energy
of
atoms
and
molecules,
a0 = a m=
×
10
cm
−9 which−9arises from the ele
!
e q≈a5.2
=
≈
5.2
×
10
2
≈
5.2
×
10
cm
−9cm
0 2
0e q
m
2
a
=
≈
5.2
×
10
cm
0 me q 2
4
m
q
e
m
q
1
e4
2
2 me q
coupling between!particles.
m
q
1
4
4
e
=
=
α
m
c
≈1 13.6
eV
me qα22 m
1ecq2eq2≈4 13.6
mem
2=
!a a= ! 2!
21 2eV 2
= 2αe m
eV
213.6
==
m
13.6
eV
a 2= !a2!2==
e c α≈
e c c2≈
2!
α
m
≈
13.6
eV
Atomic Binding Energy: Size
and
energy
of
the
ground
state
of a hydrogen a
a
e
2!
22!2 2 2
2!
2
e
α
=
q
2 /(!c) is the fine structure constant. The wavelength corresponding to !
2 wavelength corresponding to !a is
eα
= qα/(!c)
fine
structure
constant.
The
2 is the
2 is
!
Here
=
q
/(!c)
the
fine
structure
constant.
The wavelength
corresponding
t
is
the
fine
structure
constant.
2
−9wavelength
Here
α=
is is
thethefine
The
correspo
Here
α q= /(!c)
q /(!c)
finestructure
structure
constant.
The
wavelength
corresp
a0 = constant.
≈
5.2
×
10
cm
2
m
q
e
hc
2!
hc = 2! 4 ≈ 3103 A
λ
=
2!
≈
10
λ = ! = hc2α
hc
2em
m
q
1 A
hc
2!
33
c
2≈2!
23 A
=
10
λ
=
a
e
!
α
m
c
≈
10
AA
=
e =2= ==
α
m
c
≈
13.6
The wavelength corresponding ato !a =
is! λ λ
e
≈ 10eV
a 2!2α! m2
e c α2 m
2
a!a
e ce c
αm
lies in
in the
theUV.
UV.When
Whenatoms
atomsofofsize
size
aare
closely
packed:
0 are
lies
a
closely
packed:
0
2 UV.
and lies
inlies
the
When
atoms
ofstructure
are
closely
packed:
and
lies
inUV.
When
the fine
atoms
ofsize
size a0 are
closely
packed:
Here
αlies
=
qin
/(!c)
is When
the
constant.
The
wavelength
and
inthe
the
UV.
atoms
ofofsize
a0a0are
closely
packed:
and
the
UV.
When
atoms
size
are
closely
packed:correspondin
−3 −3
24 24 −3 −3
nsolid≈≈
(2a
)
≈
cm
nsolid
(2a
)
≈
10
hc
−310cm2!
24. . −3 3
nsolid 0≈0λ(2a
)
≈
10
.24A −3
−3cm
−3
−3
24
0
=
≈
10
n=
≈
(2a
)
≈
10
cm
nsolid
≈
(2a
)
≈
10
cm
..
2
solid
0
0
!a
α me c
and lies in the UV. When atoms of size a0 are closely packed:
energy scales: atomic
NGC 2237, the Rosette Nebula in Monoceros
energy scales: atomic
energy scales: molecular
The simplest molecular structure consists of two atoms bound to each other in the
form of a diatomic molecule:
energy scales: molecular
Molecular Binding Energy: In addition to the electronic internal binding energy of the
Molecular
Energy:
In addition
toto
the
binding
energy
of
the
InBinding
addition
to other
the electronic
internal
binding
energy ofinternal
the
atoms,
there
are twoVibrational
other
atoms,
there
are two
contributions
theelectronic
energy
a diatomic
molecule.
contributions to the energy of a diatomic molecule. Vibrational
energy levels
atoms, levels
there are
two other
contributions to the energy of a diatomic molecule. Vibrational
energy
separated
separated
by by
! "1/2
energy levels separated by
me
Evib = !ωvib ≈! "1/2 "a ≈ 0.25 eV.
mµe
"a ≈ 0.25 eV.
Evib = !ωvib ≈
µ
The molecule
can alsocan
rotate.
This will
with
The molecule
also rotate.
Thiscontribute
will contribute
withan
anenergy
energy of
of approximately
approximately
! "
The molecule can also rotate. This Jwill
2 contribute with an energy of approximately
me
Evrot = 2 2 ≈! " "a ≈ 10−2 eV.
Jµa
mµe
"a ≈ 10−2 eV.
Evrot = 20 ≈
µa
µ
The wavelengths of radiation from 0vibrational transitions are ∼ 40 times larger than
The wavelengths of radiation from vibrational transitions are ∼ 40 times larger than
those of electronic transitions; similarly the rotational transitions lead to radiation with
The wavelengths of radiation from vibrational transitions are about 40 times larger
those of electronic
transitions; similarly the rotational transitions lead to radiation with
than
those
3 of electronic transitions
wavelengths ∼ 10 larger.
wavelengths ∼ 103 larger.
Nuclear Energy Scales: Atomic nuclei are bound by the strong interaction force that
Nuclear Energy Scales: Atomic nuclei are bound by the strong interaction force that
produces a binding energy per particle ∼ 8 MeV, which is the characteristic scale for nuclear
produces a binding energy per particle ∼ 8 MeV, which is the characteristic scale for nuclear
energy levels. In the astrophysical context, a more relevant energy scale is the one at which
energy levels. In the astrophysical context, a more relevant energy scale is the one at which
nuclear reactions can be triggered. For two protons to fuse together, while undergoing
nuclear reactions can be triggered. For two protons to fuse together, while undergoing
energy scales: molecular
Energy levels of ammonia (NH3) in the lowest vibrational state (Wilson, T. L. et al.
1993, A&A, 276, L29). energy scales: molecular
Multi-wavelength overview of the Orion nebula.
energy scales: nuclear
Atomic nuclei are bound by the strong interaction force that produces a binding
