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This article was downloaded by: [Temple University Libraries]On: 18 November 2014, At: 03:57Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK
Journal of Nuclear Science and TechnologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tnst20
Visibility of the growth direction of an alpha-particletrack in a diffusion cloud chamberChizuo Moriaa Department of Electrical Engineering, Aichi Institute of Technology, 1247 Yachigusa,Yagusa-cho, Toyota-shi, Aichi 470-0392, JapanPublished online: 05 Nov 2013.
To cite this article: Chizuo Mori (2014) Visibility of the growth direction of an alpha-particle track in a diffusion cloudchamber, Journal of Nuclear Science and Technology, 51:2, 196-200, DOI: 10.1080/00223131.2014.854710
To link to this article: http://dx.doi.org/10.1080/00223131.2014.854710
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Journal of Nuclear Science and Technology, 2014Vol. 51, No. 2, 196–200, http://dx.doi.org/10.1080/00223131.2014.854710
ARTICLE
Visibility of the growth direction of an alpha-particle track in a diffusion cloud chamber
Chizuo Mori∗
Department of Electrical Engineering, Aichi Institute of Technology, 1247 Yachigusa, Yagusa-cho, Toyota-shi, Aichi 470-0392, Japan
(Received 1 May 2013; accepted final version for publication 8 October 2013)
The growth direction of an alpha-particle track in a diffusion cloud chamber can be observed with thenaked eye. Unlike the track of a beta-particle, an alpha-particle track appears first at the beginning of itspath and proceeds to the end of the path. However, the alpha-particle covers the path in less than 10 ns,which is inconsistent with the visibility of the track-growth direction. This inconsistency was attributed tothe following reasons: the number of ion pairs per unit length in the path is very large compared with thatof a beta particle, the ion density near the end of the path is larger than that near the beginning, and alcoholdroplets have to grow to a diameter of about 3 μm for being able to be seen. The alcohol molecules have todiffuse from a distant place to the region where the ions are created by the alpha-particle and the diffusiontime near the end of the path is longer than that near the beginning of the path. This phenomenon wasexamined by diffusion theory.
Keywords: alpha-particle; cloud chamber; track; growth direction; energy loss; diffusion; ion pairs
1. Introduction
Simple diffusion cloud chambers with an alpha-particle source and ethyl alcohol (hereafter alcohol) arefrequently used in radiation education to “see” radia-tions. The white linear line that appears after the passageof an alpha-particle is called a track.
With the naked eye, it is possible to see the track ex-tending from the vicinity of the source to the end of thetravel path of the particle. However, the alpha-particletravels the range in several nanoseconds (ns), so one ex-pects the entire track to appear simultaneously throughall the track length. This apparently inconsistent phe-nomenon is examined in this study.
2. Past studies on track formation in cloud chambers
Since Wilson invented the expansion cloud chamber[1], these chambers have frequently been used to findnew particles [2,3]. In 1939, Langsdorf developed thediffusion cloud chamber [4]. These chambers were usedwith accelerators for research into high-energy nuclearphysics [5]. However, because of the development of newelectronic radiation detectors, cloud chambers are littleused today, except for radiation education purposes.
Large expansion cloud chambers were used mainlyto study cosmic rays. The momentum and the sign of the
∗Email: [email protected]
electric charge of a new particle were determined withthe help of a strong magnetic field. To count the num-ber of individual droplets in the track for finding the en-ergy loss (−dE/dx), they had to allow ions diffuse at least0.1 s before expanding the chamber. After expanding thechamber, the droplets grew to a few micrometers and astrobe light was flashed [6,7] to photograph them.
In the study of cosmic rays, researchers observedcomparatively lighter particles such as positrons, muons,and pions. Because the ion density created along thepath of these light particles is very less and almost con-stant, the liquid droplets appear almost simultaneouslyalong the entire particle path. Therefore, the direction ofthe track growth was not accessible for researchers. Ac-tually, in large diffusion cloud chambers, beta-ray tracksappear simultaneously throughout the path and we arenot able to see the growth direction. Conversely, the di-rection of the tracks of alpha-particles can be deter-mined with the naked eye.
3. Inconsistency concerning the growthof alpha-particle tracks
3.1. Travel time of alpha-particlesThe energy loss of heavy charged particles inmaterial
is expressed by the Bethe equation [8]:
C© 2013 Atomic Energy Society of Japan. All rights reserved.
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Figure 1. Relation between the travelling distance and trav-elling time of alpha-particles with different energies in air.
