Negative muons in matter: Atomic and molecular aspects

Notleer Physics A3 74(198~607o-617c O North-Holland Publishing Co., Amsterdam Not to be reproduced by photopdnt or mitrofüm without written pomtission from the publisher.

NEGATIVE MUONS IN MATTER : ATOMIC AND MOLECULAR ASPECTS G . FIORENTINI National Institute for Nuclear Physics, INFN, Pisa and Istituto di Fisica, University of Pisa, Italy Abstract : Some of the atomic and molecular processes occurring in the study of negative moons in matter are considered . In particular, the formation of mesomolecules is discussed . We also discuss the implications for the study of mesomolecule spectroscopy . and for the possibility of producing energy by means of moon catalysed fusion . 1 . Introduction There are several problems of atomic and molecular physics which are involved in the study of negative moons in matter . Roughly, they can be divided in three classes, depending on the value of the relative moon-nucleus distance, ruZ At ruZ ~ ao one has the formation of the initial bound state, see Fig .l . Here one

Figure 1 Formation of the initial bound state has to study the emission of the Auger electron which carries away the moon kinetic energy and the properties of the initial (u, Z1, Z , . . . , e l , e2 , . . .) system . This is really a many body problem and its detailed properties are quite difficult to be studied . We have now a semi-quantitative understanding of this stage through the so-called "model of large mesic molecules" and its refinements') . Eventually, the process of the initial bound state formation might be used as a probe of the outer electron distribution of molecules, however it'seems that we need a deeper understanding than we have now before this goal is reached . me The subsequent stage of the moon history (a o > ruZ > ao ) is the atomic m,,Z cascade . The moon loses energy through e .m . radiation and AugeF processes, i .e . electron emission . The electrons involved in these reactions belong to the host atom from the beginning or have been transferred to it through charge exchange collisions with the surrounding atoms (refilling of atomic orbitals) or are emitted directly from the surrounding atoms when the muonic atom comes close to one of them (external Auger effect) . These processes are by now generally understood 807c

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G . FIORENTINI

and their effects are taken into account in the computer codes which describe the evolution of the atomic cascade . m In the region of the deep muonic atom levels, ru Z ~ ao me the value of the charge of the nucleus to which the muon is bound is the cr5tical parameter . For Z > 1 interatomic forces prevent the (uZ) to come close to the other nuclei or electrons and a regime of "isolated muonic atom" is reached . On the contrary, the muonic hydrogen isotopes (up),(ud)and (ut), being small and neutral, can diffuse through matter in a way similar to neutrons . Therefore, they can come close to the nuclei and electrons of other atoms and interact with them . These inter= actions are a specific class of atomic and molecular processes on which there have been interesting developments in the recent years and we expect to have significant progress in the future . The interest in this matter arises primarily from the fact that it has become possible to make very accurate spectroscopic studies of muonic molecules, both theoretically and experimentally . Also, some interesting speculations about practical applications of these interactions arose recently . For these reasons, I will concentrate on the interactions of muonic Hydrogen . The aim of this talk is to give an idea on the status of the field through the discussion of some selected topics .

2 . Energy levels of "mesomolecules" It is well known that an electron can bind two Hydrogen nuclei to form the simplest possible molecular system, H2 = (p e p) . Negative muons can do the same, thus forming systems (X u X') where X = p,d or t . These systems are termed "meso molecules", with a name which is twice incorrect since there is no meson and they are charged systems . The term "muonic molecular ions" would be more appropriate but it sounds so long and pompous that may be it is better to keep the other name . Formation of mesomolecules occurs through collisions ( X u X') + energy . The formation reactions are classified according to the way the binding energy of the mesomolecule is released . Besides the usual Auger process (Fig .2a) it is also possible to have energy transfer through the excitation of the vibrational and rotational levels of (ordinary) molecules (Fig .2b) 2 ' 3 ) . This latter process is

Figure 2 Formation of the mesomolecule a) through the Auger (non resonant) process b) through the resonant process

NEGATIVE MUONS IN MATTER

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only possible if the mesomolecule has weakly bound levels, with a binding energy Eb smaller than the dissociation energy of the Hydrogen molecules, a few eV . Also, due to the quantization of the vibrational and rotational energy this process can occur only if a resonance condition, between Eb, the kinetic energy Ek~ n of the muonic atom and the quantum jumps DE of the ordinary molecule, is satisfied (see Fig .3) Ekin Eres

Eres =

DE - E b

,

hence, the name "resonant formation" for this process .

