Investigating the Absorption of Radiation in Matter using G-M Counter

 

Abstract

In this experiment, we shall:

Observe and demonstrate the interaction between the magnetic moment of the electron spin with an external magnetic field.

Determine the characteristic parameter of this interaction, namely the Landé g-factor for spin (denoted gs)

Introduction

When we applied an external magnetic field and then by supplying electromagnetic energy on a paramagnetic solid material (which have unpaired electron spins), transitions can be induced between spin states. The resulting absorption spectra are described as electron spin resonance (ESR) or electron paramagnetic resonance (EPR).

ESR is a purely quantum mechanical effect. It relates to the interaction of an external magnetic field to an electron’s magnetic moment, which is a result of intrinsic spin. The spin of an electron may be up or down. This implies that in the presence of an external magnetic field, one spin state will be higher in energy than other. More specially, the spin with a moment pointing in the direction of the external field is lowest in energy. These states have an energy E, as either:

U = -µ.Bs = -µsz B =±(1/2) gs µ_B B     (1)

Where g is the Lande’ factor, µ_B is the Bohr magneton (a constant), and B is the value of the external magnetic field. Thus, the change in energy from the negative to the positive spin state can be written:

DE = g µ_BB             (2)

Therefore, because g and µ_B are constants, there are pairs of values of the energy E and of the external field B that satisfy Eq. (2) and result in a transition of spin states. At commonly attainable magnetic field strengths the corresponding energy lies in the radio-frequency (RF) regime. Where the general expression for the energy of a photon applies:

E = hv        (3)

Where E is the energy of an RF photon, h is Plank’s constant, and ν is the frequency of the photon. When a photon of energy E is incident on an electron in a field of strength B electron resonance will be observed.

Spin of the electron, The Landé g-factor.

The idea of electron spin and its associated magnetic moment was first introduced by George Uhlenbeck and Samuel Goudsmit to explain the fine structure in atomic optical spectra. A proper relativistic quantum mechanical treatment of the hydrogen atom leads naturally to the electron possessing, in addition to its mass m and charge e, an intrinsic spin angular momentum with quantum number s = 1/2. The new quantum property, the spin, obeys the same quantization rules as those governing orbital angular momentum of atomic electrons. The spin vector S has magnitude :

S = (S(S + 1))^^1/2ћ      (4)

In the presence of a magnetic field, the component of spin parallel to the field (z direction) is quantized. It has two possible values,

S_z = m_sħ = ± (1/2)ħ  (5)

Since an electron has a charge and an angular momentum, it also has a magnetic moment, µ_L,  proportional to the orbital angular momentum, given by

μ_l = g_l (e/2m) l     (6)

Where g_l is the electron orbital momentum g-factor.

The electron also has a magnetic moment, µ_s, proportional to its spin, and it is given by:

μ_s= -g_s (e/2m)S = ±(1/2) g_s (eħ /2m)   (7)

where gs is the electron spin g-factor. According to Dirac’s theory, gs is precisely 2.

Transitions between the states can therefore be excited by photons of frequency n_r, given by

_hvr = g_s μ_B B  (8)

Free Radicals

The electrons that we will manipulate are located in the paramagnetic molecule Diphenyl-Picryl-Hydrazyl (DPPH). DPPH is pictured below; notice the unpaired electron. This electron is also very weakly bound to the nitrogen atom. The combination of these two properties makes this molecule ideal for studying the magnetic properties of the electron.

Figure 3 shows a block diagram depicting the experimental setup. There are essentially two circuits: The first controls and monitors the RF signal driving of the electrons (at left), and the second controls the external magnetic field (right). You should observe the output of the absorption of the DPPH on one channel of the oscilloscope (DC couple the o-scope channel) and the voltage across the resistor on the other channel.

Helmholtz Coils

The magnetic field is provided by a pair of Helmholtz coils in a series connection with an AC and a DC power supply. The average magnetic field can be determined from the current in the coil which is measured by the DC ammeter.

The characteristic which distinguishes Helmholtz coils from other sources of magnetic field is that the distance between the coils is equal to their radius (R). 

The magnetic field midway between the coils and on their axis is given by:

B= 8μ₀NI / √125 R

where I is the current and N is the number of turns in each coil.

    The RF unit

For the typical magnetic fields used in this experiment, photons of frequencies in the range ~25-50MHz are required for the transitions referred to above. These are provided by a short coil connected to a high frequency oscillator. The magnetic field of this coil is perpendicular to that provided by the Helmholtz coils, an arrangement that results in a perturbing torque acting on the electron’s spin dipole moment. When the oscillator frequency matches the resonance condition, the impedance of the oscillator circuit decreases and the current increases. A DC voltage proportional to this current is fed to the Y-input of the oscilloscope.

Methodology

We will use:

- DPPH sample,

- ESR unit: ESR resonator with field coils and ESR power supply

- Oscilloscope

- Teslameter, digital

Procedure

* The setup was ready as the figure above.

* We inserted the sample into large RF coil, then we put the sample at the middle between Helmholtz coils.

* We turned the oscilloscope on, power supply, RF unit and the multimeter at DC amperes.