− dEdx
= 4πr 20 z2mc
2
β2NZ
[ln
(2mc2
Iβ2
)
− ln(1 − β2) − β2] , (1)
β2 = 1 − 1{E
/(M0c2) + 1
}2 , (2)
where r0 and m are, respectively, the classical electronradius and the electron rest mass; z, β, E, and M0 are,respectively, the charge number, ratio of particle velocityto the speed c of light in vacuum, energy, and restmass ofthe particle; andN,Z, and I are, respectively, the numberof atoms per unit volume, atomic number (or effectiveatomic number), and average ionization energy for thematerial.
Equation (2) gives the velocity v for an alpha-particlewith initial energy E0. From the minute travelling dis-tance �x, we can derive the minute travelling time �t(= �x/v). The minute energy loss �E [= (−dE/dx)�x] isdetermined by Equation (1). The energy E of an alpha-particle after travelling a distance �x is E = E0 − �E.Successive calculations give the relationship betweentravel time t (= ��t) and distance travelled x (= ��x),as shown in Figure 1. The travel time t is a function of thealpha-particle energy. The maximum energy of a naturalalpha-particle is 8.785MeV (for 212Pb in the thorium se-ries). The travel time is, therefore, less than 10 ns for allnatural alpha-particles.
3.2. Growth time for an alpha-particle trackThe growth of an alpha-particle track was observed
at 60 frames per second by a digital camera in moviemode. As shown in Figure 2, there is no track in the first
Figure 2. Time sequence of the growth of an alpha-particletrack in a diffusion cloud chamber observed by a movie-modecamera. There was no track at 0 s. There appeared a short track17 ms after 0 s. The track growth completed after 51 ms.
frame (0 s), then the beginning of a track appears in thesecond frame (17 ms), and it grows in the third frame(34 ms). Finally, the track is completed in the fourthframe (51 ms). Although this growth time depends onthe degree of supersaturation of the alcohol vapor [3],the time (less than 10 ns) for ion creation by the travel ofan alpha-particle differs greatly from the time (∼50 ms)for track growth. This discrepancy has not been pointedout and the reason behind it is yet to be explained.
4. Alpha-particle track-growth time and the direction
4.1. Explanation of the visibility of thetrack-growth direction
The reason why the track-growth direction can bedetermined by the naked eye is related to the extremelylarge density of ions per unit length along the pathtravelled by an alpha-particle, the ion density differ-ence along the path, and the small diffusion velocity ofethanol molecules in the cloud chamber.
Figure 3 shows the energy Eα [calculated by Equa-tion (1)] of an alpha-particle with an initial energy of
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Figure 3. Energy Eα of alpha-particles with initial energy6MeV, number of ion pairs perμm length of the alpha-particletravelling distance, and perpendicular range Rp of δ-ray elec-trons which compose a conical shape expressed by the dottedline. In each 1 μm length, n ion pairs and hence n ions are in-cluded as a function of alpha-particle travelling distance.
6 MeV as a function of the distance travelled in air un-der normal conditions. The range of an alpha-particlewith 6MeV is about 4.6 cm, as shown in Figure 3, whichagrees with the reference value 4.7 cm [9]. This agree-ment gives us confidence that the calculation using theBethe equation is correct. At the beginning of the pathof the 6 MeV alpha-particle, the energy loss calculatedby Equation (1) is 87 eV/μm. Since theW value for air tomake an ion pair is 34 eV, 2.6 ion pairs/μmare created onaverage, as shown in Figure 3. However, near the end ofthe path, the energy loss is about 200 eV/μm and about6 ion pairs/μmare created.Most of the created electronsin the ion pairs attach to oxygen molecules, which formnegative ions and become the nucleus for alcohol dropletgrowth as well as positive ions of ion pairs.
When low electric voltage is applied to an ioniza-tion chamber, it is known that the ionization currentwill be small due to the recombination of the ion pairs.When the applied voltage is zero, the current will be zero.A similar phenomenon would appear in a cloud cham-ber without applied electric voltage. However, since thedroplets actually appear, not all of the ion pairs recom-bine. It is difficult to determine the degree of the recom-bination. Here, it is supposed that the half of the ionpairs recombine. Itmeans that the initial n ion pairs leaven ions.
The droplets scatter light by Mie scattering, so to bevisible to the naked eye they must grow to over about3μm in diameter [6,7,10,11]. A droplet 3μm in diametercontains about 1.38× 1011 alcohol molecules. The tem-perature at the upper part of the diffusion cloud cham-ber is about 20◦C (saturated vapor pressure of alcohol is5870 Pa) and the temperature at the lower part is about
Figure 4. Schematic drawing of ions within a perpendicularrange Rp of δ-rays and the diffusion area of alcohol moleculesin discs with a thickness of 1 μmwith different radii, as shownby the white area. A disc with a radius of 290 μm is situated at2 cm of the travelling distance in Figure 3, which includes threeions in the perpendicular range. The other disc with a radiusof 415 μm is situated at 4.2 cm and includes six ions.
