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Supersonic heavy-ion collisions
Volume 49B, number 3
PHYSICS LETTERS
15 April 1974
S U P E R S O N I C H E A V Y - I O N C O L L I S I O N S "~ C.Y. WONG and T.A. WELTON
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA Received 20 December 1973 An illustrative one-dimensional example of a supersonic heavy-ion collision reveals important features such as shock wave velocity, density increase, energy increase, heat energy produced and the Mach number at which dissociation of nucleons occurs.
When a nucleus suffers a small density perturbation, the effect is subsequently propagated with the speed of sound. Thus, one can conveniently classify a heavyion collision as subsonic or supersonic, depending on whether the relative velocity after contact is less or greater than the sound speed. In a subsonic collision, the presence of a perturbation affects the flow in all space both upstream and downstream and the density and pressure are smooth functions o f location. In a supersonic collision, the effect of the perturbation extends only downstream and allows the formation of shock waves characterized by near discontinuities in density, pressure and temperature. We shall study in this paper the discontinuities arising from a supersonic collision with an illustrative example. The sound speed in nuclear matter is given by [ 1,2]
a = [(2pae/aPlo + p2a2e/ap21o)/mn]l/2,
(1)
where p is the nuclear density in fm -3, e is the energy per nucleon and m n is the nucleon mass. The derivatives are evaluated at a fixed entropy per nucleon o. Thus, the sound speed for cold nuclear matter at equilibrium is a o = Ko~-~--m n ,
(2)
where K o is the nuclear "incompressibility" at equilibrium. The critical kinetic energy To which a nucleon possesses when it travels with the velocity o f sound a o is [2] 1
TO - 2 m n a 2 = K o]18.
(3)
For the nuclear incompressibility, Brueckner et al. [3] '~ Research sponsored by the U.S. Atomic Energy Commission under contract with Union Carbibe Corporation.
gave K o = 288.57 MeV while Bethe [4] gave 134 MeV. The critical kinetic energy is 16 MeV with Brueckner's incompressibility and is 7.5 MeV with Bethe's incompressibility. From theoretical arguments, one expects that the incompressibility of finite nuclei is appreciably reduced from that o f the nuclear matter [5]; the sound speed in f'mite nuclei should be substantially less than that in nuclear matter. Unfortunately, there is no experimental information on K o either for nuclear matter or for finite nuclei. In order to undertake a detailed calculation of a supersonic nuclear collision, we need a plausible equation of state for nuclear matter. Brueckner et al. [3] proposed a semi-empirical equation of state for nuclear matter at zero temperature: Co(p) = bop 3/2 + blP + b2P 4/3 + b3P 5/3.
(4)
The set of coefficients as given by Brueckner et al. and reported in ref. [4] (which will be designated as Brueckner I) leads to an equilibrium density po = 0.204 fm-3, a binding energy % ( 0 0 ) = - 16.57 MeV and an incompressibility K o (Po) = 288.57 MeV. This incompressibility is considerably higher than the theoretical estimate of other workers [4]. For this reason, one also introduces another set of coefficients (Brueckner II) such that [4] Po = 0.17 fm -3, %(00) = - 15.68 MeV and Ko(Po) = 134 MeV. The coefficients in MeV are then determined to be b o = 75.06, b 1 = - 9 3 4 . 7 5 , b 2 = 1808.42 and b 3 = - 9 6 0 . 6 3 . To generalize the equation of state to small temperatures one assumes a Fermi gas model to obtain the equation of state in the following form:
e(p,o) = c o ( p ) + (6[4~r)2/3(h2c2[2mnC2)o2p 2/3,
(5)
where o is the dimensionless entropy per particle. For 243
Volume 49B, number 3
PHYSICS LETTERS
15 April 1974
t.5
(o) .u
0
L,
4.0
o -,0
0.5
/
!(b)
(o)
30 .I
3.0 f
I
2.5
f
f .u
/
2O
f;
p/p+~ I /
I,
f
lo
2.0
J
I
f
f
t
~.5 1
2
3
MACH NUMBER (b)
t.0
2
3
4
MACH NUMBER
Fig. 1. a) The shock wave velocity s÷measured in the center of mass system in units of the velocity of sound ao, as a function of relative velocity expressed in Mach numbers, b) The ratio of central density p to equilibrium density p÷ as a function of Mach number. lack of a definite determination, the effective mass m n is taken to be the same as a nucleon mass. The essential features of the discontinuity in a supersonic collision can be best illustrated by a one-dimensional example. We consider two identical slabs o f nuclear matter moving toward each other with velocities lu+l = l u - I = + (Urel/2), densities p_ = p+ = Po (equilibrium density), pressures p+ = p_ = 0, and total energy per nucleon E+ = E_ = u 2 / 2 + e o ( P o ) . After the collision has begun, the central region will have velocity u, density p, pressure p and total energy E differing from the corresponding quantities in other regions on the left and the right. To obtain a simple calculation, one is frequently justified in making the shock wave approximation in which the thickness o f any discontinuities is assumed to be zero. Such an idealization is, o f course, useless if the actual thickness o f the approximate discontinuity (shock front) is not 244
Fig. 2. a) The intrinsic energy per particle e in the central region as a function of relative velocity, b) Heat energy per particle generated in the central region as a function of relative velocity. small compared with the thickness o f the colliding slab. A simple rule is that the thickness of any such transition region is approximately equal to the mean free path of a nucleon in nuclear matter which is roughly 0.7 fm for a nucleon with the critical kinetic energy. Denote the speed with which the fight shock front and the left shock front moves as s+ and s_, respectively. Following Rankine and Hugoniot [1], we have, from conservation o f mass flux p(s+ - u ) = p+(s+ - u+),
(6)
conservation o f m o m e n t u m flux p(s+ - u ) (u - u+) = p - p + ,
(7)
and conservation o f energy flux
p(s+ - u )
(E - E. ) = pu - p+ u..
