_,g:E;3;:r 2. .. 8.1. This is to certify that the thesis entitled MODULUS 0F ELASTICITY OF SEPARATED LITHIUM ISO‘I‘OPES presented by Wayne Marvin Robertson has been accepted towards fulfillment of the requirements for MASTER OF SCIENCE degree in ME‘I'AILURGICAL ENGINEERING {/jz’ /.vr,/ Major/{g rofessor ,/ Date 20 May 1959 DJ, M 0.3;),\ 0-159 Director of Thesis 3 ELASTIC MODULUS OF SEPARATED ISOTOPES OF LITHIUM by WAYNE MARVIN ROBERTSON A THESIS Submitted to the College of Engineering, Michigan State university of Agriculture and Applied Science in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE Department of Metallurgical Engineering 1959 ACKNOWLEDGMENTS The author is grateful to Dr. A. J. Smith, Professor and Head of the Department of Metallurgical Engineering, for his direction of the graduate program and his support of the work. He is indebted to Dr. Donald J. Montgomery, Professor of Physics, for his suggestion of the problem and for his assistance in carrying out the experiments. He also wishes to acknowledge helpful discussions with Mr. D. G. Triponi, Instructor in the Department of Metallurgical Engineering, and the aid of Mr. B. Curtis, Mechanical Technician in the Department of Metallurgical Engineering, in the design and construction of the apparatus. ii TABLE OF CONTENTS Page INTRODUCTION 1 THEORY OFTHEMETHOD . 2 SOURCEANDPURITY OF MATERIALS 7 APPARATUSANDPROCEDURE ................................... 8 RESUETS .................................................. 13 ACCURACY OF MEASUREMENTS 18 DISCUSSION ............................................... 19 CONCLUSIONS . 22 REWCES CITED 0.0.0.00000000000000000000000000000000000 23 iii LIST OF FIGURES Page 10 EIECTRICALCECIIIT DIAGRAM OOOOOOOOOOOOOOOOOOOOOO0000.00.00 9 2. CONTROHAEDATWSPHERE mm 00.0.0000...0.00.00.00.00000011 iv ABSTRACT The elastic modulus of the separated isotopes of lithium was measured at room temperature. The modulus was determined from obser- vations of the resonant frequency of transverse vibration of extruded wires supported at one end. Lithium-6 (actually 96.1% Li-6, 3.9% Li-7) was found to have an elastic modulus of 7.93 i 0.21 x 1010 dynes/cme, and lithium-7 (actually natural lithium, 92.5% 1.1-7, 7.5% Li-6) an elastic modulus of 7.98 i 0.33 x 1010 dynes/cme. The values are identical within the limits of precision of the measurements, as would be eXpected on the basis of the usual theories. Results in the literature for the elastic modulus of lithium range from h.9 x 1010 to about 9.2 x 1010 dynes/cme. It is believed that the average of the values reported here, 7.96 x 1010 dynes/cmg, is within 20% of the absolute value for the modulus of polycrystalline lithium, and represents a more reliable estimate of this value than those reported previously. INTRODUCTION The elastic modulus of lithium is of interest because this metal is one of the few for which a theoretical calculation of this prOperty can be made. A primary object of the investigation was to develop a simple technique to measure the elastic modulus. The work grew out of an investigation exploiting isotopic mass as a probe for studies of the solid statel. In the course of the deveIOpment it was found that the values of the elastic modulus of natural lithium occurring in the literature had a wide variance. It seemed worthwhile to re-measure this quantity, as well asto determine it for the separated isotOpes of lithium. It appeared possible to develOp useful information at the same time that the feasibility of the method was being proved. The method deveIOped for use at room temperature would be particularly useful if it could be extended easily to higher or lower temperatures. Specifically, it could be used to investigate the martensitic transition2 known to occur in natural lithium at about 80°K. The mechanism of the transformation might be elucidated by finding the effect of isotopic mass on the temperature and perhaps the rate of the transition. The detection of the transition is not always easy experimentally, and it is believed that the discontinuity in elastic modulus likely to appear at the transition might constitute a practicable indicator. A simple technique for measuring the elastic modulus at room temperature is necessary before the property can be studied at other temperatures. 