P A P E R 2

Symmetry Property of the Electric Quadrupole Perturbation Function in Gamma Ray Angular Correlations

S C o n n e l l , K B h a r u t h - R a m , h A p i ’e l , JPF S e l l s c h o p , a n d M S t e m m e t .

P A P E R 2

Symmetry Property of the Electric Quadrupole Perturbation Function in Gamma Ray Angular Correlations

S CONNELL, K BHARUTH-RAM, II APPEL, J P F SELLSCHOP, AND M STEMMET. i

86

9^

Symmetry Property of the Electric Quadrupole Perturbation Function in Gamma Ray Angular Correlations

S C o n n e l l , K B h a r u t h - R a m + , H a p p e l ° , J P F S e l l s c h o p , a n d M S t e m m e t .

Schonland Research Centre for Nuclear Sciences,

University of the Wituiatersrand,

P 0 Wits SO50, Johannesburg, South Africa.

The experimental measurement of perturbed angular correlations (PAC) of

radiation emitted from probe nucle: implanted in a host lattice is making an

increasing impact as one of the major analytical techniques using hyperfine

interactions. Most recent measurements have involved single crystal samples

which are more informative than polycrystal ones, and the perturbing fields

have not necessarily been axially symmetric, in these cases the theoretical

treatment of the data is necessarily more complex1. We have investigated

the periurbation function applicable in the case of the hyperfine interaction

of probe ions implanted in a host matrix. It is shown that in the general

case of Tf - 7 correlations involving static electric quadrupole interactions

the perturbation function can always be expressed as a sum of cosine terms,

without having to make any assumption about the axial symmetry of the

perturbing field or employing the averaging procedure used in the case of

polycrystalline hosts. Though many have used this fact, the formal justification for it in this most general case has not been presented before.

* Physics Department, University of Durban-Westville, P Bag X54001, Durban 4000,

South Africa and Honorary Research Associate of the Univcrfi'.y of the Witwatersrand.

"Visiting Ablett Fellow from the Universitat Karlsruhe, Postfach 3640, D 7500 Karl­

sruhe, Federal Republic of Germany.

; 87

The study of the perturbed angular distribution of gamma rays emitted by probe

nuclei implanted in host material has long been used to obtain information on the

microscopic structure of crystals by investigating the hyperfine interaction between the

probe nucleus and the surrounding field configurations. This information is extracted

by analysing the frequencies which become superimposed on the normal decay spectrum

of the probe nuclei. Central to this analysis is expressing the perturbation aa a sum

of oscillatory cosine terms. While this procedure has been widely used a search of the

literature reveals that it has been justified only for polycrystalline targets2,8 and axially

symmetric local electric field gradients3,4. The formulation for non-axially symmetric

efg’s has been worked out by Rots et al5. However their treatment of the oscillatory term

in the perturbation follows tha t of Lewis®, who considers the efg of an axially symmetric

crystal with the axis of quantization parallel to the symmetry axis of th t crystal. It is

not immediately obvious tha t this reduction of the perturbation expression is valid for

non-axially symmetric efg’s in single crystal targets where the averaging procedure used

for polycrystals cannot be employed. In this report we show tha t for the perturbation

of 7 7 correlations due to static electric quadrupole interactions in general, we need

consider only the real part of the exponential term in the perturbation function.

The directional correlation function for a double 7 -ray cascade which includes a time

dependent perturbation factor G(t) describing the effect of the host lattice environment on the decaying ions is2

= £ 4b.(l).A »,(2)C jfo**(«)[(». + 1)(2*, + 1 )] '<*i<k,N,N,

where

x < mb|A(t)|ma > < m{,|A(0|mJ, >* . (2)

*2 mj, N 2

The relevant angles and directions are shown in Fig. 1 . The coefficients >1*,

t\k e into account the initial alignment of the intermediate state, and the spins and

multipolarities of the levels and the transitions involved. The interaction induces time

88

m s

dependent transitions among the magnetic substates of the intermediate level as given by

the matrix elements where A(f) is the time evolution operator of the substates. This may

be seen as producing a periodic ‘re-orientation effect* depicted by the 3- j coefficients,

giving rise to oscillations modulating the correlation with frequencies proportional to

the eigenvalue difference for the transition. Being a physically measurable quantity, the

correlation expression is of course real.

