Nuclear Instruments and Methods in Physics Research B 315 (2013) 278–282

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Nuclear Instruments and Methods in Physics Research B

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Radiation emission as a virtually exact realization of Heisenbergsmicroscope

0168-583X/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.nimb.2013.03.041

⇑ Corresponding author.E-mail address: kka@phys.au.dk (K.K. Andersen).

1 The term ‘‘latitude’’, from the danish word ‘spillerum’, was the term which Bohrused in the Introductory Survey in relation to the claim that ‘‘a certain amount oflatitude be allowed in our account of the mutual action between the object and themeans of observation’’, N. Bohr – Atomic Theory and The Description of Nature,Cambridge University Press, Cambridge, 1961.

K.K. Andersen a,⇑, S. Brock b, J. Esberg a, H.D. Thomsen a, U.I. Uggerhøj a

a Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmarkb Department of Culture and Society, Aarhus University, Jens Chr. Skous Vej 5, 8000 Aarhus C, Denmark

a r t i c l e i n f o

Article history:Received 20 November 2012Received in revised form 21 March 2013Accepted 29 March 2013Available online 8 April 2013

Keywords:RadiationBremsstrahlungFormation lengthHeisenbergs microscope

a b s t r a c t

Through the concept of ‘formation length’, recently observed directly in the radiation emission fromultrarelativistic electrons and an essential component in the interpretation of strong field radiation fromelectrons penetrating single crystals, we discuss the indeterminacy in the location of radiation emission.The analogy with the indeterminacy in the Heisenberg microscope Gedanken experiment is demon-strated from a number of viewpoints to be almost exact. The positive attitude regarding photon emissionas a process that is somehow located in space and time is emphasized. We therefore interpret the mea-surements of formation lengths in radiation emission as a practically realizable version – using virtualincident photons instead of real – of the Heisenberg microscope Gedanken experiment.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction the configuration of the meters and scales in terms of which this

This paper discusses a new interpretation of experimental re-sults, but although the main emphasis is on experiments, it doesnot report any new measurements. Instead, based on recent as wellas old findings, we show that radiation emission from energeticelectrons in several contexts can be seen as an equivalent of the fa-mous Heisenberg microscope, a Gedanken experiment that in text-books is referred to as being ‘not experimentally possible’ [1].Strictly speaking, this may be correct although experiments per-formed at much lower energies claim otherwise [2], but as weshow below a new interpretation of the indeterminacy connectedto the exact location of radiation emission makes a very closeexperimental connection. Recent investigations include neutron-optical tests of ‘‘error-disturbance uncertainty relations’’ [3].

In our context, in whatever sense the indeterminacy relationshave an epistemic implication, the significant thing is methodolog-ical and conceptual. Heisenberg – and later Bohr – wanted to dem-onstrate that the use of kinematic and dynamic concepts withinquantum theory only made sense in relation to specific modelsof physical experiments [4]. Consequently, the very distinction be-tween a parameter that could be assigned a determinate value and

value could be measured, is only possible if a certain amount of lat-itude1 is recognized. No phenomenon for which there can be specificforms of empirical evidence can be described in abstraction fromsuch latitude. This was the context of the Gedanken experiment ofHeisenberg, which combines the principal view on microphysicalphenomena – the view pointing to an unavoidable indeterminacyin the joint determination of conjugate variables – to the macrophys-ical issue of possible evidence, i.e. possible measurement of the val-ues of specific parameters. A crucial statement in Heisenbergsoriginal paper is

When one wants to be clear about what is to be understood bythe words ‘‘position of the object’’, for example of the electron(relative to a given frame of reference), then one must specifydefinite experiments with whose help one plans to measurethe ‘‘position of the electron’’; otherwise this word has nomeaning [5].

Here, the important thing is the positive attitude of both Bohrand Heisenberg. The issue is not that physical understanding andexperimental practice has reached some sort of limit. The issue isthat the imaginative design of experiments opens up, in evernew ways, for the possibility of tracing new forms of physical phe-nomena. This happens because one can analyze and configure howany such particular experimental design embodies the possibilityof determining specific values of relevant parameters. So the pointof the Gedanken experiment is not to say, negatively, that we cannever know at which spatiotemporal point a certain photon wasemitted. The point is that despite the indisputable fact that we can-

D

f

kt

ki

Fig. 1. A schematic diagram showing the Heisenberg microscope.

