Range of Alpha Particles in Air

Transcription

1 Range of Alpha Particles in Air Oliver Kelly December 6, 2000 Abstract A fixed scintillation counter was used to detect and hence allow measurement of the range in air at S.T.P. of 4.88MeV α particles emitted from a 209 P o source. The range of the particles was found to be 3 ± 0.1cm. From this the range in water was estimated at 3.1 ± cm. 1

2 Contents 1 Theoretical Background Why do we have alpha particle emission? Interactions of Radiation with Matter Heavy Charged Particles Electrons γ-rays Approximation of Variation of Solid Angle 8 3 Scintillation Counter 9 4 Apparatus 10 5 Procedure 10 6 Results 12 7 Conclusion 15 8 References 16 2

3 1 Theoretical Background 1.1 Why do we have alpha particle emission? Alpha emission is a Coulomb repulsion effect.as the Coulomb forces increase with size at a faster rate (namely, as Z 2 ) than does the nuclear binding force, which increases approximately as A, it can be seen that alpha particle emission becomes increasingly important for heavy nuclei. The alpha particle is favoured for the spontaneous emission of energy from a nucleus because it is very stable and has a tightly bound nucleus. it is probably the best known example of quantum tunneling, while the energy of the particle isn t classically large enough to be emitted - quantum mechanics predicts a finite and observable probability of emission of an alpha particle from a nucleus. 1.2 Interactions of Radiation with Matter It is known that energetic particles and photons lose energy and may be diverted from their initial direction as they go through matter. In this experiment where we have alpha particles passing through air (79% Nitrogen 21% Oxygen) at room temperature and atmospheric pressure. The low density of the air means while the probablity of the electric field due to the unshielded protons of the nucleus interacting with an individual atom of air is quite small, the chances of it interacting with the nucleus of said atom are negligible. For example the famous Rutherford scattering experiment found very few alpha particles were even slightly scattered by the gold atoms in a thin foil of gold. This is why we concetrate our discussion of the interaction of charged particles and γ-rays with gas upon their interaction with the electron shells surrounding the nucleus. 1.3 Heavy Charged Particles Now, heavy charged particles (relative to electron mass) interact with a gas mainly by making ionising collisions with the bound electrons. The amount of energy (kinetic energy) transferred is small compared with that of the particle and the particle is not deflected to any measurable extent. Therefore it continues on a straight path until its kinetic energy is gradually dissapated in a large number of collisions. Hence particles of the same type and energy passing through the same medium will have approximately equal ranges. And it is this range which we set out to measure for 4.88MeV alpha-particles in air. 3

4 These heavy charged particles may also elevate a bound electron to a higher energy level or they may provide a bound electron with sufficient energy that it not only ionises the atom but also that the electron has sufficient kinetic energy to ionise an electron from another atom. Such energetic electrons are called δ-particles. While a particle has sufficient energy to ionise an atom the faster it goes the less time it has to transfer energy to a bound electrons electric field thus increasing the chance it will just be elevated to a higher energy level. However the slower it is going the more energy which can be transferred (from Heisenberg s Uncertainty principle) and thus there is an increased chance of δ-particles being formed. The energy required to form an electron - ion pair for alpha - particles in O 2 is 32.2 ev and N 2 is 36.0 ev hence in air is 35.2 ev. So if we measure the specific ionisation (i.e. no. of ions per unit length) versus the range of alpha - particles in air. This curve rises as we move away from the radiation source dur to the incidence of δ- particle creation rising as the velocity of alpha-particles decreases and then drops down to almost zero (this is the range of alpha-particles in air ). See Fig below. However the specific ionisation may rise again as when an alpha particle tunnels out from the 209 P o source in approximately 20% of cases it does so at a lower energy level then that expected i.e. 4.88MeV in this case. In order to dispose of the excess energy remaining after this has occured a γ-ray equal to the energy of the surplus energy is 4

