""" 01-phonon-chain.py ================== Phonons of a diatomic chain: a row of masses on springs in which the acoustic mode, the optical mode and the zone-boundary mode are animated live as eigenmodes, with a small dispersion inset tracking the branch. Companion to chapter 10 (lattice vibrations / phonons). Run from the repo root: manim -qm --disable_caching dft_notes/animations/chapter_10/01-phonon-chain.py PhononChain Writes to: dft_notes/animations/chapter_10/videos/01-phonon-chain.mp4 """ import numpy as np from manim import ( Scene, MathTex, Tex, VGroup, VMobject, Write, Create, FadeIn, FadeOut, Circle, Dot, Line, Axes, ParametricFunction, Rectangle, ValueTracker, LaggedStart, UP, DOWN, RIGHT, UL, ORIGIN, PI, YELLOW, linear, ) # ---------------------------------------------------------------- palette CORAL = "#cc785c" # light atoms M1 TEALC = "#5db8a6" # heavy atoms M2 INK = "#e8e4de" # warm off-white text DIM = "#9a958c" # secondary text # ---------------------------------------------------------------- chain geometry N_PAIRS = 8 # 8 light + 8 heavy = 16 atoms N_ATOMS = 2 * N_PAIRS D = 0.82 # neighbour spacing BASE_Y = 0.75 # chain height X0 = np.array([-D * (N_ATOMS - 1) / 2 + i * D for i in range(N_ATOMS)]) RAD = np.array([0.12 if i % 2 == 0 else 0.175 for i in range(N_ATOMS)]) # eigenvector sign patterns (displacement_i = sign_i * A * sin(om * t)) SIGN_AC = np.ones(N_ATOMS) # acoustic q->0 : all in phase SIGN_OP = np.array([(-1.0) ** i for i in range(N_ATOMS)]) # optical q->0 : neighbours oppose SIGN_ZB = np.array([1.0 if i % 2 == 0 else 0.0 for i in range(N_ATOMS)]) # q=pi/a : heavy still def spring_pts(p1, p2, r1, r2, coils=4, amp=0.06): """Zigzag polyline between the rims of two atoms (a cartoon spring).""" dv = p2 - p1 L = float(np.linalg.norm(dv)) if L < 1e-6: return [p1.copy(), p2.copy()] u = dv / L nv = np.array([-u[1], u[0], 0.0]) s = p1 + u * r1 e = p2 - u * r2 seg = L - r1 - r2 pts = [s] n = coils * 2 for k in range(1, n): off = amp if k % 2 == 1 else -amp pts.append(s + u * seg * (k / n) + nv * off) pts.append(e) return pts class PhononChain(Scene): def construct(self): tau = ValueTracker(0.0) mode = {"amp": 0.0, "om": 1.6, "sign": SIGN_AC} def env(): """Soft ramp-in so each mode starts smoothly from rest.""" return float(np.clip(tau.get_value() / 1.2, 0.0, 1.0)) def atom_x(i): return X0[i] + mode["sign"][i] * mode["amp"] * env() * np.sin(mode["om"] * tau.get_value()) def make_atom_upd(i): def upd(m): m.move_to(np.array([atom_x(i), BASE_Y, 0.0])) return upd def make_spring_upd(j): def upd(m): p1 = np.array([atom_x(j), BASE_Y, 0.0]) p2 = np.array([atom_x(j + 1), BASE_Y, 0.0]) m.set_points_as_corners(spring_pts(p1, p2, RAD[j], RAD[j + 1])) return upd # ------------------------------------------------ 1. title card title = Tex("Phonons of a diatomic chain", color=INK, font_size=40).to_edge(UP, buff=0.35) leg1 = VGroup( Circle(radius=0.07, color=CORAL, fill_opacity=1, stroke_width=0), Tex(r"$M_1$ = light atoms", color=INK, font_size=27), ).arrange(RIGHT, buff=0.16) leg2 = VGroup( Circle(radius=0.10, color=TEALC, fill_opacity=1, stroke_width=0), Tex(r"$M_2$ = heavy atoms", color=INK, font_size=27), ).arrange(RIGHT, buff=0.16) legend = VGroup(leg1, leg2).arrange(RIGHT, buff=0.8).next_to(title, DOWN, buff=0.22) self.play(Write(title), run_time=1.6) self.play(FadeIn(legend), run_time=0.8) self.wait(1.5) # ------------------------------------------------ 2. the chain at rest baseline = Line(np.array([X0[0] - 0.7, BASE_Y, 0.0]), np.array([X0[-1] + 0.7, BASE_Y, 0.0]), stroke_color=DIM, stroke_width=1.5).set_opacity(0.35) springs = [] for j in range(N_ATOMS - 1): sp = VMobject(stroke_color=DIM, stroke_width=2.5, fill_opacity=0) p1 = np.array([X0[j], BASE_Y, 0.0]) p2 = np.array([X0[j + 1], BASE_Y, 0.0]) sp.set_points_as_corners(spring_pts(p1, p2, RAD[j], RAD[j + 1])) springs.append(sp) atoms = [ Circle(radius=RAD[i], color=CORAL if i % 2 == 0 else TEALC, fill_opacity=1, stroke_width=0 ).move_to(np.array([X0[i], BASE_Y, 0.0])) for i in range(N_ATOMS) ] self.play(Create(baseline), Create(VGroup(*springs)), run_time=1.2) self.play(LaggedStart(*[FadeIn(a, scale=0.5) for a in atoms], lag_ratio=0.06), run_time=1.5) self.wait(1.6) # live-motion plumbing (amp = 0, so the chain sits at equilibrium) for i, a in enumerate(atoms): a.add_updater(make_atom_upd(i)) for j, sp in enumerate(springs): sp.add_updater(make_spring_upd(j)) # ------------------------------------------------ 3. dispersion inset inset_frame = Rectangle(width=3.8, height=2.8).set_stroke(DIM, 1).set_fill(opacity=0) inset_frame.move_to(np.array([5.05, -2.35, 0.0])) axes = Axes( x_range=[0, 1, 1], y_range=[0, 1.05, 1], x_length=3.0, y_length=1.85, axis_config={"stroke_color": DIM, "stroke_width": 2, "include_ticks": False, "include_tip": True, "tip_width": 0.12, "tip_height": 0.12}, ).move_to(np.array([4.93, -2.30, 0.0])) f_ac = lambda q: 0.66 * np.tanh(2.2 * q) # acoustic branch g_op = lambda q: 0.88 - 0.16 * q * q # optical branch cur_ac = ParametricFunction(lambda t: axes.c2p(t, f_ac(t)), t_range=(0, 1, 0.02), stroke_color=TEALC, stroke_width=3) cur_op = ParametricFunction(lambda t: axes.c2p(t, g_op(t)), t_range=(0, 1, 0.02), stroke_color=CORAL, stroke_width=3) tick0 = MathTex("0", color=DIM, font_size=24).next_to(axes.c2p(0, 0), DOWN, buff=0.14) tick1 = MathTex(r"\pi/a", color=DIM, font_size=24).next_to(axes.c2p(1, 0), DOWN, buff=0.14) ylab = MathTex(r"\omega", color=DIM, font_size=24).next_to(axes.c2p(0, 1.05), UL, buff=0.10) wtp = MathTex(r"\omega_+", color=CORAL, font_size=26) wtp.move_to(axes.c2p(0.30, g_op(0.30)) + UP * 0.16) wtm = MathTex(r"\omega_-", color=TEALC, font_size=26) wtm.move_to(axes.c2p(0.82, f_ac(0.82)) + DOWN * 0.22) self.play(FadeIn(inset_frame), run_time=0.5) self.play(Create(axes), run_time=0.9) self.play(Create(cur_ac), run_time=0.9) self.play(Create(cur_op), run_time=0.9) self.play(FadeIn(VGroup(tick0, tick1, ylab, wtp, wtm), lag_ratio=0.15), run_time=0.8) self.wait(1.0) P_AC = axes.c2p(0.02, f_ac(0.02)) # acoustic : q -> 0, lower branch P_OP = axes.c2p(0.02, g_op(0.02)) # optical : q -> 