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Periodic structure and stability of nuclear matter
Volume 52B, number 3
PHYSICS LETTERS
PERIODIC STRUCTURE
14 October 1974
AND STABILITY OF NUCLEAR MATTER A.B. MIGDAL
L.D. Landau Institute for Theoretical Physics, Moscow, USSR
Received 2 September 1974 The stability of ~r-condensate in nuclear matter is considered. It is shown that the periodic structure of the condensate field leads to a periodic modulation of the nuclear density. The influence of the condensation on nuclear properties is discussed. The problem of stability o f nuclear matter was first considered in ref. [1]. It was shown that, at a sufficient density, there must arise a second order phase transition with the appearance of a 7r-condensate. In ref. [2] a method of finding the pion spectrum in nuclear matter was developed which takes account of nucleon correlations. It was also pointed out that after the appearance of the n-condensate another phase transition may take place, which corresponds to existence of superdense nuclei (for more detailed discussion see ref. [3]). According to the estimates made in ref. [2] the critical density n c turned out to be less than the nuclear one n o . Provided this estimation is valid, in nuclei, a periodical 7r-condensate field with the wave vector k o ~ 1.5 fm -1 (k o ~ PF) should exist. The periodic spin structure leads in the second order of amplitude to a periodic structure o f the neutron and proton density with the wave vector 2k o 3 fm -1. It has been suggested in ref. [4] that the distortions of the electric nuclear formfactor in a narrow interval of transferred momenta at q ~ 3 fm -1 observed in the experiments on the electron scattering [5] may be accounted for by this periodic structure (as was shown in [4] the shell-density fluctuations perturb the form factor in a broad region of q). In ref. [6] some possible theoretical objections to the calculation method developed in [2] were analysed in detail, and it was shown that there are no reasonable grounds to doubt these calculations. Some uncertainty exists only in the choice of the spin-spin nucleon interaction constant in a nucleus which determines the critical density n c and is obtained from the analysis of the nuclear experiments. It seems doubtful whether it is possible to make a more precise estimation o f this constant; that is why the final answer to 264
the question of 7r-condensation in a nucleus may be obtained only experimentally. The energy of the static pion field with the wave vector k o in nuclear matter may be written as follows (the isotopic indices are omitted): 4
) 2o+
dV
(1)
As shown in ref. [2] (h = m~r = c = 1) ~ 2 ( k o ) = 1 + k2o + II(0, ko)
(2)
where H ( ~ , k) is the polarization operator, found in ref. [2] for N = Z. The quantity ~ 2 ( k ) is negative at n > n o, k ~ k o and has a minimum at k = k o . Near the minimum ~2(k) has the form
~2(k)
(k2-ko2) 2 = ~ 2 ( k o) + 3' k~ 0
(3)
~ 2 ( k o ) = v(n c - n ) < O. In an infinite medium the minimal energy corresponds to a field of the form ~0o = a sin k o Z,
a 2 _ _ 4 5 2 (k o) 3 ;k(ko) "
(4)
It should be noticed that the laminated structure (4) minimizes the energy in the )t~04-model. The question of a space and isotopic structure o f n-condensate in a realistic mr-interaction will be discussed elsewhere. An approximate consideration using the "lhomasFermi calculation indicated the possibility o f a threeaxes periodicity (crystal structure) [7].
