Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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810 Principles of Radiation Interaction in Matter and Detection

A.7 Free Electron Fermi Gas

A number of physical properties in metals can be understood in terms of the free

gas electron model, which was put forward by Drude in 1900 to explain the metallic

conductivity. According to this model, some (conduction) electrons are free to move

around the whole conductor volume. These electrons behave as molecules of a perfect

gas. Forces among conduction electrons and ions are neglected. This classical theory

accounts, among others, for the derivation of Ohm’s law and, furthermore, for the

relation between electrical and thermal conductivity. While it fails to explain other

phenomena like, for instance, the heat capacity and the paramagnetic susceptibility

of conduction electrons.

In free classical electron model, the electron kinetic energies can have any value

and, as the temperature decreases, the average kinetic energy decreases linearly

with the temperature becoming zero at 0 K (K is the absolute temperature in units

of kelvin). In fact at thermal equilibrium, the average kinetic electron energy is

(k is the Boltzmann constant) and is derived assuming that electrons obey to the

classical Maxwell–Boltzmann statistics.

In quantum-mechanics not all the energy levels are permitted and the continuous

energy distribution is replaced by a discrete set of energies. In addition, electrons

have an intrinsic spin angular momentum of

~ and, at 0 K, they must occupy

energy levels consistent with the Pauli exclusion principle, i.e., their mean energy

is far from zero. The statistics taking into account the Pauli exclusion principle is

the Fermi–Dirac statistics. A gas is called degenerate when deviations from classical

properties occur. In 1927, Pauli and Sommerﬁeld pointed out that the electron gas

within a metal must be treated as a degenerate gas, whose properties are essentially

diﬀerent from those of an ordinary gas. It turns out that, at ordinary temperatures,

the energy distribution diﬀers very little from the one at 0 K.

Following the treatment in Section 4.2 of [Bleaney, B.I. and Bleaney, B. (1965)],

let us consider the momentum space, where the coordinates are the components

, p

) of the momentum instead of the components of the position (x, y, z). In

this space the particle momentum is represented by a point. The magnitude and

direction of the momentum are the length and the direction of the radius vector from

the origin of the coordinate system and to the momentum point. From Heisenberg’s

uncertainty relation (see page 232) we have that momentum components cannot be

determined more precisely

∗

that:

4x 4y 4z4p

= h

Therefore, if the electron is constrained to be inside a volume V , we have:

P =

, (A.1)

∗

In statistical mechanics, for a phase-space of one degree of freedom one quantum state occupies

a volume 4x 4p

= h (for the generalization to more degrees of freedom, e.g., see discussion at

page 247 of [Morse (1969)]).

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Free Electron Fermi Gas 811

where P is the size of the corresponding volume-element in the momentum

space. The sphere momentum-volume, for which the momentum is between p and

p + dp , is 4πp

dp ; once divided by the volume-element in the momentum-space

[Eq. A.1], it becomes:

el,p

= 4πp

dp. (A.2)

Due to the Pauli exclusion principle, only two electrons (with opposite spins) can

have assigned the same volume-element position in the momentum-space.

Furthermore, the total kinetic energy is the sum of the individual electron kinetic

energies W

= p

/(2m), where m is the electron rest-mass. This energy will be the

smallest when

/(2m)] has its minimum value.

The minimum energy value is represented by a sphere of radius p

and volume

πp

in the momentum-space. p

is called the Fermi momentum. The total number

of electron, N

, can be derived considering that in each momentum-volume element,

P , two electrons of opposite spins can be found. We have:

= 2

πp

From which, we get:

3π

2/3

, (A.3)

where n = N

/V is the electron density in the metal (e.g., for further treatments and

discussions, the reader can see Section 4.2 of [Bleaney, B.I. and Bleaney, B. (1965)]

and Section 11-11 of [Eisberg and Resnick (1985)]; E

is the so-called Fermi energy,

namely the energy of the highest level occupied at 0 K. The Fermi temperature is

given by:

Equation (A.3) shows that E

depends on the metal, since n varies with the ma-

terial. From Eqs. (A.2, A.3), the number of electron states per unit volume with

electron energies between W and W + dW , g(W ) dW , is:

g(W ) dW = 2

el,p

= 8πp

= 16πW m

= 16π

√

W m

√

2π

√

3/2

√

3/2

dW, (A.4)

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

812 Principles of Radiation Interaction in Matter and Detection

where g(W ) is called the density of states. The mean electron energy hW i can be

calculated as:

hW i =

W g(W ) dW

3/2

5/2

. (A.5)

The Fermi energy varies from ≈ 3 to 7 eV in usual metals and is À kT at all

ordinary temperatures. Thus, the classical gas model cannot be applied for the case

of a free electron gas.

