STRUCTURE OF MATTER
Particles and force interactions
Matter exists in two forms: particles and fields Particles can be found in four states: solid, liquid, gas, and plasma, depending on physical conditions Additionally, there are four types of force interactions associated with fields: gravitational and electromagnetic fields, which are part of the environment, and strong and weak nuclear fields, which operate at the atomic level.
Matter can undergo transformations, such as the creation of electromagnetic waves from the annihilation of particles and antiparticles An example of this process is the formation of an electron-positron pair when γ-radiation is absorbed, demonstrating the conversion of energy from a field into particle form.
Matter is composed of two fundamental particle groups: leptons and quarks, with leptons being unaffected by strong nuclear forces Each group is divided into three generations, where the first generation of leptons includes the electron and its neutrino, the second features the muon and its neutrino, and the third comprises the tau particle and its corresponding neutrino.
Flavour Charge electron electron neutrino u (up) +2/3 e ν e d (down) −1/3 muon muon neutrino c (charm) +2/3 μ ν m s (strange) −1/3 tau tau neutrino t (top) +2/3 τ ν t b (bottom) −1/3
Quarks are categorized into three generations based on their flavour: the first generation includes up (u) and down (d) quarks, the second consists of charm (c) and strange (s) quarks, and the third features top (t) and bottom (b) quarks Each quark has a unique non-integer electric charge, with +2/3 for the first quark in each pair and -1/3 for the second Additionally, quarks are characterized by their colour, which can be red, green, or blue, and they all possess a spin quantum number of ±1/2 Each quark has a corresponding antiparticle that shares the same mass and spin but has an opposite electric charge, rotation, and magnetic moment The antiparticles are identified with the prefix anti-, leading to terms like antiu, antid, antired, antigreen, and antiblue When a particle and its antiparticle meet in the right quantum states, they can annihilate each other, resulting in the creation of new particles.
Table 1.2 Selected basic characteristics of antiparticles
Identical value of spin (integer, non-integer) but opposite rotation (clockwise, counter-clockwise)
Opposite magnetic moment (positive, negative) – if half-value
Opposite charge – if not without charge
Quarks combine to form composite particles known as hadrons, which must have an integer electric charge and a color combination that is colorless or white Hadrons are categorized into two groups: mesons and baryons Mesons, made up of two quarks (a quark and an antiquark), possess an integer spin value, with examples including the pion particles (π0, π+, and π−) In contrast, baryons are composed of three quarks of different colors (red, green, and blue) and have half-integer spin; for instance, protons consist of two u quarks and one d quark, while neutrons consist of two d quarks and one u quark Elementary particles, which include fundamental hadrons and field quanta, are divided into fermions and bosons based on their spin values Fermions, characterized by half-integer spin, follow Fermi-Dirac statistics and adhere to Pauli’s exclusion principle, preventing identical energy fermions from occupying the same system Conversely, bosons, which have integer spin, are described by Einstein-Bose statistics and can exist in unlimited numbers at the same energy level.
Particles composed of quarks – have an integer value of electric charge
– are white (colourless) Mesons – 2 quarks: quark + antiquark (integer spin)
Baryons – 3 quarks: (half-value spin)
There are four fundamental types of force interactions: strong, electromagnetic, weak, and gravitational, all characterized by non-contact and exchange properties due to the transfer of quanta Basic Bose particles, such as photons for the electromagnetic field, gluons for the strong nuclear force, W ± and Z 0 particles for weak interactions, and the hypothetical graviton for gravitational interactions, represent these excitations The gravitational and electromagnetic fields have unlimited ranges, while the strong and weak interactions are limited to approximately 10^-15 m and 10^-18 m, respectively, making them saturated fields At distances around 10^-15 m, which correspond to atomic nuclei sizes, the relative strengths of these interactions can be analyzed.
In particle physics, gravitational forces are negligible compared to other fundamental interactions, with ratios of 10^-3 to 10^-40 highlighting their relative weakness; however, gravity plays a crucial role in macroscopic objects Gravitational interactions occur solely between particles with mass and are characterized by an always-attractive nature that cannot be absorbed, transformed, or shielded In contrast, the electromagnetic force operates between electrically charged particles, allowing for both attraction and repulsion, and can be neutralized Additionally, the weak interaction facilitates the transformation of one quark into another, while the strong interaction is essential for binding protons and neutrons together, forming the nucleus of an atom.
Among particles with mass, only electrons and protons are stable, while others, like free neutrons, are unstable A free neutron undergoes β-decay, transforming into a proton, electron, and electron antineutrino after about 10^3 seconds This process represents the conversion of a down quark into an up quark.
Of the particles with non-zero mass, muon μ – possesses the longest life span (2.10 −6 s) Most hadrons decay immediately after formation since they exist no longer than 10 −12 s.
Energy
Energy is a scalar physical quantity that signifies the capacity to perform work, adhering to the law of conservation of energy, which asserts that the total energy in an isolated system remains constant over time This principle indicates that energy cannot be created or destroyed, but can be transformed or transferred between forms and locations For instance, when an electron and a positron annihilate, they produce two photons, each carrying energy equivalent to 0.51 MeV The total energy (E) of a particle or a system within a force field is the sum of its rest energy (E₀), kinetic energy (Eₖ), and potential energy (Eₚ).
