Atomic & Nuclear Physics Cheat

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### Normal Zeeman Effect (10-15 Marks) The Normal Zeeman Effect refers to the splitting of a single spectral line into three components when the light source is placed in a strong external magnetic field. This effect is observed in atoms with zero net spin, i.e., $S=0$, for example, in singlet states ($^1S_0$, $^1P_1$, etc.). #### Theoretical Explanation 1. **Classical Theory (Lorentz):** Classically, an orbiting electron constitutes a current loop, generating a magnetic dipole moment $\vec{\mu}_l$. In an external magnetic field $\vec{B}$, this magnetic moment experiences a torque and precesses around the field direction with the Larmor frequency $\omega_L = \frac{eB}{2m_e}$. This precession leads to a change in the electron's energy. 2. **Quantum Mechanical Theory:** * **Orbital Magnetic Moment:** For an electron with orbital angular momentum $\vec{L}$, the orbital magnetic moment is given by $\vec{\mu}_l = -\frac{e}{2m_e}\vec{L} = -\mu_B \frac{\vec{L}}{\hbar}$, where $\mu_B = \frac{e\hbar}{2m_e}$ is the Bohr magneton. * **Interaction Energy:** When an atom is placed in an external magnetic field $\vec{B}$, the interaction energy is $U = -\vec{\mu}_l \cdot \vec{B}$. If $\vec{B}$ is along the z-axis, $U = -\mu_{lz} B = \frac{e}{2m_e} L_z B$. * **Quantization:** According to quantum mechanics, $L_z = m_l\hbar$, where $m_l$ is the magnetic orbital quantum number ($m_l = -l, -l+1, ..., 0, ..., l-1, l$). * **Energy Shift:** The change in energy due to the magnetic field is $\Delta E = m_l \frac{e\hbar}{2m_e} B = m_l \mu_B B$. * **Selection Rules:** For electric dipole transitions, the selection rule for $m_l$ is $\Delta m_l = 0, \pm 1$. #### Derivation of Zeeman Shift Consider a spectral line emitted due to a transition between two energy levels, $E_1$ and $E_2$, with orbital angular momentum quantum numbers $l_1$ and $l_2$, and magnetic quantum numbers $m_{l1}$ and $m_{l2}$. The energy of the emitted photon in the absence of a magnetic field is $E_0 = h\nu_0 = E_1 - E_2$. In the presence of a magnetic field, the energy levels shift: $E'_1 = E_1 + m_{l1} \mu_B B$ $E'_2 = E_2 + m_{l2} \mu_B B$ The energy of the photon emitted in the magnetic field is: $E' = h\nu' = E'_1 - E'_2 = (E_1 - E_2) + (m_{l1} - m_{l2}) \mu_B B$ $h\nu' = h\nu_0 + \Delta m_l \mu_B B$ $\nu' = \nu_0 + \Delta m_l \frac{\mu_B B}{h}$ The frequency shift, known as the Zeeman shift, is $\Delta\nu = \nu' - \nu_0 = \Delta m_l \frac{\mu_B B}{h}$. Since $\Delta m_l = 0, \pm 1$, we get three possible frequencies: 1. $\Delta m_l = 0 \implies \nu' = \nu_0$ (unshifted central component, $\pi$ component) 2. $\Delta m_l = +1 \implies \nu' = \nu_0 + \frac{\mu_B B}{h}$ (shifted component, $\sigma^+$ component) 3. $\Delta m_l = -1 \implies \nu' = \nu_0 - \frac{\mu_B B}{h}$ (shifted component, $\sigma^-$ component) These three lines are equally spaced, with a frequency separation of $\frac{\mu_B B}{h}$. This triplet is characteristic of the Normal Zeeman Effect. **Example:** Transition from $l=1$ to $l=0$ state (e.g., $^1P_1 \to ^1S_0$). For $l=1$, $m_l = -1, 0, +1$. For $l=0$, $m_l = 0$. Possible $\Delta m_l$: * $m_{l1}=0 \to m_{l2}=0 \implies \Delta m_l = 0$ * $m_{l1}=+1 \to m_{l2}=0 \implies \Delta m_l = +1$ * $m_{l1}=-1 \to m_{l2}=0 \implies \Delta m_l = -1$ This results in a normal Zeeman triplet. ### Anomalous Zeeman Effect and Comparison (10-15 Marks) #### Anomalous Zeeman Effect The Anomalous Zeeman Effect is the splitting of a spectral line into more than three components when an atom is exposed to an external magnetic field. This effect is observed in atoms where the net spin angular momentum $S \neq 0$, i.e., for multiplet states (e.g., $^2P_{1/2}$, $^2P_{3/2}$, $^3D_1$, etc.). It arises from the interaction of both the orbital magnetic moment and the spin magnetic moment of the electron with the external magnetic field. **Theoretical Explanation:** 1. **Total Angular Momentum:** In atoms with $S \neq 0$, the total angular momentum is $\vec{J} = \vec{L} + \vec{S}$. The total magnetic moment $\vec{\mu}_J$ is not parallel to $\vec{J}$ because the gyromagnetic ratio for spin is approximately twice that for orbital motion. $\vec{\mu}_J = -g_J \mu_B \frac{\vec{J}}{\hbar}$, where $g_J$ is the Landé g-factor. 2. **Landé g-factor:** The Landé g-factor accounts for the combined effect of orbital and spin magnetic moments. For LS coupling, it is given by: $g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$ 3. **Interaction Energy:** The interaction energy with an external magnetic field $\vec{B}$ (along z-axis) is: $\Delta E = -\vec{\mu}_J \cdot \vec{B} = g_J \mu_B J_z B = g_J \mu_B m_J B$ where $m_J$ is the magnetic quantum number for total angular momentum ($m_J = -J, ..., +J$). 4. **Energy Levels Splitting:** Each energy level characterized by $J$ splits into $(2J+1)$ sub-levels, with the energy spacing determined by $g_J \mu_B B$. 5. **Selection Rules:** The selection rules for transitions are $\Delta m_J = 0, \pm 1$ and $\Delta J = 0, \pm 1$ (but $J=0 \to J=0$ is forbidden). **Example:** Consider the D1 line of Sodium, a transition from $^2P_{1/2}$ to $^2S_{1/2}$. * $^2S_{1/2}$: $L=0, S=1/2, J=1/2$. $g_J = 1 + \frac{1/2(3/2) + 1/2(3/2) - 0}{2(1/2)(3/2)} = 1 + \frac{3/4 + 3/4}{3/2} = 1 + \frac{3/2}{3/2} = 2$. Energy levels split into $m_J = \pm 1/2$. * $^2P_{1/2}$: $L=1, S=1/2, J=1/2$. $g_J = 1 + \frac{1/2(3/2) + 1/2(3/2) - 1(2)}{2(1/2)(3/2)} = 1 + \frac{3/4 + 3/4 - 2}{3/2} = 1 + \frac{3/2 - 2}{3/2} = 1 - \frac{1/2}{3/2} = 1 - 1/3 = 2/3$. Energy levels split into $m_J = \pm 1/2$. Due to different $g_J$ values for the initial and final states, the energy shifts are different, leading to more than three components in the spectrum (e.g., 4 lines for D1 line of Na). #### Comparison with Normal Zeeman Effect | Feature | Normal Zeeman Effect | Anomalous Zeeman Effect | | :------------------- | :--------------------------------------------------- | :---------------------------------------------------- | | **Net Spin (S)** | $S=0$ (Singlet states) | $S \neq 0$ (Multiplet states) | | **Number of Lines** | Always 3 lines (a triplet) | More than 3 lines | | **Energy Shift** | $\Delta E = m_l \mu_B B$ | $\Delta E = g_J \mu_B m_J B$ | | **Magnetic Moment** | Only orbital magnetic moment contributes | Both orbital and spin magnetic moments contribute | | **Lande g-factor** | $g_J = 1$ (implicitly) | $g_J$ depends on L, S, J (calculated using formula) | | **Frequency Shift** | $\Delta\nu = \Delta m_l \frac{\mu_B B}{h}$ | $\Delta\nu = (g_{J1}m_{J1} - g_{J2}m_{J2}) \frac{\mu_B B}{h}$ | | **Simplicity** | Simpler, explained by classical theory (Lorentz) | More complex, requires quantum mechanics and spin | | **Observation** | Observed in elements like Zn, Cd, Hg (singlet states)| Observed in elements like Na, H, Li (doublet states) | ### LS Coupling and JJ Coupling (10-15 Marks) The interaction between the orbital and spin angular momenta of multiple electrons in an atom determines the total angular momentum and thus the fine structure of atomic spectra. There are two primary coupling schemes: LS (Russell-Saunders) coupling and JJ coupling. #### LS (Russell-Saunders) Coupling LS coupling is typically observed in lighter atoms (Z 70-80) where the spin-orbit interaction for individual electrons is stronger than the electrostatic interaction between electrons. **Mechanism:** 1. **Individual Spin-Orbit Coupling:** For each electron, its orbital angular momentum $\vec{l}_i$ couples with its own spin angular momentum $\vec{s}_i$ to form its total individual angular momentum $\vec{j}_i$. $\vec{j}_i = \vec{l}_i + \vec{s}_i$ The quantum number $j_i$ can take values from $|l_i - s_i|$ to $(l_i + s_i)$. 2. **Sum of Individual Total Angular Momenta:** The total angular momenta of individual electrons ($\vec{j}_i$) then combine vectorially to form the total angular momentum $\vec{J}$ of the atom. $\vec{J} = \sum_i \vec{j}_i$ The quantum number $J$ can take values from $|j_1 - j_2|$ to $(j_1 + j_2)$ for two electrons, and similarly for more electrons. **Notation:** States in JJ coupling are often represented by the configuration $(j_1, j_2)_J$. **Example:** For two equivalent p-electrons ($l_1=1, l_2=1, s_1=1/2, s_2=1/2$): * For each p-electron ($l=1, s=1/2$), $j$ can be $|1 - 1/2| = 1/2$ or $1 + 1/2 = 3/2$. * Possible combinations for $(j_1, j_2)_J$: * $(1/2, 1/2)_J$: $J$ can be $|1/2 - 1/2|=0$ or $1/2+1/2=1$. States: $(1/2, 1/2)_0, (1/2, 1/2)_1$. * $(1/2, 3/2)_J$: $J$ can be $|1/2 - 3/2|=1$ or $1/2+3/2=2$. States: $(1/2, 3/2)_1, (1/2, 3/2)_2$. * $(3/2, 1/2)_J$: Same as above. * $(3/2, 3/2)_J$: $J$ can be $|3/2 - 3/2|=0, 1, 2, 3$. States: $(3/2, 3/2)_0, (3/2, 3/2)_1, (3/2, 3/2)_2, (3/2, 3/2)_3$. #### Comparison of LS and JJ Coupling | Feature | LS Coupling (Russell-Saunders) | JJ Coupling | | :-------------------- | :----------------------------------------------------------- | :----------------------------------------------------------- | | **Applicability** | Lighter atoms (Z 70-80) | | **Interaction Strength**| Electrostatic interaction > Spin-orbit interaction | Spin-orbit interaction > Electrostatic interaction | | **Coupling Order** | $\sum \vec{l}_i \to \vec{L}$, $\sum \vec{s}_i \to \vec{S}$, then $\vec{L} + \vec{S} \to \vec{J}$ | $\vec{l}_i + \vec{s}_i \to \vec{j}_i$ for each electron, then $\sum \vec{j}_i \to \vec{J}$ | | **Good Quantum Numbers**| $L, S, J, M_J$ (approximately) | $j_i$ for each electron, $J, M_J$ (approximately) | | **Term Symbol** | $^{2S+1}L_J$ | $(j_1, j_2, ...)_J$ | | **Energy Splitting** | Small splitting due to spin-orbit interaction (fine structure) | Large splitting for individual $j_i$, then smaller splitting for $J$ | ### Quantum Numbers Associated with Vector Atom Model (10-15 Marks) The Vector Atom Model (VAM) describes the quantization of angular momenta in atoms and their interactions, leading to a better understanding of atomic spectra, including fine structure and the Zeeman effect. It uses vector sums of quantized angular momenta. The key quantum numbers associated with the Vector Atom Model are: 1. **Principal Quantum Number ($n$):** * **Description:** Determines the electron's main energy level and the size of its orbital. * **Allowed Values:** $n = 1, 2, 3, ...$ (positive integers). * **Significance:** Higher $n$ means higher energy and larger orbital. 2. **Orbital Angular Momentum Quantum Number ($l$):** * **Description:** Determines the magnitude of the electron's orbital angular momentum ($\vec{L}$) and the shape of its orbital. * **Allowed Values:** $l = 0, 1, 2, ..., n-1$. * **Magnitude:** $|\vec{L}| = \sqrt{l(l+1)}\hbar$. * **Notation:** $l=0$ (s-orbital), $l=1$ (p-orbital), $l=2$ (d-orbital), etc. 3. **Magnetic Orbital Quantum Number ($m_l$):** * **Description:** Determines the orientation of the orbital angular momentum vector $\vec{L}$ in space relative to an external magnetic field (z-axis). * **Allowed Values:** $m_l = -l, -l+1, ..., 0, ..., l-1, l$ (in integer steps, $2l+1$ values). * **Z-component:** $L_z = m_l\hbar$. * **Significance:** Explains the spatial quantization of orbital angular momentum and is crucial for the Normal Zeeman Effect. 4. **Spin Angular Momentum Quantum Number ($s$):** * **Description:** Describes the intrinsic angular momentum of an electron, known as spin. It's a purely quantum mechanical property. * **Allowed Values:** For an electron, $s = 1/2$. * **Magnitude:** $|\vec{S}| = \sqrt{s(s+1)}\hbar = \sqrt{1/2(3/2)}\hbar = \frac{\sqrt{3}}{2}\hbar$. * **Significance:** Accounts for the fine structure of spectral lines and the Anomalous Zeeman Effect. 