  We determined the spin exchanges between the Cu2+ ions in the kagome layers
of volborthite, Cu3V2O7(OH)2x2H2O, by performing the energy-mapping analysis
based on DFT+U calculations, to find that the kagom\'e layers of Cu2+ ions are
hardly spin-frustrated, and the magnetic properties of volborthite below ~75 K
should be described by very weakly interacting antiferromagnetic uniform chains
made up of effective S=1/2 pseudospin units. This conclusion was verified by
synthesizing single crystals of not only Cu3V2O7(OH)2x2H2O but also its
deuterated analogue Cu3V2O7(OD)2x2D2O and then by investigating their magnetic
susceptibilities and specific heats. Each kagome layer consists of intertwined
two-leg spin ladders with rungs of linear spin trimers. With the latter acting
as S=1/2 pseudospin units, each two-leg spin ladder behaves as a chain of S=1/2
pseudospins. Adjacent two-leg spin ladders in each kagome layer interact very
weakly, so it is required that all nearest-neighbor spin exchange paths of
every two-leg spin ladder remain antiferromagnetically coupled in all spin
ladder arrangements of a kagome layer. This constraint imposes three sets of
entropy spectra with which each kagome layer can exchange energy with the
surrounding on lowering the temperature below ~1.5 K and on raising the
external magnetic field B. We discovered that the specific heat anomalies of
volborthite observed below ~1.5 K at B = 0 are suppressed by raising the
magnetic field B to ~4.2 T, that a new specific heat anomaly occurs when B is
increased above ~5.5 T, and that the imposed three sets of entropy spectra are
responsible for the field-dependence of the specific heat anomalies.