Fig. 1 . Relevant angles and directions in the correlation expressi

For a static interaction the time evolution operator is given by

A(t) - exp [ ~ i H t / h )

where / / is the static interaction Hamiltonian. As frequencies are the observables, the

procedure is to separate the frequency term of the perturbation from the other terms

of the correlation expression by diagonalising the interaction.

a ( 0 = u - l r iBt^ u (4)

where U is the unitary matrix diagonalising H, and E is the diagonal matrix containing the eigenvalues E n .

U H U ~ l = E. (5 )

89

G

It is now possible to expand the matrix elements of A(t)

< m fc|A(t)|fn0 > = ^ < n |mb >* e ~*E»t / K < n |m a > (6 )n

and G'(t) may be rewritten

! £ * { o =* y , ( - i ) 8u "“ +m‘ [(2* i + i ) (2 fca + 1)]* ( \ 1 t i ) ( 7, 1 „\ mo ~ m« N l J \ m b ~ m b N 2

nnx < n |m b > • < n |m a > < n'|m£ > < >* e- ^ E’>-E»')t/ h . (7)

This expression holds for any static interaction involving classical fields and is in

general complex2,7. In the case of polycrystal targets the perturbation has the same

form as equation (7) except that Nj = N? — N and fcj = = it due to the averaging

procedure over all directions of the microcrystals2,8’7. In the case of axial symmetry

H is already diagonal2. The procedure justifying the exclusion of the sine terms in the

exponentip.I has already been given for the latter two special cases of the more general

expression (7)2,3,4. What follows is the the formal justification for the same exclusion

in the more general case, as we seek an expression of the form

nn'

where the (n ,n ') coefficients and the frequencies are defined implicitly byequation (7).

Physically this should be the case due to the conservation of parity and the time

reversal symmetry of the electromagnetic interaction. While such arguments may be

used to justify dropping the sine part of the exponential in equation (7), a more

rigorous argument would be to demonstrate from the formalism that equation (8 ) is

correct. To achieve this we make use of the specific symmetry properties of the electric

quadrupole interaction. The non-vanishing matrix elements of this interaction in the

m-representation are:

< m \ H \ m > — [3m2 - / ( / + 1)] • h u q

< m ± 2 \H \m > = ^ [(/ T "»)(/ ? m - 1) ( / i m + l ) ( / ± m + 2 ) ] ^ h u q rt ^

90

where

(10)

is the quadrupole coupling constant, and electric field gradient (efg) asym m etry param ­

eter is d e f in e d ks

It is clear tha t the same holdr F r̂ the m» trix elements of A(t). The effect of these

symmetries on the perturbation function of equation (2 ) ra". be seen by -labelling

m a *-* - m ' a , mfc ♦-* -m j,, interchanging '.he first two columns of the 3- j coefficients a rd using the property (12). Thus

Due to the triangle rule for the 3 - j coefficients and the non-vanishing matrix

elements of the electric quadrupole interaction, both N i + N ? and J- k j are always

even. Therefore,

nn'

and equation (8 ) follows. Note that equation (14) is interpreted classically as the nuclear

spins processing clockwise and anti-clockwise simultaneously.

A more instructive form is achieved if the sum over n ,n ' is broken up into groups with the same frequency

n = ( v „ - v yv) / v „ .(ii)

for the principal axis system of the efg. From t. is it can be shown tha t the matrix

elements have the following symmetries:

< m bj / / |m tt > = < m a \H\mb >

— < -mj,j//| - m Q > (12)

_ Vma -"*• " I / ' x < m b|A ^(0 |m a > < m{,|A^(t)|m^ >*

*2N i

(13)

(14)

(15)

91

where the prime on the summation indicates tha t it is restricted to values of n and n'

leading to the same frequency and the integer p labels the distinct frequencies. Hence

W ( k i , k 2; f ) = Y i X*i(lM *>(2 )S*!fc,pCO8(0pu;oO (16)

p

where gf, is defined implicitly by

( E n — E n ') / h = u/nn<

* 9v u o (17)

and wo is the smallest non-vanishing frequency related to the quadrupole frequency uq by

(3u>q, for integer I 6 u > g , for half-integer I.<*>o

The may be regarded as Fourier coefficients in a decomposition of the

perturbation, although this is not a proper Fourier transform as the frequencies are

not necessarily integral multiples of a fundamental frequency.

