Neutron star crust (nuclear and plasma physics)

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Neutron star crust (nuclear and plasma physics)
Neutron star crust
(nuclear and plasma physics)
Lecture 1
Pawel Haensel
Copernicus Astronomical Center (CAMK)
Warszawa, Poland
[email protected]
La̧dek Zdrój, February, 2008
Pawel Haensel (CAMK)
Neutron star crust Lect.1
La̧dek Zdrój, Poland
1 / 23
Neutron star crust
Problems and Methods
White dwarfs and neutron star envelopes (below 108 g cm−3 : plasma
and atomic physics (many-particle theory, non-ideal dense plasmas)
Neutron star crusts: plasma and nuclear
physics (108 g cm−3 -1014 g cm−3 (strongly
coupled Coulomb plasma, unstable
neutron-rich nuclei, nuclear many-body
problem, ... ). Basic constituents - n p e like in “terrestrial physics”.
Standard reference: S.L. Shapiro, S.A. Teukolsky Black Holes, White
Dwarfs, and Neutron Stars: The Physics of Compact Objects (Wiley,
Extreme physical conditions: density of the
matter, high neutron excess due to dense
electron gas.
1983). Alas, 25-years old.
New up-to-date reference: P. Haensel, A.Y. Potekhin, and D.G. Yakovlev Neutron
Stars 1. Equation of state and structure (Springer, 2007)
Pawel Haensel (CAMK)
Neutron star crust Lect.1
La̧dek Zdrój, Poland
2 / 23
Structure of envelope and crust
Schematic structure of an envelope of a neutron star with the internal
temperature ∼ 108 K.
Pawel Haensel (CAMK)
Neutron star crust Lect.1
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3 / 23
Full pressure ionization
mu =
1
12
12 M ( C)
≈ m0 =
1
56
56 M ( Fe)
= 1.66 × 10−24 g
mean distance between electrons - re
−1
4π 3
Z ρ
=
r
ne ≈
A m0
3 e
radius of the innermost electron orbit in an (A, Z) atom
a0
~2
aZ0 =
, a0 =
= 0.5292 × 10−8 cm
2
Z
me e
expect complete ionization when re . aZ0
ρci ≈ 2.7 AZ 2 g cm−3
Examples
ρci (56 Fe) ∼ 105 g cm−3
Pawel Haensel (CAMK)
ρci (4 He) ∼ 102 g cm−3
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4 / 23
Plasma parameters
Simplifying approximation: one kind of nuclei (A, Z), A = N + Z. Model - one
component plasma (OCP). Number density of nuclei (=ions) ni . Electron
number density ne = Zni .
Outer crust: all nucleons bound in nuclei =⇒ (macroscopic) nucleon (=baryon)
number density nb = A ni
Inner crust nb = A0 ni where A0 ≡ A + A00 . Here A00 is the number of free
(unbound) nucleons per one atomic nucleus. We have A00 ni = nn (1 − w), where
w is the fraction of volume occupied by atomic nuclei (be careful - neutron skin
complicates things ...) .
Outer crust A0 = A, inner crust A0 > A.
Mass density ρ = E/c2 , but in the envelope and crust ρ ≈ mu nb , where
mu = 1.6605 × 10−24 g is the atomic mass unit.
Pawel Haensel (CAMK)
Neutron star crust Lect.1
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5 / 23
Electrons
Relativity parameter for electrons
xr ≡
pF
≈ 1.00884
me c
ρ6 Z
A0
1/3
,
(1)
where
pF = ~kF = ~ (3π 2 ne )1/3 ,
6
is the electron Fermi momentum and ρ6 ≡ ρ/10 g cm
q
F = c2 (me c)2 + p2F
(2)
−3
. The Fermi energy
(3)
Electron rest energy me c2 is included. Electron Fermi temperature
TF = Tr (γr − 1) ,
(4)
where
Tr = me c2 /kB ≈ 5.930 × 109 K ,
γr =
p
1 + xr 2 ,
(5)
is the relativistic temperature unit, kB - Boltzmann constant.Non-relativistic T Tr and xr 1. Ultra-relativistic - xr 1 or T Tr . Non-degenerate T TF . Strongly degenerate - T TF .
Pawel Haensel (CAMK)
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6 / 23
Electrons - continued
p
The electron Fermi velocity vF = (∂p /∂p)pF = cxr / 1 + x2r = cxr /γr .
4π 3
a
3 e
The relative strength of the electron Coulomb interaction for ultra-relativistic
degenerate electron plasma
ne =
e2
≈ 2 × 10−3
ae F
(6)
The electron plasma temperature
Tpe = ~ωpe /kB ≈ 3.300 × 108 xr
p
βr K .
(7)
where βr = vF /c and electron plasma frequency is given by
ωpe = 4πe2 ne /m∗e
1/2
(8)
Here, m∗e ≡ F /c2 = me γr is the effective dynamical mass of an electron at the
Fermi surface.
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7 / 23
Ions - 1
The strength of the Coulomb interaction of ions is characterized by the Coulomb
coupling parameter,
Γ = (Ze)2 /(ai kB T ),
(9)
where
ai = (3ni /4π)1/3
(10)
is the ion-sphere radius.
At sufficiently high temperatures, the ions form a classical Boltzmann gas. With
decreasing T , the gas gradually, without a phase transition, becomes a strongly
coupled Coulomb liquid, and then (with a phase transition) a Coulomb crystal.
The gas and liquid constitute the neutron star ocean, the crystal is formed under
the ocean.
Pawel Haensel (CAMK)
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8 / 23
Ions - 2
The ion-sphere radius ai - ion-sphere model for strongly coupled Coulomb
systems (T . Tl ) (liquid - solid). Nearly uniform electron background. In this
model the matter is considered as an ensemble of ion spheres filled by the uniform
electron background. Ion spheres are electrically neutral, ≈ non-interacting
between themselves.
1
ni
= 43 πa3i
Ecell = WN (A, Z) + WL + We
WL = Wee + Wei
Wee =
3 Z 2 e2
5 ai
WL = −
Pawel Haensel (CAMK)
2 2
Wei = − 32 Zaie
We =
Ee (ne )
ni
9 Z 2 e2
10 ai
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9 / 23
Ions - 3
The crystallization of a Coulomb plasma occurs at a well defined temperature
Tm =
ρ 1/3 175
Z 2 e2
6
≈ 1.3 × 105 Z 2
K,
ai kB Γm
A0
Γm
(11)
where Γm is the ”melting value” of Γ. For a classical OCP, Γm ≈ 175 .
