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'? 4t* 7wX6 MDDC866 UNITED STATES ATOMIC ENERGY COMMISSION INTERPRETATION OF THE TRITON MOMENT by R. G. Sachs Argonne National Laboratory Reprinted with permission from The Physical Review, Vol. 72. No. 4, August 15, 1947 Issuance of this document does not constitute authority for declassification of classified copiesof the sameor similar contentand title and by the same author. Technical Information Branch, Oak Ridge, Tennessee AEC, Oak Ridge, Tenn., 4849750A5206 I I iiiIiiiiiiiiiiiii iiiIiiii 4 Interpretation of the Triton Moment* R. G. SACHs Argonne National Laboratory. Chicago. Illinoiu (Received April 29. 1947i In order to account for the measured magnetic moment of the triton it is necessary' to assume that the wave function in the ground state is a linear combination of 2S, 'P, 'P. and 'D functions. An attempt is made to determine the amplitudes of these functions from the magnetic moment on the assumption that the intrinsic nucleon moments are additive and relativistic effects are negligible. With certain reasonable assumptions concerning the nature nf the wave functions, it is found that the relative probabilities for finding the system in the p., *P, and 4D states satisfy! the relation shown by the curves in Fig. 1. Wherever the result would otherwise be arbitrary, the wave functions have been chosen in such a way as to mini mize the amount of P state, with the exception that onlV the lowest oneparticle configurations have been considered. If the amplitude of the 25 state is taken to be as large as possible, the wave function contains no "D state, 8 percent 'P state, and 17 percent IP state. A wave function of this form would seem to indicate that there is a spinorbit coupling other than the tensor interaction acting among nuclear particles. In the other extreme case that the wave function contains a maximum of the 'D function, the 'S state probability is zero, the 'D probability is 22 percent, the 'P is 30 percent, and the P is 48 percent. If the wave function of He' has the same form as that of H3, the He' moment would be expected to lie tin one of the curves shown in Fig. 2. 1. INTRODUCTION THE recent measurements'1 of the magnetic moment of the triton give a value about 6.7 percent greater than that of the proton. If the ground state of the triton were a pure IS, state, it would be expected that the moment would be equal to the proton moment. It is believed, of course, that the ground state is not a pure 'S state but contains an admixture of 'P, *P, and 4D states.' A theory based on simpli fying assumptions leads' to the conclusion that *This work has been carried out under the auspices of the Atomic Energy Commissinn. It was completed and submitted for declassification on March 14, 1947. LH. L. Anderson and A. Novick, Phys. Rev. 71, 372 (1947). SF. Bloch. A. C. Graves, M. Packard,'and R.I1V. Spence, Phys. Rev. 71,373 and 551 (1947). the presence of these states should result in a reduction of the moment instead of the observed increase. However, it has been pointed out' that cross terms between the various states in the expression for the magnetic moment have been neglected in the simple theory. These may be positive and could, therefore, account for the large moment. It is the purpose of this paper to obtain a general expression for the magnetic moment in terms of the amplitudes of the various wave functions and thereby to gain some information concerning the nature of the ground state wave SE. Gerjuoy and J. Schwinger. Phys. Rev. 01. 138 (1942'1. SR. G. Sachs and J. Schwinger. Phys. Rev. 70, 41 (1946). 6 R. G. Sachs, Phys. Rev. 71, 457 (1947). TR ITON MIOMEN 1 function of the triton. The expression for the moment will be found to consist of a sum of terms of three different types. The first are the diagonal elements which are uniquely given in terms of the constant amplitudes of the wave functions. The second are cross terms which involve overlap integrals between the "radial" parts of the wave functions. These "radial" ave functions actually are not purely radial but are also functions of the cosine, q. ur' . diigle between the vector connecting the nItrons and thi vector connecting the proton to the center of mass of the two neutrons. The third set of terms consists of cross terms involving overlap integrals between