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Equation of state for a degenerate baryon gas including strong forces
Volume
30B, number 2
EQUATION
PHYSICS
OF
LETTERS
15 September
STATE FOR A DEGENERATE INCLUDING STRONG FCRCES L. MARSHALL
BARYON
1969
GAS
LIBBY
The University of Colorado, Boulder, Colorado, and the RAND Corporation, Santa Monica, Calif., USA
and F. J. THOMAS The RAND Corporation, Received
Santa Monica,
Calif.,
USA
21 July 1969
‘The equation of state for a degenerate gas of the 45 known species of baryons is computed for O°K, with inclusion of the interaction energy derived from high energy proton-proton elastic scattering. A region of partial collapse is indicated for density - 104’ cmW3.
To-day the spectrum of baryon masses is fairly completely known to 2.0 GeV, and sketchily to 3.0 GeV, allowing calculation of an improved equation of state for a degenerate baryon This calculagas up to densities - 1018 g/cm3. tion is of increasing importance to relate elementary particle research to possible observation of neutron stars [ 11. In principle the behavior of such stars gives information on highly excited baryons not accessible to study by our present particle accelerators because of energy limitations, and on strong forces acting between particles at short range. Super-dense matter was conjectured first by Landau [2] before the discovery of the neutron and examined by Oppenheimer and Volkoff [3] at a time when of unstable baryons only the neutron was known. Later [4,5] the A, C, A and E were included in study of the baryon gas. In the present paper we study a degenerate gas of 45 known baryons, and assume them to interact with the baryon-baryon interaction energy recently derived [6] directly from proton-proton scattering at energies up to - 30 GeV. As pressure increases, inverse beta decay produces baryons according to N*O + v;
e-+p*n+v; A0 +
V;
Z”
+
V;
A0 + v;
co+
v;
etc.
From the neutral species, charged baryons, both strange and non-strange, are produced by exchange reactions such as 88
A*0
+ A*0
-.,
A*+
+
A*-
A*+
+ A*0
~
A*++
+
A*-
co + co - p -0 +.Z
n
-E-
+ c+p
, etc.
The equation for the pressure gas of non-interacting particles, ample in ref. 5, is given by -Pn
of a degenerate derived for ex-
=c j
1
where xj = Pj/‘mj c is computed for each species of particle at the top of its Fermi sea, and the summation is made over the presently known baryons [?I. The statistical weights, aj, which are the products of Pauli spin and isotopic spin multiplicities, can be quite large; for example aj = 16 for ~(1236), h(1670), A(l920), . . . , and Uj = 6 for .X(1190), X(1382), X(1610), . . . . We have neglected effects of mesons for simplicity. We assume all allowed baryon states to be filled and the gas to be at OoK. We assume the baryon-baryon interaction energy to be identical for all pairs of baryons *, equal to that derived [6] from proton-proton
* Footnote
see next page.
Volume
30B, number
2
PHYSICS
elastic scattering cross (see ref. 6 fig. 1 )
sections,
LETTERS
15 September
1969
given by
E(B) = Zt+ (do/dt),l)+ (2)
2= E2/VminMp(l
-COS
0)+
where the momentum transfer is t = 3 = 2c2p2( 1 - cos 0) and Vmin = + 7r‘min 3 Ymin = = 0.3 fm, and dp is the proton mass. As shown in ref.6, E(B) is approximately equal for scattering angles tI = 60° and 90°, so that we may choose E(0) = E(90°) for all 8, equivalent to setting Q = 90° for average collisions in the baryon gas. The pressure Ps from strong forces is computed according to = dE/dV
P,
= (4n~~)+lE/d-/.
(3)
The elastic scattering cross section is well fitted by exponential dependence on t, namely (d o/dt)$
= Ai exp (-ibt)
(4)
so that g
=g
5
=$
@A$(1
- bt) exp
(-$bt) .
