Resonance Analysis of the Internal of Uranium Beryllium 13 Jack Li

To cite this version:

Jack Li. Muon Spin Resonance Analysis of the Internal Magnetic Field of Uranium Beryllium 13 . 2017. ￿hal-01536617v1￿

HAL Id: hal-01536617 https://hal.archives-ouvertes.fr/hal-01536617v1 Preprint submitted on 12 Jun 2017 (v1), last revised 5 May 2018 (v5)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. µSR ANALYSIS OF THE INTERNAL MAGNETIC FIELD OF UBe13

JACK LI

Uranium beryllium 13 (UBe13) is a heavy fermion system whose properties depend strongly on its internal magnetic structure. Dierent models have been proposed to explain its magnetic distri- bution, but additional experimental data is required. An experimental method that is particularly useful is muon spin spectroscopy (µSR). In this process, positive are embedded into a sample where they localize at magnetically unique sites. The net magnetic field causes precession of the muon spin at the Larmor frequency, generating signals that can provide measurements of the internal field. This experiment specifically determines the muon localization sites of UBe13. To do so, results from muon spin experiments at various temperatures and external magnetic field strengths are an- alyzed. The experiments took place at TRIUMF in the University of British Columbia. Data from the temperature and magnetic field ramps are analyzed through ROOT. The Fourier transforms of experimental data showed peaks of muon localization at the geometric centers of the edges of the crystal lattice. These results can be used to build a rigorous model of UBe13’s internal magnetic structure and resulting magnetic field distribution. Note that data was taken from experiments conducted by another party not formally associated with this paper.

I. BACKGROUND resistivity, specific heat) change dramatically at 1-10 K, as if its outer-shell electrons were several orders of mag- A. Heavy Fermion Systems and Uranium nitude heavier. This behavioural shift can be explained Beryllium 13 by models that rely heavily on an understanding of the internal magnetic structure. There is still a lack of in- Heavy fermions have been studied extensively and are formation on UBe13, and this project aims to oer new the focus of modern physics research. These compounds insight to build a rigorous model of its internal magnetic are generally Ce, Yb, and U based intermetallics, al- structure. though Np and Pu compounds are reported. A common Under the framework of muon spin resonance spec- feature among heavy fermion systems is the also the exis- troscopy (µSR spectroscopy), this project determines the tence of a partially filled 4f or 5f electron shell. At room muon localization sites and takes advantage of the muon’s temperature, f-shell electrons remain at their atomic sites strength as a magnetic probe to map its internal field dis- and the system behaves as an interacting collection of tribution. The µSR process is discussed in more detail in f-electron moments and conducting electrons. However, the next section. f-electron moments become coupled with conduction elec- trons and with each other at low temperatures. Thus, the eective mass of conduction electrons become 10 to 1000 B. An Overview of µSR Spectroscopy times that of a bare electron. Many anomalies appear in heavy fermion systems at low enough temperatures. µSR is an abbreviation for muon spin resonance and Uranium beryllium 13 (UBe13) is a characteristic is a technique used in solid-state physics to probe the heavy fermion. Figure 1 describes the crystal structure internal magnetic structures of various materials. The of this compound. The primary cube contains 8 uranium process generally involves three components: an experi- atoms, where 8 beryllium atoms form a cube around each mental sample, a positive spin-polarized muon beam, and uranium atom. 24 more beryllium atoms arrange them- an exterior magnetic field Bext. selves into a polyhedron structure inside the beryllium Positive muons are produced though a series of nuclear cube. reactions and are collected to form a muon beam. The initial polarization of the muons in the beam is close to 100% i.e., the spin of the positive muons is opposite to the momentum. The muon beam is directed by elec- tromagnetic field guides and shot into the sample being studied, where the muons are implanted at various in- terstitial sites inside the material. Locked in these sites, the muons spin at the Larmor frequency ÊL related to the net magnetic field by ÊL = “µ Bnet , where “µ is the known of the muon.| | This relationship can be determined theoretically, and the proof is given FIG. 1: Crystal structure of UBe13. in Appendix A. While spinning, the muon undergoes a decay process, Like most heavy fermion systems, UBe13 behaves like and the muon decays into a , a positron neutrino a metal at room temperature. Chemical properties (e.g. and a muon anti-neutrino. The positron is emitted pref- 2

