MDDC-866 
 
 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 
 
 copies of the same or similar content and title 
 
 and by the same author. 
 
 Technical Information Branch, Oak Ridge, Tennessee 
 AEC, Oak Ridge, Term., 4-8-49— 750-A5206 
 
Interpretation of the Triton Moment* 
 
 R. G. Sachs 
 
 Argonnc National Laboratory, Chicago, Illinois 
 
 (Received April 29, 1947) 
 
 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 2 5, 2 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 
 of the wave functions, it is found that the relative probabilities for finding the system in the 
 1 P , *P, and i D states satisfy the relation shown by the curves in Fig. 1. Wherever the results 
 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 only the lowest one-particle configurations 
 have been considered. If the amplitude of the 2 S state is taken to be as large as possible, the 
 wave function contains no l D state, 8 percent i P state, and 17 percent i P state. A wave function 
 of this form would seem to indicate that there is a spin-orbit 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 2 5 state probability is zero, the 4 Z5 probability is 
 22 percent, the 'P is 30 percent, and the 2 P is 48 percent. If the wave function of He 3 has the 
 same form as that of H 3 , the He 3 moment would be expected to lie on one of the curves shown 
 in Fig. 2. 
 
 1. INTRODUCTION 
 
 THE recent measurements 1 - 2 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 2 S( 
 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 2 S state but contains an admixture of 
 2 P, *P, and 4 £> states. 3 A theory based on simpli- 
 fying assumptions leads 4 to the conclusion that 
 
 * This work has been carried out under the auspices of 
 the Atomic Energy Commission. It was completed and 
 submitted for declassification on March 14, 1947. 
 
 l H. L. Anderson and A. Novick, Phys. Rev. 71, 372 
 (19471. 
 
 2 F. Bloch, A. C. Graves, M. Packard.'and R.'.W. Spence, 
 Phys. Rev. 71,1373 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 6 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 
 
 8 E. Gerjuoy and J. Schwinger, Phvs. Rev. 61, 138 
 (1942). 
 
 * R. G. Sachs and [. Schwinger, Phvs. Rev. 70, 41 
 (1946). 
 
 » R. G. Sachs, Phvs. Rev. 71, 457 (1947). 
 
 1 
 
T R I TO N M () M [•: x T 
 
 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" wave 
 functions actually are not purely radial but are 
 also functions of the cosine, q. of tl e angle 
 between the vector connecting the tv -citrons 
 and the 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 momenta. How- 
 ever, it seems likely that such high configurations 
 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 energy of 
 the system. For that reason, it will be assumed 
 in the final analysis that these 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 2 P or 4 P states. 
 This conclusion appears to be at variance with 
 current ideas concerning the nature of the triton 
 wave function. 3 If it is accepted that the inter- 
 action term responsible for the mixing of states 
 is the tensor interaction, then the 4 D state would 
 be directly coupled to the 2 5 state but the P 
 states would not be. Therefore, it might be 
 expected that the D state probability- would be 
 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 2 5 state; that i* that the 
 advantage gained through the large average 
 value of the tensor interaction in the P and D 
 states might be large enough to over-compensate 
 the correspondingly large kinetic energy, in 
 
 which case the energy would be a minimum for 
 a small 5 state probability . 
 
 Further information concerning these ques- 
 tions may be obtained experimentally by means 
 of a measurement of the moment of He 3 . 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 3 . 
 
 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.'' 
 
 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 <r 3 is the Pauli spin 
 operator for the proton and on = (<ti — ff2)/2, 
 where cti and <r 2 are the Pauli operators for the 
 two neutrons, then the wave functions have the 
 following form : 
 
 2 5: *, = */„ (la) 
 
 ^2={ai2-a 3 )(T- g)\Pf-,, (lb) 
 
 "P: ^ = (7r(rX?)(r-()W 3 , (lc) 
 
 ^ 4 = [>u-rXp+i(«rj-r)(«Fis-p) 
 
 -j'(w3-p)((7i 2 r)>/4, (Id) 
 
 , = \Ol 
 
 A P: ^ s = |«Ms-rXp-r— (<Fi2-r)(» 3 -p) 
 
 -(«7 3 T)( ffi; -p)L/ s> (le) 
 