E = E 0 + E k + E p (1.1) where E 0 is the energy related to the particle mass according to Einstein’s relationship
The equation E₀ = mc² illustrates that m₀, the rest mass, relates to energy, with c representing the speed of light in a vacuum, the ultimate speed for energy propagation Photons, which have no mass and zero rest energy, cannot exist at rest and consistently travel at the speed of light, c, within any coordinate system.
The mass m of a particle moving with relativistic velocity v (almost at the velocity of light in a vacuum) increases according to the relation m m v c
, (1.3) where v is the velocity related to the observer.
Particles with non-zero mass m 0 > 0, energy E, velocity of movement v and momentum p = mv are related by the equation
Kinetic energy E k is defined by the following equation
Kinetic energy, defined as the energy of motion, can only have positive or zero values (E k ≥ 0) and is independent of direction In contrast, potential energy can be either positive or negative depending on the chosen zero level In central fields governed by Newtonian forces, such as Coulomb’s law and Newton’s law of gravitation, the zero level is set at infinity, resulting in negative potential energy (E p < 0) since a positive force is needed to move an object away from its attractive force In mechanics, the potential energy of a mass point is expressed by the equation E p = mgh, where m represents mass, g is gravitational acceleration, and h is height When the zero energy level is defined at the Earth's surface and h is greater than zero, the potential energy becomes positive, equal to the product mgh.
Energy is typically measured in joules, but in the fields of atomic and radiation physics, the electronvolt (eV) is the preferred unit One electronvolt is defined as the energy gained by an electron when it is accelerated through a potential difference of one volt.
1 J = 1 C.1 V and charge 1 C equals a total charge of approximately 6×10 18 electrons,
1 eV = 1.6×10 −19 J The relation of 1 eV to 1 J is the same as that for the charge of 1 electron to
In the realm of elementary particle physics, the electron's rest mass is approximately 0.51 MeV, while the proton and neutron have rest masses of about 938.28 MeV and 939.57 MeV, respectively The unit eV is utilized not only as a measure of energy but also as a unit of mass, following the relationship defined in equation (1.2), where the c² factor is often omitted for simplicity.
Quantum effects
Classical physics fails to adequately describe atomic-level processes, where phenomena occur independently of macroscopic interactions and involve distinct physical quantities A key aspect of this realm is the significance of Planck’s constant (h) and Dirac’s constant (ħ = 1.05×10 −34 J.s), which are interconnected by the equation ħ = h/2π In the context of circular motion, Planck’s constant signifies the minimum energy that can be radiated, highlighting the unique characteristics of quantum mechanics.
Quantum properties lead to the quantization of angular momentum, which is defined as the vector product of the position vector r and the momentum vector p, where p is the product of mass and velocity (p = mv).
The cross-product of two vectors results in a vector that is perpendicular to the plane formed by the original vectors, with its magnitude equal to the product of their magnitudes and the sine of the angle between them In regular circular motion, while the momentum vector \( p \) and the position vector \( r \) continuously change direction, the magnitude and direction of their cross-product remain constant Additionally, since the vectors \( r \) and \( v \) are perpendicular, the angular momentum \( L \) can be expressed as \( L = rmv \), given that \( \sin(\pi/2) = 1 \).
Figure 1.1: Orbital angular momentum L of a particle with momentum p at circular motion with the radius r.
Quantum mechanics dictates that the angular momentum of a particle's orbital motion can only take on specific discrete values, which are multiples of Dirac's constant Additionally, the projections of an atom's angular momentum along the coordinate axes are restricted to well-defined values.
Elementary particles exhibit angular momentum, spin, and magnetic moment resulting from their rotation Particles with half-integer spin are classified as fermions, while those with integer spin are known as bosons.
In particle physics, the spin value of a particle influences its behavior, with fermions, such as electrons and nucleons, having a spin of ½, while bosons, like photons, possess a spin of 1 This distinction explains why fermions cannot occupy the same energy level, leading electrons in heavy atoms to fill higher energy levels further from the nucleus instead of the lowest ones Conversely, bosons can occupy the same energy state, allowing for different interactions in quantum systems.
Elementary particles, including atoms and molecules, exhibit both particle-like and wave-like properties, a concept initially discovered through experiments on light The wave nature of light is evidenced by phenomena such as interference and diffraction, while the photoelectric effect illustrates that light consists of energy packets known as photons The energy (E) of a photon, measured in joules, is connected to its frequency (f) in hertz and wavelength (λ) in meters through a specific equation.
E = hf = hc/λ, (1.7) where c is the velocity of light in a vacuum and h = 6.63×10 −34 J.s = 4.13×10 −15 eV.s is Planck’s constant Therefore, Planck’s constant reflects the sizes of energy quanta in quantum mechanics.
The motion of each particle with mass m, momentum p and energy E is related to wave- lengths λ of the de Broglie wave given by the equation λ = h p h mE
2 (1.8) and to frequency f given by the equation f E
Equation (1.8) indicates that elementary particles possess extremely short wavelengths For instance, when an electron in an electron microscope is accelerated by a voltage of 1 kV, its energy can be calculated as 1 keV, which equals 10^3 eV multiplied by 1.6 × 10^(-19) J/eV, resulting in an energy of 1.6 × 10^(-16) J.
The wavelength of an electron is four orders of magnitude shorter than that of visible light, which enhances the resolving power of an electron microscope, making it significantly more accurate than an optical microscope.