one radial function and the derivative with respect to q of another such function. These may be very large if the wave functions contain very high con figurations, that is, if the individual particles have very high orbital angular moment. How ever, it seems likely that such high configurationso do not occur in the ground state, since in the ground state the wave function adjusts itself in such a way as to minimize the kinetic cnery. of the system. For that reason, it will II asuilmu in the final analysis that tlhee cross terms vanish, or, more specifically, that the radial functions do not depend on q. This assumption eliminates a great deal of the arbitrariness from the results Considering then terms of only the first two types, it is found that the observed moment can be accounted for only if the D state probability is less than that of either the 'P or 'P states. This conclusion appears to be at variance with current ideas concerning the nature of the triton wave function." If it is accepted that the inter action term responsible for the mixing of states is the tensor interaction, then the "D state would be directly coupled to the 'S state but the P states would not be. Therefore, it might be expected that the D state probability would he larger than the P probabilities. This expectation is based on the premise that the wave function is predominately: a 'S state. There is the possibility that the wave function contains little or no 'S state; that is that the advantage gained through the large average value of the tensor interaction in the P and D states might be large enough to overcompensate the correspondingly large kinetic energy in which case the energy would be a minimum for a small S state probability. Further information concerning these ques tions may be obtained experimentally by means of a measurement of the moment of He". This paper includes a discussion of the relation be tween the moment of He' and the various possible mixtures of states which are consistent with the observed moment of H'. In this discussion, no consideration is given to the possibility that the intrinsic moments of the neutron and proton are not additive. Also, the relativistic correction to the triton moment is ignored.b 2. THE WAVE FUNCTIONS The possible forms of the triton wave func tions, with respect to their dependence on the spins of the particles, have been given by Gerjuoy and Schwinger.3 We denote by p. the unit vector in the direction of the distance between the two neutrons and r, the unit vector in the direction of the distance from the center of gravity of the neutrons to the proton. If wa is the Pauli spin operator for the proton and i2=(a912I)/2, w here w1 and e are the Pauli operators for the two neutrons, then the wave functions have the following form: 'S: 4,= /f,. S= l127( ua3) rp)#f?, 'P: oj=&3 (JrXpHr p)fJ,, I = [air. rXp+i(o3r)(oi2p) ifi p)Li2 r)]4f4,, p. 4 = o,.rXpo+(wl.r)(3.e) 2 (la) (Ib) (Id) i ,1 (r, r)(9,p) f (Ile) *D: #c= [iss,:r)(7,,r) ifo,.p)(3p)] X (r. p)#c, (If) ,;= [(f,. r)(rr .r)+( ,. e)(F "p) 2 (,.2W)]3(rp)Vf,, (Ig) SH. Mlargenau. Phvs. Rev. 57, 383 (1940). P. Caldirola, Phvs. Rev. 69, 608 (1946) G Breit. Phys. Rev. 71, 400 11947). R. G Sach.. Phys. Rev. 72. 91 (1947). K. G. S,\CHS ,s = [(m( rr ) (a ep) + (wip ) (os r) (re)(asr1oa)]fd (lli) S= [(wi' rX p)(s3rX p) (rXe)Y(uraw.O)](rp) ,f. (lii The function 4 is given by 4=(4r '()'(x,+*2 XX2x a"2, (2) where the x are the spin wave functions of the indicated particles. The functions J', are functions of the distances corresponding to r and p, and they are also even functions of the quantity q=(r.p). For simplicity, they will be described as "radial" functions. The extra factors (rp) which are displayed in these equations, but not in those tabulated by Gerjuoy and Schwinger, are introduced in order to satisfy the Pauli principle for the two neutrons. The normalization conditions for the radial functions take the form ( lf I 1=I. (3a) ifq21'f212 =3b) 'fq2(1 q) Ifa1= (3c) f(Iq2)f4,2=. (3d) (ilq)fI'=1. l(3e) If (I _q2) /,i, = q(q')l1fe12 I, (3f) f(1+ 3q2)q2 Ifl=t, (3g) 3if(3+q ) fa ., (3h) I f(1q) q2if '= 1. (3i) The integrals indicated in these conditions are to be taken over the variable q (limits: I to +1) and over the magnitude of the distance between the neutrons and the magnitude of the distance from the center of the