 *D: ^c = [(ai-r)(a;rr)-(a I2 -p)(<, r p)] 
 
 X(r-p)f/- , (If) 
 V-; = [(w 12 • r) (ff : , • r) + («r, 2 • p) (a 3 ■ p) 
 -!("i2-ff3)](r-p)^/7, (lg) 
 
 6 H. Margenau, Phvs. Rev. 57, 383 (1940). P. Caldirola, 
 Phvs. Rev. 69, 608 (1946). G. Breit, Phvs. Rev. 71, 400 
 (1947). R. G. Sachs, Phvs. Rev. 72, 91 (1947). 
 
K . G . SACHS 
 
 ^8 = [(«* 12 • r) (CT3 ' p) + (WIS ' ?) (<F3 ' r) 
 
 -f(r- e )(a,,-(T.OW s> (ih) 
 
 ^ B = C(ffis-rXp)(i»a-rXe) 
 
 -i(rX»)*(*ii-*i)](r- ( ')*fg. (1i) 
 
 The function ^ is given by 
 
 * = (4tV2)" 1 (xi + X2- - xrx* + X3 M , (2) 
 
 where the x are the spin wave functions of the 
 indicated particles. The functions /,■ are functions 
 of the distances corresponding to r and j>, and 
 they are also even functions of the quantity 
 $=(r-j>). For simplicity, they will be described 
 as "radial" functions. The extra factors (r-p) 
 which are displayed in these equations, but not 
 in those tabulated by Gerjuoy and Sch winger, 
 are introduced in order to satisfy the Pauli 
 principle for the two neutrons. The normalization 
 conditions for the radial functions take the form 
 
 »JlAl--i. 
 
 (3a) 
 
 »JVi/.i , -i. 
 
 (3b) 
 
 *JW-Y)l/iI"-i. 
 
 (3c) 
 
 §/(i-<z*)l/4l*=i. 
 
 (3d) 
 
 fj"(i-rf)l/il , -i, 
 
 (3e) 
 
 y*« 2 (i-Q*)!/«i*=i, 
 
 (3f) 
 
 ij*(l+3<2*)g*|/ 7 |»=l, 
 
 (3g) 
 
 |J"(3 + <7 2 )|/ 8 | 2 = 1, 
 
 (3h) 
 
 ija-sfwi/ii'-i- 
 
 (3i) 
 
 indicated in these conditior 
 
 s are 
 
 ver the variable g (limits: - 
 
 -1 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. (1) ; that is, 
 
 ;'-i 
 
 (4) 
 
 The coefficients, a,, and the form of the func- 
 tions, fj, could only be determined by solving the 
 Schroedinger equation for the three-body prob- 
 lem. It is our purpose to express the magnetic 
 moment of the triton in terms of the ay and 
 certain integrals over the /,-. Then we can hope 
 to get some idea concerning the quantities a, 
 and fj from the observed moment. The magnetic 
 moment of the nucleus is given by 
 
 M = Mp(*, <T 3 ^)+M.,(*. [>l'+<T2*]*) 
 
 + (*,£»■*), (5) 
 
 where Lz ! is the z-component of the orbital 
 angular momentum of the proton, n r is the 
 magnetic moment of the proton, and y. n is the 
 magnetic moment of the neutron. In this ex- 
 pression, the wave function, ^, is taken to be 
 that function for which the magnetic quantum 
 number of the total angular momentum is +\. 
 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 
 
 M = L otj*a k \n p {j\<n'\k) 
 
 +P„UW+«s'\k) + (j\Lf\k)}, (6) 
 
 where the j, k refer to the wave functions \pj, ^/ : . 
 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: 
 
 ( a 5|L 3 »| s 5)=0, (7) 
 
 where the indicated -S states refer to an arbitrary 
 linear combination of the functions <pi and i/- 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 orthogonality. 
 Similarly, the matrix elements of the orbital 
 angular momentum vanish if the functions are 
 orthogonal in spin. It follows that: 
 
 ( 2 S|<r/|lP)=0, ( 2 S|<r/| 4 P)=0, 
 ( 2 S|<7/|ID)=0, ( 2 P|<r/| 4 Z>)=0, (8) 
 
 ( 4 P|<r/| 4 £>)=0, 
 
 where j = 1, 2, 3. Also: 
 
 ( 2 SjZ, 3 '| 4 P)=0, 
 C-P\L 3 '\ i P)=0, 
 
 ( 2 S!L 3 2 ! 4 £>)=o, 
 
 ( 2 P|L 3 *| 4 £>)=0. 
 
