PB 161594

^&cknlcciL vtote 93

AN EXPERIMENTAL STUDY OF

BETA DECAY USING THE RADIATIONS

FROM ORIENTED NUCLEI

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^^ote

93

AUGUST 1961

AN EXPERIMENTAL STUDY OF

BETA DECAY USING THE RADIATIONS

FROM ORIENTED NUCLEI

Dale D. Hoppes

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An Experimental Study of Beta Decay Using

the Radiations from Oriented Nuclei

Dale D, Hoppes

The angular distribution of electrons with

respect to the nuclear spin direction is shown to

furnish information about the relative contribution of the different operators in first-forbidden

beta decay. An experimental determination of two

coefficients in a Legendre polynomial expansion of

this distribution function for a transition in the

decay of cerium-141 is made by observing the radiations from both aligned and polarized nuclei. Together with calculations based on reliable shell

model assumptions this information allows the determination of the relative contribution of the

significant relativistic operator a. The result

M(a)/M(cr^ X ^) =-29.4 ± 1.5 is compared with some

general estimates of this quantity.

1. Introduction

The theory of beta decay is now sufficiently formulated that one

can use measurements of angular distributions of the particles emitted

This article was composed for the fulfillment of the publication

requirement for the degree of Ph.D. in the School of Arts and Sciences

of the Catholic University of America, Washington, D, C.
to obtain information about the initial and final nuclear states involved in the transition. In the present experiment the distribution

of emitted electrons with respect to the nuclear spin direction is

measured for a first-forbidden transition occurring in the decay of

Ce . Sufficient information is obtained from subsidiary experiments

and calculations to permit a determination of a reduced nuclear matrix

element ratio which is sensitive to the radial nuclear wave functions

and the composition of the nuclear potential.

In part 1.1 of the introduction the pertinent beta-decay formulae

and approximations are discussed, while in part 1.2 the information

necessary for a calculation of the nuclear orientation of Ce is

presented. Previous experimental results on the decay of Ce are

given in part 1.3.

1.1. Determination of the Reduced Nuclear Matrix Elements

in First Forbidden Beta Decay [1,2,3]

In order to write the interaction Hamiltonian for use in a perturbation theory development of the beta-decay transition probability

one must take selected products of the components of the spinor wave

functions of the two nucleons and two leptons that are involved such

that the result is relativistically invariant. It has been shown that

if one forms a four-dimensional polar vector^, l) and axial vector

(o", Yc) from four-by-four Dirac matrices and then contracts the result

of operating separately in the nucleon and lepton spaces with each of

these operators, the essential interaction is established. The transformation of one nucleon into another is indicated by an i-spin
operator, while the non-conservation of parity can be completely

described by writing (l + Yr) times the operator in the lepton part.

One sees that there are two types of operators present; a and Yc connect components of the nucleon wave functions which give terms of the

order (v/c) , ^ l/lO compared with the operators 1 and CT, The nucleon / r r ,w

latter give the so-called allowed transitions in which the parity of

the initial and final nuclear states is the same.

In order to see how other operators comparable to the 2_and Yc

arise, one considers the leptons as free particles and makes a multipole

expansion of the resulting plane wave, e ^^^'^*—, where ^ and q are the

momentum of the electron and the neutrino, respectively. This yields a ^ ^ ^ ^-k k

^ ^ series of terms oc 2 Z E (pr) (qr) Y. (059) where Yp (8,9)

-6=0 m=-l k=0

^ ^

is a spherical harmonic. Since p < 2, q < 2, and r < R, the nuclear

radius, (R « 0.02 in relativistic units), it is seen that the order of

magnitude of succeeding terms will be 7:^ (.04.) , since the Hamiltonian

appears squared in the transition probability. Thus the leading term

will be 1 or cr, while the next order terms for these operators are r

and or, respectively. The latter is separated into a.r, cr x r, and B..,

with respective tensor rank 0, 1, and 2. The leading term in the expansion for a, or for Ycj will be of the same order of magnitude as the

second terms for the other operators and has the same requirement of a

parity difference of the initial and final states. In table 1 the first

forbidden operators are displayed, together with their tensor rank,

selection rules, and a diagrammatic representation of the angular

momentum, L, carried away by the leptons. The meaning of [\{J.JA.) is
that the three numbers must be possible lengths of the sides of a

triangle, possibly of zero area. The orbital angular momentum may be

carried away by either the electron or the neutrino; also the wave

function of the electron for a given total angular momentum will contain both ('C'+l/2) and {i^l/2) contributions of comparable magnitude,

since v/c « 1 for the higher energy electrons. Thus an operator may

have matrix elements between several lepton states.

In order to write general expressions for any angular relation

that may be possible, one most wisely expresses the operators in

spherical tensor form and proceeds by a density matrix method [4.] that

allows separation of angular and non-angular parts for each particle

involved. This procedure makes it possible to easily sum over quantities that are not observable and obtain expressions which clearly display

the origin of the remaining observables. As a pertinent example, let

us consider the correlation of the direction of electron emission with

respect to the nuclear spin axis [5,6,7]:

W(p,j) =E (-l)''''^^^ fj^(j)Fj^(L,L',J^,j)b^(L,L')Pi^(p-tj)

K

Here we wish to specify the coefficients of the possible Legendre polynomials, P, (p»J), which have as an argument the cosine of the angle

between ^, a unit vector in the direction of electron emission, and J,

a unit vector in the direction of the initial state nuclear spin. The

degree to which the latter can be considered a unique direction in

space is given by
f.(j) =Z (-)^"'^(JMJ-M|jJkO)a„

^ M

^

where M, the magnetic quantum number, is the projection of J on the z

axis, (JMJ-MJJJkO) is a vector coupling coefficient and aw is the

fraction of the nuclei with projection M, The entire dependence on the

nuclear orientation is contained in these parameters, in which k is the

highest power of M that is considered in any f, . The F, (L,L',J.J) is

an established product of numerical factors and vector coupling coefficients involving not only the order k and initial nuclear spin J,

but also the selection rules of the term in that the two operator tensor

ranks L and L' and the final nuclear state spin J^ must satisfy certain

triangle conditions imposed by the vector coupling coefficients. These

triangle relations are A(kLL')> A(kJ.J.), A(LJ^J.) and A(L'J.J.); they

tell us just what b, (L,L') occur in the coefficient of P, (p»j) for a

given transition. The b, (L,L' ). contain the meat of the problem: the

products of all nuclear matrix elements of tensor rank L and L' times

their associated energy-dependent lepton functions. The bgClLO's, which

from A(kLL') consist of only bQ(OO), b^dl), and b^laa), multiply the

isotropic tenii P„ and give the shape factor, which is the departure of

the spectral shape from that of an allowed transition, for unoriented

nuclei. In general the higher terms of a distribution or correlation

function must be divided by this isotropic term as a step in the

interpretation of an experimental angular distribution as a function of

energy.
Let us now consider the beta distribution function as applied to

a 7/2" - 5/2 transition. From the triangle conditions this first

forbidden transition cannot involve L = operators and will give only

terms up to k = 4-* We will write instead of f. the more familiar

/2k\'"^ k (2J+k+l)' ^ —

quantity fj^ =

(^ jj J [ (2k+l)(2J-k): ^ ^k* Then, subsituting the

values of known quantities, we find:

b, (11) -I- 0,527 b, (12) + 0.0361 b, (22)

W(p,J) ^ 1 - [- 7-^- ~- ^

] fiP,(p-^) --- b^dl) +bQ(22) 1

1

0.582 b^(ll) + 0.927 b„(l2) + .0243 b,(22) , ^ ,^ ^, + r,..^ 2_.

