Frank Brascia
City College of N e w York N e w York, 10031
Fortes and Quantum Field Theory
M a n y general chemistry textbooks mention the great stability of the deuteron and the high stabilities of nuclei containing more than one proton. They also call attention to the very strong attractive forces that must be postulated to account for these stabilities. Several such textbooks mention the Hideki Yukawa (1935) postulate, namely, tbat the nucleons are hound by the rapid exchange of pi mesons (pions). These particles, which may be electrically charged or neutral, serve as one of the "glues," the forces of attraction. At least one textbook1 discusses the nature of the forces between nucleons in terms of the quantum field theory; it also indicates the common nature of nuclear, electrical, and gravitational forces. The object of this article is not to imply that this topic should be included in general chemistry, but ratherit is presented to satisfy the appetite of the reader who prefers to be I I over-prepared." Electrostatic and gravitational forces reach to astronomical distances and decrease by the square of the distance between t h e interacting bodies. Coulomb's law and Newton's law describe these forces but do not explain them. These laws are also silent on the nature of the electrostatic or gravitational "field." Quantum (non-relativistic) theory visualizes the field as composed of a "messenger"-a particle with definite properties which is exchanged between the interacting particles (see Fig. 1). The problem here is an apparent violation of the conservation laws. T h e spontaneous dissociation (not ohserved in nature) of a proton into a neutron and a vositive vion-a necessarv assumvtion of the Yukawa theory-would violate the mass-energy conservation
law. The energy of a stationary particle is given by its mass: E = me2. But the sum of the masses of the neutron and pion is much greater than the proton mass. Energy, therefore, cannot be conserved in this dissociation. Similarly, the spontaneous dissociation of a proton into a proton and a neutral pion would violate the conservation principle. However, energy balance is restored when the pion is absorbed by another nucleon. Nevertheless, the virtual steps involved do violate conservation laws and the existence of such pions is assumed in theories to explain nuclear and particle properties. This dilemma is resolved by involving another basic law of nature, the Heisenberg uncertainty principle: the product of the possible error in the measurement of the energy of a particle, AE, and At, the time interval over which the energy measurement is made, must at least equal h / 2 or ~ AE X Ai = h/2r (1) The smaller the error allowed in the energy measurement, the greater must be the time interval jar the measurement. This means tbat the energy measurement made in the time interval At cannot be more accurate than AE = h/2rAt
(2)
Equally acceptable, we can say that the energy of a particle is not conserved, but that the maximum vari* tion is given by eqn. ( 2 ) . This law makes it possible for us to talk about processes that violate the conservation laws provided that the duration of the process is not smaller than the time, At, given by eqn. (1). EXAMPLE (a) A @-particlepossesses a kinetic energy of 1.00 X lo-%'erg. What is the minimum permissible time required to measure this energy with an accuracy of 0.1070? ( b ) A ball possesses a kinetic energy of 6.00 X 108 ergs. What is the minimum permissible time required to measure this energy with an accuracy of O.lOYO? ANSWER (a) The msximum acceptable error, AE, is 1.00 X and the minimum permissible time interval becomes
Figure 1. Skater onabgy suggested b y Deny* Wilkinwn, University of Oxford, (Scientific Americon, Febwaty, 1966, p. 471, to viruolize the theoroticd model in which o force is represented as an exshmga of a third partisle between the interacting particles. la) Two skater. playing boll, the exchange of the bolt produces a repulsive force between the skaters. Each time the boll is thrown, the thmwor rscoilr, and each time the catcher coteher he recoils. Ibl Attractive force is represented b y rubstitvtino " a bwmsrana " for the boll. The thrower aims awov from the catcher and the catcher catcher while focing owoy from the thrower; thus
642
/
Journal o f Chemical Education
lo-*
erg
It is, therefore, impossible to make this measurement with the desired accuracy in s. flight time of less than about 10' sec. If we make the measurement in 10-1 sec, the error in the energy measurement must increase by a factor of about lo6, from lo-" erg. Note that the error we must make is greater than the to quantity (lo-" erg) being measured.