energy per particle about 8 MeV, which is the characteristic scale for nuclear energy
levels
lear Energy Scales: Atomic nuclei are bound by the strong interaction force th
duces a binding energy per particle ∼ 8 MeV, which is the characteristic scale for n
energy scales: nuclear
gy levels. In the astrophysical context, a more relevant energy scale is the one at
ear reactions can be triggered. For two protons to fuse together, while undergo
In the astrophysical context, a more relevant energy scale is the one at which
nuclear
reactions
can be
triggered.
Forbrought
two protons
to fuse
ear reaction,
it is
necessary
that
they are
within
the together,
range ofwhile
attractive
undergoing nuclear reaction, it is necessary that they are brought within the range of
attractive overcoming
nuclear force. This
overcoming
the Coulomb repulsion:
e. This requires
therequires
Coulomb
repulsion:
n
q2
α
"≈
=
mp c2 ≈ 1 MeV.
l
2π
, however, possible for nuclear reactions to occur through quantum-mechanical tun
n the de Broglie wavelength λdeB = h/(mp v) of the two protons overlap. This oc
n the energy is approximately
"nucl
α2
≈ 2 mp c2 ≈ 1 keV.
2π
vitational Binding Energy: In the non-relativistic Newtonian theory of gravity, t
× 10 W
order Lof= 4magnitude
26
!
L! = 4 × 1026 W
(12)
(12)
L! = 4 × 1026 W
(12)
This discussion assumes that the sunUis primary
supported
by
pressure
of
an
ideal
(13)
th ∼ kT
gas of electrons and ions. Thus, in a star, the total (thermal) kinetic energy of
Uth ∼ kT
(13)
particles is
Uth ∼ kT
k = 1.38 × 10−23 J/K
An estimate of the typical temperature
the
Sun can be estimated by
k = 1.38inside
× 10−23
J/K
combining the kinetic and gravitational potential
energy per particle:
−23
mpJ/K
kL
= 1.38
×GM
10
!26
4
×
10
W
!
kT=
∼
i
R
GM!!mp
kTi ∼
R!
GM! mp
kT
∼
7
i
Ti ∼
1.8∼×R
10
K
Uth
kT
!
Ti ∼ 1.8 × 107 K
where k is the Boltzman constant:
√
7
L!
8
10
K
R! = √Ti√∼ 1.8 ×
=
7.1
−23 × 10 m
2
k =4πσ
1.38
J/K 8
L×
T!10
!×
R! = √
=
7.1
× 10 m
4πσ × T!2
√
L!
8
R! = √
=
7.1
×
10
m
GM
m
2 ! p
4πσ
×
T
kTi ∼ !
R
(13)
(14)
(14)
(14)
(15)(12)
(15)
(15)
(16)
(13)
(16)
(16)
(17)
(14)
(17)
(17)
(15)
sun’s structure
nuclear reaction, it is necessary that they are brought within the range of attract
q2
α
2
force. This "requires
overcoming
the
Coulomb
repulsion:
≈
=
m
≈
1
MeV.
pc
energy scales:
l
2π nuclear
2
"≈
however, possible for nuclear reactions to
q
α
=
mp c2 ≈ 1 MeV.
l
2πthrough quantum-mechanical
occur
It is, however, possible for nuclear reactions to occur through quantum-mechanica
It is, however,
possible for λ
nuclear
through
quantum-mechanical
the de Broglie
wavelength
h/(mptov)occur
of the
two
protons overlap. Thi
deB =reactions
whenthe
thededeBroglie
Broglie
wavelength λdeB = h/(mp v) of of
thethetwo
tunneling when
wavelength
twoprotons
protonsoverlap. Th
overlap. This occurs when the energy is approximately
the energy is approximately
when the energy is approximately
"nucl
α2 "nucl2 ≈ α2 mp c2 ≈ 1 keV.
≈ 2 mp c ≈2π
1 2keV.
2π
Gravitational Binding Energy: In the non-relativistic Newtonian theory of grav
tational Bindinggravitational
Energy: binding
In theenergy
non-relativistic Newtonian theory of gravi
tational binding energy
Egrav
Egrav
GM 2
Gmp2 2
≈
N .
≈
R
R
GM 2
Gmp2 2
≈
N .
≈
R
R
"nucl
α2
≈ 2 mp c2 ≈ 1 keV.
2π
energy scales: gravitational
Gravitational Binding Energy: In the non-relativistic Newtonian theory of gravity, the
In the non-relativistic Newtonian theory of gravity, the gravitational binding energy:
gravitational binding energy
Egrav
Gm2p 2
GM 2
≈
N .
≈
R
R
The potential energy per particle:
The potential energy per particle varies as
Gm2p
Egrav
4π 1/3
!g =
=
N=
Gm2p N 2/3 n1/3 .
N
R
3
otential energyGeneral
per particle
varies effects
as
relativistic
become important when Rgm = Egrav /Emass ∼ 1.
GeneralErelativistic
important when
1/3
Gm2pbecome
4π
grav
Thermal
and
Degeneracy
Energy:
of the system depends on the origin of
!g =
=
N=
Gm2pThe
N 2/3behavior
n1/3 .
N
R
3
the momentum distribution of the particles. The familiar situation is the one in which
al relativistic effects become important when Rgm = Egrav /Emass ∼ 1.
short-range interactions between particles effectively exchange the energy so as to randomize
mal and Degeneracy Energy: The behavior of the system depends on the origin of
the momentum distribution. When such a system is in steady state, we can assume that the
omentum distribution of the particles. The familiar situation is the one in which
local thermodynamical equilibrium, characterized by a temperature T , exists in the system:
range interactions between particles effectively exchange the energy so as to randomize
omentum distribution. When such a system is in steady! state,
≈ kB Twe can assume that the

