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Ferromagnetism and order in nickel manganese alloys Piercy, George Robert 1952

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FERROMAGNETISM AND ORDER IN NICKEL MANGANESE ALLOYS by GEORGE ROBERT PIERCY. ' A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Physics We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF APPLIED SCIENCE.• Supervisor i n Metallurgy Department. Head of Department of Physics. THE UNIVERSITY OF BRITISH COLUMBIA OCTOBER 1952 SYNOPSIS An investigation of ferromagnetism in nickel manganese alloys up to forty atomic percent manganese has been carried out. Twenty alloys within this composition range were subjected to three heat treat-ments such that the conditions within the alloys varied from atomic disorder to a high degree of long range order. The degree of order of Ni3Mn calculated from measured saturation magnetisation using the Ising model of ferromagnetism was consistent with the value calculated from the ratio of measured integrated intensity of the (110) x-ray diffraction superlattice line to that of line (111). The relationship between saturation magnetisation and concentration for the disordered alloys can be explained adequately by the existence of short range order. A value of 3 . 4 Bohr magnetons was obtained for the effective magnetic moment of manganese atoms i n a nickel l a t t i c e . Acknowledgement The author i s grateful for financial aid i n the form of a National Research Council Bursary during the winter of 1951-52 and the summer of 1952. Without such aid this work could not have been carried out. The author i s also grateful to the staff of the Department of Mining and Metallurgy for their co-operation, encouragement, and advicej i n particular to R. Butters for the chemical analysis of the alloys and to Dr. E. R. Morgan, the supervisor of this research. TABLE OF CONTENTS page Introduction 1 Previous Work 7 Procedure . . . . . . . . 9 Results Preliminary experiments . . . . . 14 Quenched alloys . . . . . . . . . . . . . . . 14 Annealed alloys . . . . . . . . . . . . 14 Intensity measurements . . . . . . . . . . . . 15 Discussion Preliminary experiments 24 Quenched alloys 24 Annealed alloys . * 28 Conclusions . . . . . . . . . . . . . 31 Appendices 1. Vacuum melting and casting apparatus . . . . . . . . 33 2. Description and calibration of magnetic balance . . 39 3. Derivation of the general equations for the order parameter at any composition . . . 49 4. Equations for the quenched alloys . . . . . . . . . . 54 5. Calculation of order parameter of Ni3Mn - from magnetisation results . . 56 - from superlattice line intensity 58 Bibliography . . . . . . . . 62 ILLUSTRATIONS page 1. Magnetic moment per atom plotted against atomic number . . . 4 2. Saturation magnetisation plotted against percent manganese -results of Kaya and Kussmann . . • 6 3. Curie temperature of nickel manganese alloys - results of Kaya and Kussmann . . . . . . . 0 8 4. Lattice parameter of nickel manganese alloys - results of Valentiner and Becker . . . . . . 8 5. Change i n saturation magnetisation of an alloy with 23.7 percent manganese with increasing annealing time at 380, 400, 420 and 450°C . . . . . v. . . . • •. 1° 6. X-ray diffraction films showing that the diffuse lines were not produced by an oxide layer . . . . . . . . . . . . . 17 7. Change i n saturation magnetisation of an alloy with 23.7 percent manganese with increasing annealing time at 490°C 18 8. Results of saturation magnetisation of a l l three heat treatments . . . . . . 20 9. X-ray diffraction films of an alloy with 23.7 percent manganese after the f i n a l heat treatments 21 10. Results of saturation magnetisation for alloys near stoichometric composition Ni3Mn . . . . . . . . . . . . 22 11. X-ray diffraction films showing superlattice lines . . . . 23 12. Diagramatic view of vacuum pumping system 35 13. Diagramatic view of high frequency induction furnace and vacuum casting unit 36 page 14. Photograph of vacuum melting and casting unit, front view . . 37 15. Photograph of vacuum melting and casting unit, rear view . . 38 16. Front view of magnetic balance 40 17. Top view of magnetic balance, showing furnace • 41 18. Top view of magnetic balance showing dewar flask 42 19. Test of probe for electrostatic measurements 45 20. Position of potential readings . . . . . . . . . . . 45 21. Plot of electrostatic potential readings . . . . . . . . . . 46 22. Potentials along magnet pole axis . . . . . . . . . . . . . 47 23. Calibration of magnetic balance for sample weight . . . . . 48 FERROMAGNETISM AND ORDER IN NICKEL MANGANESE ALLOYS Introduction A ferromagnetic material is defined as a material that has a large force exerted on i t when placed i n a magnetic f i e l d gradient. The source of the ferromagnetism i s thought to be the spin of the electrons* Although the orbita l motion of the electrons may also have a magnetic moment, this i s thought to be small compared with the spin magnetic moment in a solid, as the gyromagnetic ratio i s nearly equal to that of an elect-ron spin. For a substance to be magnetic, i t i s necessary to have more of the electron spin moments oriented i n one direction than i n the other. If equal numbers of electrons have their spins aligned i n each direction, their magnetic moments w i l l cancel and the material as a whole w i l l not be magnetic. In ferromagnetic materials there i s a spontaneous 2 alignment of the electron spins i n the partially f i l l e d shells of each atom. Because one energy level can only contain one electron of each spin orientation, the alignment of electron spins w i l l raise some of the electrons into higher kinetic energy states. If a magnetic material does not have a high density of states i n the unfilled electron shell, then the aligning of the electrons w i l l require more energy than is available. In the Heisenberg theory^, the energy that aligns the electrons i s a result of the electrostatic exchange interaction between the electrons of the atoms. The outer, valence electrons w i l l not take part i n the magnet-ism because the density of states i s not sufficiently high. In the trans-i t i o n metals, however,' there i s an unfilled 3d shell i n the iron group and an unfilled 4f shell i n the rare earths. It is found that nickel, 2 cobalt, iron, and gadolinium are the only magnetic elements. Slater , using the Heisenberg theory, attributes the lack of magnetism i n other elements of the transition groups to the large radius of their 3d shell compared with their relatively small interatomic distance. Under this condition the 3d shells of adjacent atoms w i l l overlap too much causing a negative exchange interaction. The relative values for the saturation magnetisation of d i f f -erent elements and alloys are often compared with the number of electrons in the 3d shell. There are five quantized energy levels i n the 3d shell, each of which can hold two electrons of opposite spin. If this band i s divided into two half bands, each of which can hold five electrons of one spin direction, the maximum magnetisation w i l l occur when one of the bands i s completely f i l l e d and the other band has the remaining 3d electrons. Because the 3d and 4s bands overlap, i t is not necessary to have an integral number of electrons i n either band, but the t o t a l number of 3. electrons i n the 3d plus 4s bands must be the same as that of the free atoms. Because the experimental value for the saturation magnetisation of nickel i s 0.6 Bohr magnetons, there must be 0.6 holes i n the 3d shell of nickel. This allows 0.6 valence electrons i n the 4s shell. If we assume that the number of 4s electrons i s constant for atoms with atomic number near that of nickel, then the magnetisation w i l l decrease with atomic number as shown in Figure 1. The experimental curves are shown as a dotted line i n Figure 1. It i s seen that alloys of nickel and manganese follow the theoretical curve at high nickel concentrations but leave the curve at manganese concentrations greater than eight atomic percent, 3 The Ising approximation , i n which only the z component of the 3d spin i s used i n the Heisenberg exchange energy term, can be used to predict the magnetisation of alloys i f