−40◦C [4] (saturated vapor pressure of alcohol is 1000Pa). Therefore, the degree of supersaturation (supersat-uration ratio) is 5.8 which satisfies the condition that thedegree of supersaturation is greater than 4 [12] to growethanol droplets in air. In saturated vapor at 20◦C, thedensity of alcohol molecules is 1.55× 1018 cm−3. Thus,three droplets correspond to three ions /μmat the begin-ning of the track require 3× 1.38× 1011 (= 4.14× 1011)molecules to grow to droplets with a diameter of 3 μm.These alcohol molecules are contained in a volume of2.67× 10−7 cm3 of air, which corresponds to a circulardisk with a thickness of 1 μm and a radius of 290 μm,as shown by the white part in Figure 4. For six droplets(six ions) near the end of the track, the disk radius mustbe 415 μm.
We expect that the diffusion time of alcoholmolecules in the larger disk diameter required to growsix droplets with 3 μm in diameter near the end of thetrack is longer than that required to grow three dropletswith 3 μm in diameter near the beginning of the track.This would be the reason why we can see the directionof the track growth of an alpha-particle.
4.2. Theoretical considerations regardingthe direction of track growth
To calculate the diffusion time of the alcoholmolecule to make an alcohol liquid droplet by being at-tached to N2 or O2 ions, it is important to find the re-gion where ion pairs are created by an alpha-particle
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travelling in air. Although little information is availableon this matter, observations reveal that many electronsliberated by alpha-particles have nearly the same veloc-ity v as the alpha particle [13]. These electrons are calledδ-rays and can ionize air molecules.
When an alpha-particle with rest massM and energyEα expels a free electron with rest mass m, the energyEe(θ ) of the electron with angle θ against the direction ofthe alpha-particle path through elastic collision is givenby the classical theory:
Ee(θ ) = 4mEα(cos θ )2/M. (3)
From Equation (3), an alpha-particle of velocity v givesthe maximum velocity 2v and the average velocity 1.5vto the expelled electron. Even though the collision is ac-tually non-elastic, this calculation almost coincides withthe experimental resultsmentioned above. Therefore, thelargest distanceRp of ion pairs yielded by δ-rays perpen-dicular to the alpha-particle path is obtained by the fol-lowing calculation.
The penetration range Re(θ ) in air of this electronwith angle θ is obtained by the Bethe equation (−dE/dx)for electrons, and then the component of the displace-ment of the δ-ray perpendicular to the alpha-particlepath is given by Rp(θ ) = Re(θ )sinθ . These perpendicu-lar ranges Rp(θ ) with various angles are averaged overθ = 0∼π /2 by numerical calculation to obtain the aver-age perpendicular range Rp.
Rp =∑n
i=1 Re (θi ) sin (θi ) Re (θi ) sin (θi ) 2πRe (θi )�θ∑ni=1 Re (θi ) sin (θi ) 2πRe (θi )�θ
,
(4)
where θi = (π/2)(i/n), �θ = (π
/2)(
1/n), and n = 45.
The above summations in Equation (4) were carried outin steps of 2◦ (= π /(2 × 45) radians).
It is considered that most ion pairs are substantiallyproduced in a distance less than Rp. Examples of aver-aged values are as follows: 10.2 μm for Eα = 1 MeVand 70.5 μm for Eα = 4 MeV. Thus, the region in whichions are produced is conical in shape and may be recre-ated by summing many thin discs with different radiiRp, as shown in Figure 3 by the dotted line and also inFigure 4.
Electrons in ion pairs are considered to combinewithoxygen molecules forming negative ions O2
−. Becausehalf of the ion pairs recombine (as mentioned in Sec-tion 4.1), n ion pairs become n ions, including positiveand negative ions. Positive and negative ions combinerather quickly with alcohol molecules and the diffusioncoefficient of the droplets is very small compared withthat of alcohol molecules. For this reason, only the dif-fusion of alcoholmolecules is taken into account in whatfollows. Owing to the geometry shown in Figure 4, thediffusion equation is expressed in cylindrical coordinates
[14] to describe a disk as a part of a conical shape.