 (9) 
 
 It is now possible to resort to symmetry 
 arguments to show that other elements vanish. 
 The operators, <r/ and L 3 Z , do not involve o (see 
 Eq. (12)), 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|<x/|2)=0, (3j«r,'|4)=0, (3|ff/|5)-0, (10) 
 
 and 
 
 (1|L 3 --J4)=0, (2|L 3 '!3)=0, (3|L 3 -|4)=0. (11) 
 
 It will be noted that, apart from the factors 
 fk, each of the functions i/'r • -^ 9 is either sym- 
 metric or antisymmetric under interchange of r 
 and p. It has already been pointed out 4 that use 
 can be made of this property, since the operator 
 Lz* is given, in units of h, by 
 
 L-,' = 2[x v— J /&,' 
 
 V dy ' bxf I 
 
 (12) 
 
 while the z-component of the total angular 
 momentum is given by 
 
 L'Jfx y-) + (i v-)lA\ (13) 
 
 IA dy dx/ V d-n <3£/J/ 
 
 where x, y, z are the components of r and £, rj, j", 
 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 one-half the 
 matrix element of L z . It follows that for two such 
 functions, \}/i and \f/ m : 
 
 (l\L t '\m)=W\L'\m). 
 
 (14) 
 
 The functions \p b and ^ 6 are both antisymmetric 
 under interchange of r and p, so they satisfy the 
 condition for the validity of Eq. (14). In addi- 
 tion, ^ 6 and ^6 are orthogonal, so that the matrix 
 element of the z-component of the orbital angular 
 moment vanishes, since both functions are 
 proper functions of this operator: 
 
 (5|Z/|6)=0. 
 Then, according to Eq. (14), 
 (5|L 3 '|6)=0. 
 
 (15) 
 
 (16) 
 
 Equation (14) may also be used to evaluate 
 the diagonal elements of L 3 Z . Considering first 
 the 2 P functions, both are seen to be antisym- 
 metric under interchange of r and p, so any 
 linear combination has the same property. Thus 
 
 ( 2 P!L 3 *|lP)=!( 2 P|L'| 2 P)=2/9. (17) 
 
 Similarly, 
 
 ( 4 P|L 3 '| 4 P) = K 4 PiZ/j 4 P)=-l/9. (18) 
 
 The situation is not quite so simple for the 'D 
 functions. 4>i is antisymmetric and ^ 7 , yp%, ^- 9 are 
 symmetric under the operation being considered. 
 Therefore, there are cross terms between \f>6 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 A D' denotes an arbitrary linear 
 combination of ^ 7 , ^s, and ^ a , then : 
 
 (6|L 3 <|6) = ( 4 £>'|£3'| 4 £>') 
 
 = ( 4 £>'jL'| 4 £>') = l/3. (19) 
 
 It is also a simple matter to evaluate the 
 diagonal elements of <x/ in the quartet states, 
 since these functions are symmetric in en 1 , or, 
 
R . G . SACHS 
 and <r 3 *. Therefore, 7 1 
 
 and 
 
 ( 4 P | <r/ 1 4 P) = f( 4 P | S< | 4 P) = 5/9 (20) 
 
 (8|£ s «|5)=— f(3+8")/,*/i 
 
 1 8/ *^ 
 
 ( 4 /?|ff/|«Z))=§(*Z?|5«| 4 /?) = -l/3. (21) — {q{\-q*)h*h', (26d) 
 
 9/ J 
 The values of the diagonal elements of the 
 
 spin matrices in the states ^ and \p 3 may be (9|L 3 2 |5)= fq 2 (l—q 2 )f*f 
 
 obtained immediately by noting that both func- 18i J 
 
 tions are antisymmetric for interchange of <n* 
 
 and a 2 *. Therefore, + i fj ( i_ flW/ ,/ i (26e ) 
 