* We started with 35 MHz up to 75 for the first coil and recorded the current value (the current value was 2I not I).

* We put the second coil and repeat the previous steps starting from 80MHz up to 125MHz.

Data Analysis

Table1: The current of Helmholtz coils  (A), resonance frequency (MHz)

2I(A)	f(MHz)
0.33	20
0.43	25
0.5	30
0.58	35
0.66	40
0.74	45
0.83	50
0.92	55
1.02	60
1.08	65
1.18	70
1.24	75
1.36	80
1.43	85
1.52	90
1.59	95
1.67	100
1.76	105
1.93	110
2
2.09
2.13

Δf=±0.01MHz

ΔI=±0.01 A

N=320 ( Number of turns)

R=6.8 cm ( raduis of one coil)

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2I(A)	I (A)	f(MHz)	B  T
0.33	0.165	20	0.00047685
0.43	0.215	25	0.00062135
0.5	0.25	30	0.0007225
0.58	0.29	35	0.0008381
0.66	0.33	40	0.0009537
0.74	0.37	45	0.0010693
0.83	0.415	50	0.00119935
0.92	0.46	55	0.0013294
1.02	0.51	60	0.0014739
1.08	0.54	65	0.0015606
1.18	0.59	70	0.0017051
1.24	0.62	75	0.0017918
1.36	0.68	80	0.0019652
1.43	0.715	85	0.00206635
1.52	0.76	90	0.0021964
1.59	0.795	95	0.00229755
1.67	0.835	100	0.00241315
1.76	0.88	105	0.0025432
1.93	0.965	110	0.00278885
2	1	115	0.00289
2.09	1.045	120	0.00302005
2.13	1.065	125	0.00307785

To estimate the true value of our slope and y-intercept we use the least square method:

Where a is the intercept and b is the slope, we can find the error in a (δa) and error in b (δb) using the following equations:

where s is the standard deviation  

B  T	f(MHz)	B^2	x*y	(y-a-bx)^2
0.00047685	20	2.27386E-07	0.009537	44.14918224
0.00062135	25	3.86076E-07	0.01553375	32.18162422
0.0007225	30	5.22006E-07	0.021675	27.84430797
0.0008381	35	7.02412E-07	0.0293335	23.82080494
0.0009537	40	9.09544E-07	0.038148	20.11111516
0.0010693	45	1.1434E-06	0.0481185	21.7520854
0.00119935	50	1.43844E-06	0.0599675	23.45740378
0.0013294	55	1.7673E-06	0.073117	31.33918713
0.0014739	60	2.17238E-06	0.088434	16.41105147
0.0015606	65	2.43547E-06	0.101439	23.09679592
0.0017051	70	2.90737E-06	0.119357	10.61995211
0.0017918	75	3.21055E-06	0.134385	26.67366333
0.0019652	80	3.86201E-06	0.157216	17.58171107
0.00206635	85	4.2698E-06	0.17563975	19.11811262
0.0021964	90	4.82417E-06	0.197676	11.56561952
0.00229755	95	5.27874E-06	0.21826725	9.028293602
0.00241315	100	5.82329E-06	0.241315	10.13838582
0.0025432	105	6.46787E-06	0.267036	63.47850827
0.00278885	110	7.77768E-06	0.3067735	48.94037815
0.00289	115	8.3521E-06	0.33235	51.48222217
0.00302005	120	9.1207E-06	0.362406	20.04401341
0.00307785	125	9.47316E-06	0.38473125	15153.95468
0.03900055	1595	8.30719E-05	3.382456	15706.7891

Slope= 27.127(+-)0.251

gs= h*slope/Mᵦ =1.93818

The theoretical value of gs=2.0023, we will find the percentage error:

𝐏.𝐄 = |𝐠𝐬(𝐞𝐱𝐩𝐞𝐫𝐞𝐦𝐞𝐧𝐭𝐚𝐥𝐥𝐲)−𝐠𝐬(𝐭𝐡𝐞𝐨𝐫𝐢𝐭𝐢𝐜𝐚𝐥𝐥𝐲)| /𝐠𝐬𝐭𝐡𝐞𝐨𝐫𝐭𝐢𝐜𝐚𝐥𝐥𝐲 *100%

= 3.2023%

Error in ∆gs = gs(exp)*∆slope/slope

Error propagation in gs = (+-)0.01793

∆f=+-0.01 MHz

∆I=+-0.01 A

Conclusion

1- The relation between the frequency and the applied magnetic field is linear according to equation 8.

2- The interaction of an external magnetic field with an electron spin depends upon the magnetic moment associated with the spin.

3- Resonance is the absorption of energy from the weak alternating magnetic field of the microwave when its frequency corresponds to the difference between two spin states.

4- Free radical, such as DPPH that we used in our experiment exhibit g-factors very close to the value for free electrons = 2.0023.

 References

1- Practical physics (0342411) manual.

2- https://homepage.physics.uiowa.edu/~fskiff/Physics_132/New%20Lab%20Manual/int_lab_man/pdfs/A4_ESR_v6.pdf