5. **Magnetic Spin Quantum Number ($m_s$):** * **Description:** Determines the orientation of the spin angular momentum vector $\vec{S}$ in space relative to an external magnetic field (z-axis). * **Allowed Values:** $m_s = -s, ..., s$. For an electron, $m_s = \pm 1/2$. * **Z-component:** $S_z = m_s\hbar$. * **Significance:** Explains the two possible orientations of electron spin. 6. **Total Orbital Angular Momentum Quantum Number ($L$):** * **Description:** For multi-electron atoms, it's the resultant orbital angular momentum from the vector sum of individual electron orbital angular momenta ($\vec{L} = \sum \vec{l}_i$). * **Allowed Values:** Determined by the specific configuration and coupling scheme (e.g., LS coupling). * **Magnitude:** $|\vec{L}| = \sqrt{L(L+1)}\hbar$. 7. **Total Spin Angular Momentum Quantum Number ($S$):** * **Description:** For multi-electron atoms, it's the resultant spin angular momentum from the vector sum of individual electron spin angular momenta ($\vec{S} = \sum \vec{s}_i$). * **Allowed Values:** Determined by the specific configuration and coupling scheme (e.g., LS coupling). * **Magnitude:** $|\vec{S}| = \sqrt{S(S+1)}\hbar$. 8. **Total Angular Momentum Quantum Number ($J$):** * **Description:** The total angular momentum of the atom, resulting from the vector sum of the total orbital angular momentum and the total spin angular momentum ($\vec{J} = \vec{L} + \vec{S}$ for LS coupling, or $\vec{J} = \sum \vec{j}_i$ for JJ coupling). * **Allowed Values:** For LS coupling, $J$ ranges from $|L-S|$ to $(L+S)$ in integer steps. * **Magnitude:** $|\vec{J}| = \sqrt{J(J+1)}\hbar$. * **Significance:** Determines the fine structure of spectral lines and is crucial for the Anomalous Zeeman Effect. 9. **Magnetic Total Angular Momentum Quantum Number ($m_J$):** * **Description:** Determines the orientation of the total angular momentum vector $\vec{J}$ in space relative to an external magnetic field (z-axis). * **Allowed Values:** $m_J = -J, -J+1, ..., 0, ..., J-1, J$ (in integer steps, $2J+1$ values). * **Z-component:** $J_z = m_J\hbar$. * **Significance:** Explains the splitting of energy levels in the Anomalous Zeeman Effect. These quantum numbers, particularly $n, l, m_l, s, m_s$ for single electrons and $L, S, J, m_J$ for multi-electron atoms under specific coupling schemes, provide a complete description of the electron's state and its behavior in external fields, fully encompassed by the Vector Atom Model. ### Stark Effect with Theory (10-15 Marks) The Stark effect is the splitting and shifting of spectral lines of atoms and molecules due to the presence of an external static electric field. It is analogous to the Zeeman effect, where splitting occurs due to a magnetic field. #### Theoretical Explanation The Stark effect arises from the interaction between the electric dipole moment of the atom (or the induced dipole moment) and the external electric field. 1. **Perturbation Hamiltonian:** When an atom is placed in an external electric field $\vec{E}$, the additional potential energy of the electron is given by the interaction term in the Hamiltonian: $H' = -\vec{p} \cdot \vec{E}$ where $\vec{p}$ is the electric dipole moment of the atom. For an electron with charge $-e$ at position $\vec{r}$ relative to the nucleus, $\vec{p} = -e\vec{r}$. If the electric field is along the z-axis, $\vec{E} = E_z \hat{k}$, then: $H' = -(-e\vec{r}) \cdot (E_z \hat{k}) = e E_z z$ 2. **First-Order Stark Effect:** The first-order energy shift $\Delta E^{(1)}$ is given by the expectation value of the perturbation Hamiltonian in the unperturbed state $|nlm\rangle$: $\Delta E^{(1)} = \langle nlm | H' | nlm \rangle = \langle nlm | e E_z z | nlm \rangle = e E_z \langle nlm | z | nlm \rangle$ For states with a definite parity (like hydrogenic orbitals, which are eigenstates of parity operator), the expectation value of an odd operator like $z$ is zero. $\langle nlm | z | nlm \rangle = 0$ Therefore, for non-degenerate states like the ground state of hydrogen, there is no first-order Stark effect. However, if states are degenerate and have mixed parity (e.g., $2s$ and $2p_z$ states in hydrogen are degenerate and can mix), then $\langle nlm | z | n'l'm' \rangle$ can be non-zero. * **Linear Stark Effect:** This occurs in atoms like hydrogen, where states with different $l$ but the same $n$ are degenerate (e.g., $2s$ and $2p$ states). The electric field mixes these degenerate states, leading to a first-order energy shift that is linear in the electric field strength $E$. For hydrogen, the $n=2$ states ($2s, 2p_x, 2p_y, 2p_z$) are degenerate. The electric field causes a coupling between $2s$ and $2p_z$ states. The energy shifts are $\Delta E = \pm 3 e a_0 E$, where $a_0$ is the Bohr radius. This results in spectral lines splitting into multiple components, symmetric around the original line. 3. **Second-Order Stark Effect:** For non-degenerate states (or states where the first-order effect is zero, like the ground state of hydrogen or states of atoms with filled shells), the energy shift is proportional to the square of the electric field strength. $\Delta E^{(2)} = \sum_{k \neq 0} \frac{|\langle 0 | H' | k \rangle|^2}{E_0 - E_k}$ This effect is generally much weaker than the linear Stark effect. It leads to a quadratic dependence on $E$ and typically a red shift (shift to lower energy). In atoms heavier than hydrogen, the $l$-degeneracy is lifted by electron-electron interactions, so the linear Stark effect is usually absent, and only the quadratic Stark effect is observed. #### Features of the Stark Effect * **Splitting and Shifting:** Spectral lines split into multiple components, and the entire pattern can be shifted. * **Polarization:** The components of the Stark-split lines are often polarized, similar to the Zeeman effect. Components with $\Delta m_l = 0$ (or $\Delta m_J = 0$) are called $\pi$ components and are polarized parallel to the electric field. Components with $\Delta m_l = \pm 1$ (or $\Delta m_J = \pm 1$) are called $\sigma$ components and are polarized perpendicular to the electric field. * **Dependence on Field Strength:** * **Linear Stark Effect:** Shift proportional to $E$. Observed in hydrogen and hydrogen-like atoms (degenerate $l$ states). The splitting is symmetric. * **Quadratic Stark Effect:** Shift proportional to $E^2$. Observed in most other atoms where $l$-degeneracy is lifted. The splitting is generally asymmetric and often results in a net shift of the line. **Applications:** * **Plasma Diagnostics:** Measuring Stark broadening of spectral lines can determine electron densities in plasmas. * **Atomic Structure:** Provides insights into the electric dipole moments and polarizabilities of atoms and molecules. * **Molecular Physics:** Used to study rotational and vibrational states of molecules. ### Pure Rotational Spectra of Diatomic Molecules (10-15 Marks) Pure rotational spectra (also known as microwave spectra) arise from transitions between different rotational energy levels of a molecule. These transitions typically occur in the microwave region of the electromagnetic spectrum. For a diatomic molecule, we can model it as a rigid rotor. #### Rigid Rotor Model A diatomic molecule consists of two atoms of masses $m_1$ and $m_2$ separated by a fixed internuclear distance $r_0$. The molecule rotates about an axis perpendicular to the internuclear axis and passing through its center of mass. 1. **Moment of Inertia ($I$):** $I = \mu r_0^2$, where $\mu$ is the reduced mass of the molecule: $\mu = \frac{m_1 m_2}{m_1 + m_2}$ 2. **Rotational Energy Levels:** According to quantum mechanics, the rotational energy levels of a rigid rotor are quantized and given by: $E_J = \frac{\hbar^2}{2I} J(J+1)$ where $J$ is the rotational quantum number, which can take integer values $J = 0, 1, 2, 3, ...$. It is more convenient to express the energy in terms of rotational constant $B$: $B = \frac{\hbar^2}{2I h c} = \frac{h}{8\pi^2 I c}$ (in cm$^{-1}$) So, $E_J = B J(J+1)$ (in cm$^{-1}$) #### Energy Level Diagram The rotational energy levels are not equally spaced. As $J$ increases, the spacing between adjacent levels increases: * $J=0 \implies E_0 = 0$ * $J=1 \implies E_1 = 2B$ * $J=2 \implies E_2 = 6B$ * $J=3 \implies E_3 = 12B$ * $J=4 \implies E_4 = 20B$ ``` J=4 ----- 20B | (8B) J=3 ----- 12B | (6B) J=2 ----- 6B | (4B) J=1 ----- 2B | (2B) J=0 ----- 0 ``` *(Diagram showing rotational energy levels and transitions)* #### Selection Rules For a diatomic molecule to exhibit a pure rotational spectrum, it must possess a permanent electric dipole moment. Homonuclear diatomic molecules (e.g., H$_2$, O$_2$, N$_2$) do not have a permanent dipole moment and thus do not show pure rotational spectra. Heteronuclear diatomic molecules (e.g., HCl, CO, HF) do have a permanent dipole moment. The selection rule for pure rotational transitions is: $\Delta J = \pm 1$ This means transitions can only occur between adjacent rotational levels. * $\Delta J = +1$ for absorption (molecule absorbs energy and moves to a higher rotational state). * $\Delta J = -1$ for emission (molecule emits energy and moves to a lower rotational state). #### Spectral Lines (Frequencies/Wavenumbers) Consider an absorption transition from a rotational level $J$ to $J+1$: $\tilde{\nu}_{J \to J+1} = E_{J+1} - E_J$ $\tilde{\nu}_{J \to J+1} = B(J+1)(J+2) - B J(J+1)$ $\tilde{\nu}_{J \to J+1} = B(J+1)[(J+2) - J]$ $\tilde{\nu}_{J \to J+1} = 2B(J+1)$ Where $J$ is the rotational quantum number of the *lower* state. The wavenumbers of the observed spectral lines will be: * For $J=0 \to J=1$: $\tilde{\nu} = 2B$ * For $J=1 \to J=2$: $\tilde{\nu} = 4B$ * For $J=2 \to J=3$: $\tilde{\nu} = 6B$ * For $J=3 \to J=4$: $\tilde{\nu} = 8B$ #### Characteristics of Pure Rotational Spectra * **Equally Spaced Lines:** The pure rotational spectrum consists of a series of equally spaced lines with a separation of $2B$. * **Region:** Occurs in the microwave region (typically 1-100 GHz or 0.03-3 cm$^{-1}$). * **Information Derived:** From the spacing ($2B$), the rotational constant $B$ can be determined. From $B$, the moment of inertia $I$ can be calculated, and subsequently, the internuclear distance $r_0$ can be found. * **Intensity of Lines:** The intensity of the spectral lines is not uniform. It depends on the population of the initial rotational level, which follows a Boltzmann distribution. The intensity usually increases with $J$ to a maximum and then decreases. * **Non-Rigid Rotor:** In reality, molecules are not perfectly rigid. As $J$ increases, the molecule stretches due to centrifugal force, increasing $r_0$ and thus $I$. This leads to a decrease in $B$ at higher $J$ values, causing the spectral lines to converge slightly (i.e., the spacing becomes slightly less than $2B$). This is accounted for by adding a centrifugal distortion term to the energy equation: $E_J = B J(J+1) - D_J J^2(J+1)^2$. ### Vibrational Spectra of Diatomic Molecule with Diagram (10-15 Marks) Vibrational spectra arise from transitions between different vibrational energy levels of a molecule. For a diatomic molecule, we can model its vibrations using the harmonic oscillator and anharmonic oscillator approximations. These transitions typically occur in the infrared (IR) region of the electromagnetic spectrum. #### Harmonic Oscillator Model A diatomic molecule can be approximated as two masses connected by a spring. This is the harmonic oscillator model. 1. **Classical Vibration Frequency:** Classically, the frequency of vibration is $\nu_{osc} = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$, where $k$ is the force constant of the bond and $\mu$ is the reduced mass. 2. **Vibrational Energy Levels:** According to quantum mechanics, the vibrational energy levels of a harmonic oscillator are quantized and given by: $E_v = (v + 1/2) h\nu_{osc}$ where $v$ is the vibrational quantum number, which can take integer values $v = 0, 1, 2, 3, ...