Interpretation of the triton moment

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Title:
Interpretation of the triton moment
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United States. Atomic Energy Commission. MDDC ;
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Sachs, R. G
Argonne National Laboratory
U.S. Atomic Energy Commission
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Nuclear moments   ( lcsh )
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by R.G. Sachs.
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Manhattan District Declassified Code.
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"Reprinted with permission from The Physical Review, Vol. 72, No. 4, August 15, 1947."

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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., 4-8-49--750-A5206


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 one-particle 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 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 '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 over-compensate
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=(a91-2I)/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) r-p)#f?,
'P: oj=&3 (JrXpHr p)fJ,,

I = [air.- rXp+i(o3-r)(oi2-p)
-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)(-3-p)]

X (r. p)#c, (If)
,;= [(f,. r)(rr .r)+( ,-. e)(F "-p)
2- -(,.2W)]3(r-p)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)
(r-e)(asr-1oa)]fd (lli)
S= [(wi-' rX p)(s3-rX p)
(rXe)Y(ura-w.O)](r-p) ,f. (lii
The function 4 is given by
4=(4r '()-'(x,+*2- XX2-x 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 (r-p)
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(I-q2)f4,2=. (3d)


(il-q)|fI'=1. l(3e)
If (I _q2) /,i, =

q(-q')l1f|e12 I, (3f)


f(1+ 3q2)q2 Ifl=t, (3g)


3if(3+q ) fa ., (3h)


I f(1-q) 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)
I-I

The coefficients, a,, and the form of the func-
tions, f,, 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 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 s-component 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 s-component 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 one-half 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 z-component of the orbital angular
moment vanishes, since both functions are
proper functions of this operator:


(51L'16)=0.


Then, according to Eq. (14),


(5|L3a'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

".PIL-a'IP)=! ('PIL'lP)=2/9. (17)

Similarly,

('PIjLjI'P)= ~1PI|L 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:

(61|L3 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)= (3|1 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:

(2|as'12)= i, (21 |1'+a'2 2) =4 3,


(4j1a-14)= 1,


(4|1r, + ff2?4)= -4 9,


(5i1 34) = I (l q2)f*f4,


(51 oil + v,.14) = If 1-q2)*f
+- If(.l-q2) f*f4

for the spin elements. The elements of the
orbital angular momentum operator are:

1 (
(31|'ll1)=- (1-q')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(1-q2')'f5ff5', (26e)


9 9i
(6 L '\7) 9- fq'tl-q')fef


+ fJqS q2)fdf7',
Sq )f ,
9

(6[ L3'8) =- qi(1-q-)fe' l',

1 r
(61 La'19)= -- Iq2tl q')f-f
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( C-q)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
+- ?P--4p-+-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 il-q2)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 (1-q ')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 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


(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)
i-6 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 non-diag
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 Gerjuoy-Schwinger 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 spin-orbit 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


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