Gmorigin
Egrav
4π of 2 2/3 1/3
p
the system depends
!g = on the
=
N=
Gmp N n .
N
R
3
Generalsituation
relativistic
effects
become
important
when Rgm = Egrav /Emass ∼ 1.
amiliar
is the
one
in which
energy
scales:
thermal
Thermal and Degeneracy Energy: The behavior of the system depends on the origin of
y exchange the energy so as to randomize
the momentum distribution of the particles. The familiar situation is the one in which
The behavior of the system depends on the origin of the momentum distribution of
short-range
interactions
betweensituation
particles
the energyinteractions
so as to randomize
the particles.
The familiar
is effectively
the one inexchange
which short-range
between particles effectively exchange the energy so as to randomize the momentum
the momentum
systemstate,
is in we
steady
wethat
canthe
assume
distribution.distribution.
When such aWhen
systemsuch
is ina steady
can state,
assume
local that the
thermodynamical equilibrium, characterized by a temperature T, exists in the system:
s in steady state, we can assume that the
y a local
temperature
T , exists in the system:
thermodynamical equilibrium, characterized by a temperature T , exists in the system:
! ≈ kB T
!
"
#2 $1/2

 (2mkB T )1/2
≈
 k T /c
if mc2 # kB T
kB T
mc2
2
1/2
if
mc
$ kB T
B
kB T )
if mc # kB T
In this case, the momentum and the kinetic energy of the particles vanish when T → 0. The
2
/c mean energy
mc
$ofkelectrons
In this if
case,
the
momentum
and thewill
kinetic
particle
vanishes when
BT
of a system
notenergy
vanishofatthezero
temperature
because electrons
2kB T
p ≈ mc
+
2
mc
2
obey the Pauli exclusion principle, which requires that the number of electrons that can
of the particles vanish when T → 0. The
occupy any quantum state be two. Because the uncertainty principle requires ∆x∆px ≥ h,
h atwezero
temperature
electrons
can associate
the number because
of quantum states
with a phase space volume:
2
1/3
energy scales: thermal
#"
!"
$1/2
1/2
2 2
22

