Muonium substituted organic free radicals in liquids. Theory and analysis of μSR spectra

Chemical Physics 54 (1981) 261-276 North-Holland Publishing Company

MUONIUM SUBSTITUTED ORGANIC FREE RADICALS IN LIQUIDS. THEORY AND ANALYSIS OF pSR SPECTRA

Emil RODUNER and Harms FISCHER P?iysikolisch-Chemisc;!es

Inrtitur der

UniversitZt,

CH-8057

Switzerland

Zkich,

Received 14 July 1980

Muonium substituted free radicals are formed when spin polarized positive muons are stopped in liquid unsaturated organic compounds. They are observed by muon spin rotation (_rrSR), i.e. via the time evolution of the muon spin polarization caused by Zeeman and hypertke interactions. A theoreticd treatment of ySR spectra of muonic radicals in zero, intermediate and high external magnetic fields is given. Its predictions are verified by observations on radicals derived from tetramethylethylene and benzene. The relation of PSR to magnetic resonance techniques is discussed.

1. introduction The positive muon 01’) is now used extensively to probe the structure of matter and magnetic interactions [I]. The experimental technique, muon spin rotation (@R), rests on the following basic principles [2] : Spin polarized positive muons (properties, see table 1) are stopped in a target of the material to be investigated. The spins then undergo precession in an external magnetic field B and/or in internal fields arising for instance from magnetic nuclei and unpaired e]ectrons. Independent of their environments the muons decay, emitting positrons preferentially along the instantaneous spin directions. The time-dependence of the spin polarization can thus be detected in the form of tirnedifferential histograms of the number of decay positrons counted in a given telescope direction against the time elapsed since the stops of the corresponding muons. The uSR histograms are radioactive decay curves modulated by the muon precession frequencies. Direct fitting to corresponding theoretical expressions or Fourier analysis reveals *Thesefrequencies, their amplitudes, phases and relaxation rates. Applications of PSR in chemistry [3-51 have so far centered around muonium (Mu), the one-electron atom p+e-. It is formed after stopping positive muons in gases, inorganic and organic liquids, semiconductors and solid insulators and can be regarded as a hydrogen isotope with a mass only 0 301-0104/81/0000-0000/$02.50

@North-Holland

l/9 that of H (see table 1). Apart from studies on the muonium formation mechanism, rate constants of its reactions with molecules have been measured. Comparison with the related data of H atoms revealed large isotope effects which are the subject of interpretation within theories of reaction rates. The possible formation of muonium substituted free radicals by addition of Mu to unsaturated molecules, for instance via Mu + CHs=CRrRa + MuH$-CRr Rs,

(1)

a well known process in H atom chemistry, had early been postulated [6]. Indirect evidence was reported in ref. [7] _However, they long eluded direct observation. The reason for this was an incorrect truncation of the radical hamiltonian which lead to inap-

Table 1 Properties

I.r+

hlu

of positive muon and muonium

mass spin magnetic moment lifetime Larmor frequency mass Bohr radius ionization potential

Publishing Company

0.11261 mass of proton 112 3.18335 proton magnetic moment 2.1971 ps 13.5538 kHz/G 0.11315massofH 0.5317 A = 1.0043a.

(H)

13.539 eV = 0.9957 I(H)

262

E. Roduner, H. Fischer / Muonium substitutedorganicfree radicals

prop&e experimental procedures. Only recently, the correct theoretical basis for the observation of muonium substituted radicals was developed. Subsequently, direct evidence for their formation was obtained [8,9]. In this paper we present a theory for the prediction and the analysis of muon precession frequencies in muonium substituted free radicals in liquids under the influence of external magnetic fields. Numerical procedures lead to exact solutions for all fie!d strengths. For the analysis of high-field @R-irpectra solutions in closed form are derived by perturbation theory. Further, analytical solutions are given for simple systems in zero magnetic field. The validity of the equations, some of which were published previously [8,10,1 l], is confirmed by experimental observations on radicals formed by muonium addition to benzene and tetramethylethylene. Subrequent papers in this series deal with the assignment of about 30 muonic radicals observed so far, isotope effects on hyperfme coupling constants, the selectivity of muonium addition to molecules, and with the determination of radical reaction rates.

2. Theory 2.1. Evolution of muon spin polarizationin multispin systems

We consider the evolution of /J* spin polarization in an environment characterized by the presence of an unpaired electron which is coupled by hyperfine interaction to the muon and to other magnetic nuclei k, and by an external magnetic field B IIz. This corresponds to the structure of a muonium substhuted radical [cf. reaction (l)]. After /I+ implantation the species shall be created in a time interval wh.ichis very short compared to the inverse Zeeman and hype&e Interaction frequencies. This ensures that the system starts at f = 0 with a muon polarization given by the beam polarization P. The evolution is observed in an arbitrary direction (I defined by the observation axis of ‘ie posltron counter telescope, i.e. the observable muon polarization is P,(~)=+I,,~)=TI{~+,},

When written out in a basis of product spin functions IXi)= Iti) lb) 7

Ix:),

(3)

i and 6%are matrices of dimension N = 4Hk(2p+ I), where I denotes the spin quantum number of nucleus k. The time evolution is governed by the equrrtion of motion j = -(i/2n) [ri, $1..

(4)

his the hamiltonian for muonium substituted radicalsin non-viscous liquids in frequency units k +A,,S*i'+~Aki*ik, k

(5)

where v,, vI1and vk are the Larmor precrssion frequencies of the electron, muon and the kth nuclear spin (v, = 2802.47 kHz/gauss for g = g, , vr = 13.5538 kHz/gaussa;;d vk = 4.258 kHz/gaussfor protons). A, andAk, the isotropic Fermi contact hyperfiie coup ling constants measure the electron spin densities at the muon and nuclear sites. As in ESR of free radicals in liquids [ 121 they represent the only couplings to be considered and should be of the order of 0 < I.AlG 500 MHz.Several methods may be used to evaluate eq. (2) [13-181. As previously [lO,ll], we apply a procedure which essentially amounts to not carrying the electron and nuclear polarizations along in the analysis, and which leads to transparent expressions for Pq. In the Heisenberg representation ev. (2) is P,(r) = TrG(0) * Cq(01,

(6)

i&(r) = exp(2ntif) 21,exp(-2ntif).