0, upper branch P_ZB = axes.c2p(0.985, g_op(0.985)) # zone edge : upper branch end marker = Dot(radius=0.065, color=YELLOW).move_to(P_AC) # ------------------------------------------------ 4. acoustic mode lead_ac = Tex("acoustic:", color=TEALC, font_size=30) rest_ac = Tex(r"in phase -- $\omega \to 0$ as $q \to 0$", color=INK, font_size=30) lbl_ac = VGroup(lead_ac, rest_ac).arrange(RIGHT, buff=0.22) lbl_ac.move_to(np.array([-1.6, -2.35, 0.0])) self.play(FadeIn(lbl_ac), run_time=0.9) self.wait(0.4) self.play(FadeIn(marker), run_time=0.5) mode.update(amp=0.26, om=1.6, sign=SIGN_AC) tau.set_value(0.0) self.play(tau.animate.set_value(7 * PI / 1.6), run_time=7 * PI / 1.6, rate_func=linear) self.wait(0.5) # ------------------------------------------------ 5. optical mode # swap happens here: every atom sits exactly at equilibrium (t = k*pi/om), # so resetting the clock is invisible. mode.update(amp=0.19, om=3.4, sign=SIGN_OP) tau.set_value(0.0) lead_op = Tex("optical:", color=CORAL, font_size=30) rest_op = Tex(r"out of phase -- finite $\omega$ at $q \to 0$", color=INK, font_size=30) lbl_op = VGroup(lead_op, rest_op).arrange(RIGHT, buff=0.22) lbl_op.move_to(np.array([-1.6, -2.35, 0.0])) self.play(FadeOut(lbl_ac), Write(lbl_op), run_time=0.9) self.play(marker.animate.move_to(P_OP), run_time=0.9) self.wait(0.4) self.play(tau.animate.set_value(10 * PI / 3.4), run_time=10 * PI / 3.4, rate_func=linear) self.wait(0.5) # ------------------------------------------------ 6. zone boundary mode.update(amp=0.33, om=2.4, sign=SIGN_ZB) tau.set_value(0.0) heavy_grp = VGroup(*[atoms[i] for i in range(1, N_ATOMS, 2)]) lead_zb = Tex(r"zone boundary $q = \pi/a$:", color=YELLOW, font_size=30) rest_zb = Tex("one sublattice stands still", color=INK, font_size=30) lbl_zb = VGroup(lead_zb, rest_zb).arrange(RIGHT, buff=0.22) lbl_zb.move_to(np.array([-1.6, -2.35, 0.0])) self.play(FadeOut(lbl_op), Write(lbl_zb), run_time=0.9) self.play(heavy_grp.animate.set_fill(opacity=0.45), run_time=0.6) self.play(marker.animate.move_to(P_ZB), run_time=0.9) self.wait(0.4) self.play(tau.animate.set_value(7 * PI / 2.4), run_time=7 * PI / 2.4, rate_func=linear) self.wait(0.8) # ------------------------------------------------ 7. end card everything = VGroup( title, legend, baseline, VGroup(*springs), VGroup(*atoms), inset_frame, axes, cur_ac, cur_op, VGroup(tick0, tick1, ylab, wtp, wtm), marker, lbl_zb, ) self.play(FadeOut(everything), run_time=0.9) line1 = MathTex(r"\text{acoustic}", r"\;\to\;", r"\text{sound}", color=INK, font_size=42) line1[0].set_color(TEALC) line1[2].set_color(YELLOW) line2 = MathTex(r"\text{optical}", r"\;\to\;", r"\text{IR spectroscopy}", color=INK, font_size=42) line2[0].set_color(CORAL) line2[2].set_color(YELLOW) end = VGroup(line1, line2).arrange(DOWN, buff=0.55).move_to(ORIGIN) src = Tex("chapter 10 -- phonons: diatomic chain eigenmodes", color=DIM, font_size=24).next_to(end, DOWN, buff=0.6) self.play(Write(line1), run_time=1.1) self.play(Write(line2), run_time=1.1) self.play(FadeIn(src), run_time=0.7) self.wait(3.0)