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The creation and study of new elementary particles is an important part of contemporary physics. Especially interesting is the discovery of a very massive particle. To create a particle of mass \(M\) requires an energy \(M c^{2} .\) With enough energy, an exotic particle can be created by allowing a fast moving particle of ordinary matter, such as a proton, to collide with a similar target particle. Let us consider a perfectly inelastic collision between two protons: an incident proton with mass \(m_{p},\) kinetic energy \(K,\) and momentum magnitude \(p\) joins with an originally stationary target proton to form a single product particle of mass \(M\) You might think that the creation of a new product particle, nine times more massive than in a previous experiment, would require just nine times more energy for the incident proton. Unfortunately not all of the kinetic energy of the incoming proton is available to create the product particle, since conservation of momentum requires that after the collision the system as a whole still must have some kinetic energy. Only a fraction of the energy of the incident particle is thus available to create a new particle. You will determine how the energy available for particle creation depends on the energy of the moving proton. Show that the energy available to create a product particle is given by $$ M c^{2}=2 m_{p} c^{2} \sqrt{1+\frac{K}{2 m_{p} c^{2}}} $$ From this result, when the kinetic energy \(K\) of the incident proton is large compared to its rest energy \(m_{p} c^{2},\) we see that \(M\) approaches \(\left(2 m_{p} K\right)^{1 / 2} / c .\) Thus if the energy of the incoming proton is increased by a factor of nine, the mass you can create increases only by a factor of three. This disappointing result is the main reason that most modern accelerators, such as those at CERN (in Europe), at Fermi lab (near Chicago), at SLAC (at Stanford), and at DESY (in Germany), use colliding beams. Here the total momentum of a pair of interacting particles can be zero. The center of mass can be at rest after the collision, so in principle all of the initial kinetic energy can be used for particle creation, according to $$ M c^{2}=2 m c^{2}+K=2 m c^{2}\left(1+\frac{K}{2 m c^{2}}\right) $$ where \(K\) is the total kinetic energy of two identical colliding particles. Here if \(K>m c^{2}\), we have \(M\) directly proportional to \(K,\) as we would desire. These machines are difficult to build and to operate, but they open new vistas in physics.

Step by step solution

01

Set Up the Conservation of Energy and Momentum

The total energy of the system before the collision is \(E_{\text{total}}=E_{1}+E_{2}=K+2 m_{p} c^{2}\), where \(E_{1}\) is the kinetic energy of the moving proton, \(E_{2}\) is the rest energy of the stationary proton, and \(K\) is the kinetic energy of the incident proton. The momentum of the system before the collision is \(p_{\text{total}}=p_{1}+p_{2}=p+0=p\), where \(p_{1}\) and \(p_{2}\) are the momenta of the incident and stationary protons respectively.

02

Apply the Energy-Momentum Relation

By the energy-momentum relation of special relativity, we have \(E_{\text{total}}^2=(p_{\text{total}} c)^2+(Mc^2)^2\). Substituting the equations from step 1 gives: \((K+2 m_{p} c^{2})^2=(p c)^2+(M c^2)^2\)

03

Solve the Equation for \(M c^2\)

Simplify the equation, then isolate \(M c^2\) to find the energy available for the newly created particle. This will result in the formula: \(M c^2=2 m_{p} c^{2} \sqrt{1+\frac{K}{2 m_{p} c^{2}}}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Energy-Momentum Relation

In physics, the energy-momentum relation is a key concept derived from Einstein's theory of special relativity that connects the energy (\(E\)) of a particle to its momentum (\(p\)) and mass (\(m\)). The relationship is summarized by the famous equation \(E^2 = (pc)^2 + (mc^2)^2\), where \(c\) is the speed of light. This equation reflects the fact that a particle's total energy includes its rest energy (\(mc^2\)) and the energy due to its motion (momentum).

In the context of particle creation, the energy-momentum relation tells us that not all of the kinetic energy of a particle can be used for creating another particle upon collision because we also have to account for the energy associated with momentum. When two protons collide inelastically to form a new particle, the resulting energy available for the creation of that particle can be calculated using this fundamental relationship to obtain the result \(M c^{2}=2 m_{p} c^{2} \(\sqrt{1+\frac{K}{2 m_{p} c^{2}}}\)\). It demonstrates that to produce a particle with mass \(M\), the incoming proton's kinetic energy \(K\) must be considered alongside its rest energy.

Inelastic Collisions

When studying particle creation, we look at inelastic collisions—these are collisions in which the colliding particles do not bounce off each other but instead merge to form a new particle. The key feature of inelastic collisions is that they do not conserve kinetic energy, though the total energy and momentum still remain conserved.

In particle accelerators, inelastic collisions are used to create new and sometimes more massive particles. During these collisions, the kinetic energy of moving particles is partly converted into the mass of newly created particles, following the mass-energy equivalence principle by Einstein (\(E=mc^2\)). However, due to the conservation of momentum, not all kinetic energy can be utilized for mass creation, leading to the aforementioned energy available for particle creation.

Special Relativity

Special relativity is a theory proposed by Albert Einstein that revolutionized our understanding of space, time, and energy. This theory includes several counterintuitive ideas, such as time dilation and length contraction, that arise at speeds close to the speed of light. Furthermore, it introduced the concept that mass and energy are interchangeable, which is fundamental to the discussion of particle creation.

Special relativity plays a critical role in understanding high-speed collisions in particle physics. The theory's energy-momentum relation is crucial for calculating the outcomes of these collisions. For instance, as particles accelerate to higher velocities, nearing the speed of light, their kinetic energy significantly increases, which in turn affects the mass of particles that can be created in collisions.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system free from external forces, the total momentum before an event must be equal to the total momentum after the event. This principle is universal and applies to all forms of collisions, including the inelastic ones that result in particle creation.

When applying the conservation of momentum to particle creation processes, the system's initial momentum, carried by the incident proton, must be conserved even after the collision when a new particle is formed. This requirement means that regardless of how much kinetic energy the incoming proton has, there will always be some residual movement of the newly created particle system, which ultimately limits the mass of the particle that can be created from a given amount of kinetic energy.