(7)

i(O) is conveniently expressed in a basis corresponding to quantization of the spins along the dire&ion b of beam polarization i(0) = j@(O)@J?(O) @ F bk(0)

=,(;tp;_p).:l’ogl

l”,

(2)

w wereb is the density operator of the system, and ;14 IS the r/component of the muon Pauli spin operator.

whereas zi, is most easily expressed in a basis corresponding to quantization along the axis of observa-

E. Roduner, H. Fischer / Muonium substituted organic pee rxi!c&

Obviously, eq. (8) may be rewritten iis , j(0) = N’ {I’ + P * ~Q,)o 1% fl I”. Insertirg (10) In (6) and noting!hat obtains P.&r) = (P/N)Tr{&

* G&)) .

, (10)

Tr{ir, ] = 0, one (11)

Usually (11) has to be evaluated only for the special cases b = (I = z (lorrgitudihal field) or b = y = x (transverse field), i.e.

^ where ]m>, 1~)are the eigenkets of H with energies hw,, ho,. The magnitude of muon polarization oscillating at a transition frequency o,, is propor. tional to the transition moment of the muor, spin operator. This reveals a close relationship I -+wees! pSR and magnetic resonance, the driving hf-field being replaced by the initial preparation of the system with spins polarized alo;ib direction 4. By expressing the eigenkets j, terr.: of tf1.ebasis (3)

eq. (12) becomes for 4 = z

t6j

For Pz there are only contributions from transitions between states ]m) and In>which contain identical basis kets I&), whereas for P, there are contributions between states !n;) and In) from basis kets 1%) and Ix/> differing i.? the muon but not in the,e!ectron and nuclear parts. These iestrictions which are less obvious in previous trealr!;en:s [13-181 greatly reduce the computational. ~+or.ki;l the calculation of PSR-frequency spectra. The treatment presented above assumes instantaneous radical formakjon. However, situations are encountered where the radicals are formed by reae tions ofmuon containing precursors, for instance by addition of muonium to olefi~, (1). Then a loss of polarization occurs if the rate constant of this reaction is of the dame order of magnitude or lower than the difference between the @R frequendies of radical and precursor. The extension of the treathent to include chemical reactions is straightforward. Consider for uistance lthe case where muonium is formed at time zero with probability 11~ and decays in a pseudo-first order reaction with rate constant h to give a radical. Following the lines of ref. 1181 we obtain from (1 I) for the radical polarization at long times (At Z@1)

wherr Ik)and II) denote the e;genstatl:s of ml;onium. ‘Yhisftirmula corresponds to that giver1in table 4 of ref. J 18) for R2 (conservation case, k:, - 0). We note that 12) and II) have IG be expressed in the same basis as the radical states im) and In), i.e. one has to in&de all;uncoupled nuclei. i.2. Selection rules and the number uf 4owed p!tiR transitions For an arbitrary magnetic field stren 4th B the hamiltonian (5) mixes product kcts (3) with equil lotal magnetic qup,ntum numbers M = rr+ -1me t Z’Kmk. Thus, we obtain from (14) and I 15) the gen. eral selection rules

and forq =x

,x 6;

I

I-I sjjzcos(o,,t).(15) k

AM=0

forq =z,

tiF=+I

forq =x.

(171

ifs

E. Roduner. H. Fische? j Muonium substituted organic free radit&

In the high field limit (r+ >A,,Ak) the eigenkets Im) and In) become equal to individual product kets (3) since the spins are effectively decoupled. Thus, the eigenstates are characterized by rnp, me and all .wk and the coefficients in (13)-(15) are equal to zero or one. For this case (19 leads to

and for longitudinal fields we have

P, =P,

These equations show that in general the J&R frequency spectrum of a radical with several coupling nuclei will consist of many lines, sharing the total beam polarization P. For instance for one nucleus Qx = 15, QZ = 6 and for four nuclei Q, = 792 and QZ = 430. Consequently, muonic radicals may be difficult to detect, though many transitions may degenerate or accidentally occur in separated groups. Such a situation seems to have been observed [9] _ In the high-field limit the spectrum simplifies considerably since the transitions (22) degenerate to only two lines (vide infra); Also, in zero-field many transitions coincide since the states belonging to equal 7 and F but different M degenerate. For this case a lengthy but straightforward calculation of the number of transitions with different frequencies Y# 0

(18)

i.e. there is no evolution of muon polarization in a strong longitudinal field, whereas (15) yields the selection ruIes Am’=*l,

Ame=Amk=O

forq=x.

(19)

On the other hand, for low fields and zero-field (Us <:A,,Ak) the eigenstates are characterized by an energy quantum number r and the quantum numbe_nFandMof the total spin operatorsP andgZ (F = s + iU f zk ik). A @R transition will be allowed (12) if 0nl&Jn) = (F’, M’Ii5q,IF,Ml + 0. Via the Wigner-E&art theorem, this condition leads to AF=O,+l, &U=O,

forq =z;

&+f=*l,

fGrq=X.

(23)

(20)

selection ruIes (17), (19) and (20) allow a calculation of the number of transitions expected in the J&R frequency spectrum of a muonium substituted radical. We restrict the following evaluation to spin-$nuclei, since we have not found convenient closed expressions for general spin systems. The secular determinant of fi for a n-spin-+ system (electron, muon and n - 2 nuclei) factorizes in subdeterminants connecting basis kets with comrnonM which are of dimension

(24)

The

Dh¶

=

For the general case transitions are allowed if AiW = 0, *I for q = z and q =x, respectively_ Therefore, for transverse fieIds the number of allowed transitions becomes