(8)
We also have the equation o f state relating the total energy E with density and entropy. E = u2/2 + e(p,o).
(9)
Volume 49B, number 3
PHYSICS LETTERS
It is clear that the central velocity u is zero by symmetry. There are then four unknowns p, o, s+, and p in the four eqs. ( 6 - 9 ) . For a given relative velocity Ure1, they can be solved by Newton's method. The resuits are given in figs. 1 and 2 in which the abscissa is given in terms of the Mach number M defined as (Mach number M) = Urel/ao.
(10)
The solid curves are results obtained with Brueckner's equation of state with Brueckner I coefficients while the dashed curves are results obtained with Brueckner II coefficients. For both sets of coefficients, the shock front moves with approximately the velocity of sound (fig. la) in the center o f mass system. Thus, in the system in which the uncollided material is at rest, the shock speed is approximately a o + Urel/2. Fig. lb shows that the density ratio p/p+ in the central region is a monotonically increasing function o f the Mach number. It has the value of about 1.5 f o r M = 1 and reaches 2.6 and 3.1 for M = 4. It is significant to note that p/p+ does not in general equal 2. It is thus apparent that conservation laws requiring the balance of mass, momentum and energy do not generally give the superposition o f the two densities even at very high relative velocities. The usual "sudden" approximation [6] which assumes superposed densities in the case o f collision o f fast ions is not generally valid. The energy e o per particle in the central region is a monotonically increasing function of the Mach number (fig. 2a). Thus, at some point, the binding energy will go to zero and the nucleons become unbound. However, it is not necessary to go to such high velocity to have the occurrence o f dissociation o f nucleons. One can envisage a collision in which the temperature in the central region is high enough to free the nucleons from their binding in nuclear matter. A more quantitative estimate of the Mach number at which this "dissociation" occurs can be made by considering
15 April 1974
the total energy E of the central region and writing
E = eo(P ) + q,
(I 1)
where q is the heat energy per particle. Clearly, dissociation can occur when E is no longer negative. We show in fig. 2b the heat energy q per particle as a function of the Mach number. By comparing figs. 2a and 2b, we fred that dissociation can occur when M ~ 2 with the Brueckner I coefficients and when M ~ 3 with the Brueckner II coefficients. Our simple result in a one-dimensional model reveals some essential features of collisions of heavy ions having a kinetic energy many tens of MeV per nucleon. A realistic three-dimensional treatment of the process is called for to study these phenomena in more detail. Already we have a notion how the collision proceeds in the head-on coUision o f two identical stars from the work of Seidl and Cameron [7]. If an analogy could be made here, we might expect that shock waves would propagate along the collision axis and emerge with the dissociation of a large number o f constituent nucleons in the forward and backward direction. The detection of these dissociated nucleons might be a way of tracing out the shock waves in such a process.
References [1] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergaman Press, Ltd. London, 1958). [2] A.E. Glassgold, W. Heckr0tte and K.M. Watson, Ann. Phys. (N.Y.) 6 (1959) 1. [3] K.A. Brueckner, J.R. Buchler, R.C. Clark, R.J. Lombard, Phys. Rev. 181 (1969) 1543. [4] H.A. Bethe, Ann. Rev. Nucl. Science 21 (1971)93. [5] L. Zamick, Phys. Lett. B45 (1973) 313. [6] K.A. Brueckner, J.R. Buchler and M.M. Kelly, Phys. Rev. 173 (1968) 944. [7] F.G.P. Seidl and A.G.W. Cameron, Astrophysics & Space Science 15 (1972) 44.