The corpuscular-wave dualism of subatomic particles leads to significant implications, particularly illustrated by Heisenberg's uncertainty principle This principle states that it is impossible to precisely measure both the position vector (Δr) and momentum (Δp) of a particle at the same time, highlighting the inherent limitations in our ability to determine these properties with arbitrary accuracy.
A smaller region of motion leads to greater momentum uncertainty Similarly, there is a relationship between the uncertainty of an energy level (ΔE) and the duration of its measurement (Δt).
When an energy state persists for an extended duration, its energy can be determined with significant accuracy For instance, the average time between an atom's excitation and the subsequent photon emission during de-excitation is about 10^-8 seconds Consequently, this duration influences the uncertainty in estimating the energy level.
Quantum effects can be quantitatively described by quantum mechanics
In quantum mechanics, the motion of an electron around an atomic nucleus is depicted as a "cloud" rather than a defined trajectory, with its shape and distance influenced by parameters like orbital angular momentum, magnetic moment, and spin This movement occurs within a region known as the orbital, and the electron's state is characterized by a wave function that includes several dimensionless parameters corresponding to the degrees of freedom Specifically, the electron's rotation involves four degrees of freedom, allowing its state to be fully described by four quantum numbers These quantum numbers, which are mostly natural integers (except for spin), define the geometry and symmetry of the electron cloud, and importantly, no two electrons in the same atom can share the same set of four quantum numbers.
The principal quantum number n plays a crucial role in determining the total energy of an electron in a hydrogen atom According to quantum theory, electrons can occupy multiple energy levels, E n, which are defined by a specific equation.
The equation \( = -\frac{1}{\varepsilon} \) (1.12) incorporates key constants such as the electron's rest mass \( m = 9.11 \times 10^{-31} \) kg, the vacuum permittivity \( \varepsilon_0 = 8.854 \times 10^{-12} \) F/m, and the electron charge \( e = 1.6 \times 10^{-19} \) C The principal quantum number, represented by \( n \), is a natural number that can take values of 1, 2, 3, and so forth, indicating the electron's shell location These shells are designated as K, L, M, N, O, and P.
Hydrogen atom
The hydrogen atom, the most basic system of nucleons and electrons, features a single electron orbiting a central proton The likely distance of the electron from the nucleus can be determined using the uncertainty principle.
If the electron moves at distance r from the nucleus then the uncertainty only equals r From equation (1.10), the uncertainty of momentum p is
The total energy of an electron within an atom is the sum of its kinetic and potential energies As described by equation (1.5) in conjunction with equation (1.15), the kinetic energy (E_k) can be calculated to understand the electron's behavior in the atomic field.
, (1.16) where m e is the mass of the electron Potential energy E p of an electron with charge −e in the force field of a proton with charge +e at distance r is given by
The potential energy of an electron within a nucleus's field is negative, achieving its maximum value of zero at an infinite distance from the nucleus, where the interaction between the electron and the proton becomes negligible Consequently, the total energy E of the electron in the presence of a single proton is defined by this relationship.
The total electron energy can be calculated and graphed as a function of distance (r) from the nucleus, revealing a curve that reaches a minimum value at a specific distance, r0 In physics, systems are considered stable at their minimum energy levels, indicating that the likelihood of finding an electron is highest at this distance Furthermore, an electron in a stable state does not emit energy.
Distance r 0 calculated from equation (1.18) using dE/dr = 0 (the extreme of the function can be calculated on the condition that its first derivative equals zero) is given by r 0 m e e 0
By substituting the numerical values for electron mass (m e = 9.1×10 −31 kg), vacuum permittivity (ε 0 = 8.8×10 −12 F.m −1), electron charge (e = 1.6×10 −19 C), and reduced Planck's constant (ħ = 1.05×10 −34 Js), the Bohr radius (r 0) is calculated to be 5.29×10 −11 m This radius is significant as it represents the distance at which the energy of a hydrogen atom in its ground state can be determined by substituting r 0 into the total electron energy equation.
Schrödinger's equation provides the solution for an electron in a proton's field, specifically for the ground state (n = 1) This fundamental state is accompanied by excited states (n = 2, 3, ), which an electron can transition to upon energy absorption By substituting numerical values into the equation, we observe energy levels for these states: E1 = -13.6 eV, E2 = -3.38 eV, E3 = -1.5 eV, continuing up to E° = 0 eV.
It can be demonstrated that for each energy level, value E n corresponds to the most prob- able distance r n from the nucleus, given by the following equation r n =n r 2 0 (1.21)
The most probable distance from the nucleus increases with n 2
According to the wave theory of matter, the wavelength of an electron is determined by its momentum, as described by equation (1.8), and is equivalent to a path of 2πr₀, which measures 3.3×10⁻¹⁰ m Essentially, an electron can orbit the nucleus indefinitely without emitting energy if its path is an integer multiple of the de Broglie wavelength, represented by the equation nλ = 2πrₙ, where n is the principal quantum number Furthermore, when the hydrogen atom transitions from higher to lower energy levels, it emits photons that create a discrete line spectrum.
Figure 1.4: Total electron energy E as a function of its distance r 0 from the nucleus.
1.4.1 Spectrum of the hydrogen atom
An electron in an excited state is unstable and quickly transitions to a lower or ground state, releasing a photon in the process When the energy level changes from E_k to E_n, where k is greater than n, a quantum of radiation is emitted with energy determined by the transition.