2 2 -. f P (p.J)

^ b^dl) +bQ(22) ^ 2

2

2.358 b„(l2) + 2.721 b^(22)

The anticipated P. term does not appear because of a hidden triangle

condition in the coupling of the neutrino and electron angular momenta

to give b, (22),

Table 2 gives the b. (LL')'s as derived from Blncer [6]. Here we

write the reduced nuclear matrix elements of Bincer in the form M(ir),

M(aXr), M(^)y and M(1B. .)• They are multiplied by the appropriate

coupling coefficient) C^ or C.) which gives the Inherent strength of

the interactions. From the corresponding lepton matrix elements we get
multiplying terms evaluated completely for the neutrino and down to

the radial wave functions for the electrons. The quantities p, E,. and

q are respectively the magnitude of the electron momentum, electron

energy, and the neutrino energy, which we will measure in relativistic

units in which n, m , and c are set equal to unity. The electron

radial functions are contained in the quantities K , and M ,, which

are similar to those defined by Bincer [8], with k = -(j+g-) for j = -t+gand K = +(j+|-) for j = l-^. Here

= U_p-Wi^)+'t(ic')]/^ „ ^gi[A(K)-A(K')]

+ S{-K)S(-,<')f f .e^CAW-Al-K')]!

M^

2p

The f 's and g's are the radial functions as calculated by Bhalla and

Rose [9]; however the phase is chosen to fit the definition of Bincer

[8], The quantities l{y:) , A(i<)> and S(ic) represent the orbital quantum

number, the phase, and the sign* respectively, for a given k.

An estimate of the size of terms and a hint to the type of information that they yield is obtained if we consider the lepton wave functions
as those due to a point charge, but evaluate them at the nuclear

radius. Furthermore, we use an expansion in powers of ^ = aZ/2R «s 10,

take (oZ) « 1, and then consider only the highest power of ^ that

appears in each b, (L,L')» M(a) does not have a multiplying ^ although

it is estimated to contribute the same as elements that do, therefore

we include it in the tensor rank one combination

?rc M(ir) + C.M(crxr)l - C M(a), which is considered as a unit. This

combination carries just the energy dependence of the allowed transitions. Using this approximation, we estimate the b, (LL')'s in table 3,

from which several rough conclusions can be drawn. The respective

magnitudes of the P, , P , and P^ coefficients are of the order 1, l/lO,

and l/lOO, aside from the respective nuclear orientation parameters,

which also are of descending magnitude. Thus the higher orders are

more difficult to measure experimentally, but it is just these terras

that are more informative. This is true because the group

^[C M(ir) + C M(oxr)]- C M(a) dominates the b^(ll) and b, (ll) terms,

where it appears squared, and the b^ (12) interference term, making it

difficult to detect the individual tensor rank one matrix elements.

In the b„(ll) term, however, the combination C M(oxr) - 2C M(ir) occurs

in the leading subterm.

Also, without knowing the individual matrix elements one can

directly solve for the ratio of tensor rank two to tensor rank one

contribution from a measurement of the coefficient of the P, term.

These arguments apply also to other experiments in which information

about the nuclear orientation is deduced from correlations with a

8
following gamma ray, since exactly the same b, 's appear. Similar conclusions can be drawn about a tensor rank zero unit contributing to a

A J = transition.

In the past there have been attempts to interpret a few spectralshape [10,11] and p- Y-correlation [12, 13] measurements in which the

leading terms in the ^ expansion have been small because of cancellation

in the multiplying nuclear matrix element combination or in which this

combination is small compared to the B. . element because some secondary

quantum relation involving the nuclear states is violated. In some cases

the nuclear matrix elements have been treated as adjustable parameters

and approximations have been made on the basis of the anticipated size

of the result, with a satisfactory fit to the experiment being taken as

justification for the degree of approximation [l4-]» In some schemes of

analysis the effect of the finite nuclear size on the electron wave

functions has been partly incorporated in the nuclear matrix elements,

leading to extra parameters for each which depend on the electron

angular momentum [15, 16]. In the current experiment we measure directly

the energy dependent quantities Ni/Nq and N-/Nq appearing in the

simplified beta distribution functioa

W(p,J) N. p ^ N_ p^

9
for the AJ = 1 transition from the ground state of Ce to the ground

state of Pr . Since a relatively large Pp term is found, the data

are analyzed without the usual approximations, but with the assistance

M(ir) M(iB..)

of well-defined calculations of TTTirrT and 777—

n— for shell model

M(a)

angular wave functions. The result is a value for TTTZTTj which is

directly dependent on the nuclear radial functions.

In order to obtain information about the lower energy transition

to the first excited state of Pr , one must measure the angular

distribution for those beta particles which are in time-coincidence

with the subsequent gamma ray. This more complicated correlation

between the beta momentum, nuclear spin direction, and the gamma

momentum (hereafter ^

- J - Y correlation) [l7,18] is considered here

only in order to determine the contribution of the lower beta group to

the overall distribution, and so will not be discussed in detail. For

this AJ = transition all six matrix elements can contribute; thus

the present experiment alone allows only a conditional analysis on the

basis of the roughest approximation unless the results of theoretical

calculations are again used to eliminate some of the variables.

1.2. Nuclear Orientation

The familiar spin Hamiltonian [19] with effective S = l/2 is

applicable to Ce in neodymium ethyl sulfate at temperatures below

1 K. This is written below, with the origin of each term indicated:

10
y = [g, ipH^S^ + g|p(H^S^ + H S )] - Magnetic interaction of the

' ' J- ^ ^ ion and an external magnetic

field

+ D[S ^ - 1/3 S(S+l)] - Interaction of the ion with

the crystalline field.

+ [A S J + B(S J +S. J )] - Magnetic interaction of the

z z XX y y nucleus with the ion.

+ P[J ^ - 1/3 J(J+l)] - Interaction of the nuclear

electric quadrupole with

the crystalline electric

field.

+ other terms - ion dipole-dipole interactions, direct interaction

of the nuclear magnetic

moment with the external

magnetic field, etc.

Here the standard terminology is used, with the exception of J for

the nuclear spin.

Ideally, the constants in the Hamiltonian would be determined

directly for the particular nucleus-ion-crystal combination by a

resonance measurement, but this has not been done for the present case.