(5)
Using the relation [6] between t and ‘Y derived from the optical model Y = (&r,)/(2$
Mp c2(1 - cos e)+) y
which for 90° simplifies Y = E k/y it follows P
S
=
$ (4n
;
(6)
to
E = ro/(2t
Mp c2)
(7)
that
.2)-l (8)
We take b = 8 GeVw2, conwhere yp, = (1+x;). sistent with p-p elastic scattering. The component of pressure resulting from strong forces, P,, as given by eq. (8), depends on interparticle distance Y and hence on particle number density N = ( 97~y3)-l, according to Y = ($rN)-+ This pressure component is zero at small densities, - 1038 cm-3, grows increasingly negative as density increases, becoming dominantly * All known baryon-baryon
cross sections become exponentially peaked in the forward direction at high energies. Then their interaction energies must depend on separation distance like the curves in fig. 1, as discussed in ref. 6.
Fig. 1. Plot of pressure versus density for a degenerate baryon gas with strong forces. The heavy black line shows the total pressure, the sum of the Fermi gas pressure for a non-interacting degenerate baryon gas and the pressure caused by particle-particle forces. At low densities the latter component is negligible; at intermediate densities it is dominant and negative so that the gas suffers partial collapse, and at high densitites it is dominant and strongly positive, so that the partially collapsed baryon gas becomes incompressible, The dashed line shows pressure of a gas of 45 species of non-interacting baryons. At point A the composition is (np)(CA)/A/M(1450)/M(1550) = 35%/19%/380/c/10/c/70/c, 7 = 1.46 and at point B, the composition is (np)/(CA)/A = 47o/c/26o/c/27o/c, 7 = 1.36. Successive production of higher and higher mass baryons keeps the gas from becoming strongly relativistic.
negative at - 1 x 1039 cmm3, so that the baryon gas collapses to a density of - 3 x 1041 cm-3. Thereafter, with increasing density, the ‘strong’ pressure becomes positive and large so that the gas becomes relatively incompressible. The sum of Fermi gas pressure and ‘strong’ pressure computed from eqs. (1) and (8) with the RAND “Joss” computer is plotted in fig. 1, showing the region where ‘strong’ pressure is negligible at low densities, the region of collapse where ‘strong’ pressure is dominant and negative, and the region of relative incompressibility where ‘strong’ pressure is dominant and strongly positive. The proton-proton interaction energy, eq.(2), is not well known for very small t (equivalent to small y) nor for very large t (equivalent to large Y), because for very small t, resonance production obscures the exponential nature of cross sections, and for very large t, no information is available on differential cross sections. 89
Volume 30B, number 2
PHYSICS
LETTERS
In principle it is possible that there may be structure in E(B) at very large t, for example a small valley in the rising slope of E(e) as it approaches zero at 0.6 5 Y S 0.8 fm. Such a valley might produce a second region of negative pressure and therefore of partial collapse of a baryon gas at densities N - 1038 cm-s. But one expects then that the second region extends over a much shorter range of baryon density. Study of pulsar behavior, if indeed they are neutron stars, may give information on the baryon-baryon interaction energy not otherwise obtainable from present day accelerators.
References 1. A. Hewish, S. Bell, J. Pilkington, P. Scott and R. Collins, Nature 217 (1968) 709; T. Gold, Nature 218 (1968) 731. 2. L. Landau, Physik. 2. Sowjetunion 1 (1932) 285. 3. J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55 (1939) 374. A. G. W. Cameron, Ap. J. 89 (1959) 884; A.G. W. Cameron and S. Tsuruta, Csn. J. Phys. 44 (1966) 1895. V. A. Ambartsumian and G. S. Saakyan, Soviet Astron. AJ 4 (1960) 187. L.Marshall Libby, Phys. Letters 29B (1969) 345. N. Barasch-Schmidt, A. Barbero-Galtiero, L. Price, A. Rosenfeld, P. Sading, C. Wohl, M. Roos and G. Conforto, Rev. Mod. Phys. 41 (1969) 109.