$. In terms of wavenumbers ($\tilde{\nu}_{osc} = \nu_{osc}/c$): $E_v = (v + 1/2) h c \tilde{\nu}_{osc}$ (in Joules) Or, often expressed as $\tilde{E}_v = (v + 1/2) \tilde{\nu}_{osc}$ (in cm$^{-1}$) #### Energy Level Diagram (Harmonic Oscillator) The vibrational energy levels are equally spaced, with a separation of $h\nu_{osc}$ (or $\tilde{\nu}_{osc}$ in cm$^{-1}$). The lowest energy level ($v=0$) has a non-zero energy, known as the zero-point energy ($E_0 = 1/2 h\nu_{osc}$). ``` v=3 ----- (7/2)hν | | hν v=2 ----- (5/2)hν | | hν v=1 ----- (3/2)hν | | hν v=0 ----- (1/2)hν (Zero-point energy) ``` *(Diagram showing equally spaced harmonic oscillator vibrational energy levels)* #### Selection Rules For a diatomic molecule to exhibit a pure vibrational spectrum, its electric dipole moment must change during the vibration. This means that the molecule must have a changing dipole moment with respect to internuclear distance. Heteronuclear diatomic molecules (e.g., HCl, CO) show IR spectra, while homonuclear diatomic molecules (e.g., H$_2$, O$_2$) do not. The selection rule for vibrational transitions in a harmonic oscillator is: $\Delta v = \pm 1$ This means transitions can only occur between adjacent vibrational levels. * $\Delta v = +1$ for absorption. * $\Delta v = -1$ for emission. #### Spectral Lines (Harmonic Oscillator) Consider an absorption transition from a vibrational level $v$ to $v+1$: $\tilde{\nu}_{v \to v+1} = \tilde{E}_{v+1} - \tilde{E}_v$ $\tilde{\nu}_{v \to v+1} = (v+1+1/2)\tilde{\nu}_{osc} - (v+1/2)\tilde{\nu}_{osc}$ $\tilde{\nu}_{v \to v+1} = \tilde{\nu}_{osc}$ Thus, the harmonic oscillator model predicts only *one* absorption line at the fundamental frequency $\tilde{\nu}_{osc}$, regardless of the initial vibrational state. This single line is called the *fundamental band*. #### Anharmonic Oscillator Model The harmonic oscillator model is an approximation. Real molecules do not behave as perfect harmonic oscillators, especially at higher vibrational energies. The bond becomes weaker at larger internuclear distances, and the potential energy curve deviates from a parabola, leading to anharmonicity. 1. **Anharmonic Energy Levels:** The energy levels of an anharmonic oscillator are given by: $\tilde{E}_v = (v + 1/2)\tilde{\nu}_e - (v + 1/2)^2 x_e \tilde{\nu}_e$ where $\tilde{\nu}_e$ is the equilibrium vibrational frequency (at the bottom of the potential well), and $x_e$ is the anharmonicity constant (a small positive value). #### Energy Level Diagram (Anharmonic Oscillator) The anharmonicity causes the vibrational energy levels to converge (get closer together) at higher $v$ values, and eventually, the molecule dissociates. ``` Dissociation Limit --- \ v=3 ----- \ \ v=2 ----- \ \ v=1 ----- \ \ v=0 ----- (Zero-point energy) ``` *(Diagram showing converging anharmonic oscillator vibrational energy levels)* #### Selection Rules (Anharmonic Oscillator) Due to anharmonicity, the strict selection rule $\Delta v = \pm 1$ is relaxed. Transitions with $\Delta v = \pm 2, \pm 3, ...$ become weakly allowed. These are called **overtones**. #### Spectral Lines (Anharmonic Oscillator) * **Fundamental Band:** $v=0 \to v=1$. Frequency $\tilde{\nu}_{fundamental} = \tilde{E}_1 - \tilde{E}_0 = \tilde{\nu}_e (1 - 2x_e)$. This is the strongest band. * **First Overtone:** $v=0 \to v=2$. Frequency $\tilde{\nu}_{first overtone} = \tilde{E}_2 - \tilde{E}_0 = 2\tilde{\nu}_e (1 - 3x_e)$. This is weaker than the fundamental. * **Second Overtone:** $v=0 \to v=3$. Frequency $\tilde{\nu}_{second overtone} = \tilde{E}_3 - \tilde{E}_0 = 3\tilde{\nu}_e (1 - 4x_e)$. Even weaker. The presence of overtones and the slight shift of the fundamental band from $\tilde{\nu}_e$ are direct evidence of anharmonicity. #### Characteristics of Vibrational Spectra * **Region:** Occurs in the infrared (IR) region (typically 4000-400 cm$^{-1}$). * **Information Derived:** Determines bond strengths (force constant $k$), identifies functional groups (fingerprint region), and provides information on anharmonicity. * **Intensity:** Fundamental bands are strong; overtones are progressively weaker. Hot bands (transitions originating from $v>0$) are also observed but are generally weak at room temperature. ### Raman Effect with Experimental Arrangement and Theory (10-15 Marks) The Raman effect is the inelastic scattering of light by molecules. When monochromatic light (typically from a laser) interacts with a molecule, most of the light is scattered elastically (Rayleigh scattering) at the same frequency as the incident light. However, a small fraction of the scattered light has frequencies different from the incident light. This phenomenon is called Raman scattering. #### Theory of Raman Effect The Raman effect arises from the interaction of incident photons with the polarizability of a molecule. 1. **Polarizability:** When a molecule is placed in an electric field $\vec{E}$, an electric dipole moment $\vec{P}$ is induced in it. The magnitude of this induced dipole moment is proportional to the electric field strength: $\vec{P} = \alpha \vec{E}$ where $\alpha$ is the polarizability of the molecule. Polarizability is a measure of how easily the electron cloud of a molecule can be distorted by an external electric field. 