It is not immediately obvious why the imaginary part of the exponential in the per­

turbation expression can be neglected, aa the correlation contains other complcx terms,

viz., the spherical harmonics and, in some cases, the perturbation itself. Equation (14)

is thus the formal justification for considering only the real part of the oscillatory expo­

nential term in the perturbation expression for static electric quadrupole interactions

in general, and ro t only in the special case of axially symmetric perturbing fields or of

a polycrystalline host matrix.

The authors record with appreciation the support for the Foundation of Research

Development and Messrs de Beers Industrial Diamonds (Pty) Ltd. Discussions with V

Hnizdo of the Department of Physics, University of the Witwatersrand are acknowledged

with appreciation.

R e f e r e n c e s

[1] D Wegner, Hyp. Int. 23 (1985) 179.

92

[2] H Frauenfelder and RM Steffen, in: Alpha-, Be ta - and Gamma-Ray Spectroscopy,

Vol. 2, ed. K Siegbahn (North-Holland, Amsterdam, 1965)p. 997.

[3] RM Steffan and K Alder, in: The Electromagnetic Interaction in Nuclear Spec­

troscopyr, ed. WD Hamilton (North-Holland, Amsterdam, 1976>p. 583.

[4] K Alder. H Albers-Schonberg, E Heer and TB Novcy, Helv. Phys. Acta 26(1953)761.

[5] M Rots, J Claes and F Namavar, Hyp. Int. 7(i979) 323.

[6 ] RR Lewis, Phys. Rev. 163 (1967) 935.

[7] RM Steffen and H Frauenfelder, in: Perturbed Angular Correlations, ed. E Karlsson,

E Matthias and K Siegbahn (North-Holland, Amsterdam, 1964)p. 10.

K B h a r u t h - R a m , S C o n n e l l , JPF S e l l s c h o p , M S t e m m e t

a n d H A p p e l .

The Electric Quadiupoie Interaction at Sites of *®F Implants inthe four characteristic types of natural Diamond

The Electric Quadrupole Interaction at Sites o / 10F Implants inthe four characteristic types of natural Diamond

K B h a r u t h - R a m + , S C o n n e l l , J P F S e l l s c h o p , M S t e m m e t

AND H APPEL°.

Wits-CSIR Schonland Research Centre for Nuclear Sciences, University of the Witwa-

tersrand, P O Wits 2050, Johannesburg, South Africa.

The electric quadrupole interaction at the residence sites of l0F nuclei in

various diamond samples (types la,lb,IIa and lib) has been investigated by

in-beam TDPAD measurements. The results of t \e uis to the spin rotation

R(t) spectra may be classified into two groups: diamond types lib and lb

display a characteristic single crystal /?(*) spectrum with two unique residence

sites with efg’B parallel to the < 111 > crystal direction; diamond types Ila and

la display a mixture of single crystal and polycrystaliine characteristics. The

efg parameters for the different diamond types are compared and discussed in

the light of crystalline and mosaic structures, and impurities in the diamonds.

^Physics Department, University of Durban-Westville, P Bag X54001, Durban 4000,

South Africa and Honorary Research Associate of the University of the Witwatersrand.

°Visiting Ablett Fellow from the Universitat Karlsruhe, Postfach 3640, D 7500 Karl­

sruhe, Federal Republic of Germany.

95

l . I n t r o d u C t i o n

The electric quadrupole interaction at the residence sites of probe nuclei and at

impurities in diamond has been the subject of several investigationsl ’3,3,4. While Appel

et a l l and Raudies et al2 investigated the residence sites taken up by the probe nuclei by

means of Time Dependant Perturbed Angular Correlation (TDPAC) measurements with

implanted radioactive 111 In and 1,1 Hf ions, our earlier investigations were carried out

by exciting and recoil implanting 1®F nuclei into the host matrix in a (p,p') reaction5,6.

Our study of the 10F implanted in type lib diamond6 showed th a t the implants occupy

two residence sites, which are characterized by unique electric field gradients (efg’s) with

quadrupole coupling constants i/qi = 59 MHz and — 25 MHz. In similar Time

Dependant Perturbed Angular Distribution (TDPAD) measurements on 10F implanted

in Ge and in crystalline and amorphous Si hosts Bonde Nielsen et al. ®’7 also observe two

residence sites for the implants with efg’s having unique quadrupole coupling constants.

Further, their analysis of the Si data indicates that the 35 MHz componen* may be

assigned to a specific crystalline site, and the 23 MHz component to a polycrystalline

or amorphous structure around the 10F implants.