5 emitted. We shall later look at how these γ-rays will ionise a gas and see how this gives rise to the small specific ionisation after the range of the alpha-particles has been passed. The rate of energy loss of heavy charged particles or the stopping power of the medium through which the heavy charged particle travels is given by the Bethe-Bloch formula where; de dx = z2 e 4 [ [ 4πɛ 0 v 2 NZ (2m0 v 2 ) ln I ] ] ln(1 v2 c 2 ) v2 c 2 de dx : rate of energy loss z : the atomic no. of the heavy charged particle e : magnitude of electronic charge ɛ 0 : Permittivity of free space m : mass of electron v : the heavy charged particle velocity N : the number of atoms present per unit volume of target Z : The atomic no. of the target atoms I : The mean ionising potential, i.e. the mean excitation energy of the target atoms. (1) Note: Generally the mean ionising potential is greater then the true ionisation potential as more tightly bound inner electrons may be removed from the atom instead of the most weakly bound. It is also greater then the value to produce electron-ion pair as it includes the production of δ-particles which themselves produce secondary ionisation. Note that eqn.(1) is particularly appropriate for ions such as alpha particles or protons of a few MeV which stay( stripped ) of atomic electrons over nearly all their range. Also the ln 2mv 2 I varies very slowly ) with v as it is logarithmic and ln (1 v2 c v2 2 c is only important 2 for heavy particles at very high energies. We can see that the larger the value of z and the lower the particles velocity the greater the rate of energy loss. Bearing in mind that where A is atomic mass, for moderate mass elements, NZ NA ρ, where ρ is the density of the target material. So, inserting this into equation (1): de dx ρz2 φ(0) v 2 5

6 de d(ρx) z2 ρ(v) v 2 (2) There is clearly now, nothing linking the right hand side to the absorbing material. This means that range can be expressed in terms of ρx in dimensions of gcm 2 or kgm 2 and is independent(approximately) of the absorbing material. That is to say, ranges of a particle expressed in the above units, should be approximately equal no matter what the target. Now if we take eqn.(1) and diregard the relativistic terms (as the alpha-particles in this expt. are non-relativistic ) we get the following expression. v 2 de φ(v) dx = z2 NZ (3) where; hence if we integrate where; [ ( )] φ (v) = e2 2mv 2 ln 4πɛ I 0 v 2 R E 0 φ(v) de = z2 NZ dx (4) 0 E 0 : the initial particle energy corresponding to 0 displacement R ; the range of the heavy charged particles (i.e. position of particle at zero kinetic energy Now if we treat the particle non relativistically, So M v0 0 E = 1 2 Mv2 de = Mvdv v 3 R φ(v) dv = z2 NZ dx 0 Mf(v 0 ) = z 2 NZR R = M z 2 f(v 0 ) NZ where f(v 0 ) is simply a function of v 0. 6

7 Finally as any charged particle when it is accelerating emits energy through Bremsstrahlung or breaking radiation. Heavy charged particles are no different. However the rate of energy loss is described by; de dx ENZ2 M 2 (5) where; E : the energy of the incident particle M : incident particles mass Z : the atomic no. of the target atoms From eqn. (2) one can see that the energy loss due to Bremsstrahlung radiation is more significant (even for electrons) at high energies and for targets of large atomic number (also very small angles of deflection mean very little acceleration). Hence this effect is deregarded in this experiment. 1.4 Electrons Electrons lose energy through two principle methods of energy loss. 1. By conventional Rutherford or Coulomb scattering. 2. By Bremsstrahlung radiation. As the energy of the electron increases the likelihood of it losing energy by Coulomb scattering decreases while the likelihood of it losing energy by braking radiation increases. Because of the indistinguishability of electrons being ionised from missile electrons the convention that an electron may only lose at most half its energy is adopted. Because of the possibly large deflections in the path of the electron the formula for the range of electrons was largely derived emperically. 1.5 γ-rays γ-rays lose energy through three principle methods of energy loss. 1. The photoelectric effect. 2. Compton scattering. 3. Pair production. The photoelectric effect is when the γ-rays give all their energy to a single electron of an atom and ionise it out of the atom entirely or 7

8 else raise it to a higher energy level. It occurs mainly at low γ-ray energy. Compton scattering involves the interaction of a γ-ray with an electron but instead of all the energy of the γ-ray going to the kinetic energy of the electron another less energetic γ-ray is emitted by the electron. Hence the wavelength of the γ-ray is increased. This generally is most likely to at lower energies as well. Pair production involves the highly energetic γ-ray getting quite close to a massive body ( into order to conserve momentum ) and forming an electron-positon pair the excess energy of the γ-ray above the threshold of 1.02MeV goes into the kinetic energy of the particles. It is not strictly possible to define a range for γ-rays as they are highly penetrating and decay at various rates depending on the energy of the particle. 2 Approximation of Variation of Solid Angle It is naieve to presume that the count rate of the alpha particles is only a function of source-detector distance, d.this is as a result of the phsical dimensions of both the detector and radiation source. Also as the detector is moved away from the source, the solid angle subtended by the detector of the source,θ, decreases and obviously this must be corrected. If we initially approximate the source to be a point and say the detector is a section of fixed area, A, on the surface of a sphere with its centre at the source, and it s radius equal to the source-detector distance. Since the surface area of a sphere is given by ; S = 4πd 2 And thus the fraction of the solid angle subtended by A is given by; Which means that; A 4πd 2 θ 1 d 2 That is to say the solid angle varies as 1 d 2, and therefore, so does the count rate. So a corrected count rate of Cd 2 is used to obtain one value of the range. This approximation will be valid for large value s of d, but will cease to apply for d small. This is due to the approximatio n 8

