text:
\section{Introduction}
\label{Sc:Introduction}

The function of biomolecules such as proteins is intrinsically linked to their atomic structure, which in turn depends on highly complex, crowded, heterogeneous native cellular environments \cite{Dhar2010,Ellis2001, Elcock2010,Benton2012,Feig2017}. Current structure reconstruction techniques, X-ray crystallography \cite{Chapman2011,martin2016serial,Shi2014}, cryo-EM  \cite{Carroni2016,Li2013} and NMR spectroscopy \cite{Kay2016,Jiang2017,Ashbrook2018,Renault4863}, are exceptionally powerful, however, they have access to only average structural information of proteins in artificial environments and require large ensembles of proteins in nearly identical conformational states. Unlike crystalline physical systems, proteins have complex sets of metastable conformation states that depend on a multitude of variables, including the presence of various other molecules in their local environment \cite{Phillips2009}. There is an abundant and essential class of molecules which do not possess well-defined conformation states \cite{Sickmeier2007,uversky2016dancing}, constantly moving through a continuous conformation space, which itself is parameterised by countless environment variables \cite{Karplus2005, Baldwin2009}. As a result, imaging 3D atomic sites of individual, arbitrary molecular systems is outside the scope of the existing structure reconstruction techniques, which impacts our understanding of the behaviour of proteins on the length scales fundamental to their function.

Due to the nanometre dimensions of molecular systems, a pathway for achieving single protein imaging can be found in sensors that are atomic in scale. Recently, the nitrogen vacancy (NV) centre in diamond \cite{Doherty13,JPrev14} has shown a great promise as a biocompatible, nanoscopic qubit-probe for room temperature magnetometry \cite{Balasubramanian08,Taylor11,Cole09,Kaufmann13} and, in particular, high resolution nuclear magnetic resonance (NMR) spectroscopy \cite {Zhao2011, Cai2013, Perunicic2014,Trifunovic2015, Ajoy2015, Kost2015, Laraoui2015, Lazariev2015, Wang2016,  Schmitt2017,Boss2017,Glenn2018, Zopes2018, Abobeih2019, cujia2021arxive}. However, imaging the atomic makeup of single molecular structures requires low temperatures, as a means of arresting atomic motion and capturing desired instances of protein conformation states. Even at low temperature, the spin coherence times of NV centres are relatively short while their  fixed quantisation poses further limitations for the application considered here. Furthermore, the single-qubit deterministic fabrication specifications required for practical development of the instrumentation technology at scale are lacking for NV centres. This specific context opens up the interesting prospect of using low temperature silicon qubits as bio-sensors. Silicon donor qubits are equipped with a combination of long coherence times \cite{Tyryshkin2011,Muhonen2014} and deterministic fabrication methods \cite{Morello2010,Pla2012,Pla2013,Mahapatra2011,Fuechsle2012, Usman2016} making them uniquely positioned for practical applications such as quantum computing, and in particular the sensing concept introduced here.

In this paper, we therefore outline a platform for single-molecule structure-determination based on qubits in silicon allowing a form of magnetic resonance imaging (MRI) of individual molecular structures at the atomic level. This work extends the idea of using a spin qubit as both detector and source of localized field gradients, initially proposed in the context of nitrogen-vacancy centres in diamond in \cite{Perunicic2016}. Here we seek to capture the technological advantages of silicon donor qubits, requiring full treatment of the qubit electronic wave function. We characterise the control mechanism over the quantum interaction between the qubit electronic wave-function and individual nuclei in the targeted molecular specimen, providing the foundation of MRI at the atomic level. To overcome the exceptional challenges of spatial-frequency encoding at the single molecule level, we introduce the nuclear spin storage protocol which enables qubit sensors to be used for 3D atomic level imaging through careful sampling and extraction of structural information from a molecular specimen. Donor placement is an important fabrication question addressed by determining the optimal donor depth based on the effects of silicon surface termination. Finally, we perform a simulated demonstration by imaging the atomic structure of macro-molecular system comprising an individual trans-membrane protein and its lipid bilayer environment.

\section{Results}
\subsection{Silicon qubit-based single molecule structure imaging system}
\label{Sc:Molecular_imaging}

In this section, we outline the physical setup and principles of the device function, dedicating the remainder of the paper to the development of quantum control and 3D sampling aspects of qubit-based molecular structure imaging in the context of a highly non-trivial test-system. We develop the platform using group-V donor qubits in silicon, taking phosphorus (Si:P) \cite{Morello2010,Pla2012,Pla2013,Mahapatra2011,Fuechsle2012} as an example, noting that the method is applicable to other spin qubit systems with similar coherence times and wave function extent.

\begin{figure}[!htb]%
\centering
 \includegraphics[width=1\linewidth]{Fig1.pdf}
 \vspace{-15pt}
  \caption{\textbf{Silicon qubit molecular imaging system.}
   \textbf{a}) An array of noninteracting near-surface donor spin qubits in a silicon chip. Biological specimens, such as a native cell membrane, are positioned on the surface (red layer). The sample is snap frozen allowing each qubit to image the conformational instance of individual molecular structures that inhabit its sensing volume (blue). In-plane control structures based on standard STM fabrication techniques \cite{Fuechsle2012, Weber2014} (orange metallic leads).
   \textbf{b}) Local Single Electron Transition (SET) islands (top-down view) \cite{Morello2010,Fuechsle2012,Pla2012} facilitate read-out and initialisation of independent qubit sensors.
   \textbf{c}) Example of a single transmembrane-lipid system consisting of the M2 proton channel of influenza A virus \cite{Cady2010} in 1,2-Didecanoyl-sn-glycero-3-phosphocholine (DDPC) lipid bilayer (upper). The sample is positioned above the phosphorus donor qubit (Si:P) at depth of $4.75\,a_0$ ($\sim2.5\,\rm nm$) (orange sphere, lower) such that the sample is immersed into the qubit's magnetic dipole field (equipotential slices shown in blue) emerging from its electronic wave function (orange, lower). Water molecules suppressed for clarity, their positions can be directly determined as part of the imaging.
   \textbf{d}) The nuclear spin storage (NSS) detection protocol (high-level schematic, left). The dipole-dipole decoupling methods adapted to the qubit's gradient field environment (gray) and interleaved target nuclei control (orange), allow the qubit to probe (flow of quantum information, blue) the quantity of the specimen's nuclei present in the targeted equipotential slice of its dipole field. A dipole-adapted MRI protocol samples the specimen across a multitude of overlapping slices (middle). Measurement signals from individual slices inverted into the atomic-level 3D image of the targeted specimen's nuclear density (illustration, right).
  }\label{Fig:1}
   \vspace{15pt}
\end{figure}

In this initial work we consider a biological specimen such as a section of a cellular membrane is positioned on a silicon chip containing near-surface qubit sensors (Fig.\,\ref{Fig:1}\,a). Subsequently, the sample is snap frozen at a specific point, capturing the desired configuration of target molecular structures in vitrified ice \cite{Al-Amoudi2004, McDonald2007,Bellare2018}. The chip may contain a single sensing qubit \cite{Morello2010,Pla2012} or a grid arrangement of qubits \cite{Hill2015}. The donor qubits can be introduced using single ion implantation \cite{Morello2010,Pla2012,Pla2013}, and in the case of Si:P, they can also be deterministically fabricated by STM lithography \cite{Mahapatra2011,Fuechsle2012} and fully characterised by STM donor metrology \cite{Usman2016, Usman2017}, which is particularly advantageous for the imaging scheme. We assume the silicon surface is hydrogen-terminated, which is quite typical in such systems, noting that oxide termination would lead to reductions in the sensing volume.
The nanoscopic size and fabrication process of the platform technology opens up a pathway towards scalability in the field of protein structure determination. This feature uniquely contrasts with conventional techniques where specimen imaging parallelisation is fundamentally limited by the relatively large physical size of the system required for a single imaging site.

The sensing signal is obtained by reading out the electron spin state of each qubit independently (Fig.\,\ref{Fig:1}\,b). As each qubit addresses a molecular-sized imaging volume, we focus on describing the protocol from the perspective of a single donor qubit (Fig.\,\ref{Fig:1}\,c). The dipole-dipole coupling between the donor electronic wave function and the molecular nuclear target spins species ($\rm ^{1}H, ^{13}C, ^{14}N$ etc.) extends above the silicon surface establishing a hemispherical sensing volume (Fig.\,\ref{Fig:1}\,a,\,c blue). The method leverages this dipole-dipole field to perform 3D imaging in a manner analogous to clinical Magnetic Resonance Imaging (MRI), with dipolar slices $\Gamma$ (blue lobes, Fig.\,\ref{Fig:1}\,c) defining distinct spatial ``lobes'' containing target nuclear spins with effectively the same coupling to the donor electron.

The key enabler behind the qubit-based molecular imaging technique is a nuclear spin storage (NSS) protocol particularly focused on the silicon qubit context (Fig.\,\ref{Fig:1}\,d, left), which allows us to probe the molecular target's nuclear destiny at individual dipolar slices by reading out the state of the qubit. In order to generate spatio-frequency encoding across Angstrom distances, the NSS protocol successfully embeds dynamical dipole-dipole decoupling (such as CORY-24 sequence) into the gradient magnetic field produced by the donor wave function. Through careful interleaved and phase-matched control (Fig.\,\ref{Fig:1}\,d, left, orange), the donor's electronic spin resonantly couples only to the target nuclei in a chosen slice, while the coupling between the target nuclei is suppressed and the unwanted effects of the gradient filed canceled out. Information about the target's nuclear density in the probed slice is intermediately stored onto the state of the donor's nuclear spin (Fig.\,\ref{Fig:1}\,d, left, blue) in order to protect it against decoherence and enable efficient readout. Details of the NSS protocol are given in Section \ref{Sc:NSS_protocol}.