*****
90
15 September 1969
30B, number 2
EQUATION
PHYSICS
OF
LETTERS
15 September
STATE FOR A DEGENERATE INCLUDING STRONG FCRCES L. MARSHALL
BARYON
1969
GAS
LIBBY
The University of Colorado, Boulder, Colorado, and the RAND Corporation, Santa Monica, Calif., USA
and F. J. THOMAS The RAND Corporation, Received
Santa Monica,
Calif.,
USA
21 July 1969
‘The equation of state for a degenerate gas of the 45 known species of baryons is computed for O°K, with inclusion of the interaction energy derived from high energy proton-proton elastic scattering. A region of partial collapse is indicated for density - 104’ cmW3.
To-day the spectrum of baryon masses is fairly completely known to 2.0 GeV, and sketchily to 3.0 GeV, allowing calculation of an improved equation of state for a degenerate baryon This calculagas up to densities - 1018 g/cm3. tion is of increasing importance to relate elementary particle research to possible observation of neutron stars [ 11. In principle the behavior of such stars gives information on highly excited baryons not accessible to study by our present particle accelerators because of energy limitations, and on strong forces acting between particles at short range. Super-dense matter was conjectured first by Landau [2] before the discovery of the neutron and examined by Oppenheimer and Volkoff [3] at a time when of unstable baryons only the neutron was known. Later [4,5] the A, C, A and E were included in study of the baryon gas. In the present paper we study a degenerate gas of 45 known baryons, and assume them to interact with the baryon-baryon interaction energy recently derived [6] directly from proton-proton scattering at energies up to - 30 GeV. As pressure increases, inverse beta decay produces baryons according to N*O + v;
e-+p*n+v; A0 +
V;
Z”
+
V;
A0 + v;
co+
v;
etc.
From the neutral species, charged baryons, both strange and non-strange, are produced by exchange reactions such as 88
A*0
+ A*0
-.,
A*+
+
A*-
A*+
+ A*0
~
A*++
+
A*-
co + co - p -0 +.Z
n
-E-
+ c+p
, etc.
The equation for the pressure gas of non-interacting particles, ample in ref. 5, is given by -Pn
of a degenerate derived for ex-
=c j
1
where xj = Pj/‘mj c is computed for each species of particle at the top of its Fermi sea, and the summation is made over the presently known baryons [?I. The statistical weights, aj, which are the products of Pauli spin and isotopic spin multiplicities, can be quite large; for example aj = 16 for ~(1236), h(1670), A(l920), . . . , and Uj = 6 for .X(1190), X(1382), X(1610), . . . . We have neglected effects of mesons for simplicity. We assume all allowed baryon states to be filled and the gas to be at OoK. We assume the baryon-baryon interaction energy to be identical for all pairs of baryons *, equal to that derived [6] from proton-proton
* Footnote
see next page.
Volume
30B, number
2
PHYSICS
elastic scattering cross (see ref. 6 fig. 1 )
sections,
LETTERS
15 September
1969
given by
E(B) = Zt+ (do/dt),l)+ (2)
2= E2/VminMp(l
-COS
0)+
where the momentum transfer is t = 3 = 2c2p2( 1 - cos 0) and Vmin = + 7r‘min 3 Ymin = = 0.3 fm, and dp is the proton mass. As shown in ref.6, E(B) is approximately equal for scattering angles tI = 60° and 90°, so that we may choose E(0) = E(90°) for all 8, equivalent to setting Q = 90° for average collisions in the baryon gas. The pressure Ps from strong forces is computed according to = dE/dV
P,
= (4n~~)+lE/d-/.
(3)
The elastic scattering cross section is well fitted by exponential dependence on t, namely (d o/dt)$
= Ai exp (-ibt)
(4)
so that g
=g
5
=$
@A$(1
- bt) exp
(-$bt) .
