Composición de especies

for the reaction symbolized above.

B . G-eneral Features of Cross-sections_for Nucl e ar Reactions.

The follovrinc considerations apply to cross-sections for nuclear reactions in the absence.of resonances. Resonance phen­

omena are discussed in section D.

Conservation of energy selects the final state.

Another atomic example is the non-radia- excited tive or Aur.er transition. An excited atom may octom.no have tv7o possible modes of decay. In addition photon to photon emission, the atom may decay by emis­

sion of an electron. Suppose, for example, the excitation corresponds to one missinc electron in the K shell. The e n e r ^ made available

vihen an electron falls into this hole may be greater than the ionization eneri^y, in vihich case an electron may be emitted from the atom. Arain the final system consists of tv:o unbound particles having a continuous ranee of possible energies.

Returning, to the nuclear reaction A + a — ►-B + b , we use

transition = number of transitions per unit time = w is given by

"Golden Rule No. 2 " : * _______________

V I I I . 2

142 ' Nuclear Reactions Ch. VIII

where H ia the matrix element of the perturbation causing th.e transition, and dn/dE = energy density of final states, counting each degenerate state separately.

IHI* may be the same for all energetically possible fin a l states; more often it depends on the state. (For In stan ce, )>il may depend on the direction of emission.) Then I'Hl® In theform - ula is a suitable average over the possible final s ta te s .■»***

dn/dE = 00 for a continuum of states. But In that case )>i|— ^ 0 , so that the expression IXl* dn/dE has the indeterminate

form 0 X 00 . This d iffic u lty is removed by limiting space to a box of volume JL . )>il is then small but finite and d n /d E la r g e but f i n it e . \iL drops out of the result. The number of fin a l

states equals the number of states of the emitted p a r t ic le . T h is Is because a change in moment\im of One particle compels a change in momentum of the other, by conservation of linear and ancw iar momentum of the system.

It was shown In Chapter IV , p.

76

that the number o f s ta te s available to a free p artic le , " b " , with momentum between p and p + dp, confined to a box of volume SI, , Is

V I I I . 3 gZ-'TT-Jir

This must be multiplied by the m ultiplicity In the fin al s t a t e * * caused by spin orientation, which is given by the factor (2 1 ^ + 1 )x

(2Ib+1 ) , where I^ is the spin of the emitted particle and Ig the spin of the nucleus. I f b is a photon, (21-b+l) is put equal to two .**■»·

dE = v^ dp{3 (true relativistically) V I I I . 4 where and v-^ are the momentum and velocity in the center o f mass frame of reference of the final (B+b) state. JSlnce "B ” i s usually' masslve compared with " b " , p^ and v-i^ can usually be meas­

ured in the laboratory frame. Combining these two equations:

From this and V I I I . 2 we get ^

No. transitions per unit time V I I I . 6

Trfr 'I'b

The following equation is essentially a definition of the cross- section ^ p per A nucleus:

* Derived In S c h iff, quantum Mechanics, p. 193 <"Golden Rule No. : is on page 148 of this text ).

* * This Is discussed in greater detail in section C, th is c h a p t e r .

* * * This point Is discussed by Bethe and Placzek, P h y s .R e v . 51 4 5 0, Appendix, p. 483. Multiplicity Is caused by the two p o s s ib l e independent polarizations.

H r ^ V I I I . 7 Where A and B refer to the (A+a) and (B+b) states respectively, and ng^ Is the density of particles "a**. Take to be l/f^ cm“’3

(one particle In the volume). Then

± «W v « i H r ( z V 0 ( 2 I » » | ) V I I I .8

since nucleus "A" Is often massive compared to *'a" , ''^"a"rel .to"A"

la often nearly equal to V _ In the center of mass frame. 'I n any case, these two velocity magnitudes are related by a cons­

tant factor. 'Writing i n · I I =1 / ^ ,

V I I I . 9

In general, la unknown. It has the form U

where U Is the Interaction ener^^. I f the vrave functions used to compute "H are normalized In voliime S L , ».nj disappears from the expression IJLHI In V I I I . 9. This is seen as follows: Let T have the form, at large distances, N e x p (ik z ). Then Nlii/

