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Phase transitions (π-condensation) in nuclei and neutron stars
Volume 45B, number 5
PHASE TRANSITIONS
PHYSICS LETTERS
(n-CONDENSATION)
20 August 1973
IN NUCLEI AND NEUTRON
STARS
A.B. MIGDAL
The Landau Institute for Theoretical Physics, the Academy of Sciences of the USSR, USSR Received 2 April 1973 It is shown that in nuclear matter at Z = N for density n < n o (no nuclear density) an electrically neutral condensate of n , n , ~r° mesons arises. The results of the calculations for the case of the neutron star (Z .~N) are given. In this case there are two phase transitions: one corresponds to ~r° condensation and second to the electrically neutral n, ~r condensation. The 7r- condensate apparently does not appear even at very high densities. +
4-
--
--
In a sufficiently strong field and single-particle level may reach the critical value at which the creation of particles is possible. In the case o f fermions the stability o f the vacuum is provided by the Pauli principle. The creation stops when the "dangerous" states are filled. F o r bosons the process terminates only when the interparticle repulsion prevents further particle production. The question o f vacuum stability of bosons in external fields was investigated in [1 ]. Both, the electrical field and nuclear field were schematically considered. It was shown that in a sufficiently strong field the square o f the pion energy 6o2 for some wave vectors k ~ k o becomes negative. This instability leads to a rearrangement of the ground state and to a formation of a pion condensate. The condensate makes the vacuum stable. The pion energies for all k become positive, and the energy o f the ground state decreases. For instance, the nuclei with Z 2 137 3/2 might happen to stable due to the formation o f the pion condensate in the electrical field of the nucleus (Ze2[R ~> 2/.tc2). The gain in pion energy is of the same order as the energy of the Coulomb repulsion, n condensation in the nuclear matter was considered first in [2] and then investigated in [1]. Later the question was considered once more in [3], [4] under the assumption that the condensate in a neutron star consists o f n - - m e s o n s (the charge is compensated by the same amount o f protons). The authors came to the conclusion that for some density o f neutrons the pion condensate will have the density o f the same order. The same results were assumed in [5]. A more realistic consideration does not confirm these suppositions. In a neutron star, while the density increases, the n °448
condensate appears. About at the same density an electrically neutral condensate of n+n - - m a s o n s arises. It can be shown [6] that the condensate field increases the single pion energy; thus the neutron matter is stable according to the process n ~ p + n - and the n - - c o n d e n s a t e does not exist. A detailed consideration o f n-meson interaction with nuclear matter for Z = N gives the following results. The pion energy 60(k) is determined by the equation (h =/a = c = 1) 6o2 = 1 + k 2 + lI(k, 60)
(1)
The polarisation operator II(k, 60) for 60 and k of interest (60 ~ 1, k ~ m ~ 7, m is the nucleonic mass) is given by two types o f graphs:
n(k, 60)
+
(2)
The first diagram corresponds to the absorption o f the pion by nucleon with the formation o f a hole in the Fermi distribution. The shaded vertex means that the nucleon-nucleon interaction is taken into account. This vertex can be expressed through the constants characterising the internucleon spin-spin interaction in nuclei [7]. The second graph corresponds to the transition o f the pion and nucleon into the A33-resonance. All other graphs involve large four-moments in intermediate states and differ slightly from the corresponding graphs in vacuum. They are taken into account in the observable mass o f the pion used in eq. (1). The left vertex o f the first graph in (2) equals r~= f okra, f = g/2m ~ 1, a is the pion isotopic index. The vertices o f the second graph are determined by the amplitude o f ( n N ) resonance scattering. The energy-momentum relation for n +, n - , n ° at Z = N has the form:
Volume 45B, number 5
PHYSICS LETTERS
6OR
20 August 1973
2mPFf2k2c~(k'6o)
w 2 = 1 + k 2 - 0.3nk 2
L4.~
6662-6662
1r2(l+g-(k)~k, 6o))
(3) At Z = 0 for 7r-, rr+, rr° we have respectively (see [6]) 6662 = 1 +k 2 + 1.3nw -T-1.5nk 2
666 6662-6662 P
_ 3nk2
k2-
2nf2k 2 +_6o+k2 /2m
6oR
6662-6662
Fig. 1.