(22)

where K = 1 for even n and K = f for odd n. Eq. (24) leads to Q” = 3 for the one-proton and Q” = 141 for the four-proton case. Of course, equivalence of the nuclei may result in considerable reductions of Q_ An exarnpie for such a case will be given in section 4.3. 2.3. A numerical example To illustrate the previous conclusions we now discuss the results of a numerical calculation based on eq_ (12) for the $R transition frequencies and amplitudes of a radical formed at t‘= 0 and containing one proton and the muon as magnetic nuclei. The couplin@ constants ofproton (Au = 56 MHz) and muon (A, = 3.18 X 56 MHz) correspond to typical values expected for hydrocarbon radicals [12,19]. Fig. 1 shows the energy eigenvalues in frequency units plotted versus the magnetic field strength B. Above B z 100 G, i.e. where v, >A,, A,, alI energies vary linearly with B and the eigenstates are pure product states. Fig. 2 shows the energy level diagram for zero, low and high fields and the J&R transitions calculated for the trans-

E. Roduner. H. Fischer / Muonium substitutedorganic free radicah

accord with expectation: At low and intermediate tie!ds the polarization is shared by many transitions, whereas at zero and high fields there are only few lines with appreciable amplitudes. The variation of the @R-transition frequencies, including signs, with B is given in fig. 4a together with that of the bare muon 01’) It is seen that in extremely high fields the two radical frequencies are displaced symmetrically (numerically by &4,I) about the muon frequency. Due to level crossing one radical frequency goes through zero at B = 36.7 X L4,l G (A, in MHz). At around 150 G the additional proton spin causes a line splitting, and below about 100 G the complicated low field pattern develops. Fig. 4b shows the frequency versus field diagram for a muonic radical without additional magnetic nuclei and the same muonelectron coupling constant ofA, = 3.18 X 56 MHz. This muonium analogue exhibits the typical four muonium transitions [3-S]. The high-field sides of figs. 4a and 4b are identical. In this region V, >_4,, i.e. the additional nucleus is effectively decoupled

tE 300.

MHZ

200 -

-300.

265

A, =56MRz A, = 3.18-S6MHz

F& 1. Energy level dh_eram for a muonic radical with one proton coupling.

verse field case. Also given are the classifications of states and transitions according to the selection rules

(19) and (20). As expected from (22) and (24) there are 15 transitions for low fields and 3 transitions with Y* 0 for zero field. The relative amplitudes of the transitions are displayed in fig. 3 and are also in

from the muon-electron system. For low and intermediate fields, however, the muon-electron (fig. 4b) and the muon-electron-proton (fig. 4a) systems show distinctly different features. This was not recognized in previous treatments of muonium sub-

stituted radicals (see, for instance, ref. 1181) and lead

me rnp mp

200

100

AF=O

AF=O

AMFd

A+f

i

B=O

20

200 G

Fig. 2. pSR-transitions of a muonic radical with one proton in different transverse fields.

6 Rodurrer,If. Fischer / Muonbm substitutedorganicfree radicals

266

05

‘it ,

ii

LlJ_

2006

0

100

zoo

---xk

A, =56 MHz A, = 3.16.56 MHz

-200

Fig.3. CSH-amplitudes of Bmuonic radical with one proton In different Lransvsrsefields. ”

200

i

MHZ

I

to ,the incorrect experimental procedure of applying low fields in the search for radicals which was mentioned in the introduction. 2.4. High jikld perturbation treatments For fields high enough so that the electron Zeeman interaction v,, is much larger than the hyperfme internctions A, and Ak, i.e. usually for B + 200 gauss, conventional perturbation theory with the hyperfme interactions as perturbation leads to the first-order er:ergies(frequency units) A,, =3,16*56MHz \ \

In this approximation the eigenstates are the pure product states (3). As pointed out before [eq. (18)] there will be no time evolution of the muon polarizaI rt in strong longitudinal fields. With (25) and the selection rulas (19) tha transition frequencies become for strong transversefields 1 = f(llu f $_4J*

(26) There are only two dlfferent observable frequencies /VI,wlfb dogeneraciesof N/4 each. Inserting in (15)

Fig. 4. Field dependence of pSR frequencies in transverse field. (a) Muon-electron-proton system, f.b)muon-electron system. Broken lines indicate low intensity. p+ denotes the free muon precession frequency.

the observable polarization becomes Z’,&) = I)P[cos 27r(vM+ +A,& + cos2n(v,, - &Av) t] .

(27)

E. Roduner, tI. Fischer / Muonium substituted organic free radicals

for IA,,1 < 23 two lines with equal ampiitudes appear displaced from the free muon Larmor frequency by &&I& whereas for l.4& > 2v, they are centered at &4,] and displaced by *v,,. Eq. (26) is identical with the resonance condition for the highfield NMR transitions of a spin-&nucleus in a free radical and well known in ENDOR-spectroscopy [ 121. Most of the ,uSR-experiments to be described below have been carried out with B not fulfilling the highfield condition, and deviations from eq. (26) have been observed. Therefore, more exact solutions will now be given. Since for most radicals the muon hy perfine coupling is larger than the coupling of the additional nuclei (IA,1 > IAkl) the following treatment is appropriate for most cases: We star&with the muon-electron hamiltonian ObviouQ,

li” = v& 1 v& + A,$

+‘,

l

(28)

and treat the nuclear terms

Table 2 Zero-order eigrlvzctors, first-order encrgrs ;~cd pSR-tra,I+ tion frequencies of a muonic radical in tra,lsverse fields e‘ .-..-l_--._._. .__._ __ II) = i&re, n IK, MK, 12)= (sk9’5’) + c154xe)) nlK, MK) 13)= r/P/35 n IK, MK> 14)= (clc&=) - st5%~~))nlK, MK)

vi2 = v_

-

SZ+S~

CAJ&~K

~,~=v_+sL+c2CA&~ v,~=v_+~+A,+c~ZA,&~~ “43 = “_

(29)

