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a two component Fermi gas with an equal number of par- ticles in each state N , the structure factor consists of two components, corresponding to correlations between par- ticles in the same state and particles in different states S ( k ) = S$_{↑↑}$ ( k ) + S$_{↑↓}$ ( k ) . When the momentum is much larger than the Fermi momentum, particles in the same state will be uncorrelated and S$_{↑↑}$ ( k ≫ k$_{F}$ ) ≃ 1 so all variation in S ( k ) will then be due to changes in S$_{↑↓}$ ( k ) [15, 29]. The Fourier transform of Eq.1 yields the follow- ing for S$_{↑↓}$ ( k ) 

which has a straightforward dependence upon I , the rela- tive probe momentum k/k$_{F}$ , and the interaction strength 1 / ( k$_{F}$ a ) . At unitarity, a → ∞ , the second term vanishes and S ( k ) varies linearly with k$_{F}$ /k . Rewriting Tan’s re- lation in this way points to a method to experimentally verify Tan’s universal relation for pair correlations be- tween spin-up/spin-down fermions since the static struc- ture factor can readily be measured using Bragg spec- troscopy. 

Apart from the contact, all parameters in Eq. (2) are easily determined or set by experimental parameters. We employ new methods to calculate the contact which over- come the need for interpolation schemes between the lim- iting BEC and BCS cases used in earlier studies [19]. To find the low temperature contact we first calculate the ground state energy across the BEC-BCS crossover using a recently developed below-threshold Gaussian pair fluc- tuation theory [16], which has shown excellent agreement with thermodynamic measurements [9, 10, 21] and quan- tum Monte Carlo (QMC) simulations [22]. We then use Tan’s adiabatic theorem to find the contact. The high temperature ( T ≳ 0 . 5 T$_{F}$ ) contact is calculated from a quantum cluster expansion [23]. Figure 1 shows the cal- culated contact I for a range of temperatures through the BEC-BCS crossover, along with the results from photo- association (Fig.1a) [8, 19]. The contact increases mono- tonically from the BCS to BEC regimes and is highest at low temperatures. Combining these calculated results for the contact with Eq. (2) for high momentum transfer Bragg scattering, we obtain a direct, quantitative predic- tion that is readily tested experimentally. 

Inelastic scattering experiments are a well established technique to probe both the dynamic and static structure factors of many-body quantum systems [24]. Ultracold atoms are highly amenable to inelastic scattering through Bragg spectrosocpy which has previously been used to measure both the dynamic [25, 26] and static structure factors [27, 28] of atomic Bose-Einstein condensates. In ultracold Fermi gases Bragg spectroscopy was used to measure both the dynamic and static structure factors over the BEC-BCS crossover, albeit at a single momen- tum [29]. The ability to vary the momentum differen- tiates Bragg spectroscopy from rf spectroscopy [30, 31], 

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providing access to a broad range of the excitation spec- trum and avoiding final state interaction effects as no third atomic state is involved. 

To demonstrate the validity of Eq.(2) we need to mea- sure S ( k ) for a range of momenta ( k/k$_{F}$ ) and at different values of the interaction parameter 1 / ( k$_{F}$ a ) . The start- ing point of our experiments are clouds containing N ≈ 3 × 10 $^{5 6}$Li atoms in an equal mixture of the two lowest lying ground states | F = 1 / 2 , m$_{F}$ = ± 1 / 2 〉 evaporatively cooled in a single beam optical dipole trap ( λ = 1075 nm) at a magnetic field of 834 G [32]. Next, we adia- batically ramp up a second far detuned laser ( λ = 1064 nm) which intersects the first trapping beam at an angle of 74 ◦ forming a crossed beam dipole trap. By appro- priately selecting the intensities of each of the two trap lasers we tune the mean harmonic confinement frequency of the crossed trap ¯ ω over the range ¯ ω = 2 π × (38 → 252) s − 1 while the aspect ratio, ω$_{x,y}$/ω$_{z}$ , varies from 3.4 to 16. Controlling the trap frequencies in this way allows us to tune the atom density and hence the Fermi wavevector 