Resonant formation

Figure 3 of the (ddu) mesomolecule

It is intuitively clear that the resonant structure can be used as a tool in order to perform accurate determinations of the energy levels of mesomolecules . Also, it is clear that under suitable conditions the resonant process can consi derably increase the rate of formation of mesomolecules . The recent developments in the field of mesomolecules are all grounded on these two ideas") . The experimental study of the mesomolecule formation occurs through the detection of the nuclear fusion reaction which occurs in the mesomolecule . The nuclei bound in a mesomolecule behave as if they were in a plasma with a density me p ~ ( a )3, which is about 10~ the liquid Hydrogen density, and with a temperature

0

corresponding to their vibrational energy : KT = Evib ~ 100 eV . Under such extreme situations, comparable to tho e inside a white dwarf, the nuclei will fuse rapidly the fusion rate being 10 5 - 10~ times the muon decay rate ao in dud and dut . In conclusion, through the observation of the nuclear fusion reactions one gets informations about the energy levels of mesomolecules . In this way a few years ago it was possible to prove the existence and to measure the binding energy of the J=1, v=1 state of the dud mesomolecule s) Eb

=

(2 .196 ± .003) eV

.

Some comments are needed in order to appreciate the significance of this work . First of all, since the typical energy scale of the problem of muonic Hydrogen atom is Eo ~ 1 keV, the accuracy of the experimental result is of the order impressive figure . From the theoretical point of view, the 6 ~ 10 -6, a rather three body (X R, X') Coulomb problem is exactly soluble in the limit m~/MX = 0 . This approximation is the starting point of all the calculations of energy levels

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G . FIORENTINI

of molecular systems, the so-called Born Oppenheimer, or adiabatic approximation . Perturbation theory is then used in order to take into account the effect of

In muonic molecules one has E _ ~ ~ .1, so that a measurement with an error 10 - 6 allows to check terms of mP the order E5 . In ordinary molecules, 10-16! where E = ~ .5x10 -3 the corresponding terms give corrections of the order me P In conclusions, non adiabatic corrections are extremely enlarged in mesomolecules . These systems are really a laboratory for testing accurate methods of solution of the SchrSdinger equations . The evolution of the theoretical calculations is schematically shown in Table I, for the case of dud mesomolecule . It is remarkable that the existence of the J=1, v=1 state was proven only recently, when it was possible to achieve accu racy of the order .1 eV . The most recent calculation s ) has an accuracy of the order .05 eV, still far from the experimental accuracy . However, to achieve this result it was necessary to solve a system of some hundreds of second order differential equations with an accuracy which is at the limit of present computing facilities .

(00)

(O1)

(10)

(11)

324 .27

32 .76

226 .55

-

-

variational (1964-68)

~ 1 eV

324 .99

35 .66

226 .74

1 .83

85 .34

perturbative(1976-80)

~.1 eV

325 .04

35 .81

226 .61

1 .91

86 .32

truncation (1980) 6 )

~.05 eV

(20)

~4ethod

Accuracy

Table I : Binding energies in eV of the dud (J,v) mesomolecule, from Ref .6 . Some relativistic corrections to the binding energies of mesomolecules have been calculated quite accurately . I would like to mention the recent calculation by Bakalov et al . of the hyperfine frequency shifts') . Such tiny structures can be nicely evidenciated through the process of resonant formation . Consider, for example, the h .f.structure of the J=1,v=1 level of the dud mesomolecule .(See Fig.4) . The two arrows indicate the transitions from the doublet and the quarted state of the (ud) atom to the J=1, v=1, S=1/2 state of the (ddu) mesomolecule . The energy difference between the two transitions is about .049 eV . The russian group observed a transition with a resonant energy Eres = ~E-Eb= .053eV which gives a formation rate with a maximum at T ~ 400K . If this transition is interpreted as the F = 1/2 i S = 1/2, then the F = 3/2 -" S = 1/2 wi 11 have a resonant energy Eres = .D04eV and correspondingly the maximum of the formation rate is shifted at a temperature T' ~ 30K [see the forthcoming publications by Breunlich et al . for a more detailed analysis] . Evidence for this transition was found in an experiment performed at SIN in 1979 by Breunlich et a1 .e) . The analysis of their experiments is still in progress . . Ultimately they will be able to determine the resonant energy with an accuracy of order 1 meV and possibly they will fully clarify the hyperfine structure of the J=1, v=1 state . Clearly, in the study of the dud system we are at the second generation of