w 2 = 1 + k 2 - 3nk 2
mPFf2k2 r¢2
6oR 6o2 _ 6o2
be shown that the energy of the system in a static field 9o has the form
¢(k,6o) l+gnn(k)(~(k,6o)
=
where n is the density of the nuclear matter, 6oR the A-resonance energy (6oR --- 2, 2). The third term in the eq. for 7r* corresponds to the S-scattering. The function O(k, 666)is determined by the expression (see [ 7 1 ) 2. 2
d3p
¢(k'6o)=mP--Tf(2,~---)3
= k.~ 2pR where OF, PF Fermi duced
k
+
1 +k 2 + n(k,o) o o 2 %(k)%(-k)
(5)
r x(* ° ,o)2 ¢ T d V
Taking into account the distortion of II(k, 6o) in the presence of the field ¢o it can be seen that
n(p) - n ( p - k )
EU---k):
666
~,= 6 [I(9°2)/&p °2 ~ 1/o 2
(4)
666 6o+kV F [ irt6o 1--2---~oFhl ~ ] -2TUF O(kuF-6o)
n(p) is the distribution function of the nucleons, are the velocity and the momentum on the surface. The quantities g - ( k ) , gnn(k) are introin [7] (in the notation o f [7] g - ( k ) = gnn(k) -gnP(k)). For small k, g - = 0.8PE, gnn = 0.4p E and decreases at k ~ PF" The analysis of (2) shows that there exist two branches for 6o2 given in fig. 1. The upper curve may be called the meson branch, since it tends to 6o 2 -- 1 + k 2, when n -+ 0. The second branch will be calred spin-sound. It may be seen from the fig. that there is a solution with 6662 < 0 at n < n o. This means that the system is unstable, unless the meson-meson interaction is taken into account. Let us consider, for instance, a meson interaction of the type k94/4. We can obtain the condensate field 90 minimising the energy of the ground state. It can
(6)
i.e. much greater than the corresponding vacuum constant (kva c ~, 1). Minimising the expression (5) over ~o with the interaction following from (6) we obtain 9 ~ -- 9 °2 = 9~, i.e. an electrically neutral condensate (all the components of the electrical current iv are equal to zero). Let us omit for simplicity the isotopic index. Then from (5) we obtain the following equation for 9 ° ~ 2 ( V 2 ) 9 ° + X9°3 = 0
(7)
For small values o f g o (near the critical point n ~ nc) one obtains 6662 O
9 o = a(sdnkox + 6o2(9k2 ) Cl sin3koX + • .) where k o corresponds to the minimum of ~ 2 ( k 2 ) = 1 + k 2 + II(k, 0), ~o = ~°2(k2), Cl ~ 1. For a and E@) we obtain a 2= - ~ ~o2/X,
E (~)
=-~ot/6k
(8)
449
Volume 45B, number 5
PHYSICS LETTERS
The equation for the field o f meson excitations in the presence o f the condensate is given by --602~ '+ ~'2(~72)tp'+ 3;k(tp°)2~ ' = O,
(9) It is seen from (9) that 6o2 for all k is positive. There are two types o f solutions of (9). One type can be obtained from (9) by averaging the coefficient in the third term. This gives
~2(k) = ~2(k) - 2 ~ ° / > Ico2[. The second type is the Goldstone's solution. Differentiating (7) over x we have ~.(7 2) 3~P°
O~P° + 3X~°2 0X
the third term of (3) tends to zero at co > ko F and the crossing is impossible. I f the second phase transition occurs, a metastable state of nuclei with n ~ n o may exist. Possibly such anomalous nuclei can be found in cosmic rays. The distortion of the meson propagator in nuclear matter due to the A-resonance leads to an essential non-pair term in the internucleon forces, which cannot be found by usual methods assuming the pair interaction. A more detailed consideration will be given elsewhere. The author expresses his thanks to A.A. Migdal, V.A. Khodel, A.M. Poliakov and his collaborators O. Marion and I. Mishustin.
('7')
Introducing ~' = X ~ o / 0 x, X being a slowly varying function of 7, into (9), and comparing with (7'), we have ¢o2(k) --* O, k 2 --* ko2. The condensate with ~o= asin koX will lead to a laminated structure of nuclear matter, which can be observed by electron and meson scattering on polarized nuclei. At densitites n ~ 2nc~ another type of phase transition may happen. The meson branch co(~)(k) may cross the zero value for some k. It is easy to see from (3) that the meson branch can cross the line co = kv F only, if a region of k with g - ( k ) ~ 0 exists. If not, the denominator of
450
20 August 1973
References [1] [2] [3] [4] [5] [6] [7]
A.B. Migdal,Soviet Phys. JETP 61 (1971) 2209. A.B. Migdal, Soviet Phys. UFN 105 (1971) 775. R.F. Sawyer, Phys. Rev. Lett. 29 (1972) 382. D.J. Scalapino, Phys. Rev. Lett. 29 (1972) 386. I. Kogut and I.T. Manassah, Phys. Lett. 41A (1972) 129. A.B. Migdal, Phys. Rev. Lett. (in press). A.B. Migdal, "Theory of finite Fermi systems and applications to atomic nuclei" (Interscience Publishers Inc. 1965).