At low T the quantum effects on ion motion become important. They are
especially pronounced at T Tpi , where
Tpi ≡
~ωpi
≈ 7.832 × 106
kB
ρ6 Z 2
A0 A
1/2
K
(12)
is the plasma temperature of ions determined by the ion plasma frequency
ωpi = 4πe2 ni Z 2 /mi
1/2
.
(13)
Notice: Tpi Tei . If Tpi Tm , huge zero-point ion vibrations suppress crystallization and reduce the melting temperature. Under these
conditions, the actual melting temperature decreases with growing ρ and drops to zero at a certain ρ = ρm : not relevant for crust
Pawel Haensel (CAMK)
Neutron star crust Lect.1
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10 / 23
ρ − T diagram for the outer envelope and crust
Density-temperature diagram for the outer envelope composed of carbon (left) or iron
(right). Electron Fermi temperature (TF ), electron and ion plasma temperatures (Tpe
and Tpi ), temperature of the gradual gas-liquid transition (Tl ), and the temperature of
the sharp liquid-solid phase transition (Tm ). Shaded: typical temperatures in the outer
envelopes of middle-aged cooling neutron stars (age ∼ 104 − 106 ). Corner bounded by the dotted line:
strong electron non-uniformity or bound-state formation (atoms!) in the iron plasma. Difficult region!
Pawel Haensel (CAMK)
Neutron star crust Lect.1
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11 / 23
Pressure - WS cell, rigid e-background
P = PL + P e ,
1
PL = n i W L < 0
3
for ρ6 1 :
Pawel Haensel (CAMK)
Pe , PL ∝ ρ4/3
Neutron star crust Lect.1
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12 / 23
Crystal phase
T < Tm - an infinite ion motion replaced by oscillations near
equilibrium positions: a crystal is formed.
Internal energy at T = 0 consists of two parts,
U0 = UM + Uq ,
(14)
UM = −Ni CM (Ze)2 /ai .
(15)
where
UM - classical static-lattice energy, CM - Madelung constant.
Uq accounts for the zero-point quantum vibrations.
CM (bcc) = 0.895929 , CM (fcc) = 0.895874 , CM (hcp) = 0.895838
(Fuchs 1935)
=⇒ ion-sphere model: UM = −0.9 Ni Z 2 e2 /ai is sufficiently accurate for the ion
liquid and solid, as long as the ions are strongly coupled (Γ 1)
Pawel Haensel (CAMK)
Neutron star crust Lect.1
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13 / 23
Melting - phase transition
The solid-liquid phase transition occurs at Γ = Γm , where the free energies F (Γ)
of liquid and solid phases intersect. Difficult to find Γm with high precision:
intersection of F -s strongly affected by the thermal corrections which are small at
Γ ∼ Γm (F - curves are nearly parallel).
The phase transition is 1st order but very weak. Fix the number density of ions
and decrease the temperature =⇒ discontinuous changes of the ion energy Ui and
the ion pressure Pi at the melting point. One can show that both quantities
undergo discontinuous drops at Γ = Γm . The drop of the ion energy density
Ui /V , associated with the crystallization to the bcc lattice at Γm = 175, is
∆Ui /V ≈ 0.76 ni kB Tm , with ∆Ui /Ui ≈ 0.5%, and the drop of the ion pressure
∆Pi /Pi is approximately three times smaller. Because the ion pressure is only a
small part of the total pressure P (the leading part Pe is provided by the
electrons), the fractional drop of the total pressure ∆P/P is much smaller than
the fractional drop of the ion pressure ∆Pi /Pi .
Pawel Haensel (CAMK)
Neutron star crust Lect.1
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14 / 23
Latent heat at freezing
The latent heat released
Qlatent = Um,liq − Um,solid ≈ ∆Ui ≈ 0.76 Ni kB Tm , where and Um,liq and
Um,solid are the corresponding values of the internal energy at T = Tm .
Qlatent ≈ 5 × 10−3 Ui . Remember that Ui itself is a tiny fraction of
the total internal energy U (mainly determined by degenerate
electrons). Nevertheless, the absolute value of Qlatent (≈ 0.76kB T per
ion) is a substantial fraction of thermal thermal energy of the ions.
Note: dense matter may form a supercooled liquid and freeze at
T < Tm .
Pawel Haensel (CAMK)
Neutron star crust Lect.1
La̧dek Zdrój, Poland
15 / 23
Electrostatic corrections to electron presure
(e)
Approximate Pe = Pid (ideal fully degenerate electron gas), add
PL ≈ −0.3 ni Z 2 e2 /ai . This gives
Z 2 e2
Pr
2 2
x
−
1
γ
+
ln(x
+
γ
)
−
0.3
n
.
P =
x
r
r
r
r
i
r
8π 2
3
ai
where
(16)
Pr = me c2 /(~/me c)3 = 1.4218 × 1025 dyn cm−2
This EOS can be used in the outer envelope of NS not too close to the surface. It
is also widely used in white dwarf cores. It is temperature independent and applies
to any composition of the matter. In the ultrarelativistic electron gas (xr 1)
both Pe and PL pressures have the same (polytropic) density dependence,
Pe ∝ PL ∝ xr 4 . Therefore, their ratio is density independent being
determined only by the ion charge number Z,
PL
6
=−
Pe
5
4
9π
1/3
Z 2/3 e2
≈ −0.0046 Z 2/3 ,
~c
(17)
(e)
which gives PL /Pid ≈ −0.04 for the matter composed of iron.
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Neutron star crust Lect.1
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16 / 23
Ground state crust - T = 0 - 1
Notation: enthalpy per nucleon = h; energy per nucleon=E; energy density - E.
Neglect envelope of NS with ρ < 108 g cm−3 (it has mass < 10−8 M ) - it could
have been accreted, and the T -effects there might be significant. Consider crust
at T = 0 and P = 0, i.e., h = E = E/nb . In this case the minimum energy per
nucleon (ground state) is reached for a body-centered-cubic (bcc) lattice of 56 Fe,
and is E(56 Fe) = 930.4 MeV. It corresponds to ρ = 7.86 g cm−3 and
nb = 4.73 × 1024 cm−3 = 4.73 × 10−15 fm−3 .
Remark56 Fe is not the most tightly bound free atomic nucleus. The maximum
binding energy per nucleon b ≡ [(A − Z)mn c2 + Zmp c2 − M (A, Z)c2 ]/A in a
nucleus with the ground-state mass M (A, Z) is reached for 62 Ni,
b(62 Ni) = 8.7945 MeV, to be compared with b(56 Fe) = 8.7902 MeV. Let us
notice that b(58 Fe) = 8.7921 MeV is also higher than b(56 Fe).