Setting N^J2< = 1 , we get N = I /sIjT

I f '^Initial ''^flnal mean the un-normalized plane wave functions, the matrix element factor in V I I I . 9 becomes

Oh. VIII Nuclear Reactions I4.3

J d i : 'J T,„.tal V I I I . 10

(This may be looked upon as taking J L = lO Henceforth we use l=i for . In order to show the meaning of this expression, we write It as

= U X Valome of nycleos ^{V I I I}. 11

where I is a suitable average of the product of the wave func­

tions over the volume of the nucleus. U, and hence the integrand, is zero outside the nucleus. U = averase Interaction e n e r g y « depth of potential w e ll. Po t our purposes here the Important

feature of V I I I . 1 1 . la its dependence on the charge of the p arti­

cipating p articles. I f " a " , say, is positively charged, its wave function w ill be reduced In amplitude at the nucleus by the barrier factor exp(-(ja,/2), where, by I I I . 3 , p. SS,

^ denotes the charge of " a " times the Coulomb potential of " A " . Physically this factor represents Coulomb repulsion. The wave function of an outgoing particle at the nucleua is also reduced by such a barrier factor. The result for the squared matrix ele­

ment i s : t /— \2

For neutral p articles: IX' <^(u X V o l. of nucleus) V I I I . 13 For + charged p articles: (U X Vol.)^X

(emission of negatively charged p ar tic le s (electrons)ls treated In Ch. IV )

For endothermie reactions there Is a threshold ener<3y for the bombarding p article. For exothermic reactions In which the energy- liberated Is much larger than the energy of the bombarding particle, there are two sim plifications In equation V I I I . 9; 1) the barrier factor exp(-GH) for the outgoing p a r tic le Is almost constant be­

cause It Is a function of energy of the emitted particle " b " , which Is almost constant; 2 ) and are almost constant and therefore the statistical weight factor i n V I I I . 9 , P^/''^a''''b*

proportional to l/vg,.

These results are now applied to sp ecific cases to deduce the general features of the O' v s . energy and O' v s . velocity curves.

2^ ^

Nuclear Reactions C3h. VIII

l ) ELASTIC (n ,n ) (both particles uncharged)

V = Vb. therefore ‘

Az low energy Ml is aDproximately'n e u t ^ ^ a constant

energy )>il is appreximately constant, therefore constant at low energy.

<r E L A S T I C (n.n)

jC a ■fe'Y e.v.

2 ) EXOTHERMIC, low enerOT UNCHARGED bombarding p a r t ic le , as In ( n , a ) , ( n ,p ) , (n ,.y X , ( n , f ) . Q Is u s u a l l y M e v . ’.-rtille neut­

ron energy l s ~ e . v . , therefore v>,{=a» constant. Therefore l A a * oC exp(-Gj^-GT37. exp(-G^) Is constant, since it depends on the almost constant energy of the out- C:oin5 p artic le , or, in the case of an uncharged **b", is 1 exactly. Also exp(-Gj^) = 1 . Therefore

l / v ^ (the " 1 /v " law)

3 ) INELASTIC ( n , n · )

e x o t h e r m i c

etc

The nucleus la left In an excited state. The process Is endo- thermlc and -Q Is the e:ccltatlon energy of the nucleus. For Incident neutron energies sllgtitl3/ above the threshold, constant, sluice the fractional change In Incident energy Is small. Put v^^i changes relatively greatly In this region:

V tC exceaa of enercy above the threshold. Therefore P n 'V v . v ^ . oC ■'^'n' ^ '^energy excess'. Therefore near the threshold c5 OC energy excess'.

INELASTIC (n.Ti') QT E N D OTH ER M IC (n,oc). (v,,y)) NEÜTÎÇON

ENe^GT' Energy

A) ENDOTHERMIC, CHARGED OUTGOING p a r t ic l e , aa In ( n . a ) , ( n . p ) . Exactly as In case 3 ) , except that the factor expC-G^^) operatea and Is dominant, oC\|energy excess’ X exp(-Gb)

In document Metodologías para el seguimiento del estado de conservación de los tipos de hábitat (página 21-27)