2. **Time-Dependent Polarizability:** For Raman scattering to occur, the polarizability of the molecule must change during a molecular vibration or rotation. Consider an incident monochromatic light wave with electric field $E = E_0 \cos(2\pi\nu_0 t)$. If the molecule is vibrating with frequency $\nu_{vib}$ and its polarizability $\alpha$ changes with vibrational motion, we can express $\alpha$ as: $\alpha = \alpha_0 + (\frac{\partial\alpha}{\partial q})_0 q$ where $\alpha_0$ is the equilibrium polarizability, $q$ is the normal coordinate of vibration, and $(\frac{\partial\alpha}{\partial q})_0$ is the change in polarizability with respect to the normal coordinate. If $q = q_0 \cos(2\pi\nu_{vib} t)$, then: $\alpha = \alpha_0 + (\frac{\partial\alpha}{\partial q})_0 q_0 \cos(2\pi\nu_{vib} t)$ 3. **Induced Dipole Moment:** Substituting $\alpha$ into the equation for $\vec{P}$: $\vec{P} = [\alpha_0 + (\frac{\partial\alpha}{\partial q})_0 q_0 \cos(2\pi\nu_{vib} t)] E_0 \cos(2\pi\nu_0 t)$ Using the trigonometric identity $2\cos A \cos B = \cos(A+B) + \cos(A-B)$: $\vec{P} = \alpha_0 E_0 \cos(2\pi\nu_0 t) + \frac{1}{2} E_0 (\frac{\partial\alpha}{\partial q})_0 q_0 [\cos(2\pi(\nu_0+\nu_{vib})t) + \cos(2\pi(\nu_0-\nu_{vib})t)]$ 4. **Scattered Frequencies:** The induced dipole moment oscillates at three distinct frequencies, which correspond to the scattered light: * **Rayleigh Scattering:** $\nu_0$ (first term) - elastic scattering, no change in frequency. * **Stokes Lines:** $\nu_0 - \nu_{vib}$ (second term) - lower frequency, molecule gains vibrational energy. * **Anti-Stokes Lines:** $\nu_0 + \nu_{vib}$ (second term) - higher frequency, molecule loses vibrational energy. The difference in frequency between the incident and scattered light, $|\nu_0 - \nu'|$, is called the Raman shift ($\Delta\nu$) and corresponds to the vibrational or rotational frequencies of the molecule. #### Selection Rule for Raman Active Vibrations A vibrational mode is Raman active if there is a change in the molecule's polarizability during that vibration. This is a crucial difference from IR spectroscopy, where a change in dipole moment is required. * Homonuclear diatomic molecules (e.g., H$_2$, N$_2$, O$_2$), which are IR inactive, are Raman active because their polarizability changes during vibration. #### Experimental Arrangement A typical Raman spectrometer consists of the following components: 1. **Light Source:** A high-intensity, monochromatic light source, usually a laser (e.g., He-Ne, Argon ion, Nd:YAG), is used to illuminate the sample. Lasers provide sufficient intensity and narrow linewidth. 2. **Sample Holder:** The sample (gas, liquid, or solid) is held in a suitable container. The sample must be transparent to the laser light. 3. **Collection Optics:** Lenses and mirrors are used to collect the scattered light from the sample, typically at 90 degrees to the incident beam to minimize collection of the strong incident beam. 4. **Monochromator/Spectrograph:** This disperses the scattered light into its constituent frequencies. It consists of slits, gratings (or prisms), and mirrors. It separates the weak Raman scattered light from the strong Rayleigh scattered light. 5. **Detector:** A sensitive detector, such as a photomultiplier tube (PMT) or a charge-coupled device (CCD), records the intensity of the scattered light at different frequencies. 6. **Computer:** For data acquisition, processing, and display of the Raman spectrum. ``` +----------------+ +-------------------+ +--------------------+ +-------------+ +----------+ | Laser Source |-----> | Sample (Cuvette) |-----> | Collection Optics |-----> | Spectrograph|-----> | Detector |-----> Computer +----------------+ +-------------------+ +--------------------+ +-------------+ +----------+ ^ | | (Scattered Light) ``` *(Simplified diagram of a Raman spectrometer showing laser, sample, collection optics, spectrograph, and detector)* #### Characteristics of Raman Spectrum * **Raman Shift:** The spectrum shows peaks corresponding to the Raman shifts ($\Delta\nu = \nu_{vib}$ or $\nu_{rot}$). These shifts are independent of the incident laser frequency. * **Stokes and Anti-Stokes Lines:** * Stokes lines are more intense than anti-Stokes lines because the population of the ground vibrational state ($v=0$) is higher than that of excited states ($v>0$) according to Boltzmann distribution. * The intensity ratio is given by $I_{Anti-Stokes}/I_{Stokes} = (\frac{\nu_0+\Delta\nu}{\nu_0-\Delta\nu})^4 e^{-\Delta E / kT}$. * **Complementary to IR:** Raman and IR spectroscopy provide complementary information. Vibrations that are IR active may be Raman inactive, and vice versa. * **Applications:** * **Structural Elucidation:** Identifying functional groups and molecular structure. * **Material Science:** Characterizing polymers, semiconductors, carbon materials (e.g., graphene, nanotubes). * **Biology/Medicine:** Studying biological molecules, detecting diseases. * **Forensics:** Identification of unknown substances. ### Rotation-Vibration Spectra in Detail (10-15 Marks) Rotation-vibration (or rovibrational) spectra arise when a molecule undergoes simultaneous changes in both its vibrational and rotational energy states. These spectra are typically observed in the infrared region and provide much more detailed information about molecular structure than pure vibrational or pure rotational spectra alone. #### Combination of Vibrational and Rotational Energies For a diatomic molecule, the total energy can be approximated as the sum of its vibrational and rotational energies: $E_{v,J} = E_v + E_J$ Using the harmonic oscillator and rigid rotor approximations: $\tilde{E}_{v,J} = (v + 1/2)\tilde{\nu}_{osc} + B J(J+1)$ (in cm$^{-1}$) where $v = 0, 1, 2, ...$ and $J = 0, 1, 2, ...$. #### Selection Rules For a molecule to exhibit a rotation-vibration spectrum, it must possess a **permanent electric dipole moment** that changes during vibration (same as for pure vibrational spectra). Thus, heteronuclear diatomic molecules like HCl, CO, HBr, etc., show rovibrational spectra, while homonuclear molecules like O$_2$, N$_2$ do not. The selection rules for rovibrational transitions are: 1. **Vibrational:** $\Delta v = \pm 1$ (for fundamental transitions, assuming harmonic oscillator). Overtones ($\Delta v = \pm 2, \pm 3, ...$) are also weakly allowed due to anharmonicity. 