The steps of the 3D imaging procedure (Fig.\,\ref{Fig:1}\,d, middle) are summarised as follows. First, we choose a spin carrying nuclear species to image inside a target molecule specimen, i.e, $\rm ^{1}H, ^{13}C, ^{14}N$ etc. Next, we orient the donor's dipole field along a particular direction. The properties and control mechanisms of the dipole field are given in Section\ref{Sc:donors_dipolar_field}. We apply the NSS protocol to measure the nuclear density of the target across a series of dipole slices. The slices are measured sequentially for each of the multiple orientations of the donor's dipole field (Fig.\,\ref{Fig:1}\,d, middle, red arrows)  in order to sample the entire sensing volume (see Sections \ref{Sc:NSS_protocol} and  \ref{Sc:optimal_control_parameters}). In the final step, the measurement outcomes are inverted from dipolar slice-space to 3D space, enabling the nuclear density of the targeted molecular structure to be imaged in sub-Angstrom voxel resolution (Fig.\,\ref{Fig:1}\,d, right) (see Section \ref{Sc:protein-lipid_system}).
Sections that follow expand on each of the mentioned steps, detailing how silicon donor qubits can be used to image the nuclear spin density of the target molecule in 3D at the atomic level.

\subsection{Controlling the dipolar field produced by a spatially extensive donor wave function}
\label{Sc:donors_dipolar_field}

Traditional MRI uses a gradient magnetic field to correlate specific (usually planar) region of the sensing volume with a particular nuclear Zeeman splitting magnitude \cite{lauffer1996mri}, this concept is called spatial-frequency encoding. In contrast, we rely on the natural dipole-dipole interaction between the target nuclei and the qubit electron spin to provide spatial-frequency encoding. The feasibility of silicon-qubit based molecular imaging rests on the nontrivial properties of the dipolar field produced by the wave function describing the qubit's electron. In this section, we focus on the shape, magnitude and the spatial control of the dipole field, while the aspects of resolution and slice cross-sectional profile are addressed later.

Qubit probes such as the NV-centre in diamond have a highly confined (Angstrom scale) electronic wave function producing relatively strong point-dipole like fields with spatial inhomogeneity characterised by the magic-angle axis, which is ideal for 3D molecular imaging via frequency encoding \cite{Perunicic2016}. However, as noted NV centres have relatively limited quantum coherence times and the orientation of their dipole field is difficult to control due to the loss of readout contrast when the background magnetic field tilts away from the centre's quantisation axis. Donor qubits in silicon, on the other hand, have electronic wave functions on the nanometre scale \cite{Wellard2005}, an order of magnitude greater compared to the structural features of interest in molecular systems. It is, therefore, crucial to examine the behaviour of the above-surface dipole-dipole field produced by such delocalised donor wave functions. We use the experimentally benchmarked Tight-Binding (TB) formalism \cite{Usman2016,Usman2017} to compute a near-surface donor electronic probability density (see Method section: Tight binding formalism) and calculate its associated dipolar field from first principles (see Method section: Calculation of the Dipole-Dipole field). This formalism has been shown to reproduce the donor electron wave function to atomic precision when compared directly with STM imaging of sub-surface donors \cite{Usman2016,Usman2017}. The same approach can be used in experimental realisations of the method, where the donor TB wave function representation obtained from STM donor spatial metrology eliminates the need for the nontrivial characterisation of the dipolar field itself.

\begin{figure}[!htb]%
\centering
 \includegraphics[width=1\linewidth]{Fig2.pdf}
 \vspace{-15pt}
  \caption{\textbf{Tight-binding characterisation of the spatial-frequency encoding of the dipole-dipole field produced by delocalised wave functions of donor electrons in silicon.}
  \textbf{a}) The cross-section of the ZZ component of the dipole-dipole field $\Gamma$ (upper) produced by the donor wave function with a diameter of $\sim4\rm\,nm$ of the P donor in Si at a depth of $\sim2.5\rm\,nm$ (lower) calculated using the NEMO tight-binding framework \cite{Usman2016,Usman2017} based on $\sim3$ million Si atoms (see Methods section: Tight binding formalism).
  \textbf{b}) Tilting the background magnetic field $\bf B_0$ allows for axis ($r$) control of the dipolar lobes and maintains the magic-angle axis. This behaviour is characteristic of a point-dipole, indicating suitability for molecular imaging.
  \textbf{c}) Variations in the axial profile of $\Gamma(r)$ as a function of $\bf B_0$ orientation (solid, coloured) originate from the asymmetry of the near-surface wave function. It is important to account for these variations in the context of 3D imaging, however, they are collectively relatively close to the point-dipole field (dashed, grey) particularly for more distant targets.
  \textbf{d}) Dipole filed coupling $\Gamma$ as a function of $r$ for a range of donor depths. Donor depth is a potent way of controlling the gradient of the dipolar field, especially for closer targets as illustrated here.
  \textbf{e}) The near-surface gradient can be further engineered by decreasing the size of the wave function using bismuth donors (blue) or an ionised neighbouring pair of phosphorus donors (2P+) (purple).
  }\label{Fig:2}
   \vspace{15pt}
\end{figure}

From the TB simulations, we find that the secular ZZ component of the dipolar field (Fig.\,\ref{Fig:2}\,a upper) emerging from the donor probability density (Fig.\,\ref{Fig:2}\,a lower) effectively retains the same shape as a point-dipole, characterised by the curved lobes and magic-angle axis (for details see Appendix \ref{Si_Sc:Effects_of_spatial_delocalisation}: Effects of spatial delocalisation). Furthermore, the characteristic dependence of a point-like dipole field on the background magnetic field $\bf B_0$ orientation is maintained. This is depicted in Fig.\,\ref{Fig:2}\,b), whereupon introduction of a tilt in $\bf B_0$ ($\theta=45^\circ)$, the dipole coupling field produced by the donor orients along its direction. These key properties enable the spatial-frequency encoding to be controlled through the co-latitude and longitude ($\theta, \phi$) orientations of $\bf B_0$.

To facilitate 3D imaging, we introduce a custom spatial-frequency encoding formalism to uniquely address individual dipole slices in the presence of varying $\bf B_0$ orientation. A dipole slice is the lobe-shaped surface on which the target nuclei experience the same coupling $\Gamma$ to the donor. In 3D space, we specify this surface by three parameters $(r, \theta, \phi)$, defined as follows. The parameter $r$ is the distance between the point on substrate surface above the donor and the given slice, along the direction of $\bf B_0$ (dashed line, Fig.\,\ref{Fig:2}\,a\,and\,b). While, parameters $\theta$ and $\phi$ are the orientations of the $\bf B_0$, as noted before. Therefore, a slice specified by single parameter $\Gamma$ in frequency space, is specified by a vector ($r, \theta, \phi$) in 3D space. The vector ($r, \theta, \phi$) is used to address of the entire lobe-shaped surface, produced by the donor wave function under $\bf B_0$ (note, it should not to be confused with the spherical polar coordinate notation for a single point in space).

To achieve Angstrom level resolution, it is essential to characterise the dependence of the dipolar field on the co-latitude direction of the background field $\theta$ (Fig.\,\ref{Fig:2}\,c). We note that the dipole field remains relatively close to that of a point-dipole (Fig.\,\ref{Fig:2}\,c dotted) under a range of $\bf B_0$ tilt ($\theta\leq60^\circ)$ sufficient for 3D molecular imaging.
The dipole field can be engineered by varying the donor depth, as indicated by Fig.\,\ref{Fig:2}\,d (see Appendix \ref{Si_Sc:Engineering_the_gradient_by_donor_placement}: Engineering the gradient by donor placement). Note, we discuss the optimal depth later. The dipole field can also be produced using donors with more confined wave functions, such as bismuth (Fig.\,\ref{Fig:2}\,e blue) or an ionised pair of phosphorus atoms 2P+ (Fig.\,\ref{Fig:2}\,e purple) (see Appendix \ref{Si_SC:Engineering_the_gradient_field_through_confinement}: Engineering the gradient field through confinement).

\subsection{Nuclear spin storage protocol}
\label{Sc:NSS_protocol}

\begin{figure}[!htb]%
\centering
 \includegraphics[width=1\linewidth]{Fig3.pdf}
 \vspace{-15pt}
  \caption{\textbf{Schematic of the Nuclear Spin Storage (NSS) detection protocol.}
  \textbf{a}) Continuous application of the homonuclear decoupling sequence, CORY-24 \cite{Cory91} comprising of $\pi/2$ pulses (red/blue), ensures that nuclear target spins have sufficiently long coherence times.
  \textbf{b}) Interleaving \cite{Perunicic2016} control rotation pulses (orange) into the decoupling sequence enables dipolar slices in the sensing volume to be individually addressed.
  \textbf{c}) The structure of the NSS detection protocol features three distinct segments.The detection segment of length $t_{\rm det}\sim1/\Gamma_i$ limited by $T_2^{\rm probe}$ (purple). Nested in between is the control segment of length $t_{\rm ctrl}\sim1/\gamma B_{\rm AC}$ limited by the relatively longer $T_1^{\rm probe}$ timescale (orange), during which the nuclear storage is used to protect the information about the target magnetisation collected by the first half of the detection segment. Lastly, the signal is obtained by measuring the state of the electron (green) which allows for $n_m$ projective measurements to be collected in a relatively short time $t_m\ll t_{\rm det}+t_{\rm ctrl}$, without repeating the entire detection protocol.
  \textbf{d}) The sensing volume is sampled in the ``criss-cross" MRI fashion consisting of slice radius sweeps ($r$) repeated for an array of $\bf B_0$ orientations ($\theta, \phi$).
  \textbf{e}) Example of the signal profiles over $r$ for a selection of $\theta$ and $\phi$ orientations (right/left). The signal peaks encode the number of nuclear target spins present in a given slice enabling the atomic-level image of the sensing volume to be formed in 3D.
  }\label{Fig:3}
   \vspace{15pt}
\end{figure}

The nuclear spin storage (NSS) detection protocol (depicted in entirety in Fig.\,\ref{Fig:3}\,c.) optimally leverages the long longitudinal ($T_1^{\rm probe}$), transverse ($T_2^{\rm probe}$) and nuclear transverse ($T_{\rm nuclear}^{\rm probe}$) coherence times of the group-V donor qubits to extract the information about the number of molecular target spins in a given slice ($r, \theta, \phi$).  The NSS protocol requires several layers of careful quantum control over both the qubit and the target nuclear spins.