Volume 52B, number 3
PHYSICS LETTERS
The quantity ~ 2 ( k ) was found in [6] for N = Z and N ~ Z . To calculate X(ko) in the medium it is necessary to find the nucleon energy change in the periodical field (4) up to terms "~ a 4. The density is assumed not to exceed much the critical density, so a being small, the expansion in powers of a converges. Omitting the calculations (see ref. [7]), we give below the result for the nucleon energy density in the field (4) ~2(ko)a2
E~n) = E(n)o +
4
3?t(ko)a4
÷
3~
(5)
The function ~.(k) is given in [7]. In particular
X =f4k4/2rr203F,
k "~PF'
X ~ 90,
k = PE"
(6) Here f i s the pion-nucleon coupling constant, f =
g/2m ~ 1. The minimum of the expression (5) corresponds to
a2 _ 4~2(ko ) 3X(ko)
~:~.) = Eo,) ,
o
c¢(ko) 6X(ko) •
(7)
The account made for the nucleon correlations essentially reduces the quantity ~t(k). The main effect of the nucleon interaction consists in the fact that each of the four vertices which correspond to the interaction )ktp4 is multiplied by the factor [1 +g-(a(ko/2PF)] -1 .~ 1/2.2, where g - is the spinspin interaction constant, and ¢(x) is the function given in [2]. As a result the effective constant ~ ~ 3, k = PF- Account of the nonlinear terms in the pion Lagrangian changes this value only inessentially. A more detailed account of the nucleon correlations, as well as the influence of the vacuum pion interaction on the constant ~ is given in [7]. The periodic condensate field leads in the second order of a to the nucleon density modulation. The density modulation is determined from the graph
14 October 1974
n('r) = n o ( r ) ( l +~2 cos 2koZ )
~°2-
3f2a2k2
1
16e 2
[ l + g - ~ ( k o / 2 P v ) ] 2'
~ "~ 1"65~2o'
k'~PF
(8)
ko =PF"
The influence of the nucleon correlations in general is reduced to the multiplying of the quantity obtained in the perturbation theory by the squared factor
[1 +g-~]-l.
7r-condensation in the finite system was discussed in ref. [8]. An equation similar to that of GinzburgLandau was obtained. If 7r-condensation exists in nuclei, in the internal region of a heavy nucleus the solution (4) is realized; it transforms to the value ~0= 0 in a layer of width 6 ~ 1/l~(ko)l on the boundary of the nucleus. In heavy nuclei 6 < R and the expressions for the density modulation are valid. After averaged over the directions of the vector k o, eq. (8) gives n = no(1 +~2 sin 2kor/2kor) at r < R - 6. Such a distribution leads to an essential distortion of the elastic electron scattering form factor at transferred momenta q ~ 2k o [4]. If the form-factor distortion observed in an experiment is due to such a mechanism, for ~2 one obtains the value ~2 ~ 0.08 [4] wherefrom we get a 2 ~ 0.04 and - ~2(ko)Dt ~ 0.03. At ~, = 3 (see above) we have 1~2(ko)[ ~ 0.1. All these values are very preliminary. Note that a rough evaluation of ~ 2 ( k o) by means of the constants introduced in [2] gives [~2(ko)l ~ 1. The laminated structure leads to the appearance of moments of inertia of spherical nuclei. The estimation of the moment of inertia may be obtained using the results of [8]. The moment of inertia is of the order of
J~
(eu,m - % , m - 1)2 JS A2
where JS is the solid body moment of inertia,
eu,m - %,m-1 is the energy difference for the neighp
p + ko
p + 2ko
The following expression was obtained [7]
bouring angular moment projection onto the layer directions, A is the pairing energy or the gap in the case of magic nuclei. The energy difference is found from the expression for the nucleon energy in the periodical field ~0o 265
Volume 52B, number 3
PHYSICS LETTERS
p2 f 2a2 mnk o2 1 et.p2'p-)'~'z = ~ + 4 2 2 " n Pz - k o / 4 Substituting with a quasiclassical accuracy 2
p2 sin20 c o s 2 0 = p 2 ½ ( 1
m2/j2)
we find the dependence of the energy on m, .-2 2-2
ev,m-ev, m - l ~
fat¢ o 1 eFJ [l+g-(o(ko/2PF)] 2"
The last factor takes into account the nucleon-nucleon interaction. Using the value a 2 given above, we get for the ratio o f the moment of inertia to that of the solid body
JVT s ~ 0.1 eF/]A • For a deformed nuclei with the deformation
14 October 1974
The n-condensation, if it exists in nuclei, influences essentially the energy of 0 - , T = 1 states and M1 probabilities. With the values ~2(k) and ;~ given above, a satisfactory estimation of the energy of the level 0 - , T = 1 in 160 as well as the explanation for the Observed enlargement o f M1 transitions o f the type d3/2-Sl/2 was obtained in [9]. Near the critical density in the model k~04 the laminated structure turns out to be stable (see note after eq. (4)). It seems probable [10] that at a sufficient density } a phase transition of the first order with the formation of the crystal structure will take place. Such a transition might be the reason of the spasmodic changes in the pulsar periods, if they are neutron stars. The author is grateful to his collaborators N.A. Kirichenko, O.A. Markin, I.N. Mishustin and G.A. Sorokin for helpful discussions.