The radiation's frequency or wavelength can be determined using equation (1.7) by substituting the energy value Due to the quantized nature of electron energies, only specific energies, frequencies, and wavelengths can be emitted by the atom, resulting in the observation of a line spectrum of radiation.
The spectral lines produced during transitions from higher energy levels to a specific energy level, denoted by n, are referred to as a series In the case of the hydrogen atom, the spectral emission lines that correspond to transitions to the fundamental energy level (n = 1) are observed in the ultraviolet region, forming what is known as the Lyman series.
The Balmer series represents electron transitions to the n = 2 energy level, making it the only series observable in the visible light spectrum In contrast, the Paschen series, which corresponds to transitions to n = 3 and higher levels, falls within the infrared region The highest energy emission occurs during transitions from n = infinity to the ground state of n = 1, as described by equation (1.23).
It corresponds to the wavelength λ = = × × × × = × −
Analogously, the highest energy in the Balmer series (n = 2) is
Its wavelength is 368 nm, which reaches the region of visible light.
Electron structure of heavy atoms
The electron structure of atoms with multiple electrons is mainly determined by two rules:
1 The system of particles is stable at a minimum total energy.
2 Only one electron exists in each individual quantum state of the atom
In heavy atoms, the state of each electron is defined by four quantum numbers, similar to the hydrogen atom Each electron orbits within the central force field created by the positively charged nucleus.
The atomic number (Z) determines the number of electrons, which are shielded by other electrons closer to the nucleus Electrons sharing the same principal quantum number are located at similar distances from the nucleus, resulting in comparable energy levels and interactions with the same field intensity These electrons occupy the same shell, labeled as K, L, M, etc Each orbital quantum number (ℓ) corresponds to (2ℓ + 1) values of the magnetic quantum number, and considering that the spin number can take on two values, the maximum number of electrons in each shell is established accordingly.
The energy of electrons is influenced by the orbital quantum number, ℓ, which increases with higher values of ℓ Additionally, this dependence on ℓ becomes more pronounced as the number of electrons in the atom increases.
A closed shell or subshell, indicated by the principal quantum number (n) and azimuthal quantum number (ℓ), is fully occupied by electrons Specifically, a closed s subshell (ℓ = 0) contains two electrons, a closed p subshell (ℓ = 1) holds six electrons, and a closed d subshell (ℓ = 2) accommodates ten electrons In these closed subshells, the total orbital and spin angular momentum of the electrons sums to zero, resulting in a completely symmetrical distribution of their effective charge.
Figure 1.5: Series of spectral lines for the hydrogen atom.
The periodicity of the physical and chemical properties of elements is linked to the sequence in which electron shells are filled In heavy atoms, higher energy shells are filled before lower shells are completely occupied, driven by the system's need for minimum total energy to achieve stability The filling order for these shells is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, and 6d Notably, the 4s state is filled before the 3d state due to its lower energy, although the s and p states of the previous shell must be fully occupied before higher n shells can begin to fill.
Hund's rule is crucial for understanding electron shell filling, as it states that electrons tend to remain unpaired with parallel spins when possible This preference arises from the mutual repulsion between electrons, leading to a spatial separation that minimizes energy and enhances stability As a result, unpaired electrons occupy orbitals singly before pairing up, contributing to the overall stability of the atom.
Excitation and ionisation of atoms
The ground state refers to the quantum state of a bound electron at its minimum energy level, while higher energy levels are known as excited states An electron transitions to an excited state by absorbing energy that matches the difference between its ground state and one of the excited energy levels.
Figure 1.6: (a) Excitation, (b) emission of fluorescence radiation, (c) ionisation.
Electrons can gain the energy needed to transition to a higher energy level through the absorption of photons This occurs when the energy of the absorbed photon matches the energy difference between the electron's initial and final states If the absorbed energy exceeds the threshold, the electron is emitted as a free electron, a process known as positive ionization Ionization involves converting an atom or molecule into an ion by adding or removing charged particles Ionized atoms are unstable and seek to return to a lower energy ground state Conversely, if the electron absorbs insufficient energy, it enters a temporary excited state until the excess energy is released.
2p possible allowed 2s state for a long time and eventually transits (de-excites) spontaneously to a lower energy state
Fluorescence occurs when an electron emits a photon, with energy corresponding to the difference between its initial and final states, typically within a brief time frame of 10^-5 to 10^-7 seconds During this process, the excitation energy is released as one or more photons However, if the electron transitions to an energy level that prevents it from returning to the ground state, it enters a metastable state, resulting in delayed radiation emission known as phosphorescence.
De-excitation is followed by the emission of radiation Radiation transitions of electrons from higher to lower energy levels result in luminescence
1.6.1 Binding energy of electrons in an atom
The binding energy of a particle in a system represents the work required to remove it from that system For an electron, this binding energy corresponds to the energy needed to overcome the electrostatic forces of the nucleus and move the electron to a position of zero potential energy The total energy (E) of an electron within a nucleus's field is negative, with its maximum value being zero as the electron approaches an infinite distance from the nucleus Thus, the binding energy (E_b) is defined by the relationship E_b + E = 0.