FIG. 2: Muon precessing about the net magnetic field in the direction of the green arrow. The external field is denoted by the black arrow. The smaller red arrow denotes the spin of the muon. erentially in the direction of the spin. The can then be monitored and stored by detection electronics. In the case of this experiment, four positron detectors sur- round the sample. The four are named L(eft), R(right), FIG. 5: Typical Fourier transform of an asymmetry function B(ack), F(orward). similar to the one in Figure 4. The peaks of this graph represent the most prominent muon Larmor frequencies that make up the corresponding asymmetry function.

and ALR(t) as

NL(t) NR(t) ALR(t)= ≠ . NL(t)+NR(t) Figure 4 gives a typical asymmetry function from a set of opposing detectors. These two asymmetry functions can uncover details about the internal magnetic field. To do so, the Fourier FIG. 3: An example of a positron counts versus time histogram. Note that the experiment described in this paper uses four transformation for the two asymmetry plots is taken for detectors, meaning four of such graphs exist for each trial. each trial. The peaks of the Fourier transforms give the most prominent frequencies of muon oscillation at inter- stitial sites, or specifically, the Larmor frequencies ÊL.

C. Magnetic Properties

The internal magnetic field of a crystal Bint is related to the external magnetic field Bext and net magnetic field Bnet by the intuitive relationship

Bnet = Bint + Bext (1)

Bint can be deduced, as Bext is experimentally controlled and Bnet can be expressed in terms of the measured Lar- mor frequency.

FIG. 4: Asymmetry vs time (given in microseconds) graph of two sets of opposing detectors on the same axes. II. EXPERIMENTAL SETUP

Experiments took place on the M15 Beamline at TRI- With this data, four histogram functions can be made UMF, with a polarized µ+ beam dedicated to the pro- for each detector - NL(t),NR(t),NF (t) and NB(t) (as duction of muons. The UBe13 crystals were oriented so in Figure 3). An additional set of functions, called the that the magnetic field aligned with the [100] crystallo- asymmetry functions, can be defined for each set of de- graphic direction, anti-parallel to the muon beam. Four tectors: ABF (t) and ALR(t). These can be written as positron detectors surrounded the sample. the dierence between the pairs of histograms divided by The experiment was conducted so that two parameters their sum. ABF (t) can be expressed as were varied at a time: firstly, the temperature of the crystals and secondly, the magnetic field surrounding the NF (t) NB(t) crystals. The experiment consisted of two data sweeps ABF (t)= ≠ NF (t)+NB(t) measuring the eect of each parameter while holding the 3 other constant. Tables 1 and 2 describe the parameter III. RESULTS AND DISCUSSION variation for each series of measurements. The range of temperatures and magnetic field strengths are able to This section focuses on the determination of these provide results from a variety of conditions. muon localization sites. The anisotropy of detection events measured in the experiment correspond to the Lar- Field (T) 1.5 1.5 1.5 1.5 1.5 1.5 mor frequencies of muons localized in the sample. This Temperature (K) 275 200 125 75 25 3 can as a result give a measurement of the time averaged net magnetic field felt by the muons. Results show sev- TABLE I: Temperature change with constant field eral important properties of UBe13.