neutrons to the proton. The wave function in the ground state of the triton is expected to be a linear combination of the nine functions given in Eq. .1ll; that is, I'= a ,. (4) II The coefficients, a,, and the form of the func tions, f,, could only be determined by solving the Schroedinger equation for the threebody prob lem. It is our purpose to express the magnetic moment of the triton in terms of the a, and certain integrals over the f,. Then we can hope to get some idea concerning the quantities a, and f, from the observed moment. The magnetic moment of the nucleus is given by i = i,, ,( 4 3 U'3I,) + ,U .i ', [o I'+ 0 z' ]* ) +*l, LZ.'), (5) where L3: is the scomponent of the orbital angular momentum of the proton, u, is the magnetic moment of the proton, and p. is the magnetic moment of the neutron. In this ex pression, the wave function, I', is taken to be that function for which the magnetic quantum number of the total angular momentum is +4. By making use of Eq. (4), the magnetic moment can be expressed in terms of the matrix elements of the spin and orbital angular moment opera tors. Thus ,A= YA c,*sl ,ljI I\k) A.i +jp.(j!ai'+af2'Ik)+(jILa'ik) (6) where the j, k refer to the wave functions V,, . In the next section it will be shown that a considerable fraction of these matrix elements vanish, so that this expression is not quite so formidable as it looks on first sight. 3. THE MATRIX ELEMENTS In evaluating the matrix elements which appear in Eq. .6), it is convenient first to deter mine which of the elements vanish. The only diagonal elements that would be expected to vanish are those corresponding to the mixing of TRITON MOMENT two S states by an orbital angular momentum operator. Thus: 'SI L' I''S) = 0 (7) where the indicated 'S states refer to an arbitrary linear combination of the functions I' and 4'2. If the states to be mixed are orthogonal in the space coordinates, then the matrix element of the spin operators vanishes, since the spin operator will not remove the space o. thogonality. Similarly, the matrix elements of the orbital angular momentum vanish if the functions are orthogonal in spin. It follows that: (S Ia ,'l'P)=0, tS Ia, I' P)= 0, ('SI a,'I'D) =0, lP Ia, I 4D) = 0, 8) ('PI a/I D) = 0, where j= 1, 2, 3. Also: I~SI L'I'P)=0, (SSI L,' ID) =0, ( (PILa'I'P)=0, (PIL3; ID)=0. It is now possible to resort to symmetry arguments to show that other elements vanish. The operators, a,' and La', do not involve p (see Eq. 112)), so they are unchanged by the trans formation p p. Therefore, if the functions to be mixed by the matrix element have opposite symmetry under this transformation, the element vanishes. A study of the functions given in Eq. (1) leads to the new results: (l a,'[2 =0, 3, 13 a 4.j=0, t31o,/ 5) =0, 110) and ( LL 'I 4)=0, (21L3 !3)=01, ~.31L I 4) =0. ll) It will be noted that, apart from the factors fi, each of the functions . s is either sym metric or antisymmetric under interchange of r and p. It has already been pointed out4 that use can be made of this property, since the operator La' is given, in units of h, by L.'= 2 x (12) \ y ) / /ax while the scomponent of the total angular momentum is given by L =[( Y + ) )]i, (13) ay ax a3 a e where x, y, s are the components of r and ', those of p. The two terms in Eq. (13) clearly have the same matrix element between two wave func tions, both of which are either symmetric or anti symmetric under interchange of r and p. Thus the matrix element of either term is onehalf the matrix element of L:. It follows that for two such functions, 0, and ',,: I La'[lm)= i(IL'm). The functions i and 4'e are both antisymmetric under interchange of r and p, so they satisfy the condition for the validity of Eq. (14). In addi tion, 4, and 4s are orthogonal, so that the matrix element of the zcomponent of the orbital angular moment vanishes, since both functions are proper functions of this operator: (51L'16)=0. Then, according to Eq. (14), (5L3a'6)=0. (16) Equation (14) may also be used to evaluate the diagonal elements of La'. Considering first the 2P functions, both are seen to be antisym metric under interchange of r and p, so any linear combination has the same property. Thus ".PILa'IP)=! ('PIL'lP)=2/9. (17) Similarly, ('PIjLjI'P)= ~1PIL zIP)= 1/9. (18) The situation ; not quite so simple for the 'D functions. 