the magnetic moment of the individ-ual atoms i s known. When using this approximation, i t i s usually assumed that the exchange interaction which aligns the electron spins i s constant for small changes i n interatomic distance.' This approximation has been used by Neel^-, Vonsovsky-', Bitter^, Smoluchowsky'', and others, to explain ferromagnetism and antiferromagnetism in ferrites and alloys. The alloys of manganese and nickel are an interesting example of the Ising approximation. When the manganese atoms are added to the nickel l a t t i c e their magnetic moments w i l l l i e i n the same direction as the nickel atoms i f the exchange interaction between the manganese and nickel atoms has a positive sign. Because there are more holes i n the 3d she l l of manganese than of nickel, one would therefore expect a linear increase in saturation magnetisation with increasing manganese concentrat-ion. This was found to be true up to six atomic percent manganese by 8 Kaya and Kussmann • Their results for the values of saturation magnetis-ation extrapolated to zero degrees Kelvin, are shown i n Figure 2. The 4 CO z o Cr Mn Fe Co Ni Cu 2 4 25 2 6 27 2 8 2 9 E L E C T R O N S PER ATOM Figure 1 . A. plot of magnetic moment per atom against atomic number. The s o l i d l i n e i s the t h e o r e t i c a l value; • 1 8 the dotted l i n e s show the experimental measurements. 5. bending over of the curve at eight atomic percent manganese i s attributed 9 by Carr to the negative exchange interaction of the nearest neighbour manganese atoms causing adjacent manganese atoms to have opposite spin. The perfectly ordered face centered cubic structure would, however, have no nearest neighbour manganese atoms below twenty-five atomic percent mang-anese. One would therefore expect the completely ordered alloy to have a value of saturation magnetisation on the line tangent to the curve at zero percent manganese. The tangent i s shown dotted i n Figure 2. The ordered alloy of Kaya and Kussmann at twenty-five atomic percent manganese has a value of saturation magnetisation far below the expected value on the tangent. A preliminary heat treatment on an alloy of this composition made during the present work, showed that the saturation magnetisation of Ni3Mn was much higher than the value found by Kaya and Kussmann. An investigation of the literature on nickel manganese alloys confirmed this higher value. However, no other measurements of saturation magnetisation have been,made on alloys other than the stoichiometric composition. The other investigators used different heat treatments and thus obtained d i f f -erent values for the saturation magnetisation of Ni3Mn. If the alloy i s ordered, superlattice lines should occur on the x-ray diffraction pattern. Because the intensity of the superlattice lines depends on the difference between the atomic scattering factors of nickel and manganese, which i s small, the superlattice lines were not seen u n t i l monochromatic radiation 10 and an evacuated camera were used recently by Averbach • Superlattice lines had previously been seen by neutron diffraction"^". Because i t was apparent that the heat treatment for much of the work on these alloys has been inadequate to give maximum ordering, i t was decided to make a thorough investigation of the magnetic 6. 1 0 0 0 5 1 0 15 2 0 A T O M I C P E R C E N T M A N G A N E S E 2 5 Figure 2 . Values of saturation magnetisation extrapolated to 0°K plotted against atomic percent manganese. ., - re s u l t s of Kaya and Kussmann. properties of nickel manganese alloys. It was proposed to make alloys i n the range zero to thirty-eight atomic percent manganese, to find the heat treatment that w i l l give maximum magnetisation, to measure the saturation magnetisation, and to find the necessary conditions to explain the change i n saturation magnetisation with composition using the Ising approximation. It was also proposed to find the order parameter for alloys near the stoichometric composition by measuring the intensity of the x-ray super-Q l a t t i c e lines. Unfortunately, a paper by Carr , which explained the change of saturation magnetisation with composition for the quenched alloys, was published when the present work was half finished. Previous Work The change i n saturation magnetisation of nickel manganese alloys has been measured from zero to forty weight percent manganese by 8 Kaya and Kussmann • Their results for the values of saturation magnetis-ation extrapolated to zero degrees Kelvin are shown i n Figure 2. They also measured the Curie temperature at a l l compositions. These results are shown i n Figure 3. After a heat treatment at 430° C for three days, their value of magnetic saturation of composition Ni3Mn was 4TTI = 7800 Gauss, compared with a value of 6400 Gauss for nickel under the same conditions. The alloy Ni3Mn has been investigated by several people. 12 Thompson obtained parallel effects for the change of magnetisation and r e s i s t i v i t y on both annealed and quenched samples. He measured the magnetic Curie temperature at 46O0 C, the c r i t i c a l ordering temperature at 520° C, and the naximum saturation magnetisation at 6~- 91.5. This i s equivalent to 4TTI = 9500 Gauss i f the density of Ni3Mn i s taken to be 8.25grams per 8 i i — — - r 10 20 3 0 40 ATOMIC PERCENT MANGANESE Figure 3. Change of Curie temperature with increasing manganese concentration i n n i c k e l manganese a l l o y s . 8 - r e s u l t s of Kaya and Kussmann. 10 2 0 3 0 40 ATOMIC PERCENT MANGANESE Figure 4. L a t t i c e parameter of n i c k e l manganese a l l o y s '  » • - r e s u l t s of Valentiner and Becker. 1' 5' cubic centimeter. The saturation magnetisation of Ni3Mn was measured by 13 , Guillard after a heat treatment of three weeks at 470° C. The value of the extrapolated curve to 0°K was <5~ = 98.16 Gauss ( 4 F I = 10200). The 14 saturation magnetisation of Ni3Mn, measured by Komar and Volkenstein , was 9300 Gauss. The change i n l a t t i c e parameter of quenched alloys with 15 composition, shown in Figure 4, was measured by Valentiner and Becker. 10 The lat t i c e parameter of ordered Ni3Mn was 3.59 A° Procedure The alloys were made from the following metals: nickel powder from Fisher Scientific Company with 0.1$ Fe and less than 0,1% Co as the main impurities; electrolytic manganese donated by the Electromanganese Corporation with 0.018% S as sulphide and 0.01555 H 2 as the main impurities. The alloys were melted under an argon atmosphere because a magnetic manganese nitride forms when manganese alloys are melted under a i r . The alloys could not be melted under vacuum because manganese i s too vola t i l e . To obtain the proper argon atmosphere the vacuum melting and casting unit described i n Appendix 1 was designed and bui l t . The oxide surface on the nickel powder was reduced by heating the nickel and manganese i n the induction furnace at 600° C. for ten minutes under a hydrogen atmosphere. After the oxide had been reduced, the furnace was evacuated to 0.5 microns and argon, which had been purified by passing over magnesium chips and copper turnings at 400° C, was let i n to a pressure of one atmosphere. The metal was then melted and the alloy was cast under the argon atmosphere into the cold brass mold. This c h i l l casting produces a high concentration gradient for short distances i n the 10. cored structure, thus decreasing the homogenizing time required to remove detectable heterogeneity. To test the ingot for segregation, samples taken from the top, center, and bottom of a cast ingot were analyzed; typical results were: 22.9, 22.6, and 22.7 weight percent manganese. The ingots were homogenized for five days at 1000° C under a purified argon x atmosphere. The saturation magnetisation of the alloys was measured 16 on a Fereday magnetic balance. This instrument is described i n Appendix 2. The samples for the magnetic balance were cut and f i l e d to shape from the homogenized ingot. The adjacent piece of the ingot was used for the analysis sample. The magnetic samples were annealed at 420° C for two y weeks, then at 400° C for one week. The correct annealing treatment was found by annealing alloys of 15.35, 19.92, 23.70, and 32.07 atomic percent manganese at temperatures of 370, 400, 420 and 450° C, and x. A standard quartz tube furnace with rubber bungs at the end of the tube and a heating element sixteen inches long was used. A rubber gas analysis bag connected to the furnace, kept the pressure slightly above one atmosphere, y. Some previous investigators have used a slow cooling heat treatment on their alloys. If this i s done they must assume the alloy i s completely ordered after the heat treatment. By using a constant temperature heat treatment, the degree of order for the alloy can be found from the theoretical curve of order against temperature. 