Therefore, we make use of the information presented in a paper by an

Oxford group on the angular distribution and linear polarization of

gamma rays from oriented Ce in neodymium ethyl sulfate [20]. They

give the values gt 1 = 3.9, gi = 0.2, A = 0.108(ij^/l)cm~"'' and

B = 0,002(|j^l)cm , based on measurements in concentrated cerium ethyl

sulfate [21] and theoretical calculations [22]. The A and B values may

be somewhat ia error as stated, since the value of mean cube ionic

radius [23] has been recalculated [24] and found to be somewhat lower,

but if one uses in the A and B expressions the magnetic moment of Ce

11
as derived from resonance measurements [25] (in a double nitrate

crystal) evaluated on the basis of the old radius, no error results

from this change. A recent calculation of these A and B expressions

gives slightly different results [26], but the present experiment is

evaluated on the basis of the first values in anticipation of a more

direct measurement of the required g's and A and B for eventual use in

an accurate spin Hamiltonian.

The second term in the spin Hamiltonian is zero for S = l/2; the

fourth and fifth terms can be shown negligible for the present circumstances.

The spin Hamiltonian is now taken to have the following simple

form for a magnetic field along the c-axis:

yi :? (0.000262 H - 0.0396 J )S °K.

z z z

The sign of A is that for \l^ negative, which would be the shell

model prediction for Ce . This sign is established by the present

experiment.

The use of this simple Hamiltonian can be tested by calculating

the relative population of the magnetic substates, aj^, where M = J ,

as a function of temperature for zero external field. From these one

can calculate the orientation parameter f^ as a function of temperature

and then attempt to fit the Oxford measurements of the variation in

gamma-ray intensity with temperature to the function 1 + A_fpP_(cos9).

The Oxford measurements show that the f , contribution should be small, 4

12
and subsequent experiments have given the relationship between the

magnetic temperature derived from the magnetic susceptibility and the

true absolute temperature. The fit is good (within the error limits of

the experiment) for A^ = 0.337. We thus use this expression for the

gamma-ray distribution to relate the change in gamma-ray intensity to

the parameter f for our different experimental conditions. Furthermore, we use the simplified spin Hamiltonian with H = 205 gauss to

calculate the orientation parameters [27]

f^ = l/J2Maj^

f = 1/J^[Z M^aj^^ - 1/3 J(J+1)]

f^ = 1/J-^[S M^aj^ - 1/5 (3J^+3J-1)Z Ma^]

for J = 7/2. We then find the odd f 's, which define the nuclear

polarization, from the alignment parameter f by means of these relations. This use of the gamma-ray measurements to predict the orientation parameters is quite important, since the temperature of a thin

radioactive surface layer cannot be reliably inferred from a measurement

of the bulk crystal magnetic susceptibility. Furthermore, the relationship between the susceptibility and absolute temperature has not yet

been directly established for neodymium ethyl sulfate in a magnetic

field.

13
1.3. Experimental Information on Ce

The present information on the decay of Ce is shown in figure 1.

Several spectral shape measurements have been made, [28,29,30,31] with

some disagreement on the endpoint energies, but with no reports of other

than an "allowed" shape. In general, data taken in the region between

4-35 and 580 kev has been extrapolated and subtracted from the total

spectrum to yield an "allowed" spectrum for the 435 kev transition, although the lower region is somewhat masked by conversion and Auger electrons. In two cases P

- Y coincidences were used to separate the inner

beta branch [29,32], which was found to be "allowed" in shape in the

higher energy region for which measurements were reported. These

investigations are sjtressed because the present results indicate that

for the 580 kev spectrum there should be some deviation from an"allowed"

shape which might be observed in a careful measurement.

Both the cerium [25] and praseodymium [33,34] ground state spins

and magnetic moments have been measured. The measured magnetic moment

of the Ce must be revised, due to new values [26] calculated for the

ionic radii involved in the nuclear paramagnetic resonance determination.

As mentioned in the previous section, a simultaneous measurement

of the gamma angular distribution and linear polarization in a nuclear

orientation experiment [20] gave the ratio of electric quadrupole to

magnetic dipole amplitude in the gamma transition as 0.08 ± 0,02. This

experiment also gives information on the contribution of different

tensor rank operators in the inner beta transition, but this effect is

obscured by the uncertainty in the gamma-multipole mixing ratio.

u
2. Experiment and Analysis

2.1. Experimental Design and Procedure [35>36,37]

The requirements for the counting of beta particles at low

temperatures have been met successfully in a series of recent experiments [38,39]. Figure 2 shows a typical counting arrangement used in

the present experiment. In the inner chamber of a glass dewar system

a 12-gram single crystal of neodymium ethyl sulfate is supported by a

thin-walled bakelite tube about 10 centimeters long. Electrons emitted

from a spot of activity at the top of the crystal are detected by an

anthracene scintillator one to five centimeters above. The light

pulses from the detecting crystal are transmitted through the glass

window into a selected, shaped, lucite light pipe about 4 feet long

which terminates in a selected photomultiplier outside the dewar system.

The output of the photomultiplier is amplified and analyzed into 100

pulse-height channels.

The gamma distribution is sampled by three 2" x 2" Nal counters,

connected in parallel, lying in an equatorial plane with respect to the

c-axis of the source crystal and spaced 3.5 inches from the activity.

The 14.5~kev gamma ray is selected and counted directly and is also used,

by means of a fast-slow coincidence arrangement, to trigger a lOO-channel

beta analyzer. This provides a separate measurement of the P

- Y ~ J

correlation for the inner transition. In addition, another gamma

counter is located in a polar position below the dewar system.

A polarizing field is supplied by a copper-wound, liquid-cooled

solenoid which can be raised about the tail of the dewar system. The

15
field for the adiabatic demagnetization is supplied by a 23 kilogauss

electromagnet. The dewar system and pumping manifolds are of conventional design and permit, with the use of an oil diffusion pun^,

helium bath temperatures below 1 K. Coils for the measurement of the

bulk magnetic susceptibility of the paramagnetic crystal are wound on

the outside wall of the experimental chamber. The sample is inserted

through a tapered ground glass socket at the bottom of the chamber;

this opening is filled with a plug carrying the sample and sealed in

place by the same soap-glycerine mixture that holds the anthracene

crystal in place. An optical system using reflections from the source

crystal faces is used to check the position of the alignment axis.

The Ce used in these experiments was obtained from cerium oxide

13 2 irradiated for 3 weeks in a flux of about 10 neutrons/sec/cm . After

1Z.3 a Ce impurity had decayed a chemical separation involving the selective precipitation of cerium and zirconium iodates and cerium hydroxide

1/3 233 was performed to remove the Pr and Pa activities that remained.