4, is antisymmetric and 47, 4%, 4, are symmetric under the operation being considered. Therefore, there are cross terms between 4fs and the other three functions which depend on the more detailed properties of the functions. The other terms can be evaluated by the above method. If 'D' denotes an arbitrary linear combination of 47, 8s, and 4s, then: (61L3 16) = 'D'I LaI 'D') = 'D'IL'I4D')= 1'3. (19) It is also a simple matter to evaluate the diagonal elements of a,' in the quartet states, since these functions are symmetric in ai', as', R. G. SACHS AnMd ff.*. Therefore,7 ('PIf/I'P) PIS'I'P) = 5 (20) and ('DIa'sID)= ('DIS'I'D)= 1'3. (21) The values of the diagonal elements of the spin matrices in the states 4i and a3 may be obtained immediately by noting that both func tions are antisymmetric for interchange of ac' and az'. Therefore, (1 laf'+ff'l 1)= (31 1'+02'3)=0. (22) On the other hand, the sum of the three a' must be equal to twice the average value of S', or (1 I ,'1)=2('SI S'1[S)=1 (23) and (3 1 s13) =2('PIS'jIP)= 1,'3. (24) All other matrix elements may be obtained by direct computation. Since the required calcula tion is tedious and not at all illuminating, it will not be presented here. The results are: (2as'12)= i, (21 1'+a'2 2) =4 3, (4j1a14)= 1, (41r, + ff2?4)= 4 9, (5i1 34) = I (l q2)f*f4, (51 oil + v,.14) = If 1q2)*f + If(.lq2) f*f4 for the spin elements. The elements of the orbital angular momentum operator are: 1 ( (31'll1)= (1q')qf3*f', (26a) 9i J (41L.'12)= f (1 q)fJ(f.+qf.), (26b) 3i J 2 (71L/'5)= q'f,*f, 9i J (81L,15) = (3+q'I2fhf, 18i 1(  q q2)f.f', 9i ./ (26d) (9 L, 5) = 17 ql q:)fg*ff6 18i + q(1q2')'f5ff5', (26e) 9 9i (6 L '\7) 9 fq'tlq')fef + fJqS q2)fdf7', Sq )f , 9 (6[ L3'8) = qi(1q)fe' l', 1 r (61 La'19)=  Iq2tl q')ff 9J (26f) (26g) (26h) where f, is the derivative off, with respect to q. 4. THE MAGNETIC MOMENT In terms of these matrix elements, the mag netic moment is given by Eq. (6). In order to bring out explicitly the symmetry character of the wave functions, we set a\=aS, aI=aS, as=a3 P, a4=a,2P, ab=a i', as=aD, a;=a;D, ac=asD, 127) a = a D, where S2 is the probability of finding the system in the 2S state, (2'P) that of finding it in the 2P state, etc. The numbers S, 2p, 'P, and D are chosen to be real. They must satisfy the normal ization condition S2+2P2+4P2+D'= I. 128) The ai also must be normalized as follows: i fq( q2)ff', (26c) The factor 2 arises from the fact that the Pauli operator, g', is twice the spin. a0 '+ la2 = 1, (29) IaI'+ 0a,12= (30) !a,\"= 1, (31) TR I TON MOMENT losi'+ iail'+ lal '+ ;agi' + ta 4aY*as f qf *f8 a l 9ff I q)q '.7*ff aha, f( Cq)q.f,'.,n =1. (32) iR 'I denotes the real part of Ihe quantity con lained in the bracket. Equation (32) has a relatively. complicated form because the functions i. eB, and 49 are not orthogonal to one another. In terms of the coefficients defined by Eq. L27), the magnetic moment is: 4 A = I 2 'S(p ;) 3 4  P[ 1,,0 al,4lU,.M,.)J .1 2 2 + 'P'(5,, 2r,) D'", D ., 2 . 9 3 2 1 1 + ?P4p+D I + . 9 9 3 where u, contains the cross terms. This last quantity is given by , = D'Rj a*a 5fqq1l q)f'ff + q ( q2)f,*f;'j +2a6*fq q)f( f U fqa ilq2)f* 2 I I ( +( ,,)1 g PPt aaIa4l ( q)ffz f f 3 3 + PS 3a*a ( q2)qfff' 3 3 I I +a,*a2 f (1q ')f,*(f2 +9f2') +D DPs 2a,*a 2 fg7f,*f 9 I  f q(1 q')f*f' +ana[ rf 13+q')f.af 2f q(1 q=)fs*f' a*a ag.bl 2(d g2)f9*fa  2fq(1 q2')2f*f'] P. (34) Here, I I is the imaginary part of the expres sion in the bracket. The expression Eq. (34) is so complicated that some assumptions concerning the wave function must be made in orrer to simplify it. We assume that the functions fk are independent of q or fA'= 0. (35) This assumption appears to be reasonable, since the wave function of the ground state will have such a form that the kinetic energy of the system is as small as possible. Therefore, it should be a very smooth function, in which case Eq. (35) would be approximately valid. The functions fk also depend on the magnitudes of the distances between the particles. In ac cordance with our assumption that these func tions are smooth, it seems reasonable to assume that they all have about the same shape. Therefore, we take f i,'f=[fS I f AIf 3]' According to the wellknown Schwartz inequal ity, this equation gives an upper limit on the magnitudes of the integrals involved. Conse quently, the magnitudes of the