11. measuring the room temperature saturation magnetisation of each sample after different annealing times. A l l magnetic balance samples were heat treated i n evacuated glass tubes with a piece of pure manganese in the tube to absorb any remaining oxygen or nitrogen. The glass tube was heated before i t was sealed to remove any occluded gases. Other samples were annealed at 800° C for two hours i n evacuated vycor tubes and were quenched in water. When Debye-Scherrer powder patterns were taken on the annealed alloys, a l l alloys above eighteen atomic percent manganese had diffuse back reflection lines. This effect i s due to a change in lattice parameter i n the sample, which can be produced by any of the following: 1. an oxide layer on the surface of the powder. 2. small order domains resulting i n many Mn-Mn interactions betvreen manganese atoms with the same spin direction at the domain inter-faces. 3. internal stresses produced during f i l i n g . To check for the possibility of an oxide layer, a powder sample of 23.7 atomic percent manganese was heated at 420° C for sixteen hours. An x-ray pattern was taken showing diffuse lines. The powder was reannealed at 800° C for two hours and quenched. The pattern of the quenched sample had sharp lines. This showed that no oxide layer was present. To check for the possibility of internal stresses s t i l l being present, a powder sample of 23.7 atomic percent manganese was annealed at 550° C for sixteen hours and then at 420° G for fourteen hours. The resulting film had sharp back reflection lines. This confirms the presence of residual strain from the f i l i n g , in the alloys annealed at 420° C for two weeks plus 400° C for one week. Apparently the recrystallisation temperature of nickel 12 manganese alloys increases with manganese content because the alloys with less than eighteen percent manganese had sharp back reflection lines with the same heat treatment i n which alloys with higher manganese concentrat-ion had diffuse lines. This, however, does not, eliminate the possibility of small order domains causing the diffuse lines, as the alloy with heat treatment at 550° C for sixteen hours plus fourteen hours at 420° C may not be sufficiently ordered to produce this effect. 17 It has been shown by Sykes and Jones that the order domain size i s increased by annealing just below the c r i t i c a l order temperature,, A preliminary heat treatment was given samples of 23.7 atomic percent manganese to determine the length of annealing time at 490° C that i s nec-essary to produce large order domains. Samples of the alloy were annealed at 490° C for different times, and were then annealed at 420° C for two weeks. It was found that the saturation magnetisation of the alloy incr-eased with the time of annealing at 490° C up to a period of 200 hours* For the f i n a l heat treatment, alloys of composition 19.9, 21.8, 23.7, 24.1, 25.0, 25.7, 27.9, and 30.3 atomic percent manganese were annealed at 555° C for fourteen hours, plus 250 hours at 490° C, plus 260 hours at 420° C, plus 260 hours at 400° C. Powder patterns were taken of these alloys and their saturation magnetisation was measured. .The intensity of the (110) superlattice line on the alloy of twenty-five atomic percent manganese was measured on the Philips gonio-meter by taking the t o t a l count for an hour with the geiger counter oscillating across the lin e . The background was taken under the same conditions on both sides of the l i n e . The intensity of the (111) line was then measured i n a similar manner. 13. Results In the f i r s t set of preliminary samples for the magnetic balance, a heavy oxide formed when the samples were annealed i n •evacuated* pyrex tubes that were incorrectly sealed. The curve of magnetisation against annealing time for these samples followed the curve later found for magnetisation against decreasing manganese content. The oxide,was obviously reducing the manganese concentration of the samples. To prevent this oxide from forming, pieces of pure manganese were put i n the evacuated tubes with the magnetic balance samples. After being sealed, the pure manganese was s l i d to one end of the tube and the tube was heated at the manganese end with a gas flame. It was hoped that the hot manganese would absorb any of the remaining oxygen and nitrogen. It was found after several t r i a l s , that the local heating of the manganese introduced more gas into the tube from the softened glass than was absorbed by the mang-anese. The local heat treatment was therefore stopped' and a l l samples on the f i n a l runs were annealed i n evacuated tubes with a piece of manganese and were not contaminated. Several other investigators have noticed a decrease i n manganese content on annealing.for a few hours and attributed i t to the sublimation of the manganese atoms. This effect i s reduced by putting i n the pure manganese. To check for any gain or loss i n manganese concentration during the heat treatment, the sample for the magnetic balance with 23.7 atomic percent manganese was analyzed at 24.0 atomic percent manganese after annealing for two weeks at 420° G and one week at 400° C. The following results were obtained from the preliminary runs: 1. The saturation magnetisation of the alloy with 15.4 atomic percent manganese was unchanged by annealing at 380, 400, 420, or 450° C. 2. The curves of saturation magnetisation against annealing time at 380, 400, 420, and 450° C for an alloy with 23.7 atomic percent manganese are shown in Figure 5« The curves for the other alloys are similar. 3. In the course of the magnetic measurements for the preliminary runs, the magnetisation of the standard nickel sample was measured twenty times on different days. It gave a room temperature value of 6000 with a standard deviation of 60. 4. The x-ray diffraction pattern of the 23.7 atomic percent manganese alloy annealed at 420° C for sixteen hours is shown in Figure 6, film A. The pattern for this same alloy after an additional anneal at 800° C with a water quench, is shown in Figure 6, Film B. 5. The saturation magnetisation of samples with 23.7 atomic percent manganese annealed for varying times at 490° C plus an additional two weeks at 420° C is shown in Figure 7. For the results of the final runs, the following notation is used for the heat treatments: Heat treatment A - Two hours at 800° C followed by a water quench. Heat treatment B - Two weeks at 420° C plus one week at 400°C. Heat treatment C - Sixteen hours at 555°C, plus 250 hours at 490°C, plus 260 hours at 420°C, plus 260 hours at 400°C. The following results were obtained from the final runs: 1. The values of the saturation magnetisation for heat treatment A measured at room, dry ice, and liquid oxygen temperatures are shown in Table 1. The value of saturation magnetisation after extrapolation of the 2 magnetisation against T curve to zero degrees Kelvin is also given for 15. the samples in which the curve was linear. 2. The values of saturation magnetisation for heat treatment B are shown in Table 1. The extrapolated values of saturation magnetisation for heat treatments A, B and C are shown in Figure 8. 3. Figure 9, films A, B, and C show the x-ray diffraction patterns obtained for alloys with 23.7 atomic percent manganese after heat treat-ment A, B, and C respectively. 4. The values of saturation.magnetisation for the samples after heat treatment C are shown in Table 1. The results of the extrapolated curve to zero degrees Kelvin are shown in Figure 10. The corresponding curve for heat treatment B is drawn on the same graph. 