245
PHYSICS LETTERS
15 April 1974
S U P E R S O N I C H E A V Y - I O N C O L L I S I O N S "~ C.Y. WONG and T.A. WELTON
Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA Received 20 December 1973 An illustrative one-dimensional example of a supersonic heavy-ion collision reveals important features such as shock wave velocity, density increase, energy increase, heat energy produced and the Mach number at which dissociation of nucleons occurs.
When a nucleus suffers a small density perturbation, the effect is subsequently propagated with the speed of sound. Thus, one can conveniently classify a heavyion collision as subsonic or supersonic, depending on whether the relative velocity after contact is less or greater than the sound speed. In a subsonic collision, the presence of a perturbation affects the flow in all space both upstream and downstream and the density and pressure are smooth functions o f location. In a supersonic collision, the effect of the perturbation extends only downstream and allows the formation of shock waves characterized by near discontinuities in density, pressure and temperature. We shall study in this paper the discontinuities arising from a supersonic collision with an illustrative example. The sound speed in nuclear matter is given by [ 1,2]
a = [(2pae/aPlo + p2a2e/ap21o)/mn]l/2,
(1)
where p is the nuclear density in fm -3, e is the energy per nucleon and m n is the nucleon mass. The derivatives are evaluated at a fixed entropy per nucleon o. Thus, the sound speed for cold nuclear matter at equilibrium is a o = Ko~-~--m n ,
(2)
where K o is the nuclear "incompressibility" at equilibrium. The critical kinetic energy To which a nucleon possesses when it travels with the velocity o f sound a o is [2] 1
TO - 2 m n a 2 = K o]18.
(3)
For the nuclear incompressibility, Brueckner et al. [3] '~ Research sponsored by the U.S. Atomic Energy Commission under contract with Union Carbibe Corporation.
gave K o = 288.57 MeV while Bethe [4] gave 134 MeV. The critical kinetic energy is 16 MeV with Brueckner's incompressibility and is 7.5 MeV with Bethe's incompressibility. From theoretical arguments, one expects that the incompressibility of finite nuclei is appreciably reduced from that o f the nuclear matter [5]; the sound speed in f'mite nuclei should be substantially less than that in nuclear matter. Unfortunately, there is no experimental information on K o either for nuclear matter or for finite nuclei. In order to undertake a detailed calculation of a supersonic nuclear collision, we need a plausible equation of state for nuclear matter. Brueckner et al. [3] proposed a semi-empirical equation of state for nuclear matter at zero temperature: Co(p) = bop 3/2 + blP + b2P 4/3 + b3P 5/3.
(4)
The set of coefficients as given by Brueckner et al. and reported in ref. [4] (which will be designated as Brueckner I) leads to an equilibrium density po = 0.204 fm-3, a binding energy % ( 0 0 ) = - 16.57 MeV and an incompressibility K o (Po) = 288.57 MeV. This incompressibility is considerably higher than the theoretical estimate of other workers [4]. For this reason, one also introduces another set of coefficients (Brueckner II) such that [4] Po = 0.17 fm -3, %(00) = - 15.68 MeV and Ko(Po) = 134 MeV. The coefficients in MeV are then determined to be b o = 75.06, b 1 = - 9 3 4 . 7 5 , b 2 = 1808.42 and b 3 = - 9 6 0 . 6 3 . To generalize the equation of state to small temperatures one assumes a Fermi gas model to obtain the equation of state in the following form:
e(p,o) = c o ( p ) + (6[4~r)2/3(h2c2[2mnC2)o2p 2/3,
(5)
where o is the dimensionless entropy per particle. For 243
Volume 49B, number 3
PHYSICS LETTERS
15 April 1974
t.5
(o) .u
0
L,
4.0
o -,0
0.5
/
!(b)
(o)
30 .I
3.0 f
I
2.5
f
f .u
/
2O
f;
p/p+~ I /
I,
f
lo
2.0
J
I
f
f
t
~.5 1
2
3
MACH NUMBER (b)
t.0
2
3
4
MACH NUMBER
Fig. 1. a) The shock wave velocity s÷measured in the center of mass system in units of the velocity of sound ao, as a function of relative velocity expressed in Mach numbers, b) The ratio of central density p to equilibrium density p÷ as a function of Mach number. lack of a definite determination, the effective mass m n is taken to be the same as a nucleon mass. The essential features of the discontinuity in a supersonic collision can be best illustrated by a one-dimensional example. We consider two identical slabs o f nuclear matter moving toward each other with velocities lu+l = l u - I = + (Urel/2), densities p_ = p+ = Po (equilibrium density), pressures p+ = p_ = 0, and total energy per nucleon E+ = E_ = u 2 / 2 + e o ( P o ) . After the collision has begun, the central region will have velocity u, density p, pressure p and total energy E differing from the corresponding quantities in other regions on the left and the right. To obtain a simple calculation, one is frequently justified in making the shock wave approximation in which the thickness o f any discontinuities is assumed to be zero. Such an idealization is, o f course, useless if the actual thickness o f the approximate discontinuity (shock front) is not 244
Fig. 2. a) The intrinsic energy per particle e in the central region as a function of relative velocity, b) Heat energy per particle generated in the central region as a function of relative velocity. small compared with the thickness o f the colliding slab. A simple rule is that the thickness of any such transition region is approximately equal to the mean free path of a nucleon in nuclear matter which is roughly 0.7 fm for a nucleon with the critical kinetic energy. Denote the speed with which the fight shock front and the left shock front moves as s+ and s_, respectively. Following Rankine and Hugoniot [1], we have, from conservation o f mass flux p(s+ - u ) = p+(s+ - u+),
(6)
conservation o f m o m e n t u m flux p(s+ - u ) (u - u+) = p - p + ,
(7)
and conservation o f energy flux
p(s+ - u )
(E - E. ) = pu - p+ u..