coefficients of the cross terms in the expression for the magnetic (36) R. G. SACHS unment are no greater than the values given by Eq. (36), so the estimates obtained below of the amount of admixed 'P, 4P, and 4D states are lower limits. If we now set a = Xt+iyk, (37) the cross terms in the magnetic moment become 2 r5 /7 P,=D'2 ((xr+ysy7) (x6X9 +y6,) 93 L[/7 2 8v2 +(, I,,) 1p 2P(xX:,+yb^), 9 2%/5 r 2 +PD (X7Yb xby7) + (jXY XC5.V) (xyV xbY) (38) 10 1 Here, use has been made of the normalization conditions given by Eq. (3) as well as the assumption Eq. (36). The normalization condi tion expressed by Eq. (32) now has the form 4 E (x,+y,')+(x7,x+y7sy) (xx, +y7yo) i6 V"7 /7 (xax,+ysy) = 1. (39) 5 The magnetic moment is still given by Eq. with la,12 =x,'+y,' and la4l'=x4'+yY. constants, xi, yi, are to be chosen in sui manner that they satisfy the conditions of (29) to (31) and Eq. (39), and that they givi correct value of the magnetic moment, i.e.,' u= 1.067p,p. The eighteen constants are clearly not d, mined by these five conditions, so some fur assumptions may be made in order to make final results somewhat more specific. The fact that the coefficient in Eq. (44 larger than unity does lead to a consider limitation on the choice of the constants, s the diagonal terms in Eq. (33) tend to rei the moment below the proton moment. Tl fore, it is necessary to take the nondiag terms to be positive and rather large. In o to minimize the negative diagonal terms in Eq. (33) we are led to choose a==0, \a1 = 1. (41) Since it seems likely that the amount of S state will be as large as possible, we might require that the constants xi, y, be chosen in such a way as to lead to the largest possible value of S. There is also some reason to guess that the D state probability will be large com pared to the 'P and 'P probabilities.' Although it will be found that this condition cannot be satisfied, we will choose the values of the con stants in such a way as to make D2 as large as possible just to see how closely we can approach the desired result. No simple analytical method was found for choosing the constants x,, y, in such a way that D' would turn out to be a maximum. For this purpose, it is desirable to make the coefficients of the terms containing D in Eq. (38) as large as is consistent with Eq. (39). It was found by examination that the maximum amount of D state resulted when x6e = ye = X7 =y? = xg = y, = 0, xA= x4= ys, k42) 'i = y4 = X. With these values of the constants, the relation between the amplitudes of the P and D states which is given by Eq. (40) is 2.12'PP + 0.1784PD [0.50'P +1.25'P2 +0.759D']=0.067, (43) ch a where we have taken Lp=2.79 and t,'/p, Eqs. = 0.685. The values of 'P', "P2, and DS which e the are given by this equation are shown in Fig. 1. It should be emphasized that these are not the (40) only possible combinations of these constants which will agree with the triton moment because eter the choice of the x,, y, which has been made is rther rather special. Sthe It is to be noted that the S state probability will be a maximum for D2=0, 'P2=8 percent )) is and P = 17 percent. D' is never as large as IP', 'able and it is at most equal to 4P1 in spite of the since fact that the coefficients xj, yi have been chosen duce in such a way as to lead to as large a value of D' lere as possible. The largest value of D' is 22 percent onal with 4P2=30 percent and 2P2=48 percent. In irder this case the wave function contains no S state. TRITON MOMENT It is generally believed that the properties of the wave function of He' are the same as those of the wave functions of the triton. Therefore, the conclusions drawn here concerning the ad mixture of states in the triton may be assumed to hold also for He'. This makes it possible to make certain predictions concerning the mag netic moment of He'. It has been shown that the moment of He' is given in terms of the moment of H' by the relation' p(He') +i(H) = Up,+P.2(.a ,+pI ) X (3D2 'P' + 2'P')/3. (44) The consequences of this equation are demon strated in Fig. 2 which shows the relation be tween the moment of He' and the amounts of 2P, 'P, and 4D states which satisfy the relation ship shown in Fig. 1. These are not the only possible values for the He3 moment, since certain specific assumptions have been made concerning the wave function in order to obtain Fig. 1. The value of the He' moment to be expected on the basis of the 4 percent of 'D state and 0 percent P state found by Gerjuoy and Schwingerl is p(He');,p= 0.763, a value which seems to be well out of the range of possibilities allowed by the considerations put forth here.' Therefore, a measurement of the moment of He' should prove to be a definitive experiment for distinguishing between the two cases. If the results are in agreement with expectations, it would then be possible to obtain another relation between the probabilities of the various states by taking the horizontal intercept of the observed moment with the.various curves in Fig. 2. 