NEGATIVE MUONS IN MATTER

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Figure 4 p rfine s ructure of the du atom and of the (ddu) J=1, v=1 state . ~ _ ~ + ~d and ~~ + ~d + ~d, . Additional subsplittings arising from spin-orbit interactions are negléctea . experiments and we are close to a full understanding of the mesomolecule formation . On the other hand, the study of the dtu ion, which I discuss in the next section, is still in a primitive stage and deserves a substantially new experimental effort') . 3 . Energy production from muon catalysed fusion ? Theoretical calculations predict the existence of a weakly bound level also in the dut mesomolecule . It is again the J=1, v=1 state . The most recent calculations6 ) give a binding energy Eb = .64 ± .05 eV . The resonant formation mechanism can occur also in this system and it was estimated that the formation rate, adtu, can be hundreds times larger than the muon decay rate'°) . Really, the actual value of the rate cannot be predicted from the theory, as it involves crucially the resonant energy, which is of the same order as the theoretical error on the binding energy . Depending on very tiny effects, which are not controlled by the theory, the formation rate can change wildly . Ultimately it has to be d termined experimentally . So far only a lower limit has been obtained, adtu ~ 210~s-1,11) which supports the idea that a muon can catalyse, during its lifetime, hundreds of nuclear fusion reactions . This has revived an old dreamlZ ) of particle physicists, i .e . the possibility of using muon catal,~sed fusion for energy production . An ingenious scheme has been proposed by Petrov s) in order to optimise the efficiency of energy production . His ideal machine, called the "mesocatalytic reactor", is an hybrid system where electric breeding is used in conjunction with muon catalysed fusion . The muon catalysis is used as a source of fast neutrons which implement (by a factor ~ 2) the efficiency of the electric breeding . A scheme of the machine is shown in Fig .5 . A positive energy balance is obtained, but some of the assumptions used are rather optimistic . In particular it seems to me that it is very hard to be able to use for muon catalysis .8 muon per produced pion .

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out +-~

ACCELERATOR (IßeV/nl

deuterium

TARGET

tit nucleons

~~U BLARKET =~FISSIORS +

Pu

heat

LECTRICIT9

s

~6EIIERATOR

nl ATOMIC ERER611 PLAGT

Figure 5 Scheme of Petrov's mesocatalysis reactor However, in order to say anything definite pro or against this scheme it is necessary to have measurements, not only limits, on the mesomolecule formation rate, which is also interesting for the study of the mesomolecular energy levels . Also, it may be interesting to study, theoretically and experimentally, the formation of mesomolecules under the action of electromagnetic fields . These fields can change the energy levels of ordinary molecules and thus modify the resonance condition . As we discussed before, small variations of Eres can induce quite large variations in the formation rate . This effect could be used in order to enhance the formation rate if necessary for practical applications . Also it could be used in order to improve the accuracy of the energy-levels measurements . It has to be added that having large values of the formation rate is not a guarantee that there will be many fusions . There are processes which can limit the number of fusions . Some of these are discussed in the next section . 4 . Muon sticking and reactivation A simplified picture of the chain of muon catalysed reactions in the d-t system is shown in Fig .6 . There is some probability, we - 1 .2x, that the muon sticks to the He nucleus . If this occurs, and tiô the muon stays bound to the nucleus, it is lost for the chain of fusion reactions . For .adtu > ao the sticking process occurs, on the average, after the muon has catalysed 1/ms fusions . Ni'= Without sticking you would expect a muon catalyses adtu~ao fus~ons before it decays . The sticking is then a serious problem if one can reach, as it seems possible, very high values of the formation rate . Clearly it is necessary to study ti6 the muon stays bound to He after it sticks to it . At the formation, the (ua) has a kinetic energy Ei n = 3.5 MeV and then it slows down through a series of ionizing collisions with the surrounding molecules . On the other hand, during the slowing

NEGATIVE MUONS IN MATTER

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Figure 6 Simplified picture of the fusion chain down, before the kinetic energy has become smaller than the appropriate thresholds, stripping reactions can occur which strip the (ua) and bring the moon again into the cycle of nuclear reactions (ua) +d