Collectively, the targeted nuclei are subjected to strong homonuclear decoupling \cite{Cory91} with spectrally broad pulses resonant with the background Zeeman splitting $\omega_0=\gamma_tB_0$, (Figs.\,\ref{Fig:3}\,a\,and\,c top register - denotes numerous nuclear spins). Decoupling suppresses the dipole-dipole interactions between the nuclear spins, narrowing their spectral broadening thus enabling the spatial-frequency encoding to be facilitated. Simultaneously with target nuclear decoupling, the NSS protocol features three segments that perform different functions. It starts by the detection segment (Fig.\,\ref{Fig:3}\,c mid register, blue) where the probe's electron spin is allowed to evolve for a time period $t_{\rm det}/2$ after such that its phase encodes the information about the state of the current molecular target magnetisation. This information is stored on the probe's nuclear state protected by its long transverse coherence time $T_{\rm nuclear}^{\rm Probe}$ (Fig.\,\ref{Fig:3}\,c, bottom register, dashed blue).

In the middle of the detection segment is the control segment of duration $t_{\rm ctrl}$  (Fig.\,\ref{Fig:3}\,c, mid register, red) where the electron spin remains protected by the relatively long $T_1^{\rm Probe}$. A dipolar slice of choice is addressed resonantly using by application of interleaving \cite{Perunicic2016} (Fig.\,\ref{Fig:3}\,c, top register, orange) which ensures that only the spins in that dipolar slice are control rotated based on the state of the probe spin by the phase $\Omega t_{\rm ctrl}$, where is the control Rabi frequency $\Omega=\gamma_tB_{\rm AC}$ (Fig.\,\ref{Fig:3}\,b).  The target nuclei experience gradient Zeeman splitting, $\omega=\gamma_tB_0+\Gamma$, while the nuclear quantum storage is subject to contact hyperfine, both require strategic refocusing by $\pi$-pulses. Details can be found in the Methods section: Details of the NSS protocol.
At the end of the control segment, the phase is brought back from storage onto the probe electron spin, followed by the second half of the detection segment (Fig.\,\ref{Fig:3}\,c, mid register, blue, right). The states from the first and the second half of the detection segment interfere with the resulting state encoding only the relative change in magnetisation of the target molecule, which is by construction proportional to the number of nuclear spins present at the probed slice.

In the final segment, the resulting phase is projected onto population difference and placed in nuclear storage, enabling measurement statistics ($n_m$ electron spin projections) to be acquired independently, without the need to repeat the entire sequence (Fig.\,\ref{Fig:3}\,c, green).

Detailed numerical analysis of the NSS protocol and illustration of its behaviour can be found in the Appendix \ref{Si_Sc:Testing-spatial-frequency-encoding-and-stability-of-NSS-detection protocol}: Testing spatial-frequency encoding and stability of NSS detection protocol. The signal produced by a single target nucleus with coupling $k$ to the probe and in the spectral vicinity if the given slice $\Gamma$ ($k\approx\Gamma$) can be captured sufficiently well for our purposes by the following relations:
\begin{align} \label{eq:Signal_from_sigle_nucleus}
S(\Gamma,k)=\left(1-\cos\left(ak/2\,t_{\rm det}\right)\right) \frac{(\gamma_{\rm t}B_{\rm AC})^2}{\Omega_{\rm k}^2 + (2\pi/T_{\rm \rho\_target})^2} \sin^2(\Omega_{\rm k}/2\,t_{\rm ctrl}),
\end{align}
\begin{align} \label{eq:fine_Rabi_rate}
\Omega_{\rm k}&=(a(\Gamma-k))^2 + (\gamma_{\rm t}B_{\rm AC})^2,
\end{align}
where $a\sim0.3$ is an effective scale factor of the coupling between the probe and the target nuclear spins dependent on the specific decoupling sequence used, and $T_{\rm \rho\_target}$ is the effective coherence time of the target nuclear spins under NSS protocol, which we estimate can reach up to $100\,\rm ms$ (extensive analysis is featured in Appendix \ref{Si_Sc:Target_coherence_under_NSS_detection_protocol}: Target coherence under NSS detection protocol).

The net signal $S_{\rm net}(\Gamma)$ detected from a slice $\Gamma$ that contains $n_{\rm net}$ nuclear spins (of the same nuclear species) is the product of the individual signals $S_i$, assuming each nucleus $i$ is in a thermally mixed state:
\begin{eqnarray} \label{eq:Net_Signal_from_all_nuclei}
S_{\rm net}(\Gamma)=\frac{1}{2}e^{-t_{\rm det}/T_{\rm 2\_ probe}}\prod_{i=1}^{n_{\rm net}}S(\Gamma,k_i),
\end{eqnarray}
where $S(\Gamma,k_i)$ is given by Eq.\,\ref{eq:Signal_from_sigle_nucleus} and $k_i$ is the coupling to the $i^\text{th}$ nuclear spin. Therefore, linearisation:
\begin{eqnarray} \label{eq:scaled_Net_Signal_from_all_nuclei}
L_{\rm net}=-\log(2S_{\rm net}),
\end{eqnarray}
ensures that the processed signal is positive definite and directly proportional to the number of target nuclear spins present in the slice $\Gamma$.

\subsection{Optimal control parameters for 3D sampling}
\label{Sc:optimal_control_parameters}

The resolution and the net experimental running time intrinsically depend on the real space cross-sectional profile of dipolar slices and the total number of the slices $n_{\rm slices}$ used to probe the sensing volume. To account for the nonlinear, nonplanar dipole-dipole field, we develop dynamic control over parameters $t_{\rm det}$, $B_{\rm AC}$ and $t_{\rm ctrl}$ as the function of slice frequency $\Gamma$. We follow by outlining a procedure for 3D sampling of the sensing volume by slice $\Gamma$ parameters $r$, $\theta$ and $\phi$.

The amplitude of the signal response per target nucleus is kept uniform by varying the detection time segment with respect to slice frequency $\Gamma$, as follows:
\begin{eqnarray} \label{eq: varing_t_det}
 t_{\rm det}= \frac{\pi\rho_{\rm det}}{a\Gamma},
\end{eqnarray}
where $\rho_{\rm det}\in[0,1]$ is the response per target nucleus ratio ($\rho_{\rm det}\approx0.2$ for dense molecular samples). For details see Appendix \ref{Si_Sc:Optimal_detection_segment_length}: Optimal detection segment length.

Similarly, the width of the slice cross-sectional profile is kept uniform by varying the strength of the resonant interleaving field $B_{\rm AC}$ with respect to the dipolar gradient:
\begin{eqnarray} \label{eq: varing_B_AC}
 B_{\rm AC}(\Gamma)= \frac{B^{\rm AC}_0}{\partial \Gamma/\partial r(r=0)} \frac{\partial \Gamma}{\partial r},
\end{eqnarray}
where, $B^{\rm AC}_0$ is the interleaving field strength used to address surface slices ($r=0$). This enables for the control segment $t_{\rm ctrl}$ to be further optimised:
\begin{eqnarray} \label{eq: varing_t_cntl}
 t_{\rm ctrl}(\Gamma)= \frac{\pi \rho_{\rm ctrl}}{\gamma_t B_{\rm AC}(\Gamma)},
\end{eqnarray}
where the factor $\rho_{\rm ctrl}$ is the nuclear target spin control rotation ratio. The detailed description is illustrated in Appendix \ref{Si_Sc:Controlling_the_cross-sectional_slice_profile_in_real_space}: Controlling the cross-sectional slice profile in real space.

There are a multitude of ways to probe the sensing volume, for demonstration purposes we adopt uniform sampling over variables ($r, \theta,\phi$) (Fig.\,\ref{Fig:3}\,d) which define each slice of the dipolar field in the sensing volume. The NSS protocol is performed over the set of slices (combinations of $r, \theta,\phi$) that spans the entire sensing volume, thus producing signal profiles proportional to the number of target nuclei present in each slice (Fig.\,\ref{Fig:3}\,e). For the methodology of converting such a collection of signal profiles into a 3D image of the nuclear target density, see Method section: 3D density image processing.

\subsection{Surface termination and optimal donor depth}
\label{Sc:Surface_termination}

The hydrogen-terminated silicon (Si-H) has favourable physical thickness and fabrication techniques, however, it contains surface nuclear spins which, unlike molecular hydrogen, is subjected to contact hyperfine interaction with the qubit electron spin. The Si-H can become a source of decoherence as they follow profoundly different dynamics under decoupling sequences designed to address dipole-dipole coupling only. The contact hyperfine interaction creates the clustering of Si-H above the donor that effectively experience different Zeeman splitting. We perform extensive analysis over the number of spins in such clusters, their maximum hyperfine interaction and map their coherence when subjected to the NSS protocol - as a function of the donor depth and the strength of the decoupling pulses, see Appendix \ref{Silicon_surface_and_the_optimal_qubit_depth}: Silicon surface and the optimal qubit depth. The results enable us to find the optimal qubit depth, which for phosphorus donors in silicon is around $2-2.5\rm\,nm$. We note that recent studies based on STM fabrication and metrology of donor qubits have found phosphorus atoms at a similar depth range below the H-passivated surface \cite{Usman2016, Voisin2020}.