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Neutron star crust Lect.1
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17 / 23
Ground state crust - T = 0 - 2
Ion-sphere model used - Γ 1. Terminology: ion sphere ≡ (spherical) unit
cell, (spherical) Wigner-Seitz cell
Charge neutrality of the unit cell, under pressure P ,
ne = Zni ,
P = Pe (ne , Z) + PL (ni , Z) ,
(18)
where Pe is the electron pressure and PL is the “lattice” contribution (also called
the electrostatic correction) resulting from the Coulomb interactions.
Spherical ”unit cell” per one nucleus in a OCP. At T = 0 enthalpy of the
cell=Gibbs free energy (G = H − T S) of the cell =Gcell .
Gcell (A, Z) = WN (A, Z) + WL (Z, nN ) + [Ee (ne , Z) + P ]/ni ,
(19)
where WN is the energy of the nucleus (including rest energy of nucleons), WL is
the lattice (Coulomb) energy per cell, and Ee is the mean electron energy density.
Neglecting quantum and thermal corrections and the nonuniformity of the
electron gas, we have
WL = −CM Z 2 e2 /rc ,
Pawel Haensel (CAMK)
Neutron star crust Lect.1
CM ≈ 0.9.
(20)
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Ground state crust - T = 0 - 3
The lattice (Coulomb) contribution PL = 13 WL ni .
The Gibbs free energy per nucleon h = g = Gcell /A is just the baryon
(=nucleon) chemical potential µb (A, Z) for a given nuclide. To determine the
ground state at a given P , one has to minimize µb (A, Z) with respect to A and Z.
To a very good approximation, a density jump, at ground-state values (A, Z)
change into (A0 , Z 0 ), is given by
∆ρ
∆nb
Z A0
≈
≈
−1 .
ρ
nb
A Z0
(21)
This equation follows from the continuity of the pressure P ≈ Pe .
A sharp discontinuity in ρ and nb is a consequence of the assumed
one-component plasma model.
Problem Derive formula for ∆ρ/ρ, Eq. (21)
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19 / 23
Neutronization - simple argument
g = Gcell /A
T =0
Pawel Haensel (CAMK)
g = h , µe = F (ne )
Neutron star crust Lect.1
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20 / 23
A model of ground state crust - 1
Use experimental masses if they are available. Because of pairing
effect (nucleons in nuclei are superfluid!) only even-even nuclei are
relevant for the ground-state problem. For the remaining isotopes, up
to the last one stable with respect to the emission of a neutron pair,
use theoretical masses.
Example 1 (Haensel & Pichon 1994) The equilibrium nuclides
present in the ”cold catalyzed matter” are listed in Table. In the fifth column one
finds the maximum density ρmax at which a given nuclide is present. The sixth column gives the electron chemical potential µe at ρ = ρmax . The
transition to the next nuclide has a character of a first-order phase transition. The corresponding fractional density jump ∆ρ/ρ is given in the last
column. The last row above the horizontal line, which divides the table into two parts, corresponds to the maximum density, at which the ground state of
dense matter contains a nucleus with mass measured in laboratory. The last row of Table corresponds to the neutron drip point which is determined
theoretically.
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Neutron star crust Lect.1
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21 / 23
Ground state crust - T = 0 - nuclei - 1
Table: Nuclei in the ground state of cold dense matter (after Haensel & Pichon 1994),
with slight modification. Upper part is obtained with experimentally measured nuclear
masses. Lower part: from mass formula of Möller. The last line corresponds to the
neutron drip point.
element
Z
N
Z/A
ρmax
(g cm−3 )
56 Fe
62 Ni
64 Ni
66 Ni
86 Kr
84 Se
82 Ge
80 Zn
26
28
28
28
36
34
32
30
30
34
36
38
50
50
50
50
0.4643
0.4516
0.4375
0.4242
0.4186
0.4048
0.3902
0.3750
7.96
2.71
1.30
1.48
3.12
1.10
2.80
5.44
78 Ni
126 Ru
124 Mo
122 Zr
120 Sr
118 Kr
28
44
42
40
38
36
50
82
82
82
82
82
0.3590
0.3492
0.3387
0.3279
0.3167
0.3051
9.64 × 1010
1.29 × 1011
1.88 × 1011
2.67 × 1011
3.79 × 1011
(4.32 × 1011 )
Pawel Haensel (CAMK)
×
×
×
×
×
×
×
×
106
108
109
109
109
1010
1010
1010
Neutron star crust Lect.1
µe
(MeV)
∆ρ/ρ
(%)
0.95
2.61
4.31
4.45
5.66
8.49
11.4
14.1
2.9
3.1
3.1
2.0
3.3
3.6
3.9
4.3
16.8
18.3
20.6
22.9
25.4
(26.2 )
4.0
3.0
3.2
3.4
3.6
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Ground state crust - T = 0 - nuclei - 2
Example 2 (Rüster, Hempel, & Schaffner-Bielich 2006)
Table:
Sequence of nuclei in the ground state of the outer crust of neutron star calculated by Rüster, Hempel, & Schaffner-Bielich (2006) using
experimental nuclear data (upper part), and the theoretical mass table of the Skyrme model BSk8 (lower part). Table from Chamel & Haensel 2008
ρmax [g/cm3 ]
8.02 × 106
2.71 × 108
1.33 × 109
1.50 × 109
3.09 × 109
1.06 × 1010
2.79 × 1010
6.07 × 1010
8.46 × 1010
9.67 × 1010
1.47 × 1011
2.11 × 1011
2.89 × 1011
3.97 × 1011
4.27 × 1011
Pawel Haensel (CAMK)
Element
56
Fe
62
Ni
64
Ni
66
Ni
86
Kr
84
Se
82
Ge
80
Zn
82
Zn
128
Pd
126
Ru
124
Mo
122
Zr
120
Sr
118
Kr
Z
26
28
28
28
36
34
32
30
30
46
44
42
40
38
36
Neutron star crust Lect.1
N
30
34
36
38
50
50
50
50
52
82
82
82
82
82
82
ai [fm]
1404.05
449.48
266.97
259.26
222.66
146.56
105.23
80.58
72.77
80.77
69.81
61.71
55.22
49.37
47.92
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Neutron star crust
(nuclear and plasma physics)
Lecture 2
Pawel Haensel
Copernicus Astronomical Center (CAMK)
Warszawa, Poland
[email protected]
La̧dek Zdrój, February, 2008
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
1 / 10
Approximate calculation of neutron drip point - 1
”Simplified nuclear mass formula”. Neglecting Coulomb, surface and
all other finite-size terms and keeping only quadratic term in
δ = (N − Z)/A.