Our current investigations were undertaken, firstly to determine whether the

residence sites of the l0F implants in the four different natural diamond types retain

their crystalline structure after recoil implantation, and secondly, to determine how the

quadrupole coupling frequencies of the 10F implants might vary in each of the different

types of natural diamond hosts. Hence a comprehensive study was made of the electric

field gradient:; in diamond types Ia,Ib, Ila and lib by in-beam TDPAD measurements

in which t \* (p,p’) reaction was used to excite and recoil implant l0F probe nuclei into

the host an nplef.

2 . E x p e r i m e n t a l P r o c e d u r e

The targets used in this study consisted of a 30 ng cm - 2 layer of C aF 2 evaporated

onto the surface of thin wafers of the four diamond host samples under investigation.

The diamond wafers were 5mm wide x 10mm long x 2 mm thick, cut and polished very

close to the < 111 > plane. The 10F nuclei were excited via the l0F (p ,p ')10F reaction

and recoil implanted into the host matrices by a 4 MeV proton beam. A puhed proton

beam of 500 ns repetition period and cs 3 ns pulse width and average intensity 3 nA

was employed. Delayed time spectra of the 197 keV -y-rays from the 10F, §* —►

transition were observed with the start signals being obtained from two 2.5 cm x 4.0 cm

Nal(Tl) dctectors positioned at 0 ° and 90" to the beam direction, and the stop signal

from a fast plastic scintillator ‘halo’ detector positioned 2m upstream from the f

In order to generate the spin rotation spectra the detector angles were intercL

alternate runs.

Since the TDPAD spectrum is strongly dependent on target orientation for single

crystal targets, a special target orientation was chosen for these experiments. The beam

and detector directions were both very close to the < 111 > crystallographic axes in the

crystals. This geometry5 is both easy to achieve experimentally, and ensures a maximum

qualitative difference in the TDPAD spectra for the polycrystalline and single crystal cases (see next section).

3 . D ata E v a l u a t i o n

The observed normalized count ratet> N(0°,f) and N ( 9 0 r‘, t ) were used to generate

the experimental spin rotation spectra, which are defined by the function

R M - ? * ( 0 O, * ) - * ( « 0 ° , 0 ' xp 1 N ( 0 ‘\ t ) + 2 N ( 9 0 ‘\ t) (1)

The theoretical function is

R <•' - - w y . v . w . t )' h U + W

The function in equation (2) is the angular correlation function

W(k i , k a; £) of ref. (8 ) but has been redefined in ref. (5) to show the explicit dependence

on the angles (t?,£>) (ie. the polar and azimuthal angles of the efg relative to the detector

plane), and 0 , the angle between the propagation vectors k t and ka for the beam and

detector directions respectively.

In the case of a single crystal host matrix, equation (2 ) becomes5

< % ,(* .*> ;<) = - C & J t . v . O O * ; ! ) . (4)

The and G ^ k] are respectively the A t coeffic:ents and the attenuation factors

defined in ref. (5). In practice there may be more then one distinct residence site for

the probe ions. Then

A *ki*. /(*» * ) ' » ' * » < • 1w0<5:0.ki.kt

where /» is the fraction of implants at the site », and f ( o , 6i) is an amplitude attenuation

factor that takes account of the finite time resolution a of the detection system and a

possible spread £ for the efg at that site. The frequency u><k is related to the quadrupole

coupling constant of the nuclear quadrupole moment Q with the efg at the implant site

l/qi —

where the diagonalized efg tensor has components jV*t | > |V*v | > |V,’, | and the efg

asymmetry parameter is defined as

i = w - - y M ) / y *

For a truly polycrystalline host matrix, summation over all the randomly oriented

microcrystals means that the direction of the efg principal axis is not observed, and the

theoretical 6pin rotation function reduces to

(»)

Thus each site is characterized by the fraction f i of the probe ions occupying the

site, the frequency (*/(*, asymmetry parameter rj* and the spread 6i of the efg at that

site. In the case of a single crystal, the crystallographic orientation of the efg derived

through the angles 0 and <p is an additional parameter.

98

The main deference between theoretical /2(f) spectra for the polycrystalline case

and single crystal cases for various crystal orientations is manifested in different

proportional contributions from the harmonics of the quadrupole in* 'c tion frequency.

