Introduction to Nuclear Forces

Transcription

1 Intoduction to Nuclea Foces One of the main poblems of nuclea physics is to find out the natue of nuclea foces. Nuclea foces diffe fom all othe known types of foces. They cannot be of electical oigin since they act between chaged paticles as well as neutal paticles (say, between a neuton and a poton in a deuteon). These foces cannot be magnetic eithe because the inteaction between the magnetic moments of the nucleons is extemely weak. Thus nuclea inteactions cause by foces, which diffe fom all known types of foces and ae called nuclea foces. Let us ecall the main popeties of nuclea inteactions. Most of the infomation about foces among nucleons is obtained fom the study of a simple twonucleon system like deuteon. The gound state of the deuteon is chaacteized by the following measued quantities. Binding enegy: E =. MeV Nuclea spin: J = 1 Even paity Magnetic dipole moment: µ =. 857 µ n Electic quadupole moment: Q = m Chage distibution half-value adius: a =. 1 fm The fact that the deuteon has an electic dipole moment Q indicates that its pobability density function is not spheically symmetical. This immediately tells us that the nuclea potential, which specifies the foce acting between the two nucleons is, itself, not spheically symmetic. The point is that all spheically symmetic potentials have l = eigenfunctions fo thei gound states, indicating a zeo quadupole moment fo thei chage distibutions. Calculations show that the measued electic quadupole moment is obtained if the gound state if the deuteon is a mixtue in which 96% is a l = state and 4% is a l = state. In spectoscopic notation, the dominant state is 3 S 1 and the less pobable state is 3 D 1. Calculations also show that this mixtue of states lead to the measued magnetic dipole moment µ =. 857 µ n. The value diffes by 3% fom what would be obtained if the deuteon wee in a pue 3 S 1 state, i.e. µ = µ P µ N =.7896 µ n 1.913µ n =. 8793µ n. We conclude fom all these consideations that the nuclea potential is not pecisely spheically symmetic, since it does not lead to a pue S gound state fo the deuteon. Summay of Popeties of Nuclea Foces 1. Nuclea foces ae foces of attaction, as can be seen fom the existence of stable nuclei consisting of potons and neutons.

2 . Nuclea foces ae shot ange. Ruthefod s expeiment on the scatteing of alpha paticles by nuclei showed that up to distances of about 1-1 cm, the expeimental esults can be explained by assuming that the inteactions between alpha paticles and nuclei is puely of Coulomb type. This means that the nuclea foces ae shot ange and thei ange can be estimated as the aveage distance between nucleons that ae bound within the nucleus by nuclea foces. a # ( V / A) but R = R A 4 a # ( % R 3 3 1/ 3 1/ 3 / A), with R 1/ 3 1 = 1.1 fm $ 13 cm = fm 3. The constant value of the aveage binding enegy pe nucleon fo most nuclei indicates that the nuclea foces have the popety of satuation. 4. Nuclea foces ae spin dependent. We know that thee ae no bound deuteon with nucleon spins essentially antipaallel, i.e. in a state of S 1. This indicates that the nuclea potential is spin dependent, being appeciably weake when two nucleons inteact with thei spins antipaallel (in a singlet state). 5. The similaity in the level stuctue of some light nuclei leads to the hypothesis of chage independence (isotopic invaiance) of nuclea foces. The concept of isotopic invaiance will be discussed late. The Concept of the Meson Theoy of Nuclea Foces The meson theoy of nuclea foces is constucted in analogy with quantum electodynamics. It is well known that in quantum electodynamics the electomagnetic field is consideed jointly with the paticles (photons) associated with it. The field as if consists of photons which ae the quanta of this field. The field enegy is equal to the sum of the enegies of the quanta. Photons ae ceated (annihilated) duing emission (absoption) of electomagnetic adiation (say, light). The electic chage is the souce of photons. The inteaction between two chages is esponsible fo the emission of a photon by one chage and its absoption by the othe. Such an appoach makes it possible to conside new phenomena associated with the inteaction of adiating systems with the intinsic adiation field. It explains, fo example, the anomalous magnetic moment of the electon and the muon, Lamb's shift of levels in the fine stuctue of the hydogen atom, and many othe fine effects. The basic idea of quantum electodynamics, viz. the quantum natue of inteactions, can be also extended to othe types of inteaction, including nuclea inteaction. This idea was fist put foth by L. Tamm in Tamm's idea povided a clea gaphic intepetation fo such popeties of the nuclea inteaction as its exchange natue which can be explained by assuming that the poton and the neuton exchange chage duing thei inteaction and that this leads to satuation. It seemed quite natual to assume that the exchange mechanism involves the tansfe (at the instant of nuclea inteaction) of some light paticles fom one nucleon to the othe. These paticles can be, fo example, electons o neutinos. Howeve, it was shown late by Tamm that