Energy per nucleon in an atomic nucleus (A, Z) (subtracting the rest
energy and neglecting neutron-proton mass difference)
EN (A, Z)/A ' E0 + S0 δ 2 ,
E0 = energy per nucleon in symmetric nuclear matter and S0 =
symmetry energy, both calculated at saturation density. The neutron
and proton chemical potentials (without rest energy contribution)
µ0n = (∂EN /∂N )Z = E0 + (2δ + δ 2 ) S0 ,
µ0p = (∂EN /∂Z)N = E0 + (−2δ + δ 2 ) S0 .
Pawel Haensel (CAMK)
Neutron star crust Lect.2
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(1)
2 / 10
Approximate calculation of neutron drip point - 2
δ corresponding to ρND - calculated from µ0n = 0,
p
δND = 1 − (E0 /S0 ) − 1 .
(2)
Experimental values E0 = −16 MeV, S0 = 32 MeV =⇒ δND = 0.225.
Beta equilibrium condition: µn = µp + µe , therefore
µe = µn − µp ' 4S0 δ .
(3)
Using µe = 0.516 (ρ6 Z/A)1/3 MeV (valid for ρ6 1), we get
bulk approximation :
ρND ≈ 2.2 × 1011 g cm−3 ,
(4)
quite close to the refined theoretical value.
Pawel Haensel (CAMK)
Neutron star crust Lect.2
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3 / 10
Inner crust - bulk approximation - 1
Coulomb and surface effects, etc. - neglected. At a given
(macroscopic, mean) nucleon density nb , nucleons in “i” and “o”
coexisting fluids. Densities: nn,i , np,i , nn,o . np,o 6= 0 only in the
proton drip regime (nb > nPD ). Equilibrium “i” - “o” ensured by
(strong) NN interaction. Implies equality of the chemical potentials of
nucleons and the equality of the nucleon pressures,
µn,i = µn,o ,
µp,i = µp,o ,
PN,i = PN,o ,
(5)
N - nucleon contribution. Condition on µp applies only at nb > nPD .
At a given nb , Eqs. (5) =⇒ nn,i , np,i , nn,o , and np,o .
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Neutron star crust Lect.2
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4 / 10
Inner crust - bulk approximation - 2
w - volume fraction occupied by the “i” phase. At a given mean
nucleon number density nb = w ni + (1 − w) no , the total energy
density is
E = w EN,i + (1 − w) EN,o + Ee ,
(6)
where Ee is the electron energy density. Beta equilibrium implies:
µn = µp + µe .
(7)
ne = w np,i + (1 − w) np,o .
(8)
Overall charge neutrality
Eqs. (5)–(8) determine the equilibrium state at a given nb .
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Inner crust - bulk approximation - 3
The two-fluid state: stable at nb at
which the energy per nucleon is lower
than in the uniform npe matter.
Stability lost at n2↔1 . While
nb −→ n2↔1 from lower density side,
w → 1. Therefore 2 ↔ 1 transition is
continuous (no density jump). Surface
and Coulomb contributions increase E
=⇒ the crust-core occurs at a baryon
number density ncc < n2↔1 upper
bound. Problem: Propose a procedure
for numerical calculation of bulk
equilibrium approximation of dense
matter from a given function E(nn , np )
at densities 108 − 3 × 1014 g cm−3
Example: For the SLy4 effective NN interaction: nPD = 0.085 fm−3 ,
n2↔1 = 0.091 fm−3 , ncc = 0.078 fm−3 . w(ncc ) ≈ 0.3 (agreement with
Compressible Liquid Drop Model - CLDM)
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Neutron star crust Lect.2
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6 / 10
Hartree-Fock calculation of ground state crust
Put A = N + Z nucleons and Z electrons in a box of volume Vcell (=unit
cell) and calculate ground state of the system at fixed nb = A/Vcell . The
A − body problem - solved using nuclear many-body theory. A ∼ 50 − 1000.
”Effective nuclear” Hamiltonian
eff
ĤN
=
A
X
j=1
t̂j +
X
eff
v̂jk
,
(9)
k<j≤A
eff
where t̂j is the kinetic energy operator of j-th nucleon, while v̂jk
is an operator of
an effective two-body interaction between a jk nucleon pair.
eff
ĤN
has to reproduce – as accurate as possible within the Hartree-Fock
approximation – relevant properties of the ground state of a many-nucleon
system, particularly, the ground state energy E0 . This last condition can be
eff
written as hΦ0 |ĤN
|Φ0 i ' hΨ0 |ĤN |Ψ0 i , where Φ0 and Ψ0 are, respectively, the
Hartree-Fock and exact wave functions, and ĤN is the exact nuclear Hamiltonian.
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Neutron star crust Lect.2
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7 / 10
Effective nucleon-nucleon interaction
Most popular: Skyrme interaction
The basic assumption which justifies a Skyrme-type effective NN
interaction (Skyrme 1956) is that its range is small as compared with
the internucleon distances. This means, that in momentum
ˆ eff (k, k0 ) can be
representation the effective NN interaction ve
approximated by a momentum independent term plus terms quadratic
in the initial and final relative momenta of an interacting nucleon pair,
k and k0 , with the appropriate spin dependence. More in Ring & Schuck 1980
eff
Parametrizing ĤN
Numerical values of the parameters of effective interaction are to be
determined from fitting masses of laboratory nuclei in ground states
and low-lying excited states. The complete Hamiltonian of the unit
eff
cell is Ĥcell
= ĤNeff + VCoul + Ĥe , where VCoul describes Coulomb
interaction between protons and electrons, and Ĥe corresponds to a
uniform electron gas.
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
8 / 10
Nucleon wave functions - 1
Box=unit cell=ion sphere. The Hartree-Fock approximation for the
many-body nucleon wave function is
h
i
(n)
ΦN Z = CN Z det ϕ(p)
(ξ
)
det
ϕ
(ζ
)
,
k
l
αi
βj
(n)
(10)
(p)
where ϕβj (ζl ) and ϕαi (ξk ) are single-particle wave functions
(orbitals) for neutrons (j, l = 1, . . . , N ) and protons (i, k = 1, . . . , Z),
respectively, and CN Z is a normalization constant. The space and spin
coordinates of a k-th proton are represented by ξk ; ζl is the same for
an l-th neutron; {αi } and {βj } are sets of quantum numbers of
occupied single-particle states for protons and neutrons, respectively.