-

8) Frequency

52

-A,+s” units

C/$#K

for 4%

c = 2-1’2 (I+ (ue + @/[A;

as perturbation. Furthermore, as usual in higher-order analysis of magnetic resonance spectra [20], equivalent nucleij are collected in groups described by total nucle?r ang~la_r momentum operators I;-= Zi ikj and Kz = Xi @, where K = C/ Iki, ZZ,Ikj - 1, ... 0 (or 4). The zero-order eigenfunctions are then the well known muonium functions [IS] multiplied by nuclear wavefunctions TlI K, MK)., Table 2 gives their structures together with the first-order energies and the @R-transition frequencies for transverse field. The spectrum now consists of the four mus~C.rmtype transitions (fig. 4b) split into the symmetric multiplet patterns characteristic for high field magnetic resonance. The amplitudes of the transitions v12 and v43 are proportional to c* and those of the transitions ~23 and v14 are proportionA to s*.Since c + 1 and s + 0 for B + -, only viz and v4s carry appreciable intensities at fields for which the treatment is valid. The splitting of these transitions decreases with increasing field, as was also shown in the numerical example (figs. 3 and 4a). The second-order corrections to v12 and vg3 are Avr2 =-a&k{[K(K+ X (c~/~,,/v;~v~~)-

l)-MK@fx+ [K(K +

I)]

1) - M;] 2s’ Iv:*},

267

s=2-l/*{1 Y_ = i(“e

Av43

--

+ (v(: * ‘@* 1”‘)

I”.

- (ue+ v~j/[AZ*(ue+VC(~~)“‘;“‘, - Uflj’

+;$ CA’k

{

[K(K + 1) ---MKQIK -- I ,!

(W

where vjj denote’the zero-order transition frequencies. They distort the symmetry of the first-order multipld pattern. To evahrate the relative imp zrtance of the first and second-order contributions consider the splitting for the muon-electron--proton system of sccfinrl 2.3. For B = 200 G the value ootatned by numerical calculation is 1.65 MHz (fig. 3). From the cquatiorrs of table 2 the first-order contribution becomes I.:!‘) MHz and eq. (30) gives an additional 0.41 MHz for the second+order contributions: i.e. there is good agreement. At higher fields the second-order terms become much less significant. For the analysis ofySR-frequency spectra in high transverse fields we note that rhe muon cclupliq con. stant may be convcnientljr extracted frum the fw

.

268

E_ Roduner. H. Fischer 1 Muon&n substituted oeanic free radicals

frequencies beIonging to the same set ofMK via 1A,I = IQZI.~

Iv43L

(31)

as long as first-order pertitrbation theory applies. Here the negative sign applies for fields high enough that vrz and v43 have the same*sign (cf. fig. 4). The nuclear coupling constants lAkl may be extracted from the line splittings, though not very accurately, since s* is small under conditions where spectra are observable_ 2.5. An analytical solution for zero jield As has been mentioned previously the number of ySR-transitions of a muonic radical is smaller in zero field then in low or intermediate fields (see figs. 2 and 3) and further reductions are expected for the case of equivalent nuclei. For such favorable conditions fiR spectra of radicals may be and have been observed experimentally in zero field, and the appropriate theory is given here. We consider a radical with N equivalent nuclei coupled to the unpaired electron, i.e. we start with the hamiltonian

Introducing the total nuclear spin operator I? = Ck i”, as before, with quantum numbers K = NIk, Nlk - 1, . .. . $ or b and degeneracies g(K), the problem reduces to that of a 3.spin system, and eq. (32) becomes fi=A,$$‘A,.&ik

(33)

This hamiltonian commutes with the operators g* and F=, where F = S f P + k, and the eigenstates may be denoted as IF, Ml, where M = MF_ Since S = I& = $ the quantum number F can take the values F = K f 1, K. K - 12 0. In the basis of kets Imp, me, MK) the

secular determinant factorizes into two subdetenninants of rank 1 corresponding to M = ‘(K -t I), two subdeterminants of rank 3 corresponding to M = iK and 2K - 1 subdeterminants of rank 4 corresponding to -K + 1 GM < K - 1. The solutions for the energies are independent ofM. They have been found already earlier [21,221] and are given in table 3. Calculation of the expansion coefficients,of states IF, M) in thebdsim~,me,M~)withMK=Ml,M,M+l

269

E. Roduner, H. Fischer /Muonium substituted organic free radicals

Table 4 BSR-transition amplitudes for a radical with equivalent nuclei in zero-field a)

0

0 0

(K + 2)(2K f 3)/3(K + I) [(W-B- 1/2~z/3W2(W--B)2~K(K+ 1)(X+ 1) [W’+B+ 1/2~2/3W2(~Yi8)2]K(Ki1)(2K+1)

;@$Ak(W-K-:)

(K - lI(2K - 1)/3K

[(IY-B+K)*/6h’(hl

3)

[(WIB-K)‘I~W(W+B)](~E:+~)

AkW ~A,&4k(iV+K+~)

[(B+ 1/2)*/3W*l(x+ 1) [(W-B -K - 1)2/6W(W - B)] (X

$AA,+$Ik(Wa) For numbering

-B)](2K+

i~~++~k(w+K+f-)

of states and abbreviations,

K-

i)

- 1)

[(WiB+K+1)2/6~Y(W+B)](2K-1)

see table 3.

is lengthy but straightforward. Table 3 displays the solutions. It is correct for all existing states, i.e. for F> 0 and JMI
Due to conservation of angular momentum the muons are spin poIarized with the spin antiparallel to the momentum in the rest frame of n*. Normally, muons from backward decaying pions are selected with a momentum of about 115 MeV/c and a spin polarization of typically 70% in the forward direction. They are stopped in the sample where they decay with the mean lifetime of 2.2 us by p+&e++rp

They are given in table 4. The positions and amplitudes of Lines with non-zero frequencies depend on the relative signs ofA, and A&_Thus, as in magnetic resonance 1231 these relative signs may be obtained from zero-field&R spectra. An example will be given along with the experimental results in section 4.3.

3. Experimental All experiments were carried out in the $2 and ~tE4 areas of the Swiss Institute for Nuclear,Research (SIN, Vtiigen). Spin polarized positive muons are produced in the following way: Protons are accelerated to 590 MeV and focussed onto a target (usually Be). Positive pions resulting from nuclear interactions are collected and transferred to an 8 m long superconducting solenoid where they decay in-flight with a mean lifetime of 26 ns according to

fve.