5. The x-ray diffraction patterns of alloys with 23.7, 25.0, and 25.7 atomic percent manganese after heat treatment C are shown in Figure 11, films A, B, and C. The prints were over-exposed to bring out the order lines. This is the reason why the. doublets are not resolved, 6. The results for the line intensity measurements are given below: 1 Line oscillating angle min. i max. 1 Time Counts i Total os dil a t i n g | angle(20) i 1 background ' 1 (no ) 42.96 44.02 1 1 hour 1 72450 \ i (no ) 44.28 45.49 1 1 hour 1 74314 , 1.21° | 1 background 1 i (no ) ; 45.61 j 46.58 1 hour 1 72889 I 1 background 1 i ( i n ) ; 53.95 [ 54.92 1 20 min.1 26584 ! i (L I D ; 54.83 | 56.36 1 20 min. 85387 ; 1.53° 1 1 background 1 1 (111) ; 56.71 J 57.70 ' 20 min.1 28079 [ 16. 9000 -o rO CD CVI cn CO < <3" 8000 7000 6000 < 5000 4000 3000 2 000 -1000 4 2 0 ° G 380°C 0 100 200 300 400 ANNEALING TIME IN HOURS Figure 5. Change i n saturation magnetisation f o r an a l l o y of 23.7% Mn with increasing annealing time. A l l measurements were done at room temperature. Figure 6, Film A - 23.1$ Mn after 16 hours at 420°C. - unfiltered Cu radiation Film B - same sample as f i l m A with an additional 2 hours at 800°C followed by a water quench. - unfiltered Cu radiation. « 8 . o ro cn CM CO CO < 9 3 0 0 9 2 0 0 -9 1 0 0 9 0 0 0 8 9 0 0 -8 8 0 0 -8 7 0 0 -8 6 0 0 8 5 0 0 1 0 0 A N N E A L I N G T I M E A T 4 9 0 ° C I N H O U R S 2 0 0 Figure 7. Room temperature values of saturation magnetisation f o r samples of 23.7/5 Mn annealed f o r varying times at 490°C with an a d d i t i o n a l 400 hours at 420°C. 19. Saturation Magnetisation 4TTI i n Gauss Heat Treatment A Heat Treatment B '293°K 200°K • : 900 K : O°K £93° K :200°K : 90 0 K : O°K £93 °K :200°K : 90° K : O°K : : 0 : 6203 : 6400 : 6400 : 6400 : 3 - 9 : 6880 : 7250: (7690 * . 7810 . 6940 . 7340' ; 7530 : 7600 .' 4.8 : '6940 : 7440. : 7730 . 7810 : 6960' 7510; : 7750 : 7800 : 6.4 . 7120 : 7700 : sioo: 8200 : 7190( 7810' : 8240 ' .8330 j 7.4 ' 7060 ' '7740 ( : 8130 . : 8210' : 7320" 7980: : 8380 : 8480. : i i . i : 6680 : 7610: 8410: ( 8550: : 6700: 7730: 8290 : 8470. : 12.7: 5860 : 7070: 7800: 8000: 6050: 7180: 7990 : 8170" : 15.4. 4740 . 6210: . 7020: 7220: 5470: 6830: . 7660 : 7850: : 16.0: 4270 : 5900' . 6647' . 6890' 4720: 6230' 7090 : 7340: : is.3: 1984; 4460. ' . 5 5 2 0 : ' 5820: 4250: 5840: 6702 ' 6870: : 1 9 , 9 : 740 : 3670: 4850: 5140: 3860: 5450: 6210 ' : 6350: 3750: 5360: ' 6186 ' 6540 : : 21.8. 100 ' 1030 2780. 8540. 9050. 9270 : 9320: 7630: 8300: ' 8540 : 8590 : : 2 3 * 7 : 100 . 370. . 1 4 5 0 : 9440: . 9500: 9450' : 9480] . 9890] 9880: 9850 : 9880 : 60 . 180 ' [ 890' 9370: 9340; 9330; : 9 3 5 0 : 10730: 11010: 10910 i :nooo : \ 25.0 50 : 110 : 400: ( 7830: 8240: 8350; [ 8380: 10130: 10260: 10340: 10350 : *: 25.7 40 ' 100 | 300: 9510: 9750: 9910: 10020 l : 28.0 50 ; 70 130: . 7360: 7710: 7800: . 7820: 7070: 7950: 7840: 7900 : ; 30.3 32 50 64: • 7 1 i ' 6850: 7160: 7270] 7310: 6710: 6890: 7010: 7070 : : 32.1 . 32 : 46 : 28. 0 . 590: 640: 620; 620 j *: 37.0 : 1 4 : 0 :  k\ 0 \ 340: 320] 330: 330: Heat Treatment C Table 1. The above values were obtained relative to the standard nickel sample, with a correction for the change i n weight of the average atom and the change in lattice parameter as shown in Figure 4, HEAT TREATMENT C 5 10 15 20 25 30 35 4 0 ATOMIC PERCENT MANGANESE Figure 8. Values of saturation magnetisation extrapolated to zero degrees Kelvin f o r heat treatments A, B, and C. 21. Figure 9. X-ray diffraction patterns for an alloy with 23.7 atomic percent manganese after heat treatment A, B, and C are shown i n films A, B, and C respectively. Unfiltered iron radiation was used. 22-. ' 1 1 1 I I L L_ 1 I I 20 25 30 A T O M I C - P E R C E N T M A N G A N E S E Figure 10. Values of saturation magnetisation extrapolated to zero degrees Kelvin f o r heat treatments B and C. Figure l l . X-ray diffraction patterns for alloys with 23.7, 25.0, and 25.7 atomic percent manganese after heat treatment C are shown in films A , B, and C respectively. Unfiltered iron radiation was used. 24. Discussion 1. Preliminary Runs. From Figure 5 showing the change of magnetisation with annealing time at 420°C, i t i s evident that saturation magnetisation incr-eases progressively with annealing time for periods up to two weeks. Most other investigators have only used times i n the order of f i f t y hours. The value of the saturation magnetisation after annealing at 420°C i s influenced by the length of the previous anneal at 490°C. This change, shown i n Figure 7, is due to the size of the order domains increasing at the high temperat-ure anneal. 2. Final Runs. a. Disordered case after heat treatment A. The t a i l above twenty-five atomic percent manganese in the curve of saturation magnetisation against concentration for the quenched alloys i s probably due to the- slow quenching rate, as the vycor tubes were not broken when they were quenched into the water. The curve of Kaya and Kussmann, shown in Figure 2, has no t a i l . 2 It was noticed that the curve of magnetisation against T was not linear for the quenched samples between twenty and thirty atomic percent manganese. These alloys had much too low a value of saturation magnetisation at 200°K to f i t the T 2 curve. This effect i s also present m the work of Kaya and Kussmann0. It i s probably due to the magnetic f i e l d strength not being sufficient to saturate the samples. If the samples had a coercive force that was small at 90°K and 293°K, and was large at 200°K, then i t would be possible for the magnetic f i e l d to saturate the samples 25. t at 90.K and 293°K but not be strong enough to saturate them at 200°K. The only alloys that have non linear curves are the quenched alloys near the order composition. If l o c a l order exists, then these alloys w i l l have an order domain size that i s larger than the domain size of the other quenched alloys and smaller than the domain size of the annealed alloys near the order composition. It i s a known fact that the magnetic coercive force reaches a maximum at one magnetic domain size and is smaller for either larger or smaller domains. Because the maximum size of the magnetic domains i s less than the size of the order domains, the non linear curves for the quenched alloys can be explained by assuming that the order domains of these alloys are the proper size to produce the c r i t i c a l magnetic domain size for maximum coercive force. Carr , in a recent paper, qualitatively explains the behav-iour of the saturation magnetisation of the quenched alloys i n terms of the number of Mn-Mn interactions. He proposes the following theory: •For a disordered alloy with an appreciable percentage of manganese atoms, the Mn-Mn interaction begins to outweigh the weaker Ni-Mn interaction, and one eventually finds the manganese spins start to cancel one another. This behaviour becomes clearer i n the energy calculations which follow and which indicate that the bending over of the curve as observed i n Figure 2, should occur for q relatively small fraction of manganese. Beyond this point, the magnetic moment is derived principally from the nickel ions and diminishes with increasing manganese content because of the replacement of nickel and to the f i l l i n g up of the nickel 3d shell. Assuming the manganese ions to have five electrons i n the 3d shell, one finds that the nickel shells become completely f i l l e d at about 30 percent manganese. The 26. moment thus goes to zero at this composition i n approximate agreement with experiment.' If we assume that for low manganese concentrations there are no Mn-Mn interactions, then from the i n i t i a l slope of the curve i n Figure 8, we find that the addition of manganese atoms to a nickel l a t t i c e produces a rise i n magnetisation similar to that produced by an atom with a magnetic moment i n the order of 3.4 Bohr magnetons. Other values for the magnetic 18 moment of manganese are: MnAs - 3.40 Mn2Sb - 0.94 MnBi - 3.52 MnSb - 3.53 Mhz^ N - 0.24 Mn2Sn - 0.86 . MnP - 1.2 The value obtained i n this alloy i s 5.7 times the magnetic moment of a 19 nickel atom. If we use the Zener assumption that the manganese atoms w i l l retain five of their 3d electrons and the rest w i l l be given to the nickel, then each manganese atom w i l l give 10-5-3.4 - 1.6 electrons to the nickel atoms. This w i l l make the magnetic moment of the nickel atoms equal zero at twenty-seven atomic percent manganese. 