 9i J 
 
 (l|<7i 2 -f-<r 2 *|l) = (3|oV + <r 2 <|3)=0. (22) 
 
 5 r 
 
 On the other hand, the sum of the three er;« must (6|L 3 ! | 7) =- I <? 2 (1 -g 2 )/«*/ 7 
 be equal to twice the average value of S z , or ^ 
 
 (l|«r 3 <|l) = 2( 2 S|5*| 2 S) = l (23) ^_ 
 
 (3 | ffI * 1 3) = 2( 2 P|S*| 2 P) = -1/3. (24) 
 
 (6|L,«|8) = 
 9 
 
 tion is tedious and not at all illuminating, it will 1 
 
 equal to twice the average 
 and +^JV(W)/<*//, (260 
 
 2 r 
 
 All other matrix elements may be obtained by ( 6 I ^1 8) =- J g(l -q 2 )f t *fi, 
 direct computation. Since the required calcula- 
 
 (26h) 
 
 tion is teaious ana not at all illuminating, it will l r 
 
 not be presented here. The results are : (6 1 L 3 * 1 9) = -- J e 2 (l -q 2 )f t *f. 
 
 (2 | o-3-l 2) = — i, (2 | «ri« + «r,' | 2) =4/3, where// is the derivative of /,- with respect to q. 
 
 (4| ff ,«|4) = i, (4|<r,' + <r,«|4)=-4/9, 
 
 4. THE MAGNETIC MOMENT 
 
 (25) 
 
 /ci *|4\_2 f (i _ 2\f*f ^ n terms 0I these matrix elements, the mag- 
 
 V \*i\ )-tJV q)U U, netic moment i s g iven by Eq . (6) . In order to 
 
 bring out explicitly the symmetry character of 
 the wave functions, we set 
 
 5|»i'+»y|4)--|J"(l-g*)/ I */4 
 
 ai = a\S, a 2 =a 2 S, a3 = as 2 P, a i = a i ~P, 
 
 for the spin elements. The elements of the a i = a i i P, a^ — a^D, a 1 =a 1 D, a i = a i D, (27) 
 orbital angular momentum operator are : _ 
 
 (3\L 3 '\l) = | (1 — a 2 )af *f ' (26a) where S 2 is the probability of finding the system 
 
 9iJ ' in the 2 S state, ( 2 P) 2 that of finding it in the 2 P 
 
 state, etc. The numbers 5, 2 P, 4 P, and D are 
 
 /a\t ,\ns C n ,,,4„ , , , x ,„,, „ chosen to be real. They must satisfy the normal- 
 
 (4 L 3 2 2)=— I (l-5 2 )/4*(/ 2 +g/ 2 , (26b) . . ... 
 
 $i J ization condition 
 
 2 r 5 2 + 2 P 2 + 4 P 2 +Z? 2 =l. (28) 
 
 (7|L S *[5)=- L 2 /,*/, 
 9i J 
 
 9i J The «,. also must be normalized as follows: 
 
 |a,|»+1oi|* = l', (29) 
 
 |a 3 | 2 +|a 4 | 2 =l, (30) 
 
 9iJ 
 
 .q(\-q 2 )f-*h', (26c) 
 9i 
 
 7 The factor 2 arises from the fact that the Pauli operator, 
 a', is twice the spin. !a 5 | 2 =l, (31) 
 
T K I T O N M OMEN T 
 
 I a, . I + l fl7; 5 +!fl8i 2 +!a,|* 
 
 + f<R 4a 7 *a 8 JV/vVs 
 
 -a,*a,f(l~g 3 )g 2 / 7 */« 
 
 -a g ^ 9 |\l-g 2 )g s / g 7,j = l. (32) 
 
 |R | ! denotes the real part of the quantity con- 
 tained in the bracket. Equation (32) has a 
 relatively complicated form because the functions 
 i/'t, ^8, and \f/t are not orthogonal to one another. 
 In terms of the coefficients defined by Eq. 
 (27), the magnetic moment is: 
 
 M = Mp - - 1 «2 1 2 S 2 (n P - n„) 
 
