The Range of Alpha Particles in Air. Lennon Ó Náraigh, Date of Submission: 5 th April Abstract:

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1 The ange of lpha Particles in ir Lennon Ó Náraigh, Date of Submission: 5 th pril 4 bstract: This eperiment assumes that the notion of ange is applicable to alpha particles an sets out to calculate sai range. In the course of the eperiment, the epenence of the range on the geometry of the apparatus will be investigate. In the eperiment, it was foun that 3.4 ±. cm for air at room temperature an atmospheric pressure. This compares well with the accepte value 3. 4cm. n estimate of the range of alpha particles in water was obtaine from the Bragg-Kleeman rule, an was foun to be ~ (.6 ±.) %. Water [ ] ir H.. Bethe, U.S. tomic Energy Commission, Document BNL-T-7, 949. Energy of alpha-particles: 4.88MeV. These were the conitions of the eperiment: Temperature of ir: 5 o C; Pressure: 76mm.

2 Basic Theory an Equations: realization of the foregoing theory is by 4.88MeV alpha particles (ions with z ) travelling through air. These are N ( v ~ 7 m / s ). lpha particles are highly ionizing an we epect these to interact strongly with matter. Thus, we epect the intensity of -raiation to fall off steeply with source-etector istance an this will efine the range the istance at which the intensity becomes zero. This situation is contraste with electrons, which o not interact so strongly with matter an thus o not ehibit the same strong range-intensity epenence. The rate of a particle s (ion s) energy loss, per unit path length, as it passes through a meium, is known as the stopping power, S of the meium. quantum-mechanical erivation incluing relativistic effects was first carrie out in 93 by Bethe an Bloch, an their formula is quote here: S E 4πZρ N mv ( ze ) log log( β ) β mv I () Where v βc is the ion velocity an ze is its electronic charge in e.s.u. s, m the mass of the electron,, Z, ρ are the atomic mass number, the atomic number an the ensity of the stopping material, respectively. I is the mean energy require to ionize a particle of the material an is about 86eV for air. s usual, terms in β can be roppe in the non-relativistic (N) limit. In a naïve treatment of equation 3 (), we might procee in the following way: Write E C I mv I mv log I o 4 C 4 ( β ) logξ o( β ) I ξ Where the terms in β are self-cancelling. Then, E I ξ E E E C, in the N limit. logξ E Use E M v to change the variable of integration: ξ v mv I m M I M 4m IM E John Lilley, Nuclear Physics, Principles an pplications, (Wiley, ), pp (Oliver Kelly)

3 4m ξ E IM I M 8C m ξ log ξ I 8C Hence, ξ f ( E ) M m (3) IM However, this analysis is wrong. The integran in (3) has a singularity at E, 4m i.e. when the travelling alpha particle has its energy reuce to ~57 KeV. This reflects the fact that Bethe s formula is vali only in a high-energy limit. For, in the low-energy limit, the slow-moving alpha particles can themselves capture electrons, yet Bethe s formula oes not consier this. The range, therefore, must always be calculate semi-empirically: E T E T E S S Spline E E B T T E E S (4) S B is the stopping power accoring to Bethe s equation, vali in the range [ E, T ]. S E is a function etermine eperimentally by fitting a curve to a stopping power versus energy ataset in the interval [, T ], while spline is a cubic spline function fitte on the interval [ T,T ] to ensure continuity between S B an S E. Typical values are T. MeV for helium ions, an T. MeV for helium ions. 4 This semi-empirical approach gives a power law for : 5..., E in MeV (5) (t STP). Thus, for a 5 MeV alpha particle, reference for power law

4 Figure. Stopping Power, S, as a function of initial energy, E. The ecreasing tail represents the energy interval in which Bethe s formula is applicable. The lowenergy limit is given by the empirical calculation, while the two regimes are joine continuously by a cubic spline. (From This compares with the stanar, eperimental value for a 5 MeV alpha particle at STP got from the following calculation: g / cm ( g / cm ) ( ρ.3 g / cm ) 3. cm air