9 Carr assumes that the magnetisation near twenty-five atomic percent manganese is due to the nickel atoms alone. If this is true, then the curve near twenty-five atomic percent manganese w i l l have a slope 640O . equal to 25" =~250.6 using the units on Figure 8. The actual slope i s -S6P4 . . This discrepancy can be eliminated by having the manganese atoms contribute to the magnetism up to twenty-five atomic percent manganese and above this concentration having a l l the manganese atoms paired off. It can be accomplished i n either of two ways: 1. by assuming that the strength of the Mn-Mn interaction i s such that 27. i t f i n a l l y overpowers the Mn-Ni interaction at twenty-five atomic percent manganese. 2. by assuming that local order exists that tends to keep the mang-anese atoms from becoming nearest neighbours i n the quenched alloys. The degree of order i s such that below twenty-five atomic percent manganese there are insufficient Mn-Mn interactions to make the contribution of the manganese atoms to the magnetic moment of the alloy equal to zero. Of these two assumptions, the latter is preferable for the following reasons: a. It i s shown i n Appendix four, that when the f i r s t assumption i s used, the value obtained for the f i n a l slope of the curve near twenty-five atomic percent manganese i s only half the experimental value. b. The Mn-Mn interaction should outweigh the Ni-Mn interaction at a manganese concentration lower than twenty-five atomic percent because the strength of the Mn-Ni interaction becomes weaker as the 3d shells of the nickel atoms become f i l l e d . c. In a previous section of the discussion, i t was shown that the non-linearity of the saturation magnetisation against T^ curve for some of the quenched alloys can be explained by assuming short range order. 20 d. Some degree of short range order i s present i n a l l alloys. Examples of this are the local order i n AgAu for which there i s no long range order, and the local order i n quenched samples of Cu3Au. e. The saturation magnetisation i s the same for quenched and annealed alloys up to nine atomic percent manganese. The Curie temperature i s the same for quenched and annealed alloys up to seventeen atomic percent manganese. Because the annealed alloys w i l l be ordered, then so w i l l the quenched alloys i f their magnetisation and Curie temperatures are the same. 28. 2b. Ordered Alloys. The maximum i n saturation magnetisation for the ordered alloys occurs at a concentration less than twenty-five atomic percent manganese because the alloy i s not completely ordered. An incompletely ordered alloy at twenty-five atomic percent must have some of i t s mangan-ese atoms with.left spin, but an incompletely ordered alloy at less than twenty-five atomic percent manganese may accomplish i t s disorder by having nickel atoms i n manganese sites and therefore does not need to have any manganese atoms with l e f t spin. It is seen from figures 8 and 10, that heat treatment C produces a maximum in the graph of saturation magnetisation versus composition at 24.1 atomic percent manganese and heat treatment B, which produces less long range order than heat treatment C, results i n a niaximum i n saturation magnetisation at 23.7 atomic percent manganese. The drop i n magnetisation of Ni3Mn from the ordered value of 13"1?00 to the value of 8380 after heat treatment B, can be explained by assuming that the magnetic moment of a manganese atom with reversed spin, w i l l cancel the magnetic moment of one manganese atom with right spin. To account for the magnetic moment found experimentally i n heat treatment B using the above assumption, we must have 0.187of the manganese atoms with l e f t spin. Therefore these atoms must occupy face centered positions. Using this value we get S = 0.8J from the Bragg and Williams theory and <T = 0.63 from equations 14 of Appendix 3, using the local order theory, i f we assume that each manganese atom with l e f t spin prod-uces three Mn-Mn interactions. The calculations are shown i n Appendix 5. The short range order parameter of heat treatment B gives a value of S = O.f? for the long range order parameter. Similar values for heat 29. treatment C were:S = 0.8$ for the Bragg and Williams, and 6" = 0.77 for the short range order theory which makes S = 0.88 for this theory. References 21 and 22 give the following equilibrium values of S at A00°C with a c r i t i c a l order temperature of 520°C: Bragg and Williams theory S = 0.79 Bethe and Peierls theory S=.0.95 Cu3Au experimental S=0.98 The low value of S obtained from the results of heat treatment B may be partially due to the small order domain size. Neither of the theories includes a factor for the domain size. Another possible reason for the low value of S, i s the assumption that a manganese atom in a face centered position reverses i t s spin and does not influence the spin of i t s nearest neighbour manganese atoms. If the reversed spin of one manganese atom reduced the effective spin of i t s three adjacent mang-anese atoms, then a small amount of disorder would produce a larger decr-ease i n the magnetisation than that calculated. This would raise the value of the order parameter calculated from the experimental results of saturation magnetisation. The values of saturation magnetisation for the annealed alloys above thirt y and below twenty-one atomic percent manganese are not as high as those expected, because long range order i s not present and thus there are more Mn-Mn interactions than i f i t were present. The Curie temperature of a magnetic material i s directly proportional to the strengths of the interactions between the magnetic moments of the atoms of the material. From the decrease i n Curie temper-ature with the addition of manganese atoms in the nickel l a t t i c e , shown in Figure 3, i t is concluded that the Mn-Ni interaction i s much smaller 30. than the Ni'-Ni interaction at this composition. The high Curie temper-atures of the ordered alloy i s probably due to the second nearest neighbour interaction between-t<he manganese atoms on the cube corners. This would account for the rapid rise i n Curie temperature hear twenty ' atomic percent manganese when long distance order i s increasing rapidly. It also accounts for the maintaining of the high Curie temperature for compositions greater than twenty-five atomic percent manganese because there are s t i l l the same number of second neighbour Mn-Mn interactions that are present i n the alloy with twenty-five atomic percent manganese. The value of the order parameter of the alloy with twenty-five atomic percent manganese after heat treatment C was calculated from the measured intensity of the (110) planes compared with the (111) planes. The calculations are shown in Appendix 5. The value obtained was S =0.84 * 0.07. The order parameter for the same alloy calculated from the saturation magnetisation was S = 0.88. This agreement i s f a i r l y good and j u s t i f i e s the assumption that the Mn atoms i n the face centered position invert their spins and thus cancel the magnetic moment of one of the manganese atoms at the cube corner. Probably the weakest point i n the explanation of the change in saturation magnetisation with concentration is the assumption that the 3d shells of the nickel atoms are being f i l l e d with increasing manganese 19 content. This principle i s i n accord with Zener's theory but does not necessarily occur i n a l l alloys, as evidenced by neutron diffraction experiments on FeCo. If we do not assume that the 3d shells of the nickel atoms are f i l l e d , then there i s no explanation for the decrease i n saturation magnetisation to zero at twenty-five atomic percent manganese for the quenched alloys. Unfortunately, there is no way to test this 31. assumption for the quenched alloys. In the case of the ordered alloys i t could be checked by an evaluation of the magnetic moments of the manganese and nickel atoms by neutron diffraction on a completely ordered alloy. Conclusions The annealing times used by most other investigators to produce complete order i n alloys near the composition Ni3Mn have been inadequate. To produce the equilibrium degree of order at 420°C, at least two weeks annealing time i s necessary. The maximum value for the saturation magnetisation of any alloy of nickel and manganese less than forty percent manganese, after heat treatment C, is 4 THE = 1Q5J80 Gauss. This i s the highest recorded value for any nickel manganese alloy after any heat treatment. The order domain size i n Ni3Mn increases with annealing time at 490°C up to periods of 2 0 0 hours. When this annealing treatment i s followed by an additional 2 5 0 hours at 420°C, and 2 5 0 hours at 400°C, the order lines can be seen on an x-ray powder pattern using iron radiat-ion. For these long heat treatments, i t i s necessary to have a piece of pure manganese with the sample to prevent the sublimation of the manganese atoms from the sample. The order parameter of Ni3Mn after the heat treatment given i n the paragraph above i s S = 0 . 