1 THEORY OF THE METHOD In measuring the elastic moduli of the materials under consider- ation, problems arise from the high degree of chemical reactivity of lithium and the small quantities of material available. Moreover, if the apparatus is to be suitable for investigations of temperature coefficient of the elastic modulus, differential measurements between isotopes should be made simultaneously on two specimens of different isotOpic composition contained in the same chamber. To get to liquid air temperatures, several experimental complications arise. Observation of the resonant frequencies of vibration of a rod was chosen as a promising method, and as one that would be simple to extend to low temperatures. The method used was the transverse vibration of a wire clamped at one end and free at the other. Thus, the wire behaved as a cantilever beam or bar. Consider a straight, homogeneous bar of uniform cross-section, symmetrical about a central plane. The differential equation for the motion of the bar perpendicular to this plane is A") :—.€_A 3‘3 (1) 326‘ E1 :9?"— displacement from equilibrium of the plane of symmetry, where y X ll distance from clamped end of bar, density (mass per unit volume) of the material, A = cross-sectional area of the bar, E = elastic modulus (Young's modulus) of the material, I = area moment of inertia about neutral plane, t = time. ‘The flexural rigidity E1 is constant along the bar. This equation is derived and solved for several boundary conditions in standard texts.3 For the solution, one writes y(x, t) as a product of a function of x alone, Y(x), and a periodic function of t, em 2171‘!) t, where i = H, and 0 is the frequency of vibration. Thus, 50"“ (2) On substituting equation (2) into (1), it is found that Y(x) must satisfy the equation BQY _ W " lbnq’“ 4Y’ ”he" ”u”: 2%);21: (3) The general solution of this equation is ch $317th 63mm, + £3 £3 ‘7'!” + C4, éauiflx 0 0 h __ A c.sh(a'“x)+85mlq(2flufl{Ccosmfl'flllt DSWI‘KA ) For a given problem, the boundary conditions determine the values of A, B, C, D, (or alternatively c 1’ c2, c3, ch) and/(that are allowable. For a bar clamped at one end and free at the other, both the deflection and the SIOpe must be zero at the clamped end, and both the bending moment and the shearing force must be zero at the free end. With a bar of length L having the clamped end at x = o, the boundary conditions are : "Y : _..—-— '-'—’ O Y(‘)‘=° 0 d ’0 x:L 3 (5) 3! 1:0 3X x=L On applying these boundary conditions, it is found that}... must satisfy the transcendental equation Cosh(1}T/4L) cas(afi,uL\ 4.] = 0 This equation must be solved numerically to get actual values for AIL. The lowest three roots are, (2lr/utlL) = 1.875, (2")12L) = “.6914, (2 Wfi3L) = 7.855. If these numbers are indicated by the notation Sn’ then [4n corresponding to a given 511 is An = .52— . <6) 217l— From equations (3) and (6), ”n4 = 7P 47"; _A_ r. S n4 (vial-El (2v)“L" For a wire of given dimensions, this equation gives the natural frequencies (7) of vibration, where am has now been substituted for ‘D . Equation (7) may be rearranged to give the elastic modulus E in terms of the dimensions of the bar and the frequency: 2. E =- 4 vr‘ p A '3’ 1’: sf,’ I For a bar of circular cross section, A = Trde/h and I = fl'dh/6h. Thus, . (8) E = 4,4 wsz‘Wfi. (9) 514'- Equation (9) applies to a bar of circular section; it also applies to an elliptical bar if the motion is perpendicular to one of the axes of symmetry of the ellipse. In this case d is the principal diameter in the direction of vibration. Numerical values were substituted into this equation for the calculation of E. For a bar subject to damping forces and excited by some external force of frequency‘a , two terms must be added to the differential equation. The first is proportional to a y/é t and takes damping into account; the second represents the driving force, assumed periodic in time, conveniently written c5 exp 211' i‘Dt, where c5 is some constant. If the damping forces are small the frequency of maximum amplitude of vibration, the resonant frequency, will be the natural frequency for free vibration of the bar, JD n' Thus, “on can be measured by varying 4D and noting its value when the amplitude is a maximum. In the above analysis the