Field (T) 0.5 0.75 1.0 1.5 2.0 2.5 Temperature (K) 3 3 3 3 3 3

TABLE II: Magnetic field change with constant temperature

The UBe13 crystals were located at the end of the beamline, under vacuum, and cooled by liquid nitrogen. The crystals were surrounded by a plastic scintillator, which generated photons when struck by decay positrons. The photons traveled through a light guide to a photo- multiplier, generating an electrical pulse to the surround- FIG. 6: The Fourier transform of the asymmetry data for 5 ing positron detectors. Scintillators on opposite sides of temperatures all at the same magnetic field of 1.5 T. At some the crystal then produced an anisotropic signal that os- temperature between 202 K and 126 K, muons are able to localize cillates with muon decay. Two additional scintillators within the lattice, shown by the Larmor frequency splitting. The smaller peaks on the right represent the parallel interstitial lattice were in the beam-line to track the muons. These scintil- positions while the larger peaks on the left represent the lators were made of plastic, and had a thickness of ap- perpendicular positions. Parallel lattice positions remain proximately 0.25 mm that allowed muons to pass through unaected by the temperature change, while perpendicular while being counted. The first scintillator was placed in positions are aected greatly. the beamline before the crystals, and the second located just before the UBe13 crystals. Muon localization in the From the Fourier transform of asymmetry data of each sample depended on the intensity of the muon beam at trial, runs for high temperatures (> 150 K) gave a single the sample. A superconducting solenoid called Helios was peak Larmor frequency as seen in Figure 6. The reason used to apply a large magnetic field to the sample. The for this is that the muons at these temperatures possess Helios controlled the width of the beamline close to the high amounts of thermal energy, and cannot be localized sample, and limited the number of muons impinging on to any particular spot in the material lattice. Muons in- the UBe13 crystals. The goal was to detect 1 muon every stead possess enough thermal energy to move between 10 µs. If two muons were stopped within that time in- lattice sites and through the crystal. The result is a dis- terval, the measurement was discarded to ensure that an tribution of Larmor frequencies for muons propagating emitted positron could be correctly matched to a single through the lattice. Additionally, thermal fluctuations implanted muon. within UBe13 can distort some of the net magnetization The UBe13 sample was cooled to extremely low tem- induced by the externally applied magnetic field. For peratures. Such cooling was made possible by liquid he- both these reasons, the results indicate a single peak cor- lium cooling and the length of the shaft that the crystals responding to the average Larmor frequency of all muons were mounted on. The dierence in temperature between in all lattice spots (as in the 202 K case in Figure 6). the end of the shaft and the sample resulted in a large As temperature is lowered below the mobility limit, temperature gradient. The temperatures were controlled two distinct peaks begin to appear, as seen in Figure 6. by both the liquid helium cooling system and an auto- As mentioned earlier, the dual peaks may be a result of matic heater. The sample was cooled slightly below the the result of muons coming to rest at two magnetically desired temperature so that the heater could fine-tune unique crystal lattice sites. Observing all possible muon the samples to the desired temperature within an ac- sites, the most plausible locations are described in Figure ceptable range of fluctuation. Data was analyzed using 7. These sites can correspond to either site I, in which ROOT developed by the European Council for Nuclear the edge it is resting on is perpendicular to the external Research (CERN). field, or site II, where the edge is parallel with the exter- 4 nal field. The orientations of the dipole moments cause From Section 1.2, the Larmor frequency ÊL equals two magnetically unique sites at I and II. As the ratio be- “µ Bnet ,where“µ is the gyromagnetic ratio. Thus, | | ÊL tween the magnitude of the two peaks for each tempera- Bnet is equivalent to . From Equation 5, an expres- | | “µ ture in the Fourier transform (Figure 6) is approximately sion for the internal field is simply 2:1, this provides evidence for the existence of multiple localization sites; there are 8 perpendicular sites (like in ÊL 2fi fL Bint = j Bext = · j Bext (2) I) and 4 parallel sites (as in II), which form a similar 2:1 “µ ≠ “µ ≠ ratio. The frequency splitting can thus be attributed to the magnetic environments of the parallel and perpen- where j is the unit vector in the same direction as Bint dicular muon sites, the higher peak corresponding to the and Bext. muon Larmor frequency at the perpendicular sites and Fit results for the temperature ramp are listed be- the lower peak to that at the parallel sites. low. Table III gives results for the perpendicular sites, Another possible localization site seen in other ura- while Table IV gives results for the parallel sites. The nium based crystals is at the geometric center of the lat- calculated magnitude of the internal magnetic fields are tice results do not support this however. If muon local- given as well. ization≠ does occur at this central site, an additional third Bext (T) Temp. (K) fL (MHz) Bint (T) peak would have been traced in the Fourier transform. ‹ ‹ 4 1.4958 3 202.8113 5.3370 10≠ As there are two well-defined peaks and that localization ◊ 3 1.4958 25 203.0593 2.3632 10≠ at site I implies localization at site II (and vice versa) ◊ 3 1.4958 75 203.3027 4.1587 10≠ ◊ 3 due to crystal symmetry, there is no evidence that the 1.4958 125 203.4158 4.9940 10≠ ◊ 3 geometric center serves as a muon site. 1.4958 200 203.6529 6.7430 10≠ ◊ TABLE III: External magnetic field B and Larmor frequency | ext| fL data for the perpendicular lattice sites during the temperature‹ ramp. The magnitude of the internal magnetic field Bint is calculated. | ‹|

