# Energy, Center-of-Mass

# ENERGY, CENTER-OF-MASS

The center-of-mass energy of a system of particles is the energy measured in the center-of-mass reference frame (see below). This energy constitutes all the energy that is available to create new particles or to explore the internal structure of particles, since the energy of the motion of the center of mass itself stays with the center of mass and cannot change the internal properties of the system.

## Center-of-Mass Reference Frame

The center of mass is the weighted average position of all the mass in the system and is the point that moves with the total momentum of the system, as if the total mass of the system were concentrated there. The center-of-mass reference frame is then a set of coordinates centered on and moving with the center of mass of the system being studied. In the center-of-mass frame, by definition, the center of mass of the system is at rest, and the total momentum is zero.

In a laboratory reference frame, on the other hand, in which a particle approaches an identical particle at rest, the center of mass is half-way between the two particles. In this case, it is not at rest but moves toward the target particle as the incoming particle approaches, always staying half-way between the particles. The moving center of mass carries its own momentum and energy, in addition to the momentum and energy of the relative or internal motions of the system of particles.

For example, consider the process in which the antiproton was discovered in 1955: a high-energy proton was fired at a stationary proton target, producing a proton-antiproton pair in addition to the two original protons. With the notation p for protons and p̄ for antiprotons, this reaction is written as p + p → p + p + p + p̄. In the laboratory frame, where the experiment was carried out, the center of mass of the two initial protons lies midway between them and approaches the target as the moving proton come closer (Figure 1(a)). After collision, the center of mass continues to move with the same momentum

**FIGURE 1**

as before, as a consequence of the conservation of momentum (Figure 1(b)).

If this same event is viewed in the center-of-mass reference frame, the two initial protons are seen to approach each other, while the center of mass remains at rest (Figure 2(a)). After the collision, the four particles scatter, and the center of mass is still at rest (Figure 2(b)).

## Relation of Center-of-Mass Energy to Laboratory Energy

Conservation of energy means that the energy before and after a collision will be the same. It does not mean that the energy in the center-of-mass reference frame is the same as the energy in the laboratory frame. The laboratory frame includes the energy of center-of-mass motion that does not appear in the center-of-mass frame.

The most direct way to relate laboratory energy and center-of-mass energy is to use a relativistic invariant. A relativistic invariant is a quantity that has the same value in all reference frames related by a constant relative velocity, that is, the reference frames related by special relativity. The quantity *E*^{2} - **p** ^{2}*c*^{2} is a relativistic invariant, where *E* is the total energy of the system of particles, **p** is the total vector momentum, and *c* is the speed of light. Using conservation of momentum in the collision along with this invariant and considering the case of two

**FIGURE 2**

**FIGURE 3**

initial particles, *a* and *b,* with *b* initially at rest in the laboratory frame, one finds

Thus, the center-of-mass energy depends on the square root of the laboratory energy for high enough kinetic energies. This means that at higher energies it is less and less effective to increase the laboratory energy in stationary target experiments, as shown in Figure 3 for the case where *m _{a}* =

*m*=

_{b}*m*. Notice that the larger

*E*

^{lab}, the more the curve showing

*E*

^{CM}bends over and falls behind

*E*

^{lab}.

Another way to express the relationship between stationary target laboratory energy and center-of-mass energy is through the minimum kinetic energy *T*^{lab} needed by particle *a* to produce particles of total mass *Μ* by colliding with stationary particle *b.* Since the rest energies of the new particles produced in the collision come from the center-of-mass energy, *E*^{CM} = *Mc*^{2}, and using Equation (1), one finds The antiproton experiment, p + p → p + p + p + p̄, carried out in the laboratory frame, requires at minimum *E ^{CM}* = 4

*m*

_{p}

*c*

^{2}(4 × 0.938 = 3.75 GeV) to supply the rest energies of the three protons and the antiproton, with the assumption that they are formed with no kinetic energy in the center-of-mass frame. Equivalently, the total mass of the products of the collision is observed to be

*Μ*= 4

*m*

_{p}. Using Equation (1), one finds for the incoming proton

*E*

^{lab}= 7

*m*

_{p}

*c*

^{2}, which means that the accelerator must supply 6

*m*

_{p}

*c*

^{2}of kinetic energy (6 × 0.938 = 5.6 GeV), since

*E*

^{lab}=

*T*

^{lab}+

*m*

_{p}

*c*

^{2}. Using Equation (2), one also find

*T*

^{lab}= 6

*m*

_{p}

*c*

^{2}.

To illustrate the effect of the square root energy dependence, consider the laboratory energy required to provide *E ^{CM}* = 100

*m*

_{p}

*c*

^{2}(94 GeV) in a proton-proton collision. Then Equation (1) yields

*E*

^{lab}= 4,999

*m*

_{p}

*c*

^{2}, requiring the accelerator to supply 4,998

*m*

_{p}

*c*

^{2}of kinetic energy (4.7 TeV). A 25-fold increase in the center-of-mass energy requires an accelerator more than 800 times more powerful.

## Colliding Beams

It is advantageous to do high-energy experiments in the center-of-mass frame because of the square root dependence of center-of-mass energy on stationary target laboratory energy illustrated above. This is accomplished with colliding beam configurations. When it is possible to produce beams of both kinds of particles needed for a planned collision, these beams are accelerated and kept in storage rings until they can be brought together with equal and opposite momentum. Then the resulting collision occurs with the system's center of mass at rest.

For example, one could perform the antiproton experiment by accelerating protons to a kinetic energy of *m*_{p}*c*^{2} (0.94 GeV) and storing them in two storage rings that will allow the beams to be brought back into head-on collisions. The total energy of each proton in a collision is then 2*m*_{p}*c*^{2}, including the rest energy, and the total center-of-mass energy is 4*m*_{p}*c*^{2}, the minimum needed to produce antiprotons. This approach produces the reaction at much lower accelerator energy at the price of a more complicated accelerator and beam arrangement.

The approach of colliding beam accelerators makes all of the accelerator energy available, in principle, to the rest energies of the collision products, with any excess energy going to the kinetic energies of the products. Unfortunately, there is sometimes no possibility of carrying out an experiment in collider configuration because of the problems in producing beams of the particles needed for a particular experiment.

The most powerful accelerator operating in 2001 is the Fermilab Tevatron in Batavia, Illinois, where collisions have been produced between 900-GeV protons and 900-GeV antiprotons. Equation (1) or (2) again allows the calculation of the kinetic energy that would have to be provided to produce the same *E ^{CM}* in a stationary target experiment. One finds

*T*

^{lab}=

*E*

^{lab}-

*m*

_{p}

*c*

^{2}= 1,730 TeV, nearly 2,000 times more energy than the 900 GeV supplied in the collider configuration. This would require an accelerator about 2,000 times as large as the Tevatron, that is, an accelerator diameter of about 2,500 miles!

*See also:*Conservation Laws; Symmetry Principles

## Bibliography

Frauenfelder, H., and Henley, E. M. *Subatomic Physics,* 2nd ed. (Prentice Hall, Englewood Cliffs, NJ, 1991).

National Research Council. *Elementary Particle Physics: Revealing the Secrets of Energy and Matter* (National Academy Press, Washington, D.C., 1998).

*William E. Evenson*

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