/3 x/J/J s "/3eF/]A. The laminated structure is thus equivalent in the sense o f the moment o f inertia to the deformation/3 -'- 0.1. Of course, these estimations are very approximate. Owing to condensation, the binding energy increases. With the above values of 5 2 and ~, the additional binding energy ~ 0.2 MeV per nucleon. In a finite system due to the boundary conditions the condensate field cannot possess the wave vector, precisely equal to that, corresponding to the minimum ~2(k). The lack of coincidence of these vectors Ak 1/R would lead to a marked decrease of the condensation energy. It is energetically more favourable to make a little change of the nucleus shape along the direction of layers leading to the maximal coincidence of these vectors. This phenomenon may be a source of certain fluctuations of the condensate field from one nucleus to another.
266
References
[1] A.B. Migdal, ZhETF 61 (1971) 2210. [2] A.B. Migdal, ZhETF 62 (1972) 1993; Nucl. Phys. A210 (1973) 421. [3] A.B. Migdal, preprint, Chernogolovka, 1974. [4] A.B. Migdal, Phys. Rev. Lett., to be published. [5] J. Bellicard et al., Phys. Rev. Lett. 19 (1967) 527; I. Sick, Nucl. Phys. A208 (1973) 557. [6] A.B. Migdal, O.A. Markin and I.N. Mishustin, ZhETF 66 (1974) 443. [7] A.B. Migdal, O.A. Markin and I.N. Mishustin, ZhETF, to be published. [8] A.B. Migdal, N.A. Kirichenko and G.A. Sorokin, Pis'ma v ZhETF 19 (1974) 326. [9] M.A. Troitskii and E.E. Sapershtein, Yadern. Fiz., to be published. [10] R.G. Palmer, E. Tosatti and P.W. Anderson, Nature Phys. Sci. 245 (1973) 119.
PHYSICS LETTERS
PERIODIC STRUCTURE
14 October 1974
AND STABILITY OF NUCLEAR MATTER A.B. MIGDAL
L.D. Landau Institute for Theoretical Physics, Moscow, USSR
Received 2 September 1974 The stability of ~r-condensate in nuclear matter is considered. It is shown that the periodic structure of the condensate field leads to a periodic modulation of the nuclear density. The influence of the condensation on nuclear properties is discussed. The problem of stability o f nuclear matter was first considered in ref. [1]. It was shown that, at a sufficient density, there must arise a second order phase transition with the appearance of a 7r-condensate. In ref. [2] a method of finding the pion spectrum in nuclear matter was developed which takes account of nucleon correlations. It was also pointed out that after the appearance of the n-condensate another phase transition may take place, which corresponds to existence of superdense nuclei (for more detailed discussion see ref. [3]). According to the estimates made in ref. [2] the critical density n c turned out to be less than the nuclear one n o . Provided this estimation is valid, in nuclei, a periodical 7r-condensate field with the wave vector k o ~ 1.5 fm -1 (k o ~ PF) should exist. The periodic spin structure leads in the second order of amplitude to a periodic structure o f the neutron and proton density with the wave vector 2k o 3 fm -1. It has been suggested in ref. [4] that the distortions of the electric nuclear formfactor in a narrow interval of transferred momenta at q ~ 3 fm -1 observed in the experiments on the electron scattering [5] may be accounted for by this periodic structure (as was shown in [4] the shell-density fluctuations perturb the form factor in a broad region of q). In ref. [6] some possible theoretical objections to the calculation method developed in [2] were analysed in detail, and it was shown that there are no reasonable grounds to doubt these calculations. Some uncertainty exists only in the choice of the spin-spin nucleon interaction constant in a nucleus which determines the critical density n c and is obtained from the analysis of the nuclear experiments. It seems doubtful whether it is possible to make a more precise estimation o f this constant; that is why the final answer to 264
the question of 7r-condensation in a nucleus may be obtained only experimentally. The energy of the static pion field with the wave vector k o in nuclear matter may be written as follows (the isotopic indices are omitted): 4
) 2o+
dV
(1)
As shown in ref. [2] (h = m~r = c = 1) ~ 2 ( k o ) = 1 + k2o + II(0, ko)
(2)
where H ( ~ , k) is the polarization operator, found in ref. [2] for N = Z. The quantity ~ 2 ( k ) is negative at n > n o, k ~ k o and has a minimum at k = k o . Near the minimum ~2(k) has the form
~2(k)
(k2-ko2) 2 = ~ 2 ( k o) + 3' k~ 0
(3)
~ 2 ( k o ) = v(n c - n ) < O. In an infinite medium the minimal energy corresponds to a field of the form ~0o = a sin k o Z,
a 2 _ _ 4 5 2 (k o) 3 ;k(ko) "
(4)
It should be noticed that the laminated structure (4) minimizes the energy in the )t~04-model. The question of a space and isotopic structure o f n-condensate in a realistic mr-interaction will be discussed elsewhere. An approximate consideration using the "lhomasFermi calculation indicated the possibility o f a threeaxes periodicity (crystal structure) [7].