Our simulations5 have shown that a geometry where the beam and detector directions

are very close to the < 111 > crystallographic axes yields a marked difference in

the contribution expected from the second harmonic for the polycrystalline and single

crystal cases. Figs. 1(a) and 1(b) display the Fourier transforms of calculated R(t)

spectra for 10F ions impi^nted in di<unund at i/, = 60 MHz, rj = 0 for a polycrystaliine

r*nd a single crystal host, respectively. In the polycrystalline case there is a large

contribution from the second harmonic relative to the substantially smaller contribution

in the single crystal case for this geometry. We have used this differing contribution

from the second harmonic to extract qualitative information on the extent to which

the residence sites for the l0F probe ions are characterized by single crystal and/or polycrystaliine features.

The data analysis thus proceeds by least dquares fitting of the experimental spin

rotation spectra with the theoretical functions of equation (5) or (9). The fits were

achieved with the aid of the CERN code MINUiT®.

4 . R e s u l t s

The experimental spin rotation spectra R(t) and the theoretical fits are displayed

in Figs. 2(a)-(d). Table 1 lists the efg parameters determined from the analysis. The

results may be classified into two groups whose characteristics are now presented.

I. Single c ry s ta l b e h a v io u r . The diamond types lib and lb show unambiguous single

crystal TDPAD patterns. Thus the minimum \ 7 was obtained using a single crystal

simulation for the population of two residence sites by the 10F implants, each with

unique electric field gradients. The principal (highest population) site, characterized by

a quadrupole coupling frequency of 58 MHz, was occupied by 50% of the 10F implants.

The principal axis of the efg for this single crystal site has been well established to be

along a < 111 > crystal direction5. In addition this site has very little distribution in

the quadrupole frequency (i es 7%) and the electric field gradient is almost symmetrical

[r] is small), as is expected for a site on the < 111 > crystallographic axis. The second

L.<u*oCL

L-0)L.3Ou.

FLRfQUtNCv (GH l)

(RfOUtNCV (GH/)

Fig 1. Fourier transforms of the theoretical R(t) spectra for a quadrupole coupling

frequency i/q = 60 MHz : (a) single crystal host (b) polycrystalline host.

residence site for the implants, with a lower quadrupole coupling frequency of about

15 MHz, is characterized by large values of 6 , 60% for the lib and 45% for the lb

100

* . /»„ PI n

w Vi

T ab ic 1. E v a l u a t e d Parame te rs

Diamond 1 e f * Site f Vq 5 n X2Type | «.nnn> ( # ) (%) (M Hz ) ( %) (%)

l i b i <111> 1 49(3) 58(2) 7(1) 20(2) .70| <I11> 2 19(1) 14(1) 59(1) 1.0(4)

! b | <!11> 1 42(2) 58(2) 7(1) 8(1) 1.5I <1U> 4b 39(2) 15(2) 45( 2 ) 9(1)

la | <111> 1 48(2) 58(2) 7(1) 16(2) .761 < m > 2 13(1) 25(2) 31(2) 5(1)1 <U1> 3 11(1) 111(3) 7(1) 2(1)

11 a I <1U> 1 55(2) 60(2) 8(1) 8(1) .51l < m > 2 37(2) 40(2) 89(2) 9(1)1 < i n > 3 28(1) 114(3) 4(1) 3(1)

respectively and is again consistent with a single crystal site with an efg alignment along the < 111 > direction. •

II. M ix ed single c r y s ta l a n d p o ly c ry s ta l l in e b e h a v io u r . The spin rotation

spectra of diamond types Ila and la were found to be characterized by a Mixture of

single crystal and polycrystalline features. A purely single crystal analysis yielded three

fractions at unique sites. The principal site was clearly single crystal in nature, and

identical to that in diamond types I Ib and Ib. The second site, of lower frequency, was

a diffuse site. For diamond type la only 13% of the l0F implants took up this sitv* which

had a distribution in quadrupole coupling frequencies of 6 = 31% around a mean value

i/q = 25 MHz. For diamond type Ila a larger fraction took up the diffuse site, for which the average i/q was higher (40 MHz) and 6 much larger (90%).

The third frequency component (« 112 MHz) for these two samples is almost

exactly twice the quadrupole frequency of the principal site. In the geometry used for

this experiment, the relative size of the second harmonic of the quadrupole frequency

for single crystals is nearly at a minimum when compared to the polycrystaUine case,

101

' * s n - .

Author Connell S H Name of thesis Internal Fields in Diamond and Related Materials 1988

PUBLISHER: University of the Witwatersrand, Johannesburg

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