\subsection{Simulated 3D imaging of a single membrane protein-lipid system}
\label{Sc:protein-lipid_system}

We demonstrate the overall qubit-based imaging protocol by simulating the atomic-level structure determination of a single transmembrane protein-lipid molecular system, the M2 proton channel of the influenza virus \cite{Cady2010} embedded in the DDPC lipid bilayer(Fig.\,\ref{Fig:1}\,c). Initially, we use a relatively fast sectional imaging mode that focuses only on the part of the protein-lipid system using the $^{13}$C nuclear sub-density - as an example of the method's nuclear species selectivity. Finally, we demonstrate the extended imaging mode capable of atomic-level reconstruction of the whole system, i.e. imaging through the entire lipid membrane. We will selectively image the $^{14}$N nuclear sub-density in the extended imaging mode.

\begin{figure}[!htb]%
\centering
 \includegraphics[width=1\linewidth]{Fig4.pdf}
 \vspace{-15pt}
  \caption{\textbf{Simulated qubit-based molecular imaging of the surface section of the protein-lipid system with the focus on $^{13}$C sub-density.}
  \textbf{a}) To facilitate a relatively fast imaging mode, the sensing volume is initially restricted to the $\rm 40\times40\times10\AA$ region of the external part of the M2 proton channel-lipid system (only neighbouring lipids shown, water molecules suppressed for clarity).
  \textbf{b}) The image of the nuclear sub-density (simulation), featuring 3D peaks (red +), the $50\%$ (FWHM) and $25\%$ contours. The imaging parameters reflect presently achievable experimental conditions for Si:P qubit systems: $B^{\rm AC}_0=0.5\rm\,\mu T$, $T_2^{\rm probe}=T_{\rm \rho\_target}=100\rm\,ms$, $r_{\rm max}=1.4\,\rm\AA$, $dr=0.25\,\rm\AA$, with $\theta_{\rm max}=63^\circ$, $d\theta=1.5^\circ$, $d\phi=6^\circ$, $n_m=1000$, $\rho_{\rm det}=\rho_{\rm ctrl}=0.2$. Experiment run time estimate $\sim 2$\,hours.
  \textbf{c}) Density peak locations are automatically paired (circled) with their nearest true atomic coordinates (marked with *). The average radial deviation from true coordinates is $\epsilon=0.36\,\rm\AA$.
  \textbf{d}) Top-down view (z-projection) of the density image shoves projections of $\alpha$-helix segments (circled in purple), with helical guidelines superimposed (green dashed).
  }\label{Fig:4}
   \vspace{15pt}
\end{figure}

In the sectional imaging mode, the sensing volume is restricted to the external (Fig.\,\ref{Fig:4}\,a, lower) $\rm 40\times40\times10\AA$ segment of the protein-lipid system. The imaging mode is optimal for identifying proteins, their conformation states as well as their near-surface reactants and ligands. It is relatively fast as the nuclei in the sensing volume possess stronger coupling to the probe, enabling for both shorter $t_{\rm ctrl}$ and $t_{\rm det}$ timescales. The 3D $^{13}$C nuclear sub-density (Fig.\,\ref{Fig:4}\,b) for the section of the molecular system can be attained within $\sim2$\,hours of signal acquisition under presently achievable experimental conditions (see Appendix \ref{Si_Sc:Carbon(13)_nuclear_density_imaging}: Carbon(13) nuclear density imaging).

Individual 3D density peaks feature prominently in the image, interpreting their maxima (Fig.\,\ref{Fig:4}\,c, +) as reconstructed atomic sites without optimisation (i.e. for bond lengths and angles) proves a close match to the true coordinates (Fig.\,\ref{Fig:4}\,c, *) with average deviation $\epsilon=0.36\,\rm\AA$ in line with the voxel grid spacing of $d=0.5\,\rm\AA$. The top-down view of the density image (Fig.\,\ref{Fig:4}\,d) provides a more intuitive insight into the protein's symmetric structure. Notice the four $\alpha$-helices (Fig.\,\ref{Fig:4}\,d, circled in purple), with their helical guidelines superimposed to aid visual interpretation (Fig.\,\ref{Fig:4}\,d, green dashed). Furthermore, the extensive overlapping of dipolar slices makes the imaging method relatively resistant to noise and quantum errors (see Appendix \ref{Si_Sc:Impacts_of_signal_noise_in_density_imaging}: Impacts of signal noise in density imaging). For comparative studies of sectional imaging mode applied to hydrogen and nitrogen sub-densities, see Appendix \ref{Si_Sc:Nitrogen(14)_density_imaging}: Nitrogen(14) density imaging and Appendix \ref{Si_Sc:Hydrogen_density_imaging}: Hydrogen density imaging.

\begin{figure}[!htb]%
\centering
 \includegraphics[width=1\linewidth]{Fig5.pdf}
 \vspace{-15pt}
  \caption{\textbf{Simulated qubit-based molecular imaging across the entire lipid membrane.}
  \textbf{a}) The imaging is illustrated for the $^{14}$N sub-density of M2 proton channel-lipid system in the sensing volume ($\rm 40\times40\times40\AA$).
  \textbf{b}) The full across-the-membrane image of the $^{14}$N density of the M2 proton channel-lipid system (simulation), featuring 3D peaks (red +), the $50\%$ (FWHM) and $25\%$ contours. The sampling parameters are: $d=0.75\,\rm\AA$, $B_0^{\rm AC}=1.25\rm\,\mu T$, $dr=0.5\,\rm\AA$, $r_{\rm max}=4\,\rm\AA$ $d\phi=5^\circ$, $\theta_{\rm max}=45^\circ$. Experimental run time estimate $\sim 6$\,hours.
  \textbf{c}) The average deviation between the density peaks (red +) and the true coordinates (marked with *) is $\epsilon=1.3\,\rm\AA$.
  }\label{Fig:5}
   \vspace{15pt}
\end{figure}

The extended imaging mode expands the sensing volume to the ($\rm 40\times40\times40\AA$) region encompassing the entire M2 proton channel and the neighbouring lipids (Fig.\,\ref{Fig:5}\,a). The nitrogen sub-density 3D image of the entire protein-lipid system (Fig.\,\ref{Fig:5}\,b) would experimentally take around $6$\,hours of signal acquisition. Here the average deviation from the true coordinates (Fig.\,\ref{Fig:5}\,c) is $\epsilon=1.3\,\rm\AA$. The lower parts of the molecule show close matching to the original coordinates, while the upper, being further away from the donor, take longer to sample allowing decoherence to lower the signal contrast and increase the deviation from true coordinates. For additional information see Appendix \ref{Si_Sc:Imaging_the_entire_transmembrane_protein-lipid system}: Imaging the entire transmembrane protein-lipid system.

\section{Discussion}
\label{Sc:Conclusion}

In this paper, we proposed a platform technology based on near-surface qubits in silicon for noninvasive \textit{in situ} structure determination of individual molecules, well-suited to lipid membrane-protein systems. In this qubit-based molecular imaging technique, each qubit functions as a probe site with a sensing volume above the silicon surface commensurate with the nanoscopic size of the targeted protein stature. The technique is enabled by the carefully devised NSS protocol that controls the effective quantum interactions between the qubit's electronic wave function and the nuclei in the sensing volume. This allows the spin qubits in silicon to facilitate MRI of the target protein's individual atomic sites.
Our platform technology leverages established atomic fabrication techniques developed for silicon-based quantum computing over the last decade. It opens up a unique avenue towards scale-up in the field of protein structure determination, which is at preset fundamentally limited by exclusive reliance on macroscopic equipment, allowing incumbent technologies to achieve up to a handful of imaging sites per research facility.

Using a single proton channel of the influenza virus as an archetypal example of a lipid membrane protein system, we showed through simulation that under achievable experimental conditions the molecular imaging protocol has the ability to directly image $^1$H, $^{13}$C and $^{14}$N nuclear densities of individual protein-lipid systems in sub-Angstrom resolution.
Experimental realisation of the qubit-based molecular imaging platform would provide noninvasive access into atomic structure of arbitrarily conformational instances of individual inhomogeneous molecular systems. This ability uniquely complements the present methods: X-ray crystallography, NMR and cryo-EM, that rely on indirect protein stature determination by averaging over conformationally uniform ensembles of molecules. By offering such an unprecedented level of insight into the function of proteins, the proposed platform technology model charts a course of significant implications to biomedical field.

\section*{Materials and Methods}
\label{Sc:Methods}

\subsection{Tight-binding description of the donor electron wave function}
\label{Methods_Sc:Tight_binding}

We investigate the effects of a donor's wave function spherical symmetry loss on the dipole field by utilising the Tight Binding (TB) description of the wave function \cite{Usman2016,Usman2017}. TB is a benchmarked numerical formalism that enables the entire wave function to be represented based on silicon lattice sites acting as continuous, local sources of the wave function following the Slater-orbital description. To reconstruct the complete wave function at any point, the orbital contributions from all of the lattice sites are summed up - for practical purposes, this approach is tractable because Slater-orbitals have an exponential drop off on the Angstrom scale.

TB formalism enables the wave function to be numerically modelled over a relatively large volume cell comprising of several million silicon atoms. That allows for the impact of the proximate silicon surface on the wave function shape to be comprehensively represented. Given the wave function description using TB has been thoroughly benchmarked experimentally via STM donor spatial metrology \cite{Usman2016,Usman2017, Usman2020}, it is an optimal choice for our purposes. Therefore, it is a viable pathway to calculating the realistic donor's dipole field to high precision suitable for applications in molecular imaging.