The energy distribution at a ﬁnite temperature, T , is derived by the Fermi–Dirac

statistics as mentioned above. Thus, the number of electron per unit volume with

an energy between W and W + dW is:

dn = f(W, T ) g(W ) dW = f(W, T )

2π

√

3/2

dW, (A.6)

where f(W, T ), the probability that an electron occupies a given state, is given by

f(W, T ) =

exp

W −E

+ 1

. (A.7)

The function f(W, T ) is called the Fermi factor. The total number of electron per

unit volume n is:

n =

∞

f(W, T ) g(W ) dW

2π

3/2

∞

√

exp

W −E

+ 1

dW.

From Eq. (A.7), we can see that E

is the energy at which the probability of a state

of being occupied by an electron is f(W, T ) =

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Gamma-ray Energy and Intensity Standards 813

A.8 Gamma-Ray Energy and Intensity Standards

The table lists some γ-ray energies and intensity standards, recommended by

the IAEA Co-ordinated Research Programme for calibration of γ-ray measure-

ments [IAEA (1991, 1998)]. The γ-ray energy is in keV, the half-life in days and P

is the emission probability. Uncertainties in the data are presented in the format

123(x), where x is the uncertainty in the last ﬁgure or ﬁgures quoted in the prime

numb er, expressed at the 1σ conﬁdence level; thus, 12.132(17) means 12.132±0.017,

and 0.0425(8) means 0.0425 ± 0.0008. X-ray sources can be found in [IAEA (1991,

1998)]. The data are the result of the work of an IAEA Coordinated Research

Project 1986 to 1990. Further X- and γ-ray data are available from [ToI (1996,

1998, 1999); Helmer (1999); Helmer and van der Leun (1999, 2000)].

Source Half-life (days) E

(keV) P

Na 950.8 ± 0.9 1274.542(7) 0.99935(15)

Na 0.62356 ± 0.00017 1368.633(6) 0.999936(15)

2754.030(14) 0.99855(5)

Sc 83.79 ± 0.04 889.277(3) 0.999844(16)

1120.545(4) 0.999874(11)

Cr 27.706 ± 0.007 320.0842(9) 0.0986(5)

Mn 312.3 ± 0.4 834.843(6) 0.999758(24)

Co 77.31 ± 0.19 846.764(6) 0.99933(7)

1037.844(4) 0.1413(5)

1175.099(8) 0.02239(11)

1238.287(6) 0.6607(19)

1360.206(6) 0.04256(15)

1771.350(15) 0.1549(5)

2015.179(11) 0.03029(13)

2034.759(11) 0.07771(27)

2598.460(10) 0.1696(6)

3201.954(14) 0.0313(9)

3253.417(14) 0.0762(24)

3272.998(14) 0.0178(6)

3451.154(13) 0.0093(4)

3548.27(10) 0.00178(9)

Co 271.79 ± 0.09 14.4127(4) 0.0916(15)

122.0614(3) 0.8560(17)

136.4743(5) 0.1068(8)

Co 70.86 ± 0.07 810.775(9) 0.9945(1)

Co 1925.5 ± 0.5 1173.238(4) 0.99857(22)

1332.502(5) 0.99983(6)

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814 Principles of Radiation Interaction in Matter and Detection

continued from previous page

Source Half-life (days) E

(keV) P

Zn 244.26 ± 0.26 1115.546(4) 0.5060(24)

Se 119.64 ± 0.24 96.7344(10) 0.0341(4)

121.1171(14) 0.171(1)

136.0008(6) 0.588(3)

264.6580(17) 0.590(2)

279.5431(22) 0.250(1)

400.6593(13) 0.115(1)

Sr 64.849 ± 0.004 514.0076(22) 0.984(4)

Y 106.630 ± 0.025 898.042(4) 0.940(3)

1836.063(13) 0.9936(3)

Nb 7.3 ± 0.9 × 10

702.645(6) 0.9979(5)

871.119(4) 0.9986(5)

Nb 34.975 ± 0.007 765.807(6) 0.9981(3)

109

Cd 462.6 ± 0. 7 88.0341(11) 0.0363(2)

111

In 2.8047 ± 0.0005 171.28(3) 0.9078(10)

245.35(4) 0.9416(6)