8 8 , calculated;from the saturation magnetisation, and S = 0 . 8 4 * 0 . 0 7 , calculated from the order line intens-i t y . The addition of manganese atoms to a nickel lattice prod-uces a rise i n saturation magnetisation that i s similar to that produced 32. by an atom with a magnetic moment of 3#4 Bohr magnetons. To explain the change i n saturation magnetisation of alloys quenched from 800°C, i t i s necessary to assume that some of the 3d electrons from the manganese atoms go into the 3d shell of the nickel atoms. Once this assumption has been made, the curve of saturation magnet-isation against composition for the quenched alloys can be explained most satisfactorily by the existence of short range order. 33. APPENDIX 1 Vacuum Melting and Casting Apparatus To prevent the contamination of the metals by nitrogen and oxygen;, a high frequency vacuum melting apparatus was designed and b u i l t . The apparatus consisted of two parts; a general vacuum pumping unit shown in Figure 12, and a high frequency induction furnace shown i n Figure 13. Photographs of the complete unit are shown in Figures 14 and 15. The brass pumping unit consisted of a Welsh duoseal fore-pump with a pumping speed of 0.37 l i t e r s per second at 0.1 millimeters pressure, a VMF 50 o i l diffusion pump, a cold trap, a Kinney vacuum valve, a pirani gauge, and an ionization gauge. A three inch brass tee was connected to the end of the unit. A l l parts of the unit were connected with eight-bolt companion flanges using greased rubber gaskets. The gasket groove had a beveled edge to prevent the gasket from being pinched when the bolts were tightened. For argon melting, a bourdon gauge was connected to the pumping unit. This was necessary to measure the pressure when the argon was let into the system. The high frequency induction furnace was attached to the brass tee of the pumping unit with a companion flange. The quartz tube (fourteen inches long and one and one half inches i n diameter) was connected to the brass by using a mixture of graphite, o c t o i l , and digested rubber i n vacuum wax. This mixture had physical properties similar to plasticene except that i t s vapour pressure was lower. The seal at the top end of the furnace was water cooled. The power was supplied by a Lepel 34. 7.5 KVA high frequency unit. The x^ater cooled high frequency c o i l had sixteen turns i n four inches length. The crucibles were Norton extraction thimbles made of alundum, with powdered alumdum packed between them. The inner crucible, which could be used for several melts, held f i f t y grams of powdered nickel. The loaded crucible was inserted into the top of the quartz tube when the mold was removed. The crucible was then lowered into the middle of the high frequency c o i l by lowering the rod through the vac-uum compression seal at the bottom of the furnace. The mold was replaced at the top of the furnace and the forepump started. It took approximately seven minutes pumping time to reduce the pressure to 0.1 millimeters, which is the forepressure of the o i l diffusion pump. The to t a l pumping time to reach 0.5 microns pressure was from thir t y to forty minutes. After the metal was melted i n the crucible, the rod at the bottom of the furnace was pushed up, thus l i f t i n g the crucible containing the molten metal to the height of the brass mold. The whole furnace was then rotated through one hundred degrees about the 0-ring seal and the metal flowed into- the cold mold. 35. Figure 12. Diagramatic view of the vacuum pumping system. 36. f i g u r e 13. Diagramatic view of the high frequency-induction furnace and vacuum casting u n i t . 37. Figure 14. Photograph of vacuum melting and casting unit - front view. 38. Figure 15 Photograph of vacuum melting and casting unit - rear view. 39. APPENDIX 2 Magnetic Balance 1. Description. The saturation magnetisation of the alloys was measured on a 16 magnetic balance similar to that of Fereday . 'The electromagnet had 7000 turns and produced a f i e l d of 3000 Oersteds. The special magnet poles designed by Fereday produced a magnetic f i e l d such that the force on a sample was constant along the pole axis near the center of the gap. The samples were — x — x i inches. The sample holder was attached to the end * 4 20 20 of the suspension arm shown i n Figure 16, When the current was on i n the electromagnet, the sample was displaced toward one of the poles. It was then restored to i t s i n i t i a l position by twisting the suspension. The angle through which the suspension was twisted was controlled and measured by a worm gear at the top of the suspension. The i n i t i a l position of the arm could be found by shining a light to the mirror on the suspension and back to a scale. The angle through which the suspension was turned was directly proportional to the force on the sample caused by the magnetic f i e l d gradient. The saturation magnetisation for a sample was found by comparing the force on i t to the force on a standard nickel sample. To measure the saturation magnetisation at high and low temperatures, the furnace shown i n Figure 17 and the dewar flask shown in Figure 18 were used. Figure 16. Front view of magnetic balance showing magnet, suspension and sample holder. Figure 17. Top view of magnetic balance showing worm gear control and furnace arrangement• 42. Figure 18. Top view of magnetic balance showing dewar flask arrangement. 43. 2. Calibration. a. Plotting of magnetic f i e l d i n the air gap. The relative change of the magnetic f i e l d i n the air gap was investigated using the analogy between magnetostatic and electrostatic f i e l d s . Because the magnetic f i e l d between the. pole pieces i s constant with time, the magnetic potential satisfies Laplace's equation. Therefore, i f the boundary conditions for the magnetic and electric fields are the same, the potentials w i l l be relatively the same over a l l points i n the f i e l d . Using this principle, relative values of the magnetic f i e l d i n the a i r gap of the magnetic balance were found by measuring electrostatic potentials. To obtain similar boundary conditions, the pole pieces of the magnet were removed and. placed the same distance apart i n a tank of water. An electric potential was put on one of the pole pieces and the other pole was grounded} the potential was then measured at a l l points on a horizontal plane through . the center of the pole pieces. The vertical probe was made from a piece of insulated wire with a t i p bare. To ensure that the insulated section of the probe did not distort the f i e l d , readings were taken on a line along the axis of the probe, perpendicular to the axis of the magnet poles and midway between them. If the insulated part of the probe was distorting the f i e l d , the readings for which the probe point was below the pole axis would not be the same as the readings above the pole axis. Figure 19, a plot of these readings, shows that the insulated wire does not distort the f i e l d . Figure 21 gives a plot of the potentials for the positions i n the f i e l d shown in Figure 20. The potential along the pole axis i s shown in Figure 22. The results of this investigation were: 44. 