K.K. Andersen et al. / Nuclear Instruments and Methods in Physics Research B 315 (2013) 278–282 279

not talk of such a point, it still makes very determinate sense totalk about an emission that is somehow located and associated withcertain items present within the physical situation at issue. This istrue even if the Gedanken experiment of Heisenberg could not beperformed in practice. Accordingly, part of what we want to showin this paper is that the idea of the formation length in connectionwith radiation emission can be interpreted in the same positiveway. This idea guides and helps us in the design of experimentsthat we can perform today. In the present context we focus mainlyon bremsstrahlung which is emission of radiation when an ener-getic electron interacts with a medium. The formation length setsthe scale over which this process occurs.

In this context, it should be noted that the present paper implic-itly is critical of the view, which is expressed for instance by Pringe[6] and Camilleri [7] to the effect that Heisenberg provided thedescription of the microscope in order to meet a criterion imposedby semantic operationalism. It is true that Heisenberg agreed withthe latter position, that a fixation of the meanings of theoreticalconcepts requires a description of possible forms of measurement.However, we believe that this was just a necessary, not a sufficient,criterion for Heisenberg. For the issue is not only to determine themeaning of theoretical concepts, but also to determine the refer-ence of these concepts, i.e. to specify in what sense the conceptsare applicable to, precisely, atomic processes characterized by theQuantum Postulate of Planck. We think that, despite other dis-agreements, Heisenberg and Bohr were in agreement already inearly 1927 about the view that (a) to construct theoretical modelsof atomic systems and processes and (b) to construct models ofpossible experiments in order to measure relevant parameters inrelation to such systems and processes, are two tasks that couldnot be separated [8]. Indeed, in this paper we argue that the notionof ‘‘formation length’’ is of importance both concerning the task (a)and the task (b) and that Heisenbergs description of the micro-scope included an attempt to fulfill both of these tasks.

It is important to mention that one can find similar formationlength related effects in other fields of physics. The effect of nuclearshadowing where the finite size of the nucleus has to be taken intoaccount in photoproduction of mesons is one example (see e.g. [9]).Nuclear shadowing is a suppression effect similar to those to bediscussed in this paper. Oppositely there exist constructive effectsin connection to photoproduction of mesons from bulk matter athigh energies [10]. One can also find several coherence related ef-fects in atomic physics. One very important effect in quantuminformation storage is loss of coherence due to interactions withthe surroundings.

Our paper is organized as follows: first a recapitulation of theHeisenberg microscope, followed by a discussion of the conceptof formation length in connection to bremsstrahlung. We showthe equivalence between the lack of knowledge of the exact loca-tion of the electron at the radiation event imposed by the indeter-minacy relations in the two cases: Heisenbergs microscope andhigh energy radiation emission.

2. The Heisenberg microscope

In the following, we recapitulate the main contents of the Hei-senberg microscope and the restrictions imposed on the observedphenomena in this Gedanken experiment. In the original paper byHeisenberg on indeterminacy [5], an ‘imaginary c-ray microscope’is described to illustrate the indeterminacy in the momentum ofthe electron imposed by a measurement of its position. The mea-surement proceeds by means of the scattering of radiation off theelectron, see Fig. 1. As a consequence of the ‘discontinuous changein momentum’ – as understood through the Compton effect – theindeterminacy relation DqDp � h was obtained and shown to be

a consequence of the commutation relation pq� qp ¼ �i�h, whereq denotes the position and p the mass times the velocity. However,as was realized by Bohr (and added in proof by Heisenberg), it isthe ‘divergence of the bundle of rays’ scattered, i.e. the classicalresolving power of the microscope k=2� with � ’ D=2f , see Fig. 1,that leads to the indeterminacy [5,11,12]. As later put by Bohr:

In fact, any attempt at locating atomic objects in space and timedemands an experimental arrangement involving an exchangeof momentum and energy, uncontrollable in principle, betweenthe objects and the scales and clocks defining the referenceframe [13].

from which it is clear that – in the specific example of Heisenberg’smicroscope – although the energy and momentum imparted to theelectron may be well-defined and controllable, the exchange of thesame quantities with the classical apparatus is not.

This paper does not attempt at resolving the issue whether ornot this ‘resolving power’ argument is correct (see e.g. [14,15] fora critique), but aims to show that this possible interpretation canbe closely imitated by experiment, i.e. that the Heisenbergmicroscope as described originally [5] to a large extent is realiz-able. Other interesting comments on the philosophical standpointsof both Bohr and Heisenberg in this connection can be found in[16].