PHASE TRANSITIONS
PHYSICS LETTERS
(n-CONDENSATION)
20 August 1973
IN NUCLEI AND NEUTRON
STARS
A.B. MIGDAL
The Landau Institute for Theoretical Physics, the Academy of Sciences of the USSR, USSR Received 2 April 1973 It is shown that in nuclear matter at Z = N for density n < n o (no nuclear density) an electrically neutral condensate of n , n , ~r° mesons arises. The results of the calculations for the case of the neutron star (Z .~N) are given. In this case there are two phase transitions: one corresponds to ~r° condensation and second to the electrically neutral n, ~r condensation. The 7r- condensate apparently does not appear even at very high densities. +
4-
--
--
In a sufficiently strong field and single-particle level may reach the critical value at which the creation of particles is possible. In the case o f fermions the stability o f the vacuum is provided by the Pauli principle. The creation stops when the "dangerous" states are filled. F o r bosons the process terminates only when the interparticle repulsion prevents further particle production. The question o f vacuum stability of bosons in external fields was investigated in [1 ]. Both, the electrical field and nuclear field were schematically considered. It was shown that in a sufficiently strong field the square o f the pion energy 6o2 for some wave vectors k ~ k o becomes negative. This instability leads to a rearrangement of the ground state and to a formation of a pion condensate. The condensate makes the vacuum stable. The pion energies for all k become positive, and the energy o f the ground state decreases. For instance, the nuclei with Z 2 137 3/2 might happen to stable due to the formation o f the pion condensate in the electrical field of the nucleus (Ze2[R ~> 2/.tc2). The gain in pion energy is of the same order as the energy of the Coulomb repulsion, n condensation in the nuclear matter was considered first in [2] and then investigated in [1]. Later the question was considered once more in [3], [4] under the assumption that the condensate in a neutron star consists o f n - - m e s o n s (the charge is compensated by the same amount o f protons). The authors came to the conclusion that for some density o f neutrons the pion condensate will have the density o f the same order. The same results were assumed in [5]. A more realistic consideration does not confirm these suppositions. In a neutron star, while the density increases, the n °448
condensate appears. About at the same density an electrically neutral condensate of n+n - - m a s o n s arises. It can be shown [6] that the condensate field increases the single pion energy; thus the neutron matter is stable according to the process n ~ p + n - and the n - - c o n d e n s a t e does not exist. A detailed consideration o f n-meson interaction with nuclear matter for Z = N gives the following results. The pion energy 60(k) is determined by the equation (h =/a = c = 1) 6o2 = 1 + k 2 + lI(k, 60)
(1)
The polarisation operator II(k, 60) for 60 and k of interest (60 ~ 1, k ~ m ~ 7, m is the nucleonic mass) is given by two types o f graphs:
n(k, 60)
+
(2)
The first diagram corresponds to the absorption o f the pion by nucleon with the formation o f a hole in the Fermi distribution. The shaded vertex means that the nucleon-nucleon interaction is taken into account. This vertex can be expressed through the constants characterising the internucleon spin-spin interaction in nuclei [7]. The second graph corresponds to the transition o f the pion and nucleon into the A33-resonance. All other graphs involve large four-moments in intermediate states and differ slightly from the corresponding graphs in vacuum. They are taken into account in the observable mass o f the pion used in eq. (1). The left vertex o f the first graph in (2) equals r~= f okra, f = g/2m ~ 1, a is the pion isotopic index. The vertices o f the second graph are determined by the amplitude o f ( n N ) resonance scattering. The energy-momentum relation for n +, n - , n ° at Z = N has the form:
Volume 45B, number 5
PHYSICS LETTERS
6OR
20 August 1973
2mPFf2k2c~(k'6o)
w 2 = 1 + k 2 - 0.3nk 2
L4.~
6662-6662
1r2(l+g-(k)~k, 6o))
(3) At Z = 0 for 7r-, rr+, rr° we have respectively (see [6]) 6662 = 1 +k 2 + 1.3nw -T-1.5nk 2
666 6662-6662 P
_ 3nk2
k2-
2nf2k 2 +_6o+k2 /2m
6oR
6662-6662
Fig. 1.