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
9 / 10
Nucleon wave functions - 2
The Hartree-Fock equations for ϕ(p) and ϕ(n) were derived by
minimizing the Hartree-Fock energy functional at a fixed volume
Vc of the unit cell,
h
i
(n)
(p)
eff
Ecell ϕα , ϕβ = hΦN Z Φe |Ĥcell
| ΦN Z Φe i = minimum ,
(11)
where Φe is the plane-wave Slater determinant for an ultra-relativistic
electron gas of constant density ne = Z/Vc . The minimization
performed at fixed average neutron and proton densities, nn = N/Vc ,
np = Z/Vc = ni Z.
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
10 / 10
Determining the ground state of inner crust
unit cell = ion sphere
Ground state at (N, Z), β-equilibrium
(n)
(p)
Once the H-F orbitals ϕβ and ϕα are determined, one finds the
minimum (ground state) value of Ecell (N, Z), filling the lowest N
neutron and Z proton states. Then, the absolute ground state
configuration is found by minimizing Ecell (N, Z) at a fixed
A = N + Z.
Neutron drip in H-F
αZ and βN correspond to “Fermi levels” for protons and neutrons,
respectively. In terms of the single-nucleon orbitals, the neutron drip
point corresponds to the threshold density at which the neutron Fermi
(n)
level becomes unbound, i.e., ϕβN extends over the entire unit cell.
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
11 / 10
Nucleon density distribution in the inner crust - HF
Density profiles of neutron and protons, at several average densities ρ, along a line
joining the centers of two adjacent unit cells. Based on Fig. 3 of Negele & Vautherin
(1973). Remains to be added: nucleon pairing =⇒ HF+Bogoliubov ≡ HFB
Pawel Haensel (CAMK)
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12 / 10
Semi-classical - Thomas-Fermi method - 1
∼ 103 nucleons in unit cell - full quantum description is ”too detailed” for
getting ground state energy - semiclassical approximation is quite precise
In HF method energy was a functional of nucleon wave functions.
In TF (more refided: Extended TF, ...) the energy is a functional of
nucleon densities
Z
EN =
EN [nn (r), np (r), ∇nn (r), ∇np (r)] + mn c2 nn (r)
cell
+mp c2 np (r) d3 r .
(12)
Z
1
ECoul =
e [np (r) − ne ] φ(r) d3 r ,
(13)
2 cell
where φ(r) is the electrostatic potential.
Used in condensed matter, and recently adapted to nuclear structure - EDFT - Energy
Density Functional Theory based on Kohn-Sham theorem.
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
13 / 10
Thomas-Fermi method - 2
φ(r) should be calculated from the Poisson equation,
∇2 φ(r) = 4πe [np (r) − ne ] .
(14)
To determine the ground state at a given nb , one has to find nn (r)
and np (r), which minimize Ecell /Vc under the constraints
Z
Z
3
[nn (r) + np (r)] d r ,
[np (r) − ne ] d3 r = 0 .
Vc nb =
cell
cell
(15)
The problem is simplified assuming spherical symmetry; in this case
the unit cell is approximated by a sphere of the radius
rc = (3Vc /4π)1/3 . The boundary conditions are such that far from the
nucleus surface the nucleon densities are uniform. This requires the
nuclear radius to be significantly smaller than rc .
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
14 / 10
Thomas-Fermi method - continued
Neutron and proton density profiles at two average mass densities along a line
joining the centers of adjacent unit cells. Based on Fig. 5 of Cheng et al. (1997),
with the permission of the authors. No wiggles in profiles - because no quantum
interference between nucleon wave functions.
Pawel Haensel (CAMK)
Neutron star crust Lect.2
La̧dek Zdrój, Poland
15 / 10
Neutron star crust
(nuclear and plasma physics)
Lecture 3
Pawel Haensel
Copernicus Astronomical Center (CAMK)
Warszawa, Poland
[email protected]
La̧dek Zdrój, February, 2008
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
1 / 16
H-F uniform nuclear matter - Hamiltonian - 1
Put N neutrons, Z protons (A = N + Z nucleons) in a large box of
volume V and calculate ground state of the system at fixed
nn = N/V , np = Z/V . Coulomb interactions switched-off. Assume
that nuclear matter is uniform =⇒ translational invariance.
Approximation: mn ≈ mp = m
Effective nuclear Hamiltonian
ĤNeff
=
A
X
j=1
t̂j +
X
eff
v̂jk
,
(1)
k<j≤A
eff
where t̂j is the kinetic energy operator of j-th nucleon, while v̂jk
is an
operator of an effective two-body interaction between a jk nucleon
pair.
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
2 / 16
H-F - uniform nuclear matter - Hamiltonian - 2
The effective nuclear Hamiltonian ĤNeff has to reproduce – within the
Hartree-Fock approximation – relevant properties of the ground state
of a many-nucleon system: the ground state energy E0 , etc. This last
condition can be written as
hΦ0 |ĤNeff |Φ0 i ≈ hΨ0 |ĤN |Ψ0 i
(2)
where Φ0 and Ψ0 are, respectively, the Hartree-Fock and exact wave
functions, and ĤN is the exact nuclear Hamiltonian.
Single-particle wave-functions - 0
Nuclear matter is uniform and spin non-polarised. Single-particle state
labels α, β = p , sz .
Coordinate-space representation:
r ), ϕ(p)
r) ,
ϕ(n)
α (r
β (r
(3)
Isospin formalism is not used - we distinguish n and p.
Pawel Haensel (CAMK)
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La̧dek Zdrój, Poland
3 / 16
Single-particle wave-functions
Eigenfunctions of p̂p and t̂
Coordinate representation. Spin sz - omitted.
∇,
p̂p = −i~∇
r ) = p ϕ(n)
r ) =⇒
p̂pϕ(n)
α (r
α (r
r ) = Ceipp·rr /~ . (4)
ϕ(n)
α (r
It is also eigenfuction of t̂ = −~2∇ 2 /2m:
r) =
t̂ϕ(n)
α (r
p2 (n)
ϕ (rr ) .
2m α
(5)
(n)
Generalization: ϕα (rr ) will be also the eigenfunctions of
ĥ = t̂ + Ûn (p̂p2 ), with eigenvalue p2 /2m + Un (pp2 ).
Normalization in periodicity box V = L3
Allowed values of px = pjx are j × 2π~/L, j = 1, 2, 3, . . ., same for py
√
(n)
and pz . ϕα (rr ) are orthogonal and normalized to 1 with C = 1/ V .
Pawel Haensel (CAMK)
Neutron star crust Lect.3
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4 / 16
Miscellaneous
Replacing
P
p
by
R
d3 p
Summation of allowed p can be replaced by integration over p , because for
L −→ ∞
Z
X
dpx
F (px ) =
F (pjx ) .