-"

u+a+d

(ua) + d

1

(ud) + a

The competition between these collisions results in a reactivation coefficient R such that the effective sticking probability at the end of the slowing down is reduced with respect to the initial value weff

-

(1-R)ws

off has been calculated by Gerstein et a1 . 14 ) and by Bracci et al . l s) . The results of the two calculations, weff = " 86% and weff = " 91% are in agreement within the errors of the approximation used by the two groups . It follows that the reactivation processes (1) are of some help in increasing the number of possible fusions, however it is clear that the sticking is the severe, ultimate problem for having more than one hundred of catalysed reactions per moon . A warning is needed : all the informations we have so far on the sticking effect are just theoretical . It is nice that two groups found the same results independently, nevertheless, since the chain of competing reaction is rather com plicated, some important point might be missed . It is therefore extremely desirable to have experimental results on this important point . Is it possible to avoid the limitations imposed by sticking problem ? A possible way out would be to provide energy to the (ua)in order it stays a longer time at energies higher than the thresholds for (1) . For example, by applying an electric field of the order E/p ~ 150 K volts/(cmxAtm) the value of R is increased by a factor two with respect to the zero field case . Again, this is not a substantial gain and, more important, breakdown occurs before you reach so high electric field . In conclusion, one needs some clever idea in order to solve the sticking problem .

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G . FIORENTINI 5 . Transfer reactions This is another class of reactions which are specific of mesic Hydrogen (Pu) + Z i (uZ) + P

On this subject there has been some recent theoretical work which needs experimental confrontation . The russian group' s '" ), as a result of their accurate calculation on mesic molecules, have discovered that molecular effects are possible in the transfer to He " They prove the existence of bound, unstable states in the X-u-He system (see Fig .7), with X=p,d,t and He = 3 He, 4 He . Formation of these states occurs

0

Figure 7 Molecular terms in the p-u-a system through Auger emission . For example (uP) + ~ . ~

((PUa)e) + e .

During their vibrational motion, the proton and the He nucleus can radiate, making a transition to the dissociated (ua) + p state (Pu«) + (ua) + P + Y It is interesting to observe that the wavelength of the photon depends on the relative distance of the nuclei at the emission time, thus we have a photon spectrum like in Fig " 8 . This molecular effect substantially increases the transfer rate to Helium (see Table II) . The experimental data on the transfer reactions are very old and poor . New experiments, which could use the emitted photon as a direct signature of the transfer reaction, would be very interesting . Transfer reactions can be interesting for quite different purposes . Some authors' 8 '' s ) have considered the possibility of using the transfer reaction as a probe for the study of surfaces . ~~ idea is very simple . Transfer reaction can have cross-sections as large as 1 .Therefore when a muonic hydrogen atom comes in contact with a solid sample the transfer reaction will occur close to the

NEGATIVE MUONS IN MATTER

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Figure 8 Photon spectrum in the pu+a -" ua+p+y

Authors *'c v

Schiff (1961)

â x w

Zamidoroga et al .(1963)

E "i

pu+ 3 Ne

pu+4 He

L

v r

< 1

tu+ 4 He

-

-

< 0 .1 ~ 10-3

(1962)

Matveenko and Ponomarev (1972) Aristov et al .(1980)

tu+ 3 He

< 10

Placci et al . (1962) Gerstein

du+4 He

du+3 He

I

0 .063

0 .055

0 .87

0 .44

Table II : u-Transfer rate to H e , a tr

X

-

I

-

0 .013

0 .010

1 .48

2 .03

10-8,[s-1],

5 .62

1 .98

from ref .16 .

surface . Different depths of the sample can be explored by using muons, pions, and by varying the temperature . Also, the transfer reaction has a specific signature, the X-ray of the muonic atom which is formed . In this way, for example, the chemical composition of the sample close to its surface could be studied .

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G. FIORENTINI 6.

