F-Praktikum Physikalisches Institut (PI) Stark korrelierte Elektronen und Spins Report Experiment Schallausbreitung in Kristallen (Sound Propagation in Crystals) Intsar Bangwi Physics bachelor 5. term Sven Köppel Physics bachelor 5. term Date of execution: Montag, 10.01.2011, 09:00-11:00 Date of paper submission: Montag, 17.01.2011 Assigned tutor: Protocol author: Language: Daily report: Length of report: Thanh Cong Pham Room No..:.323, Phone: 47244 cong@physik.uni-frankfurt.de phamtanhcong@gmail.com Sven Köppel English 1 page (1 sheet), attached 6 pages
Abstract This experiment is about sound propagation in solid state bodies and isotropic media. The sound velocity and elastic constants of LiF are measured by using piezo transducers. Theoretical Background Simple lattice applications: Density of LiF Lithium fluoride is an inorganic compound with a Face-centered cubic crystal structure (fcc). Knowing the lattice parameter a=0,401 nm 19 6,94 and the atomic masses of 9 F and 3Li, we can easily calculate the density of LiF: An elementary unit consists of 4 Li + 4 F atoms (see ill. 1) in the volumina V =a 3 =64,48 10 30 m 3, so we get the density : = 4 M 4 M Li F 4 19 u 4 6,04u = =2,679 g V N A V cm 3 (This matches very well the experimental value of exp =2.635 g cm 3 ) The elastic constants Sound propagation in solid states can be explained with the simple model of linking adjacent atoms in the lattice structure together with springs, that is, harmonic oscillators. A sound wave (with 10 6 cm ) thus can be expressed with infitesimal displacements. This model is called linear elasticity theory and is basically described with Hooke's law, T =C, having T (Cauchy stress tensor), C (Stiffness tensor) and (infinitesimal strain tensor) all being higher order mathematical objects. Starting from this equation (or rather formalism), I want to derice the different elastic constants in a fcc crystal lattice. We can write the stain tensor like =[ xx xy xz yx yy yz zx zy zz] and describe general infinitesimal movements (whenever the body is expanded or compressed) with a vector field R r which describes the movement of every point in the lattice structure: R r =u r x v r y w r z This movement can also be expressed with the stain tensor as R r = r. Now we can identify the components to express the stain tensor through the u,v,w scalar fields which yields equations like xx = u x or xy yx xy = u y v x (9 equations in total). Illustration 1: fcc system Source: English Wikipedia, (File:Lattice_face_centered_cub ic.svg), BSD license, Copyright 2010 Baszoetekouw F-Praktikum (PI), Schallausbreitung in Kristallen: The elastic constants Page 2 of 6
The mixed terms are first-order taylor approximations which are sufficient in an infitesimal displacement. We notice that we got a symmetrical tensor = T and thus only 6 independent dimensionless coefficients which fully describe the movement of the solid state body. We can introduce the stress tensor T; the component T xy describes the force in x direction on a per unit area which normal shows into y direction. The 9 components of tensor can be directly reduced to 6 independent components by requiring the angular acceleration to vanish. Using the symmetry of the fcc lattice, we can reduce the 36 equations which are expressed in T =C to only a very few independent ones, yielding the equation set C12 C12 0 0 0 C 12 C 11 C 12 0 0 0 C T 12 C 12 C 11 0 0 0 =[C11 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44] Inversing this matrix immediately gives us C 44 =1/T 44, C 11 C 12 = T 11 T 12 1, C 11 2C 12 = T 11 2T 12 1 Following this, we just need a way to express the stress tensor throught the external sound waves (in other words, their velocity). This can be performed by using Newton's Second Law which combines mass density and acceleration t 2 u to the offending forces T, e.g. in x direction: 2 u t = T xx 2 x T xy y T xz z Thus we can study waves in different [hkl] directions. In our experimental setup the sound waves propagate in [110] direction, yielding these simple equations: v L 2 = 1 2 C 11 C 12 2C 44 v 2 T 1 =C 44 v 2 T 2 = 1 2 C 11 C 12 F-Praktikum (PI), Schallausbreitung in Kristallen: The elastic constants Page 3 of 6
Measuring setup The crystal probe has a trapezoidal profile (ill. 2). There are three piezo elements (quartz crystal ultrasonic transducers) embedded (numbered 1, 2 and 3). They all emit ultrasonic waves parallel to the [110] layer, but with different polarizations (two transverally and one longitudinally polarized ones). These probes are coupled with BNC connectors to power amplifiers both on sending and recieving sides. A pulse generator creates very short 1,2us square-waves. With a diode circuit the high level ramp is used to cut out short wave packages from a high frequency generator which are inserted amplified into the quartz crystal transducer. Using another diode blender and a low-pass filter, we get an output voltage level which can be compared to the pulse generator. On this scale the original signal, the transit signal and various numbers of echos should be visible. The times between these signals are meant to be measured (with an oscilloscope) and can be used to calculate the sound veolcity in the LiF crystal. Execution Illustration 2: Cut of the LiF crystal probe Source: Experimental manual In this experiment there are two methods to measure the sound veolcity. The first is called waytime-method and is fully described by the most simple formula of mechanics: v= L T 0 where L is the length of the way the signal has travelled in time T 0. In our setup, the wave propagates throught the whole crystal, so we measure L=13,7 mm. In the second way we don't measure time distances on the oscilloscope, but look directly at the output signal of the crystal (before low-pass). The propagating wave can be described at the origin with the ansatz x 1 = A sin t. After transmitting throught the crystal, it looks like x 2 = A sin t L/v. So there is a phase difference of = 2 1 = L v =2 f L v If we tune the output frequency from the HF generator slowly, we change the wave form to anything like x = A sin 2 t. By looking closely to the oscilloscope, =2 can be archived (the waves look like each other), which yields =2 =2 f 1 f 2 L v This gives us another nice and quite easy way to read the sound velocity v from the oscilloscope: v= f 1 f 2 L F-Praktikum (PI), Schallausbreitung in Kristallen: Execution Page 4 of 6
Measurement retults Connection Piezo 1 1. method v=l/t 0 T 0 = 4±0.5 µs v 1 =3425 m/s 2. method v= f 1 f 2 L f =0,23Mhz v 1 =3151 m/s Average c= v 2, =2635kg/m 3 v 1 =3288m/s c T 2 =28GPa Piezo 2 T 0 = 3±0.5 µs v 2 =4566 m/s f =0,33Mhz v 2 =4521 m /s v 2 =4544 m/s c T 1 =54GPa Piezo 3 T 0 = 2±0.5 µs v 3 =6850 m/s f =0,51Mhz v 3 =6987 m/s v 3 =6919m/s c L =126GPa Inserting these values in the equations v L 2 = 1 2 C 11 C 12 2C 44 v 2 T 1 =C 44 v 2 T 2 = 1 2 C C 11 12 immediately gives us the results: C xy experimental Literature value Quality ( C lit C exp /C lit ) C 44 54 GPa 62 GPa 12% C 11 100 GPa 112 GPa 10% C 12 44 GPa 42 GPa 1% Conclusions This was a quite short and interesting experiment. Of course the results are not very exact, since the values were just read with the naked eye, and there was a lot of noise, espacially at the phase difference method. However it is quite incredible how well such material constants can be determined using these methods. F-Praktikum (PI), Schallausbreitung in Kristallen: Conclusions Page 5 of 6
APPENDIX: Handwritten report of the day F-Praktikum (PI), Schallausbreitung in Kristallen: APPENDIX: Handwritten report of the day Page 6 of 6