The binding energy of an electron within a nucleus is positive and corresponds to its total energy, as described by equation (1.12) In heavy atoms, additional factors influence this equation, as total energy also depends on the atomic number Z Consequently, the binding energies of electrons in these atoms are Z² times greater for the same principal quantum number n For instance, the binding energy of an electron in the K-shell of a hydrogen atom is 13.6 eV, while in a uranium atom (Z = 92), it reaches approximately 10⁵ eV, reflecting a Z² increase.
Binding energy, often referred to as ionisation potential, varies among electrons in heavy atoms due to their differing total energy levels Typically, valence electrons exhibit the lowest ionisation potential values, reflecting their unique energy characteristics.
When an electron absorbs a quantum of radiation with energy hf that exceeds its binding energy, part of this energy is used to overcome the work needed to eject the electron from the system The leftover energy is converted into the kinetic energy of the emitted electron This process exemplifies the law of conservation of energy and aligns with Einstein's equation for the photoelectric effect: hf = E_b + (1/2)mv².
Electrons are released from various materials, including metals and non-metals, when they absorb energy from electromagnetic radiation, such as visible or ultraviolet light This process allows a current to be generated in a circuit by simply shining light on a metal surface, as each photon transfers its energy to a single electron The rate of electron emission is influenced by the intensity of the radiation, and photoelectron emission occurs only when the radiation frequency exceeds a specific threshold Once the surface is irradiated, photoelectron emission begins immediately, and higher radiation frequencies result in increased kinetic energy of the emitted electrons The photoelectric effect has significant applications in medicine, such as in scintillation counters that generate light flashes upon radiation exposure, which are then used to measure radiation intensity and count events These devices play a crucial role in nuclear tracer analysis and in technologies like computed tomography (CT) scanners for x-ray detection.
A positively charged ion is created when an atom undergoes ionization, leading to a dominance of the nucleus's positive charge This ionization process increases the system's total energy because the presence of an electron typically lowers total energy, resulting in a negative energy state Consequently, ionized atoms are unstable and tend to revert to a lower energy ground state by emitting fluorescence radiation The changes in the electron cloud surrounding the nucleus after energy absorption depend on the energy level absorbed; for energies around eV, slightly bound electrons may be excited or ionized, while inner shell electrons in heavy atoms, bound by tens to hundreds of keV, can lead to the emission of ultraviolet light or X-rays Overall, excited atoms can emit infrared, visible, ultraviolet light, or X-rays, depending on the energy difference between their excited and ground states.
Principle of mass spectroscopy
Mass spectrometry is an analytical technique used to determine the mass, elemental composition, and chemical structures of molecules This method involves converting molecules into ions, allowing them to be manipulated by external electric and magnetic fields The technique relies on the principle that ions of different isotopes within an element possess distinct charge-to-mass ratios (q/m), resulting in varied trajectories when subjected to a magnetic field.
Mass spectroscopy is a process that begins with the conversion of a sample into a gas state, followed by ionization using an electron beam Next, a longitudinal electric field accelerates these ions, which are then separated in a magnetic field based on their specific charge values Finally, the intensities of the separated ion beams are detected and analyzed.
Ions with mass m and charge q are generated from an ion source and are accelerated in an electric field characterized by a potential difference U The resulting kinetic energy of these accelerated ions can be expressed as a function of their charge and the potential difference they experience.
Accelerated ions enter a magnetic field with induction B that is perpendicular to their direction of movement This interaction causes the ions to experience a magnetic force, known as the Lorentz force, which is determined by the cross-product of the velocity vector (v) and the magnetic field vector (B) Given that the angle between these vectors is 90 degrees, the sine of this angle equals 1, maximizing the magnetic force experienced by the ions.
In a magnetic field, ions follow a circular path with a radius (r) determined by the balance between centrifugal force and magnetic force This relationship can be expressed by the equation mv²/r = qvB, illustrating how the mass (m), velocity (v), charge (q), and magnetic field strength (B) interact to define the trajectory of the ions.
From (1.34) it follows that r mv=qB (1.30)
Calculating velocity v from equation (1.32) and substituting it into equation (1.35), r U
The constant A is determined solely by the accelerating voltage and magnetic induction, remaining consistent across different isotopes The radius of the circular path for a specific isotope is influenced by its specific charge, resulting in distinct trajectories for various isotopes (see Fig 1.7).
When ion detectors are arranged in a straight line, ions will impact at various locations, such as m1, m2, etc., despite being subjected to the same accelerating voltage By measuring the relative abundance of these isotopes, one can determine the isotopic composition of the sample.
The quadrupole mass analyser is a contemporary and extensively utilized instrument that employs an oscillating electric field generated by four electrodes to filter ions This device features an ion trap that collects ions for a specific duration, subsequently releasing them into a detector based on their mass-to-charge ratio (q/m).
A time-of-flight (TOF) analyser measures the time it takes for ions to reach a detector, allowing for the identification of ions based on their mass Since ions with the same charge possess identical kinetic energies, lighter ions arrive at the detector before heavier ones This analyser operates in pulse mode, calculating the time taken to detect ions in relation to their charge-to-mass ratio (q/m).
Atomic nuclei
Atomic nuclei are incredibly dense structures composed of nucleons, specifically protons and neutrons Key characteristics of a nucleus include the atomic number (Z), which indicates the number of protons, the mass number (A), representing the total count of nucleons, and the neutron number (N), which signifies the number of neutrons The relationship between these quantities is expressed as A = Z + N Additionally, the total electric charge of a nucleus is given by Z multiplied by 1.6×10^−19 C.