3. Formation length

The formation length is loosely speaking the length of travel foran electron during the time interval in which an emitted photonseparates from it by a reduced wavelength. This distance becomesvery long due to the small difference in velocity cð1� bÞ ’ c=2c2

which appears at high energies. Here v ¼ bc is the speed of theelectron and c ¼ E=mc2 the Lorentz factor related to the energyof the electron, E. The aim is to show, that in analogy to the Heisen-berg microscope case, a reduction of the indeterminacy in the ac-tual location of emission (i.e. the formation length) isunavoidably accompanied by radiation of higher energies, i.e. thescattering by the electron of virtual photons of shorter wavelength.Thus – as in the Heisenberg microscope case – to pinpoint the loca-tion of emission requires short wavelength (virtual) radiation,emitted as high energy quanta, whereby the uncertainty in the

Fig. 2. The normalized photon power spectrum level from 149 GeV electrons penetrating Ta foils, as a function of the independent foil thickness (in units of the radiationlength X0 on lower scale, equivalent tantalum foil thickness on upper scale). With squares (and error bars denoting the statistical uncertainty) are shown the measured valuesfor a photon energy bin with the centroid value of 1.05 GeV. The transition from the LPM to the BH level, going to thinner targets, proceeds according to a logarithmic functionof the thickness. The thickness where the deviation from the LPM level sets in, corresponds to the formation length of a 1.05 GeV photon emitted from a 149 GeV electron,indicated by a vertical dash-dotted line. Adapted from [24].

[GeV]ωh

-210×2 -110 -110×2 1 2

m /

refe

renc

eμ

45

0.2

0.4

0.6

0.8

1

1.2

1.4

BGO data

δm MC BD-μ45

Fig. 3. The ratio between a measurement with lg ¼ 45 lm and the reference measurement. Both spectra have been background subtracted. The measurements are shownwith stars and a Monte Carlo simulation using the theory of Blankenbecler and Drell is shown. The horizontal bar is the bin width and the vertical bar is the statistical error.

280 K.K. Andersen et al. / Nuclear Instruments and Methods in Physics Research B 315 (2013) 278–282

momentum of the electron becomes large. This of course must beso following the present understanding of the radiation emissionprocess which is in accordance with quantum theory. The aim istherefore not to display the internal consistency, but to show thestrong similarities with the Gedanken experiment of the Heisen-berg microscope.

3.1. Classical formation length

As for the c-ray microscope that has a classical resolving power,in classical theory there is a formation length, resulting from thetime it takes for a photon to separate from the electron by one re-duced wavelength, k

��¼ k=2p. The corresponding distance of travel

of the electron, lf , is

lfv ¼ lf þ

k2p

� �1c

ð1Þ

which for v ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� 1=c2Þ

pc ’ ð1� 1=2c2Þc yields

lf ¼2c2cx

ð2Þ

Other classical approaches can be found in [17].Ter-Mikaelian was the first to discover that it takes relatively

long time and therefore a long distance for an energetic electronto create a photon [18]. The interactions of the electron over this‘formation zone’ affect the radiation spectrum decisively and maylead to enhancement or reduction of total intensity as well aschanges in the spectral shape. The original claim by Ter-Mikaelianwas at first strongly opposed by Landau who found the idea coun-ter-intuitive and repulsive [19]. As noted by Akhiezer, student andsuccessor of Landau as director at the Theoretical Physics center inKharkov [20]: ‘It is quite remarkable that the collective phenomenaappear at arbitrarily high energies, although at first glance it seemsthat if the particle wavelength is less than the average distance be-

K.K. Andersen et al. / Nuclear Instruments and Methods in Physics Research B 315 (2013) 278–282 281

tween the atoms of the material, collective phenomena should notappear and the material should behave as a gas of independentatoms’.

The quantum mechanical description of the formation lengthcan be obtained by substituting x in Eq. (2) with x� ¼ x E

E��hxand thus including recoil of the electron. For the interested readera more recent article about the formation length is [21].