2π~/L
j
(6)
It is sufficient that V is macroscopic (but much weaker conditions can be fine see Thomas-Fermi). In 3-D we get
Z
X
d3 p
L −→ ∞ =⇒
F (pp) = V
F (pp) .
(7)
(2π~)3
p
(n)
fα
(p)
, fβ
Single-particle wavefunctions, occupation numbers
Z
?
1 ipp·rr /~
√
r
r )ϕ(n)
r ) = δm,n .
ϕ(n)
(r
)
=
e
,
d3 rϕ(n)
α
αn (r
αm (r
V
P (n)
P (p)
= 0, 1
α fα = N ,
β fβ = Z .
Pawel Haensel (CAMK)
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(8)
5 / 16
Single-particle energies
Do not depend on sz - two-fold degenerate.
tα = hα|t̂|αi =
p2
.
2m
(9)
HF single-particle energies:
X (n)
fα0 [hαα0 |v nn |αα0 i − hαα0 |v nn |α0 αi]
enα = tα +
0
+
epβ
= tβ +
α
X
β
X
(p)
fβ hαβ|v np |αβi ,
(p)
fβ 0 [hββ 0 |v pp |ββ 0 i − hββ 0 |v pp |β 0 βi]
β0
+
X
fα(p) hβα|v np |βαi .
(10)
α
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
6 / 16
Skyrme interaction - matrix elements
∇1
v̂12 - in spatially uniform nuclear matter. k 1 = p 1 /~ , k̂k 1 = −i∇
hα1 α2 |v̂|α1 α2 i =
V
−2
Z
3
d r1
Z
3
d r2 e
k 1r 1 +k
k 2r 2 )
−i(k
v̂(rr 1 − r 2 )ei(kk 1r 1 +kk 2r 2 )
Zero-range A0 δ(rr 1 − r 2 ). Matrix elements of δ(rr 1 − r 2 )
Z
−2
V
d3 r1 = V −1 .
(11)
(12)
Momentum-dependent gradient-terms:
A1 δ(rr 1 − r 2 )(k̂k 1 − k̂k 2 )2 , . . . matrix element ∝ (kk 1 − k 2 )2 /V , . . .
Helpful: r 1 , r 2 −→ R = (rr 1 + r 2 )/2 , r = r 1 − r 2 ,
k 1 , k 2 −→ k = (kk 1 − k 2 )/2 , P = k 1 + k 2 ,
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
7 / 16
eα , E for Skyrme
e(n)
α1 =
X
p21
+
hα1 α2 |v̂|α1 α2 i + · · · =
2m
α
2
p2
2 X (n)
= 1 +
f (C1 + C2 p22 + C3 p21 . . .) + . . .
2m V p p 2
(13)
2
(n)
Ground state occupation numbers: fp 2 = Θ(p2 − pFn )
P(n)occ
3
/(3π 2 ), same for protons.
2 p
= N =⇒ nn = kFn
(n)
2/3
eα contains terms with factors: nn , nn , np , nn p2 , ...
Ground state energy density - kinetic part:


(n)occ
( p)occ
2
2
X
1 X
 p = ~ (τn + τp )
Ekin = 2
+
V
2m
2m
p
p
τn =
Pawel Haensel (CAMK)
(14)
3
3
2
2
nn kFn
, τp = np kFp
,
5
5
Neutron star crust Lect.3
(15)
La̧dek Zdrój, Poland
8 / 16
HF - EN,int
Spin factors omitted for simplicity (only one spin state included):
hΦ0 |
(n)occ (n)occ
A
1 X X
1 X
nn
|pp1p 2 ia
hpp1p 2 |v̂12
v̂ij |Φ0 i =
2 i6=j=1
2 p
p
1
2
(p)occ (p)occ
+
1 X X
pp
hpp1p 2 |v̂12
|pp1p 2 ia
2 p
p
1
2
(p)occ (n)occ
+
X X
p1
pn
hpp1p2 |v̂12
|pp1p2 i
(16)
p2
|pp1p 2 ia ≡ |pp1p 2 i − |pp2p 1 i
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
9 / 16
E(nn , nn ) for Skyrme - 1
Final form
E(nn , np ) =
~2
(τn + τp )
2m
1 + t0 (2 + x0 )(nn + np )2 − (2x0 + 1)(n2n + n2p )
4
1
+ t3 (nn + np )γ (2 + x3 )(nn + np )2 − (2x3 + 1)(n2n + n2p )
24
1
+ [t1 (2 + x1 ) + t2 (2 + x2 )] (τn + τp )(nn + np )
8
1
+ [t2 (2x2 + 1) − t1 (2x1 + 1)] (τn nn + τp np ) .
8
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
(17)
10 / 16
E(nn , np ) for Skyrme - 2
Parameter values of the Skyrme forces. SLy4 used in these lectures.
force
SLy4
SLy7
SkM∗
Sk10
t0 (MeV fm3 )
t1 (MeV fm5 )
t2 (MeV fm5 )
t3 (MeV fm3+3γ )
γ
x0
x1
x2
x3
W0 (MeV fm5 )
-2488.91
486.82
-546.39
13777.0
-2482.41
457.97
-419.85
13677.0
-2645.0
410.0
-135.0
15595.0
-1057.3
235.9
-100.00
14463.5
1
0.2885
0
0
0.2257
120.0
1
6
1
6
1
6
0.834
-0.344
-1.000
1.354
123.0
0.846
-0.511
-1.000
1.391
126.0
0.09
0
-1.000
0
130.0
Applications to bulk equilibrium and surface energy in neutron star crust: in Douchin,
Haensel, & Meyer (2000).
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
11 / 16
Equation of state for bulk matter - general
Thermodynamics at T = 0. Chemical potential without rest energies.
∂E
∂E
µn =
, µp =
.
(18)
∂nn np
∂np nn
Pressure of nucleons
P = −E + nn µn + np µp .
(19)
Mass-energy density and baryon (number) density:
ρ = nn mn + np mp + E/c2 ,
Pawel Haensel (CAMK)
Neutron star crust Lect.3
nb = nn + np .
La̧dek Zdrój, Poland
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12 / 16
Equation of state of symmetric nuclear matter
m = (mn + mp )/2
~2
(~c)2
=
= 20 .276 MeV fm2
2m
2mc2
energy per nucleon (without rest
energy) = E/nb
SLy4
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
13 / 16
Saturation parameters of nuclear matter
Self-bound symmetric nuclear matter.
P = 0 at nb = n0 .
nn = (1 + δ)nb /2, np = (1 − δ)nb /2.