5 For any material, we see from equation (), or from intuitive consierations, that E ρ. Diviing equation () by ρ an consiering one kin or projectile only, we get ρ E Z log I mv ( Z ) Material (6) Here, the factors Z an the logarithm vary slowly. The Bragg-Kleeman rule estimates the ensity-epenence of the range through the following empirical relationship: ρ ρ (7) Where the inices an refer to the first an secon meia, respectively. s well as the I E an ρ epenence of the stopping process outline, there is a statistical element involve. For, there is a variation in the energy transfer per collision an in the amount of ionization prouce by a given energy loss istance travelle. Thus, there is a sprea in the observe range of monoenergetic particles, calle straggling. However, this is not significant for heavy charge particles like alpha particles for 5MeV alpha particles, the stanar eviation of the range istribution is about % 7 Figure. elative intensity as a function of istance travelle for alpha particles an electrons. fter Lilly, p. 36. This suggests that we can moel the relative intensity as a function of istance as a Fermi function: λ y( ; λ, λ, λ3 ) ( λ ) λ3 e Detection of lpha Particles: (8) 7 Lilly, p 34.

6 In this eperiment, we use a photomultiplier tube as a counter, whose entrance winow is coate with ZnS scintillator power. The pulses from the photomultiplier pass via an amplifier to a pulse counter. There are two types of scintillator powers. One is organic, an such a material is not a semiconuctor. Such a material has two wiely space electronic energy levels E an E, say, an aroun these levels there are many vibrational levels V, V, ; V, V, n incient particle interacts with the scintillator power an raises the molecules of the material from a level to a level. The material then eëcites an emits a scintillation photon in the visible range. Because of the level scheme, very few scintillation photons can raise the material itself to an ecite level. Thus, the scintillator power is transparent to its own raiation. The other type of scintillator power is inorganic an many of these are alkali halies, such as NaI. Such materials crystallize an have semiconuctor properties. Thus, there is a fille ban (a group of ajacent electronic states in k-space) an an unfille ban, separate by a bangap. Incient, energetic particles can cause the ecitation of a valence electron (electron in the fille ban) into the conuction ban (the empty or part-fille ban). On eëcitation, the electron emits a scintillation photon. However, in general this photon is in turn, capable of eciting a valence electron, an so such a material is not transparent to its own raiation. It is only through the aition of a small amount of impurity to the crystal that the transparency can occur. For, the impurity provies the electron with allowe states in the bangap, an eëcitation from these states will not lea to a photon capable of reëciting electrons. Figure 3. Energy level schemes for organic an inorganic scintillator powers.

7 Metho an esults: (i) To measure the count rate C as a function of the source-etector istance,. The following graph was obtaine: 3.5 Plot of Count ate against istance / cm Figure 4. Plot of Count rate, C, against the source-etector istance,. This curve is not foun to ecrease to zero as sharply as the Fermi function. This is because the probability of etection by the counter is proportional to the area of the etector (a constant), an the source-etector istance only. Thus, in later sections, C we stuy the quantity f (,, ), while here we eamine C ( ) f (,, ). Ω 4π

8 (ii) To measure the count rate per unit soli angle ( Ω C ) against the sourceetector istance, assuming that the source is point-like an the etector can be moelle as a spherical cap. Now the spherical cap is the curve surface area of a segment of sphere, although this is approimate by π (.533cm), where aperture is the raius of the cap aperture cap etector aperture. Thus, we approimate Ω by The eact result is obtaine from integration an the erivation is in appeni : Ω (, ) 4arctan (9) This epenence is ehibite in the following graph: 7 B Plot of Soli ngle against istance / cm (istance) Figure 5. Plot of Soli angle against the istance for a point-like source an a spherical cap. (Geometry illustrate in appeni ).