The majority of an atom's mass is found in its nucleus, as nucleons are roughly 2,000 times heavier than electrons Atomic mass is typically measured in mass units, with one mass unit defined as 1/12 of the mass of the carbon isotope 12 C, equating to 1.66×10 −27 kg and an energy equivalent of 931 MeV The determination of atomic mass is commonly performed using a mass spectrometer.
Isotopes are types of nuclei with identical charge but different mass (identical Z, different A) There are approximately 280 stable isotopes in nature Together with those produced arti- ficially, their number exceeds 1100.
Isobars are atoms whose nuclei contain an identical number of nucleons but a different number of protons (different Z, identical A).
Isomers are atoms that possess the same number of protons and neutrons but exhibit different nuclear energy levels Due to their instability, isomers eventually release their excess energy as radiation over time.
The radius of a proton is 1.23×10 −15 m Radius R A of the nucleus of a heavy atom can be calculated by
Mass spectrometry operates on the principle of accelerating ions using a voltage (U) to achieve uniform kinetic energy Once accelerated, these ions are subjected to a magnetic field (B), which deflects them based on their mass; lighter ions experience greater deflection This process effectively separates ions of different masses (m1, m2, m3), enabling detailed analysis of their composition.
For example, for the nucleus of the 238 U isotope, R U = 1.23×10 −15 ×238 1/3 = 1.23×10 −15 ×6.2 ≅
≅ 7.6×10 −15 m From hydrogen to uranium, the radii of nuclei increase only by a factor of 6.2 This can result in extremely high values for nuclear mass density (ρ ≅ 10 14 kg/m 3 ).
Nuclear forces, arising from strong interactions, operate solely within the nucleus and have a range of approximately 10^-15 meters These forces are independent of the charge of nucleons and are recognized as the strongest forces in nature at this scale.
The stability of a nucleus is determined by its binding energy, which can be estimated using the total mass defect This is calculated by summing the masses of protons and neutrons, represented as Zm p + Nm n However, experimental measurements often reveal that the actual mass of the nucleus (m nucleus) is lower than this calculated value The discrepancy between the calculated mass and the measured mass is known as the mass defect, expressed as Δm = (Z.m p + N.m n ) – m nucleus.
A portion of the rest energy of nucleons, indicated by their rest mass, is transformed into binding energy that maintains the integrity of the nucleus The energy required for a nucleus to disintegrate into individual nucleons is expressed by the equation ΔE = Δm.c².
A greater mass defect results in a higher binding energy.
The binding energy of a nucleus, denoted as ΔE, is associated with all its nucleons For instance, helium-4 (4 He) has a mass defect of 0.030 mu, which corresponds to its binding energy value.
Uranium-235 has a mass defect of 1.908 atomic mass units (u), which is equivalent to a binding energy release of ΔE = 1741 MeV The unified atomic mass unit is defined as one twelfth of the mass of a carbon-12 atom in its lowest energy state.
Binding energies per nucleon, ΔE/A, vary across different nuclei, with light and heavy nuclei exhibiting values between 7 and 7.5 MeV, while mid-range nuclei in the periodic table show approximately 8.5 MeV (refer to Fig 1.8) In contrast, the binding energies of chemical systems, such as molecules, are typically in the range of several electron volts (eV), indicating that mid-mass nuclei possess the highest stability.
Unlike electromagnetic forces, which affect all charges, nuclear forces are saturated, meaning that each nucleon interacts primarily with only one other nucleon or a very limited number of neighboring nucleons within the nucleus.
At short distances, strong interactions significantly outweigh electromagnetic interactions The nucleus, with an electric charge of Ze, generates an electrostatic force field characterized by a potential U(r) that varies with the distance r from the nucleus Consequently, a potential barrier is created by electromagnetic interactions, preventing positively charged particles, such as protons, deuterons, and α-particles, from easily penetrating the nucleus.
Protons and neutrons, like electrons, have their own angular momentum (spin) and associated spin dipole magnetic moments While the spin vectors of protons align in the same direction, those of neutrons are anti-parallel The spin value is a half-integer multiple of ħ, meaning nuclei with an odd number of protons or neutrons exhibit a net spin that is non-zero These nuclei also have a non-zero magnetic moment due to the presence of electrically charged quarks in protons and neutrons Examples of such nuclides include 1H, 2D, 7Li, 13C, 14N, 19F, 23Na, and 127I In total, there are over one hundred stable atoms that possess a magnetic moment and non-zero spin Conversely, nuclei with even numbers of protons or neutrons, such as 12C, 16O, 32S, and 40Ga, exhibit zero spin.
Figure 1.8: Binding energy per nucleon as a function of mass number A.
Figure 1.9: Potential barrier of an atomic nucleus The maximum potential barrier is at the surface of the nucleus (distance R), while increasing distance r is weaker.
The nuclear spin number I, in conjunction with the Dirac constant (Iħ), determines the spin of the nucleus Nuclei with even mass numbers exhibit integer spin values in ħ, whereas those with odd mass numbers display half-integer spin Nuclear magnetic resonance effects are observed in nuclei where I is greater than 0.