Spectra taken with scintillation spectrometers and a thick-lens beta-ray

spectrometer confirmed the purity of the final Ce fraction. Neodymium ethyl sulfate was prepared by stirring together, in a large excess of water, di-ethyl sulfate and neodymium hydroxide and was then

purified by successive recrystallization. The active crystals were prepared by touching the desired spot on a mounted inactive single crystal

into a pool of about 10 drops of saturated neodymium ethyl sulfate

solution containing about l/2 millicurie of the separated Ce . After

about one hour the contact was broken and the active area of the crystal

16
was trimmed with a scalpel to a disk of about 1/4. inch diameter. The

resultant activity was about IO-5O |ic, which was sufficient to give

reasonable counting rates in the experimental geometry without excessive heating.

The crystal was then mounted in the apparatus, the dewars were

slipped on, and the assembly was slowly cooled to the temperature of

liquid nitrogen. At this point the helium space and experimental space

were evacuated, helium gas was introduced and liquid helium then added.

Pumping on the liquid helium reduced the temperature to about 1 K, at

which time the assembly was placed in the large magnet and the 23,000

gauss field applied. After thermal equilibrium had been established

the exchange gas was pumped out and the field removed. The temperature

of the sample fell to about 0.02 K as indicated by the magnetic susceptibility and as predicted by previous measurements which also gave

the relation between the susceptibility and the temperature.

In one series of runs the ratio N„/n„ was measured alone by observing the counting rates in each of the ^ channels and the Y peaks

for about 15 minutes while the sample was aligned, then briefly admitting a small amount of helium gas into the experimental space to

heat the sample so that a measurement of the isotropic radiations could

be made for a corresponding length of time. The ratio of the counting

rates "cold" and "warm", when properly corrected for attenuating effects, thus gives the ratio of the P- term to the isotropic term, for

the particular angles at which the ^ and Y counts are taken. In this

particular case the P, gamma-ray distribution term contribution is

4

17
negligible, both from the very small value of f , and from the relatively 4

small contribution that the necessary quadrupole operator makes to the

transition. The uncorrected cold- to-warm ratio was typically 1.06 for

the upper beta channels and 0.968 for the equatorial gamma counters.

For this particular set of runs the gamma anisotropy did not correspond to that anticipated from the demagnetization conditions,

indicating that local heating occurred or that the cerium was not all

properly located in the crystalline lattice. Since only f , which is

given correctly in either case by the gamma measurements, is desired,

the attenuation is not important. In the polarization measurements,

in which f^ is determined from f. with the supposition of full crystal

acceptance of the cerium ions, spot checks without field showed that

indeed the gamma anisotropy was in agreement with the predicted temperature for the source on which the present data are based.

For the polarization experiments a field of 205 gauss was applied

along the c-axis of the crystal. This resulted in an increase in the

temperature of the sample and consequently a decrease in the orientation

parameter f _, but maximized the parameter f , as evidenced by the

greatest cold-warm difference in the beta counting rate. In this case,

the rate of warm-up of the sample was greatly diminished, due to the

increased specific heat of the neodymium ethyl sulfate in a magnetic

field. Counting was continued for periods up to one hour before the

sample was warmed and the normalizing counts taken. During this time

the uncorrected beta-ray cold-to-warm intensity ratio was typically 1.30

for the field in the direction of the electron detector and 0.76 for the

18
opposite field. The uncorrected equatorial gamma ratio was about 0.975.

The data in this latter type experiment give twice the ratio of

the pA^»J) term to the isotropic term if the corrected quantities

I

(•= l) for the two field directions are summed and twice the ratio of

the P, (^'J) term to the isotropic if the difference is taken.

The above runs also give information on the ^ - y - J correlation

[17] for the 435-kev transition in the present simplifying geometry in

which the gamma rays are detected at right angles to the electron

momentum-nuclear spin axis. This measurement is less accurate due to

the reduced counting rate available in the coincidence system, but the

I difference in the uncorrected quantities (t l) for the two field

w directions, plus 0.22, is still determined to about 10^, as measured

by the standard deviation of 7 runs. The sum of these two ratios,

which measures the P^ term, is slightly negative, but zero within the

accuracy of measurement.

2.2. Experimental Corrections and Evaluation of Data

Since the coefficients N,., N. , and N- contain energy dependences

which in theory are characteristic of the individual matrix element

combinations, a measurement of N./Nq and N^/N-. as a function of energy

is desirable. Usually the P^ coefficient, like the spectral shape

factor Npj itself, is dominated by a combination of matrix elements with

a common p/E energy dependence. For the Pp coefficient the characteristic energy dependence is expected to be p /E unless there is

cancellation among the matrix elements or unless the B . element is

dominant.

19
In order to make a best determination of the energy dependences

several corrections to the data are necessary. We will consider these

in turn, together with the limitations on the accuracy imposed by each.

(1) Background

In this respect Ce is quite good compared to other materials

which have been used for low temperature beta spectroscopy [38,39]j for

in the present case internal conversion electrons and electrons from

Compton scattering of the gamma ray occur only in the lower-energy

region of the beta distributions, where electron scattering prevents

analysis. Background due to gamma rays and cosmic rays striking the

beta scintillator are minimized by the small crystal size; these and

other contributions from phototube and amplifier noise and electrons

scattered from the walls were shown to be quite negligible by measurements made with the detector just barely shielded from the direct beta

radiation from the source.

For the gamma detectors the background was small and reproducible;

it was subtracted as the first step in the analysis of the gamma distribution.

(2) Solid angle corrections

For the alignment (no magnetic field) experiments in which coincidences were not recorded, the source-beta detector distance was 2.5

centimeters. In order to correct for the fact that angles other than

the assumed or tt were involved, a multiplicative correction to the

P^ term of 1.025 was applied, following the calculations of Rose [4-0].

For the polarization runs the source- beta detector distance was 1 cm

20
with a source radius of 0.125 inches and a detector radius of 0.219

inches - a geometry too poor to permit the use of available correction

formulae. A Monte Carlo computer calculation for this geometry gave

the attenuation coefficient 0.925 for the P, term and 0.788 for the P„

term. Application of the Rose formula to the gamma detectors gave attenuation coefficients of 0.944 to 0.988, depending on the location.

All these multiplicative corrections (the reciprocals of the attenuation

coefficients) were applied in a final step.

(3) Electron backscattering [4-1 >42]

The backing for the beta source is a crystal of Nd(C H-SO. )^H 0,

whose electron backscattering properties have not been investigated

directly. Moreover, there is very little quantitative information on

the angular and energy distribution of electrons backscattered from any

material in the region of 100 to 600 kev. In view of this lack of

information, the following scheme of analysis was adopted:

a) From the available date [43»44>45>46] a composite spectrum of

backscattered electrons as a function of fraction of the incident energy

for all incident angles scattered into the detector in our geometry was

constructed,

b) The data of Bothe [43] and Seliger [47] was used to normalize

the above spectrum to the total relative backscatter expected from a

crystal of the present composition.

c) Theoretical allowed spectra for the two beta groups were constructed, and divided into energy bins. The backscatter contribution

of each energy bin in all other bins was calculated for each spectrum

21
from the data of part (b).

d) Only the upper part of each spectrum was considered in the

final analysis, such that the backscattered contribution was less than

5^ of the total in the lowest bin.

e) In the case of the P, term, the correction was applied to each

^c I + b

P channel by multiplying each {- l) by the quantity ' _ ,/^ where b w

' is the fractional contribution of the backscattered electrons to that

channel. For the Pp term the correction used was of the form (l + b).