(5)
Using the relation [6] between t and ‘Y derived from the optical model Y = (&r,)/(2$
Mp c2(1 - cos e)+) y
which for 90° simplifies Y = E k/y it follows P
S
=
$ (4n
;
(6)
to
E = ro/(2t
Mp c2)
(7)
that
.2)-l (8)
We take b = 8 GeVw2, conwhere yp, = (1+x;). sistent with p-p elastic scattering. The component of pressure resulting from strong forces, P,, as given by eq. (8), depends on interparticle distance Y and hence on particle number density N = ( 97~y3)-l, according to Y = ($rN)-+ This pressure component is zero at small densities, - 1038 cm-3, grows increasingly negative as density increases, becoming dominantly * All known baryon-baryon
cross sections become exponentially peaked in the forward direction at high energies. Then their interaction energies must depend on separation distance like the curves in fig. 1, as discussed in ref. 6.
Fig. 1. Plot of pressure versus density for a degenerate baryon gas with strong forces. The heavy black line shows the total pressure, the sum of the Fermi gas pressure for a non-interacting degenerate baryon gas and the pressure caused by particle-particle forces. At low densities the latter component is negligible; at intermediate densities it is dominant and negative so that the gas suffers partial collapse, and at high densitites it is dominant and strongly positive, so that the partially collapsed baryon gas becomes incompressible, The dashed line shows pressure of a gas of 45 species of non-interacting baryons. At point A the composition is (np)(CA)/A/M(1450)/M(1550) = 35%/19%/380/c/10/c/70/c, 7 = 1.46 and at point B, the composition is (np)/(CA)/A = 47o/c/26o/c/27o/c, 7 = 1.36. Successive production of higher and higher mass baryons keeps the gas from becoming strongly relativistic.
negative at - 1 x 1039 cmm3, so that the baryon gas collapses to a density of - 3 x 1041 cm-3. Thereafter, with increasing density, the ‘strong’ pressure becomes positive and large so that the gas becomes relatively incompressible. The sum of Fermi gas pressure and ‘strong’ pressure computed from eqs. (1) and (8) with the RAND “Joss” computer is plotted in fig. 1, showing the region where ‘strong’ pressure is negligible at low densities, the region of collapse where ‘strong’ pressure is dominant and negative, and the region of relative incompressibility where ‘strong’ pressure is dominant and strongly positive. The proton-proton interaction energy, eq.(2), is not well known for very small t (equivalent to small y) nor for very large t (equivalent to large Y), because for very small t, resonance production obscures the exponential nature of cross sections, and for very large t, no information is available on differential cross sections. 89
Volume 30B, number 2
PHYSICS
LETTERS
In principle it is possible that there may be structure in E(B) at very large t, for example a small valley in the rising slope of E(e) as it approaches zero at 0.6 5 Y S 0.8 fm. Such a valley might produce a second region of negative pressure and therefore of partial collapse of a baryon gas at densities N - 1038 cm-s. But one expects then that the second region extends over a much shorter range of baryon density. Study of pulsar behavior, if indeed they are neutron stars, may give information on the baryon-baryon interaction energy not otherwise obtainable from present day accelerators.
References 1. A. Hewish, S. Bell, J. Pilkington, P. Scott and R. Collins, Nature 217 (1968) 709; T. Gold, Nature 218 (1968) 731. 2. L. Landau, Physik. 2. Sowjetunion 1 (1932) 285. 3. J. R. Oppenheimer and G. M. Volkoff, Phys. Rev. 55 (1939) 374. A. G. W. Cameron, Ap. J. 89 (1959) 884; A.G. W. Cameron and S. Tsuruta, Csn. J. Phys. 44 (1966) 1895. V. A. Ambartsumian and G. S. Saakyan, Soviet Astron. AJ 4 (1960) 187. L.Marshall Libby, Phys. Letters 29B (1969) 345. N. Barasch-Schmidt, A. Barbero-Galtiero, L. Price, A. Rosenfeld, P. Sading, C. Wohl, M. Roos and G. Conforto, Rev. Mod. Phys. 41 (1969) 109.
*****
90
15 September 1969