1/2
#
(2mk
T
)
if
mc
#
kBk2TB#
1/2
2

B
2k
T
k
T
(2mk
T
)
if
mc
#
T kB T
1/2
2 2k kB
B
B
T
(2mk
T
)
if
mc
B
B
B
T
k
T
2mkBppT≈
)≈ mc
if
mc
#
k
T
B
B
mc p ≈ mc
+
≈≈
B+
+
≈
2
2
2
2

mc
mc
2 2
2
2

mc

mcmc
mc
k
T
/c
if
mc
$
kBk2TB$
B
k
T
/c
if
mc
T kB T
kB T /c
if $
mc
B
2
energy
scales: degeneracy
kB T /c
if mc $ kB T
his case,
case,
thecase,
momentum
and
kinetic
vanish
when
Twh
In this
the momentum
the energy
kinetic
energy
of
the particles
vanish
his
the
momentum
and the
theand
kinetic
energyof
ofthe
theparticles
particles
vanish
when
T→
rgy of the particles vanish when T → 0. The
an
energy
ofenergy
amean
system
will
not
vanish
zero
temperature
because
energy
ofelectrons
a system
of
electrons
willnot
not at
vanish
atat
zero
temperature
meanThe
of a of
system
of electrons
will
vanish
zero
temperature
bee
n energy
of
a
system
of
electrons
will
not
vanish
at
zero
temperature
becaus
because
electrons obey the
Pauli exclusion
principle, which requires that the number
anish at zero
temperature
because
electrons
ofthe
electrons
thatexclusion
can
occupy principle,
anywhich
quantumrequires
state be two.
y
the
Pauli
exclusion
principle,
that
number
ofofelectrons
th
obey
Pauli
which
requires
that
the number
of electr
y the Pauli exclusion principle, which requires
thatthe
the
number
electrons
res that the number of electrons that can
occupy
anythe
quantum
berequires
two. Because
the uncertainty principle
require
Because
uncertainty
principle
upy any
quantum
state
be state
∆x∆
py any quantum
state
be two.
two. Because
Becausethe
theuncertainty
uncertaintyprinciple
principlerequires
requires
∆
uncertainty
principle
requires
∆x∆p
x ≥ h,states with a phase space volume:
we
can
associate
the
number
of
quantum
can associate the number of quantum states with a phase space volume:
an associate the number of quantum states with a phase space volume:
we can associate
the number of quantum states with a phase space volume:
with a phase
space volume:
2
1/3
1/3
pF 2=
!(3π
pF = !(3π
n)
. n) .
2
1/3
pF = !(3π n) .
n)1/3 . The The
the
quantum
mechanical
scale of the
energy
!F sets
sets
the
quantum
mechanical
of the energy
quantity quantity
!F quantity
sets the
quantum
mechanical
scale ofscale
the
energy
 scale
)of 2the
* energy
quantity !F sets the quantum mechanical
2

)
*
p
!
2
2/3
2
(
F 2
2 
p
=
(3π
n)
if
mc
# !F

!
2
2/3
2
)
*
cale of the(energy
F

2m
2m
2
=
(3π
n)
if
mc
#
!
2 2
2
F
2
pF c2 4+ m2 c42 − 
mc p2m
≈
!2m
2
2/3
2
(
F
2 !F2
(3π n) 2 1/3 if mc # !2F
!F = pF c + m c − mc
≈ 2m =
*