Bext (T) Temp. (K) fL (MHz) Bint (T) Î Î 3 1.4958 3 203.9058 9.4088 10≠ ◊ 2 1.4958 25 203.9863 1.0000 10≠ ◊ 2 1.4958 75 204.0007 1.0109 10≠ ◊ 2 1.4958 125 204.0010 1.0111 10≠ ◊ 3 1.4958 200 203.6529 6.7430 10≠ ◊ TABLE IV: External magnetic field B and Larmor frequency | ext| fL data for the parallel lattice sites during the temperature Î ramp. The magnitude of the internal magnetic field Bint is | Î| calculated.

Fit results for the magnetic field ramp are listed FIG. 7: I and II are possible localization sites for implanted below. Table V gives results for the perpendicular sites, muons in the UBe13 crystal, represented by blue spheres. Red while Table VI gives results for the parallel sites. The spheres represent uranium atoms with their dipole fields aligned calculated magnitudes of the internal magnetic fields are with Bext.Notethereare8muonsitesperpendiculartothe magnetic field (symmetrical to I) and 4 parallel to the field given as well. (symmetrical to II). Beryllium atoms are not shown for simplicity. Bext (T) Temp. (K) fL (MHz) Bint (T) ‹ ‹ 3 0.50 3.061 67.6263 -1.0558 10≠ Identifying the muon sites of UBe13 can allow for the ◊ 3 0.75 3.125 101.4496 -1.5109 10≠ calculation of the internal magnetic field B from Equa- ◊ 3 int 1.00 3.160 135.2578 -2.0736 10≠ ◊ 3 tion 1: 1.50 3.110 202.7998 -3.7511 10≠ ◊ 3 2.00 3.135 270.4000 -5.0000 10≠ ◊ 3 Bnet = Bint + Bext 2.50 3.149 337.9396 -6.6950 10≠ ◊ 3 3.00 3.181 405.4530 -8.5839 10≠ In the presence of an external magnetic field Bext,the ◊ muon spins at their lattice sites align parallel with Bext TABLE V: External magnetic field B and Larmor frequency | ext| (see sites I and II from Figure 7). Assuming perfect muon fL data for the parallel lattice sites during the magnetic field Î ramp. The magnitude of the internal magnetic field Bint is spin alignment with Bext, tracing these magnetic dipole | ‹| lines reveal that the magnetic field at the various lattice calculated. sites Bint is directly parallel with the external field. 5

2 10≠ 1.2 · shown in Figure 7, the magnetic dipole moments super- impose each other in the opposite direction of Bext at the perpendicular lattice sites (at Site I). 0.9

0.6

0.3 Internal Field Strength (T) 0 0 50 100 150 200 Temperature (K)

FIG. 8: The internal field strengths plotted against temperature. Data is from the temperature ramp. The blue curve denotes FIG. 10: Map of the internal magnetic field of UBe13 at its muon localization sites. Red spots are weaker magnetic sites and blue Bint and the red curve denotes Bint ‹ Î spots are stronger magnetic sites.