Summary and outlook

From the foregoing discussion it appears that the theoretical and experimental work in the last years has opened the door of a new, refined spectroscopy . In addition to the determination of the energy levels of mesic molecules it is now also possible to detect transitions between the mesomolecular states . The recent result of Bardin et al . e °), wh~ m~asured the ortho-para transition rate in the has pup molecule ,l op = (4 .1± 1 .4)10 san intrinsic interest, besides being important for the study of weak interactions . There are still many open problems . For the dud ion the recent result by Breûnlich et al . has shown that large effects on the formation rates can arise from the hyperfine structure of levels . Now we have to understand the hyperfine structure in detail . May be that these experiments also give some light on the presumably more complicated (dtu) system . There has not been any substantial new development on the study of the (dtu) ion since the Vancouver conference aqd we still are faced with the problems menti oned i n" ) . It seems to the author that there is not much space for theoretical progress after the impressive work of Ponomarev's group ee ), unless new experimental results are available . There is really a need to have new experiments with intense muon beams . Only some new devoted experiments, as those outlined in 9 ), can clarify the formation of dtu molecules and ultimately answer the question : is the idea of getting energy from muons ingenious or ingenuous ? Furthermore, there is some interesting work to be done on the transfer reactions, most of the questions being again for the experimentalists . Finally, let me mention that there is a disagreement between the experimentale ° ) and theore tical value e ') of aop , which deserves further investigation . References

2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

For reviews on this subject see L .I . Ponomarev and S .S . Gerstein , in Muon Physics fC .S . Wu and W . Hughes edsl, Academic Press, N .Y . (1975) H.Schneuwly,in Exotic Atoms 1979 [K . Crowe, J . Duclos, G . Fiorentini, G. Torelli eds] Plenum N .Y . (1980) . See also the contribution by M . Boschung et al ., I18, and by H . Daniel et al ., Y3, presented at this conference E.A . Vesman, J .E .T .P . Letter s _5 (1967) 113 S .I . Vinitski et al ., J .E .T .P . 47 (1978) 444 See the relatively recent reviewby J . Rafelski in Exotic Atoms 1979 V .B . Bystritsky et al . J .E .T .P . _49 (1979) 232 S .I . Vinitski et al ., Zh . Ek . Te Fiz . _79 (1980) 698, to be translated in J .E .T .P . D .D . Bakalov et al ., Zh . Ek . Te . Fiz . _79 (1980) 1629, to be translated in J .E .T .P . W .H . Breunlich, Nucl . Physics A353 (1981) 201 W .H . Breunlich et al . Paper I15, presented at this conference S .S . Gerstein et al ., J .E .T .P . 51 (1981) 1053 S .S . Gerstein and L .I . Ponomarev, Phys . Lett . 72B (1977) 80 V .M . Bystritsky et al ., Phys . Letters 94B (198476 V .M . Bystritsky et al ., Dubna preprint D~80-788 (1980) See, for example, L .W . Alvarez, Adventures in Experimental Physics a 92(1972) Yu . Petrov, Nature, 285 (1980) 466 and, by the same author, in "Proceeding of the XIV Leningrad-Tifinter School" S .S . Gerstein et al ., Dubna preprint p4-80-632 and contribution I19 presented at this conference L. Bracci and G . Fiorentini, Pisa preprint 1980, in publication on Nucl . Phys .A

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NEGATIVE MUONS IN MATTER

16) 17) 18) 19) 20) 21) 22)

Yu . A . Aristov et al ., Dubna preprint p4-80-378 Yu . A . Aristov et al ., Leningrad preprint 635 P . K . Haff, Phys . Letters 62A (1977) 301 L . Bracci and G . Fiorentini, Phys . Letters 78A (1980) G . Bardin et al ., CERN preprint EP/81-55 andcontribution M35, presented at this conference D .D . Bakalov et al ., Contribution I17 presented at this conference One should also mention that the russian group, besides discussing the molecular physics problem, recently reconsidered also the study of nuclear interactions in mesic molecules . See the paper by L .N . Bogdanova et al ., I16 and by V .B . Belyaev et al ., I20 presented at this conference . Discussion

A. I. Yavin (Tel Aviv) : What about the possibility (apart from orbiting arormd nuclei) in solids ?

of trapping u + 's and u

's

G . Fiorentini : The study of u + 's trapped in interstitial positions in solids, and also the study of their diffusion between different interstitial positions, is now a widely used technique for the study of proton-like impurities, termed uSR . For a recent review see Exotic Atoms 1979 (ref . 1) and the proceedings of the 1980 uSR conference in Vancouver . _ I do not know of any possibility of trapping u 's apart from orbiting in muonic atoms . Clearly it would be very interesting to have such a possibility think for example about trapping ~'s !!