9 The following plot was obtaine for Ω C versus. Plot of Count ate per Soli ngle as a function of istance (Smooth Curve).6 B /-.cm / cm Figure 6. Plot of count rate per soli angle as a function of source-etector istance, for the assumption of point-like source. smooth curve has been fitte. This gives a value for the range of value of 3. 4cm 3.4 ±. cm. This compares with Bethe s

10 (iii) Eamining the behaviour of Ω C as a function of assuming that the source is finite in etent: The geometry of this situation is consiere in appeni. We get the following calibration curve for Ω as a function of /, an we estimate the functional form as a power series, truncating after five terms..5 B Plot of Soli ngle against.5.5 Y M M*... M8* 8 M9* 9 M.3484 M M M M / cm Figure 7. Plot of Soli angle against source-etector istance. (Geometry inicate in ppeni B).

11 The following plot of Ω C as a function of source-etector istance is obtaine:.6 B Plot of Count ate per Soli ngle as a function of istance (Smooth Curve) /-.cm / cm Figure 8. Plot of count rate per soli angle as a function of source-etector istance, for the assumption of non-point-like source geometry. smooth curve has been fitte. gain, we fin that 3.4 ±. cm.

12 Further Questions: We coul estimate the range of alpha particles in water using the Bragg-Kleeman rule, mentione in the Theory section: ρ ρ (7) Where is the mass number of the meium in which the ions travel. Taking air 3. ±. 5cm from the first eperiment, 7, air Nitrogen (efining an effective mass number for water as 8 ) an we get Water ~ (.6%) ir Water ρ 3 air.99kg / m 8, It is more ifficult to arrange a similar eperiment for electrons (say). This is because electrons interact less with matter an are more likely to pass through a material unaffecte. Thus, we woul epect a more graual fall off in electron intensity as a function of source-etector istance, making the notion of range ifficult to apply. The anomaly in figures 6 an 8: In both figures 6 an 8 we see that the count rate per unit soli angle starts to increase once the range has been reache. This may be ue to gamma emission by ecite Po nuclei. For, an alpha particle may tunnel out of the nucleus at a lower energy than the epecte 4.88 MeV. The unstable nucleus must then ecay to the groun state by the emission of a gamma ray. It is possible that this gamma raiation eplains the nonzero intensity observe once the travelling alpha particles have eceee their range. Uncertainty in Measurements: In the secon eperiment, the sources of error are the following: The usual measurement errors (istance, count rate C: δ N N ). The error incurre by reaing the range from figure In the thir eperiment, the sources of error coul be foun among the following: 8 t STP

13 The error incurre in fitting a polynomial to the atapoints of Ω ( ). This manifests itself in the large error bars in figure These were erive from the following consierations: Ω N n N n δ n δ ( ) δω( ) n n n n n δ n an by assuming that ~ 5% n The sparsity of ata points in the critical region where the intensity falls to zero (or to backgroun) rapily. Conclusions: In this eperiment it was foun that the range of alpha particles in air is 3.4 ±. cm, which agrees with Bethe s result. This measurement was accomplishe by first of all assuming a point-like source an subsequently, by correcting for this assumption. In both cases, the result was the same. The Bragg-Kleeman rule was use to estimate the range of alpha particles in water, an this was foun to be 54 ± 3µ m. This is to be compare with 38µm for 5 MeV alpha particles in liqui form. 9 This suggests either that the Bragg-Kleeman rule is capable only of preicting relative ranges only to orers of magnitue, or that we fin a more sophisticate way of efining effective mass numbers for molecules such as water. This question shoul certainly be investigate. 9

14 ppeni : The erivation of the equation ( ) 4arctan, Ω (9). ssume a point-like source raiating into a spherical cap an viewe in the coorinate system shown below: ( ) cos sin sin cos cos cos cos cos sin Ω Ω Ω 4arctan 4 Ω (9)

15 ppeni : The source of finite etent: It is not legitimate to moel the apparatus as a spherical cap-like etector an a point-like source. Inee, the source is finite in etent an this becomes important for small source-etector istances. In the case of a circular isc of raius S facing another smaller circular isc of raius (figure ), the soli angle at one isc subtene by the other is S Ω, () S Ω (,, ) S S We tabulate Ω Ω by means of a graph (figure 7). With this assumption, the source an etector have the same surface area., for the special case where S an fin all values of ( )