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Physics of Radioactive Beams
Chapter 1
Production of secondary beams of rare isotopes
Carlos A. Bertulani, Texas A&M University-Commerce, TX 75429, USA
(^1) These notes consist of a series of lectures presented by the author at the Geselschaft f¨ur Schw-
erionenforschung, Darmstadt, Germany in the Spring of 1994. GSI-Report 1994-11. This material was latter extended and published in the book “Physics of Radioactive Beams”, C.A. Bertulani, M. Hussein and G. Muenzenberg, Nova Science, Hauppage, NY, 2002, ISBN: 1-59033-141-
0.1. INTRODUCTION
0.1 Introduction
The study of nuclear physics demands beams of energetic particles to induce nuclear reactions on the nuclei of target atoms. It was from this need that accelerators were born. Over the years nuclear physicists have devised many ways of accelerating charged particles to ever increasing energies. Today we have beams of all nuclei from protons to uranium ions available at energies well beyond those needed for the study of atomic nuclei. This basic research activity, driven by the desire to understand the forces which dictate the properties of nuclei, has spawned a large number of beneficial applications. Amongst its many progeny we can count reactor- and spallation-based neutron sources, synchrotron radiation sources, particle physics, materials modification by implantation, carbon dating and much more. It is an excellent example of the return to society of investment in basic research. All of these achievements have been realized by accelerating the 283 stable or long-lived nuclear species we find here on Earth. We see them in Fig. 1, the black squares, plotted as a function of the numbers of protons (Z) and neutrons (N) that they contain. In recent years, however, it has become evident that it is now technically possible to create and accelerate unstable nuclei and, as we see in Fig. 1, there are some 6-7,000 distinct nuclear species which live long enough to be candidates for acceleration. They are the nuclei within the so-called drip-lines, the point where the nucleus can no longer hold another particle. It needs little imagination to see that this development might not only transform Nuclear Physics but could lead to many new, undreamed of, opportunities in industry, medicine, material studies and the environment. Fig. 2 shows schematically the two main methods of radioactive beam production which have been proposed. They are commonly known as the ISOL-Isotope separation on line - and In-flight techniques. In the ISOL method, we must first make the radioactive nuclei in a target/ion source, extract them in the form of ions and, after selection of mass by an electromagnetic device, accelerate them to the energy required for the experiments. In contrast, the in-flight method relies on energetic beams of heavy ions impinging on a thin target. Interactions with the target nuclei can result in fission or fragmentation, with the nuclei which are produced leaving the target with velocities close to those of the projectiles. A cocktail of many different species is produced which, since the ions have high velocities, does not need further acceleration to transport it to the secondary target. On route to the target the reaction products can be identified by mass, charge and momentum in a spectrometer (fragment separator). Thus a pure beam is not separated out from the cocktail. Instead each ion is tagged and identified by these primary characteristics and the secondary reactions are studied on an event-by-event basis. Another possibility is a combination of the two methods in which the in-flight reaction products are brought to rest in a gas cell, sucked out and separated by mass and then re-accelerated to the required energy. For reviews of experimental and theoretical developments involving production, acceleration, and reactions with unstable nuclei, see, e.g., Refs. [1, 2, 3, 4, 5, 6, 7, 8].
0.1. INTRODUCTION
Figure 2: A schematic view of the basic methods of producing radioactive nuclear beams. At the left we see the ISOL method with and without a post-accelerator. At the right we see the In-flight method and the proposed hybrid in which fragments are caught in a gas cell and then re-accelerated.
in an almost central collision, as shown in Fig. 3. A part of the projectile (participant) is scrapped off and forms a highly excited mixture of nucleons with a part of the target. A piece of the projectile (spectator) flies away with nearly the same velocity of the beam. The neutron-to-proton-ratio of the spectator is nearly equal to that of the projectile. Since the N/Z - ratio of light nuclei (stable) is close to one, the fragment is far from the stability line. Statistically, a large number of fragments with different N/Z - ratios is created and several new exotic nuclei have been discovered in this way. Experiments with secondary-beam are limited by reaction cross section and luminosity. The luminosity L is defined as the product of beam intensity i and target thickness t:
L = i · t. (1)
The reaction rate N is the product of luminosity and reaction cross section σr:
N = σr · L. (2)
0.1. INTRODUCTION
Figure 3: (a) Schematic description of a nuclear fragmentation reaction producing rare isotopes. The lower fragments are called participants, while the upper one is called by spectator. Using uranium projectiles (N/Z ∼ 1 .6) one expects to produce (light) spectator nuclei of about the same N/Z ratio. (b) Coulomb fission of relativistic projectiles leading to the production of rare isotopes. For a heavy unstable projectile an exchanged photon with the target can give it enough energy to fragment into several types of isotopes.
