bei48482_FM

Nuclear Transformations 449

so that

V  (12.24)

In most nuclear reactions,  cand a nonrelativistic treatment is sufficient.

In the laboratory system, the total kinetic energy is that of the incident particle only:

KElab^12 mA^2 (12.25)

In the center-of-mass system, both particles are moving and contribute to the total

kinetic energy:

KEcm^12 mA(V)^2 ^12 mBV^2

^12 mA^2 ^12 (mAmB)V^2

KElab^12 (mAmB)V^2

KEcm KElab (12.26)

The total kinetic energy of the particles relative to the center of mass is their total

kinetic energy in the laboratory system minus the kinetic energy 2 ^1 (mAmB)V^2 of the

moving center of mass. Thus we can regard KEcmas the kinetic energy of the relative

motion of the particles. When the particles collide, the maximum amount of kinetic

energy that can be converted to excitation energy of the resulting compound nucleus

while still conserving momentum is KEcm, which is always less than KElab.

mB



mAmB

Kinetic energy in

CM system

Kinetic energy in

lab system

mA



mAmB

Speed of center

of mass

(a) Motion in the laboratory coordinate system before collision

(b) Motion in the center-of-mass coordinate system

before collision

(c) A completely inelastic collision as seen in laboratory and

center-of-mass coordinate systems

Before

collision

After

collision

Laboratory

coordinate system

Center-of-mass

coordinate system

mA v– V Center of mass mB

mA v

Center of mass mB

V =

mAv

mA + mB

• V

Figure 12.16Laboratory and center-of-mass coordinate systems.

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