The nuclear magneton (nm), defined as 1 nm = eħ/2m p, represents the magnetic moment of a nucleus, analogous to the Bohr magneton for electrons The nuclear magneton is approximately 1/1836 times lower than the Bohr magneton, with a value of 1 nm equating to 5.05×10 −27 J.T −1 (or A.m 2) The magnetic moment of a proton is measured at 2.8 nm, translating to 1.41×10 −26 J.T −1, which is about 658 times weaker than that of an electron This significant difference explains the weak magnetic phenomena of nuclei, necessitating advanced devices for observation These magnetic properties are fundamental to the sophisticated imaging technique known as nuclear magnetic resonance imaging.
Forces acting between atoms
Electrons can be rearranged in external shells to yield higher stability, whereby a chemical bond between atoms is formed creating valence electrons Chemical bonds mostly do not affect internal electrons.
Bonds between neutral atoms cannot be explained within the framework of classical phys- ics However, forces that create a covalent bond can be quantitatively calculated using quan- tum mechanics.
Molecules achieve stability when the energy of the combined atoms is lower than that of individual, separated atoms When atoms interact and their total energy decreases, they can form a molecule As two atoms approach each other, they may either form a covalent or ionic bond, or they may not bond at all.
When electron shells overlap, they create a single physical system; however, the Pauli exclusion principle dictates that two electrons cannot occupy the same quantum state Consequently, if interacting electrons are compelled to occupy higher energy states than those in isolated atoms, the system becomes unstable and acquires higher energy This overlap leads to Born repulsive forces, which have a limited range and diminish with distance according to the inverse cube law, proportional to r^(-13).
Ionic bonds are formed through Coulomb attractive forces, which have a greater range than covalent bonds and involve a larger distance between atomic nuclei than the sum of their radii In these bonds, charge is transferred from one atom to another, resulting in spherically symmetrical bonds with no preferred direction The non-saturated nature of ionic bonds allows for multiple mutually attracted ions, leading to a stable spatial configuration where oppositely charged ions occupy regular positions, typically resulting in crystalline structures Common examples of ionic bonds include compounds formed between metals and halogens, such as sodium (Na+) and chlorine (Cl-).
Figure 1.10: Ion bond Electrons are donated by one atom to another, resulting in positive and negative ions which attract each other.
Covalent bonds involve the simultaneous sharing of one or more pairs of electrons between bound atoms, a phenomenon influenced by the quantum-mechanical exchange effect A key feature of covalent bonds is their reliance on electron spin, with stronger bonds forming when spins are anti-parallel Additionally, covalent bonds exhibit a saturation effect, requiring only a limited number of electrons for stability Only electrons from external, incompletely filled shells can participate in these bonds, leading to an asymmetrical distribution of electric charge within the molecule, which in turn results in dipole-like electrical properties.
The covalent bond is recognized as the strongest type of chemical bond, exemplified by the bond energy of 4.3 eV between two hydrogen atoms (H–H) This energy release translates to 417 kJ when one mole of hydrogen molecules (H2) is formed, calculated as 6.02×10²³ × 4.3 × 1.6×10⁻¹⁹ Atoms connected by covalent bonds are positioned closely together, with the distance between protons in a hydrogen molecule measuring 0.074 nm, while the Bohr radius, representing the internal path of an electron in a hydrogen atom, is approximately 0.054 nm.
Figure 1.11: Covalent bond Two atoms share a pair of electrons to complete the outer shell.
Molecules are formed when atoms are at a specific distance where attractive and repulsive forces are balanced, corresponding to the system's minimum potential energy Moving atoms further apart requires an external energy increase, as they cannot be separated naturally The binding energy is defined as the energy released during the molecule's formation, and additional details on bonds can be found in section 2.1.
Figure 1.12: Potential energy E of the system as a function of distance r W – binding energy, r 0 – dis- tance of minimal potential energy. molecule sharing of electrons magnetic field r 1
Physical basis of nuclear magnetic resonance tomography
Nucleons, including protons and neutrons, have half-integer angular momentum spin, which contributes to their magnetic moments of 2.8 nm and 1.9 nm, respectively These magnetic properties of atomic nuclei are fundamental to the phenomenon of nuclear magnetic resonance (NMR), a crucial technique used in imaging applications.
Most atomic nuclei have a magnetic moment, which is essential for identifying the presence and quantity of specific nuclei in matter samples and for diagnostic imaging techniques Every chemical element contains at least one isotope with a non-zero spin number in its nucleus, resulting in a corresponding magnetic moment.
Nuclear magnetic resonance (NMR) is a non-invasive technique that analyzes the distribution and behavior of magnetic moments from specific isotopes in a magnetic field Due to its ability to provide insights into biochemical processes within living tissues, it has emerged as the leading non-invasive radiological method In clinical settings, this technique is commonly referred to as magnetic resonance (MR) imaging.
Magnetic Resonance (MR) imaging is a safe diagnostic tool as it does not cause ionizing radiation damage, utilizing low-energy radiofrequencies absorbed by nuclei in a strong static magnetic field While various isotopes with a non-zero magnetic moment can be employed, hydrogen nuclei are predominantly used in clinical settings due to their superior MR sensitivity Hydrogen, abundant in organic compounds and present in high concentrations in water—the primary component of biological tissue—allows for effective imaging The distribution of water molecules in tissues not only reflects their structural composition but also closely correlates with pathological changes, contributing to the high success rate of MR imaging in medical applications.