(8)
We also have the equation o f state relating the total energy E with density and entropy. E = u2/2 + e(p,o).
(9)
Volume 49B, number 3
PHYSICS LETTERS
It is clear that the central velocity u is zero by symmetry. There are then four unknowns p, o, s+, and p in the four eqs. ( 6 - 9 ) . For a given relative velocity Ure1, they can be solved by Newton's method. The resuits are given in figs. 1 and 2 in which the abscissa is given in terms of the Mach number M defined as (Mach number M) = Urel/ao.
(10)
The solid curves are results obtained with Brueckner's equation of state with Brueckner I coefficients while the dashed curves are results obtained with Brueckner II coefficients. For both sets of coefficients, the shock front moves with approximately the velocity of sound (fig. la) in the center o f mass system. Thus, in the system in which the uncollided material is at rest, the shock speed is approximately a o + Urel/2. Fig. lb shows that the density ratio p/p+ in the central region is a monotonically increasing function o f the Mach number. It has the value of about 1.5 f o r M = 1 and reaches 2.6 and 3.1 for M = 4. It is significant to note that p/p+ does not in general equal 2. It is thus apparent that conservation laws requiring the balance of mass, momentum and energy do not generally give the superposition o f the two densities even at very high relative velocities. The usual "sudden" approximation [6] which assumes superposed densities in the case o f collision o f fast ions is not generally valid. The energy e o per particle in the central region is a monotonically increasing function of the Mach number (fig. 2a). Thus, at some point, the binding energy will go to zero and the nucleons become unbound. However, it is not necessary to go to such high velocity to have the occurrence o f dissociation o f nucleons. One can envisage a collision in which the temperature in the central region is high enough to free the nucleons from their binding in nuclear matter. A more quantitative estimate of the Mach number at which this "dissociation" occurs can be made by considering
15 April 1974
the total energy E of the central region and writing
E = eo(P ) + q,
(I 1)
where q is the heat energy per particle. Clearly, dissociation can occur when E is no longer negative. We show in fig. 2b the heat energy q per particle as a function of the Mach number. By comparing figs. 2a and 2b, we fred that dissociation can occur when M ~ 2 with the Brueckner I coefficients and when M ~ 3 with the Brueckner II coefficients. Our simple result in a one-dimensional model reveals some essential features of collisions of heavy ions having a kinetic energy many tens of MeV per nucleon. A realistic three-dimensional treatment of the process is called for to study these phenomena in more detail. Already we have a notion how the collision proceeds in the head-on coUision o f two identical stars from the work of Seidl and Cameron [7]. If an analogy could be made here, we might expect that shock waves would propagate along the collision axis and emerge with the dissociation of a large number o f constituent nucleons in the forward and backward direction. The detection of these dissociated nucleons might be a way of tracing out the shock waves in such a process.
References [1] L.D. Landau and E.M. Lifshitz, Fluid mechanics (Pergaman Press, Ltd. London, 1958). [2] A.E. Glassgold, W. Heckr0tte and K.M. Watson, Ann. Phys. (N.Y.) 6 (1959) 1. [3] K.A. Brueckner, J.R. Buchler, R.C. Clark, R.J. Lombard, Phys. Rev. 181 (1969) 1543. [4] H.A. Bethe, Ann. Rev. Nucl. Science 21 (1971)93. [5] L. Zamick, Phys. Lett. B45 (1973) 313. [6] K.A. Brueckner, J.R. Buchler and M.M. Kelly, Phys. Rev. 173 (1968) 944. [7] F.G.P. Seidl and A.G.W. Cameron, Astrophysics & Space Science 15 (1972) 44.
245