(4.) Resolution

In this particular set of runs the resolution, (^ ) at I/2 the

137 peak maximum, of the beta detector was 30^ for 624 Cs conversion

electrons. This comparatively poor resolution meant that the counts

collected in a given channel of the pulse height analyzer at a nominal

energy were due to electrons of a range of energies.

This correction was treated quite like that of the backscattered

electrons. Each of the theoretical and backscatter spectra was divided

into bins, and the area of each bin was apportioned over neighboring

bins according to a Gaussian distribution of the correct width. The

change of resolution with energy and the absence of a low energy tail

significant for the type and range of spectra analyzed was checked by

measuring the response of the system to monoenergetic electrons from

the exit port of a beta-spectrometer. Thus, in the end, the upper part

of the experimental spectrum was reproduced from the synthesis of the

two theoretical spectra and the backscatter spectrum of each. Each

energy bin was assigned a mean energy for each of the four components,

22
based on an intensity-weighted average of original bins from which the

contributions came. The effect of contributions from the lower spectrum

were considered in evaluating the upper spectrum results.

(5) Thick-source scattering [4-8]

For angular correlations involving electrons, one usually chooses

to make the source as thin as possible. In the present case there is

the conflicting requirement that one must also be certain that the Ce

ions are included in the crystalline lattice, especially if one is to

infer the f, orientation parameter from the fp. With the specific

activity available it was not possible to prepare satisfactory sources

by dissolving some of the mother crystal and allowing it to recrystallize

with a small amount of active solution, as has been done with other materials [39]. Therefore, a crystalline layer of the order of l/lO millimeter thick was grown on the mother crystal, the thickness being somewhat uncertain because it was not known to what extent the base crystal

was dissolved. An attempt to determine the thickness by slowly removing

the active material with a microtome did not prove successful, mostly

due to difficulty in holding the crystal securely.

Calculations such as those suggested by Frankel [48] indicated

that for our approximate thickness scattering should become quite important for the lower part of the spectrum. For this reason also only

the upper part of the spectrum is considered, where for Co grown

crystalline layers and "thin" sources gave the same polarization results, namely the maximum predicted by theory.

23
(6) Analyzer dead time losses

Although the beta pulse height analyzer does not distort the

spectrum as the counting losses increase, the cold to warm intensity

ratio is in error by 5 to 15^» The losses were determined by finding

the dead-time of each channel and integrating over all channels; this

correction was applied to all data before ratios were taken.

(7) Corrections to the ^

- Y - J coincidence data

In general, the corrections were as for the non-coincidence data,

with the difference that accidental coincidences were subtracted from

all data. These coincidences were obtained by duplicating the counting

rates of the real experiment, but with the gamma detectors receiving

radiation only from a separate source of Ce . The ratio of true

coincidences to accidental coincidences was in all cases greater than

10.

When all the above corrections are applied one obtains a value of

-nVNq as a function of beta channel number as shown for the 5B0-kev

beta transition in figure (3), where the nominal energies for the first

and last points shown are 392 and 566 kev respectively. The error in

the early channels stems mainly from uncertainty in the inner spectrum

contribution and backscattering corrections and in the effect of thick

source scattering, while in the upper channels the uncertainty arises

mostly from counting statistics.

The energy dependence over this restricted region is very slight

compared to the realistic limits of error shown on the points. For

this reason the above values are averaged to a value of -1.11 ± 0.10

24
2 for N /n^ at an average energy of I.85 m c = 434- kev. The above limits

included a nominal 5% uncertainty in f. in addition to the oth«r sources

already mentioned.

Similar arguments about the lack of visible energy dependence within

the error of measurement leads to the averaging of the alignment and

polarization data to give an average value of +0.36 ± 0,10 for nVn^^.

This- value, vghile derived from a rather small experimental change compared to that due to the P. term, is still almost an order of magnitude

larger than anticipated in the § approximation and provides the basis

for the extended analysis in the next section.

The p

- Y - J runs represent a total of three hours of counting

with the nuclei oriented. This cumulated data, after correction for

accidental coincidences and the effect of the pure gamma distribution

term, yields for an average over the upper channels the experimental

quantity l/2[

^^ '

1 = +0.113 ± 0.015. The angle of or n is

warm

that between p and J. When corrected as described above and evaluated

2

for an average energy of 1.68 m c with f. =-0.373, this value gives

-0.4.2 ± 0.06 for N./Nq in the distribution function

1+ Ni/Nq p/E f, + nVNq P /e fp for the angles selected in the

experiment. A similar treatment for l/2[—

LJ^j yields a value

warm

for N^/Nq of -0.13 ± 0.15.

2.3. Interpretation of data and conclusions

In order to illustrate the magnitude of terms and indicate the type

of information that can be obtained from an approximate analysis, we

first consider the beta angular distribution for the 580-kev transition

25
in an approximation using the highest power of § multiplying each

matrix element combination in each b, (LL'). Using this procedure one

makes no assumptions about the relative size of the matrix elements,

but makes considerable error m the estimation of numerical coefficients.

The approximation does assume that the tensor rank one operator matrix

element combination §[C.M(CTXr) + CJW(ir^)] - CJi<(a^) is not appreciably less

than its largest term.

Comparing the result of this approximation to the experimental

nVNj^, it is seen that the ratio of the tensor rank two matrix element

to the grouping of tensor rank one matrix elements is as shown in

figure (4). The alternate choice for this ratio in the solution of a

quadratic expression would lead to a predominance of M(iB, .) incompatible with the spectral shape. A similar comparison of the approximation for the bp(LL')'s with the experimental N^/n^^ gives a value for

the ratio of C^M(CTXr) - ZC^iir) to 5[C^M(axr) + C^(ir)] - C^(a) as

shown in figure (5)» where the results of figure (4.) are also used.

The effectiveness of such an approximation for a given case is

necessarily checked by considering the importance of higher terms in

the § expansion. When all terms down through order zero in § are considered, the usefulness of the grouping of tensor rank one elements is

lost, and one must consider the problem at least as one in the ratio of

three matrix elements to a fourth. The present experiment then does not

yield sufficient equations to determine the three unknowns, unless one

uses the limits on the energy dependence from the N-i/No term or N^

(spectral shape measurement) as a third equation. Such a technique

26
does not lead to well-defined matrix element ratios in the present case,

but does show that for possible values of the parameters the neglected

terms in the § expansion may be larger than those used in the first

approximation.

