d increases to Tm = Z (m0 + Dm)1/2, which may be detected if Tm - T0 > Dtmin ~ 1 nanosec, the minimum convenient time interval (Section 10.1). Hence the minimum detectable molecular weight DMW, expressed in proton masses (mp = 1.67 x 10-27 kg), is
In Figure 4.6, beam and rod are coiled in two dimensions (although three-dimensional coiling, more difficult to depict, provides maximum compactness). Lbeam and lrod are chosen to minimize total coil area Acoil = 2 Rbeam Lbeam + 2 rrod lrod. Using aluminum for the rod and beam (E = 6.9 x 1010 N/m2, G = 2.6 x 1010 N/m2, r = 2700 kg/m3), Lbeam = 1200 nm, lrod = 1920 nm, rrod = 0.5 nm, Vsample = 1 nm3, and Dtmin = 10-9 sec, then DMW ~ 1 proton for Lspring = 1 nm >> Dxmin = 0.01 nm, the minimum measurable displacement (Section 4.3.1). (For Lspring > 1 nm, physiological ~1 g accelerative loads on a ~1 micron nanorobot in which the sensor is embedded produce undetectable accelerative beam displacements < Dxmin.)
Moving the suspension rod closer to the sample holder reduces device sensitivity. For Lspring = Lbeam, minimum DMW ~ 1200 protons, permitting moderately large molecules to be weighed in minimum time (~10-6 sec). Using a very thin sensor configuration (hsensor~20 nm) and coils occupying ~50% of sensor cavity space, sensor dimensions are (79 nm)2 x 20 nm giving a massometer volume ~105 nm3 or ~0.01% of the total volume of a 1 micron3 nanorobot. Sensor mass is ~ 3 x 10-19 kg, including diamondoid housing and prorated share of support mechanisms. If one measurement cycle including sample loading/unloading operations can be completed every ~10 microsec, then the device processes its own mass of glucose molecules in ~10 sec. Note that biological molecules weighed in air or in vacuo may lose water of hydration, whereas weighing in water presents other complications that merit further investigation.
J. Soreff points out that for a molecule on a spring, the precision of the measurement is linked to the sharpness of the resonance, which is optimized to the degree the vibrational frequency of the spring/molecular mass system is kept well below the vibrational resonance frequencies of the molecule itself. The following discussion reports the results of a simple lumped-element analysis of the effects of internal vibration on the massometer completed for the author by J. Soreff, which confirms the previously described performance estimates for the massometer.
Suppose that it is desired to weigh some mass M to a precision DM by attaching it to a weak measuring spring (spring constant kweak) and measuring the shift in resonant frequency. The shift in frequency due to coupling to an internal mode of the sample must be less than (1/2) (DM / M) (1 / 2 p) (kweak/M)1/2. If the sample has some lowest internal mode frequency nint, then approximating the sample as two M/2 masses separated by xstrong connected by a strong spring (kstrong) gives nint = (1/p) (kstrong/M)1/2. If this sample is attached to a weak measuring spring of length xweak, which is in turn attached to a rigid support, then the equations of motion are:
Setting xweak = est and xstrong = R est where t = time, s = time constant, and R = -(1 + (2 kweak / M s2)), and since the angular frequencies of the two springs are wstrong = 2 p nint = 2 (kstrong/M)1/2 and wweak = (kweak/M)1/2, then the lower frequency solution for wweak << wstrong may be approximated by s = i wweak (1 - wweak2 / 2 wstrong2) and so the (undesired) fractional change in frequency due to the flexibility of the object to be weighed is
For a molecule sample size of Lsample, nint ~ vsound / Lsample ~ 500 GHz for vsound ~ 1500 m/sec and Lsample ~ 3 nm; M = r Lsample3 = 2.7 x 10-23 kg for r = 1000 kg/m3. To detect the mass difference of a single proton in a sample molecule, sensor Q = M / mp = 16,000; from Eqn. 4.28, the sensor may operate at a factor of Q1/2 below nint, or nspring ~ nint / Q1/2 ~ 4 GHz. A period of 0.25 nanosec is slightly too short to allow a phase shift of Dtmin ~ 1 nanosec, since the period is only one-quarter of this. However, a phase shift of p radian