( b ) The maximum acceptable error is 6.00 X 10' ergs and the minimum permissible time interval becomes At =
6.63 X lo-" erg set = 2 X 3.14 X 6.00 X lo5 ergs
This means the energy measurement cannot be mado in less than 1.76 X 10-a aec. Thus, in a reasonable flight time, the energy of the ball can be measured with extraordinary accuracy. If we make the measurement in see, the minimum error in the energy measurement will decrease by a factor about 10".
A photon is believed to be the particle exchanged between charged particles and a "graviton" is believed to be the particle exchanged between neutral bodies. One electron creates a photon and a second electron at a distance absorbs it and then the process is reversed. The photon exchange between the two electrons is the description of electrostatic forces in terms of the quantum field theory. The detection of gravitational radiation, predicted by Albert Einstein, has been r e p ~ r t e d . ~ The results may be interpreted in terms of gravit,y waves (Einstein classical gravitation theory) or gravitons. Hideki Yukawa recognized from the experimentally determined range of nuclear forces, about 1.5 X 10-l3 cm, that the "messenger" would have to be a very energetic particle with a short life ( t extremely small). He calculated (Appendix) that these particles moving at practically the speed of light should have a rest mass about 275 times t,he electron mass. Such particles are now called pions, a. The exchanges between pions and other nucleons are proton-neutron e neutron proton-neutron z3 proton
++ r+ + neutron neutron-proton r- + proton 8 neutron-proton
Between like nucleons, the exchange forces involve only neutral pions proton-proton e proton neutron-neutron i;t neutron
++ no ++ neutron proton z3 proton-proton 8 neutron-neutron
*"k.
*icrr
Skater analogy to i!lurtrote the short range of nuclear forces. Skoterr now represent nucleons and the boomerang represents the pion.
Figure 2.
from which it originated (see Fig. 2). Under these conditions, repulsive elect.rostatic forces between protons will rip a nucleus apart. The pion exchange between two nucleons is the description of nuclear forces in terms of the quantum field theory. I t should be emphasized that nucleons, not nucleons and pions, come together to form nuclei; similarly, atoms are formed from nuclei and electrons, not from nuclei, electrons, and photons. Photons are materialized when atoms return to a lower state after acquiring an energy hu; pions are materialized when nuclei return to a lower state after acquiring an energy >m,c2. Appendix If a pion travels with (almost) the speed of light, the distance i t can travel in 5.0 X 10-a4sec is 1.6 X 10-15 cm. Note that this distance corresponds to the range of nuclear forces. The rest mass, ma, of a. pion is its minimum mass (if not a t rest, its mass would increase with speed according to relativity theory). This rest mass may be represented as moee. Then AE = moe' represents the minimum error allowed in creating a pion. If we let A1 = 5 X 10-2' see, this will correspond to the maximum error and therefore the maximum time nature will allow the pion to be apart from a.nucleon. The rest mass, in electron messes (electron g), of the pion csleulated from mass = 9.11 X
tro
I n terms of our skater analogy, if two nucleons are less than 1.5 X 10-la cm apart, the pion can be exchanged. However, if the separation distance is greater than 1.5 X 10-l3 cm, the life of the pion is too short to reach the second nucleon; instead, it returns to the nucleon
' Wmwn, JOSEPH, Phys. Rev. Letters, 22, 1320 (1969) and the meeting of the New York Academy of Sciences, New York, N. Y., October 17, 1969.
is 268 electron masses. Powell and his eo-workers discovered (1947) the particle that fulfilled the predictions of Yukawa.
Suggested Readings BABTON. G.. "Introduction to Advanced Field Theory," Interscience Publishers (division of John Wiley & Sons, Inc.), New York, 1963, p. 1-2. ELTON.L. R. B.. "Intr~ductoryNuclear Theory" (2nd Ed.), W.H. Saundsrs Co.. Philadelphia. 1966, Chapter 10. HENLET,E., A N D TIXIRRINO. W., "Element~ryQuantum Field Theory," MoGmw-qfll Book Co.. New York. 1962. Chapter 21. KAPCAN, I.. Nuclear P h y ~ i w "Addison-Wesley Publishing Co.. Reading, Maas.. 1955, Section 17-5. LAPP, R.. A N D ANDREWS, H., "Npelear Radiation Phyaios" (2nd Ed.), PrentiepHsll Ino., Ewlewood Ciilis, N. J., 1954, Chapter 3.
Volume 47, Number 9, September 1970
/
643