(36)

Conservation of spin and momentum during this process causes the angular distribution of positrons to be anisotropic with respect to the muon spin direction. The e* emission probability is proportional to 1 t Q cos 0, where 0 is the angle between the muon spin and the positron momentum’airection and a is the asymmetry coefficient with an average value over all energies of 4. If the muon spin direction evolves in time and positrons are counted in a fixed direction this results in a time dependence of the positron counting rate reflecting the evolution of muon spin polarization. The experimental arrangement for gSR observations, developed and used at SIN by various groups, is shown schemaiically in fig. 5. The sample (S) is set in the center of a pair of Helmholtz coils which produces a magnetic field B between 0 and 6 kG. Three additional pairs of coils, perpendicular to each other, serve to compensate the earth magnetic field and any stray field in zero-field experiments. Scintillation counters are grbuped about the sample in the incoming (a, b, c), the forward (fi, f2) and the

210

E. Roduner, H. Fischer /hfuonium substitutedorganicfree radicals

Pig. 5. JAR ;~ppc~rrtus. S: sltrnpls, D,b? c, fl, f2, pr, ps: scintiIlution counters, D: dsgradcr;b mugnetic field.

Events where a second muon has stopped within a data gate of a few muon lifetimes (typically set to 6 ps) before or after a stop event are rejected to avoid background and distortions in the histogram. This sets an upper limit to the sampling rate and the acceptable muon beam current. Typically IO6 to lo* good events are accumulated in one experiment at a sampling rate of 2500 s-’ . They are stored in two histograms corresponding to the forward and perpendicular telescopes in the memory of an on-line PDP-I 1 computer. Each histogram is divided into 4 K bins with a width of 0.4- 1.7 ns, depending on the time or frequency resolution to be obtained. For unpolarized muons a histogram would depict a single exponential that decays with the muon lifetime r&. Time evolution of muon spin polarization leads to superimposed oscillations. The general form of a histogram is [2,S] H(r) = No {Bc + e-‘/3

perpendicular (pr , p2) direction to detect the passage of individual charged particles. The resultant fluorescence pulses are converted to electrical pulses which are loglcally analyzed in a fast electronics setup. The beam of incoming muons 01’) is usually reduced to a diameter of 15-20 mm by passage through lead collimators. Further the muons are slowed down In polyethylene (in the collimator hole) und in a water degrader (D) of continuously variable thickness to maximize the number of muon stops in the sample. The logical procedure of obtaining a MSR-histogram is as follows: A stopped muon has passed through the incoming counters a, b, c but not through f (fr or fs) or p (pr or p2). This event is recognized by simultaneous pulses in a, b, c in anticolncidcnce with f and p, i.e. by a logical signature P * b * c * f+, Correspondingly the srgnature fr * fs * E * @represents o decay positron detected in the forwnrd, and p1 - pX * 2 - f represents a decay posltron detected in the perpendicular telescope. Reco,uMon of a muon stop and its associated decay positron lnltiate the start and the stop of a lifetime measuring apparatus. This consists of a time-to-amplitude converter (TAC), n fast analog-to-digital converter (ADCj urtd II rnuiticharmal memory which accumulates uccepted positron events in the form of a histogram.

[ 1 t F(f)] ) ,

(37)

where No is a factor depending only on the total number of counts and is roughly equal to the number of counts in the first channel, Bc is the background fraction (usually <1%) and F(t) reflects the time dependcnce of the muon spin polarization. If several muonic species contribute and/or if one species shows more than one frequency, F(r) is a sum of contributions of the form F,(r) = Ai e-&it

COS(Ujt

+ 4,),

(38)

each describing a precession on a specific frequency wr with its amplitude (asymmetry) A,, damping constant hi and initial phase &. The asymmetries At depend on the beam polarization P, the asymmetry coefficient a, the solid angle of the telescope, the partitioning of P between different species and frequencies and on reaction or relaxation rates. A, represents the reaction or relaxation processes, and @fdepends on experimental factors as well as on reaction 0~ relaxation terms. Quantitative analysis of all parameters is possible [3-524,251 but in this paper only frequencies oi are discussed. The conventionai way of /.tSR data analysis is by direct tit of the theoretical expression (37) to the histogram; an example is given in section 4.2. This method involves four parameters per frequency and is expensive and often difficult for multiline spectra

E, Roduner, H. Fischer / Muonium substituted organic *freeradicals

of muonic radicals, Therefore in most of this work a fit in Fourier space is preferred where each frequency can be treated separately [26]. The MINUIT fit programme [27] was used in combination with a fast Fourier transform routine. The analyses were performed off-line at the CDC 6400/6500 installation of the Eidgeniissische Technische Hochschule, Ziirich. Tetramethylethylene and benzene of commercial grade (Fluka) were used. These liquids were degassed on a vacuum line via three freeze-pump-thaw cycles and sealed in thin-walled spherical sample cells of 40 mm diameter. All experiments were carried out at ambient temperature. Apart from signals due to muonium substituted radicaIs all histograms revealed the presence of muons in diamagnetic environment. These are probably due to muonium substituted diamagnetic molecules formed by epithermal reactions or fast thermal processes in the radiation spur [3,4,28]. Nothing more specific can be said about the chemical nature of these species since the frequency resolution of pSR spectra is too low to discern chemical shift effects.

271

‘0

5kG

.i-;-.__L__ II0

u__j_&_!-L_ ._ 2kG

0

IkG

jj”

0.6kG

CR,

,

t’

__0

50

__.*. 100

lb

230

MHz

Fig. 6. pSR precession frequencies obtained with tetramcthylethylene at various transverse magnetic fields. D: muons in diamagnetic environment. R: muonium substituled radical.

4. ,uSR-spectra of radicals 4.1. A h&h-fieldcase As shown in section 2.4 muonium substituted radicals should be observed most easily in high transverse magnetic fields since there only two different frequencies (26) should appear. Following the original suggestion of Brodskii [ 6) our search for radicals concentrated on samples of unsaturated compounds where the radicals should be formed by rapid muonium addition, cf. eq. (1). Fig. 6 shows Fourier spectra obtained from tetramethylethylene with magnetic fields between 0.6 and 5 kG. The line denoted D obeys the relation v = 13.55 kHz/G and is due to muons in diamagnetic environment. The two other lines, denoted .R, are attributed to the radical (CH&MuC-C(CH&. All frequencies are given in table 5, and average fit errors from the forward and perpendicular histograms are given in parentheses. Due to a more refined analysis the frequencies deviate slightly from those published previously [S]. Inspection of fig. 6 and table 5 shows that the sum of the two radical frequencies is independent of B.