Nuclear matter parameters at saturation point
SkI0
SkM∗
SLy4
SLy7
n0 [fm−3 ]
0.1553
0.1603
0.1595
0.1581
E0 [MeV]
-15.99
-15.77
-15.97
-15.89
K0 [MeV]
370.4
216.6
229.9
229.7
S0 [MeV]
32.64
30.03
32.00
31.99
Pawel Haensel (CAMK)
Charge symmetry of nuclear forces=⇒
E(nb , δ) = E(nb , −δ)
Neglecting higher order terms:
2
nb − n0
,
n0
(21)
K0 - incompressibility, S0 - symmetry
energy.
K0
E(nb , δ) ' E0 +S0 δ +
9
Neutron star crust Lect.3
2
La̧dek Zdrój, Poland
14 / 16
Energy-density for different δ
E/nb vs. nb
δ = (nn − np )/nb .
Pawel Haensel (CAMK)
SLy4
δ = 1 − 2xp , xp = (1 − δ)/2 .
Neutron star crust Lect.3
La̧dek Zdrój, Poland
15 / 16
Further study - Problems
Notation r = r 1 − r 2 , ∇ = ∂/∂rr
nb =
3
2kF
.
3π 2
Problem Consider a model of nuclear matter with effective interaction
v̂(rr ) = [a + b nb ] δ(rr ) .
(22)
Derive formula for energy per nucleon as a function of kF . Show that nuclear matter can
have a stable bound state (saturation). Find values of a and b that yield saturation at
E0 = −16 MeV and at n0 = 0.16 fm−3 .
Problem Consider a model of nuclear matter with effective interaction
←
−
→
−
∇
∇
v̂(rr ) = a0 δ(rr ) − a1 δ(rr )
.
i
i
(23)
Derive formula for energy per nucleon as a function of kF . Show that nuclear matter can
have a stable bound state (saturation). Find values of a0 and a1 that yield saturation at
E0 = −16 MeV and at n0 = 0.16 fm−3 .
Problem Using E(nn , np ) for SLy4 force, calculate P (nb ) for symmetric nuclear matter
and check it with Table. Calculate n0 , E0 , S0 and K0 for SLy4.
Pawel Haensel (CAMK)
Neutron star crust Lect.3
La̧dek Zdrój, Poland
16 / 16
Neutron star crust
(nuclear and plasma physics)
Lecture 4
Pawel Haensel
Copernicus Astronomical Center (CAMK)
Warszawa, Poland
[email protected]
La̧dek Zdrój, February, 2008
Pawel Haensel (CAMK)
Neutron star crust Lect.4
La̧dek Zdrój, Poland
1 / 15
Compressible Liquid Drop Model - basics
The model is classical (non-quantum) par excellence.
Nucleon subsystems: bulk nuclear matter inside nuclei - (i), neutron gas outside
nuclei (o), nucleons in the transition layer between (i) and (o) - the surface (s) or
(surf).
Ecell = EN,bulk + EN,surf + ECoul + Ee ,
(1)
EN,bulk = Vc [w EN,i + (1 − w) EN,o ] .
(2)
EN,surf and ECoul depend on sizes and shapes of nuclear structures.
Input quantities calculated using many-body microscopic theories. Unknown
parameters: nuclear radii - rp , rn , density of neutron and protons inside nuclei nn,i , np,i , density of neutrons outside nuclei - nn,o , surface tension at the nuclear
surface - σs , number of neutrons ”adsorbed at the nuclear surface” - Ns , thickness
of ”neutron skin” - sn , radius of the unit (ion) cell- rc , . . . .
Equilibrium conditions: mechanical and ”chemical” equilibrium, charge
neutrality of the unit cell =⇒ structure of the ground state of the inner
crust at a given nb .
Pawel Haensel (CAMK)
Neutron star crust Lect.4
La̧dek Zdrój, Poland
2 / 15
Surface tension - plane ”i-o” interface - 1
V = L3 . Fixed Nn ,Np , L. Variables:
µn , µp (adjusted to get correct
Nn ,Np ). Then position of the
interface (z0 ), densities nn (z), np (z)
and pressure P will result from
minimizing
Ω(µn , µp ) = E − Nn µn − Np µp .
dV = L2 dz
Z
Ω=
dV ω(z) = Ωbulk + Ωsurf
ω(z) = E − µn nn (z) − µp np (z)
Ωbulk = −P L3
−P = Eo − nn,o = Ei − nn,i µn,i − np,i µp,i
Pawel Haensel (CAMK)
Neutron star crust Lect.4
La̧dek Zdrój, Poland
3 / 15
Surface tension - plane ”i-o” interface - 2
Ω = min. ⇔ ωtot = ωbulk + ωsurf = Ω/L2 = min.
ωbulk = −P L does not depend on surface properties
ωtot = min. ⇔ ωsurf = min
Z
ωtot
0
[ω(z) − Ei + nn,i µn,i + np,i µp,i ] dz
=
−∞
+∞
Z
[ω(z) − Eo + nn,o µn,o ] dz
(3)
0
ωref (z) = Θ(−z) [Ei − nn,i µn,i − np,i µp,i ] + Θ(z) [Eo − nn,o µn,o ]
Z
(4)
+∞
(ω − ωref ) dz .
ωsurf =
(5)
−∞
Plane interface: ωref = −P is z0 -independent =⇒ ωsurf does not depend on
z0 .
Pawel Haensel (CAMK)
Neutron star crust Lect.4
La̧dek Zdrój, Poland
4 / 15
Plane interface - surface layer - 1
Example of parametrization of densities:
nn (z) = nn,o + [nn,i − nn,o ] f (z; an , bn ) ,
np (z) = np,i f (z; ap , 0) ,
h
i
z−b
f (z; a, b) = 1/ 1 + e a .
We have to find ωsurf (ap , bp , an , bn ) = min. The
value at minimum is equal σs . It is a function of
only one variable, e.g. xi or nn,o .
Definition z0 just within “surface layer”. Only zn − zp has physical meaning.
Z
+∞
[f (z; ap ) − Θ(zp − z)] dz = 0 ,
−∞
Z +∞
[nn (z; an , bn ) − nn,i Θ(zn − z) − nn,o Θ(z − zn )] dz = 0 .
(6)
−∞
Pawel Haensel (CAMK)
Neutron star crust Lect.4
La̧dek Zdrój, Poland
5 / 15
Plane interface - surface layer - 2
Neutron skin, surface density (=column density) of adsorbed neutrons
Neutron skin thickness sn = zn − zp .