3 these paticles cannot be quanta of the nuclea field; since they cannot simultaneously explain the small ange of nuclea foces and the high binding enegy. No othe light paticles wee known at that time. Tamm's idea was late developed by the Japanese physicist Yukawa who assumed (in 1935) that the ole of nuclea quanta is played by unstable chaged o neutal paticles, the mesons, which had not been expeimentally discoveed at that time, but which wee supposed to have a mass m m e. Yukawa's aguments can be gaphically pesented as follows. Accoding to quantum mechanics, thee exists the uncetainty elation E t Putting E = mc, we can assume that the enegy E = / t may be esponsible fo the ceation of a vitual meson with mass m = E/c = / c t fo a shot time t in the immediate vicinity of the nucleon. Unlike odinay paticles that can move feely in space and in time, vitual paticles exist only fo a shot time t duing which they must be sepaated fom the nucleon by a distance a not exceeding a = c t. Afte the passage of time t, the vitual paticle is captued once again by a nucleon. Thus, it can be assumed that a nucleon is suounded by a cloud of vitual mesons that ae continuously being ceated and annihilated. The adius of this meson cloud is given by a = c / E = / mc. A vitual meson can be absobed not only by its own nucleon, but also by some othe nucleon if it happens to be in the meson cloud of the latte. It is this tansfe of a vitual meson fom one nucleon to anothe that is esponsible fo the nuclea inteaction. Quantitative estimates fo the nuclea inteaction time nuc and the vitual meson mass m can be easily obtained by equating a to the ange of nuclea foces. Assuming this value to be 1 13 cm (latest estimates put this value at cm), Yukawa obtained nuc = # t = a/c =.7 $ 1 3 s, E t #1 MeV, m # m e This is how the nuclea quantum o the Yukawa meson was pedicted. If Yukawa mesons do exist in actual pactice, they can be detected only if they ae ceated in a fee state and not vitually, i.e., if they ae sepaated fom the place of thei oigin by a distance exceeding the ange of nuclea foces. Such a pocess is possible only when the law of enegy consevation is obeyed. Hence, the ceation of mesons equies a lage kinetic enegy of the colliding nucleons, a pat of which may be tansfomed into the est enegy of the ceated mesons.

4 The discovey of nuclea quanta is associated with an inteesting and instuctive couse of events. It was fist decided that the µ -mesons (now called muons) with a mass m = 7 m e, which wee detected in 1938 in cosmic ays, ae the nuclea quanta. Howeve, it was soon found that muons do not paticipate in a stong nuclea inteaction. Late, in , pions o mesons wee detected fist in cosmic ays and then in acceleatos. Pions (,, and -mesons ) ae stongly inteacting paticles with a mass of appoximately m = 7m e. It is the pions that play the ole of nuclea quanta (pobably togethe with some othe stongly inteacting paticles). It can be easily seen that fo m = 7m e (coesponding to E 14 MeV ) # t $ # E = s, a = mc = cm The discovey of π-mesons stimulated the development of specific vesions of meson theoies taking into account the popeties of nucleons and π-mesons. We cannot go into details of these theoies, and shall confine ouselves to just the ough semiqualitative concepts of the meson theoy obtained in analogy with quantum electodynamics. Applying Quantum Mechanics to Mesons It was mentioned above that accoding to quantum electodynamics, the mechanism of electomagnetic inteaction involves the tansfe of a photon fom one chage to anothe. The equation of motion fo a feely moving photon can be witten in the fom E = p c. In ode to obtain the equation fo the potential field of a unit chage, we must make the substitution # E ; i t p. i The equation fo the potential in empty space will then assume the fom 1 # $ $ =. c t Fo the time-independent case, = and the solution of the above equation is the function t # = $ e 1 4 This of couse can be veifies by substituting the solution in the diffeential equation and taking note of

5 = 1 d ( d d ) d e 1 The solution # = $ is the expession fo the inteaction potential enegy of a unit chage 4 e (-e) in the potential V given by V =. 4 It follows fom the above analysis (which obviously coincide with the coesponding expessions in electostatics) that the electomagnetic inteaction has an infinitely long ange. We have mentioned ealie that accoding to meson theoies, the tansfe of inteaction takes place though a π-meson that is a paticle with a nonzeo mass (m ). The equation fo a feely moving paticle with m is witten in the fom E = p c m c 4. Afte the substitution fo the enegy and momentum opeatos, the equation fo the meson potential field of a nucleon in empty space assumes the fom 1 m c c t # $ $ = Fo the time-independent case ( / t = ), the solution of this equation has the fom e = g N g In the above equation, = and N is the stength of the Yukawa potential. It plays a ole mc c e simila to the dimensionless quantity fo electomagnetic inteactions. The magnitude of 4 c the chage g N can be detemined fom a compaison with the expeiment. The wave function φ is elated to the meson field suounding a nucleon. This apidly deceasing function V = g N e is called the Yukawa potential. The Compton wavelength = / mc of the meson can seve as a measue of the ate at which the function φ deceases (i.e. a measue of the adius of the meson cloud). Fo m = 7m e we get Com cm. = #.