We now consider a method of analysis almost specific for Ce

It is based on the fact that with well-defined shell model angular wave

functions one can calculate precisely the ratios of any two of the set

M(ir), M(iB..), and M(o-Xr) to the third [4.9,50], with no assunptions

about the nuclear radial functions. The ratio of the matrix element of

M(a) to any of the set, however, depends directly on the radial functions.

The correctness of our evaluation procedure depends strongly on the

firmness with which the shell model [51] predictions for the initial and

final state wave functions can be established. The cerium ground state

consists of 58 protons and 83 neutrons, and is with some certainty a

pure 2f«/p neutron state. The prasejodymium ground state with 59 protons

and 82 neutrons has most simply the proton configuration

8 1 137

(ig^/p) ^^^t^/o^ ' ^^ would be indicated by the erCsg„ ground state

5 configuration of (g^/p) ^^c/n) > but other configurations are not ruled

out. Recent discussions [52,53]» have pointed out that the configuration (97/0) ^'^t:/^) roust be considered and that (97/p) (cic/o) ^ay also

contribute. Higher seniority configurations can be expected to be much

less important. However, in the calculation of the ra t ip, s M(ir)/M(axr)

and M(iB. .)/M(CTxr) all the first mentioned configurations give exactly

Ij *v MM. the same value, and the higher seniority configurations give zero

contribution.

27
In order to calculate the nuclear matrix elements, let us define

the operators in their spherical tensor form, using the terminology of

Edmonds. These definitions are the equivalent of those of Morita and

Morita [7] and Bincer [6]. We include an i with certain operators so

that all ratios of matrix elements vgill be real. We define the

spherical tensor

m, m_ 1 " 2

~*

in which \1p^ (t) is a solid spherical harmonic, {km.lmAktM) is a

vector coupling coefficient, and B, is a spherical tensor characteristic of each operator. We also make use of the canonical transformation in the case of the operator a, in order to transform it into a

form that can be used with the non-relativistic nuclear wave functions.

In table 4 we give each matrix element with its equal in tensor

operator form.

We next use the appropriate formulae of Edmonds [50], chapter 7,

for operators in a coupled scheme, and find the reduced matrix elements

of the individual operators as shown in his chapter 5. The results of

these calculations are:

M(aXr) ^

M(iB,.) . —?—H- = -3 V 2/5 M(axr) ^ ^ -i/?

28
M(a) , J r^ u/(r)(^ +^)U.(r)dr

1 i f dr r 1 M(axr) M r 3 „ ^/ n„ / n ,

Thus the first two ratios are pure numbers while the third depends on

the radial functions of the initial and final state and their derivatives. For this reason the experimental determination of this third

ratio is deemed most interesting and we use the first two ratios, together with the value C = -1.21 C„ from the decay of the neutron [54-]>

to calculate theoretical values for our two experimental results as a

function of M(a)/M(aXr).

One must next consider the best values to use for the electron

radial wave functions. These would be solutions for the Dirac equation

for the nuclear charge distribution, evaluated at a radial distance

corresponding to the mean radius of the decaying nucleon in the correct

shell. There are available calculations by Bhalla and Rose [9] in which

l/3 -13

a spherical charge distribution of radius 1.2A ' 10 centimeters is

assumed; however, the functions are only evaluated at the same radius.

We have followed the scheme of the above authors in calculating, for an

average energy of 1.85 m c , all necessary electron radial functions for

that value of the radius and for a radius 8^ greater. It is felt that

the larger radius is probably more representative of the point of interaction, while not seriously changing the general radial solutions.

These values were then used to calculate the necessary b, (LL')'s as a

function of M(a)/M(aXr), and thus to farm the quantities N./n^ and No/Nq

which are compared with experiment in figure (6).

29
Taking the region where there is a common solution to represent

the possible values of

p ^/rV

' y

' )

' > ^^ arrive at the values shown in

A — —

table 5» where the approximate solution in which the § expansion is

considered in its three highest powers, but with the neglect of (ccz)

terms I is shown by way of contrast. While the two more precise calculations give very similar values for M(a)/M(axr) it is seen from figure 7

that it is somewhat accidental that the agreement. is so close, since a

slightly smaller radius would not give a solution at all, while the

1/3 -13

value would shift somewhat for a radius of 1.4A 10 cm. Moreover,

the shape correction factor is dependent on this choice of radius as

well as the value of M(a)/M(CTXr). Calculations of the shape factor

using the radial wave functions of Bhalla and Rose [9], but using different explicit radial values in forming the K's and M's of Bincer [6]

indicate that higher values of the nuclear radius and of the ratio of

M(a)/M(CTXr) give a more nearly allowed shape. However, more careful

execution of this rather difficult shape measurement would be of considerable interest.

3. Discussion

The experimental value of approximately -29 for the ratio

M(a)/M^Xr^ is to be compared with the values for

-r jrV(r)(^+^)U.(r)dr

"TT Q ii for possible initial and final state ^

J r\ (r)U^(r)dr

radial functions. These calculations are beyond the scope of this

discussion, but would appear to be of considerable theoretical

interest.

30
The relative contribution of a and Yc "to the first forbidden

transitions has been discussed by Pursey [55]> Ahrens and Feenberg

[56], and Rose and Osborne [49]. Pursey adopts the single particle

model and a specific nuclear Hamiltonian containing short-range twoparticle ordinary, charge exchange, and spin-orbit forces. He then derives a relation between M(a) and M(ir) by calculating the commutator

of r (times the correct i-spin operators) with the complete nuclear

Hamiltonian. By averaging over a spherical core plus one transforming

particle and assuming particular potentials, he finds:

M(a) = M{iT), where

X = (W.-wp - (M^-Mp) + (^)

- 15a"'^/^(n-z)(i-3a"^/^)

- 3(K^-KpA"^/^(l-3A"2/^).

For the case at hand X = +25. The nveasured value for this

quantity is +29.4.

Ahrens and Feenberg [56] use the commutator relation between r and

the complete nuclear Hamiltonian to establish a relation between M(a)

and M(ir), but evaluate the necessary commutators with the Coulomb and

specifically nuclear parts of the Hamiltonian without assuming a

definite nuclear potential. There is an assumption that off-diagonal

31
terms in the matrix elements of H^ and H , are small for both the

C nuclear

initial and final states or that the nuclear beta decay matrix element

is relatively small for states other than the specified initial and final

ones. The specifically nuclear contribution is evaluated from the semiempirical energy formula, with reference to the treatment of the nucleus

as a degenerate gas of free particles to apportion the terms between potential and kinetic energy. Under the circumstances they find

M(a)/M(i^) - ^ A» where A = 1 + (Wq-2.5)A-'-/^/z for ^ emission. For

Ce , this would give M(a)/M(ir) aa 12 instead of the measured 29.

Ahrens and Feenberg point out that if they make the assumption of only

short range, two-particle nuclear interactions (equivalent in effect to

the Hamiltonian of Pursey and contrary to their first evaluation), they

l/3 would calculate A ?« 2. A + (W^-2.5)A ' Z. This would give a value of

about 29 for Ce-"-"^-^.