2m
2
2
2 + m2 c4 −
F c = (!c)(3π
2
1/3 n)
2if mc $ !F
!2
2 p2/3
2 mc ≈  p c = p(!c)(3π
!
c
F
F
n)
if
mc
$ !F
(3π n)
if mc # !F  F
m
2
1/3
2
= (!c)(3πifn)
mc $ !F E
F c dominant
Quantum-mechanical effects willpbe
!F # kB Tif(degenerate).
antum-mechanical
will!Fbe dominant if !F # kB T (degenerate). Electron
!c)(3π 2 n)1/3
ifeffects
mc2 $
ntum-mechanical
!2F ∼ me c2 foreffects will be dominant if !F # kB T (degenerate). Electr
∼f m
for
!Fe c#
kB T (degenerate). Electrons"have#−3
2
2
1/3
pF = !(3π 2 n)1/3 .
p
=
!(3π
n)
.
F
2
1/3
pF = !(3π n) .
energy
scales:
degeneracy
sets the quantum mechanical
scale
of!thesets
energy
The quantity
the
quantum mechanical scale of the ener
F

) 2mechanical
*
quantity !F sets the quantum
scale of the energy
 2
2
) 2*
p
2 2/3
2
 F = ! 
p
(3π
( 2n) ) *if mc # !F  F = ! (3π 2 n)2/3
2m
2m
2
2
2
4
2m
pF2 2
c + m c(− mc ≈
2
!2 2 4
2mc2/3
2 2m