In most of the reactions the usable target thickness is limited by the width of the exci- tation function (i.e., the cross section as a function of the excitation energy). Production reactions with a wide excitation function covering a broad energy range can profit in lumi- nosity by the use of thick targets.
The condition for fragmentation of heavy ions is that the projectile should move faster than the nucleons move inside the nucleus. The projectile energy should be sufficiently above the Fermi domain, e.g., above 100 A MeV. The usable target thickness for these high energies is of the order of several grams per square centimeter, corresponding to 10^23 atoms/cm^2. The excitation function for complete fusion of heavy ions, however, has a width of only 10 MeV. This corresponds to a usable target thickness of the order of one milligram per square centimeter or 10^18 atoms/cm^2. Consequently beam intensities for the investigation of complete fusion reactions must be by four to five orders higher to achieve the same luminosity as for fragmentation.
Fig. 4 shows as an example the production cross sections for the tin isotopes from complete fusion (dotted line), nuclear fragmentation (solid line), and Coulomb fission of 238 U (dashed line). The symbols represent experimental data. The fragmentation cross-sections (solid line) have been calculated with a semi-empirical code [10].
It is very pedagogical and useful at his stage to discuss the production of nuclei in the nuclear fragmentation region of Fig. 4. We develop some mathematical tools to understand them. The simplest theoretical model to describe the isotopic distribution of fragments in heavy ion collisions at high energies is the abrasion-ablation one of [11]. In the model’s abrasion stage, the nucleons in the overlap volume of two energetic heavy ions are scrapped
0.2. PROBABILITY APPROACH TO HIGH ENERGY SCATTERING
as t (b) db, where t (b) is known as the thickness function. It is defined in a normalized way, i.e., ∫ t (b) db = 1. (3)
For unpolarized projectiles t (b) = t (b). In most practical situations one can use t (b) ' δ (b), which simplifies the calculations considerably. Since the total transverse area for nucleon-nucleon collisions is given by σN N , the probability of having an inelastic nucleon-nucleon collision is given by t (b) σN N. The probability of finding a nucleon in dbB dzB is given by ρ (bB , zB ) dbB dzB , where the nuclear density is normalized to unity: ∫ ρ (bB , zB ) dbB dzB = 1. (4)
Using these definitions, it is easy to verify that the probability dP of occurrence of a nucleon- nucleon collision is given by
dP = ρ (bB , zB ) dbB dzB. ρ (bA, zA) dbAdzA. t (b − bA − bB ). (5)
Thus, as in the case of free nucleon-nucleon collisions, we define T (b) σN N as the probability of occurrence of a nucleon-nucleon collision in nucleus-nucleus collisions at impact parameter b. This is obtained by multiplying dP by σN N and integrating it over all the projectile and target volumes, i.e.,
T (b) σN N =
ρ (bB , zB ) dbB dzB ρ (bA, zA) dbAdzA t (b − bA − bB ) σN N. (6)
The thickness function for nucleus-nucleus collisions, T (b), can thus be related to the corresponding thickness function for nucleon-nucleon collisions as
T (b) =
ρ (bB , zB ) dbB dzB ρ (bA, zA) dbAdzA t (b − bA − bB ). (7)
We notice that our definition immediately implies that T (b) is also normalized to unity: ∫ T (b) db = 1. (8)
We can also define the individual thickness functions for each nucleus. That is, for nucleus A,
TA (bA) =
ρ (bA, zA) dzA , (9)
and similarly for the nucleus B. In terms of these definitions
T (b) =
dbAdbB TA (bA) TB (bB ) t (b − bA − bB ). (10)
0.2. PROBABILITY APPROACH TO HIGH ENERGY SCATTERING
Now we are able to describe more specific aspects of nucleus-nucleus collisions in terms of nucleon- nucleon collisions. For example, we may want to calculate the probability of occurrence of n nucleon-nucleon collisions in a nucleus-nucleus collision at impact parameter b. If for simplicity we call A(B) the number of nucleons in nucleus A(B), this probability is given by
P (n, b) =
AB
n
[T (b) σN N ]n^ [1 − T (b) σN N ]AB−n^. (11)
The first term is the number of combinations for finding n collisions out of AB possible nucleon- nucleon encounters. The second term is the probability of having exact n collisions, while the last term is the probability of having AB − n misses. The total probability, or differential cross section, is given by
dσ db
∑^ AB
n=
P (n, b) = 1 − [1 − T (b) σN N ]AB^ , (12)