 3 
 
 — *p^>,-1I«4|»(m,-m-)D 
 
 2 2 
 
 9 3 
 
 2 1 1 
 
 9 9 3 
 
 where ji, contains the cross terms. This last 
 quantity is given by 
 
 u x = -£»*(RJa,*aJ 5 Jg*(l -g*)/.*/* 
 + 2JV(l-g=)/ 6 *//] 
 
 + 2a i *a i fq(l-r)f e *f s ' 
 
 + - i PsA-a 3 *a i j(\-q-)qf 3 *f l ' 
 
 +a i *a i J(l-q')f i *(f, + qf 2 ') 
 +-D*Ps\2a 7 *al2 fg*/ 7 */s 
 
 - fq(i-g-)fi*fA 
 
 4-o„*<J f 
 
 (3+g 3 )/.*/* 
 
 -2 J" a (l -*)/••/•'] 
 
 -a 9 *a 6 |r(l-c7 2 )/ 9 */5 
 
 -2jg(l-g*)V,*/ 8 ']|. (34) 
 
 Here, .') | J 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 order to simplify it. We assume 
 that the functions /* are independent of q or 
 
 /*' = 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 f k 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 
 
 //#.-[/ l/fl'/l/il 1 ]. W 
 
 According to the well-known 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 
 
R . G . SACHS 
 
 moment are no greater than the values given by 
 Eq. (36), so the estimates obtained below of the 
 amount of admixed IP, 4 P, and A D states are 
 lower limits. 
 If we now set 
 
 a k =Xk+iyk, (37) 
 
 the cross terms in the magnetic moment become 
 
 2 r 5 y/7 -i 
 
 Mx>= D 2 \ (xtXi+ygyi) (.v 6 .v 9 + v 6 v 9 ) 
 
 3v3~ Lv/7 2 J 
 
 8v2 
 
 + (Hp-Hn) *P i P(x t x i +y t y i ) 
 
 9 
 
 2\/5 r 2 
 9 LV7 
 
 V7 -i 
 
 (Xiyi-Xnys) . (38) 
 
 10 J 
 
 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 
 
 9 4 
 
 L (x, 3 +yj 2 )-\ (x 1 x i +y 7 y t ) - (xv-VQ+yvjo) 
 
 ;-6 y/7 
 
 (*»x»+y«y,) = l. (39) 
 
 The magnetic moment is still given by Eq. {33), 
 with | a 2 1 2 = x 2 2 +3'2 2 and |a 4 | 2 = .r 4 2 +y4 2 . The 
 constants, Xj, yj, are to be chosen in such a 
 manner that they satisfy the conditions of Eqs. 
 (29) to (31) and Eq. (39), and that they give the 
 correct value of the magnetic moment, i.e., 1 
 
 to minimize the negative diagonal terms in 
 Eq. (33) we are led to choose 
 
 M=1.067/i„. 
 
 (40) 
 
 The eighteen constants are clearly not deter- 
 mined by these five conditions, so some further 
 assumptions may be made in order to make the 
 final results somewhat more specific. 
 
 The fact that the coefficient in Eq. (40) is 
 larger than unity does lead to a considerable 
 limitation on the choice of the constants, since 
 the diagonal terms in Eq. (33) tend to reduce 
 the moment helow the proton moment. There- 
 fore, it is necessary to take the non-diagonal 
 terms to be positive and rather large. In order 
 
 fl 2 = 0, j a-4 1 2 = 1 - 
 
 (41) 
 
 Since it seems likely that the amount of 5 
 state will be as large as possible, we might 
 require that the constants Xj, yj be chosen in 
 such a way as to lead to the largest possible 
 value of 5"-. There is also some reason to guess 
 that the D state probability will be large com- 
 pared to the -P and 4 P probabilities. 3 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 D" 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 Xj, yj in 
 such a way that Z> 2 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 
 
 x 6 =y 6 = x 7 =y 7 = .v 9 = y 9 = 0, 
 Xi = x 4 = —ys, 
 yi = yi = x&. 
 