9 that the source is a single point which obviously cannot be assumed closer in. Also the detector itself can no longer be approximated by a sphere. We also used a table of solid angles subtended by a circular disk based on the following integration where; ( S Ω app h, R ) = h Ω app : is the approximated solid angle ( ) 2 S S 2 Ω (h, R, ρ) ρdρ 0 S : is the radius of the subtending circular disk R : the radius of the detector screen ρ : the distance from the centre of the subtending circular disk to a point on that disk. h : the distance between the source and the detector The table was generated using a standard Simpsons rule numerical method on a computer. This had an estimated accuracy of the values of 0.1%. 3 Scintillation Counter The scintillation counter used operated in the following way (see Fig below). We had ZnS scintillator powder on the entrance window which emits photons of light when radiation falls on it. These photons from the scintillator fall onto the photocathode of the photomultiplier.the photocathode surface is coated with a material which ejects photoelectrons when photons fall on it quite easily (photoelectric effect). These electrons are then accelerated by the dynodes which on average emit 3 or 4 secondary electrons depending on the voltage. Hence if we have 10 or more dynodes the result is a multiplication of the electrons produced at the cathode by a factor of perhaps a million. Such a large multiplication factor has the effect that the output voltage is usually quite large. Large enough that only a rather small amount of external amplification is required before the signal enters the single channel analyser. 9

10 6 Characteristic Curve of Vacuum Diode Apparatus The experiment uses a 4.88 MeV alpha particle source mounted on the end of a vertically movable rod, and a fixed scintillation counter as a detector. An EG&G Ace-Mate supplies the high voltage to the photomultiplier The scintillation counter consists of A photomultiplier tube with ZnS scintillation powder on the entrance window An amplifier through which the pulses from the photomultiplier pass A single channel analyser(discriminator), SCA, which receives from the amplifier. A pulse counter which relieves from the SCA An EG&G Ace-Mate supplies the high voltage to the photomultiplier. 5 Procedure 1. The appparatus was set up as shown in the Diagram. 2. The pulses from the photomultiplier were passed via the single channel analyser to the pulse counter. 3. The source was lowered down close to the scintillator. With the 10

11 HV potentiometer turned off it was turned down to zero. It was then turned on and the voltage brought slowly up to 0.85 kv. 4. Coarse and fine grain settings were set at 20 and 5 respectively. 5. On the discriminator the lower level was set to 0.5V and the upper level to 9V. This was to prevent as many anomalous electron currents registering as radiation detected as possible. 6. We used the amplifier output connected to an oscilloscope to check that there were pulses and lay in the range set on the discriminator. 7. The distance shown in the diagram as d was set at 1cm and the count rate was then measued at htis distance. 8. This was then repeated at various values of d from 2cm to 6cm. The graph of Cd 2 versusd where C is the count rate was then plotted.(see graph 1). It was possible to approximately determine the range of alpha particles in air from this graph and hence more readings were taken in the 2.5cm to 3.5cm region. 9. This furthur data combined with the earlier data and the values of R and S (see discussion of scintillator counter) to calculate the values R h and S h, where h was given to be d + 3.2mm ± 0.1mm. Hence the values of the solid angle calculated by Gardner and Verghese were used to calculate the value of the solid angle and therefore the actual reading of the rate by calculating the count rate per unit solid angle. 10. A graph of count rate per unit solid angle versus h was then plotted. The range and mean range were then calculated from this graph. 11

12 6 Results 12

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15 7 Conclusion It was found from the graph that the range of alpha particles in air is 3 ± 0.1cm, the published value is 3.4cm. The mean range was found to be 2.5 ± 0.1cm. While these results are in good agreement with those of others this experiment proved to be alarmingly susceptible to external electric fields. Also the rise after the range of the alpha particles has been passed was mildly shocking, however I believe this can be explained in terms of the γ-rays having lost some energy after travelling a distance through the medium now giving all their energy to the scintillating powder and hence causing a rise in perceived specific ionisation. From the range of the particles in air it was possible to calculate a range for them in water. The figure obtained was (3.1 ± 0.5)x10 3 cm. 15

16 8 References 1. K. S. Krane Introductory Nuclear Physics 2. S. E. Hunt Nuclear Physics for Engineers and Scientists 3. A. BeiserModern Physics 4. Methods of Experimental Physics,Vol. 5A 5. Gardener and VergheseNuclear Instruments and Methods 6. E. FinchRadiation Detectors 16