113

Sn 115.09 ± 0.04 391.702(4) 0.6489(13)

125

Sb 1007.7 ± 0.6 176.313(1) 0.0685(7)

380.452(8) 0.01518(16)

427.875(6) 0.297(3)

463.365(5) 0.1048(11)

600.600(4) 0.1773(18)

606.718(3) 0.0500(5)

635.954(5) 0.1121(12)

125

I 59.43 ± 0.06 35.4919(5) 0.0658(8)

134

Cs 754.28 ± 0.22 475.364(3) 0.0149(2)

563.240(4) 0.0836(3)

569.328(3) 0.1539(6)

604.720(3) 0.9763(6)

795.859(5) 0.854(3)

801.948(5) 0.0869(3)

1038.610(7) 0.00990(5)

1167.968(5) 0.01792(7)

1365.185(7) 0.03016(11)

137

Cs 1.102 ± 0.006 × 10

661.660(3) 0.851(2)

133

Ba 3862 ± 15 80.998(5) 0.3411(28)

276.398(1) 0.07147(30)

302.853(1) 0.1830(6)

356.017(2) 0.6194(14)

383.851(3) 0.08905(29)

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Gamma-ray Energy and Intensity Standards 815

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Source Half-life (days) E

(keV) P

139

Ce 137.640 ± 0.023 165.857(6) 0.7987(6)

152

Eu 4933 ± 11 121.7824(4) 0.2837(13)

244.6989(10) 0.0753(4)

344.2811(19) 0.2657(11)

411.126(3) 0.02238(10)

443.965(4) 0.03125(14)

778.903(6) 0.1297(6)

867.390(6) 0.04214(25)

964.055(4) 0.1463(6)

1085.842(4) 0.1013(5)

1089.767(14) 0.01731(9)

1112.087(6) 0.1354(6)

1212.970(13) 0.01412(8)

1299.152(9) 0.01626(11)

1408.022(4) 0.2085(9)

154

Eu 3136.8 ± 2.9 123.071(1) 0.412(5)

247.930(1) 0.0695(9)

591.762(5) 0.0499(6)

692.425(4) 0.0180(3)

723.305(5) 0.202(2)

756.804(5) 0.0458(6)

873.190(5) 0.1224(15)

996.262(6) 0.1048(13)

1004.725(7) 0.182(2)

1274.436(6) 0.350(4)

1494.048(9) 0.0071(2)

1596.495(18) 0.0181(2)

198

Au 2.6943 ± 0.0008 411.8044(11) 0.9557(47)

203

Hg 46.595 ± 0.013 279.1967(12) 0.8148(8)

207

Bi 1.16 ± 0.07 × 10

569.702(2) 0.9774(3)

1063.662(4) 0.745(2)

1770.237(9) 0.0687(4)

228

Th decay chain 698.2 ± 0.6 84.373(3) 0.0122(2)

238.632(2) 0.435(4)

240.987(6) 0.0410(5)

277.358(10) 0.0230(3)

300.094(10) 0.0325(3)

510.77(10) 0.0818(10)

583.191(2) 0.306(2)

727.330(9) 0.0669(9)

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816 Principles of Radiation Interaction in Matter and Detection

continued from previous page

Source Half-life (days) E

(keV) P

228

Th decay chain 698.2 ± 0.6 860.564(5) 0.0450(4)

1620.735(10) 0.0149(5)

2614.533(13) 0.3586(6)

239

Np 2.35 ± 0.004 106.123(2) 0.267(4)

228.183(1) 0.1112(15)

277.599(2) 0.1431(20)

241

Am 1.5785 ± 0.0024 × 10

26.345(1) 0.024(1)

59.537(1) 0.360(4)

243

Am 2.690 ± 0.008 × 10

43.53(1) 0.0594(11)

74.66(1) 0.674(10)

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Appendix B

Mathematics and Statistics

817

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818 Principles of Radiation Interaction in Matter and Detection

B.1 Probability and Statistics for Detection Systems

This Appendix reproduces, with the permission, the Sections 27 and 28, pages 191–201,

from Groom, D.E. et al. (2000), Review of Particle Physics, Particle Data Group, The

Eur. Phys. Jou. C 15 , 1;

° by SIF, Springer-Verlag 2000. Complementary informa-

tion on elements from the theory of statistics, errors and their propagation can be found

in Chapter 10 of [Melissinos and Napolitano (2003)] (see also Chapter 10 of [Melissinos

(1966)]).

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Probability and Statistics for Detection Systems 819