 (42) 
 
 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 4 P 2 P + 0.178 4 PZ>-[0.50 2 P 2 
 
 + 1.25 4 P 2 +0.759Z> 2 ] = 0.067, (43) 
 
 where we have taken /i p = 2.79 and n„/cp 
 = -0.685. The values of 4 P 2 , 2 P 2 , and Z> which 
 are given by this equation are shown in Fig. 1. 
 It should be emphasized that these are not the 
 only possible combinations of these constants 
 which will agree with the triton moment because 
 the choice of the xy, y,- which has been made is 
 rather special. 
 
 It is to be noted that the 5 state probability 
 will be a maximum for Z> 2 = 0, 4 P 2 = 8 percent 
 and ! P 2 = 17 percent. D- is never as large as 2 P 2 , 
 and it is at most equal to 4P 2 in spite of the 
 fact that the coefficients Xj, yj have been chosen 
 in such a way as to lead to as large a value of D 2 
 as possible. The largest value of U 1 is 22 percent 
 with 4 P 2 = 30 percent and 2 P 2 = 48 percent. In 
 this case the wave function contains no 5 state. 
 
8 
 
 TRITON MOMENT 
 
 It is generally believed that the properties of 
 the wave function of He 3 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 3 . This makes it possible to 
 make certain predictions concerning the mag- 
 netic moment of He 3 . It has been shown that 
 the moment of He 3 is given in terms of the 
 moment of H 3 by the relation 1 
 
 M(He 3 )+M(H 3 )= M p + M n -2(Mr + M,-^) 
 
 X(3£> 2 - 4 P 2 -|-2 2 .P 2 )/3. (44) 
 
 The consequences of this equation are demon- 
 strated in Fig. 2 which shows the relation be- 
 tween the moment of He 3 and the amounts of 
 \P, 4 P, and *D states which satisfy the relation- 
 ship shown in Fig. 1. These are not the only 
 possible values for the He 3 moment, since certain 
 specific assumptions have been made concerning 
 the wave function in order to obtain Fig. 1. The 
 value of the He 3 moment to be expected on the 
 basis of the 4 percent of k D state and percent 
 P state found by Gerjuoy and Schwinger 3 is 
 yu(He 3 )/ju p = —0.763, a value which seems to be 
 well out of the range of possibilities allowed by 
 the considerations put forth here. 8 Therefore, a 
 measurement of the moment of He 3 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 
 
 ■ The Gerjuoy-Schwinger assumption of a small amount 
 of 'D function and even smaller amounts of the P functions 
 is not consistent with the results obtained here because 
 of the condition, Eq. (35). However, if 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 2 P , l P, and 'D probabilities 
 are small. In this sense the measurement of the moment 
 of He 3 might be considered as a test of the assumption 
 expressed by Eq. (35). A large average value of // would 
 be rather surprising, since the usual assumption that /, 
 be a function of (r 1 2 2 -}- r , 3 2 -f- r ea 2 ) . where r,„„ is the distance 
 between particles m and n, would lead lo /V = 0. 
 
 i 1 1 1 1 1 : 1 : 1 1 
 
 "\ 10 X 
 
 "\ 
 
 \ 
 
 \ 
 
 us 
 
 \ 
 
 - 
 
 \ , 
 
 / 20% 
 
 
 
 \ 6 ?> 
 
 / / 40V' 
 
 . 
 
 i i 1 1 1 1 1 1 1 1 1 
 
 Fig. 1. Relation between i P, *P, and 'Z> state proba- 
 bilities required to account for experimental moment of 
 the triton. The special assumptions made in obtaining 
 these curves are expressed by 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 /,-. 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 
 ft 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 momenta 
 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 . G . SACHS 
 
 with the measured magnetic moment. The 
 amount of 5 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 spin-orbit cou- 
 
 -i : 1 1 r 
 
 10 12 
 <p*r eutl' 
 
 ■ ■ I l_ 
 
 20 22 24 
 
 Fig. 2. The magnetic moment of He 3 in units of the 
 proton moment. These curves have been obtained on the 
 assumption that the relations shown in Fig. 1 applv also 
 to He 3 . 
 
 pling plays an important role in determining nu- 
 clear structure. The other alternative is that there 
 is little or no 5 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 expected 6 
 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. 
 Moszkowski. 
 
UNIVERSITY OF FLORIDA 
 
 3 1262 08910 5299 
 
 [