1, The f i e l d was constant over the cross-section of the sample per-pendicular to the pole axis. 2. The force on the sample, which is proportional to grad H, was constant for a distance of approximately one half inch along the f i e l d axis in the center of the gap. The choice of rest point for. the suspension i s therefore not c r i t i c a l . b. Calibration for a change i n sample weight. The Alhico nickel samples used for this calibration were of two different shapes. One shape was cylindrical with the ratio of length to diameter equal to f i v e . The other shape was the same except i t had a square cross-section instead of a c i r c l e . The ratio of width to length was the same for a l l samples. The results are shown i n Figure 23. The circles are values for samples with a round cross-section; the dots are for samples with a square cross-section. It was therefore shown that the deflection on the magnetic balance divided by the weight of the sample was constant. The constant was the same for both shapes and a l l weights of the samples investigated. ro 1 1 1 1 X •VOLTS • : ' PROBE - mm 2 O i Z POTE 1 3 i 1 1 1 1 1.5 2.0 . 2.5 3.0 3.5 4.< DEPTH OF PROBE INTO TANK IN INCHES Figure 1 9 . Test of probe used i n measurement of the electrostatic • potentials between the magnet poles. 98765432 98 7654 32 Figure 2 0 . Position i n a i r gap between the magnet poles for the potential measurements shown i n Figure 2 1 . I I I I I I I I I I I I I I I I I I I I 22 -20 18 16 14 ro x co I 2 o > 10 ,o Q- 6 I I I I I I !• I • I '» I' 'I' I j I I I I •» SMALL POLE LARGE POLE DISTANCE PARALLEL TO POLE FACE Figure 21. Pot e n t i a l at various points i n the a i r gap between the magnet poles. The p o s i t i o n of the readings is' shown i n Figure 20. 5 6 7 8 DISTANCE ON POLE AXIS FROM CONCAVE POLE Figure 22. Potentials along the magnet pole a x i s . f 1 1 I 1 1 T WEIGHT OF SAMPLE IN MILLIGRAMS Figure .23. C a l i b r a t i o n of magnetic balance f o r sample weight. The dots are measurements on samples with square cross-section; the c i r c l e s on samples with round cross-section. 49. APPENDIX THREE S t a t i s t i c a l Derivations 1. Definitions„ Let the face centered cubic l a t t i c e be divided into four simple cubic lattices L]_, L?, L3, and L^, such that an atom in one sublattice has four nearest neighbours i n each of the other three sublattices but no nearest neighbours i n i t s own sublattice. Let the order parameters of L]_, L2, L 3 , and L4 be a, b, c, and d, respectively. If :A i s the fraction of L, sites occupied by A atoms Let a = 2 f i A - l 1 when a = 1 a l l atoms on L^ are A atoms when a = 0 one half the atoms on L j are A atoms when a = -1 no atoms on L l are A atoms similarly: b = 2 ^ A " 1 c - 2 ISA" 1 d = 2 f 3A- 1 We therefore get the equations / 1 * A { = 1 - -P = 1 - a J 1 A " 2 J I B J l A 2 with similar equations for the other sublattices. Let (a + b + c + d) = R . . . . . . 2 Let be the t o t a l fraction of A atoms i n the alloy. FA = f IA + j~2A + f 3A + fkA 4 = 1/2 + 1/8 ( a + b + c + d) 50. F A = 1/2 + 1/8 R 3 F B = 1/2 - 1/8 R 4 Let z. be the coordination number of the l a t t i c e . 2. Derivation of Equations. Consider the nearest neighbour interactions of the atoms i n sublattice L i with the atoms in L2, L3, and L4. The to t a l number of A atoms and B atoms in L i i s ^i_±_§_^ I a n d ^ ~ a ^ ^ respectively. The.probab-i l i t y of any one atom in lat t i c e L]_ being an A atom, i s equal to the fraction of A atoms i n L^. Using these last three facts, the number of A-A interactions where one of the A atoms i s in L]_ i s : . / l + aA N f Z: (l + b\ ' 2 Y1 + c\ , Z: / l + d\ V—; 4 [ 3 \ — ) + 3 { — ) + 3 vrr-/ N? f -l which equals J^ g I 3 + 2a + R + ab + ac + ad I Similarly, the number of B-B interactions, where one of the B atoms i s in L]_ i s : (H*) i [ fM-f M ' f M ] which equals jj^ £ 3 - 2a - R + ab + ac + adj The number of A - B interactions i n which the A atom i s i n L^ i s : ^ [3 + 4a - R - (ab + ac + ad)J 51. The number of B - A interactions i n which the B atom i s i n L i i s : J 3 - 4 a + R - (ab + ac + ad)J Since A-B interactions are equivalent to B-A interactions, the sum of the last two equations w i l l be the number of A-B interactions in which one of the atoms i s on L^. This equals j-jp" ' (6 - 2 (ab + ac + ad)~] 7 Equations similar to 5> 6, and 7 are derived using L?, L 3 , and in turn as the reference axis. The tot a l number of A-A interactions i s the sum of equation 5 with the three similar equations obtained by considering the A-A interactions using 1%, L-j, and as the reference l a t t i c e , a l l divided by 2. Let Q,pj^, Q gg and Q^B be the t o t a l number of A-A, B-B, and A-B interactions, respectively. QAA - [ 1 2 + 6R + 2 (ab + ac + ad + be + bd + cd)j but Zr = 12 for the •$•<*-. l a t t i c e . QAA = I £ l 2 + 6R + R2 - (a 2 +b2 +c 2 +d 2)] . . 8 similarly QBB = I [L2 - 6R + R2 - ( a 2 + b 2 + c 2 + d 2)] . . . . . . . . 9 QAB. = I Q24 - 2R2 + 2 ( a 2 + b 2 + c 2 + d 2) J 10 It is seen that the t o t a l number of interactions obtained by adding equation 8, 9 and 10 is 6N. This i s in agreement with the coordination number of twelve. Case 1. Disordered l a t t i c e . When the lattice i s completely disordered, 52. a = b = c = d = 2j-lk - 1 = 2 F A - 1 R = a + b + c + d = 4 a = 4 ( 2 F A - l ) Putting these values of a, b, c, d, and R into equations 8, 9 and 10 we get 2 11 ( Q A A ) d i s = 6 FA N ( Q B B ) D . S = 6 N [ F / - 2 F A + 1 ( Q A B ) d i s - 12 H (P A - F A 2 ) ] 12 .13 Case II. Ordered Lattice, Let L]_ be the sublattice containing the B atoms when f u l l y ordered. Assume: (1) that a l l B atoms l i e in the one sublattice L^ ( i t i s immaterial which sublattice they are i n as long as they a l l l i e i n the same one) (2) the other three sublattices are indistinguishable, i.e., b = c = d Case I l a . F A > \ A 4 . . . . then b = c = d = 1 as these lattices have no B atoms b u t a + b+ c + d = R = 4 ( 2 F A - l ) a = 8 F A - 7 putting these values of a, b, c, d into equations 8, 9, 10, we get (QAA) . = (96 F A - 48) I - N (12F A - 6) ord Q A toB>ord" 0 (QAB) - (-96FA + 96) N = 12N (1 - F A ) ord 8 14 l i b . Assume lq_ lattice i s f i l l e d with B atoms, i Let b = c = d b u t a + b + c + d = R = 4 (2FA - l ) b = | F A - 1 putting these values into equations 8, 9, 10 we get *ord J (QAA)^ - ^ F / (QBB) ord i6 . F A 2 - 12F A + 6 N 12F, - I2- F, 2 N A 3 A FA £ these check with case H a for F. = 2 4 54 APPENDIX 4 Equations for the Quenched Alloys The object of this Appendix i s to find the slope of the curve of saturation magnetisation of the quenched alloys near twenty-five atomic percent manganese assuming the alloys are disordered. Near twenty-five atomic percent manganese the 3d shell of the nickel w i l l be nearly f i l l e d . One would therefore expect the Mn-Mn interaction to be much stronger than the Mn-Ni interaction, which i n turn w i l l be much stronger than the Ni-Ni interaction. Therefore the fraction of manganese atoms with l e f t spin should be proportional to QBg, the number of Mn-Mn interactions. Because the Mn-Ni interaction i s much stronger than the Ni-Ni interaction, one would also expect the spin of a nickel atom to be i n the same direction as the spin of i t s adjacent manganese atom. This is equivalent to saying the fraction of nickel atoms with l e f t spin i s the same as the fraction of manganese atoms with l e f t spin. If we assume the i)»m'bcr of manganese atoms with l e f t spin i s KQBB, then the fraction with l e f t spin i s K^BB . when the value F BN of Qgg for the disordered alloy i s substituted from equation 12, Appendix 3, the fraction of manganese atoms with l e f t spin i s 6KFg. For the mang-anese contribution to the magnetic moment to equal zero at twenty-five atomic percent manganese, 6KFg must equal 1/2 at FB equal to 1/4. There-fore K = ^  . Therefore the fraction of manganese and nickel atoms with l e f t spin i s 2Fg. 55. If the 3d electrons of the manganese go into the nickel 3d shell, then the magnetic moment of a manganese atom is five Bohr magnetons, and that of a nickel atom i s - ^ j j i = 0.6 - Fg . When the ordinate i s changed to makex(Nj_ = 1 - 3.8 Fg, then ^i.^ = 8.3 . Therefore the manganese contribution to the magnetic mom-ent of the metal i s 8.3 FgN(l - 4Fg) and the nickel contribution to the magnetic moment i s F AN(1 - 3.8Fg)(l - 4 F g ) . T n e t o t a l magnetic moment i s the sum of these and equals: M T = [8-3F B + (1 - Fg)(l - 3.8Fg)] t 1 - 4Fg]N , - - 8 . 