According to the high-field treatment of section 2.4 this implies IA,/ > 2v, = 2~” for all fields. and for positive A, a negative sign of the lower radical frequency. From (3 1) the sum of the radical frequeucics is equal to the absolute muon-electron hyperfinc interaction. The average value is IA~1 = (I 6 I .3 1 L 0.03) MHz, if the lowest field v;rlue is not taken into account. At that field line-broadening indicates the onset of splittings. Also given in table 5 are the radical frequencies calculated from A,, .,u,, = Y” iId g=g, via the high-field limit equation (26) and t’rom the equations of table 2 derived by the more exact high-field perturbation treatment based on the hamiltonian (28,29). As is clearly seen (26) does not predict the frequencies sufficiently correct even for R = 5 kG, and the deviations increase with decreasing B. On the other hand, the use of the equations of table 2 leads to a most satisfying agreement hetwaen predicted and experimental data. The error limit of lAPI shows that muon coupling constants of radicals may be obtained by pl;R with an accuracy which is comparable with that insulted in the determination of nuclear hyperfine irrtcraclions

E. Roduner, H. Fischer lhfuonium substituted organic free radicals

,272

Table 5 rrSR-transition ‘B(G)“)

frequencies

(MHz) for tetramethylethylene

in high fields

-

.D

nk

UH

UP + UP

IV121b,

Iv431

999.4 2004.6 3004.0 4008.0 50092

8.074(3) 13.546(2) 27.170(3) 40.715(5) 54.323(3) 67.894(3)

68.88(13) 64.81(4) 52.31(2) 39.17(6) 25.73(3) 12.33(4)

92.62(13) 96.52(4) 108.97(2) 122.09(S) 135.56(2) 149.04(3)

16150(18) 161.33(6) 161.28(3) 161.26(a) 161.29(4) 161.37(5)

7258 67.11 53.49 39.94 26.33 12.76

88.73 94.20 107.83 121.37 134.98 148.55

‘1 Calculated b, Calculated ‘) Calculated

from nD = vfi = 135538 kHz/G. via eq. (26). via equations given in table 2 with s* = 0.

595.7

by high resolution ESR spectroscopy [ 12,191. The radical (CH3)5CH-C(CH3)2 has been studied by ESR and exhibits a couplir~g ofA, = 33.4 MHz at room temperature [29]. Corrected with the ratio of magnetic moments ol,/.u, = 3_18335)A, leads to a predicted value of A, = 106.13 MHz for the muonium substituted analogue. The observed value of lAPI = 161.31 MHz is considerably larger. We do not believe that this could indicate an incorrect assignment of the muonium substituted species since there is independent proof from the zero-field spectrum of section 4.3. The large isotope effect may rather reflect a different averaging of the angular dependence of the 0-H and &Mu couplings by the hindered internal rotation about the Q-CD bonds. Similar isotope effects have been found in a variety of other radical; [8,1 I] and will be discussed in detail elsewhere_

b,

Iqzl

c,

68.71 64.80 52.33 39.17 25.76 12.30

Iv431 =)

a

92.60 96.51 108.98 122.14 135.55 149.01

13.55 kHz/G. It belongs to muons in diamagnetic environment and is not displayed_ The positions of the radical precession frequencies exhibit a field dependence similar to that observed in fig. 6. Consequently lAPI can be obtained by addition of the frequencies (32). From the spectrum at 4 kG, Id,\ = 5 14.6 MHz. For B = 1 kG the radical transitions are split by (1.5 f 0.2) MHz. Based on the following considerations we attribute this splitting to the hyperfine interaction of one proton: The coupling constants of the protons in cyclohexadienyl have been reported repeatedly [19] and are near room temperature [31] 134 MHz for the methylene, 25.3 MHz for the ortho, 7.8 MHz for the meta and 37.1 MHz for

4.2. Line splitting in a moderate fieti As shown in section 2.4 a splitting of the two fiR transitions of a radical is expected in moderate transverse tields if the radical contains magnetic nuclei in addition to the muon which couple to the unpaired electron. Such a splitting is observed here for the cyclohexadienyl radical. Tt is formed by addition of muonium to liquid ‘benzene in analo,~ to the known addition of hydrogen atoms [30] _Fig. 7 shows @3R frequency spectra obtained for B = 4 kG and B = 1 kG in the frequency region v > 150 MHz. The two groups of transitions are attributed to the radical. A third transition occurred at lower frequencies u =

LSUM

7kG

Y 150

2w

250

Fig. 7. PSR precession frequencies transverse fields of 4 and 1 kG.

.

MO

3SONH.?

obtained

with benzene at

E. Rodunet, Fi. Fischer /Muonium substituted organic free radicals

the para protons. From the largest ctinstant a muon hyperfiie coupling of 427 MHz is estimated for muonium in the methylene position as it would follow from the addition process. The experimental value is of the correct order of magnitude though larger by about 20%. This reflects an isotope effect which is probably caused by the difference in vibration amplitudes of the muonium substituted radical and its’proton analogue. If now the proton coupling constants of the muonium substituted radical are equal to those reported for its H analogue the equations of section 2.4 [table 2, (30)] predict for B = 1 kG a splitting to a doublet of 1.38 lMHz by the methylene proton. This agrees with observation. All other protons would lead to much smaller splittings which are not resolved but cause a line broadening of aO.3 MHz. The vaiues of /A,1 and of the splitting thus confmm the assignment of the spectrum. The B = 1 kG case also serves to demonstrate the direct fitting of a histogram to the theoretical equation (37). The top part of fig. 8 shows the raw histogram containing 12 X 10’ events in 3600 time bins of