We summarise the modelling procedure as follows. The $sp^{3}d^{5}s^{*}$ TB Hamiltonian is optimised for silicon band structure \cite{Boykin2004} while the central-cell-correction model is used to include intrinsic strain and non-static dielectric screening effects \cite{Usman_JPS2015}. The donor atom is represented by a Coulomb-like potential that has been cut-off to a specific value and screened by a k-dependent dielectric function \cite{Nara1965}. Its nearest neighbour bond lengths (P-Si) have been adjusted to reflect the strain as provided by DFT calculations \cite{Overhof2004}. In accordance with the often-used experimental setups, the sample surface is set to have (001) orientation, consisting of dimer rows of Si atoms and follows the $2\times1$ surface reconstruction scheme. The impact of the surface strain due to the reconstruction is included in the tight-binding Hamiltonian by scaling the inter-atomic interaction energies in accordance with variations in the lattice bond lengths.

Hydrogen termination is highly effective in saturating the silicon surface dangling bonds. We model H passivation implicitly by shifting the energy of dangling bonds in the model, effectively neutralising their electrostatic potential~\cite{Lee2004}. The silicon volume cell around the P donor is chosen to be a square box of $40\,\rm nm^3$ (approximately 3 million atoms) with closed boundary conditions, which has been shown to be sufficient for wave function benchmarking \cite{Usman2017}. The real-space Hamiltonian is solved in parallel by the Lanczos algorithm to calculate the single particle energies and wave functions of the donor atom. The tight-binding Hamiltonian is implemented within the framework of NEMO-3D. For further details see Usman et al.\cite{Usman2017, Usman2021}.

The tight binding model provides a means to evaluate the full wave function at any point in space, however, to calculate the dipole-dipole coupling field (next Method section), it is sufficient to consider the probability density at the individual silicon and phosphorus atomic sites.

\subsection{Calculation of the Dipole-Dipole field}
\label{Methods_Sc:Dipole-Dipole_field}

The Si:P dipolar field responsible for the spatial-frequency encoding depends on the orientation of the background magnetic field $\bf B_0$. To construct the dipole-dipole coupling field above the silicon surface, we calculate the weighted sum of contributions from each lattice site:
\begin{align} \label{Methods_eq:wfn_sum}
&\Gamma(x,y,z,{\bf B_0})= \sum_{i}^{N_{\rm lattice}}|\Psi_i|^2D_{\rm point}^{\rm ZZ}(x-x_i,y-y_i,z-z_i,{\bf B_0}),
\end{align}
where $\Gamma$ is the net dipole coupling at the point $(x,y,z)$ above the Si surface for a given background magnetic field $\bf B_0$, $|\Psi_i|^2$ is donor electron's probability at the $i^{\rm th}$ site location $(x_i,y_i,z_i)$ inside a lattice consisting of $N_{\rm lattice}$ atoms describing the TB wave function. Here, $D_{\rm point}^{ZZ}(x-x_i,y-y_i,z-z_i,{\bf B_0})$ is the point-to-point dipole-dipole coupling term between the lattice site $i$ and the point $(x,y,z)$. For ${\bf r}=(x-x_i,y-y_i,z-z_i)$, $D_{\rm point}^{ZZ}$ is given by:
\begin{align} \label{Methods_eq:pointDD}
&D_{\rm point}^{ZZ}({\bf r, B_0})=\gamma_{\rm e} \gamma_{\rm n} \frac{\hbar\mu_0}{4\pi} \frac{1}{|{\bf r}|^3} \left(1-3(\hat{\bf B}_{\rm 0}\cdot\hat{\bf r})^2\right),
\end{align}
which has the shape characterised by the dipolar lobes (see Appendix \ref{Si_Sc:Effects_of_spatial_delocalisation}: Effects of spatial delocalisation) aligned according to $\bf B_0$. In principle, the probability density $|\Psi_i|^2$ can be calculated to a higher resolution by using additional coordinates in between the lattice sites, however this requires significant numerical resources while providing a relatively modest improvement to the gradient field accuracy. Furthermore, to optimise numerical calculations, we threshold the lattice sites by selecting those that satisfy $|\Psi_i|^2/|\Psi_{\rm max}|^2\geq10^{-4}$. We are thus able to reduce $N_{\rm lattice}$ from several million to around 50 thousand, while accounting for $98\%$ of the net probability density.

\subsection{Details of the NSS protocol}
\label{Methods_Sc:NSS_protocol}

We detail the internal function of the NSS protocol in the context of the phosphorus donor which possesses a spin-$1/2$ nuclear spin, other group-V elements or qubit systems equipped by highly coherent nuclear spins would follow an equivalent procedure. The outline of the (NSS) detection protocol is illustrated in Fig.\,\ref{Fig:3}\,c, and we will now describe its construction in detail here.
\\
\noindent
{\bf Electron spin phase accumulation during the first detection segment:} The detection segment of NSS protocol is responsible for the encoding of the state of the targeted nuclear density onto the phase of the probe. In contrast to \cite{Perunicic2016}, we separate it from the lengthy process of spectrally narrow control rotations interleaved into the dipole-dipole decoupling. This is achieved by splitting the spin echo sequence on the electron spin in \cite{Perunicic2016} and inserting the interleaved control section in the middle. We begin by preparing both the donor's electron and nuclear spins in the ``up'' state. Reading from the left (Fig.\,\ref{Fig:3}\,c), the pulse sequence applied to the probe spin starts off by a $\pi/2$ pulse equivalent to the echo sequence. During the course of the initial detection segment (purple) only the decoupling pulses are run on the molecular target nuclei. The decoupling pulses are fast relative to detection time $t_{\rm det}$ and their effect suppresses the dipole-dipole coupling between target spins. Collectively this renders the target's magnetisation effectively static, allowing the probe to acquire the phase associated with the entire nuclear spin environment.
\\
\noindent
{\bf Storing the electron spin state to the nuclear spin storage:} Instead of immediately interfering the phase acquired in the first detection segment with that of second, we swap the probe's electron spin state width the donor's nuclear spin state. Therefore the information about the molecular magnetisation is now stored in the state of the probes nuclear qubit, protected by minutes long $T_2$ of the donor's nuclear spin. The probe spin is now polarised ``up'' and the slice-selective control interleaved protocol (orange) is initiated for $t_{\rm ctrl}$ duration.
\\
\noindent
{\bf Select control during nuclear spin storage:} At a half mark ($t_{\rm ctrl}/2$) the probe spin is flipped. This is an important step, and it is useful to consider in detail. Decoupling pulses are resonant with the background Zeeman frequency ($\gamma B_0$), while the target nuclear spins near the probe are shifted away from it by the probe's dipolar field which is effectively comprised of many lobes each with a different coupling $\Gamma_i$. The spectral width of the decoupling pulses is orders of magnitude higher than any of the $\Gamma_i$, therefore the pulses will flip the nuclear spins without difficulty, however, errors in nuclear spin phase will accumulate on the timescale of order $\sim 1/\Gamma_i$. Since $t_{\rm ctrl}$ needs to be longer than that timescale (to allow for narrow spectral width of lobe slices), these errors will become significant, ultimately changing the apparent magnetisation of the entire molecule. By introducing the $\pi$ flip on the probe in the middle of the sequence, the coupling frequency changes sign, allowing for the gradient induced phase errors in the nuclear spins to gradually refocus. The interleaved protocol needs to be carefully phase-matched after the $\pi$ pulse since the fine control frequency changes from $\sim\gamma B_0+\Gamma_i$ to $\sim\gamma B_0-\Gamma_i$.
\\
\noindent
{\bf From nuclear spin storage to second detection segment:} By the end of the interleaved protocol, only the selected dipole lobe slice is targeted. During this period, the probe spin (red) has stayed in the polarised state, protected by seconds long $T_1^{\rm probe}$. In the meantime, the information stored (purple bars) on the P nuclear spin needs to be curated. There are multiple ways to do this - our choice in this instance is straightforward, featuring three $\pi$ pulses. The first and last are to cancel the hyperfine phase imparted by the probe electron spin on to the P nuclear spin during each of the two control segments (red). The middle $\pi$ pulse serves the role of the traditional echo pulse that causes interference between the two detection segments. At the end of $t_{\rm ctrl}$ the state is swapped back from storage to the probe spin, and allowed to interfere with the new phase acquired during the second detection segment (purple). This phase contains the information about the targeted nuclear spins located on slice $\Gamma_i$.
\\
\noindent
{\bf Electron spin measurement segment:} At the end of the second detection segment, the state is stored back on the donor's nuclear spin. Instead of repeating the entire sequence numerous times, we can interrogate the state in storage by re-initialising the electron spin and performing a control-not operation based of the nuclear spin. This process is relatively fast ($\rm\mu s$ time scale) and is repeated on its own $n_m$ times, therefore substantially improving the total time of the protocol.
\\
\noindent
{\bf The signal:} Thus far, the NSS detection protocol provides several advantages. The detection segment duration can be tailored independently to match the lobe being interrogated $t_{\rm det}\sim1/\Gamma_i$. The detection segments evolution is limited by $T_2^{\rm probe}$ and made distinct from the control segment. This allows the control segment (orange/red) to be longer, $t_{\rm ctrl}\sim1/B_{\rm AC}$, in order to achieve higher spectral resolution while being protected by a relatively lengthy $T_1^{\rm probe}$. Keeping the operations in the NSS protocol segmented enables us to capture the behaviour of the signal originating from individual target nuclei in a simple analytic form given by Eq.\,\ref{eq:Signal_from_sigle_nucleus} (note, carefully interleaved dipole-dipole decoupling underpins all the segments). It is crucial to note that the cumulative effect of the detection segments and $\pi$ pulses on donor nuclear spin produce an echo effect. This leaves the signal proportional only to the effective interaction rate between donor's electron spin and the target nucleus during the course of their integration $t_{\rm det}$, as captured by the first term in Eq.\,\ref{eq:Signal_from_sigle_nucleus}). The net phase accumulated from this integration on the donor's electron spin depends on the change in the target's effective magnetisation from the first to the second detection segments. In the NSS protocol, the change in magnetisation is effected by the control rotation of the targeted nuclear spin during the control segment, as captured by the second term in Eq.\,\ref{eq:Signal_from_sigle_nucleus}.