2 10≠ 2 · A contesting theory attempting to explain the mag- netic structure of UBe13 is based on the concept of the Bint Bint ‹ ≠ Î polaron. Polarons are quasiparticles they are not a 1.6 physical particles but interactions between≠ electrons (or

erence (T) muons) with atoms in crystals. Polarons are induced in

1.2 UBe13 by the introduction of muons in the crystal sam- ple. As muons are negatively charged, they attract pos- 0.8 itive ions and repel negative ions, causing a temporary region of asymmetry. This region is the polaron, which 0.4 can then diuse through the crystal. The polaron has the ability to change the orientation of the magnetic dipole Internal Field Di 0 moments of surrounding magnetic ions - uranium ions in 0 0.25 0.5 0.75 1 this case. The polaron can align these dipole moments, Bext/T (T/K) causing a larger magnetic moment and a greater internal field as a result. FIG. 9: Bint Bint vs. Bext/T .Dataistakenfromthe Results of this experiment do not support the exis- ‹ ≠ magnetic field ramp. AnÎ interesting note from Curie’s Law is that tence of magnetic polarons. If polarons did appear, fre- magnetization is proportional to Bext/T . quency distributions seen in Figure 6 would reveal ap- proximately equal distributions between the frequencies,

Bext (T) Temp. (K) fL (MHz) Bint (T) given the random nature of the polaron’s movement. The Î Î 3 0.5 3.061 67.9974 1.6821 10≠ 2:1 ratio as seen from the experiment does not agree with ◊ 3 0.75 3.125 102.0027 2.5719 10≠ this theory. ◊ 3 1.00 3.160 135.9950 3.3657 10≠ ◊ 3 1.50 3.110 203.9092 4.4433 10≠ ◊ 3 2.00 3.135 271.8340 5.5804 10≠ IV. SUMMARY AND CONCLUSION ◊ 3 2.50 3.149 339.762 6.7505 10≠ ◊ 3 3.00 3.181 407.6690 7.7766 10≠ ◊ The experiment examined the internal magnetic struc- TABLE VI: External magnetic field B and Larmor frequency ture of UBe13 using µSR at TRIUMF. The data reveals | ext| fL data for the parallel lattice sites during the magnetic field two primary Larmor frequencies consistent with muons Î ramp. The magnitude of the internal magnetic field Bint is occupying magnetically unique lattice sites that are par- | Î| calculated. allel and perpendicular to the external field. The Fourier power of the two peaks correspond roughly to the ex- pected distribution in muon stopping sites. At temper- An interesting note from these results is that the in- atures higher than the mobility limit, thermal energy ternal field at perpendicular sites is anti-parallel to the causes the muon to travel between muon sites, forming a externally applied field at low temperatures and weak single peak. However at lower temperatures, such ther- magnetic field strengths, as shown in Table V. Look- mal energy was removed and the oscillation between lat- ing at the superposition of magnetic dipole moments of tice locations decreased, forming two distinct peaks in uranium atoms can reveal the reasons behind this. As the spectrum. Increased field splitting between sites at 6 higher levels of magnetization could also verify the loca- where a and b are numerical values with the restriction tion of these muon stopping sites. The applied magnetic that a2 + b2 = 1. Measurable quantities such as position, field was used to induce a net magnetization in the UBe13 velocity, angular momentum are represented by a corre- sample, which allowed for sensitive inter lattice measure- sponding operator. Operators act on wavefunctions to ments. produce defined quantities. In special cases, the result of UBe13 has a cubic structure and is therefore one of such an operation can be written as the most straightforward heavy fermions to study. As a O ◊ = ⁄ ◊ characteristic heavy fermion, UBe13’s magnetic proper- | Í | Í ties can readily be applied to other heavy fermion sys- where O is the operator with the original wavefunction tems. Although applications of UBe13 are not known as ◊ . If this case were to arise, the wavefunction ◊ is of yet, recent years have seen the identification of quan- | Í | Í tum critical points in a host of antiferromagnetic heavy- an eigenfunction of O with a scaling factor ⁄, called an fermion compounds. The ability to harness the behav- eigenvalue. Then, the eigenvalue ⁄ will be observed if iors of heavy fermions could lead to rapid development in a measurement of the observable operator O is made on quantum technology and new generations of technology. the state ◊. In vector form, operators are in some cases 2 2 matrices. ◊A special type of operator is the Hamiltonian, which corresponds to the observable that is the system’s en- V. ACKNOWLEDGEMENTS ergy.TheZeeman interaction, the energy between the magnetic moment and the applied magnetic field, is a The author would like to thank Eric Woolsey for pro- Hamiltonian and is perhaps the most important inter- viding the data used here and his mentorship during this action in magnetic resonance. The Zeeman interaction project. He would also like to thank Prof. Andrew Mac- written as a 2 2 matrix is: Farlane and Prof. Robert Kiefl for their openness to dis- ◊ cussion and helpful comments during the course of writ- ~“µB 10 H = (A1) ing. ≠ 2 0 1 5 ≠ 6 where B is the exterior magnetic field. Appendix A: Derivation of the Larmor Frequency A particle’s state at any time can be found by using the time evolution operator, defined as