2. **Rotational:** $\Delta J = \pm 1$ (for diatomic molecules). The transition $\Delta J = 0$ is forbidden for most diatomic molecules (except for molecules with electronic angular momentum, which is rare for ground state diatomics). #### P, Q, and R Branches Consider a fundamental vibrational absorption transition from $v=0$ to $v=1$. The energy of the absorbed photon is $\tilde{\nu}_{abs} = \tilde{E}_{v=1,J'} - \tilde{E}_{v=0,J''}$. Here, $J''$ refers to the rotational quantum number of the lower (initial) state ($v=0$), and $J'$ refers to the rotational quantum number of the upper (final) state ($v=1$). We have three possibilities for $\Delta J$: 1. **R-branch ($\Delta J = +1$):** Transitions where $J' = J'' + 1$. $\tilde{\nu}_R(J'') = [(1 + 1/2)\tilde{\nu}_{osc} + B(J''+1)(J''+2)] - [(0 + 1/2)\tilde{\nu}_{osc} + B J''(J''+1)]$ $\tilde{\nu}_R(J'') = \tilde{\nu}_{osc} + B[(J''+1)(J''+2) - J''(J''+1)]$ $\tilde{\nu}_R(J'') = \tilde{\nu}_{osc} + B(J''+1)(J''+2-J'') = \tilde{\nu}_{osc} + 2B(J''+1)$ where $J'' = 0, 1, 2, ...$. The R-branch consists of lines at frequencies higher than the pure vibrational frequency $\tilde{\nu}_{osc}$. 2. **P-branch ($\Delta J = -1$):** Transitions where $J' = J'' - 1$. $\tilde{\nu}_P(J'') = [(1 + 1/2)\tilde{\nu}_{osc} + B(J''-1)(J'')] - [(0 + 1/2)\tilde{\nu}_{osc} + B J''(J''+1)]$ $\tilde{\nu}_P(J'') = \tilde{\nu}_{osc} + B[(J''-1)J'' - J''(J''+1)]$ $\tilde{\nu}_P(J'') = \tilde{\nu}_{osc} + B J''[(J''-1) - (J''+1)] = \tilde{\nu}_{osc} - 2B J''$ where $J'' = 1, 2, 3, ...$ (since $J'$ cannot be negative, $J''$ must be at least 1). The P-branch consists of lines at frequencies lower than the pure vibrational frequency $\tilde{\nu}_{osc}$. 3. **Q-branch ($\Delta J = 0$):** Transitions where $J' = J''$. $\tilde{\nu}_Q(J'') = [(1 + 1/2)\tilde{\nu}_{osc} + B J''(J''+1)] - [(0 + 1/2)\tilde{\nu}_{osc} + B J''(J''+1)]$ $\tilde{\nu}_Q(J'') = \tilde{\nu}_{osc}$ For most diatomic molecules, $\Delta J = 0$ is forbidden, so the Q-branch is absent. If present (e.g., in molecules with angular momentum about the internuclear axis), it would appear as a single strong line (or a very narrow band of lines if $B_v$ varies with $v$) at the center frequency $\tilde{\nu}_{osc}$. #### Rovibrational Spectrum Diagram The rovibrational spectrum typically shows a series of lines on either side of the "missing" central pure vibrational frequency ($\tilde{\nu}_{osc}$). ``` ^ Intensity | | | R-branch (ΔJ = +1) | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ----------------------------------|-----------------|-----------------|----------------------------------------------------> Wavenumber (cm⁻¹) ... P(3) P(2) P(1) R(0) R(1) R(2) R(3) ... ### Regions of Electromagnetic Spectrum and Their Applications (10-15 Marks) The electromagnetic (EM) spectrum is the range of all types of EM radiation, ordered by frequency or wavelength. It spans from very long radio waves to very short gamma rays. All EM waves travel at the speed of light in a vacuum. Different regions of the spectrum interact with matter in distinct ways, leading to various applications. #### 1. Radio Waves * **Wavelength:** $> 1$ mm (up to thousands of kilometers) * **Frequency:** $ 30$ EHz (highest energy) * **Interaction with Matter:** Highly penetrating, ionizes atoms, causes nuclear transitions. Produced by radioactive decay and nuclear reactions. * **Applications:** * **Medical Treatment:** Radiotherapy for cancer. * **Sterilization:** Sterilizing medical equipment and food. * **Industrial Gauges:** Thickness monitoring, density measurement. * **Astronomy:** Studying extreme cosmic events (supernovae, active galactic nuclei). Understanding these regions and their interactions with matter is fundamental to various scientific and technological advancements. ### Cyclotron: Construction, Working, Limitations (10-15 Marks) A cyclotron is a type of particle accelerator that accelerates charged particles (like protons, deuterons, alpha particles) to high energies using a combination of a constant magnetic field and an alternating electric field. It was invented by Ernest O. Lawrence and M. Stanley Livingston in 1934. #### Construction The main components of a cyclotron are: 1. **Dees:** Two large, D-shaped hollow metal electrodes (D$_1$ and D$_2$) placed in a vacuum chamber. These are open along their diameter and slightly separated from each other. 2. **Oscillator (AC Voltage Source):** A high-frequency alternating voltage source is connected across the dees, creating an electric field in the gap between them. The frequency of this oscillator is critical and must be matched to the cyclotron frequency (resonant condition). 3. **Electromagnet:** A strong, uniform magnetic field is applied perpendicular to the plane of the dees by a large electromagnet. This magnetic field curves the path of the charged particles. 4. **Ion Source:** A source (e.g., an arc discharge) located at the center of the dees, which produces the charged particles to be accelerated. 5. **Deflector Plate:** An electrostatic deflector plate positioned near the outer edge of one of the dees, which steers the accelerated particles out of the cyclotron onto a target. 6. **Vacuum Chamber:** The entire setup is enclosed in a vacuum chamber to prevent collisions of accelerated particles with air molecules. ``` N Pole +-------------------------------------+ | | | +-------------------------------+ | | | +-----+ +-----+ | | | | | | | | | | | | | D1 | | D2 | | | | | | | | | | | | | +-----+ +-----+ | | | +-------------------------------+ | | ^ ^ ^ | | | | | | | Ion Source Deflector | | (Center) (Outer Edge)| | | +-------------------------------------+ S Pole ``` *(Simplified top-down diagram of a cyclotron showing Dees, ion source, and deflector, with magnetic field perpendicular to the plane)* #### Working Principle The working of a cyclotron is based on two key principles: 1. **Lorentz Force:** A charged particle moving perpendicular to a uniform magnetic field experiences a Lorentz force, causing it to move in a circular path. $F_B = qvB = \frac{mv^2}{r}$ From this, the radius of the circular path is $r = \frac{mv}{qB}$. The angular frequency of this circular motion (cyclotron frequency) is $\omega_c = \frac{v}{r} = \frac{qB}{m}$. The period of rotation is $T = \frac{2\pi m}{qB}$. Crucially, the period of rotation is independent of the particle's speed and the radius of its path. 2. **Resonance:** The alternating electric field must be synchronized with the cyclotron frequency of the particles. **Steps:** 1. **Ion Injection:** Positive ions are generated at the center by the ion source. 