\subsection{3D density image processing}
\label{Methods_Sc:density_image}

A set of $\bf B_0$ orientates is selected, for $\theta: [0,\theta_{\rm max}]$, with spacing $d\theta$ and $\phi: [0,2\pi]$, with spacing $d\phi$. For each orientation the signal profile is recorded by sweeping across slice lobe radius $r: [0,r_{\rm max}]$ with the uniform spacing steps $dr$. Figure\,\ref{Fig:3}\,e depicts the signal profile as a function of $r$ for a selection of $\theta$ and $\phi$ orientations. All of the signal profiles are concatenated in an ordered vector $\underline{S}$ of length $n_{\rm slices}$. The sensing volume is represented in the discretised format of grid dimensions ($l,w,h$) with spacing $d$ containing $n_{\rm voxels}$ voxels such that $n_{\rm slices}\approx n_{\rm voxels}$. This discretisation format is used to define the unknown nuclear target spin density, which is stored in a vectorised form $\underline{\rho}$. To convert data from $\underline{S}$ to $\underline{\rho}$ we carefully construct a matrix transform $\underline{\underline{M}}$, by computing each slice in 3D space over the discretised sending volume, vectorising it and row-wise concatenating all of the slice vectors into a matrix $\underline{\underline{M}}$. It follows that (pseudo)inversion of the transform provides the 3D nuclear density voxel image $\underline{\rho}=\underline{\underline{M}}^{-1}\times\underline{S}$. This approach is robust however, it causes the numerical resources to scale quadratically with the number of voxels ($O(n_{\rm voxels}^2)=O(l^6)$, for section volume of side length $l$). Therefore, focusing on smaller volume segments is an efficient way to lower the computational requirements.

The inversion protocol used here is general and resource intensive limiting the voxel grid resolution, however, this is physically inconsequential. The gains can be made in developing more advanced sampling procedures. Uniform spacing in ($r, \theta,\phi$) undersamples radially more distant locations in 3D space, leading to density peak splitting in distant corners of the sensing volume. This can be fixed by inserting more slices in between radial directions for larger $r$, instead of a uniform sampling grid. Adaptive sampling can make even more significant advances. Here, the 3D conversion process could run in parallel to data acquisition, choosing slice parameters $(r, \theta, \phi, B_{\rm AC}, t_{\rm det}, t_{\rm ctrl}, n_m)$ iteratively on the run to minimise a chosen cost function, e.g. voxel entropy. Such an approach would optimise the sampling of unknown molecular structures during runtime.

\section*{Data Availability}
All relevant data are available from the authors on reasonable request.

\section*{Acknowledgements}
The authors wish to thank David Simpson, Julia McCoey and Alastair Stacey for useful discussions. L.C.L.H. acknowledges support of an ARC Laureate Fellowship (FL130100119). L.C.L.H. and M.U. acknowledge support from the Australian Research Council (ARC) funded Center for Quantum Computation and Communication Technology (CE170100012). The computational resources were provided by the National Computing Infrastructure (NCI) and Pawsey Supercomputing Center through National Computational Merit Allocation Scheme (NCMAS).

\section*{Author contributions}
V.S.P., C.D.H. and L.C.L.H. conceived the idea of using low-temperature Si qubits for bio-molecular imaging. V.S.P. and C.D.H. developed the NSS quantum control protocol and V.S.P. developed the 3D measurement protocol for the qubit-based molecular imaging. V.S.P., C.D.H., M.U. and L.C.L.H. contributed to the development of the theoretical framework of the protocols. M.U. designed and carried out tight binding wave function modelling. V.S.P. developed analytical and numerical Hamiltonian models and carried out simulations. V.S.P. analysed the results and wrote the manuscript in consultation with all the authors.

\section*{Additional information}

Competing financial interests: The authors declare no competing financial interests.

Correspondence and requests should be addressed to  V.S.P. (email: vpe@unimelb.edu.au) or to L.C.L.H. (email: lloydch@unimelb.edu.au).

redpajama_set_name: RedPajamaArXiv


Dhaka, 28 January 2019: The Inter-Department Badminton and Volleyball Competition 2019 and the prize giving ceremony organized by Physical Education Centre of Bangladesh University of Professionals (BUP) held on 28 January 2019. This competition was started on 08 January 2019 and 16 departments of this university participated in both competition.
In the Badminton Competition, the Department of Public Administration became champion and the Department of Business Administration in Marketing became runner up in Single (Male) Category and the Department of Business Administration in Management became champion and the Department of Information Communication and Engineering became runners up in Double's (Male) Category in their respective matches. On the other hand, the Department of Development Studies became champion and the Department of Business Administration in Finance and Banking became runner up in Single (Female) Category and the Department of Development Studies became champion and the Department of Economics became runners up in Double's (Female) Category.
In Volleyball Competition, the Department of Marketing defeated the Department of Disaster and Human Security Management in Male Category and the Department of International Relations defeated the Department of Business Administration in General and became champion in their respective matches.
In the prize giving ceremony, General Md Emdad-Ul-Bari, ndc, psc, te, Vice-Chancellor of BUP, was present as chief guest and distributed prizes among the winners. Moreover, there were also present BUP high officials, officers, teachers and students.


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Q: Antilinear adjont of operator How to prove that the following statements are true?
Let $H$ be a Hilbert space, $C$ be a conjugation in $H$. Operator $T$ is linear and $X,Y$ are antilinear.

*

*$(XT)^{\#}=T^{*}X^{\#}$

*$(TX)^{\#}=X^{\#}T^*$

*$(XY)^{*}=X^{\#}Y^{\#}$

*$(X^{\#})^{\#}=X$

*$(\alpha X+\beta Y)^{\#}=\overline{\alpha}X^{\#}+\overline{\beta}Y^{\#}$
I have:

*

*$T^{*}$ is adjoint of $T$ (a linear operator) if
$\langle Tx,y \rangle = \langle x,T^{*}y \rangle$.

*$X^{\#}$ is antilinear adjoint of $X$(an antilinear operator) if
$\langle Xx,y \rangle = \overline{\langle x,X^{\#}y \rangle}$.

A: It's just straightforward application of the definitions.
$$
\langle (XT)^\#x,y\rangle=\overline{\langle x,XTy\rangle}=\langle X^\#x,Ty \rangle=\langle T^*X^\#x,y\rangle.
$$
As this occurs for all $x,y$, you get $(XT)^\#=T^*X^\#$. The rest are similar.

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Kaley Cuoco and Karl Cook: Why Are They REALLY Getting Divorced?
by Simon Delott at September 9, 2021 4:01 pm .
Last week, Kaley Cuoco and Karl Cook confirmed their split after three years of marriage.
They're not wasting any time on getting a divorce, either.
Kaley and Karl initially released only a vague statement about their "paths" leading them away from each other.
Now, however, it looks like there's a more specific explanation for the end of their marriage.
This week, an inside source spoke to People about what led to be breakup.
"She has a big career," the insider acknowledged, "and puts that first in her life."
The source summarized: "It's basically a growing apart and each pursuing separate interests."
According to the insider, Kaley and Karl "haven't spent that much time together recently."
Karl is a dedicated equestrian, and spent a lot of his time this winter training in Florida.
Meanwhile, Kaley spent that time in Los Angeles.
"She has a lot going on in her career now," the source continued.
"And," the insider affirmed, Kaley "loves her work."
"Their relationship is just following different paths," the source reasoned.
"A marriage doesn't work if you rarely see the other one," the insider commented.
That actually aligns very closely with what Kaley and Karl said themselves.
They just weren't as detailed about it.
"Despite a deep love and respect for one another," the former couple said in a statement last week.
Kaley and Karl shared: "we have realized that our current paths have taken us in opposite directions."
They also asked for privacy at this time.
"We have both shared so much of our journey publicly," Kaley and Karl acknowledged at the time.
"So," they explained, "while we would prefer to keep this aspect of our personal life private."
Their statement continued "we wanted to be forthcoming in our truth together."
"There is no anger or animosity," Kaley and Karl emphasized.
They shared that the reality of the situation is "quite the contrary."
We hope that they will be able to keep that attitude.
Sometimes divorces remain amicable.
Sometimes they start that way and end in bitterness and mutual loathing.
It's easy to sound upbeat and mature before you start dividing up marital property and even social ties.
Kaley and Karl spoke frankly about how they did not actually move in together after marrying.
"We have a very unconventional marriage," Kaley acknowledged freely in a 2019 interview.
"We have different locations that we are at a lot," she explained. "We are not together every single day."
However, the COVID-19 pandemic took that decision out of their hands.
"This quarantine has forced us to actually move in together," Kaley shared last year.
"It's been great for our relationship," she assessed at the time. "And we like each other, we realized, which is even better!"
It may be that they would still be married now if it weren't for the COVID-19 pandemic.
Plenty of relationships, including marriages, fell apart when people were forced to actually spend time together -- especially under stressful circumstances.
In their case, perhaps it wasn't being forced together, but then parting ways after that quality time that made their marriage untenable.
Kaley Cuoco might be best known for her role on The Big Bang Theory, but she also stars on a genuinely funny comedy, Harley Quinn.
The animated HBO Max series is a loving tribute to the character and to an array of DC Comics fan-favorites, including Batman himself.
We wish Kaley -- and Karl, for that matter -- all of the best as their conscious uncoupling continues.
Tags: Kaley Cuoco
Kaley Cuoco Rushes Divorce From Karl Cook Amid Rumors of Messy Separation
Kaley Cuoco and Karl Cook Split After 3 Years of Marriage
Kaley Cuoco Photos
Kaley Cuoco Videos
Kaley Cuoco Engagement Video
4 Celebs Who Are Proud Of Their Breast Implants
Kaley Cuoco on Karl Cook Relationship

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Q: Tree component with table-like "cells"? I need to have several, separately editable "cells" for a tree node. A TreeTable is not suitable, as I have several node types which have different record structures.
Is there already a (preferably free) component for this? If not, do you have some implementation hints for me? Especially it would be important that I can assign different "CellRenderers" and "CellEditors" to different cells, and to be able to resize a certain cell for all nodes of a certain node type.