!
p
c
+
m
c
−
≈
=
(3π
n)
if
mc
# !F
F

Quantum-mechanical
effects
will be
22mFdominant
1/3 2m if
2
2
2 2
2
4

p
c
=
(!c)(3π
n)
if
mc
$
!
2
1/3
F mc ≈
F
!F pF c + m c −
p
c
=
(!c)(3π
n)
F
 p c = (!c)(3π 2 n)1/3
if mc2 $ !F
F
nical effects will be dominant if !F # kB T (degenerate). Electrons have
Quantum-mechanical effects will be dominant if !F # kB T (de
ntum-mechanical effects will be dominant if !F # kB T (degenerate). Electrons hav
2
Electrons have !F ∼ me c for
for
"
#
−3
me c2 for
"
#−3
!
31
−3
!
n=
≈ 10
cm#
31
−3
"
−3
n
=
≈
10
cm
me c
!
me c
31
−3
n
=
≈
10
cm
7
−3
ρ ∼ nmp ≈ 10 gmcm
.
ec
ρ ∼ nmp ≈ 107 g cm−3 .
ρ ∼ nmp ≈ 107 g cm−3 .
!F ∼ kB T occurs
occursatat
∼ kB T occurs at
nT −3/2
3/2
(mk
)
B
−3/2
16
2
3/2
nT
=
=
3.6
×
10
(cgs)
for
mc
# !F .
(mkB )
3
16
2
=
= 3.6 × !10 (cgs) for mc # !F .
3
!
The energy sclae of the indiviudal particles also characterizes the energy i
e energy sclae of the indiviudal particles also characterizes the energy involved in the
!
!3 particles also characterizes the energy involved
nergy sclae of the indiviudal
ons between particles.
If thisscales:
quantity isionization
larger than the binding energy of th
energy
energy
sclae
of the indiviudal
particles
also
he energy sclae of the indiviudalThe
particles
also
characterizes
the energy
involved
in cha
the
m, the atoms will be ionized and the electrons will be separated:
particles.
If this energy
quantity
is larger
ollisions between particles. If thiscollisions
quantity between
is larger than
the binding
of the
atom
ystem,
The energy scale of the individual particles also characterizes the energy involved in
collisions
particles.
Ifthe
quantity
larger
the binding
energy
system,
the
will
be
ionized
and
the ofelectrons
kthis
≥ atoms
!a .is will
thethe
atoms
will between
be ionized
and
electrons
bethan
separated:
BT
the atomic system, the atoms will be ionized and the electrons will be separated:
w
kB T ≥
!a . w
kB T ≥ !half
a.
ransition temperature at which nearly
of the atoms are ionized
occurs
4
temperature
atThe
which
nearly
half ofof
thethe
atoms
areatoms
ionized
occurs
when
≈he!atransition
/10 ∼The10transition
K. The
atoms
can
be
stripped
off
the
in another
contex
transition
temperature
at which
nearly
of
temperature
at which
nearly
half
atoms
are
ionized
occurshalf
when
≈ !a /10
104 K. The
off4are
the
atoms
in together,
another
Thi
s when
the∼density
is soatoms
highkcan
that
atoms
with elec
T be
≈ the
!stripped
∼ 10
K. so
Theclose
atoms
can becontext.
stripped
o
B
a /10
BT
ccurs
whenThe
the
density
is stripped
so highoccurs
the
atoms
so close
with
atoms
can be
offthat
the atoms
in another
context.
occurs
when
the
when
the are
density
isThis
sotogether,
high that
theelectrons
atoms a
ng a common
pool
rming a
density is so high that the atoms are so close together, with electrons forming a
common pool
common
pool
forming a common pool
!F ≥ !a .
!F ≥ !a .
In this case, the electrons will be quantum mechanical.
s case, the electrons will be quantum mechanical.
!F ≥ !a .
n this case, the electrons will be quantum
mechanical.
In this case,
the electrons will be quantum mechanic
Plasmas and Magnetic Fields: To treat a plasma as ideal, it is necessary tha
deal Plasmas and Magnetic Fields:
To Plasmas
treat a plasma
as ideal, Fields:
it is necessary
that
the
Ideal
and Magnetic
To treat
a plas
mb interaction energy of ions and electrons be negligible. The typical Coul
!F ≥ !a .
this case, the electrons will be quantum! mechanical.
F ≥ !a .
this case,and
the Magnetic
electrons
will
be
quantum
mechanical.
energy
scales:
ideal
plasma
alInPlasmas
Fields:
To treat
a plasma
as ideal, it is necessary tha
In this case, the electrons will be quantum mechanical.
Ideal Plasmas and Magnetic Fields: To treat a plasma as ideal, it is necessary that th
ulombIdeal
interaction
energy of ions and electrons be negligible. The typical Coulo
Plasmas and Magnetic Fields: To treat a plasma as ideal, it is necessary that the
CoulombTointeraction
energy of ions and electrons be negligible. The typical Coulomb
treat a plasma as ideal, it is necessary that the Coulomb interaction energy of ions
tentialCoulomb
energy
interaction
between
ions
and
the
electrons
in the
is given b
interaction
energy
of ions
and
electrons
be negligible.
Theplasma
typical
and electrons
be negligible.
The
typical
Coulomb
potential
energy interaction
betweenCoulomb
potentialions
energy
between
and
and theinteraction
electrons in the
plasma ions
is given
by the electrons in the plasma is given by
potential energy interaction between ions and the electrons in the plasma is given by
!Coul ≈ Zq 22n1/3
1/3 .
!Coul ≈ Zq 2n 1/3.
!Coul ≈ Zq n
.
If the high temperature
plasma
is tobe
betreated
treated as as
an ideal
gas, this
energy
should
be should
heIfhigh
temperature
plasma
is
to
an
ideal
gas,
this
energy
the high temperature plasma is to be treated as an ideal gas, this energy should be
If the high temperature plasma is to be treated as an ideal gas, this energy should be
!Coul
≤ kkBT,
!Coul
T,
! ≤≤kB
T,
Coul
B
which
requires
requires
which
requires
ichwhich
requires
!!!k k""3"3 3
BB
88 −3
−3−3
2.2
×
10
×
10
(cgs).
nT
8Z −3(cgs).
−3nT ≤ ≤kB 2 ≈≈2.2
2
≈
2.2
×
10
Z
(cgs).
nT ≤
Zq
Zq
2
Zq
energy scales: ideal plasma
On the
other to
hand,
to treat
thehigh-density
high density quantum
gas as ideal
n the other
hand,
treat
the
quantum
gas as ideal
the other hand, to treat the high-density quantum gas as ideal
hich requires
which requires
ich requires
!Coul ≤ !F = (!2 /2m)n2/3 ,
!Coul ≤ !F = (!2 /2m)n2/3 ,
26 3
−3
n ≥ 8Z 3 a−3
≈
10
Z
cm
0
3 −3
n ≥ 8Z a0 ≈ 1026 Z 3 cm−3
ρ ≈ nmp ≥ 102 g cm−3 .
ρ ≈ nmp ≥ 102 g cm−3 .
t us now go back to the tacit assumption we made in the above analysis, vi
t us now go back to the tacit assumption we made in the above analysis, viz., th
ysical interactions between the particles of ths system are capable of mantai
ysical interactions between the particles of ths system are capable of mantaining
ermal equilibrium. Determining the precise condition that will ensure this is n
rmal equilibrium. Determining the precise condition that will ensure this is not a s
sk; but naively we would require that (i) the mean free path for particles