Figure 1.13 MR image of a rat brain (coronal plane) Here, the distinct hyperintense lesion in the brain is a lesion caused by stroke.
In the presence of a non-zero magnetic field (B0), the magnetic moments of protons exhibit a macroscopic magnetization vector (M), resulting from their anti-parallel alignment with B0 Conversely, when B0 is zero, the magnetic moments are randomly aligned, leading to a net magnetic moment of zero The precession of these moments suppresses all transverse components, further emphasizing the significance of B0 in determining the overall magnetic behavior.
Magnetic Resonance (MR) is best understood through the interaction of radiofrequency radiation with atomic magnetic moments When nuclei are placed in a strong external magnetic field (B0), they align and begin to precess around an axis parallel to B0 This precession causes the transversal components to cancel out due to the random phases of each proton's precession, resulting in a macroscopic magnetization vector (M) that rotates around the same axis as the external magnetic field The frequency of this precession (ω) is influenced by the intensity of the external magnetic field (B0) and the specific type of nucleus, which is characterized by the gyromagnetic constant (γ).
Larmor formula (1.34), the basic equation used in MR ω γ= ⋅B 0 (1.34)
To calculate the frequency value of the rotating magnetic field for 1 H protons in an external magnetic field of 1 T, we utilize the magnetic moment of 1 H proton, which is 1.41×10^(-26) J.T^(-1) According to equations (1.34) and (1.31), the frequency can be determined using the relationship f = B × γ.
J.s 66MHz The frequency value for the protons is 42.6 MHz At B 0 = 1.5 T, which is the most widely used magnetic field in medical imag- ing, ω is 63.6 MHz Since other nuclei have different values of gyromagnetic ratios γ, the resonance effects occur at different frequencies For B 0 = 1 T and 13 C nuclei, the frequency is
10.7 MHz, for 19 F the frequency is 40.1 MHz and for 31 P nuclei the frequency is 17.2 MHz These examples demonstrate that the values of resonance frequencies are very different for various nuclei The gyromagnetic ratio is defined as the ratio of the magnetic moment to its own angular momentum and spin, and is given by γ = à
Resonance frequencies are influenced by the chemical state of a compound, as the nucleus of an atom experiences a magnetic field (B0) that is modified by a weaker field (B), calculated as (1 − σ)B0, where σ represents the shielding constant related to the electron density around the nucleus This variation in magnetic moments arises from differences in electron envelopes, leading to distinct resonance frequencies for different chemical compounds of the same element This principle is crucial for analyzing sample compositions, exemplified by the observation of three distinct peaks in the NMR signal of ethyl alcohol, corresponding to protons in the CH3, CH2, and OH groups.
When an element is placed in a static external magnetic field (B0), its magnetic moment interacts with the field, leading to the splitting of energy levels For hydrogen, each nucleus has two distinct energy states: protons occupy lower energy levels when aligned parallel to B0 and higher energy levels when aligned anti-parallel The distribution of protons across these energy levels is determined by the Boltzmann distribution, while the energy difference between adjacent levels (∆E) is defined by a specific formula.
The occupancy difference of protons at lower and higher energy levels is minimal, with only 7 protons per million at 1 T, yet this slight variation is crucial for manipulating vector M, given that 1 cm³ contains about 10²³ hydrogen nuclei As the magnetic field increases, this difference grows linearly, necessitating enhanced sensitivity in MR methods for higher magnetic fields Vector M is influenced by the external high-frequency magnetic field B₁, which is perpendicular to the static field B₀ and matches the Larmor frequency Consequently, the nuclei system absorbs energy from the electromagnetic field B₁, leading to the phenomenon known as nuclear magnetic resonance.
From a quantum perspective, the transition of an element occurs between adjacent stationary energy levels, with equal probability in both directions, particularly noted in hydrogen (1 H) At a constant temperature, the population of lower energy levels is more significant due to the Boltzmann distribution, leading to a steady state As a result, magnetic resonance is observed through energy absorption, as the transition to higher energy states is more prevalent.
An electromagnetic field B1 is generated by a brief radiofrequency impulse, typically ranging from 10 MHz to 1 GHz, emitted by transmission coils The intensity and duration of this pulse determine the orientation of the resulting magnetic moment M, which can be flipped to align either perpendicularly or parallel to the external magnetic field.
Figure 1.15: Splitting of energy levels in hydrogen protons 1 H The distance between adjacent energy levels ∆E is dependent on magnetic field strength.
Figure 1.16: Flipping of magnetisation M onto a perpendicular plane (ϕ = 90°) in the direction of exter- nal static magnetic field B 0 by applying electromagnetic field B 1 transmitted from the transmission coils.
B 0 (at a 90° or 180° pulse, respectively) Immediately after the RF pulse is applied, all excited nuclei are at the same phase and start to return to equilibrium
The relaxation process, known as the return, involves nuclei releasing absorbed energy as electromagnetic radiation, which is detected by receiver coils This phenomenon is grounded in Faraday’s law of electromagnetic induction, as nuclear magnetization M precesses during its return in accordance with Larmor precession The resulting induced alternating electromotive force, characterized by angular frequency ω, is referred to as the MR signal or free induction decay (FID) signal The FID signal captures contributions from various compounds containing hydrogen nuclei, such as water, ice, protein, and tissue, with its amplitude directly proportional to the number of nuclei involved in its formation.