5. CONCLUSION The conclusion that the amount of D function is small compared to the amount of P function SThe GerjuoySchwinger assumption uf a small amount of 'D unction and even smaller amounts of the P functions is nor consistent with the results obtained here because of the condition, Eq. (35). However, ii the average value of f/' happens to be large enough, in contrast to the requirement of Eq. (35), the term Eq. (26a) would give a contribution to the moment sufficient to account for the measured value even if the 'P, 'P, and 'D probabilities are small. In this sense the measurement of the moment of He' might be considered as a test of the assumption expressed by Eq. (35). large average value of Ja' would be rather surprising, since the usual assumption that f, be a function of (ri,'+r,,'+rvn'). where r.. is the distance between particle. ni and a. would lead o0 f,'=0 Fil. 1. Relation between 'P, 'P, and 'D state proba bilities required to account for experimental moment of the triton. The special assumptions made in obtaining these curves are expressed hb Eqs. (35), (41), and (42). in the ground state of the triton is somewhat surprising if one believes that the tensor inter action is responsible for the admixture of states since, then, it would appear that the D state should play a predominant role. It is possible, of course, that this conclusion is a consequence of erroneous assumptions concerning the nature of the radial wave functions fi. The derivatives of these functions have been neglected, and it can be seen that important terms could be introduced if the derivatives were not negligible. However, it has been found that the values of f.' required to make these terms appreciable are quite large To assume that it has such a large value would imply that the wave functions consist of products of one particle wave functions corresponding to high orbital angular moment of the individual particles. This seems most unlikely. For the present, it seems reasonable to drop such terms. There appear to be two essentially different wave functions of the triton which are consistent R. C. SACHS 1 the measured magnetic moment. The amount of S state may be large and the amount of D state very small or zero. In this case, one might be forced to assume that a spinorbit cou FIG. 2. The magnetic moment of He' in units of the proton moment. These curves have been obtained on the assumption that the relations shown in Fig. 1 apply also to He'. pling plays an important role in determining nu clear structure. The other alternative is that there is little or no S state. This would imply that the tensor interaction has a sufficiently high average value to compensate the increase in kinetic energy which would appear for such a wave function. There would probably still be some difficulty in understanding the saturation of nuclear forces. A better understanding of this point could be obtained by carrying through a calculation of the binding energy of the triton on the assump tion that there is little or no S state. The possibility that the simple theory is entirely wrong should not be overlooked. The intrinsic moments of the neutrons and proton may be sufficiently perturbed by their mutual interaction to account for an appreciable fraction of the difference between the triton moment and the proton moment. Finally, relativistic correc tions to the triton moment would be expected6 and these may not be negligible compared to the effects under consideration. The numerical work required for the con struction of Figs. 1 and 2 was carried out by S. MIoszkowski. UNIVERSITY OF FLORIDA S iiuiiiiii iii UI I 1 N1 ..' . i! "0~i 
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