4 The experimental curve in the same units has a slope of - 15.00. Therefore, the theoretical curve for the saturation magnetisat-ion near twenty-five atomic percent manganese using these assumptions has a slope of approximately one half the experimental value. 56. APPENDIX FIVE Calculations of Order Parameters 1. Calculation of the order parameter of Ni3Mn after heat treatments B and C. If we assume that a l l manganese atoms i n face centered sites w i l l reverse their spin without affecting the spins of the adjacent manganese atoms, and that the number of nickel atoms with reversed spins is equal to the number of manganese atoms with reversed spins, then the observed drop i n magnetisation from 14700 on the tangent to 9500 for the experimental value of heat treatment B can be explained by saying that the fraction of manganese atoms i n face centered sites i s ^  / ' -9^y^^— I =.187 When any la t t i c e site is chosen at random, the probability for i t to have the right atom i s r, and the wrong atom i s w. The Bragg and Williams long range order parameter i s defined as S « r - w. For the cube corner s i t e w • .127 r = 1 _ w = .8.13 For the face center s i t e w = . 062 r » 1-w = .936 s = r - w = i (r - w) a + | (r - w) b s = .8.1 This can be checked by finding the short range order parameter. 57. from Appendix 3' Q B B = | £ l 2 - 6R + R 2 - ( a 2 + b 2 + c 2 + d 2 ) J . . . . 9 Q a b = | £24 - 2R2 + 2(a 2 + b 2 + c 2 + d 2 ) J . . . .10 F . - i + i f a + b + c + d l 3 A 2 8 L J R - 8F A - 4 Putting a = b = c = d ^ a and equation for R into 9 and 10 QAB - § [24 - 2R2 + 2(a 2 +3b2)] N r 2 2 2 1 Q BB - 8 L 1 2 - 6 ( 8 F A " 4 ) + R ~ ( a + 3 b U 2 2 Substituting a + 3b from QBB into QAB we get H But the number of manganese atoms i n face center sites i s (J86)M 4 and each of these has three manganese atoms as nearest neighbours. Therefore, Q B B - 3 ( « ^ ) N . 0.14 N-Substituting this into equation 16 and putting F A = ^ QAB = 2.72. N but = 9 - 9 rand 9 max - 9 rand a t F a = 4 9 max = ( QAB)ord 6N ' 1 9 r a n d - ( QAB)dis 6N Therefore, putting i n the values of 9j max and 9 rand found from equation 13 and 15 of Appendix 3, we get o" = 0.66 and S = = ,m This checks with the value of s = .81! found from the theory of Bragg and Williams. When these calculations were repeated for heat treatment C, 58 the results were: s = 0.88 from Bragg and Williams & = 0.774 and s = 0.88 from the short range order. 2. The calculation of the order parameter of Ni3Mn from x-ray-diffraction line intensities. In this calculation the order domain size i s assumed to be very large. For smaller domain sizes, this calculation would not be valid. The intensity of an x-ray diffraction line i s given by 23 the equation: I = const..-p.L.e""2M A(e)F 2 . . . . . . . . . 17 where I = integrated intensity F = structure amplitude factor p = multiplicity factor L = Lorenz and polarisation factor -2M e = Debye-Waller temperature factor A( e) = absorption factor The individual factors w i l l be calculated separately. 1. Multiplicity factor. This factor is equal to the number of equivalent sets of planes with the same Miller indices, p = 12 for the 110 plane p = 8 for the 111 plane. 2. Absorption factor. 23 Taylor gives the following formula for the angle between the incoming x-rays and the compact powder surface equal to (f t 59. A ( ^ ) - const, s i n ^ e l % + sin <p In the Philips spectrometer $ - O , therefore the absorpt-ion factor i s independent of the Bragg angle Q . 3. Lorentz and polarisation factor. 23 Taylor gives this factor equal to 2 - • ' L = 1 + cos 2 & for a polychromatic beam. cos sin 6 For the ( i l l ) planes L = 6.925 For the (110) planes L = 11.086 4. Temperature factor. 3 h2  M " mK ® ff(x), 1 x 4 s i n 2 0 H = Planck's constant i n erg.sec m = mass of atom i n grams K = Boltzmann's constant i n erg deg" 1 ® = Debye characteristic temperature x = @ where T i s the absolute temperature T A = wavelength i n cm. = Debye function © = Bragg angle. From the consideration of the Debye temperature for nickel and manganese, atoms with similar atomic number, and the melting points of these elements, the Debye temperature of Ni3Mn was taken to be 3406K. .749. This gave a value for the Debye f u n c t i o n 2 4 of (p-Q = 0.' 2M The value of e • for (110) plane i s 0.9650 The value of e _ 2 M for (111) plane i s 0.9485 60. 5. Structure amplitude factor. a. For face centered cubic lattices i n general: F = JE £ n cos 2 7T (hxn + ky n +4%) where x, y, s are the co-ordinates of the atoms in the c e l l , hkl are the Miller indices for the plane under consideration, and -$~n i s the scattering factor of the atom at X n y n Z h . For the 110 and 111 planes we get: F110 - fcfi -F l l l " (*L + 3* 2) where 6, i s the mean atomic scattering factor for the cube corner sites, and ^ is the mean atomic scattering factor for the face center sites. The corrections for the atomic scattering factors of nickel and manganese 25 for dispersion due to the k electrons were taken from James. b. Effect of ordering on the structure factor. The mean atomic scattering factor for any of the sublat-tices can be obtained from the order parameter and the values of the scattering factors corrected for dispersion due to k electrons. Let the fraction of manganese atoms i n face centered sites be P. At cube corner sites w r At face center sites w r s - \ ( r - w ) a + I (* " *)b = 1 " P From the values of r and w in terms of P, the values of ^, and £; are: * » - | > - p ) * M n + P * N i ] [ ^ - f ^ N i * f r ^ M n j 61. h2D - | l - i l f j [.012303 18 [ m L 3 J Therefore F l l l ~h + 3*2 • *Mn + 3^1 F 1 1 0 = ^ l " *2 = ( W ^ d " ^> 6. Values for I n n i m From the preceding yalues for the various factors i n equation 17, we can find the ratio of I J I Q i n terms of P . *LL1  I l i £ l 2 I l l l Taking the experimental value of I as the average number of counts when scanning across the line once, minus the average.background, the following values are obtained for I-QQ ^ d I]j_l from the measurements li s t e d i n the 'results*. I n n = t o t a l no. counts - background no. oscillations where K = angular velocity of scanning = 2° (26) per minute 1.21 = angle (2©) per osc i l l a t i o n 60 = time of scanning i n minutes. In a similar manner i s found and I11D = .0075 * .0016 I l l l This i s equated to the theoretical value i n equation 18 to find P and hence S. The value of S obtained i s S = O . 8 4 + 0.07 where the probable error was calculated from the uncertainty i n the results of the experimental intensity. It was assumed that a l l constants used were accurate. 62. BIBLIOGRAPHY 1. W. Heisenberg, Z. Phys. 49, 619 (1928) 2. J . C. Slater, Phys. Rev. 3.6, 57 (1930) 3. E. Ising, Z. Phys. 31, 253 (1935) 4. L..Neel, Le Magnetisme, II Ferromagnetisme (Strasbourg 1939) 5. S. V. Vonsovsky, J. Tech. Phys. (U.S.S.R.) 18, 131 (1948) 6. F. Bitter, Phys, Rev. _5Jt, 79 (1938) 7. R. Smoluchowsky, J. Phys. et Rad. 12, 389 (1951) 8. S. Kaya and A. Kussmann, Z. Phys. 7.2, 293 (1931) 9. W. J. Carr, Phys. Rev. 8J>, 590 (1952) 10. B. L. Averbach, J. App, Phys. 22, 1088 (1951) 11. C. G. Shull and S. Siegal, Phys. Rev. 7_5_, 1008 (1949) 12. N. Thompson, Proc. Phys. Soc. J52, 217 (1940) 13. C. Guilland, Compte Rendu 219, 614 (1944) 14. A. Komar and N. Volkenstein, J. Exp. Theor. Phys. (U.S.S.R.) 11, 723, (1941) 15. S. Valentiner and G. Becker, Z. Phys. 93_, 795 (1935) 16. R. A. Fereday, J. Phys. Soc. 2j2, 251 (1930) J. Phys. Soc. 46, 214 (1934) 17. C. Sykes and F. W. Jones, P. R. S. A 166, 376 (1938) 18. R. M. Bozorth, ,Ferromagnetism» (D. Van Nostrand Company, Inc., New York, 1951). 19. C Zener, Phys. Rev. 8j>, 324 (1952) 20. A. F. Guinier, Proc. Phys. Soc. _5JL, 310 (1945) 21. C. Sykes and F. W. Jones, P.R.S. A 157. 213 (1936) . 63 22. F. C, Nix and W. Shockley, Rev. Mod. Phys, 10, 1 (1938) 23. A. Taylor, An Introduction to X-ray Metallography (Chapman and Hall Ltd., 1949). 24. A. H . Compton and S. K. Allison, X-rays i n Theory and Experiment (D. Van Nostrand Co. Inc. 1949) 25. R. W. James, The Optical Principles of the Diffraction of X-rays. (G. Bell and Sons, 1950). 

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