Rose and Osborne [49], in addition to deriving the ratio of radial

integrals that we have calculated for M(a)/M(ir), also use a manipulation of the radial equations to convert the numerator into an integral

of the product of the initial and final radial wave functions plus

another integral involving the nuclear potential.

The latter integral is taken to have contributions from only

Coulomb and spin-orbit terms. In general, they predict

M(a)/M(ir) f« 12 to 37 for the present circumstances. In their derivation they also make assumptions about the commutation properties of the

nuclear potentials.

32
In all the above calculations the arguments are of a somewhat

general nature, with a possibility of some variation in specific cases.

For this reason other measurements of the relative contribution of the

relativistic operators are important. One other ratio which has been

measured and analyzed in a straight-forward manner is that of the ratio

M(Yr)/M(ia.r) in the O" ^

0"*" transition of Pr^'^ [10,57]. Although

? «•• turn there is some question as to the sign of the ratio, the absolute magnitude seems to be considerably lower than that which would be inferred

from the present tensor rank one analogue.

4-. Summary

In the present experiment, the use of the nuclear orientation technique permits a significant matrix element ratio measurement for a

particularly interesting but otherwise rather inaccessible transition.

While the experimental techniques are difficult. and the final value of

-29.4 for M(a)/M(aXr) may change by five to ten percent when the correct

hyperfine constants are known, the accuracy is still sufficient to provide a good check for the general expression discussed in section 3, and

to serve as a goal in the determination of radial shell model wave

functions.

5. Acknowledgments

I wish to thank R. S. Kaeser for much of the cryogenic design and

assistance with the experiments; Dr. R« W. Hayward for his assistance

in the interpretation of the results; Dr. E. Ambler and Dr. R. P. Hudson

for much advice on the nuclear orientation aspects of the problem and

for the use of their equipment; and the members of the Radioactivity

Section and the Cryogenic Physics Section who have assisted me in many

ways

.

I also wish to thank Dr. F. L, Talbott for directing this thesis and

Dr. J. G. Brennan for advice on the theoretical aspects of the problem.

33
References

[l] E. J. Konopinski, Annual Reviews of Nuclear Sciencej edited by

E. Segre and L. I. Schiff (Annual Reviews, Inc., Palo Alto,

Calif., 1959) 1, p. 99.

[2] M. Deutsch and 0. ^Kofoed-Hansen, Experimental Nuclear Physics,

edited by E. Segre (Wiley, New York, 1959) 1, p. 427.

[3] M. E. Rose, Handbook of Physics, edited by E. U. Condon and

H. Odishaw (McGraw-Hill, New York, 1958) p. 9-90.

[4] S. Devons and L.J.B. Goldfarb, Encyclopedia of Physics, edited

by S. Flugge (Springer-Verlag, Berlin, 1957) ^2, p. 362.

[5] K. Alder, B. Stetch, and A. Winther, Phys. Rev. 107, 728 (1957)

[6] A. Bincer, Phys. Rev. 112, 244 (1958).

[7] M. Morita and R. S. Morita, Phys. Rev. 109 , 20^8 (1958).

[8] A. Bincer, Op. Cit., p. 248.

[9] C. P. Bhalla and M. E. Rose, ORNL-2954j Oct. 3, I960

(unpublished)

.

;iO] F. T. Porter and P. P. Day, Phys. Rev. 114 , 1286 (1959).

11] Bincer, Church, and Weneser, Phys. Rev. Letters 1., 95 (1958).

12] G. Hartwig and H. Schopper, Phys. Rev. Letters 4, 293 (i960).

;i3] R. M. Steffen, Phys. Rev. Letters Aj 290 (i960).

;i4] T. Kotani, Phys. Rev. 114 , 795 (1959).

;i5] Z. Matumoto, Progr. Thepret. Phys. (Kyoto) 2^, 643 (i960).

16] T. Kotani and M. Ross, Progr, Theoret. Phys. (Kyoto) 20, 643

(1958).

17] M. Morita and R. S. Morita, Phys. Rev. 110, 461 (1958).

18] A. Z. Dolginov, Nuclear Phys. 7, 591 (1958).

19] E. Ambler, Progress in Cryogenics, edited by K. Mendelsson,

(Heywood, London, I960) 2^ p. 235.

3^
[20] Cacho, Grace, Johnson, Knipper, Scurlock, and Taylor, Phil.

Mag. /i^) 1287 (1955). A private communication from C. E.

Johnson to E. Ambler gives a revised gamma anisotropy versus

temperature curve based on subsequent measurements relating

the magnetic and absolute temperatures. [H. Meyer, Phil. Mag.

2, 521 (1957)].

Bogle, Cooke, and Whitley, Proc. Phys. Soc. (London) A64., 931

(1951).

R. J. Elliot and K. W. H. Stevens, Proc. Roy. Soc. A215 . 437

(1952); A^l^, 387 (1953).

B. Bleany, Proc. Phys. Soc. (London) A68, 937 (1955).

B. R. Judd and I. P. K. Lindgren, Theory of Zeeman Effect in

the Ground State, UCRL-9188, May 9, 1960 (unpublished).

Kedzie, Abraham, and Jeffries, Phys. Rev. 108, % (1957).

Haag, Johnson, Shirley, and Templeton, Phys. Rev. 121 t 591 (l96l)

H. A. Tolhoek and J. A. M. Cox, Physica 1^^, 101 (l953).

M. S. Preedman and D. W. Engelkemier, Phys. Rev. 7^, 897 (l950).

J. T. Jones and E. N. Jensen, Phys. Rev. 27, 1031 (1955).

Joshi, Subba Rao, and Thosar, II Nuovo Cimento % 601 (1958).

E. Kondiah, Arkiv fur Fysik 4, 81 (l95l).

R. L. Robinson and L. M. Langer, Phys. Rev. 109 , 1255 (1957).

J. E. Mack, Rev. Modern Phys. 2£, 64 (1950).

H. Lew, Phys. Rev. 21, 619 (1953).

See, for example. Chapters 5» 7, and 9, Beta and Gamma Ray

Spectroscopy, edited by K, Siegbahn (interscience. New York,

1955).

R. W. Hayward and D. D. Hoppes, IRE Trans, on Nuclear Science

(IRE, New York, 1958) NS5 , No. 3.

H. Postma, Thesis, Groningen (i960).

35
[38] Wu, Ambler, Hayward, Hoppes, and Hudson, Phys. Rev. 105 % 14-13

(1957); Phys. Rev. 106, 1361 (1957).

[39] Ambler, Hayward, Hoppes, and Hudson, Phys. Rev. 108, 503 (1957);

Phys. Rev. 110, 787 (1958).

[4.0] M. E. Rose, Phys. Rev. £1, 610 (1953).

[4,1] R. D. Birkhoff, Encyclopedia of Physics, edited by S. Flugge

(Springer-Verlag, Berlin, 1958) ^, 53.