Surface density of adsorbed neutrons νn = sn (nn,i − nn,o )
Number of adsorbed neutrons - plane surface of area A:
Ns = Aνs .
Bookkeeping of neutrons and protons - L3 box + plane surface
Surface at z = zp defines Vi ≡ L2 Lp :
Z +L
2
np (z)dz = L2 Lp np,i ,
Np = L
−L
Z +L
Nn = L 2
nn (z)dz = Nn,i + Ns + Nn,o , Nn,i = Vi nn,i ,
−L
3
Nn,o = (L − Vi )nn,o , Ns = L2 sn (nn,i − nn,o ) .
Pawel Haensel (CAMK)
Neutron star crust Lect.4
(7)
La̧dek Zdrój, Poland
6 / 15
Relation between σs and es
Separation of surface contribution
EN = Ebulk + L2 es = Ωbulk + L2 σs
+(Nn,o + Nn,i )µn + Ns µn + Np µp ,
(8)
es,zp = σs + sn (nn,i − nn,o ) µn .
(9)
Third surface of reference:
Z
+∞
[nn (z) + np (z) − Θ(zN − z)(nn,i + np,i ) − Θ(z − zN )nn,o ] dz = 0 .
−∞
Pawel Haensel (CAMK)
Neutron star crust Lect.4
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7 / 15
σs and es - results for SLy [Douchin,Haensel,Meyer 2000]
Crossing of σs and es at neutron drip where µn = 0. Before neutron drip, nn,o = 0 and
µn < 0 =⇒ σs < es . For small δi < 0.3 both σs , es ∝ δi2 . Strong dependence of es on
zref . Monotonic decrease of σs . For small δi < 0.3
sn ∝ δi .
Pawel Haensel (CAMK)
Neutron star crust Lect.4
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8 / 15
Coulomb energy - simplest approx. - spherical nuclei - 1
Constant charge densities, sharp proton surface.
Charge units: e = qp = −qe . For r < rc
ρel (r) = Θ(rp −r)(np,i −ne )−Θ(r−rp )ne . (10)
Coulomb energy of a W-S cell
ECoul =
Z
I=
Z
d r
r<rc
Pawel Haensel (CAMK)
3
Neutron star crust Lect.4
r 0 <rc
1 2
e I ,
2
d3 r0
(11)
ρel (r)ρel (r0 )
.
|rr − r 0 |
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(12)
9 / 15
Coulomb energy - simplest approx. - spherical nuclei - 2
Problem: Calculate ECoul (rp , npi , w)
Aswer:
rp5 n2p
f3 (w) , w = (rp /rc )3
6
1
f3 (w) = (2 − 3w1/2 + w)
5
ECoul = 2πw(enp,i rp )2 f3 (w)
ECoul = (4π)2 e2
Hint Use multipole expansion
1
r − r0|
|r
Pawel Haensel (CAMK)
= 4π
∞
X
m=l
X
1
l
r<
l+1
l=0 m=−l 2l + 1 r>
Neutron star crust Lect.4
? n0
n) ,
Ylm (n̂
)Ylm (n̂
(13)
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Variable, constraints, equilibrium
For the sake of simplicity - no neutron skin
Five independent variables. Possible set ne , nn,i , np,i , nn,o , rc rp
Constraints - charge neutrality and nb = const.:
ne = wnp,i , w(nn,i + np,i ) + (1 − w)nn,o = nb = const.
Thermod. potential density to be minimized:
E = wEi + (1 − w)Eo +
3σs w
+ ECoul + Ee
rp
w = (rp /rc )3
ECoul = 2πw(ene rp )2 f3 (w)
four independent variables
Pawel Haensel (CAMK)
Neutron star crust Lect.4
Ee = a(ne )4/3
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Minimize in rc
Choice of variables: [nn,i np,i nn,o ] rc
E = wEi + (1 − w)Eo +
w=
nb −nn,o
nn,i +np,i −nn,o
[.....] fixed.
3σs w
+ ECoul + Ee
rp
rp = w1/3 rc
∂ECoul
2ECoul
∂Esurf
Esurf
=
,
=−
rc
rc
rc
rc
”Virial Theorem”
Esurf = 2ECoul
Pawel Haensel (CAMK)
Neutron star crust Lect.4
La̧dek Zdrój, Poland
12 / 15
Minimize in nn,i
Choice of variables: nn,i [np,i rp rc ]
E = wEi + (1 − w)Eo +
[.....] fixed.
3σs w
+ ECoul + Ee
rp
w = (rp /rc )3 is also fixed. From nb -constraint:
nn,o =
nb − w(nn,i + np,i )
−w
=⇒ δnn,o =
δnn,i
1−w
1−w
∂E
∂E
w
=−
∂nn,i
1 − w ∂nn,o
∂E
= w(µn,i − µn,o ) = 0 =⇒
∂nn,i
µn,i = µn,o
Pawel Haensel (CAMK)
Neutron star crust Lect.4
La̧dek Zdrój, Poland
13 / 15
Minimize in np,i
Choice of variables: np,i [nn,o rp rc ]
E = wEi + (1 − w)Eo +
[.....] fixed.
3σs w
+ ECoul + Ee
rp
w = (rp /rc )3 is also fixed. From nb -constraint:
nn,i =
nb − (1 − w)(nn,0 )
− np,i
w
∂E
∂E
=−
∂nn,i
∂np,o
∂E
= 0 =⇒
∂np,i
Pawel Haensel (CAMK)
µn,i − µp,i − µe =
Neutron star crust Lect.4
8π 2
e np,i rp2 f3 (w) .
5
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Minimize in rp
Choice of variables: rp [Np Nn rc ]
[.....] fixed.
E = wEi + (1 − w)Eo +
3σs w
+ ECoul + Ee
rp
rp varied =⇒ w varied. nb fixed by construction.
nn = (1 − w)nn,o + wnn,i , np = wpn,i
∂E
∂E
=−
∂nn,i
∂np,o
∂E
= 0 =⇒
∂np,i
Pawel Haensel (CAMK)
Pi − Po =
2σs 4π 2 2 2
−
e np,i rp (1 − w) .
rp
15
Neutron star crust Lect.4
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SLy4 - curvature included - inner crust
Spherical unit cell radius rc , the proton
radius of nuclei rp , and the fraction of
volume w filled by protons (in percent) vs.
average matter density ρ. Douchin & Haensel (2000)
Pawel Haensel (CAMK)
Nuclei: A and Z vs. ρ. The dotted line
gives number of the nucleons in nuclei after
subtracting the neutrons belonging to
neutron skin. Douchin & Haensel (2000)
Neutron star crust Lect.4
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