6 Appaently, the quantity Expeimental Veifications Com coincides with the ange a of the nuclea foces intoduced above: c a = c t = = = E m c Expeimental evidence fo the exchange of pions between two inteacting nucleons is found in neuton-poton scatteing. The fist high enegy expeiment was pefomed with incident d neutons of enegy 9 MeV. The measuements show that the diffeential coss section is d o appoximately symmetic about a scatteing angle of 9. Com. Thus, thee is an equally ponounced pefeence fo lage scatteing angles. The physical intepetation of the obseved pefeence of lage angles is that in appoximately half the scatteing, the neuton changes into a poton and the poton changes into a neuton, when the two nucleons ae vey close. One way this can happen is indicated by the set of eactions: n p # then # p n That is, the neuton emits a negatively chaged meson into its field, becoming a poton. Then the meson joins the field of the poton, and it is absobed by the poton, which becomes a neuton. The same scatteing pocess can also happen though the set of eactions p n then n p In this case, the poton emits a positively chaged meson, which is subsequently absobed by the neuton. Thus, in about half the neuton-poton scatteings, a meson tansfes chage as well as momentum between the two inteacting nucleons.

7 In about half of the scatteings, the neutons and potons do not exchange identities when they inteact but they still must exchange a meson that caies the tansfeed momentum. The two sets of eactions that occu ae n n p p then then p p n n The neutal - meson tansfes momentum but no chage between the inteacting nucleons. This pictue implies that an isolated poton should be suounded by a meson field which will sometimes contain a - meson and sometimes contain a - meson. Of couse, the nucleon must absob the meson it has emitted within a vey shot time, but then it can emit anothe one. Similaly, the meson field suounding an isolated neuton should sometimes contain a - meson and sometimes a - meson. But the poton files cannot contain a - meson and the neuton field cannot contain a - meson. Expeimental veification of these pedictions is povided by electon scatteing measuements of the chage distibution of the poton and of the neuton. The following figue shows the adial dependence of the chage densities of the two species of nucleons. The chage density of the poton is eveywhee positive, and extends out to a distance of about fm. At lage, this chage is caied by a - meson. Fo the neuton, the chage density is not eveywhee zeo. At smalle, it is positive and at lage it is negative. The volume integal of the chage density is howeve zeo since the neuton is neutal and has no net chage. At values of appoaching fm, the nucleon chage densities ae popotional to some measue of the intensity of thei meson field. Both poton and neuton chage densities ae deceasing faily gadually as inceases. The nucleon foce, that acts between two nucleons when thei meson fields ovelap, also theefoe deceases gadually as thei sepaation inceases. Thus the onset of the attactive pat of the nuclea potential, descibing the nucleon foce acting when the two nucleons ae beginning to get close enough to inteact, is faily gadual and it is not as depicted in the following figue. Nevetheless, this is a good appoximation fo desciption of many featues of nuclea potential.

8 Meson theoy also povides an explanation of how the neuton can have an intinsic magnetic dipole moment, even though its net chage is zeo. The neuton sometimes become a poton plus a - meson. The poton has an intinsic magnetic moment, and the - meson can poduce a cuent that makes an additional contibution to the magnetic dipole moment. Questions 1. Why is 3 P 1 not a component of the gound state of the deuteon? What about 1 S?. Explain why a stable system of two neutons has not been obseved? 3. What paticle would emain if a poton emitted a - meson? If a neuton emitted a - meson? Why is it that a poton field cannot contain only a - meson and a neuton field contains only a meson? 4. Estimate the maximum time that a meson can exist in the field of an isolated nucleon befoe it is absobed by that nucleon. Estimate how many mesons thee can be at any instant in the field of a nucleon at a distance of fm and the distance of.5 fm. 5. The lifetime has been detemined by studying the decay K fom est. The aveage distance taveled by the in the block of photogaphic emulsion befoe it $ decays in the obsevable mode # e e is measued, and fom the calculated velocity of flight of the, its lifetime is obtained. Given that the lifetime is.8 pedict the aveage distance taveled by a befoe it decays s,