Volume 52B, number 3
PHYSICS LETTERS
The quantity ~ 2 ( k ) was found in [6] for N = Z and N ~ Z . To calculate X(ko) in the medium it is necessary to find the nucleon energy change in the periodical field (4) up to terms "~ a 4. The density is assumed not to exceed much the critical density, so a being small, the expansion in powers of a converges. Omitting the calculations (see ref. [7]), we give below the result for the nucleon energy density in the field (4) ~2(ko)a2
E~n) = E(n)o +
4
3?t(ko)a4
÷
3~
(5)
The function ~.(k) is given in [7]. In particular
X =f4k4/2rr203F,
k "~PF'
X ~ 90,
k = PE"
(6) Here f i s the pion-nucleon coupling constant, f =
g/2m ~ 1. The minimum of the expression (5) corresponds to
a2 _ 4~2(ko ) 3X(ko)
~:~.) = Eo,) ,
o
c¢(ko) 6X(ko) •
(7)
The account made for the nucleon correlations essentially reduces the quantity ~t(k). The main effect of the nucleon interaction consists in the fact that each of the four vertices which correspond to the interaction )ktp4 is multiplied by the factor [1 +g-(a(ko/2PF)] -1 .~ 1/2.2, where g - is the spinspin interaction constant, and ¢(x) is the function given in [2]. As a result the effective constant ~ ~ 3, k = PF- Account of the nonlinear terms in the pion Lagrangian changes this value only inessentially. A more detailed account of the nucleon correlations, as well as the influence of the vacuum pion interaction on the constant ~ is given in [7]. The periodic condensate field leads in the second order of a to the nucleon density modulation. The density modulation is determined from the graph
14 October 1974
n('r) = n o ( r ) ( l +~2 cos 2koZ )
~°2-
3f2a2k2
1
16e 2
[ l + g - ~ ( k o / 2 P v ) ] 2'
~ "~ 1"65~2o'
k'~PF
(8)
ko =PF"
The influence of the nucleon correlations in general is reduced to the multiplying of the quantity obtained in the perturbation theory by the squared factor
[1 +g-~]-l.