A particle in a quantum system is represented by a iH t U(t)=exp≠ . wave function ~ x Â(x,t,s) Via Equation A1 and Taylor expansion of e , the opera- tor can be written as a 2 2 matrix as ◊ where Â2(x,t,s) determines the probability of finding a exp i“µBt 0 particle with spin s at position x at time t. In some U(t)= 2 0 exp i“µBt cases, the wavefunction Â(x,t,s) can be written in the 5 2 6 form „(x,t) ◊(s,t), where the „ component is called the position state and ◊ is called the spin state.Thediscus- Thus, calling a state at time t =0 sion hereon focuses on this spin factor. A common way a of expressing the spin state is through bra-ket notation, Â(0) = denoted by the ”ket” vector ◊ . As the rest of this paper | Í b | Í 5 6 deals exclusively with fermions, which has only two spin 2 2 states ( 1 ), ◊ can be expressed as a linear combination where a and b are numerical values such that a +b = 1, ± 2 | Í the time evolution operator U(t) can be used to give the of the + 1 state + and the 1 state .Thesetwo 2 2 state at any later time t: states can also be| writtenÍ as column≠ vectors,|≠Í with exp i“µBt 0 a a 1 0 Â(t) = 2 = ei“µBt (A2) + = , = | Í 0 exp i“µBt b b | Í 0 |≠Í 1 5 2 65 6 5 6 5 6 5 6 This product can be written as a trigonometric ex- meaning that for the general spin state, pression via Euler’s formula. The periodic motion of the muon is given an associated value Ê = “ B, the Larmor a L µ ◊ = frequency. | Í b 5 6

[1] S. Sakarya, Magnetic Properties of Uranium Based Fer- [2] M. Liu, Properties of Heavy Fermion Systems, (2010). romagnetic Superconductors, thesis, (2007). 7

[3] A. L. Yaouanc and D. R. Pierre, Muon Spin Rotation, P.Brison, F. Hardy, A. Huxley, S. Raymond, B. Salce, Relaxation, and Resonance: Applications to Condensed andI. Sheikin, Comptes Rendus Physique7, 22 (2006). Matter, (2010). [9] A. Ramirez, P. Chandra, P. Coleman, Z. Fisk, J. [4] J. Jang, T. Uittenbosch, E. Woolsey, N. Zacchia, Analysis Smith,and H. Ott, Physical review letters 73, 3018 of the Internal Magnetic Field of UBe13,(2016). (1994). [5] R.H. Hener, D.W. Cooke, Z. Fisk, R.L. Hutson, M.E. [10] A. MacFarlane, Introduction To Magnetic Resonance Schillaci, J.L. Smith, J.O. Willis, D.E. Maclaughlin, Probes Of Materials. 1st ed. (2017). C. Boekema, R.L. Lichti, A.B. Denison, and J. Oost- [11] V. Storchak, J. Brewer, D. Eshchenko, P. Mengyan,O. ens, Muon Knight shift and zero-field relaxation in Parfenov, A. Tokmachev, P. Dosanjh, Z. Fisk, andJ. (U,Th)Be13 (1986). Smith, New Journal of Physics 18, 083029 (2016). [6] J. L. Smith, Z. Fisk, J. O. Willis, B. Batlogg, & H. R. Ott. Impurities in the heavy fermion superconductor UBe13. JACK LI, St. George’s School, 4175 W 29th Ave, Vancou- (1984) ver, BC V6S 1V1 [7] C.-T. Lin, Microwave Surface Impedance Measurements. E-mail address, [email protected] (1999) [8] J. Flouquet, G. Knebel, D. Braithwaite, D. Aoki, J.-