2. **Acceleration in the Gap:** An alternating electric field is applied across the gap between the dees. When D$_1$ is negative and D$_2$ is positive, ions are attracted to D$_1$ and accelerate across the gap, gaining kinetic energy. 3. **Circular Path in Dee:** Once inside D$_1$, the electric field is zero (due to Faraday shielding). The uniform magnetic field perpendicular to the dees forces the ions to move in a semicircular path. 4. **Re-acceleration:** By the time the ions complete a semicircle and reach the gap again, the polarity of the electric field is reversed (D$_1$ becomes positive, D$_2$ becomes negative). The ions are again accelerated across the gap towards D$_2$, gaining more kinetic energy and increasing their speed. 5. **Increasing Radius:** With increased speed, the radius of their semicircular path inside D$_2$ increases ($r \propto v$). 6. **Repetitive Acceleration:** This process repeats. The ions continuously gain energy as they cross the gap and move in increasingly larger semicircles, spiraling outwards. 7. **Extraction:** When the ions reach the maximum radius (and thus maximum energy), they are deflected by a deflector plate and directed towards a target. #### Limitations 1. **Relativistic Mass Increase (Cyclotron Resonance Condition Breakdown):** As particles accelerate to very high speeds (approaching the speed of light), their mass increases according to Einstein's theory of special relativity ($m = \frac{m_0}{\sqrt{1 - v^2/c^2}}$). This increase in mass leads to a decrease in the cyclotron frequency ($\omega_c = \frac{qB}{m}$), making the particles fall out of synchronism with the constant frequency of the alternating electric field. This limits the maximum achievable energy. 2. **Magnetic Field Uniformity:** Maintaining a uniform magnetic field over a large area and at high strengths is challenging and costly. 3. **Space Charge Effects:** At high beam intensities, the mutual repulsion between charged particles (space charge) can cause the beam to spread and become defocussed, reducing efficiency. 4. **Power Dissipation:** A significant amount of power is required to maintain the strong magnetic field and the high-frequency electric field. 5. **Energy Limit for Electrons:** Electrons are much lighter than protons and reach relativistic speeds much more quickly, making cyclotrons unsuitable for accelerating electrons to high energies. 6. **Cost and Size:** High-energy cyclotrons are very large and expensive to build and operate. **Modifications to overcome relativistic effects:** * **Synchrocyclotron:** The frequency of the accelerating electric field is decreased as the particle's mass increases, keeping it in resonance. * **Isochronous Cyclotron:** The magnetic field is designed to increase with radius, compensating for the relativistic mass increase and maintaining a constant cyclotron frequency. * **Synchrotron:** The magnetic field is increased in time to keep particles at a constant orbit radius, and the electric field frequency is also varied. ### Van de Graaff Generator: Construction and Working (10-15 Marks) A Van de Graaff generator is an electrostatic generator that uses a moving belt to accumulate electric charge on a hollow metal sphere, creating very high electrostatic potentials (up to several million volts). It was invented by Robert J. Van de Graaff in 1929. #### Construction The main components of a Van de Graaff generator are: 1. **Hollow Metal Sphere (Collector):** A large, hollow, spherical conducting shell mounted on an insulating column. This is where the charge accumulates. 2. **Insulating Column:** A tall, sturdy column made of an insulating material (e.g., plexiglass, ceramic) that supports the metal sphere and separates it from the ground. 3. **Endless Insulating Belt:** A continuous belt made of a flexible insulating material (e.g., silk, rubber, plastic) that runs between two pulleys. 4. **Pulleys:** Two pulleys, one at the base and one inside the hollow sphere, that drive the insulating belt. The lower pulley is often driven by an electric motor. 5. **Spray Comb (Lower Comb):** A set of sharp metal needles (or brushes) positioned near the lower pulley, close to the outer surface of the belt. This comb is connected to a high-voltage DC source (like a transformer or rectifier, or simply rubbed against a material to generate charge). 6. **Collector Comb (Upper Comb):** Another set of sharp metal needles positioned inside the hollow sphere, near the inner surface of the belt, and connected directly to the inner surface of the sphere. 7. **High-Voltage DC Source (for charging the lower comb):** Typically a few kilovolts (e.g., 10-50 kV) to initiate the charge transfer. 8. **Discharge Electrode (optional):** A grounded sphere or rod used to demonstrate the high potential and discharge the accumulated charge, often producing visible sparks. ``` +-------------------------------+ | | | Hollow Metal Sphere | | (Collector) | | | | +-----------------------+ | | | Upper Comb | | | | (Collector) | | | | | | | | Upper Pulley | | | | (Insulating) | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | 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