δCδt
= 1r
δ
δr
(r D
δCδr
). (5)
In Equation (5), C is the density of alcohol molecules atradius r, t is the time, and D is the diffusion coefficient.In the steady state, the flow rate Q (molecules/(s× unitaxial length)) from radius b with density C2 to radius awith density C1 is expressed by the following equation[14]:
Q = 2πD (C2 − C1)ln (b/a)
, (6)
where C2 = 1.55× 1018 cm−3 and C1 = 0 in the presentcase. Several ions are distributed within the disk of ra-dius Rp, so it is appropriate to consider the radius a =Rp. The diffusion coefficient D for an alcohol moleculein air is 0.088 cm2/s which is the average value of the ex-perimental value and the theoretically determined valueextended to −40◦C by using values from the literature[15].
When the alpha-particle energy Eα is considered inFigure 3, the corresponding number of ion pairs (ac-tually the number of ions) and the radius Rp of thedisk with thickness 1 μm are obtained as shown in Fig-ure 3. When Eα = 4 MeV, we find n = 3 and Rp =70.5 μm, as shown in Figure 3. Then the 3× 1.38× 1011
(= 4.14× 1011) alcohol molecules (obtained in Sec-tion 4.1) within the radius b= 290μmmust diffuse to ra-dius a=Rp = 70.5μm and the required diffusion time iscalculated by Equation (6) under the simple assumptionthat the molecules are supplied by a steady-state currentfrom radius b to radius Rp. The actual diffusion, how-ever, is not a steady-state current but a transient current.The quasi-transient state was considered by doing suc-cessive numerical calculations overminute regions underthe steady-state assumption. In the first step, we calcu-late the steady-state current from b = 150 μm to Rp for1 ms. Next, the steady-state current from b= 200 μm toRp for 1 ms is calculated, then from b = 250 μm to Rp,and so on.
Curve (A) in Figure 5 is for Eα = 4 MeV making3 ion pairs at about 2 cm of the alpha-particle track byreferring to Figure 3. The required diffusion time (timeafter the passing of an alpha-particle) to have 4.14× 1011
alcohol molecules is found to be 6.5 ms, as shown bythe arrow (a) in Figure 5, which corresponds to the timerequired to grow each of the three droplets to a diam-eter of 3 μm. Similarly, curve (B) is for Eα = 1 MeVmaking 6 ion pairs at around 4 cm of the alpha-particletrack by referring to Figure 3. For six ions, 8.28× 1011
molecules must diffuse to the radius Rp = 10.2 μm,which gives a diffusion time of 42.5 ms shown by the ar-row (b). The difference of 36 ms (= 42.5–6.5 ms) is closeto the experimental value of 35 ms (= 51–17 ms) shownin Figure 2. Thus, the reason why the direction of the
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Figure 5. Number of alcohol molecules per μm length of thealpha-particle track, which diffuse into the area where ionsare present in the perpendicular range Rp for growing dropletsto form droplets. Curve (A) is the case where three ions arepresent per μm and they need 4.14× 1011 molecules for grow-ing three droplets with a diameter of 3 μm in 6.5 ms, as shownby the arrow (a). Curve (B) is the case where six ions are presentand they need 8.28× 1011 molecules for growing six droplets in42.5 ms, as shown by the arrow (b).
alpha-particle track growth can be seen by the naked eyeis that the diffusion time of alcohol molecules at the be-ginning of the track is shorter than that at the end of thetrack.
5. Summary
We examined why it is possible to see the growth di-rection of an alpha-particle track in a diffusion cloudchamber with the naked eye despite alpha-particle traveltime of less than 10 ns.
Alcohol droplets must grow to a diameter of about3 μm to become visible by Mie scattering of light. Theenergy loss (−dE/dx) of an alpha-particle and thus thenumber of ions per unit length of the alpha-particle pathare very large compared to the corresponding quantitiesfor beta-particles. In addition, the number of ions perunit length near the end of the path is larger than thatnear the beginning of the path. Therefore, near the endof the path, a larger number of alcohol molecules have
to diffuse from a farther distance than that near the be-ginning of the path. The time required to grow dropletsto a diameter of about 3 μm near the end of the path islonger than that near the beginning of the path. For thisreason, the alpha-particle track appears first at the be-ginning of the path and then at the end of the path, andthereby the direction of the track growth may be seen bythe naked eye.
AcknowledgementsThe author acknowledges theRadiation Education Forum
(NPO), Chubu Atomic Power Conference, and Japan AtomicEnergy Relations Organization, which have allowed him toattend various radiation seminars where he could find someinteresting phenomena such as the subject of this paper. Heis also grateful to the valuable discussions with Prof. TetsuoIguchi and Prof. Akira Uritani of NagoyaUniversity and Prof.Hiroyuki Takahashi of the University of Tokyo.
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