A: From what I understand, what you want is not a tree in a table, but a table in a tree. Am I correct ?
If so, you can take a look at this old example in comp.lang.java.gui, which achieves what you want in the most typical fashion : by using as JTree renderer a Jtable painter

A: You might look into org.netbeans.swing.outline.Outline, mentioned here.


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Pac-Attack / Pac-Panic
Posted by Bobinator on August 18, 2019
Pac-Attack/Pac-Panic - Super Nintendo, Sega Genesis, Game Boy, Game Gear, Game Boy Color, Philips CD-i (1993)
This entry is part 7 of 10 in the series Pac-Man
Super Pac-Man
Pac-Man Plus
Jr. Pac-Man
Professor Pac-Man
Pac & Pal
Pac-Mania
Pac-Man 2: The New Adventures
Pac-Man Arrangement
Ms. Pac-Man Maze Madness
American Super Nintendo Cover
American Sega Genesis
European CD-i Cover
American Game Boy Cover
One of the greatest signs of success when you're comparing video game mascots is just how many spinoffs they've had. All the greatest mascots will have a kart racer, party game, or puzzle games somewhere in their history. Pac-Man, possibly the greatest video game mascot of all, obviously has all three. Pac-Attack is hardly the most inspired or the most recommendable when it comes to the falling block genre, but it's serviceable. It's no Tetris, or truthfully, even a Klax, but it's far from offensive.
Calling it a quick cash-in wouldn't be far from the truth, as the game actually started life as a 1992 arcade game called Cosmo Gang: The Puzzle. The arcade game itself was a spinoff of a series of mechanical redemption games released by Namco, which only had a small presence in North America. In 1993, the game was released to American consoles with a Pac-Man reskin, with the Japanese release taking its original Cosmo Gang form. Aside from the changes in graphics and sound, however, the two games are mostly identical in gameplay.
The game's main mode asks you to gather as high of a score as you can before your field fills up entirely with objects, much like in Tetris. You're constantly given sets of three objects, each of which can contain one of three things. Wooden blocks disappear when you make a whole row of them, with more lines earning you more points. Ghosts can't be cleared once placed however, unless you place a Pac-Man icon somewhere on the field. It's Pac-Man himself that adds the biggest wrinkle to the proceedings, and knowing how to use him is where most of the game's strategy comes in. He'll generally try to move towards the bottom of the screen, disappearing if he can no longer move, but he'll also turn if he hits a wall, meaning sometimes you'll have to make 'pathways' for him to reach the ghosts you want to remove.
In typical falling block game fashion, the game keeps picking up speed the further you go, until you only have a split second to decide where to place things. The only mercy you're given is a fairy that sometimes appears that'll destroy all the ghosts beneath her once you place her. On the easiest difficulty level, you're given one of these for free when you're in trouble, while on harder levels, you'll have to destroy enough ghosts to fill a meter first. There's little else to the main gameplay beyond that, with the only real change being the gradual increase in speed.
There's also a puzzle mode that takes place over a hundred stages, where you're given a partially filled board, a number of ghosts, and a limited number of Pac-Men to clear them all. Not only do you have to get rid of all the ghosts that you're already shown, however, but you'll also be given more to place onto the field. It can be tough even from the early stages figuring out where to place Pac-Man so he can actually reach the ghosts, so this helps to add a lot more replay value than the somewhat simplistic main mode. There's also a versus mode for two players, in which the ghosts you clear send more towards your opponent to hassle them. It's fun for a round or two, but the best two out of three matches take so long to play out that it quickly starts to get a little dull.
Overall, there's not really terribly much to recommend Pac-Attack over the games it's trying to imitate so hard. The only thing that really helps distinguish it is the puzzle mode, which, while challenging enough, only adds so much replay value. There's nothing the game really does wrong, but against games like Dr. Robotnik's Mean Bean Machine and Dr. Mario, there's a lot less on offer.
The Super Nintendo and Genesis versions are nearly identical, aside from having a different menu screen, for some reason. The Game Boy & Game Gear versions are solid handheld ports, with the only real change being the forgivable omission of the versus mode. The game can also be found as a bonus feature on Game Boy Color's Pac-Man: Special Color Edition, which is a colorized but no-frills port of the Game Boy version. The CD-i version, released only in Europe, features new art that replaces almost all of what little text the original game had. It appears on assorted compilations, including Namco Anthology Vol. 2 for the PlayStation.
It's not really an improvement.
Series Navigation << Pac-ManiaPac-Man 2: The New Adventures >>
Nontan to Issho: Kuru Kuru Puzzle
Captain America and the Avengers (Arcade)
Hebereke's Popoon
Hebereke's Popoitto
Quarth
Tag: Falling Blocks, Namco, Nintendo Game Boy, Puzzle, Sega Game Gear, Sega Genesis, Super Nintendo Entertainment System
Trap Gunner
Top 47,858 Games of All Time Episode 142: Lunar: The Silver Star


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Gum Plant Health Benefits
The gum plant contains phenolic and flavonic substances, which give it antispasmodic properties, as well as saponins, to which it owes its expectorant properties. The resin of the gum plant is formed by terpene acids, among which the most important is the grindelic acid, with antitussive, antispasmodic, and bradycardic (slows the heartbeat rate) properties.
1 The use of the gum plant is recommended for the following:
2 Gum Plant Scientific Facts
3 How to use Gum Plant
4 Another Gum Plant Species
The use of the gum plant is recommended for the following:
Bronchial asthma because of its expectorant and antispasmodic effect.
Acute bronchitis and bronchial catarrh, since it soothes the respiratory mucous membrane and promotes their regeneration.
Whooping cough and persistent bronchial cough because of its antitussive properties.
Heart arrhythmia, especially tachycardia.
WARNING! When taken in high doses, the gum plant has toxic effects and can even produce heart failure.
Gum Plant Scientific Facts
Other names: August flower, gumweed, resin-weed.
French: Grindelia.
Spanish: Grindelia.
Environment: Native to the northwest American coast, it grows in salty soils and marshes. Very common in California.
Description: Vivacious plant of the Compositae family, growing up to 80 cm high, with flower clusters resembling those of daisies. Its stem and leaves are impregnated with a sticky resin. The plant has a soothing aroma and mildly sour taste.
Parts of the plant used medicinally: The flower clusters.
How to use Gum Plant
Infusion with a teaspoonful of flower clusters per cup of water. The usual dose for adults is three cups a day and for children half the adult dose.
Syrup: It is usually prepared in pharmacies with 5% of fluid extract. Drink two or three spoonfuls a day.
Another Gum Plant Species
Besides the species robusta, there is another species of gum plant with the same medicinal properties: Grindelia squarrosa Pursch. Both of them are native to the Pacific coast of North America; however, their interesting properties have made them famous outside the United States, and at present, they can be found in many herb shops worldwide.
George D. Pamplona-Roger, M.D. "Encyclopedia of Medicinal Plants." George D. Pamplona-Roger, M.D. Encyclopedia of Medicinal Plants. Ed. Francesc X. Gelabert. Vols. 1 San Fernando de Henares: Editorial Safeliz, 2000. 310. Print.
grindelia for asthma grindelia homeopathy uses grindelia medicinal uses grindelia robusta homeopathy uses gum plant medicinal uses


Henry Davids has still got it, even at age 39, he can produce an innings players 20 years younger can only dream about.
The Multiply Titans thrashed the WSB Cape Cobras by seven wickets in Wednesday night's CSA T20 Challenge match in Benoni thanks to another blistering innings from their veteran opener.
"It was very clinical, all round team performance. The intensity we played at, with both bat and ball, was impressive," said Davids.
It was Davids' 39-ball innings of 57, which flattened the visitors from the Cape, a knock which featured his trademark lofted cover drive but also some punishing blows straight down the ground.
Besides Davids the match also featured outstanding displays from the Titans' two spinners Gregory Mahlokwana, who bowled with both his left and right arms and leg-spinner Shaun von Berg, who got a number of deliveries to spin viciously passed the Cobras batsmen. That pair both conceded less than 20 runs in their respective four-over spells, never allowing the Cobras to break free after they'd battled in the opening power play and scored just 38 runs losing the wickets of Hashim Amla and David Bedingham in the process..
The Cobras did recover after that bad start to post 150/5, with Kyle Verreynne holding the innings together before retiring hurt after scoring 44, while Vernon Philander (22*) and Rory Kleinveldt (12*) hit some big blows towards the latter stages of the innings.
Davids then shared partnerships of 51 for the second wicket with Theunis de Bruyn (29) and 56 for the third with Diego Rosier (43*) as the Titans sailed to victory with 26 balls to spare.
"The way (Rosier) came in, the intensity he showed was very good. I was going well at that stage and he just kept it up and that puts pressure on the opposition bowlers, it's great for us going forward. A lot of teams go conservative in that middle period but we are showing real intensity, it allows to go up a notch," Davids added.
He and the Titans will return to Willowmoore Park on Friday night to face the Hollywoodbets Dolphins, who lost at home to the bizhub Highveld Lions in midweek.


More 1955 alumni from Buchtel HS have posted profiles on Classmates.com®. Click here to register for free at Classmates.com® and view other 1955 alumni.
If you are trying to find people that graduated in '55 at Buchtel in Akron, OH, check the alumni list below that shows the class of 1955.