[4.2] W. Paul and H. Steinwedel, Beta and Gamma Ray Spectroscopy,

edited by K, Siegbahn (interscience. New York, 1955).

[4.3] W. Bothe, Z. Naturf. ^a, 54-2 (194-9).

[44-] H. Frank, Z. Naturf. Ua , 247 (1959).

[4.5] H. Kulenkampf and K. Ruttiger, Z. Phys. 152, 249 (1958).

[46] H. Kanter, Annalen der Phys ik 20, 144 (1957).

[47] H. Seliger, Phys. Rev. 88, 4O8 (l952).

[48] S. Frankel, Phys. Rev. 8^, 673 (1951 ).

[49] M. E. Rose and R. K. Osborne, Phys. Rev. 21, 1326 (1954).

[50] A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton

Univ. Press, Princeton, 1957) Chapters 5 and 7.

[51] M. G. Mayer and J. H. D. Jensen, Elementary Theory of Nuclear

Shell_ Structure (Wiley, New York, 1955).

[52] Noya, Arima, and Horie, Pror. Theoret. Phys. (Kyoto)

Supplement to No. 8, 33 (1959).

[53] N. Zeldes, Nuclear Physics 2, 1 (1956/1957).

[54] 0. C. Kistner and B. M. Rustad, Phys. Rev. 114, 1329 (1959).

[55] D. L. Pursey, Phil. Mag. 42, 1193 (1951 ).

[56] T. Ahrens and E. Feenberg, Phys. Rev. 86, 64 (1952).

[57] Graham, Geiger, and Eastwood, Can. J. Phys. i6, 108 (1958).

36
Table 1. The nature of the operators

in f irst-forbidcien beta decay

(The quotation marks indicate that the 'femall" components

of the relativistic wave functions must be used.)

Operator Tensor Rank Selection Rule

Lepton

Angular Momentum

Orbital or Resultant

Y5

a»r

a

r

aXr

hi

1

1

1

2

A(J.J^O)

A(J^J^O)

A(J^Jfl)

A(J^J^l)

A(J^Jfl)

A(J^Jf2)

T

T

r

T

T

T

37
Table 2. The pertinent b {L,L ) 's of Bincer

n

K^''^ = ^/2 ^^

{ [c^V(axr) + 2C^C^MW)M(ir) + C^V(ir)]K^^

-[2/3 qC^^ M^(axr)-2/3 qC^^^(ir) + 2C^C^(axr)M(a)

+ 2C..^(ir)M(a) M

1-1

[1/2 C^^^(£xr) - 2C^C^(axr)M(i^) + 2C^^^(ir)]K_2_,

1/6 q^C, V(axr) + l/3 q^C..V(ir) + 2/3 qC,C.JVl(axr)M(a)

- 2/3 qC^^M(ir)M(a) + C^^^Co^) -1-1

bo(22) = 3/8^ C^V(iB,.)[K.2.2 + 1/9 q2 K.^.J

38
Table 2. The pertinent b (L,L )'s of Bincer (Contd)

- [2/3 qC^V(oyr)-2/3 qC,,V(ir) + 2C,,C.M(CTxr)M(a)

+ [c^V(o;xr) - C^C^M(ir)M(oxr) - 2C^V(ir^)] M^^^

- [1/3 qC^VCaxr) - qC^C^M(ir)M(o><r) + 2/3 qCyV(ir)

1^11

+

^A^/Si^i.^^^^'^ " ^^M^^^l}^[2;}'] ^-1-:

+ [1/12 q\V{axr) - 1/3 CyC^q^(ir}M(axr)

+ 2/3 q C^S^(2><r)M(a) - 2/3 q C^^(i£)M(a)

39
Table 2. The pertinent b (L,L ) 's of Bincer (Contd)

bj(l2) - 1/2/0/3 p|^{|;-C^2M(crxr)M(iB..)-C^C^(ir)M(iB..)]M^.,

+ [9/3 C,^(axr)MCiB, .) - 9/3 C,C.*l{ir)M(iB, .)

+ 1/10 [c^2^(o;xr)M(iB^.) - 2 C^C^(ir)M(iB. .)] M_ 2-2

- 1/18 [q^C^^(oxr)M(iB^j) + 2q^C^C^(i£)M(iB^^ )] M_j^_^|

^1^22) - -3/8

^

{c>2(,3^_)^3/3 ^_^^ ^ ,/g ^2

^_^_J |

40
Table 2. The pertinent b (L,L )'s of Bincer (Contd)

b,(ii) =

Re

(F(Z,E) C,V(orxr) - C,C,iA(ir)M(a><r) - 2C..^^(ir) 1-2

+ 2/3 qC^V(ir) + C^C^(^^)M(a) " 2C^^(ir)M(a)]

- flA C, V(oxr) - C,C.14(ir)M((P<r) + C.,V(ir) -2-2

"2(12) = //l°?fe{[cA'w(f<i)«(iBij) +C^C^M(ii)M{iB..); 1-2

- [1/3 qC^\{aXT)Ni{iB^.) - 1/3 qC^C^(ir^)M(iB^^)

+ [1/2 C^^{aXT)M{iB^.) - C^C^(i^)M(iB.j) -2-2

b2{22) = 21/40 j^ C^a^) K.2.,

41
Table 3. The approximate magnitude

of the b (L,l') 's n

H

<'''.

Parameter Highest power of ^ RelativeMagnitude

bo(ll) 2 100

bQ(22) 1

b^(ll) 2 100

b^(l2) 1 10

b^ (22) 1

b,(n) 1 10

b2(l2) 1 10

b2(22) 1

b3(22) 1

^2
M

Table 4. Matrix Elements in Spherical

Tensor Form^

(li) =^n2^ (J,LjSf||T^^(l.r)l|j.4S,)

i

"(«) = ^ajjr (JfVfll^io'-^'i^llJiSSi)

o
Table 5. M(a)/M(aXr) for three evaluations

Wave Function Employed ITfmc)

"Exact", R i: 1.3 A-"-/-^ 10"-^"^cm -29./, ±1.-4

"Exact", R ± 1.2 A^/^ 10"^-^cm -29.5 ± 2.3

"Approxima te", R = 1.3 A^/^ 10"^-^cm -36.6 ± 1.0

U
If)

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y

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LIGHT PIPE

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GAMMA DETECTOR

Figure 2. Experimerital apparatus.
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(0c

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(TJOM

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aramete

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IIo (U Z o 3 \ — • CD z^ T •H
+1.0

CAm(C7xr)

Figure 6. A comparison of the experimental coefficients with the

predictions of an "exact" analysis in which the ratio

M(a)/^(axr) is the only free parameter. The solid curve

is for a" nuclear radius of 1.3 x A^/^ x 10"^^ cm; the

dotted line is for a radius 8^ smaller.
U. S. UKPAKTMKNT OF C0MMP:K(

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

rhe scope of activities of the National Bureau of Standards at its major laboratories in Washington, U.C, and

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