Elastic response of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ to spontaneous and field-induced phase transitions

${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ is a uniaxial ferromagnet that displays a first-order magnetization process (FOMP) at 25 T for field applied along the hard $b$ axis. Here, we report on ultrasound and magnetostriction measurements of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ in static and pulsed magnetic fields up to 40 T. The FOMP causes stepwise anomalies in the magnetoelastic properties. ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ elongates along the $a$ and $c$ axes and shrinks along the $b$ axis, leading to an almost zero volume effect. The sound velocities of the longitudinal and transverse acoustic waves decrease sharply at the FOMP, whereas the sound attenuation shows pronounced peaks. An analysis of the ultrasound data using a mean-field theory suggests the existence of quadrupolar interactions and crystal-electric-field effects. The $5f$ electronic states are between itinerant and localized, typical of uranium-based intermetallic compounds.

Figure 1

Temperature dependencies of (a) the magnetic susceptibility, $\chi$, for an excitation field of 1 mT applied along the $a$ axis; (b) the relative sound-velocity changes, $\mathrm{\Delta }v/v$; and (c) the sound attenuation, $\mathrm{\Delta }\alpha$, for the longitudinal, $C_{33}$, and transverse, $C_{44}$ and $C_{55}$, acoustic modes for ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$. The $\mathrm{\Delta }v/v$ curves are vertically offset for clarity. The $\mathrm{\Delta }\alpha$ values for $C_{33}$ and $C_{55}$ are scaled by a factor of 40. The ultrasound frequencies were 34.7, 17.1, and 30.6 MHz for the acoustic modes $C_{33}$, $C_{44}$, and $C_{55}$, respectively. The inset in panel (a) shows the crystal structure of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$.

Figure 2

Temperature dependencies of [(a)–(c)] the relative sound-velocity changes, $\mathrm{\Delta }v/v$, and [(d)–(f)] the sound attenuation, $\mathrm{\Delta }\alpha$, for the longitudinal, $C_{33}$, and transverse, $C_{44}$ and $C_{55}$, acoustic modes, respectively, for field applied along the $b$ axis of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$. The curves are vertically offset for clarity. The ultrasound frequencies were 34.7, 17.1, and 30.6 MHz for the acoustic modes $C_{33}$, $C_{44}$, and $C_{55}$, respectively.

Figure 3

Experimental and calculated temperature dependencies of [(a), (b)] the relative sound-velocity changes, $\mathrm{\Delta }v/v$, and [(c), (d)] the sound attenuation, $\mathrm{\Delta }\alpha$, for the longitudinal acoustic mode $C_{33}$ of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$. The ultrasound frequencies were 34.7, 99.8, and 168.2 MHz.

Figure 4

Field dependencies of (a) the magnetization, $M$; (b) the relative sound-velocity changes, $\mathrm{\Delta }v/v$; (c) the linear magnetostriction, $\mathrm{\Delta }l/l$, along the $a$, $b$, and $c$ axes; and (d) the sound attenuation, $\mathrm{\Delta }\alpha$, for the longitudinal, $C_{33}$, and transverse, $C_{44}$ and $C_{55}$, acoustic modes for field applied along the $b$ axis of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ at 2 K. In panels (b) and (d), the curves are vertically offset for clarity. The insets in panels (b) and (d) show $\mathrm{\Delta }v/v$ for $C_{33}$ and $\mathrm{\Delta }\alpha$ for $C_{44}$ and $C_{55}$, respectively, in the vicinity of the FOMP. The ultrasound frequencies were 38.4, 57, and 92 MHz for the acoustic modes $C_{33}$, $C_{44}$, and $C_{55}$, respectively.

Figure 5

Field dependencies of the magnetostriction, (a) $\mathrm{\Delta }l/l_a$, (b) $\mathrm{\Delta }l/l_b$, and (c) $\mathrm{\Delta }l/l_c$, along the $a$, $b$, and $c$ axes for field applied along the $b$ axis of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ at selected temperatures.

Figure 6

Field dependencies of [(a)–(c)] the relative sound-velocity changes, $\mathrm{\Delta }v/v$, and [(d)–(f)] the sound attenuation, $\mathrm{\Delta }\alpha$, for the longitudinal, $C_{33}$, and transverse, $C_{44}$ and $C_{55}$, acoustic modes, respectively, for field applied along the $b$ axis of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ at selected temperatures. In panels (b) and (e), some curves are vertically offset for clarity. The ultrasound frequencies were 38.4, 57, and 92 MHz for the acoustic modes $C_{33}$, $C_{44}$, and $C_{55}$, respectively.

Figure 7

Phase diagram of ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ for field applied along the $b$ axis. The gray region shows the temperature-field window where ultrasonic dispersion is observed in the static-field measurements.

Figure 8

Temperature dependencies of (a) the relative sound-velocity changes, $\mathrm{\Delta }v/v$, for the longitudinal, $C_{33}$, and transverse, $C_{44}$ and $C_{55}$, acoustic modes for ${\mathrm{U}}_3{\mathrm{Cu}}_4{\mathrm{Ge}}_4$ and (b) calculated quadrupolar susceptibility, $-{\chi }_{O_{\gamma }}$, for the quadrupolar operators $O_{20}$, $O_{yz}$, and $O_{zx}$. The ultrasound frequencies were 34.7, 17.1, and 30.6 MHz for the acoustic modes $C_{33}$, $C_{44}$, and $C_{55}$, respectively. (c) CEF level scheme of a ${\mathrm{U}}^{4+}$ ion up to 350 K obtained from the CEF parameters given in the text.