"A Broken Covenant and the Curse of Death: A Study of Hebrews 9:15-22."
"A Broken Covenant and the Curse of Death: A Study of Hebrews 9:15-22." The Catholic Biblical Quarterly 66.3 (July 2004), pp. 416-436.
"Christ, Kingdom, and Creation: Davidic Christology and Ecclesiology in Luke-Acts."
"Christ, Kingdom, and Creation: Davidic Christology and Ecclesiology in Luke-Acts."Letter & Spirit: A Journal of Catholic Biblical Theology 3 (2007), pp. 113-138.
"Covenant, Cult, and the Curse-of-Death: διαθήκη in Hebrews 9:15-22."
"Covenant, Cult, and the Curse-of-Death: διαθήκη in Hebrews 9:15-22." Hebrews: Contemporary Methods - New Insights, ed. Gabriella Gelardini (Leiden: Brill, 2005).
"Covenant in the Old and New Testaments: Some Current Research (1994–2004)."
"Covenant in the Old and New Testaments: Some Current Research (1994–2004)."Currents in Biblical Research 3.2 (2005), pp. 263-292.
"Kingdom and Church in Luke-Acts: From Davidic Christology to Kingdom Ecclesiology."
"Kingdom and Church in Luke-Acts: From Davidic Christology to Kingdom Ecclesiology." Reading Luke: Interpretation, Reflections, Formation Scripture and Hermeneutics Series VI; eds., Craig Bartholomew, Joel Green, Anthony Thiselton (Grand Rapids, MI: Zondervan, 2005), pp. 294-326.
"Scripture and the Liturgy: Inseparably United."
"Scripture and the Liturgy: Inseparably United." Origins 35.39 (March 16, 2006), pp. 648-653.
"Temple, Sign, and Sacrament: Towards a New Perspective on the Gospel of John."
"Temple, Sign, and Sacrament: Towards a New Perspective on the Gospel of John."Letter & Spirit: A Journal of Catholic Biblical Theology 4 (2008), pp. 107–143.
"What Laws Were Not Good: A Canonical Approach to the Theological Problem of Ezekiel 20:25-26"
"What Laws Were Not Good: A Canonical Approach to the Theological Problem of Ezekiel 20:25-26" with John Bergsma. Journal of Biblical Literature 123.2 (2004), pp. 201-218.
"Worship in the Word: Toward a Liturgical Hermeneutic."
"Worship in the Word: Toward a Liturgical Hermeneutic." Letter & Spirit: A Journal of Catholic Biblical Theology 1 (2005), pp. 101-136.


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ComputerLanguage.com
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Definition: WAAS
(1) See Windows as a service.
(2) (WaaS) (Workspace As A Service) A thin client architecture that is delivered from a cloud service provider. The difference between WaaS (workspace) and DaaS (desktop) services is that the workspace model offers a complete computer environment that is fully populated with antivirus, backup and more. Although both terms may be used synonymously, the desktop model is typically just the OS desktop. See DaaS and XaaS.
(2) (Wide Area Augmentation System) A system of earth stations and satellites that improves the tracking accuracy of the GPS navigation system to approximately 10 feet. Deployed in 2003, WAAS comprises a group of 25 reference stations and two satellites that cover North America. The stations track the GPS satellites and send correction signals to the WAAS satellites, which transmit them to WAAS-enabled receivers. See GPS augmentation system and GPS.
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known as the Moth Orchid, abbreviated Phal in the horticultural trade, is an orchid genus of approximately 60 species. Phalaenopsis is one of the most popular orchids in the trade, through the development of many artificial hybrids. It is native to southern China, the Indian Subcontinent, Southeast Asia (Thailand, Philippines, Malaysia, Indonesia, etc.), New Guinea, the Bismark Archipelago, and Queensland.
A Phalenopsis orchid. One of the many cultivated species.


Dr. Marilyn Sandor is one of Southwest Florida's favorite pediatric dentists. She wants to make each visit to Naples Pediatric Dentistry a successful and enjoyable experience for your child. She understands that children are curious, energetic, and always learning. This is what makes her work so rewarding. She enjoys teaching children the importance of oral health care and how good care will benefit them throughout their lives.
Dr. Sandor received her Bachelor's degree in Biology from the University of Chicago, and her Doctor of Dental Surgery degree from the University of Illinois at Chicago College of Dentistry. She completed her pediatric dental residency at the University of Illinois at Chicago Department of Pediatric Dentistry. In addition, Dr. Sandor received her Master degree in Oral Sciences at the University of Illinois at Chicago Graduate College. Dr. Sandor is an active member of the American Academy of Pediatric Dentistry, the American Dental Association, the Florida Academy of Pediatric Dentistry, the Florida Dental Association, and the West Coast Dental Association.
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As much as we love the flames, the mechanics are embarrassed by them. Apparently, on these big V8 cars, it means there is an air leak somewhere in the front of the exhaust!
"Some additional info; It started life as one of the Moe Carter's Camaros, I believe it finished 4th in the 1981 24 hours of Daytona. Moe rolled it at Lime Rock. Jerry Simmons bought it and bodied it as a C4 Corvette, raced it in T-A. Jerry sold it to Charlie Webster, who sold it to Butch Kummer who sold it to Eric Taylor who sold it to Jeff Bailey(the South Atlantic Road Racing Championship mentioned in the article). It still holds the track record at Watkins Glen short course, but that configuration is gone so it will always have that track record. Butch Kummer named it Godzilla. Whew!"
Totally agree, it will be great to see this on-track.


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Home > Journals > Canadian Journal of Plant Science > Volume 91 > Issue 6 > Article
17 October 2011 Soil treatments and amendments for amelioration of clubroot of canola
S. F. Hwang, S. E Strelkov, B. D. Gossen, G. D. Turnbull, H. U. Ahmed, V. P. Manolii
S. F. Hwang,1 S. E Strelkov,2 B. D. Gossen,3 G. D. Turnbull,1 H. U. Ahmed,1 V. P. Manolii2
1Alberta Agriculture and Rural Development (AARD), Crop Diversification Centre North, Edmonton, Alberta, Canada T5Y 6H3 (e-mail: ) sheau-fang.hwang@gov.ab.ca
2Department of Agricultural, Food and Nutritional Science, University of Alberta, Edmonton, Alberta, Canada T6G 2P5
3Agriculture and Agri-Food Canada, Saskatoon Research Centre, Saskatoon, Saskatchewan, Canada S7N 0X2
Canadian J. of Plant Science, 91(6):999-1010 (2011). https://doi.org/10.1139/CJPS2011-028
Hwang, S. F., Strelkov, S. E., Gossen, B. D., Turnbull, G. D., Ahmed, H. U. and Manolii, V. P. 2011. Soil treatments and amendments for amelioration of clubroot of canola. Can. J. Plant Sci. 91: 999-1010. Clubroot, caused by Plasmodiophora brassicae, has become a long-term challenge to canola production in central Alberta, Canada. Trials were established in P. brassicae infested commercial fields near Leduc, Edmonton, and St. Albert, AB, to examine the effects of fungicidal soil treatments and soil amendments on crop damage caused by clubroot. Treatment of the soil with Terraclor (quintozene) reduced clubroot severity and increased plant growth (canopy coverage and plant height) and seed yield in severely infested soils. Ranman (cyazofamid) at 7.5 L ha-1 increased canopy coverage and height in 2 of 4 site years, and yield in 1 of 4 site years. Among the soil amendment treatments, wood ash at 7.5 t ha-1 or ground limestone (calcium carbonate) at 5.0 or 7.5 t ha-1 increased plant height and yield in severely infested soils at Leduc. Limestone at 7.5 t ha-1 reduced clubroot severity at both sites and at 5.0 t ha-1 reduced clubroot at St. Albert. Wood ash at 7.5 t ha-1 reduced clubroot severity at Leduc in both years and all rates of wood ash reduced clubroot severity at St. Albert in 2008. Canopy coverage was increased by limestone at 7.5 t ha-1 in both years at Leduc and by wood ash at 7.5 t ha-1 in one of the site-years at Leduc. In 2009 and 2010, in-row treatments with Terraclor (quintozene), calcium carbonate, wood ash or quintozene calcium carbonate or wood ash reduced clubroot severity at the Edmonton site, but yield was unaffected. We conclude that quintozene and/or high levels of limestone or wood ash can reduce the impact of P. brassicae on canola. However, in-row assessment at application rates that might be economically feasible showed little potential to increase yield in infested commercial fields.
S. F. Hwang, S. E Strelkov, B. D. Gossen, G. D. Turnbull, H. U. Ahmed, and V. P. Manolii "Soil treatments and amendments for amelioration of clubroot of canola," Canadian Journal of Plant Science 91(6), 999-1010, (17 October 2011). https://doi.org/10.1139/CJPS2011-028
Received: 9 February 2011; Accepted: 1 July 2011; Published: 17 October 2011
Canadian Journal of Plant Science
cendre de bois
Plasmodiophora brassicae
terrachlor
Terraclor
Seedling age and inoculum density affect clubroot severity and seed...
Effects of fungicide, seeding date and seedling age on clubroot...
Evaluation of liming properties and potassium bioavailability of three Atlantic...
Chernozemic soils of Canada: Genesis, distribution, and classification
Distribution and potential effects of water beetles in lakes recovering...
Oyster Shell Dissolution Rates in Estuarine Waters Effects of...
Factors determining plant species richness in Alaskan arctic tundra
S. F. Hwang, S. E Strelkov, B. D. Gossen, G. D. Turnbull, H. U. Ahmed, V. P. Manolii "Soil treatments and amendments for amelioration of clubroot of canola," Canadian Journal of Plant Science, 91(6), 999-1010, (17 October 2011)