7r-condensation in the finite system was discussed in ref. [8]. An equation similar to that of GinzburgLandau was obtained. If 7r-condensation exists in nuclei, in the internal region of a heavy nucleus the solution (4) is realized; it transforms to the value ~0= 0 in a layer of width 6 ~ 1/l~(ko)l on the boundary of the nucleus. In heavy nuclei 6 < R and the expressions for the density modulation are valid. After averaged over the directions of the vector k o, eq. (8) gives n = no(1 +~2 sin 2kor/2kor) at r < R - 6. Such a distribution leads to an essential distortion of the elastic electron scattering form factor at transferred momenta q ~ 2k o [4]. If the form-factor distortion observed in an experiment is due to such a mechanism, for ~2 one obtains the value ~2 ~ 0.08 [4] wherefrom we get a 2 ~ 0.04 and - ~2(ko)Dt ~ 0.03. At ~, = 3 (see above) we have 1~2(ko)[ ~ 0.1. All these values are very preliminary. Note that a rough evaluation of ~ 2 ( k o) by means of the constants introduced in [2] gives [~2(ko)l ~ 1. The laminated structure leads to the appearance of moments of inertia of spherical nuclei. The estimation of the moment of inertia may be obtained using the results of [8]. The moment of inertia is of the order of
J~
(eu,m - % , m - 1)2 JS A2
where JS is the solid body moment of inertia,
eu,m - %,m-1 is the energy difference for the neighp
p + ko
p + 2ko
The following expression was obtained [7]
bouring angular moment projection onto the layer directions, A is the pairing energy or the gap in the case of magic nuclei. The energy difference is found from the expression for the nucleon energy in the periodical field ~0o 265
Volume 52B, number 3
PHYSICS LETTERS
p2 f 2a2 mnk o2 1 et.p2'p-)'~'z = ~ + 4 2 2 " n Pz - k o / 4 Substituting with a quasiclassical accuracy 2
p2 sin20 c o s 2 0 = p 2 ½ ( 1
m2/j2)
we find the dependence of the energy on m, .-2 2-2
ev,m-ev, m - l ~
fat¢ o 1 eFJ [l+g-(o(ko/2PF)] 2"
The last factor takes into account the nucleon-nucleon interaction. Using the value a 2 given above, we get for the ratio o f the moment of inertia to that of the solid body
JVT s ~ 0.1 eF/]A • For a deformed nuclei with the deformation
14 October 1974
The n-condensation, if it exists in nuclei, influences essentially the energy of 0 - , T = 1 states and M1 probabilities. With the values ~2(k) and ;~ given above, a satisfactory estimation of the energy of the level 0 - , T = 1 in 160 as well as the explanation for the Observed enlargement o f M1 transitions o f the type d3/2-Sl/2 was obtained in [9]. Near the critical density in the model k~04 the laminated structure turns out to be stable (see note after eq. (4)). It seems probable [10] that at a sufficient density } a phase transition of the first order with the formation of the crystal structure will take place. Such a transition might be the reason of the spasmodic changes in the pulsar periods, if they are neutron stars. The author is grateful to his collaborators N.A. Kirichenko, O.A. Markin, I.N. Mishustin and G.A. Sorokin for helpful discussions.
/3 x/J/J s "/3eF/]A. The laminated structure is thus equivalent in the sense o f the moment o f inertia to the deformation/3 -'- 0.1. Of course, these estimations are very approximate. Owing to condensation, the binding energy increases. With the above values of 5 2 and ~, the additional binding energy ~ 0.2 MeV per nucleon. In a finite system due to the boundary conditions the condensate field cannot possess the wave vector, precisely equal to that, corresponding to the minimum ~2(k). The lack of coincidence of these vectors Ak 1/R would lead to a marked decrease of the condensation energy. It is energetically more favourable to make a little change of the nucleus shape along the direction of layers leading to the maximal coincidence of these vectors. This phenomenon may be a source of certain fluctuations of the condensate field from one nucleus to another.
266
References
[1] A.B. Migdal, ZhETF 61 (1971) 2210. [2] A.B. Migdal, ZhETF 62 (1972) 1993; Nucl. Phys. A210 (1973) 421. [3] A.B. Migdal, preprint, Chernogolovka, 1974. [4] A.B. Migdal, Phys. Rev. Lett., to be published. [5] J. Bellicard et al., Phys. Rev. Lett. 19 (1967) 527; I. Sick, Nucl. Phys. A208 (1973) 557. [6] A.B. Migdal, O.A. Markin and I.N. Mishustin, ZhETF 66 (1974) 443. [7] A.B. Migdal, O.A. Markin and I.N. Mishustin, ZhETF, to be published. [8] A.B. Migdal, N.A. Kirichenko and G.A. Sorokin, Pis'ma v ZhETF 19 (1974) 326. [9] M.A. Troitskii and E.E. Sapershtein, Yadern. Fiz., to be published. [10] R.G. Palmer, E. Tosatti and P.W. Anderson, Nature Phys. Sci. 245 (1973) 119.