w 2 = 1 + k 2 - 3nk 2
mPFf2k2 r¢2
6oR 6o2 _ 6o2
be shown that the energy of the system in a static field 9o has the form
¢(k,6o) l+gnn(k)(~(k,6o)
=
where n is the density of the nuclear matter, 6oR the A-resonance energy (6oR --- 2, 2). The third term in the eq. for 7r* corresponds to the S-scattering. The function O(k, 666)is determined by the expression (see [ 7 1 ) 2. 2
d3p
¢(k'6o)=mP--Tf(2,~---)3
= k.~ 2pR where OF, PF Fermi duced
k
+
1 +k 2 + n(k,o) o o 2 %(k)%(-k)
(5)
r x(* ° ,o)2 ¢ T d V
Taking into account the distortion of II(k, 6o) in the presence of the field ¢o it can be seen that
n(p) - n ( p - k )
EU---k):
666
~,= 6 [I(9°2)/&p °2 ~ 1/o 2
(4)
666 6o+kV F [ irt6o 1--2---~oFhl ~ ] -2TUF O(kuF-6o)
n(p) is the distribution function of the nucleons, are the velocity and the momentum on the surface. The quantities g - ( k ) , gnn(k) are introin [7] (in the notation o f [7] g - ( k ) = gnn(k) -gnP(k)). For small k, g - = 0.8PE, gnn = 0.4p E and decreases at k ~ PF" The analysis of (2) shows that there exist two branches for 6o2 given in fig. 1. The upper curve may be called the meson branch, since it tends to 6o 2 -- 1 + k 2, when n -+ 0. The second branch will be calred spin-sound. It may be seen from the fig. that there is a solution with 6662 < 0 at n < n o. This means that the system is unstable, unless the meson-meson interaction is taken into account. Let us consider, for instance, a meson interaction of the type k94/4. We can obtain the condensate field 90 minimising the energy of the ground state. It can
(6)
i.e. much greater than the corresponding vacuum constant (kva c ~, 1). Minimising the expression (5) over ~o with the interaction following from (6) we obtain 9 ~ -- 9 °2 = 9~, i.e. an electrically neutral condensate (all the components of the electrical current iv are equal to zero). Let us omit for simplicity the isotopic index. Then from (5) we obtain the following equation for 9 ° ~ 2 ( V 2 ) 9 ° + X9°3 = 0
(7)
For small values o f g o (near the critical point n ~ nc) one obtains 6662 O
9 o = a(sdnkox + 6o2(9k2 ) Cl sin3koX + • .) where k o corresponds to the minimum of ~ 2 ( k 2 ) = 1 + k 2 + II(k, 0), ~o = ~°2(k2), Cl ~ 1. For a and E@) we obtain a 2= - ~ ~o2/X,
E (~)
=-~ot/6k
(8)
449
Volume 45B, number 5
PHYSICS LETTERS
The equation for the field o f meson excitations in the presence o f the condensate is given by --602~ '+ ~'2(~72)tp'+ 3;k(tp°)2~ ' = O,
(9) It is seen from (9) that 6o2 for all k is positive. There are two types o f solutions of (9). One type can be obtained from (9) by averaging the coefficient in the third term. This gives
~2(k) = ~2(k) - 2 ~ ° / > Ico2[. The second type is the Goldstone's solution. Differentiating (7) over x we have ~.(7 2) 3~P°
O~P° + 3X~°2 0X
the third term of (3) tends to zero at co > ko F and the crossing is impossible. I f the second phase transition occurs, a metastable state of nuclei with n ~ n o may exist. Possibly such anomalous nuclei can be found in cosmic rays. The distortion of the meson propagator in nuclear matter due to the A-resonance leads to an essential non-pair term in the internucleon forces, which cannot be found by usual methods assuming the pair interaction. A more detailed consideration will be given elsewhere. The author expresses his thanks to A.A. Migdal, V.A. Khodel, A.M. Poliakov and his collaborators O. Marion and I. Mishustin.
('7')
Introducing ~' = X ~ o / 0 x, X being a slowly varying function of 7, into (9), and comparing with (7'), we have ¢o2(k) --* O, k 2 --* ko2. The condensate with ~o= asin koX will lead to a laminated structure of nuclear matter, which can be observed by electron and meson scattering on polarized nuclei. At densitites n ~ 2nc~ another type of phase transition may happen. The meson branch co(~)(k) may cross the zero value for some k. It is easy to see from (3) that the meson branch can cross the line co = kv F only, if a region of k with g - ( k ) ~ 0 exists. If not, the denominator of
450
20 August 1973
References [1] [2] [3] [4] [5] [6] [7]
A.B. Migdal,Soviet Phys. JETP 61 (1971) 2209. A.B. Migdal, Soviet Phys. UFN 105 (1971) 775. R.F. Sawyer, Phys. Rev. Lett. 29 (1972) 382. D.J. Scalapino, Phys. Rev. Lett. 29 (1972) 386. I. Kogut and I.T. Manassah, Phys. Lett. 41A (1972) 129. A.B. Migdal, Phys. Rev. Lett. (in press). A.B. Migdal, "Theory of finite Fermi systems and applications to atomic nuclei" (Interscience Publishers Inc. 1965).