4. Bremsstrahlung

Direct measurements of the formation length have howeveronly appeared quite recently. In this context, ‘direct’ means mea-surable by means of macroscopic, classical devices such asmicrometer screws or a caliper. The technique behind the mea-surements is based upon the observation of radiation emissionspectra ‘kinks’ arising at different photon energies for targets ofdifferent thicknesses. These ‘kinks’ are due to the fact that oncethe targets become sufficiently thin and the formation length suf-ficiently long, the formation length extends out of the target andsuppression due to multiple Coulomb scattering cease. This is theso-called Landau–Pomeranchuk–Migdal (LPM) effect. An initialstudy of this effect was performed at SLAC, see [22], with an expla-nation by the Kharkov group [23], while more in-depth investiga-tions were done at CERN [24]. Even earlier channeling radiationstudies showed such effects [25]. It turns out that for targets thin-ner than the formation length, a transition to the incoherentbremsstrahlung regime (Bethe–Heitler (BH)) follows quite closelya simple logarithmic thickness dependent power spectrum

dNc

d ln �hx¼ a

X0� lnðb� Dt þ 1Þ

bð3Þ

where X0 is the so-called radiation length of the target material, Dtis the target thickness and a and b are fitting parameters related tothe total cross-section. A measurement of the thickness of the foil,Dt, is thus a measurement of the formation length for the photonenergy at the ‘kink’ between the LPM and Ternovskii-Shul’ga-Fomin(TSF) regimes. For details on the theories and measurements, see[24]. An example of the expected shape and measured values to-gether with theoretical curves is shown in Fig. 2.

As seen in Fig. 2, the transition from the LPM level towardsthe BH level appearing around a Ta thickness of 30 lm(Dt=X0 ’ 7 � 10�3), is in good agreement with measurements. Manymore examples of such an agreement is found in [24]. So a mea-surement of the foil thickness is effectively a measurement of theformation length of the photon, i.e. an expression of the indetermi-nacy with which we know the location of the electron at the eventof photon emission, exactly as in the Heisenberg microscope. Asmaller thickness corresponding to a smaller uncertainty in thelocation of the electron at the emission event, must be connectedwith the electron interacting with virtual photons of higher energy,leading to a depletion of the less energetic photons observed in thelaboratory, in agreement with observations.

A further modification of the above experiment was performedin 2011 where two thin gold foils with a variable separation lg wasused [26]. Radiation emission is LPM suppressed in both targetsand one can consider this a suppression–alleviation–suppressioneffect that produces a small peak in the radiation spectrum. Theenergy at which this peak appears is directly related to the distancebetween the foils and the formation length of the emitted photon.In Fig. 3 is shown the ratio between a measurement wherelg ¼ 45 lm and one where the separation is large and the targetscan be considered as independent (called reference). A small peakis seen around 400 MeV which is directly related to the separationof the target foils. The original article [26] should be consulted formore informations.

5. Finite spacetime region

Of particular relevance to the formation length concept dis-cussed above, is Niels Bohrs comments on the impossibility ofevents perfectly localized in both space and time:

‘‘According to the classical concepts, the description of the scat-tering requires a finite extent of the radiation in space and time,while in the change of the motion of the electron demanded bythe quantum postulate one seemingly is dealing with an instan-taneous effect taking place at a definite point in space. Just as inthe case of radiation, however, it is impossible to definemomentum and energy for an electron without considering afinite spacetime region. In consequence, according to relation(2), the associated spacetime regions can be given the same sizefor both individuals in interaction [11]’’.

Here, Bohr is referring to ‘relation (2)’: DtDE ¼ DxDIx ¼DyDIy ¼ DzDIz ¼ h, where E and I are the energy and momentumrespectively, i.e. the indeterminacy relations. The ‘finite spacetimeregion’ is in the present context equal to the formation zone, andwe therefore interprete the experimental results discussed above,as corroborating the ideas of Bohr to the effect that the recognitionof this zone is a precondition for the possibility of detecting specificoutcomes of the given experiments.

6. Conclusion

We have shown how the principal ideas of both Heisenberg andBohr – put forward in the context of Gedanken experiments – im-ply a necessary amount of latitude in the determination of the val-ues of conjugate variables. This is illustrated and put to actual usein real experiments in modern radiation physics which invoke theidea of formation length – a direct demonstration of such latitude.We have shown how to interprete the measurements of certainfeatures of bremsstrahlung emission in the light of an associationbetween the indeterminacy of Heisenberg and the idea of forma-tion length. Similar conclusions can be drawn from radiation emis-sion by electrons in strong fields. Such effects have recently beenobserved in crystalline fields [27].

Acknowledgment

We thank A.H. Sørensen for constructive criticism to an earlyversion of the manuscript.

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