, 0

2.0

IO

273

0.87 ns width. The clearly visible oscillations have a frequency of 13.5 MHz and belong to muons in diamagnetic environment. The radical frequencies are in the range 200-300 MHz (fig. 7). They are superimposed on the diamagnetic oscillations and not directly noticeable. The middie part of fig. 8 displays an expansion of the first 0.6 PS of the histogram after adding 8 subsequent bins and removing the exponential muon decay. Addition of bins averages out all frequencies above 145 MHz, i.e. the fast radical oscillations, and leads to a clearer display of the diamagnetic signal. The solid line in this part of fig. 8 is the best fit to the theoretical expression (38) (AD = 0.027, UD = 13.64 MHz,.QD = 0.75 rad and hD = fuc = 0). To display the radical precessions, the range of 45 to 90 ns of the histogram is expanded without addition of bins but removal of the muon decay (bottom of fig. 8). It shows a beat behaviour typical for a superposition of frequencies. The solid line corresponds now to a sum of three terms (38), one for the diamagnetic signal with parameters as given above and two radical precessions (ARK = 0.028, vR1 = -221 MHz, @RI= 0.6 rad, AN = 0.022, ZJ~ = 294 MHz, QR? = 0.8 rad and ham = hm = 0). To reduce the number of parameters in the fit the radical frequencies were taken from the Fourier transform (fig. 7) with a negative sign for ~~~ in accord with a positive A, > 2u~. Line splitting and relaxation are unimportant for the 45 ns considered and were neglected. The initial phases of all three signals agree within experimental errors. 4.3. A zero-field case

3Op5

tA

As was pointed out before, a zero-field study of muonium substituted radicals seems feasible if they contain only few coupling nuciei, preferably magnetically equivalent, in. addition to the muon. The radical (CH&MuC-C(CH& of section 4.1 is chosen as exaInpie here, since the proton analogue was shown

1

.4x

.06

.07

38 PS

Fig. 8. JAR histogams obtained with benzene in a transverse field of 1 kG..Top: raw data. Middle: Oscillations due to muons in a diamagnetic environment. Bottom: Oscillationsof muons in radicals and diamagnetic environment.

to possess 6 equivalent CHs-fl-protons with appreciable coupling of A P = 64.2 MHz [29,33-l whereas the coupling to the CH3=y-protons was unresolvably small. The theory of section 2.5 was applied taking A, = 64.2 MHz andA, = 161.3 MHz (section 4.1) and neglecting the coupling to y-protons. The six equivalent proton spins couple to total nuclear spins of K = 3,2,1 and 0 with statistical probabilities g(K) of 1,5,9 and 5, respectively [20]. From the formulae of table 3 the

214

E. Rodutter,H. Fischer /Muon&m substitutedorganicfree radicals

energies of the resulting states of total spin angular momenta F with F ranging from 4 to 0 follow. They are shown in fig. 9 together with the g2Wansitions allowed for AF= 0, *I and AK = 0 (20). There are 16 allowed transitions with v # 0. For 6 inequivalent protons 1688 transiiions would be expected from (24), i.e. equivalence reduces the number of lines considerably. . The Fourier power spectrum calculated from the amplitude expressions of table 4 is shown in the upper part of fig. 10. Accidentally two lines coincide at 161 MHz. 27% of the muon polarization do not evolve in time. Fig. 1Ocgives the experimental Fourier power spectrum obtained from a high statistics run (23 X lo6 events in the forward histogram) with magnetic fields compensated to 6200 mG. In the peak amplitcdes iC strongly deviates from the theoretical spectrum but ’ the observed frequencies agree remarkably well. The three peaks predicted to be the most intense are clearly observed. Further three weak peaks agree in position with predrcted lines but are of insignificant intensities. In table 6 the experimental frequencies of the slrony?r transitions are listed and compared with those calculated for three sets ofA, andAp. The agreement is excellent fo: A, = 16 1.3 1 MHz and A, = (63.75 f 1S)MHz. Note, that I.4,1 = 161.31 MHz was also obtained from the high-field spectl’a (section 4.1) and that the proton analogue of the ra6ical hasA, =

a

A, = 161.3 MHz A,, = 63.8 MHz

I

I

I)

I,

I.

I

b II

e 5

I k!&$

-

0

J;‘Q I * _*Nllrl..JU*~I-L SO 100

Ld..

150

..,-

.._-

D

200

250MHz

Fig. 10. @R precession frequenciesobtained with tetramethylethylene in zero-field. (a) Calculatedpower spectrum for instantaneous radical formztion. (b): Czlculated power spectrum for muonium addition with a rate of lOlo s-l. (c) Experimental. Cu: cyclotron frequency.

64.2 MHz [29,32]. Thus, the zero-fielc! spectrum conand likewise equal signs of A, and A,, as is reasonable for nuclei in /L&positions. Further, in agreement with prediction no precession signal was found in the perpendicular telescope except for a background peak at 50.7 MHz which corresponds to the SIN cyclotron frequency. As obvious from the figure the calculated intensity firms the assignment to (CH&MUC-C(CH~)~

Table 6 &R transition frequencies (MHz) for tetramethylethylene in zero-field Experimental

49.62(7) a) 58.89(7) 102.40(30)

Calculated for A,, = 161.31 AP= 63.60

161.31 63.75

161.31 MHz 63.90 MHz

49.52 58.87 102.45

49.60 58.93 102.38

49.68 59.00 102.31

I:lp.9. Lnqy levelsund ullowcd NSR-transitionsof M tI,),Mu(‘-j’(C&,)2

I3 zero field.

a) Error in units of 0.01 MHz.

E. Roduner, H. Fischer / M~onium mbstituted organic free radical

distribution (fig. I Oa) does not reproduce the experimental finding. Lines predicted above 150 MHz are not observed at all. This effect may be explained by a non-instantaneous radica? formation and loss of polarization ln a muon con talnlng paramagnetic precursor. The theory for this case has been given in section 2.1. As radical precursor muonium may reasonably be assumed. AFpliCatiOn of eq. (16) with a muonium reaction rate of A = ~O”‘S-~ then leads to the theoretical power SFectrum of fig. lob which indeed is in striking agreement with the experimental one. Formation of the radical by addition of muonium to the olefin with a rate competing with the muonium precession frequ:ncies implies a thermal addition reaction. Consequently it can be excluded that a significant fraction cd the radicals is formed by epithermal processes. Finally, it should be mentioned that the intensities c,f the high-field radical spectra are also affected by the incomplete transfer of polarization from precur.;or to radical, though less drastically [24].