and the total nucleus-nucleus cross section is given by
σ =
db
1 − [1 − T (b) σN N ]AB^
For nucleus-nucleus collisions one may ask what is the average number of nucleon-nucleon collisions at a given impact parameter b. One has
〈n(b)〉 =
∑^ AB
n=
nP (n, b) =
∑^ AB
n=
n
AB
n
[T (b) σN N ]n^ [1 − T (b) σN N ]AB−n
= α
∂α
∑^ AB
n=
AB
n
[ασN N ]n^ [1 − T (b) σN N ]AB−n
α=T (b)
= α
∂α
[1 − T (b) σN N + ασN N ]AB^
α=T (b)
αABσN N [1 − T (b) σN N + ασN N ]AB−^1
α=T (b)
or 〈n(b)〉 = ABT (b) σN N. (15) One can also calculate the standard deviation in the number of nucleon-nucleon collisions. First, we need to calculate 〈n^2 (b)〉. One can use the same trick as in the derivation above, replacing the sum over n^2 P (n, b) by the application of twice the operator α∂/∂α. The net result is 〈 n^2 (b)
= ABT (b) σN N + AB (AB − 1) [T (b) σN N ]^2. (16)
Using 14 and 15 we find for the standard deviation 〈 n^2 (b) − 〈n(b)〉^2
n^2
− 〈n〉^2 = ABT (b) σN N [1 − T (b)]. (17)
0.3. ISOTOPE YIELD IN HIGH ENERGY COLLISIONS
Figure 5: (a) Geometry of nucleus-nucleus collisions at high energies. (b) The average primary fragment excitation energy obtained from the densities of states is shown as a function of the fragment mass by the solid points. The variances of the distributions are displayed as error bars. The solid line shows the excitation energy obtained using the surface energy estimate of Ref. [11].
Using t (b) ' δ (b) we get
P (n, m, bA, bB ) =
A
n
B
n
[TB (bB ) σN N ]n^ [1 − TB (bB ) σN N ]B−n
× [TA (|b − bB |) σN N ]m^ [1 − TA (|b − bB |) σN N ]A−m^. (24) The abrasion-ablation model [11] used in Section 3 is based on this equation. It is extended to account for the isospin dependence of the nucleon-nucleon collisions in a trivial way. In that Section m is interpreted as the number of holes created in the nucleon orbitals in the target. These equations can also be derived quantum-mechanically using the eikonal approximation ). This was shown by H¨ufner, Sch¨afer and Sch¨urman [17]. The derivation presented in this Supplement is much simpler and only requires the use of probability concepts.
0.3 Isotope yield in high energy collisions
Using the probability approach to high energy scattering, described in Supplement A, we can develop a simple model to calculate the isotopic yield in high energy collisions of heavy
0.3. ISOTOPE YIELD IN HIGH ENERGY COLLISIONS
nuclei. This is know as the abrasion-ablation model. According to Carlson, Mastroleo and Hussein [18], the differential primary yield can be written as the product of a density of states ω (ε, Zf , Af ) and an integral over impact parameter,
dσ 0 dε
(ε, Zf , Af ) = ω (ε, Zf , Af )
×
d^2 b [1 − Pπ(b)]ZP^ −Zf^ Pπ(b)Zf^ [1 − Pν (b)]NP^ −Nf^ Pν (b)Nf^.
(25)
The integral gives the cross section for each primary fragment state as the sum over impact parameters of the probability that Zf projectile protons and Nf = Af − Zf projectile neutrons do not scatter, while the remaining ones do. The distinction between protons and neutrons generalizes the expression 5 of the Supplement A and permits one to account for the differences in their densities. We can use Eq. 18 and write the probability that a projectile proton does not collide with the target as
Pπ(b) =
d^2 s dz ρPπ (z, s)
× exp
[
−σppZT
dz ρTπ (z, b − s) − σpnNT
dz ρTν (z, b − s)
]
where ρPπ and ρTν are the projectile and target single-particle proton and neutron densities while σpp and σpn are the total (minus Coulomb) proton-proton and proton-neutron scatter- ing cross sections, respectively. The neutron probability can be expressed likewise as
Pν (b) =
d^2 s dz ρPν (z, s)
× exp
[
−σpnZT
dz ρTπ (z, b − s) − σppNT
dz ρTν (z, b − s)
]
where we have identified the total neutron-neutron scattering cross section σnn with the proton-proton one. At lower energies it is important to account for the Pauli blocking of the nucleon-nucleon scattering. This tends to reduce the values of σN N entering Eqs. 26 and 27. A geometrical model of Pauli blocking is discussed in Supplement B. Note that Eqs. 25, 26 and 27 are slightly different than those that we have derived in Supplement A. The difference is the replacement of the probability 1 − T (b) σN N by
0.3. ISOTOPE YIELD IN HIGH ENERGY COLLISIONS
Figure 6: Calculated secondary yields in the reaction 16 O +^208 Pb at Elab = 20 Mev/nucleon are shown as open circles for the isotopes of (a) lithium, beryllium, and boron, and (b) of carbon and nitrogen. The data of ref. [20] are shown as full circles.