various damping effects - air damping, internal friction and energy loss to the support - were considered to be small. If damping is large, the resonant frequency will be lowered and the calculated value of E will be too low. Therefore, in applying equation (9) to the calculation of B, it is necessary to verify that the damping actually is small. The theory of the effect of air damping has been considered by Stokes“. Karrholm and Schroder5 applied this theory to the vibration of fibers supported at one end. They concluded that the theory applied to this situation and that air damping was negligible except for very 6 fine fibers. Stauff and Montgomery reached similar conclusions for a stretched wire. Application of Stokes' theory to these experiments shows that, because of the large wires used, air damping was completely negligible. The effect of internal damping in determination of E by transverse vibration is generally considered to be exceedingly small (Richards7, p. 86). Fine8 indicates that if damping is small, the curve of amplitude of vibration versus frequency will be very sharp, having its peak at the resonant frequency. In the experiments the actual band width due to damping was not determined, but it was observed that the resonant frequency appeared quite sharp. This would indicate that the effect of internal damping was indeed small and could safely be neglected. The third possible source of damping was loss of energy at the support, the steel extrusion die. Because the wire was extruded directly and left attached to the die, the wire was clamped quite tightly by the die. The support was very massive compared to the wire sample, so that no appreciable inertial energy was transferred to the support. SOURCE AND PURITY OF MATERIALS The lithium-6 used was purchased from Oak Ridge National Laboratory and was purified by redistillation. The material had the following analysis: 1.1-6 Lot No. 535(31 - redistilled 1015 g Isotopic analysis (mass number and atomic percent):6, 96.1 i .l; 7, 3.9 + .l. Spectrographic Analysis (element and weight percent, precision : 50%): Ag T Fe 0.5 Ni (0.01 T A1 <0.01 T K (0.02 T Pb (0.02 T Ba (0.02 T Mg 0.02 Si (0.05 Ca 0.05 Mn (0.01 Sn <0.01 Cr (0.01 T Mo (0.01 Sr (0.01 Cu 0.01 Na 0.03 v (0.02 The natural lithium.was produced by the Iithium.Corporation of America. It was their low sodium grade, in the form of 3/8 inch diameter rods with the following Specifications: Na 0.005% K 0.01 Ca 0.02 N 0.06 Fe 0.001 Preparation of Samples The samples were stored in.mineral oil but nevertheless acquired a coating, probably mostly nitride. This coating could be removed by scraping or cutting and after removal a reasonably clean surface could be retained by coating with oil. The cleaned sample was formed into a small slug that would fit into the extrusion die. 7 APPARATUS AND PROCEDURE The electrical apparatus was Similar to that used by Stauffg. It was essentially a string electrometer as described by Montgomery and Millowaylo. Because the wires used were stiffer than the stretched fibers used by Stauff, it was necessary to modify the apparatus somewhat to get adequate driving force. The electrical circuit diagram is shown schematically in Figure 1. Brass electrodes diametrically Opposed were connected through a step-up transformer to an amplifier fed by a variable-frequency audio oscillator. The oscillator was adjusted until resonance of the fiber was observed. The frequency was read directly from the oscillator dial. The oscillator was calibrated by observing Lissajous figures of the oscillator output against line frequency on a cathode-ray oscillo- scope. The calibration was checked periodically, and was found to be quite stable. The wires were extruded from a small steel die directly into a glass chamber. Because lithium reacts with oxygen and nitrogen in the air, rapidly forming a dense coating of oxide and nitride on the freshly extruded, bare wire, it was necessary to fill the glass chamber with a gas inert with respecttn lithium. Carbon dioxide was found to be quite satisfactory. The chamber was evacuated and purged several times with carbon dioxide; then the wire was extruded. The lithium wire had a tendency to curl as it was extruded. To 8 WUDOQFU m A“ .yl zuzauam Edy—04.0 ”II > can taoEO .._