275

time. Here we add that the specific time-window of ,u+%of about lo-’ to about 10m5s combined with the experimental requirement that not more than one individual muon may be present in the sample at any the Offers Certain advantages over other methods in the determination of radical reaction rate constants: The kinetics is always fitst or pseudo-first order, sim. PlifYing the analysis. It can be observed in a rather wide time range.

Acknowledgement We gratefully, acknowledge support from the Swiss Nntional Foundation for Scientific Research and the Swiss Institute for Nuclear Research.

References [ 1J F.N. Gygax, W. Kiindig and P.F. Me&, eds., Muon spin rotation (North-Holland, AmeterdP-t, 197Y).

[ 2J V.W. Hughes and C.S. Wu eds., MI/XI p!:ysics, Vols. I 5. Concluding remarks The principles of/&R as ;Ipplied to multi-spin systems are very similar to thos: of magnetic resonance, the driving high-frequency rs diation being replaced by the evolution of the system prepared initially in a noneigenstate. Consequently, the information obtained is quite analogous. For muonium substituted radical!; high-field experiments yield lhe muon-electron hyperfine interaction in much the rame way as ENDOR gives this quantity for a selected nucleus. In zero-field, as in zero-field NMR and ESF: of radicals all hyperfine interactions are obtained simultaneously. The accuracy of the data also comparer well with that obtainable by the more conventional techniques. In this paper only radicals in liquids were considered. Exten.. sion of the theory to single crystals or isotropic solid systems is straightforward in principle and involves the change of the hamiltonian only. Obviously, application of /,SR in free radical chemistry is restricted to species wldch a-e formed by the implantation of muons in chemical systems, for instance via muonium addition (1). In fact, all of the hitherto observed muonium substituted radicals are formed in this way, and we will report on them in due

3 (Academic Press, New York, 1975). [3J D.G. Fleming, D.M. Garner, L.C. Vaz, D.C. Walker, 1.11. Brewer ?nd K.M. Crowe, Advan. Chem. Ser. 175 (1979) 279 ff. [4) P.W. Percival,E. Roduncr and fj. Fischer, Advan. Chcm. Ser. 175 (1979) 345 ff. [S] P.W. Percival, Radiochim. Acta 26 (‘1979) 1. [6] A.M. Brodskii, Zh. Exp. Teor. Fiz. 43 (1963) 1612 Ifinal. Trans. Soviet Phys. JETP 17 (1963) 1085). [7] J.H. Brewer, K.M. Crowe, F.N. Gygnx, R.I‘. Johns.% D.G. Fleming and A. Schenck, Phys. Rev. fi9 (!974) 495. (81 E. Roduner, P.W. Percival, D.C. Fleming, J. Hochmunn and H. Fischer, Chem. Phys. Letters 57 (1978) 37. [9] C. Bucci, G. Guidi, GM. de’hlunuri, M. Manfred;, P. Podini, R. Tedeschi, P.R. Grippnand A. Vecli. Chem. Phys. Letters 57 (1978)41. [ 1OJ E. Rod’uner and H. Fischer, Chem. Phys. Letters 65 (1979) 582. [ 111 E. Roduner, in: Exotic atoms ‘79, eds. K. Crowe, J. Duclos, C. Fioreni. li and G. Torelli (Plenum Press, NCW York, 1980). [ 121 J.E. Wertz and J.R. Bolton, Electron spin resonnnce (McGraw-Hill, New York, 1972). [ 131 V.G. NOSOV and I.V. Yakovleva. Zh. Eksp. Tcor. I+. 43 (1962) 1750 1Eogl. Trans. Soviet Phys. JETP I6 (I 963 1 1236). [ 14) LG. banter and V.P. Smilga, Zh. Eksp. Tear. I:iL. 54 (1968) 559 [ Engl.Trans. Soviet Phys. JETP 17 (I!%8 301).

276

E. Rodunet, H. Fischer/Muon&m substitutedorganicfree radicals

[15] J.H. Brewer, FH. Gygax and D.G. Fleming, Phys Rev. A8 (1973) 77. [16] WE. Fischer, HeIv. Phys. Acta 49 (1976) 629. [17] R Beck, PF. Meier and A. Schenck, Z. Physik B22 (1975) 109. [ 181 P.W. ?eicivaiand H. Fischer, Chem. Phys. 16 (1976) 89. [19]landolt-Barnstein, Numerical data and functionaI relationships, New Series, Group II, Vol. 9b, eds. H. Fischer and K.-H. HeIIwege (Springer, BerIiu, 1977). 1201 R.W. Fessenden, J. Chem. Phys. 37 (1962) 747. [21] R.A. Me&I, Phys. Rev. 46 (1934) 487. 1221 P-F. Meier and A. Schenck, Phys. Letters 50A (1974) 107. [23] H.C. Helier, J. Chem. Phys. 42 (1965) 2611. 1241 P-W_Percival and J. Hochmarm, Hyperfme Interactions 6 (1979) 421.

&5]_ J. Hochmann, Hypertime Interactions 6 (1979) 431. [26] E. Roduner, J. Hochmann, B.C. Webster and H. Fischer, Chem. Phys., to be published. 1271 F. Jamesand M. Roos, Comp. Phys. Commun. 10 (1975) 343. [28] P-W. Percival, E. Rodurier and H. Fischer, Chem. Phys. 32 (1978) 353. 1291 D. Griller and K-U.Ingold, I. Amer. Chem. Sot. 96 (1974) 6203. [30] H. Fischer, 2. Naturforsch. 17a (1952) 693. 1311 M.B. Yim and D.E. Wood, J. Amer. Chem. Sac. 97 (1975) 1004. $321 PJ. Krusic, P. Meakin and J.P. Jesson, J. Phys. Chem. 75 (1971) 3438.