is displayed as an error bar on each point. Also shown, as a solid line in Fig. 5(b), is the excitation energy that would be obtained using a surface energy estimate [11]. The latter yields about half the average energy of the hole distribution but remains within its variance for all but the largest mass losses. As can be seen in Fig. 5(b), the average excitation energy of fragments that undergo little abrasion remains below the particle emission threshold, although their energy distribution extends above it. In these cases, use of the energy distribution rather than an average value is essential for describing the decay. One can resort to several statistical models to calculate the particle evaporation during the ablation stage as a function of the primary fragment charge Zf , mass Af , and excitation energy ε. For the calculations presented here the limiting Weisskopf-Ewing evaporation formalism was used [19]. The result of the evaporation calculation can be expressed as the probability, P (Z, A; ε, Zf , Af ), of yielding a residue of charge Z and mass A, given a primary compound nucleus of charge Zf mass Af and excitation energy ε. In terms of this quantity and the differential primary yield dσ 0 (ε, Zf , Af ) /dε, one can calculate the observed secondary yield, σ 0 (Z, A), as
σ 0 (Z, A) =
Zf , Af
dε P (Z, A; ε, Zf , Af )
dσ 0 dε
(ε, Zf , Af ) (29)
We show the calculation for 16 O +^208 Pb together with the experimental data [20, 21] at 20 MeV/nucleon and 2 GeV/nucleon in Figs. 6 and 7, respectively. In particular we note
0.3. ISOTOPE YIELD IN HIGH ENERGY COLLISIONS
Figure 7: Calculated secondary yields in the reaction 16 O +^208 Pb at Elab = 2 Gev/nucleon are shown as open circles for the isotopes of (a) lithium, beryllium, and boron and (b) of carbon, nitrogen, and oxygen. The data of ref. [21] are shown as full circles.
the agreement of the calculation with the projectile-like isotope data. The agreement with the data, especially that at 2 GeV/nucleon, is quite reasonable.
The microscopic calculation of the absorption probabilities and cross sections permits the use of more realistic collision geometries. A natural step in this direction is to next replace the average single-particle projectile densities in Eqs. 26 and 27 by the probability distributions of the individual projectile orbitals. This allows one to take into account differences in the abrasion probabilities or the different orbitals. In 16 O for example, we expect the removal of a p-orbital nucleon to be more likely than that of an s-orbital one, since the former will tend to be at a larger radius than the latter. Thus generalized, the expression for the differential primary yield, dσ 0 (ε, Zf , Af ) /dε, becomes a sum over all the possible combinations of orbital transmission and absorption factors that result in a given primary charge Zf and mass Af. The popu- lation of primary fragment states will no longer be evenly distributed but will depend on the absorption probabilities of the single-particle states on which they are based. As the nucleons that are less bound are the more superficial ones and, thus, also the ones more likely to be absorbed, the average primary fragment excitation energy will be lower in this more realistic model. The model presented in this Section shows that we can understand the main features of the isotopic fragmentation yield in heavy ion collisions at high energies in terms of simple Glauber calculations and with statistical decay models for the spectators and participants. Later on we will discuss the production of exotic nuclei in fusion reactions and in Coulomb
0.4. PAULI BLOCKING OF NUCLEON-NUCLEON SCATTERING
q′^ is a vector which can only rotate around a circle with center at p = (k 1 + k 2 + k)/ 2. These conditions yield an allowed scattering solid angle given by [22]
ΩP auli = 4π − 2(Ωa + Ωb − Ω)¯ , (31)
where Ωa and Ωb specify the excluded solid angles for each nucleon, and Ω¯ represents the intersection angle of Ωa and Ωb (see Fig. 8). The solid angles Ωa and Ωb are easily determined. They are given by
