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| 16 Disintegration of particles |
In many cases the laws of conservation of momentum and energy alone can be used to obtain important results concerning the properties of various mechanical processes. It should be noted that these properties are independent of the particular type of interaction between the particles involved.
Let us consider a "spontaneous" disintegration (that is, one not due to external forces) of a particle into two "constituent parts", i.e. into two other particles which move independently after the disintegration.
This process is most simply described in a frame of reference in which the particle is at rest before the disintegration. The law of conservation of momentum shows that the sum of the momenta of the two particles formed in the disintegration is then zero; that is, the particles move apart with equal and opposite momenta. The magnitude p_0 of either momentum is given by the law of conservation of energy:
E_i=
E_{1i}+\frac{{p_0}^2}{2m_1}
+E_{2i}+\frac{{p_0}^2}{2m_2}
here m_1 and m_2 are the masses of the particles, E_{1i} and E_{2i} their internal energies, and E_i the internal energy of the original particle. If \epsilon is the "disintegration energy", i.e. the difference
16.1
which must obviously be positive, then
16.2
which determines p_0; here m is the reduced mass of the two particles. The velocities are v_{10} = p_0/m_1, v_{20} = p_0/m_2.
Let us now change to a frame of reference in which the primary particle moves with velocity V before the break-up. This frame is usually called the laboratory system, or L system, in contradistinction to the centre-of-mass system, or C system, in which the total momentum is zero. Let us consider one of the resulting particles, and let \v{v} and \v{v}_0 be its velocities in the L and the C system respectively. Evidently $\v{v} = \v{V}+\v{v}_0, or \v{v}-\v{V} = \v{v}_0, and so
16.3
where \theta is the angle at which this particle moves relative to the direction of the velocity \v{V}. This equation gives the velocity of the particle as a function of its direction of motion in the L system. In fig14 the velocity \v{v} is represented by a vector drawn to any point on a circle1 of radius v_0 from a point A at a distance V from the centre. The cases V \lt v_0 and V\gt v_0 are shown in fig14a and fig14b respectively. In the former case \theta can have any value, but in the latter case the particle can move only forwards, at an angle \theta which does not exceed \theta_\max, given by
16.4
this is the direction of the tangent from the point A to the circle.
14
The relation between the angles \theta and \theta_0 in the L and C systems is evidently
16.5
If this equation is solved for \cos\theta_0, we obtain
16.6
For v_0 > V the relation between \theta_0 and \theta is one-to-one fig14a. The plus sign must be taken in 16.6, so that \theta_0 = 0 when \theta = 0. If v_0 \lt V, however, the relation is not one-to-one: for each value of \theta there are two values of \theta_0, which correspond to vectors \v{v}_0 drawn from the centre of the circle to the points B and C fig14b, and are given by the two signs in 16.6.
j
In physical applications we are usually concerned with the disintegration of not one but many similar particles, and this raises the problem of the distribution of the resulting particles in direction, energy, etc. We shall assume that the primary particles are randomly oriented in space, i.e. isotropically on average.
In the C system, this problem is very easily solved: every resulting particle (of a given kind) has the same energy, and their directions of motion are isotropically distributed. The latter fact depends on the assumption that the primary particles are randomly oriented, and can be expressed by saying that the fraction of particles entering a solid angle element \dd{o}_0 is proportional to \dd{o}_0, i.e. equal to \dd{o}_0/4\pi. The distribution with respect to the angle \theta_0 is obtained by putting \dd{o}_0 = 2\pi \sin\theta_0\dd{o}_0, i.e. the corresponding fraction is
16.7
The corresponding distributions in the L system are obtained by an appropriate transformation. For example, let us calculate the kinetic energy distribution in the L system. Squaring the equation \v{v}=\v{v}_0+\v{V}, we have v^2={v_0}^2+V^2+2v_0V\cos\theta_0, whence \dd{(\cos\theta_0)} = \dd{(v^2)}/2v_0V. Using the
kinetic energy T = \mfrac{1}{2}mv^2, where m is m_1 or m_2 depending on which kind of particle is under consideration, and substituting in 16.7, we find the required distribution:
16.8
The kinetic energy can take values between T_\min = \mfrac{1}{2}m(v_0-V)^2 and T_\max=\mfrac{1}{2}m(v_0+V)^2. The particles are, according to 16.8, distributed uniformly over this range.
When a particle disintegrates into more than two parts, the laws of conservation of energy and momentum naturally allow considerably more freedom as regards the velocities and directions of motion of the resulting particles. In particular, the energies of these particles in the C system do not have determinate values. There is, however, an upper limit to the kinetic energy of any one of the resulting particles. To determine the limit, we consider the system formed by all these particles except the one concerned (whose
mass is m_1, say), and denote the "internal energy" of that system by E_i'. Then the kinetic energy of the particle m_1 is, by 16.1 and 16.2, T_{10} = {p_0}^2/2m_1 = (M-m_1)(E_i-E_{1i}-E_i') where M is the mass of the primary particle. It is evident that T_{10} has its greatest possible value when E_i' is least. For this to be so, all the resulting particles except m_1 must be moving with the same velocity. Then E_i' is simply the sum of their internal energies, and the difference E_i-E_{1i}-E_i' is the disintegration energy \epsilon. Thus
16.9
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More precisely, to any point on a sphere of radius vo, of which
fig14shows a diametral section. ↩︎