Ωa = 2π(1 − cos θa) , Ωb = 2π(1 − cos θb) , (32)
where q and p were defined above, b = k − p, and
cos θa = (p^2 + q^2 − K F^2 1 )/ 2 pq , cos θb = (b^2 + q^2 − K F^2 2 )/ 2 bq , (33)
The evaluation of Ω¯ is tedious but can be done analytically. The full calculation was done by Bertulani [22] and the results have been reproduced in the Appendix of Ref. [23] (see also [24]). To summarize, there are two possibilities:
(1) Ω = Ω¯ i(θ, θa, θb) + Ωi(π − θ, θa, θb) , for θ + θa + θb > π (34) (2) Ω = Ω¯ i(θ, θa, θb) , for θ + θa + θb ≤ π , (35)
where θ is given by cos θ = (k^2 − p^2 − b^2 )/ 2 pb. (36) The solid angle Ωi has the following values
(a) Ωi = 0 , for θ ≥ θa + θb (37)
(b) Ωi = 2
[
cos−^1
cos θb − cos θ cos θa sin θa(cos^2 θa + cos^2 θb − 2 cos θ cos θa cos θb)^1 /^2
cos θa − cos θ cos θb sin θb(cos^2 θa + cos^2 θb − 2 cos θ cos θa cos θb)^1 /^2
− cos θa cos−^1
cos θb − cos θ cos θa sin θ sin θa
− cos θb cos−^1
cos θa − cos θ cos θb sin θ sin θb
)]
for |θb − θa| ≤ θ ≤ θa + θb , (39) (c) Ωi = Ωb for θb ≤ θa, θ ≤ |θb − θa| , (40) (d) Ωi = Ωa for θa ≤ θb, θ ≤ |θb − θa| , (41)
The integrals over k 1 and k 2 in Eq. 30 reduce to a five-fold integral due to cylindrical symmetry. Two approximations can be done which greatly simplify the problem: (a) on average, the symmetric
0.4. PAULI BLOCKING OF NUCLEON-NUCLEON SCATTERING
situation in which KF 1 = KF 2 ≡ KF , q = k/ 2 , p = k/ 2 , and b = k/ 2 , is favored, (b) the free nucleon-nucleon cross section can be taken outside of the integral in Eq. 30. Both approximations are supported by the studies of Refs. [23] and have been verified numer- ically [22]. The assumption (a) implies that Ωa = Ωb = Ω¯, which can be checked using Eq. 41. One gets from 31 the simple expression
ΩP auli = 4π − 2Ωa = 4π
K F^2
k^2
Furthermore, the assumption (b) implies that
σN N (k, KF ) = σN Nf ree (k)
ΩP auli 4 π
= σf reeN N (k)
K F^2
k^2
The above equation shows that the in-medium nucleon-nucleon cross section is about 1/2 of its free value for k = 2KF , i.e., for E/A ' 150 MeV, in agreement with the numerical results of Ref. [23]. Since the effect of Pauli blocking at these energies is very large it is important to calculate the in-medium nucleon-nucleon scattering cross section according to Eq. 30, including the energy dependence of the free nucleon-nucleon cross sections. The connection with the nuclear densities is accomplished through the local density approxi- mation, which relates the Fermi momenta to the local densities as
K F^2 =
[
3 π 4
ρ(r)
] 2 / 3
ξ (∇ρ/ρ)^2 (44)
where ρ(r) is the sum of the nucleon densities of each colliding nucleus at the position r. The second term is small and amounts to a surface correction, with ξ of the order of 0.1 [23]. Inserting Eq. 44 into Eq. 43, and using E = ℏ^2 k^2 / 2 mN , one gets [24] (with ρ¯ = ρ/ρ 0 )
σN N (E, ρ) = σN Nf ree (E)
1 + α′^ ρ¯^2 /^3
with α′^ = −
E (MeV)
where the second term of Eq. 44 has been neglected. This equation shows that the local density approximation leads to a density dependence proportional to ρ¯^2 /^3. The Pauli principle yields a 1 /E dependence on the bombarding energy. This behavior arises from a larger phase space available for nucleon-nucleon scattering with increasing energy. The nucleon-nucleon cross section at E/A. 200 MeV decreases with E approximately as 1 /E. We thus expect that, in nucleus-nucleus collisions, this energy dependence is flattened by the Pauli correction, i.e., the in-medium nucleon-nucleon cross section is less dependent of E, for E. 200 , than the free cross section. For higher values of E the Pauli blocking is less important and the free and in-medium nucleon- nucleon cross sections are approximately equal. These conclusions are in agreement with the ex- perimental data for nucleus-nucleus reaction cross sections [25], and is explained in Ref. [23].
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